Handbook of Dynamical Systems : Volume 2

Preface This Handbook is Volume 2 in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems help to clarify mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. The authors and editors have made an effort, however, to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopaedic completeness, but present selected paradigms. There is a web of relationships among the chapters, allowing them to be organized in different ways, each somewhat arbitrary. The chapters are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and

partial differential equations. Two of the three articles in the section onfinite-dimensional methods deal with dynamics of networks of neurons. One of them focuses on a variety of techniques, geometrical and analytical, for investigating interactions of pairs of oscillators. A complementary review uses geometrical ideas to discuss further aspects of small network interactions, and some interesting consequences for particular neural applications, including sleep rhythms and visual scene segmentation. Many of the mathematical features carry over to more general coupled oscillator networks, ranging from electronics to solid state physics, while other ingredients are specific to the particular applied field, here neurophysiology. This interplay of general mathematical structure with specific applied context is an important underlying theme of the volume, and is demonstrated by the paradigms presented. Addressing fluid dynamics on geographic scales, the third survey illustrates an application of lobe dynamics of transverse planar heteroclinic tangles to global transport phenomena by ocean currents. Methods are geometrical and are based on the analysis of invariant manifolds of hyperbolic trajectories and invariant tori. Numerical simulation has become an omnipresent- and sometimes naively appliedtool in bridging the gap between universal mathematical theory and specific applied context. It is one of the goals of the section on numerics to indicate the level of sophistication necessary to properly deal with various aspects of dynamics, numerically. We start with a thorough survey of state-of-the-art continuation methods, which seek to determine the parameter dependence of particular types of solutions, such as critical points or periodic orbits. This includes computations of normal forms, of codimension two bifurcations, and of homoclinic bifurcations. Stable and unstable manifolds, which may intersect along homoclinic curves for flows, will typically split under time discretization and exhibit exponen-

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Preface

tially small separation. This effect and the associated invisible chaos mark a fundamental difference between continuous time autonomous flows and their time discretizations. Both upper and lower bounds for exponentially small splitting effects are provided. More generally, another survey aims at assessing the capability of theory, algorithms, and software to elucidate the structure of dynamical models in mathematics, science, and engineering. Specific dynamical objects include classical initial value problems, periodic solutions, invariant tori, stable and unstable manifolds, and their bifurcations. In the context of chaotic dynamical systems, hyperbolicity-based shadowing estimates provide a tool to diagnose whether it is possible to achieve numerical solutions that are close, over very large time spans, to actual solutions. Finally, set oriented numerical methods are presented which provide a robust tool for the approximation of low-dimensional invariant objects, such as attracting sets or invariant manifolds. These methods are also capable to derive statistical information about the dynamical behavior via the computation of approximate SRBmeasures and almost invariant sets. Topological methods are an important tool for the analysis of dynamical systems. Conley index, in this context, stands for Conley's successful idea of extending Morse theory to the requirements and challenges posed by dynamical systems which, a priori, do not possess a variational structure. Several examples in the first article apply Conley index to corroborate numerical input into computer assisted proofs. While the exposition here, not the method, limits itself to finite-dimensional dynamics, the survey on functional differential equations demonstrates applications of topological fixed point theory to infinite-dimensional dynamics. Partial differential equations are a fascinating infinite-dimensional source o f - and challenge f o r - dynamics. They are represented by eight articles, mostly parabolic, constituting almost half of the entire volume. While three surveys are related to fluid flows, we first comment on the five surveys which address general PDE topics. The intimate relations to finite-dimensional dynamics, in particular when studying global attractors and large-time behavior, are investigated from different perspectives in two surveys. In the non-parabolic case, even mere existence of a global attractor is not always obvious. In addition to dimension estimates and inertial manifold reductions, these articles extract geometrical information about the associated global attractors from structural PDE properties like Lyapunov functions and comparison principles. Such structures can be found in a variety of equations, including reaction diffusion equations, wave equations with damping, Navier-Stokes, Cahn-Hilliard, Kuramoto-Sivashinsky equations and certain integro-differential equations. Partial differential equations involve space coordinates as well as time coordinates. This allows for richer behavior, including spatial and spatio-temporal patterns. For example, travelling waves are solutions of associated ordinary differential equations. Depending on the type of wave, the desired solutions take a particular geometric form such as homoclinic orbits (for pulses), heteroclinic orbits (for fronts), or periodic orbits, as well as more complicated objects corresponding to multiple pulses and spirals. Their bifurcations and their PDE stability analysis pose formidable challenges addressed here. These results are complemented by a survey of some first steps towards understanding the spatio-temporal complexity of higher-dimensional stable and metastable patterns in nonlinear evolution equations of gradient type. The singularities generated by finite time blow-up of solutions

Preface

vii

are yet another PDE phenomenon which illustrates the close interaction of temporal dynamics with spatial profiles. Some of these singularities can be investigated using related equations that come from changes of variables motivated by self-similar blow-up solutions. The article on blow-up emphasizes the unifying role of abstract center manifolds in this spatio-temporal analysis. Two articles in the section on partial differential equations are closely related to questions of fluid flows. We begin with a large survey on the state of the art for existence, uniqueness, and model derivation of the central Navier-Stokes equation itself. Starting from the molecular level with Hamiltonian systems, the survey proceeds to Boltzmann equations, where the original time reversibility is lost, and arrives at the Navier-Stokes equations, which themselves lead to macroscopic models for turbulent flows. This spans a large hierarchy of dynamical systems, and the various limiting processes are made precise. Ginzburg-Landau equations, on the other hand, are almost universally relevant model equations and at the same time closely related to specific fluid flow phenomena. They appear as reduced, albeit infinite-dimensional, modulation equations. In particular they describe bifurcations in unbounded domains and govern the associated dynamics of spatiotemporal patterns. Full mathematical justification of the reduction process from the underlying PDE to the Ginzburg-Landau approximation is addressed. Like the Ginzburg-Landau approximation, the nonlinear Schr6dinger equation can be viewed as an envelope equation for certain idealized fluid flows, but also possesses major applications in nonlinear optics and in plasma physics. It is surveyed here to illustrate sample behavior and phenomena for a general class of PDEs: one-dimensional, nonlinear, dispersive wave equations. These provide examples of infinite dimensional dynamical systems which exhibit diverse and fascinating phenomena, including solitary pulse waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive wave turbulence, and the propagation of spatio-temporal chaos. Classes of nonlinear Schr6dinger equations thus provide prototypical PDEs which illustrate the use of geometrical, analytical, and computational methods to capture and to describe the rich behavior of nonlinear dispersive waves. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to name just a few, are ubiquitous dynamical concepts throughout the articles. It is this conceptual unity, and its ability to capture and mirror an ever-changing world, which is the innermost source of strength of the dynamical systems approach. Many friends and colleagues have helped bringing this volume to life. I am indebted to Floris Takens for the original idea of the Handbook series, and, with Henk Broer, Boris Hasselblatt, and Anatol Katok, for sharing the dynamics of their volumes with me, to Stefan Liebscher for his untiring care for the web services, to Regina L6hr for expertly keeping track ever so patiently of ever so many versions and revisions, and to Arjen Sevenster at Elsevier for efficient collaboration. Nancy Kopell and G6rard Iooss have helped significantly

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Preface

with editing this volume and have very generously shared their stimulating criticism with me. Most of all, of course, I am indebted to all authors for their contributions. Berlin, August 2001

Bernold Fiedler

List of Contributors Auerbach, D., University of Maryland, College Park, MD (Ch. 7) Bardos, C., University Denis Diderot and University Pierre et Marie Curie, Paris (Ch. 11) Beyn, W.-J., Bielefeld University, Bielefeld (Ch. 4) Cai, D., New York University, New York, NY (Ch. 12) Champneys, A., University of Bristol, Bristol (Ch. 4) Dellnitz, M., University of Paderborn, Paderborn (Ch. 5) Doedel, E., Concordia University, Montreal (Ch. 4) Ermentrout, G.B., University of Pittsburgh, Pittsburgh, PA (Ch. 1) Fife, EC., University of Utah, Salt Lake City, UT (Ch. 13) Fila, M., Comenius University, Bratislava (Ch. 14) Gelfreich, V., The Steklov Mathematical Institute at St. Petersburg, Russia and Institut fiir Mathematik I, FU, Berlin (Ch. 6) Grebogi, C., Universidade de $6o Paulo, S6o Paulo (Ch. 7) Govaerts, W., University of Gent, Gent (Ch. 4) Guckenheimer, J., Cornell University, Ithaca, NY (Ch. 8) Jones, C., Brown University, Providence, RI (Ch. 2) Junge, O., University ofPaderborn, Paderborn (Ch. 5) Kopell, N., Boston University, Boston, MA (Ch. 1) Kuznetsov, Y.A., Utrecht University, Utrecht (Ch. 4) Matano, H., University of Tokyo, Tokyo (Ch. 14) McLaughlin, D.W., New York University, New York, NY (Ch. 12) McLaughlin, K.T.R., University of Arizona, Tucson, AZ (Ch. 12) Mielke, A., Universitiit Stuttgart, Stuttgart (Ch. 15) Mischaikow, K., Georgia Institute of Technology, Atlanta, GA (Ch. 9) Mrozek, M., Uniwersytet Jagiellohski, Krak6w (Ch. 9) Nicolaenko, B., Arizona State University, Tempe, AZ (Ch. 11) Nussbaum, R.D., Rutgers University, Piscataway, NJ (Ch. 10) Polfi6ik, E, Comenius University, Bratislava (Ch. 16) Poon, L., University of Maryland, College Park, MD (Ch. 7) Raugel, G., CNRS et Universit6 de Paris-Sud, Orsay (Ch. 17) Rubin, J.E., University of Pittsburgh, Pittsburgh, PA (Ch. 3) Sandstede, B., Ohio State University, Columbus, OH (Chs. 4, 18) Sauer, T., George Mason University, Fairfax, VA (Ch. 7) Terman, D., The Ohio State University, Columbus, OH (Ch. 3) Yorke, J.A., University of Maryland, College Park, MD (Ch. 7) Winkler, S., Brown University, Providence, RI (Ch. 2) ix

CHAPTER

1

Mechanisms of Phase-Locking and Frequency Control in Pairs of Coupled Neural Oscillators*

N. Kopell Department of Mathematics and Center for BioDynamics, Boston University, Boston, MA 02215, USA E-mail: nk@bn, edu

G.B. Ermentrout Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Phase oscillators and averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Derivation of phase-difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Computing frequency, phase lags and stability for pairs of coupled oscillators . . . . . . . . . . . . 2.3. Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Frequency effects and the rise/fall time of synapses . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Interaction near a Hopf bifurcation or a homoclinic orbit . . . . . . . . . . . . . . . . . . . . . . . 3. Interactions of spiking cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Spike response method and periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Networks with inhibitory connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Spike response method and electrical coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Oscillators with multiple spiking cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Long-range synchronization of the g a m m a rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. G a m m a to beta transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Long-distance synchronization of the beta rhythm . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Changing ionic currents changes synchronization properties . . . . . . . . . . . . . . . . . . . . . . 4.5. Synchronization of excitatory cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Interactions ofbursting neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Fast threshold modulation (FTM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Electrical coupling of cells or compartments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Electric coupling of heterogeneous cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 7 7 10 12 15 16 20 20 23 27 31 31 34 35 38 40 41 42 49 51 51

*Work partially supported by NIH grant RO1-MH47150 to NK and GBE, NSF grant 9706694 to NK and a NSF grant to GBE. We thank J. Ritt and S. Epstein for careful readings and helpful comments. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved

Mechanisms of phase-locking andfrequency control

5

1. Introduction Oscillations occur in many networks of neurons, and are associated with motor behavior, sensory processing, learning, arousal, attention and pathology (Parkinson's tremor, epilepsy). Such oscillations can be generated in many ways. This chapter discusses some mathematical issues associated with creation of coherent rhythmic activity in networks of neurons. We focus on pairs of cells, since many of the issues for larger networks are most clearly displayed in that context. As we will show, there are many mechanisms for interactions among the network components, and these can have different mathematical properties. A description of behavior of larger networks using some of the mechanisms described in this chapter can be found in the related chapter by Rubin and Terman. For reviews of papers about oscillatory behavior in specific networks in the nervous system, see Gray [21], Marder and Calabrese [41], Singer [55], and Traub et al. [62,64]. The chapter is organized by mathematical structure, focusing on some of the techniques that have been found to be useful to understand the behavior of networks of neurons. The behavioral repertoire of two-component networks includes synchronized oscillations, antiphase oscillations, phase-locked oscillations with phase difference other than zero or 7r, as well as non-phase-locked solutions and steady states. We concentrate on the oscillatory phase-locked solutions, including some that occur when neither of the cells is itself an oscillator. One aim of the chapter is to provide methods for understanding what aspects of the cells and their interaction determine which of these behaviors are found, especially in complicated contexts. We pay particular attention to how changes in various time scales associated with the cells and their connections affect the behavior of the network. Though the focus is on pairs of components, we discuss implications for larger networks throughout the chapter. The component cells we have in mind are modeled by voltage-gated conductance equations, which have the form 61)

Cd-t -- -- ~

lion + IA

(1.1)

(Rinzel and Ermentrout [49]). Here lion represents an ionic current; it is the product of a driving force of the form v - VR and a conductance (inverse of resistance). Here VR denotes the "reversal potential" of the current, which depends on the ion carried by the current. The value of VR determines whether the current is depolarizing (inward, moving the voltage toward the threshold and making it easier for the cell to fire an action potential) or hyperpolarizing (outward, moving the voltage away from threshold for firing an action potential). The conductance depends in a dynamic way on the voltage, adding further differential equations; there are voltage-dependent "gates" that open or close with kinetics that also depend on the particular current. For example, for a sodium current, the conductance is usually written in the form ~,m3h, where ~ is the maximal conductance, and m and h satisfy the equations dx

at

=

-

6

N. Kopell and G.B. Ermentrout

where x = m or h. m ~ ( v ) is a monotone-increasing sigmoidal function (saturating for large and small v), while h ~ (v) is a monotone-decreasing sigmoidal function). The kinetic functions r (v) are different for h and m, and also differ between currents in the same cell; in general, the gating equations have slower kinetics than the voltage equation. Thus, the full equations are highly nonlinear, and have a range of time scales. The term IA is the applied current, sometimes referred to as the drive to the cell. It can model current injection by an experimenter, or a modulatable (time-independent) quantity representing processes slow enough to be treated as constant, and affecting the overall excitability of the cell. None of the techniques we present deals directly with the full Hodgkin-Huxley equations, though the last deals with some equations of this type. Rather, in each of the succeeding sections, we first present reduction techniques that allow us to deal with a class of simpler equations. We then present methods of analysis for that class, and some contexts in which the analysis gives answers about phase and frequency of the networks. In some cases, such as electrical coupling between cells or parts of cells, we visit the subject in several different sections, producing complementary insights with complementary methods. Section 2 concerns weakly coupled oscillators. If the oscillators have robust limit cycles, the full equations reduce (to lowest order) to ones whose interactions are through the differences of their phases. Though much has been written about such equations, e.g., (Hoppensteadt and Izhikevich [24], Kopell [31], Kopell and Ermentrout [34]), we focus here on some novel uses of these equations, including analysis of interactions with conduction delays. Another regime we discuss concerns oscillators near a Hopf bifurcation. In this case, the equations can be analyzed using normal forms that take into account amplitudes as well as phases. Using analysis and geometry, we apply these ideas to showing how interactions via diffusion can lead to non-synchrony between oscillators. Section 3 describes the interaction of spiking neurons, for which other approximations can be made that enable analysis. In this section, we treat the cells as "integrate-and-fire" (I&F) neurons. We describe the "spike-response method", a formalism that describes the time-dependent response of the voltage of the cell to the history of the pulsatile inputs that it has received. Though this method can in principle be used quite generally, it is most useful when the dynamics are fairly simple in the interval between spikes (as in I&F neurons). We use this method to see how the dynamics of the synapses affects the network behavior, including synchronization and emergent frequency; we focus mainly on inhibitory networks. We also apply the method to electrically coupled neurons to show how the shapes of the spikes can affect whether or not the electrical (diffusive) coupling is synchronizing, and how, in turn, that depends on the frequency of the network. The spike-response method can sometimes be applied to networks of excitatory and inhibitory neurons. But when there are small numbers of cells, different kinds of cells in the network, and complicated dynamical patterns, another kind of reduction, considered in Section 4, can be more effective for analysis. This reduction looks at Poincar6 maps in a neighborhood of a particular periodic solution. Though the full equations (and the Poincar6 map) can be high-dimensional, the spread of time scales in the system can allow it to be well approximated by low-dimensional maps. We use these ideas to discuss precise synchronization in the presence of conduction delays.

Mechanisms of phase-locking and frequency control

7

Section 5 deals with bursting oscillators. A bursting neuron is one that emits a fast sequence of spikes, interspersed with a well-defined inter-burst interval. Such a neuron has at least three time scales even without coupling: the time scale of the voltage change within a spike, the scale of the recovery variable within a spike, and some slower process that starts and/or ends the bursts. There has been much work on mechanisms that produce such bursting. (See Izhikevich [26] for many references, and the chapter by Rubin and Terman [52] of this volume.) In this section, we shall simplify the bursting neurons by working with the envelope of the spikes, ignoring the fast currents that produce the individual spikes. The work in this section is complementary to related work in Rubin and Terman [52]; we focus on network behavior when there are fast excitatory or inhibitory synapses, with no extra time scales associated with the coupling. We introduce geometrical ideas associated with "fast threshold modulation" and time metrics for computing synchronizing effects, and show how synaptic thresholds can affect the frequency of the coupled system. We also discuss how strong electrical coupling of "compartments" of a single cell can have some unexpected mathematical properties.

2. Phase oscillators and averaging It is not possible to analyze the general behavior of a pair of coupled nonlinear oscillators. However, if the coupling is sufficiently weak, then it is possible to reduce the behavior of a general coupled system to something much simpler. In most of this section, we work with an identical pair of oscillators that have stable limit cycles. In that case, the full system can be reduced to a single equation for the difference in the phases of the oscillators. The special structure of neural equations and the kinds of coupling between them translates into constraints on the reduced equations that can be used to understand the circumstances under which the oscillators synchronize or stabilize at other phase differences. In this section, we derive the equation, and show how to make use of the special structure. We then apply the ideas to oscillators with delays in the signals. If the limit cycle of the oscillators is only weakly stable, i.e., comparable in strength to the coupling, then the reduced equations must take into account not only the phases, but how the deviation of the trajectories from the limit cycle affects the phases. Such a situation occurs near a Hopf bifurcation. In that case, there is another reduction procedure, using a normal form, that encodes the behavior of the system. We discuss that derivation and apply it to show why coupling via discrete diffusion need not lead to stable synchrony. The analytical computations have a geometric counterpart, as we show.

2.1. Derivation o f phase-difference equations Consider a system of two similar coupled nonlinear oscillators: dXi dt

= Fi(Xi) + eGi(Xi,

Xj),

X i ~ R'", i -- 1,2, i :/: j,

(2.1)

8

N. Kopell and G.B. Ermentrout

with 0 < e << 1. Suppose dXi = Fi(Xi)

dt

has an orbitally asymptotically-stable limit cycle, X~ (t), with period Ti = T + Oi (6). (For this we assume that IF2 - Fll ~- O(e).) Then for sufficiently small e the solutions to (2.1) have the form Xi (t) = Xo(Oi) + Oi (e) and, up to lowest order in e, dO1

- - 1 + e l l 1 (02 -

dt dO2 -~ 092 -+- 8H2(01 dt

(2.2)

01),

02),

where 01,02 a r e coordinates on a toms and represent the phases of the two oscillators. 1 and 032 are the frequencies of the uncoupled oscillators, with I1 - o921 - - 0 ( 6 ) . A rigorous derivation (Ermentrout and Kopell [ 14]) uses center manifold ideas to get equations on a toms; the use of averaging then shows that the interactions are (to lowest order) through the difference of the phases. A formal method producing Equation (2.2) in a computationally simpler way is given in Ermentrout and Kopell [16]. The functions Hi are T-periodic in each of their arguments and can be explicitly computed from the original equations. Let 4~ = 02 - 01. Then

1 f o T X , (t) . Gl (Xo(t) , Xo(t + O ) ) d t . Hi (0) = -~

(2.3)

A similar formula holds for H2 with the ~b replaced by -q~. The function X* (t) is the normalized T-periodic solution to the adjoint equation"

dX* = -DF(Xo(t))TX dt

*,

X* (t) . X~o(t) -- 1.

The average in (2.3) is easy to calculate; the only difficulty lies in the computation of X* (t). Williams and Bowtell [73] devised a simple way to compute X* (t) for a stable limit cycle: Start with random initial conditions and integrate

dy dt

--

-

DF(Xo(t)) T y

backwards. Since all Floquet multipliers of the periodic solution are outside the unit circle (except for a multiplier of 1), backward integration will converge to the periodic solution. Thus, one can calculate the adjoint and the interaction function (2.3) for any given differential equation with a stable limit cycle. So far, the form of the coupling between oscillators has been very general. Coupling between neurons is typically very special and is usually through only one component in the equations, the membrane potential, Vi. The coupling is generally additive and has one

Mechanisms of phase-locking and frequency control

9

of two different forms depending on whether the synapse is chemical or electric. In the first case the coupling from cell j to cell i has the form: /coupling-- gijsj(t)(Vi(t) - Vij,syn).

The variable sj(t) is related (perhaps in a very complicated fashion) to the potential Vj (t) and gij, Vij,syn are constants. For example, sj (t) may satisfy an auxiliary equation, such as dsj

-- k ( g j ) ( 1

- sj ) - ;s--L-J.

dt

r

Let V* (t) be the voltage component of the adjoint and Vo(t) the voltage component of the periodic trajectory of the uncoupled system. We let so(t) be the periodic solution to the equation for s when V is set equal to Vo(t). Then, from (2.3), the coupling function of cell 2 to cell 1 becomes

HI (dp) = -g12 7 f0T V* (t)so(t -Jr-~b) (Vl2,syn - v o ( t ) ) d t . A similar formula holds for H2, which is a function of - r by replacing r by - r and interchanging the indices. There is a nice intuitive interpretation of the voltage component of the adjoint: it is an infinitesimal phase response curve (see Section 4.5 for more details). That is, it gives the effect on the phase of the oscillator when the latter is perturbed with a brief positive current pulse, i.e., a delta function. Thus, the function H1 (r is obtained as the convolution of the infinitesimal phase response curve with the current so(t + q~)(VlZ,syn -- Vo(t)) that carries the coupling signal. The latter can be experimentally measured so that, even in the absence of an explicit membrane model, it is possible to derive a phase equation description for weakly coupled neurons. The other kind of coupling considered here, through electrical synapses or "gap junctions", is much simpler in form. The coupling is directly dependent on the difference of the potentials, i.e., the coupling from cell j to cell i is a discretized form of the Laplacian: /coupling = gij ( Vj - Vi ).

In this case, (2.3) implies that the associated coupling function is

gij fo T V*(t)(Vo(t + r Hi (r -- --T-

- Vo(t))dt.

One important distinction between gap junctions and general synapses is that for gap junctions, H1 (0) - - 0 whereas for synapses, Hi (0) will not generally vanish. As we will see, the size and sign of Hi (0) is relevant to the frequency of the coupled system, which need not be the same as that of the uncoupled oscillators. Furthermore, if Hi (0) ~- 0, for at least one of the coupling terms, then larger arrays of oscillators need not synchronize (Kopell and Ermentrout [33,34]).

10

N. Kopell and G.B. Ermentrout

2.2. Computingfrequency, phase lags and stability for pairs of coupled oscillators The main reason for the development of the present theory is to study the synchronization properties of coupled neural oscillators. We start by considering a pair of mutually coupled identical oscillators with symmetric coupling. Then Equations (2.2) hold with H1 : H2 and col = 1. Let ~p = 02 - 01 as above. We can rewrite this system as

d~

dt

dO1 dt

=

--26Hodd(~b),

(2.4)

-- 1 + eH(q~),

where Hodd is the odd component of H. Phase-locked solutions are solutions for which the phase difference 4~ is constant at a root of Hodd(4~). Since Hodd is an odd periodic function, there are always at least two roots, synchronous (tp -- 0) and antiphase (4) = T/2). The frequency of these solutions is s = (1 + eH(~))/T. Note that, for the synchronous solution, synaptic coupling changes the frequency by ell(O)~ T, but electrical coupling keeps the frequency the same. One of the goals of the analysis is to determine the stability of the synchronous and antiphase solutions and how it depends on the parameters in the models. It follows from (2.4) that a phaselocked solution ~b is stable if and only if H~dd(tp) > 0. In particular, the synchronous solution is stable if and only if H' (0) > 0. For synaptically coupled neurons, stability of synchrony can be investigated by looking at the sign of the integral

Hi(O) = ~1 fo T V*(t)s~(t)(Vsyn- Vo(t))dt

(2.5)

For neurons coupled with gap junctions,

lf0

. ' (o) = ~

v* (t) v/~(t) dt.

The key to understanding situations that lead to synchrony for weakly coupled neuronal oscillators is found in the evaluation of these integrals. In Figure 2.1 we illustrate these integrals for excitatory synaptic coupling, in which the driving term (Vsyn - Vo(t)) is positive. (For inhibitory coupling, the driving term is negative, changing the sign of H' (0).) In Figure 2.1A, the adjoint is non-negative, as is the case in many simple cortical neural models. For an excitatory synapse that persists for a few milliseconds, the function so(t) rises quickly and then decays more slowly; thus the derivative, s~ (t) is positive only very briefly during the rise time and is then negative after that. This implies that the product, V* (t)s~ (t), is negative for most of the time (bottom of Figure 2.1A) and so the integral (2.5) is negative and synchrony is unstable. On the other hand, in cortical models that have spike-frequency adaptation (an additional outward ionic current that tends to slow the firing down), the PRC/adjoint can be negative for some time after the spike. The intuition behind this is that,

Mechanisms of phase-locking and frequency control

r

V*(t)

-- "

s J

I

I

V*(t)

I

I

I

I

,,"

S

I

I

I

s'(t)

's~.

I I

11

I l 1

' , ~ J s'(t)

I I I

s .A

t

T

A

V*(t)s'(t)

-

B

t

/"

A2

t

Fig. 2.1. The adjoint, V* (t) plotted along with the derivative of the synaptic conductance, s' (t) (upper panels) and the resulting product (lower panels). If the sign of the integral of the product is positive (negative) then synchrony is stable (unstable). A. A negative product in a neuron without adaptation; clearly the area A1 is larger than A2. B. A positive product in a neuron with adaptation; the adjoint has a significant negative region to the fight of the origin.

when there is such a current, stimulation after the spike creates more outward current, and thus delays the onset of the next spike. In Figure 2.1B, we depict this situation. The result is that the product V* (t)s~ (t) is now primarily positive and thus the integral (2.5) is positive and the synchronous solution stabilizes. A similar analysis can be used to show that gap junction coupling is stable for systems with a PRC shaped like V* in Figure 2.1A. PRCs of this shape occur when the spike is very narrow; V~(t) is positive for most of the cycle corresponding to a slow depolarization (increase in voltage) leading to the next spike. Thus the product of V~(t)V*(t) is mainly positive resulting in a positive integral and a stable synchronous state. Adaptation does not change this picture. The key observation is that, for very narrow spikes, the timewindow in which the potential is decreasing is small and occurs at a time when the PRC is small in magnitude. Thus, even though this derivative is quite large and negative, its effect is strongly diminished by the neuron's inability to respond to stimuli at that point in time. In Section 3.3. we use other methods to show that changing the shape of the spike can change this conclusion, especially at high frequencies. In Section 2.5 we also consider oscillations that are more gradual (less spikelike), and show why gap junctions can destabilize synchrony. If the oscillators are not identical, but are close in frequency, a phaselocked solution can be found from (2.2). We may also relax the assumption that the coupling is symmetric. If the oscillators are to be phaselocked, we must have dO]/dt = dO2/dt, so the analogue

N. Kopell and G.B. Ermentrout

12 of (2.4) is

dcb/ dt = e ( A + H2(-q~)

-

nl (~)),

where eA = 1 - 092. It can be seen from this equation that the result of unequal frequencies and/or differences in the coupling is that a phase-locked solution does not necessarily lock at synchrony, but with a phase-lag that compensates for the difference in frequency and/or coupling. In Section 4, we will contrast this behavior with that of relaxation oscillators, which can synchronize even when the oscillators are different and the coupling is asymmetric.

2.3. Delays As we mentioned above, the function sj(t) can be quite complicated in its relationship to the potential, Vj. For example, suppose that the synapse occurs on a dendrite some distance from the cell soma where the oscillations are generated. Then the effect of this distant synapse must be filtered through the dynamics of the dendrite. In the case of passive dendrites, the filtering is linear and

f0 cX3R ( t ' ) s o ( t - t ' ) d t ' ,

sR(t) =

(2.6)

where so(t) is directly related to the potential of the presynaptic neuron and R(t) is the filter function. In the simplest case, the filter can be a simple delay, R(t) = 6(t - r) so that SR (t) = so(t -- r). Because the delays are linear, it is very simple to compute the effect of these on the averaged coupling functions. Let

Ho(,)

-

l for V*(t)so(t + ~b)(Vo(t)- Vsyn) dt

be the interaction function in absence of the filter R, as in (2.3). Then, using (2.6) and changing the order of integration, we get that the full interaction function is just HR(*) --

R(t')Ho(c~-t')dt'.

f0 ~176

(2.7)

That is, the true interaction function is obtained just by averaging the restricted interaction function against the functional R. In particular, delays lead to a phase translation of the interaction function: H ~ ( O ) = n0(4~ - r ) .

In many neural systems, there exists the possibility of delays at many levels. There are delays due to axonal propagation and synaptic transmitter diffusion, and delays encountered when synaptic input occurs far from the source of the oscillation and must be communicated through dendrites. A delay, whether explicit or via a functional relationship

Mechanisms of phase-locking and frequency control

13

like (2.6) can cause the rising phase of the synapse to occur during the maximal sensitivity of the phase-response curve, and thus the integral will be positive (negative) for excitatory (inhibitory) synapses and synchrony will be stable (unstable). This is the principle that underlies the results of van Vreeswijk et al. [65], Crook et al. [ 11,12], and many others. This is illustrated in Figure 2.2 for the case of a pure delay in the synapse. As we mentioned above, one source of the delay can be the propagation time down the axon. In this case, r = d / v where d is the distance between the two neurons and v is the axonal conduction velocity. Thus, neurons that are connected at different distances can have different synchronization properties solely due to this conduction delay. These ideas were exploited in Crook et al. [ 11 ] in a one-dimensional spatial network of cells. In this model, the excitatory synapses were synchronizing with short delays but desynchronizing with long delays. Thus, if most of the connections are local, synchronized activity is stable. If connections between cells have a longer range, then the delay destabilizes the synchronous solution and waves result. Similar models involving inhibitory synapses show that delays can change synchronization properties. Another source of delay occurs due to the passive propagation of potential down the dendrite of a neuron. Crook et al. [ 12] show that the position of a synapse on a dendrite (Figure 2.3) can alter the stability of the synchronous solution. For simplicity, we assume

V (t)

T s~(t +I:) Fig. 2.2. An appropriate delay r can shift the positive part of sf(t) toward a region where the adjoint is large and positive, thus stabilizing synchrony since the area of the product will be positive (see Figure 2.1).

Fig. 2.3. Diagram of two neurons that make synaptic connections along the dendrite.

14

N. Kopell and G.B. Ermentrout

that the d e n d r i t e c a n be m o d e l e d as an infinite c a b l e satisfying: OV rm m Ot

- -

--

-~- ~ 2 02 V

V

Ox 2 '

w h e r e rm is the m e m b r a n e t i m e c o n s t a n t a n d )~ is the s p a c e c o n s t a n t o f the m e m b r a n e . To c o m p u t e the effect o f a s y n a p s e that is a d i s t a n c e d a w a y f r o m the a c t i v e l y o s c i l l a t i n g s o m a , w e use the G r e e n ' s f u n c t i o n for p a s s i v e cables: 1

G(x,t)

-

e

--t /rm--(x /Z)2/ (2t /rm)

v/4zc z z t / r m

R(t)

n o

25

40

Soma

,

,

,

,

20

-25

0 -20

-75

,

I

,

10 '

I

I

i

I

20 '

I

-50

i

30 '

I

I

i

40 '

I

-40

50

0

2

'

Dendrite (1)

10

20

,

,

1 0

-60

-1

-70

,

I

,

10 -63

.

I

,

I

,

20 ,

-64

,

I

,

30 ,

,

-2

i

40

50 ,

0

10

20

10

20

0.7

Dendrite (5)

-65

0

-66 -67 '

-68 0

10

20 t (msee)

' 30

'

J 40

'

-0.7 50

0

t (msee)

Fig. 2.4. Left: The response function R (t) for synapses on the soma or different compartments of a dendrite of the post-synaptic cell. Right: The odd part, H0, of the corresponding interaction function. Somatic synapses lead to a small stable phase-lag between identical cells and both the synchronous and anti-phase solutions are unstable. A dendritic synapse near the soma results in bistability of the synchronous and anti-phase states. If the synapse is further away, only synchrony is stable.

Mechanisms of phase-locking and frequency control

15

In this case, our filter is simply R(t) = G(d, t). Figure 2.4 illustrates R(t) for three different distances from the soma and the corresponding function H (40. From (2.7), we have

H~(O) =

R(t')H~(-t')dt'.

f0 G

Thus, the effect of the dendrite is similar to the delay. For example, suppose that H6 (0) < 0 so that synchrony of a directly coupled synapse is unstable. Then, by putting the synapse farther out on the dendrite, we can weight the above integral to a point where H~ is positive and thus stabilize the synchronized state. By adding additional structure to our simple point neurons (such as passive dendrites or axonal delay) we are able to alter the synchronization properties of a pair of mutually coupled cells. Any linear manipulation of the timing directly translates into a weighted average of the original interaction function through (2.7). Bressloff and Coombes [5] offer a complete analysis of coupled integrate-and-fire neurons with dendritic delays. We discuss the general ideas later in the chapter.

2.4. Frequency effects and the rise~fall time of synapses In the arguments above, we have altered either the phase-response curve (through adding an adaptation current) or the timing of the synapse. If we hold the timing of the synapses fixed and vary instead the frequency of the underlying oscillations, we see that the effect is almost as if the synaptic timing is varied. This is illustrated in Figure 2.5 for a biophysicallybased cortical neuron model with a fast excitatory synapse (rise time is 0.05 msec and

1

Ho(~) 0.5

0

-0.5

-1 0

0.2

0.4

0.6

0.8

Fig. 2.5. The odd part of the interaction function for a fixed synaptic time at three different frequencies. At high frequencies, only the anti-phase state is stable.

16

N. Kopell and G.B. Ermentrout

.5

o

-.5

m

m

i

m

m

i

m

m

m

=n

Frequency Fig. 2.6. Schematic bifurcation diagram with frequency as the parameter showing stability of the synchronous state at low frequencies and stability of the anti-phase state at high frequencies.

decay is 2 msec). The odd part of the interaction function is illustrated. Zeros are phaselocked solutions and those with a positive slope are stable and those with a negative slope are unstable. As the frequency decreases the antiphase state loses stability through the appearance of an intermediate locked state. This stably persists until the frequency is quite low and the synchronous state becomes stable. One can summarize this behavior via the bifurcation diagram shown in Figure 2.6. The analogous behavior is seen for inhibitory coupling but the stability is reversed. Van Vreeswijk et al. [65] obtained a similar result with I&F neurons but the frequency was held fixed and the timing of the synapse was varied. As we show in Section 3, similar results can be obtained using the spike-response methods, even when the coupling is strong. In that case, however, the frequency of the coupled system is determined only implicitly.

2.5. Interaction n e a r a H o p f bifurcation or a h o m o c l i n i c orbit Near a Hopf bifurcation, the resulting periodic orbits are only weakly stable. Thus, trajectories that have initial conditions off the limit cycle may take a significant amount of time to return to the latter; during this time, the dynamics can act on the system to stabilize or destabilize a configuration. For most kinds of neural coupling, the assumption that the system is near a Hopf bifurcation is not very useful. The reason is that supercritical (i.e., stable) Hopf bifurcations are of low amplitude, and these are not large enough to trigger chemical synaptic interactions, which are via all-or-nothing spikes. (But low-amplitude oscillations can contribute to interactions of spikes. See Section 4.5.) For coupling involving electrical synapses (gap junctions), small amplitude oscillations can be sufficient to carry coupling signals. In the rest of this subsection, we discuss the normal form associated with coupled oscillators, each near a Hopf bifurcation. We then use these and some geometric ideas to show why the discrete version of diffusion need not be stabilizing. Near a Hopf bifurcation, the equations for a single oscillator can be expressed compactly using complex notation z (instead of a 2-dimensional real system), and neglecting terms

Mechanisms of phase-locking and frequency control

17

higher than the third power. With appropriate change of coordinates, and linear coupling between the oscillators with variables z l and z2, the truncated coupled equations take the form (Aronson et al. [2]): dzl = lzI(a + bi)zl - (p + q i ) z 2 z l + v(d, + id2)(z2 - zl), dt dz2 = #2(a + bi)z2 - (p + iq)z2z2 4- v(dl + id2)(Zl 22). dt -

-

The parameters here are real, and we will consider only the case #1 = #2 - - #, a small parameter measuring the distance to the Hopf bifurcation. There are two conceptually important parameters to consider here. One is rl = q / p , which measures the "twist" in the system for an individual oscillator. This parameter describes how much the local angular velocity at any point off the limit cycle deviates from that on the limit cycle. The second important parameter is ot = d2/dl, which measures how much the coupling deviates from being scalar (a real number times the difference between the positions of the two oscillators). To understand why these parameters matter, we consider a simple case motivated by an example of Sherman and Rinzel [54], who showed numerically that weak gap-junction coupling (in cells of the pancreas) could lead to antiphase solutions. The equations have limit cycles that are relatively sinusoidal in shape, without spikes or well-defined plateaus as in the relaxation regime of the van der Pol equation. For illustration, we use the MorrisLecar equations (Morris and Lecar [46]), in the non-relaxation regime. These equations (uncoupled) have the form 61)

C-dt dw dt

=

--[gcamoc(V)(1)

-- [ w o c ( v ) - w ] / r .

--

VCa) + gKW(V -- VK) + gL(1) -- VL)],

(2.8a) (2.8b)

The following idea shows why the twist and the deviation from scalar coupling can conspire to desynchronize the synchronous state. Consider a limit cycle with two pairs of initial conditions, and the direction those trajectories are pushed by the coupling (Figure 2.7). Recall that neural oscillators interacting through gap junctions have a coupling term only for the v-component; hence the coupling pulls each trajectory horizontally. Consider the bottom pair of initial conditions, where the limit cycle has increasing v and increasing w. The leading trajectory is pulled inward and the lagging one is pulled outward. Here is where the twist is relevant: the cells are pulled somewhat off the limit cycle, and in the new position moves faster or slower compared to the limit cycle. The change in speed cannot be seen from the phase-plane diagram, but can be seen from the second equation, which says that the rate of change of w is proportional to the vertical distance to the w-nullcline. The leading oscillator is pulled inward, and hence closer to the nullcline, while the lagging oscillator is pulled outward and hence farther from the nullcline. Thus, in that portion of the trajectory, the leading oscillator slows down and the lagging one speeds up, creating a synchronizing effect. A similar analysis shows that the opposite is true for the pair of conditions in the region in which v is decreasing while w continues to increase. Thus, there

18

N. Kopell and G.B. Ermentrout

Fig. 2.7. Diffusive coupling can be synchronizing or desynchronizing in different portions of the limit cycle. For the lower pair of initial conditions, the coupling pulls nearby trajectories closer together, while the opposite is true for the upper pair. See text for explanation.

are synchronizing and desynchronizing effects along different portions of the trajectory, and whether the synchronous state is stable depends on a balance of those effects. The normal form contains the essential information about this balance. For the equations of Sherman and Rinzel, deVries et al. [66] carried out the normal form analysis. As in Aronson et al. [2], they do a change of coordinates to polar form, with

zj -- ~/lZJ 7 r ja

eiOj.

If #2 = #l = # the equations for the polar variables are given by /'1 = rl (1 -- r 2) + y (cos4~ - o t sin ~b)r2 - y r l , /'2 -- r2(1 - r 2) + y(cosq~ + c~ sin~b)rl - yr2,

-- + ~ ( r 2 . A r 2).

g ( s. i n ~ +. ot cos~b) r2

y (sin 4~ - ot c o s r rl

rl

r2

where 4' = 02 - 01 and y = v d l / # a . Note that y may be interpreted as the strength of the coupling compared with the attraction of the limit cycle. It was shown in Aronson et al. [2] that the synchronous solution is stable if and only if

•

+ (1

> o.

This implies that if c~O > 0, the synchronous solution is always stable; if this product is sufficiently negative, then y must be large enough to produce stability.

Mechanisms of phase-locking and frequency control

19

The condition for stability of the antiphase solution is ~, < 1 / 4 , 0 < V(1 +or 2) + (1 + c~r/)(2V- 1). This implies that, if (1 + or0) < 0, then for small enough coupling, the antiphase solution is always stable. DeVries et al. [66] computed the values of the parameters in the normal form, and showed that the antiphase solution is stable, and the synchronous one is unstable. We note that the parameters c~ and 0 also turn out to be important in creating spatiotemporal chaos in the complex Ginzburg-Landau equations (Kuramoto [37]) zt -- )~z - (1 + i o)z2-~ + y(1 + iot)zxx.

A situation in which the desynchronization can be seen immediately from the geometry, without a normal form analysis, is given in Han et al. [22]. In this situation, the limit cycle is near a homoclinic orbit and thus spends a great amount of time near the saddle point. Suppose that the trajectory of each oscillator comes close to the stable manifold of the saddle point (Figure 2.8). Consider initial conditions in which both trajectories are approaching the saddle point. The trailing oscillator will be pulled outside of the limit cycle while the leading oscillator will be pulled inside the limit cycle. Thus, the trailing oscillator will be pulled closer to the stable manifold of the saddle while the leading oscillator is pulled further away. The oscillator on the trajectory nearer to the stable manifold moves slower, and hence the phase-difference widens. A similar analysis holds for initial conditions in which the trajectories are moving away from the saddle point.

W

>

u

Fig. 2.8. Illustration of why diffusive coupling between voltages near a homoclinic destabilizes synchrony. The stable and unstable manifolds and the limit cycle are illustrated. On the left part of the limit cycle (near the stable manifold) diffusion pulls the slower oscillator closer to the stable manifold and the faster oscillator away from the manifold. Thus the faster cell goes even faster and the slower one goes slower, so that they separate in phase. The same reasoning holds for initial conditions on the right hand side of the rest point.

N. Kopell and G.B. Ermentrout

20

A final remark is that the destabilization of the synchronous state by twist and non-scalar coupling happens in a range well described by a normal form near a Hopf bifurcation; it fails when the oscillations are in a relaxation regime. The reason is that the trajectories are held so closely to the nullclines that the twist has too minor an effect to change stability.

3. Interactions of spiking cells 3.1. Spike response method and periodic solutions Most neurons fire all-or-nothing action potential (spikes), and the interactions of the neurons are via signals induced by the receipt of the spikes. In some simple cases, this leads to a different kind of reduction than the weak-coupling limit. This subsection of the review is taken from Chow [8]. The simplest version of such a reduction assumes that the nonlinearities of currents in (1.1) are localized in time to the action potential. That is, away from the spike, the voltage equation has the form

dl)

dt

--l--v.

As described in Section 2, this is known as an integrate-and-fire model. When the potential reaches threshold, a delta function pulse is subtracted to reset the potential to v = 0 and we say that the cell has spiked. Though this may appear too highly abstracted to be useful, it is a reasonable approximation when the behavior between the spikes is essentially linear. A biophysical model that behaves in a very similar way is one reduced from a compartmental model of Traub and Miles [61 ]. (See appendix of Ermentrout and Kopell [ 17] for the reduction.) With synaptic current, the equation for the i th cell is

dvi = Ii- Ui- Z6(tdt l

i tl)-kJ ~

S (t - t / ) ,

i=/=j,

(3.1)

l

where tIi are the times that cell i reaches threshold, J > 0 for excitatory interactions and J < 0 for inhibitory interactions. The functions 3 (the Dirac delta function) and S above are each defined only when the argument is non-negative. The synaptic current S can be modeled in various ways. One simple and explicit version has the current induced from each spike as a double exponential

S(t) - e x p ( - f l t ) - exp(-c~t),

(3.2)

where fl is the synaptic decay rate and ot is the synaptic rise rate. Note that this describes the synaptic current explicitly as a function of t, rather than obtaining it as a solution to an auxiliary differential equation as in Section 2. With ot >/3, there is a rapid rise, followed by a slower decay, as in solutions to auxiliary equations described in Section 2.

Mechanisms of phase-locking and frequency control

21

A central idea in working with some networks of spiking cells is to consider the effects of the spikes, rather than the differential equation (3.1). This leads to a formalism known as the spike-response method (Gerstner [ 19], Chow [8]). The dynamics are encoded in two sets of kernels, representing the effects on a cell of its own spikes and those of the other cell. That is 7 L ~ r l i ( t - - t l ) - + - J Zi 6 1

ui(t)--Ii

i (t --

t/).

(3.3)

l

The kernels are zero for t - t~ < 0. The kernel r/i (t) describes the recovery dynamics of neuron i after it fires, while the kernel 6i (t) represents the response to synaptic inputs (see Figure 3.1). Arguments using the kernels for integrate-and-fire, which can be explicitly computed, generalize to other kernels having similar qualitative properties. But the theory has some constraints. For example, it assumes that the form of the synaptic kernel does not depend on the time of arrival or phase of the synaptic input. Thus the interaction kernel has the form t

e(t)--

Z

L(s) S ( t - s ) d s ,

(3.4)

where S(t) is the synaptic input and L(t) is the linear response function. With a prescribed synaptic signal, it is possible to address questions about existence and stability of periodic solutions in the network. One such way is to look for self-consistent solutions, taking into account all the history of the previous spikes. Suppose that the cells have been firing with (so far unknown) period T, with neuron 2 lagging neuron 1. We pick the origin in time so that cell 1 has fired at t/ - - l T, and cell 2 has fired at t~ -- (4> - l)T, l = 1, 2 . . . . . where 0 ~< 4) < 1. Cell 1 fires again at T = 0, and then self-consistency requires that 1 -- vl (0) -- II + Z

rl(IT)+ Z

l>/1

3.0

1.5

~T).

Je(IT

(3.5a)

l>/1

-

0.3

-

0.2

"

"~

~Ib

"~ 0.1 0.0 -1.5

I 0

,

I 2 t

,

.......

I

I

4

0

,

I

,

5

I I0

,

I 15

t

Fig. 3.1. Examples of kernels O(t) and e(t). The two curves for e(t) correspond to two choices for the form of the synaptic input S (t).

22

N. Kopell and G.B. Ermentrout

Cell 2 fires again at t = r T, and self-consistency then implies that 1 = v2(r

=/2 + E

rl(1T) + E

Js(1T + ~T).

(3.5b)

The difference in the index of the second sum in (3.5a) and (3.5b) arises because cell 1 fires before cell 2. Hence, the input to cell 2 (from all negative times) includes the 1 = 0 input from cell 1, which occurs at t = 0. However, the 1 = 0 firing of cell 2 is at t = ~bT > 0, which is not included as an input to cell 1. One can derive a relationship for the phase ~b by subtracting (3.5b) from (3.5a) to get 0-= Ii - 12 - Gr(q~),

(3.6a)

GT (~b) = J ~-"~(s(IT - T + q~T) - s ( Z T - dpT)).

(3.6b)

where

l>~ l

Note that, from (3.4), s(0) = 0; this implies that G r ( 0 ) = 0 = GT(0.5). Hence, if the neurons are the same (I1 = I2), there is a synchronous solution (4~ = 0) and an antiphase solution (~b = 0.5). The difficult part is determining stability. The consistency equations do not give a map that takes a phase difference 4~ between the cells at some time to the phase difference at the next cycle, only a condition to be satisfied at steady state. Hence, other arguments are needed to deal with stability. We refer the reader to Chow [8] and Bressloff and Coombes [4] for the stability argument. This is subtle because the system is infinite-dimensional; one needs to know information about all past cycles. Chow [8] looks at small perturbations around a periodic phaselocked solution; assuming that the kernels decay very fast, so that they are negligible after some time, he derives a high-dimensional return map. The analysis shows that a necessary, but not sufficient, condition for stability is that G} (~b) > 0. In related work, van Vreeswijk et al. [65] argued heuristically that this condition was associated to stability. Gerstner et al. [20] did a related analysis with N ~ cx~ neurons; this allows the population to force a single neuron without itself being affected by that neuron, creating a somewhat easier problem. The methods of Chow [8] can also be used for larger networks. We can now see how this model is related to the weak coupling theory. Indeed, if Il = I2 = I (that is, the oscillators are identical), then we must solve a pair of equations for the ensemble period, T and the phase, r 0 = -Gr(r 1 = I + O(T) + Jg(T, dp), where O(T) and g(T, ~b) are the first and second sums in (3.5). This set of equations should be compared to phase-locked solutions to (2.4): 0 -- --28Hodd(~b),

Mechanisms of phase-locking and frequency control 1 --

T

=

23

1 + ell(g)).

Note that for weak coupling, the phase is determined independent of the period. For the spike-response model, GT (4~) changes with T, and so the non-synchronous solutions have frequency-dependent phases. (GT(0) = 0 for all T.) In the case of weak coupling for the spike-response model (J small), we can approximate T by the uncoupled period and obtain a function G T that is essentially (to lowest order) independent of T. Furthermore, since the period does not vary to lowest order, the stability is reduced to that of a 1-dimensional system, as in the first equation of (2.4). The condition G~(~b) > 0 is a sufficient condition (and, as above, necessary) for stability for the weak coupling case and, up to a scaling factor, GT -- Hodd. Bressloff and collaborators have extended the theory of coupled integrate-and-fire models to arrays and networks where they then use the results of Kopell and Ermentrout on large systems of phase models (Bressloff and Coombes [6]). This is possible since the resulting equations are functions of phase-differences only and fixed points correspond to periodic phaselocked solutions. As in the case of two oscillators, the main technical difference is that the ensemble period is implicitly defined while in the weakly coupled case, it is explicit. Bressloff and Coombes [5] have applied the techniques of this section to a pair of integrate-and-fire neurons coupled through passive dendrites in a manner similar to the work of Crook et al. [ 12]. In their work they can explicitly compute the stability boundaries for synchrony as a function of the position of the synapse on the dendrite for arbitrary coupling strength since the model is essentially linear. They have also studied the role of synaptic depression and other effects on the phase-locked behavior of coupled integrateand-fire oscillators. They have developed elegant machinery for analyzing arbitrary networks of integrate-and-fire neurons that goes beyond the special cases considered in this section.

3.2. Networks with inhibitory connections Many theoretical papers have remarked that inhibition may play a more important role in synchronization of networks than excitation (van Vreeswick et al. [65], Gerstner et al. [20], White et al. [71]). Hence we focus on inhibition; similar methods work for excitatory connections and show, as in the case of weak coupling, that such an interaction is not synchronizing. For inhibitory networks, we show below that the inhibition provides both the synchronizing signal and determines the frequency of the coupled network. The work is motivated by experimental work on a slice preparation of the hippocampus, in which the action of the excitatory cells is blocked and the relevant part of the network contains only inhibitory cells (Whittington et al. [72]). 3.2.1. Synchronization by inhibition. For inhibitory connections, we set J = - 1 . (The size of J is taken into account in the definition of e.) Starting from formula (3.6) for

24

N. Kopell and G.B. Ermentrout

the function GT(Ck) and an explicit form for the synapses, we can compute the necessary conditions for stability of synchrony, namely G~ (0) > 0. We first note that, from (3.6), G'~(O) - - T s ' (O) - 2 T ~

s' (IT).

Thus, the synchronous solution is unstable if s'(0) > - 2 Y~4>~t s'(1T). For a particular form of the synapse, such as the double exponential (3.2), the s kernel can be explicitly l computed, and used to compute G T (0). It can be shown that if the rise rate ot is infinite, then the necessary condition is violated; otherwise the necessary condition is always satisfied for integrate-and-fire neurons. It can be shown that some sufficient conditions are also satisfied except possibly for some very small periods. Van Vreeswijk et al. [65] made the same argument for the special case of the standard integrate-and-fire model with synapses of the form s(t) = at exp(-ott). The same conclusion is reached by working with weak coupling methods, as in Section 2.2. Indeed, the function s(t)(Vsyn - V) in that section plays the same role as the explicit synaptic function S(t) plays in computing s(t) in (3.2). The integrate-and-fire formalism captures some essential behavior of what has been called "type 1" neurons (Rinzel and Ermentrout [49]). These are neurons for which the transition between stable equilibrium and periodic firing (as applied current is increased) is via a saddle node. "Type 2" neurons are associated with Hopf bifurcations near the onset of periodicity with increasing drive. As we will see below, the differences between type 1 and type 2 affect synchronization properties. We remark here that type 2 cells have behavior that is not captured by the integrate-and-fire formalism; in particular, such neurons need not synchronize with inhibition. For a kernel approach to that behavior, see Kistler et al. [30]. For type 1 neurons, the stability analysis shows that the synchronous solution for homogeneous cells is stable over a very wide range of periods (changeable by modulating the applied current). However, the stability is fragile when there is heterogeneity, e.g., in the drive I. (Relevant simulations are in Wang and Buzs~iki [68] and White et al. [71 ].) For heterogeneous neurons, a similar analysis can be done (Chow [8]). When the heterogeneity is too large, there is no locked solution. For smaller heterogeneity, there is a locked but not synchronous solution. The stability of the latter turns out to depend partly on the emergent frequency of the network, which in turn depends on applied current as well as the synapses (White et al. [71 ], Chow et al. [ 10], Chow [8]). The close-to-synchronous solutions are stable only in a parameter regime in which the network frequency is governed by the decay time of the inhibition (see Section 3.2.2 below). The system can be driven so that individual cells fire at much higher frequencies, but in that case the effect of the heterogeneities is to produce phase drift between the oscillators. In this regime, the synaptic current does not have enough time to change significantly within a cycle, and hence acts effectively as a DC current. This regime has been called the "tonic" regime, for the tonic (i.e., constant) behavior of the coupling current (Chow et al. [10]). If the cells are driven at much lower levels, the effect of the heterogeneity is to cause the faster cell to suppress all firing in the slower one. The highest degree of synchrony occurs when the drive is such that the frequency of the coupled system is closely related to the decay time of the inhibition, a parameter regime

Mechanisms o f phase-locking and frequency control

25

that has been called the "phasic" regime. As described in Section 3.2.2, the dependence of the network frequency on the decay time of the synapse is different in the tonic and phasic regimes. We can get more insight into the way inhibition synchronizes if we consider the case in which only the previous spike from a pre-synaptic neuron is significant in affecting the timing of the post-synaptic neuron. This is a reasonable assumption in the phasic regime. Let tl and t2 be the times of spiking of a pair of coupled cells on some cycle. Assume t2 > tl, but sufficiently close that the firing of cell 1 does not prevent the firing of cell 2 in that cycle. We start with the analogue of (3.3), namely v i ( t ) -- Ii + ~ ( t - ti) - ~(t - t j ) ,

j =7t=i.

(This ignores the effect of cell 1 on cell 2 in the current cycle, which is permissible if the cells are close together, since ~(0) -- 0.) Cell i fires again when vi (t) -- 1, at t = i/. We wish to compute t2 - t i = A and compare this to t2 - t l -- A to check for stability of the synchronous solution. For simplicity, we again assume that the driving currents Ii a r e equal. The definitions of ti imply that _

,(i.

-,.)

-

1 -

To linearize the equation around the values associated with one cycle later, we let Tp be the period of the coupled synchronous solution, and define pi via ti -- ti -- Tp + Pi. Using the relations il - t2 - Tp + Pl - A and {2 - tl -- Tp + P2 + A, we get r l ' ( T p ) ( p 2 - P l ) -- s'(Tp) [(p2 - Pl) + 2A].

Using P2 - Pl = A -- A, we have A--

~ ' ( T p ) + r/'(Tp)

e'(Tp) - r l ' ( T p )

A.

Note that r/' (Tp) > 0 and we are assuming that Tp is such that e' (Tp) < 0, i.e., the period is high enough that the synapse is decaying when the next spike appears. (See Gerstner et al. [20].) Hence the coefficient of A is less than 1 in absolute value. Thus, if the rise of the synapse is not infinitely fast, we recover the result from the above theory that the condition e' (Tp) < 0 is a necessary condition for inhibition to be synchronizing. But the calculation also shows that the condition e'(Te) < 0 is sufficient for synchrony, and that the size of r?' (Tp) helps determine the rate of synchronization near the synchronous state; if r/' (Tp) is small, as it would be if the period is too long relative to the recovery period of the neuron, the rate of synchronization would be low. Note that the r/(t) in Figure 3.1 decays quickly; however, if the spike of the cell triggers self-inhibition (via an "autapse", or self-synapse), the resulting kernel O(t) decays over a much longer time, and the two cells synchronize much more quickly.

N. Kopell and G.B. Ermentrout

26

A related treatment is done in Chow [8], assuming that the neuron that fires second is influenced both by the last recent firing of the leading cell and the spike on the previous cycle. 3.2.2. F r e q u e n c y regulation in inhibitory networks. In the full Hodgkin-Huxley equations, there can be many different time scales associated with the synapses and the intrinsic dynamics. In order to understand how some of these may affect the p e r i o d of the network solution, we use a slightly more complicated version of the integrate-and-fire neurons, one that is not pre-scaled. It is dV C~ = I - gm(W - Wr) -- g S ( t ) , dt

(3.7)

where gm is an effective membrane recovery conductance and V,. is an effective membrane reversal potential. V (t) is reset to V0 whenever it reaches the threshold potential VT. Note that we are assuming identical cells (11 = 12) and a synchronous solution, so we may drop subscripts. This section is taken mainly from Chow et al. [ 10]. In these equations, there are three natural time scales. The first is the effective membrane time constant g m / C = rm associated with the passive return to the rest potential after a spike. The second is the decay time r of the synapse. (Though the rise time is very important for synchronization, it does not play an important role in frequency determination.) The third is the period T when two identical cells are coupled. This time scale is an emergent property of the system, and can be altered by changing the driving current I. Note that, in the absence of coupling, the frequency of an individual neuron can be very high hundreds of Hz - and the reduced spiking models in White et al. [71 ] and Ermentrout and Kopell [ 17] also behave this way. When coupled, however, the frequency is affected by time scales in the coupling. The aim of the analysis is to understand the circumstances under which different time scales in the equations dominate the determination of the network frequency. As in Section 3.2, we consider different parameter regimes, which we define asymptotically by relationships between T, r and rm. (As usual with asymptotic analysis, the conclusions hold in some regime larger than the strict definition of the regimes.) The tonic and phasic regimes have been introduced less precisely in the previous section. -

Tonic regime: Phasic regime: Fast regime:

T << rm, rm << T, r << T,

T << r. rm << r. r << rm or r ~ rm.

The phasic and tonic regimes are applicable for slowly decaying inhibition from the synapses, where "slow" here denotes the second inequality defining each of these regimes. The differences in the two are seen in the first inequality; in the tonic regime, the network period is small compared to the membrane time constant, while in the phasic regime it is large. Thus, in the tonic regime, there is significant filtering of the synaptic signals from the cell dynamics; in the phasic regime, the cell time constants are too fast to accomplish significant filtering. The fast regime denotes the parameter ranges for which the network

Mechanisms of phase-locking and frequency control

27

period is long compared to the synaptic decay time. In this regime, the intrinsic membrane time constant may be large or comparable to the synaptic decay constant; the coupling is close to pulsatile. The names "tonic", "phasic" and "fast" refer to the effects of the synaptic currents on the voltage of the post-synaptic cell. Thus, in the tonic regime, the synaptic current varies slowly compared with the other time constants, and hence acts effectively like a constant current. In the phasic regime, the current leads to significant changes in voltage within a network period, and so the signal has a significant phasic component. In the fast regime, the signal decays quickly relative to the period and the filtering properties of the network. The dependence of T on r and rm can be determined in a way similar to that used to compute the phase relationship between the coupled cells: one assumes a period T and then shows that T is determined uniquely in a self-consistent way. For the simple equations of the form (3.7), one can deduce a general but implicit formula for T. The computations are most transparent for fast synaptic rise times, in which S(t) = e x p ( - t / r ) , where r = 1//~ is the decay time of the inhibition. (See Chow et al. [10] for generalizations.) One finds that for the phasic regime one gets T = r In A, where A is a function of all the parameters, including r. Thus, the period is essentially proportional to the synaptic decay time, and weakly (logarithmically) dependent on other parameters. For the tonic regime, the dependence is quite different: the period is independent of r (or weakly dependent for other kinds of synapses). In the fast regime, T depends logarithmically on r and other parameters, and is proportional to the membrane time rm. One of the significant differences between the phasic and tonic regimes is the network behavior in the presence of inhomogeneities; it is only in the phasic regime that the network can remain coherent in the presence of even mild heterogeneities (see Section 3.2.1). Hence, when there are inhomogeneities and coherence is observed, the theory implies that the frequency should be proportional to the decay time of the inhibition. This appears true for inhibitory networks in the hippocampus (Whittington et al. [72]). The main inhibitory transmitter associated with this (and many other inhibitory networks) is GABAA, whose decay time leads to a frequency that is in the range of 30-80 Hz, depending on excitatory drive. Rhythms outside this frequency range require a different mechanism for the creation of coherence.

3.3. Spike response method and electrical coupling In the section on phase-difference coupling, the analysis was applied to spiking neurons with very thin spikes, and a heuristic explanation was given to show that electrical coupling need not be synchronizing. We now show that, with the techniques of this section, we can examine the properties of electrical coupling in more detail. The main results (Chow and Kopell [9]) are that the ability to synchronize depends on the shape of the spike, and is highly influenced by the period of the coupled network (which is in turn influenced by the drive to the cells). The reason that the spike shape matters is that the coupling current is a scaled difference of voltages, and therefore depends on the voltage during the spike as well as between spikes. In order to investigate the role of the shape of the spikes, it is necessary to include the spike shape in the equations; the standard I&F equations just reset the voltage as soon as

28

N. K o p e l l a n d G.B. E r m e n t r o u t

the threshold for spiking has been reached. Here, the spike is "pasted on" via an extra term in the voltage equation that changes the voltage in ways that depend on the rise rate of the spike, the width of the spike and a constant to calibrate the maximal voltage that the spike creates. All these parameters (or rather some combination of them) affect the voltage of the cell during a spike, which in turn affects the coupling current between the cells. The scaled equation for an uncoupled cell is du

dt

= I - v + A(t-

tt),

where A(t) -

for0 < t < A , for A ~< t.

I VA e ~-t

I -(1 + Vm)6(t-

A)

Here the dimensionless parameters are the applied current I, the spike rise rate ~, the spike width A, and the spike potential scale VA. The threshold is set at v - 1. The term Vm is computed from the parameters in A (t) in order that, just after t = A, v ( t ) is reset to v = 0. For a fast rising and narrow spike, Vm can be computed to be approximately VAe ~za/ (1 + ~) (Chow and Kopell [9]). The coupled equations have the form dvi

dt

I

(3.8)

Ii - 1)i -+- A ( t - tl) + g(1Jj -- Vi),

where g is the scaled strength of the electrical coupling (gap junction synapse). As in the case of chemical synapses, it is possible to construct an integrated version of these equations written in terms of kernels that encode the effects of the spikes on the neuron that produced that spike and on the other neuron. This is facilitated by working with the sum and difference of the two equations of (3.8). The analogue of (3.3) is then

1Ji(') -- li + Z 1

Ys(' -- 'I' ) + Z

l

Yc (, --

,I).

Here the kernel Ys refers to the effect of a spike on the cell that produced it, and is analogous to the kernel r# in (3.3); the kernel Yc encodes the effects of the coupling, and is analogous to J e . (See Figure 3.2.) Both of these are dependent on the shape of the spikes, as given in the formula for A ( t ) . The analysis is similar to that for inhibitory coupling described in earlier sections. In particular, the formula for the voltage at the next spike is given by (3.5) with r# and J e replaced by Ys and Yc. The formula for determining the phase difference 4, is again (3.6), with the same substitution. Thus, the outcome depends critically on the shape of the coupling kernel yc. The necessary condition for a solution with a phase lag 4, to be stable is that G~ (05) > 0. For a fixed T, this can be computed directly from the analogue of (3.6). As for the inhibitory coupling, Yc(0) = 0, reflecting the fact that the cells do not immediately affect one

29

Mechanisms of phase-locking and frequency control

2

0.2 t _

1

7~ 4 "

0

t

' 5

t

-o.2 ri .." 1".."

-1

-0.4 [;':

Fig. 3.2. Kernels Vs(t) and Vc(t) showing dependence on size of gap junction conductance. The dashed lines correspond to a conductance 10 times as large.

0.5

= i

i

9 i

i

i

iiiiiiiilliiiii

B

@~ @@@

0

=

a

m

m

m

m

I

m

m

m

m

Fig. 3.3. Bifurcation diagram showing dependence of stable solutions (synchrony and antiphase) on the level of drive to the cells. Higher drive corresponds to lower period.

another on firing. Thus, the necessary condition for stability of synchrony is y~ Vc(l T) > 0. Now Yc is monotone-increasing in a tiny range near t = 0, then decreasing for t in some range, and then increasing back up to zero. So if T is sufficiently large, all the terms in the sum are positive; if it is small enough (high frequency), the sum can be negative. This suggests that for high enough frequency, the synchronous solution can be unstable. (It always exists.) This is verified numerically in Chow and Kopell [9] for a biophysically-based model (White et al. [71 ]). A nonsynchronous solution, such as antiphase, may not exist if the coupling is too strong; with strong enough coupling, a spike of one cell quickly produces a spike in the other one, preventing an antiphase solution. How large "strong-enough" is depends on the frequency; for low frequency (high period), the cell receiving the coupling current is close to fully recovered when the spike arrives, and hence a small coupling strength may induce the spike. For fixed coupling strength, antiphase is not possible for low enough frequencies. Figure 3.3 summarizes the existence and stability of the synchronous and out-of-phase solutions as the drive is decreased, which, in turn increases the network period. The diagram was computed from analysis of the necessary condition for stability for the

30

N. Kopell and G.B. Ermentrout

integrate-and-fire models, and matches the behavior found for a biophysical model (White et al. [71]). At very low periods (high drive), only antiphase is stable (or neither is stable). At a higher period, the antiphase solution loses stability (to a third mode we have not discussed). At still higher periods, the synchronous state becomes stable. With drive decreasing more, antiphase becomes stable again and there is bistability of synchrony and antiphase. At the lowest drives, only synchrony is stable. We note that antiphase solutions doubles the network frequency. The frequencies that can be produced by coherent activity mediated with electrical coupling are much higher (even in the synchronous mode) than can be supported by inhibition, which is tied to the rates of the synapses. With antiphase solutions, the electrical coupling can support frequencies over 200 Hz for biologically reasonable parameters. We can summarize a role of the spike shape and the strength of the gap junctions on the network behavior by discussing how these parameters change the endpoints of the frequency regimes in which there is synchronous and antiphase behavior. Computation of GT(q~) shows that this function depends on a combination 6c of parameters including and VA; for fast rising spikes and weak coupling, it can be shown that 6c is approximately 1 + 2gym~(1 + ~), where Vm = VAexp(~A)/(1 + ~). From this, it is possible to compute how the endpoints of the stability regions change with parameters. Increasing 6c, which can be accomplished by increasing the spike amplitude or increasing the spike width A, has several effects. First, it raises the period necessary to have stable synchrony, i.e., makes it harder to have synchrony in high frequency regimes. Second, in the lower frequency regimes, increasing 3c or A increases the lower bound on period for stable anti-synchrony; this diminishes the frequency regime over which the antiphase solution is stable, thus enhancing synchrony. Thus, changing the spike shape can have different effects over different frequency regimes. One way to understand why the shape of the spikes might affect the ability to synchronize is to realize that, during the spike of one cell, the coupling current to the other cell increases the voltage, as in excitatory chemical synapses. During the reset portion and return to threshold of one cell, the current experienced by the other cell acts like an inhibition. Thus, there is tension between the effects of the two parts of the cycle. Since the excitation tends to be desynchronizing, one would expect spikes that are larger (in amplitude and duration) to be more destabilizing than smaller ones. The dependence on network period can be understood in a similar way: changing the drive to the cells mainly changes the interspike interval, rather than the spike itself. Thus, higher frequencies are associated with a large relative amount of excitation, and hence are more vulnerable to destabilization of synchrony. The spike response formalism is the same for both chemical synapses and electrical synapses (gap junctions). Because of the linearity between spikes, the formalism can also be used when there are both chemical and electrical synapses; in that case, the relevant kernels are the sum of the kernels of each kind. Though they do not interact in a nonlinear way (within this class of models), they can produce some nonintuitive effects. For example, if the individual cells are firing at frequencies too high to sustain synchrony even for electrical coupling, addition of inhibition can synchronize the cells. This does not happen by the mechanism described in the section on chemical synapses, which requires that the inhibition be strong enough to dominate the time scale of the emergent rhythm. Even inhi-

Mechanisms of phase-locking and frequency control

31

bition too weak to do the latter can help, however; at the high frequencies (the tonic regime for inhibitory coupling), the coupling term acts as a constant inhibition, pulling down the frequency enough that the gap junction can perform the synchronization (White, Kopell, Chow, unpublished).

4. Oscillators with multiple spiking cells The work in Section 3 concerns a pair of cells, interacting via spikes. The spike-response formalism allows one to deal analytically with dependence on many cycles in the past. In this section, we deal with oscillators that are composites of spiking cells, with excitatory and inhibitory interactions within a single unit oscillator. To handle this analytically, we work in frequency regimes in which the response to a cell receiving a signal depends mainly on the timing of the last cycle, not many previous cycles. In this regime, we have another technique, that of low dimensional maps. There are two different kinds of maps that we will discuss here. One is derived from the classical "phase-response curve (PRC)" theory, which describes how a limit cycle oscillator reacts to a periodic short pulse that perturbs the oscillator (Winfree [75]). The PRC gives the size of a perturbation in phase at the end of a cycle as a function of the time at which the pulse is received. From the PRC, one can read off the possible stable phase lags or leads between the forcing oscillator and the forced one. The theory depends on having the periodic solution of the forced oscillator be sufficiently attracting that the effect of the perturbation is only on the phase, and there is no effect beyond one cycle. This theory can be extended to work with a pair of oscillations whose interactions are via short pulses and is useful in describing interactions via excitatory synapses. The second kind of map is useful in situations in which the time of firing of a cell depends on decay time of the coordinating current, usually an inhibitory current; we call these "hold-and-fire" systems. For such maps, the firing time of each of the cells involved in the circuits is calculated as a function of previous firing times; since the period of the coupled system is usually significantly different from that of the uncoupled local circuit, the maps are not described in terms of changes of phase, and are calculated directly, not from a PRC. We will refer to these as spike-time maps. Though the spike times of all the cells come into the calculation, the maps can sometimes be approximated by ones that are of very low dimension, even one-dimensional. The essential reason has to do with the multiple time scales in the Hodgkin-Huxley equations: on the time scale of the signals, many intrinsic variables relax to a predictable position, decreasing the number of real degrees of freedom. This collapse is associated with different time scales in different problems, and will be illustrated below. It has not been rigorously proved that the remaining degrees of freedom correspond to a center manifold, but we conjecture that this is true.

4.1. Long-range synchronization of the gamma rhythm We now give a case study of a spike-time map analysis. We use the gamma rhythm described above in Section 3.2, as seen in a hippocampal slice. In this case, however, the

32

N. Kopell and G.B. Ermentrout

rhythm is induced by stimulation instead of pharmacological agents that block the excitatory cells; hence the network contains both excitatory and inhibitory cells. At issue is the ability to have precise synchronization of spikes over distances such that the conduction delays are a significant fraction of the period of the rhythm (e.g., 8 ms in a 25 ms period). It was noticed by Traub et al. [63], in both data from hippocampal slices and large-scale simulations with thousands of neurons (each with many compartments and conductances) that synchronization over distances was found in conjunction with a stereotypic pattern of activity: the inhibitory cells fire doublets on each cycle, the first spike (statistically) at the same time as the firing of the excitatory cells. The goal of the map analysis is to understand the role, if any, of the doublets in the synchronization process, and to use that to predict the parameter ranges in which such a mechanism could work rapidly and robustly in the presence of heterogeneity and/or noise. A minimal model has two oscillating local circuits, each composed of one excitatory (E) and one inhibitory (I) neuron (Ermentrout and Kopell [17]). Within each local circuit, the E- and I-cells have inputs to one another. Between the local circuits, the most important connection are the long-distance ones from an E-cell to its distant I-cell (Figure 4.1). Generalizations to a network with a more complex set of connections are in Ermentrout and Kopell [ 17]. Each of the cells is described by a set of Hodgkin-Huxley equations, with minimal nonlinear currents corresponding to spiking; since there are the usual three gating variables (m, n, h) of the Hodgkin-Huxley equations, each cell is four-dimensional. Furthermore, the synaptic currents have their own dynamics, described as differential equations dependent on the voltage, rather than currents with prescribed time courses. Hence the total number of equations for even a pair of 2-cell circuits is quite large. Nevertheless, under some broad set of hypotheses, it is possible to analyze the stability of the synchronous solution with a one-dimensional map. The hypotheses can be checked numerically and shown to hold for the Hodgkin-Huxley equations in some parameter regimes. Some of the hypotheses describe properties of each E-I circuit, i.e., give constraints on ranges of parameters for the Hodgkin-Huxley equations. The rhythm is driven by excitation from the E-cells, so the latter must be excitable enough to fire rhythmically; the populations of I-cells can be either excitable (do not fire without input) or oscillatory, but may not fire faster than the E cells. (If they do, they suppress the E-cell activity.) The I cells have the property (shared by some "type 1 cells") that they can have a long latency to firing after being stimulated because of the proximity of a saddle-node point. This provides

Q

Q DELAY

Fig. 4.1. Minimal model for synchronization of the gammarhythm across distances. Each E-I circuit is an oscillator, and there are long E to I connections between the two circuits.

Mechanisms o f phase-locking and frequency control

33

a form of "relative refractory period", which will be critical to the analysis. We note that this relative refractory period is not encoded in simple integrate-and-fire neurons and is not needed in the contexts discussed in the previous sections" a model that has this property is important only in contexts in which a cell receives excitation shortly after it spikes, as in the case of doublet firing. As discussed below, the second spike of the doublet is provoked by excitation from the distant circuit, arriving before the local I-cell has fully recovered. The construction of the map follows from the above hypotheses. The E-cells are able to fire when synaptic inhibition has worn off sufficiently. Thus, we treat E-cells as linear between spikes, firing when an inhibition-determined threshold has been reached. The Icells have an extra property, associated with relative refractory period. Suppose a cell fires at t - 0, receives excitation at t -- 7r > 0 and fires again at t - TI (7r). The more recovered the cell, the shorter the time to fire. Hence TI decays with 7r. For ~p very short, the cell may not fire at all. As will be seen below, the behavior of the system follows almost entirely from the properties of the map Ti. This map, and its dependence on network properties, can be found by simulation. It can be shown that TI is related to the intrinsic refractory period of the I-cells, but is influenced by the rest of the network. For example, increasing the self-inhibition in the local network increases the values of TI. We now analyze the dynamics of the network in a neighborhood of the synchronous solution. The following is a simplification of an analysis in Ermentrout and Kopell [ 17], using as long connections only excitatory synapses to the I-cells (Figure 4.1). Let tl and t2 be the times of firing of cells El and E2 on some cycle, with A -- t2 -- tl. If the initial conditions are close enough to synchrony (A - - 0 ) , we can construct a map that gives the times il and t2 of firing in the next cycle, from which we can determine the conditions under which there is a stable solution. The time at which the inhibition for an E-cell wears off enough for it to fire is approximated by the time at which the inhibitory conductance has decayed enough; the latter threshold depends on the drive to the E-cells. We assume that there is saturation of the synapse from the I- to the E-cell, so that it is only the timing of the second spike of the doublet that matters. Cell I1 is essentially recovered from firing in the previous cycle when it receives excitation from El at t l. (We assume no or minimal local conduction delay. We are also implicitly assuming that the membrane time of the inhibitory cells is shorter than the inhibitory decay time, which implies the recovery of an I-cell by the end of a cycle.) Hence it fires a short (history-independent) time tel later. (This time can be arbitrarily small, depending on the drive to the E-cells, but the latter drive may not be so large that the Icells fire before the E-cells.) Cell Il receives excitation from E2 at time t2 + 6, where is the delay time between the circuits. Cell I l then fires the second spike of its doublet at t2 + 3 + Ti(t2 § ~ - tl - tei) = TI, inducing an inhibitory current in the E-cell at that time. El fires at t -- il in the next cycle when the inhibitory conductance decays to some threshold level g,, i.e., gie e x p [ - ( { , - ~ ) / r ] -

g,,

(4.1)

where r is the time constant for decay of inhibition and gie is the maximal conductance of the inhibitory current, g, is related to the drive to the cell via the intrinsic period p• induced by that drive in an uncoupled cell, namely: gie exp[(-p~, + t e i ) / z ] = g , . From the

34

N. Kopell and G.B. Ermentrout

above, we can compute the time [l in terms of tl and t2 and similarly for [2. The ability to do the analyses given below depends on the fact that both il and i2 are closely approximated by functions only of t2 - tl. This is true for TI because the I-cells are sufficiently simple that there are no internal degrees of freedom besides the time after firing. Similarly, since the E-cells do not have currents whose gating variables have long time-constants, the time to firing of the E-cell depends only on the times of receipt of inhibition, not other historydependent variables. We now show how we can use the above to analyze the range of parameters in which the gamma configuration is stable. It is easy to solve for t-~.explicitly. Subtracting [l from t2 we get m

m

,4 ~ [2 -- t-I -- --,4 + TI(-,4 + 6 - tei) - TI(,4 -+- 6 - tei) =~ F y ( , 4 ) .

(4.2)

Standard stability analysis implies that the synchronous solution (A = 0) is stable if - 1 < F~ (0) < 1, i.e., - 1 < T{ (6 - tei) < 0. TI is a decaying function, so its derivative is negative for all 6. However, the rate at which the system synchronizes is given by IF~ (0)1 - 1, and hence becomes small as 6 becomes large. Indeed, (4.2) tells us that the conduction delay for which the speed of synchronization (near the synchronous solution) is largest is the one for which F~ (0) = 0, i.e., I { ( 6 ) = - 1 / 2 . The function TI can be numerically computed for a given biophysical model; for the reduced Traub-Miles model, T[ (6) is close to zero (and so F~ (0) is close to 1) for 6 about 8 ms or larger (Ermentrout and Kopell [17]). This effectively places a limit on the possible conduction delay that can be tolerated and still produce fast synchronization of gamma by this mechanism. We note that when the rate of convergence to the fixed point is small, small differences in the two local circuits can produce significant phase-lags between the circuits; this is another reason why longer conduction delays disrupt synchronization in this mechanism. Modifications of the parameters in the circuit can change the range of conduction delays in which there is effective synchronization, but not significantly. (See Ermentrout and Kopell [ 17] for more details.) The doublet configuration is robustly stable for the minimal circuit involving two sites, each giving excitation to each 1-cell. The mathematics describing the minimal network identifies the essential nonlinearity (the relative refractory period) involved in the synchronization, but does not address the question of how this might work in a large, distributed network in which each 1-cell gets excitation at many different times from many different E-cells. Such a distributed network was analyzed in Karbowski and Kopell [29]. Using methods of statistical physics, it was shown that the same nonlinearity works to produce synchronization with the doublet configuration, and that the latter is stable in a larger parameter region than the configuration having triplets in the I-cell within each cycle. Furthermore, the analysis showed that a small amount of disorder in the positions of the E-cells (which determine the conduction delays to the 1-cells) can increase the parameter range in which the doublet configuration is stable.

4.2. G a m m a to beta transition Traub et al. [62,64] noticed that with more stimulation given to cells in a hippocampal slice, the network first produced a gamma rhythm, and then spontaneously switched into

Mechanisms of phase-locking and frequency control

@

@

1 -(bII"

(])-

@,

35

'@

1 -(Z)II

q)-

Fig. 4.2. The change from local gamma rhythms to local beta rhythms requires adding connections between the E-cells.

a lower frequency rhythm (12-30 Hz) called beta. (See Kopell et al. [35] for references to papers that discuss the potential significance of such a switch in living animals.) In the hippocampal slice, it is possible to look at the detailed timing of the spikes in the cells, and it was discovered that the lower frequency rhythm has a different structure from that of the gamma rhythm: the inhibitory cells continue to fire at a gamma frequency, while the excitatory cells miss cycles. Furthermore, the excitatory cell population is essentially synchronous, with almost all E-cells missing the same cycles. Experimental work reveals at least two kinds of changes in the components and coupling of the network in going between the two rhythms. When the network is displaying beta, the E-cells have an extra ionic conductance, a so-called after-hyperpolarization (AHP) current that slows down the E-cells. Furthermore, the effective coupling of the network changes during the transition: the coupling between E-cells, which is sparse in the network before the large stimulation, grows stronger (Figure 4.2). The mathematical question is whether those changes are enough to account for the changes in the network behavior. It is clear that slowing down the E-cells can ensure that the latter are unable to keep up with the I-cells, and hence miss cycles. Thus, the balance of excitability of the E- and I-cells is critical to producing the beta rhythm. The question to be investigated, for a local collection of E- and I-cells, is what facilitates the synchronization of the E-cells, so they fire on the same cycles. This is not a straightforward consequence of the E - E coupling, since it is known that E-E coupling can actively prevent synchronization (see Sections 2 and 3). Indeed, in the absence of an I-cell network, typical models of cortical neurons, when coupled with typical synaptic time constants, tend to oscillate in antiphase with one another. However, sometimes slow outward currents can help with synchronization of excitatory cells. This is discussed in Section 4.5, which uses a PRC kind of analysis. 4.3. Long-distance synchronization of the beta rhythm This section is a variation on Section 4.1, showing here how a different rhythm can be analyzed in a similar way, yielding a different conclusion. The minimal circuit for considering long-distance synchronization of the beta rhythm also uses two networks, each having one E- and one I-cell; since the cells of each type are synchronous in the rhythm whose stability we wish to analyze, we can combine the cells of each type into a single cell. In this formulation, we are assuming that the E-cell represents a synchronous local population, and that this synchrony relies on local E-E connections.

36

N. K o p e l l a n d G.B. E r m e n t r o u t

50

|

m~ --T~

I

40

% 30

20

0

i

0

i

5

i

/

i

lO

i

i

15

20

8 Fig. 4.3. The maps TI and T/~ as a function of the delay.

For the long-distance synchronization of the two separated circuits, we first consider only E-to-I connections as above for gamma. The map for analysis of the synchronization of two sites, each displaying beta, is a variation of (4.2), with two modifications. The cell I i now fires three spikes per beta cycle, two during the gamma cycle in which the E-cells fire, and one during the cycle in which the E-cells are silent. The excitation from the distant cell is, as before, received by an I-cell after it has fired its first spike of the period. The map TI for the gamma rhythm is replaced by T/~, which is defined to be the interval between the time cell Il receives excitation from cell 2 and the time it fires its third (not second, as before) spike; T~ depends on the times tl, t2, defined as in Section 4.1. For a graph of T/~, see Figure 4.3. The time at which the third spike of the I-cell fires is A

t2 + 6 + T/~ (t2 + 6 -- t l -- tei ) ~ T~.

The second modification introduces another, intrinsically-based, source of inhibition, namely the slowly-decaying AHP current of the E-cells, with time constant "rahp, triggered by a spike of that cell. As in formula (4.1), the time il of the next El spike is then defined implicitly as the time at which the inhibitory conductance has decayed to some threshold level; in this case the formula takes into account both extrinsic and intrinsic inhibitors:

gieexp[-([l-

T'B)/r] + gahp e x p [ - - ( / l - tl)/rahp] = g/~,.

(4.3)

Here gahp is the maximal effective conductance of the slow current, which takes into account the actual maximal conductance gahp, but scales it using the maximal value of the gating variable for that current and the ratio of the driving force of the AHP current to that of the synaptic current. A formula for g~, analogous to the one for g, can be found in Kopell et al. [35]. The analysis of (4.3) is not as straightforward as that of (4.1) since there is no simple explicit formula for il to obtain a formula analogous to (4.2). However, it is possible to do the stability analysis with the implicit formula (4.3) by linearizing around the solution

Mechanisms of phase-locking and frequency control

37

in which the E-cells are synchronous (while missing beats of the I-cells). To do this, we define pi ---- {i -- ti -- pfi, the variation in a given cycle from the periodic rhythm with period Pt~. A is defined as before. We can rewrite (4.3) as g~, -- g i e e x p [ - ( p ~

-+- Pl - A - 6 -

T~(A -+-6- tei))/r]

-q- gahp exp[--(p/~ -+- Pl)/rahp].

(4.4)

To lowest order in the small quantities A, Pl, P2, the above can be expressed as A pl + B ( p l - A(1 + T ~ ( 6 - tei)/T))--0,

where A = (gahp/rahp)eXp(--p~/rahp) and B = ( g i e / r ) e x p ( - ( p ~ - 6 - T~(6 - t e i ) ) / r ) . A similar formula holds for p2 with A replaced by - A . Subtracting these two formulae, and noting that P2 - Pl = A -- A, we have that (A + B ) ( A -

A)---2BA(1

+ T~(6- tel)/r),

i.e., A - B - 2BT~(6 - tei)/r A =

A+B

A ---- D A .

(4.5)

Synchrony is stable if the coefficient D of A in (4.5) lies between - 1 and + 1. The fastest synchronization occurs when D - - 0 and, as above, this is also the range in which there is the most robustness to heterogeneity. As shown in Kopell et al. [35], the size of D can be analyzed as a function of the conduction delay 3, and for physiological ranges, it turns out to be significantly within the stability range [ - 1 , 1] for 6 = 20 ms or more. Thus, by combining the properties of the new slow current (encoded in the definition of A) and the extra spike of the I-cells in each cycle (affecting both the resulting period and the definition of the key non-linearity Tt~), we get that the beta rhythm can allow stable synchronization across much longer conduction delays than can the gamma rhythm in the appropriate physiological parameter regime. This has been confirmed by simulations of the Hodgkin-Huxley equations for the minimal network and for large-scale biophysically accurate representations (Kopell et al. [35]). So far, the analysis of this subsection concerns only the long-range synchronous solution, and makes no use of long-range E-E connections (though local E-E connections are critical to having a local representation using only a single E-cell). We have shown that long E-E connections (see Figure 4.4) are not necessary to the stability of the long-range synchronous solution. Nevertheless, the long E-E connections do play a critical role in the network behavior when the parameters are in the regime to create a beta rhythm. The above analysis does not rule out bistability, and does not address the issue of domain of attraction. As can be seen numerically, without the long-range E-E connections, the solution in which the two E-cells fire in antiphase is stable and has a large domain of attraction. The main function of the long E-E coupling is to insure that the antiphase solution is not a stable alternative to synchrony (Kopell et al. [35]). Adding E-E connections when the parameters

38

N. Kopell and G.B. Ermentrout I

0

I

DELAY

Fig. 4.4. Long distance synchronization of the beta rhythm requires connections between the distant E-cells to remove solutions competing with the synchronous one.

are appropriate for the gamma rhythm does not increase the range of conduction delays that are tolerated for synchrony, so it is the difference in the other biophysical factors that creates the advantage for the beta rhythm.

4.4. Changing ionic currents changes synchronization properties The beta rhythm is significantly slower than the gamma rhythm, partly due to the extra outward current in the E-cells, and partly due to the extra inhibition from the extra I-spike. One might conjecture that the greater robustness of the beta rhythm to conduction delays may depend only on the relative periods, rather than the biophysical details. The following case study shows that the latter interpretation is not correct. Again the analysis uses a spike-timing map, this time to highlight how differences in time constants other than the ones used in Sections 4.1 and 4.3 can lead to a l-dimensional map. The case study concerns the alpha rhythm (8-12 Hz) in the neocortex, which is slower than gamma or beta. This rhythm is found easily in the neocortex (indeed, much more robustly than is gamma or beta), but its cellular origins are more mysterious. It has been shown that (at least some versions of) this rhythm do not synchronize well over distances, producing variable phase lags instead of synchrony (Roelfsema et al. [50], von Stein et al. [67]). To explore how this could come about, Jones et al. [28] built a minimal model of a local alpha rhythm using ionic currents known to exist in cells in the layers of the cortex where the alpha rhythm is thought to originate. The key currents are the so-called h- and T-currents, which react differently to inhibition than do most other currents: the hcurrent is activated by inhibition (its activation current m ~ is monotone-decreasing instead of increasing), while the T-current has a voltage dependence that requires it to be at a low voltage in order to de-inactivate before firing. With these currents, it is possible to create a rhythm in which the inhibition does not determine the frequency. Instead, the time constants of the h- and T-currents of the E-cell are the critical determinants. The I-cells in the network act to provide the essential inhibition needed to turn on those currents in the E-cells. In this rhythm there is no beat skipping, and the I-cells fire almost synchronously with the E-cells. With only the long E-to-! coupling, the analysis can be done as in the case of the gamma or beta rhythms. Again there are doublets which provide phasing information; the key

Mechanisms of phase-locking and frequency control

39

question is whether the information acts to synchronize. As in the case of Section 4.1, the full equations reduce to analysis of a 1-dimensional map, but the reasons are not the same. If there is to be only one essential degree of freedom, the general hypotheses of PRC theory should be at least approximately satisfied. That is, the local oscillations, when perturbed, should return approximately to their periodic wave-forms within one cycle; also, the effect of the stereotypic signal from the other local circuit should depend only on the time in the cycle of the recipient oscillator that the perturbation is received. In the case of the circuit displaying gamma, there is no slow gating variable, and the frequency is determined by the decay of the inhibition; there is no slow variable to hold any memory beyond the end of a cycle. Furthermore, the timing of the I-cell doublet depends essentially only on the time of receipt of the excitation from the circuit and the time of the first spike in the doublet. By contrast, in the case of the alpha rhythm, some of the gating variables of the T- and h-currents are slow (the activation of the h-current, de-inactivation of the T-current). However, special properties of the h- and T-currents allow the hypotheses of the PRC theory to hold. The gating variables of the h- and T-currents are strongly voltage-dependent, and are quickly reset to almost fixed values when the E-cell spikes. Though other gating variables, such as that of the fast-spiking potassium current, do not reset with an E-cell spike, their effects are gone by the end of the long cycle. Thus, for the E-I oscillations in the alpha rhythm, the timing of the E-cell spike paces all of the other variables, and there is no history dependence in the other variables. As above, the time of the doublet spike of the I-cell depends only on time of receipt of the excitation and the time of the previous I-spike, both paced by the spike times of the E-cells. Thus, in spite of the many dimensions in the Hodgkin-Huxley equations, the only essential degree of freedom is the difference in timing of the two E-cells. The resulting map is an analogue of Fy of (4.2). In the current case, however, the properties of the analogous function are different: the slopes show that the connection is desynchronizing for some conduction delays, and essentially neutral for small enough delays. The change of sign can be traced to the different processing of inhibition in the network; in the previous rhythms, an inhibitory pulse to an E-ceU slows down the rhythm, while for this rhythm an inhibitory pulse speeds up the rhythm by turning up currents that hasten the next spike of an E-cell. Adding long E-E connections does not change this behavior, and can even exacerbate the problem. It is possible to build networks with flexible synchronization properties by taking advantage of the variation in behavior in different voltage regimes. For example, starting from the simple network model of an alpha rhythm and raising the voltage (by increasing the applied current) brings the network to a range in which the h- and T-currents no longer operate. In that parameter regime, the same network will then display gamma or beta (depending on other parameters). Raising the voltage, which switches the local behavior, then also switches the synchronization behavior: when the local behavior is beta instead of alpha, the two separated networks synchronize rapidly and robustly (Jones et al. [28]). Changes in synchronization properties can also be created by activity-dependent plasticity of the synaptic connections. In simulations with large biophysically based neurons, Bibbig et al. [3] showed that local gamma activity, in two separate networks not oscillating in phase, could give rise to changes that procedure long-range E-E connections and E-I

40

N. Kopell and G.B. Ermentrout

connections" in turn, this helps transform an antiphase relationship in the beta rhythm to synchrony, without the prior existence of adequate E - E connections.

4.5. Synchronization o f excitatory cells In previous sections, we used spike-timing maps to investigate synchrony. In this section, we use PRC-derived maps to show that some intrinsic properties of cells can lead them to synchronize when they are coupled by excitation, in contrast to integrate-and-fire cells described earlier. The PRC, to be denoted by P(t), describes the advance in time of the next spike of the receiving cell from the time it would otherwise spike next, if the input arrives at time t. Such a PRC may be found experimentally, computed numerically from biophysical equations or sometimes computed analytically from equations. We can use the function P (t) to create a map for a pair of cells that are mutually coupled. Let t l be the time at which cell 1 last fired and let t2 be the time at which cell 2 last fired. We will assume without loss of generality that t2 > tl. When t2 fires, cell 1 is advanced by an amount P (t2 - t l ) so that tl -- tl + T - P(t2 - tl), where T is the natural period of both cells. (Note that one could have different periods but this complicates the algebra considerably.) By a similar argument, it is clear that t-2 --t2 + T - P(il - t2) m

_

_

is the time that cell 2 fires again. As usual let A -- t2 - tl and A -- t2 -- tl. We then see that A-- A - P(T - P(A) - A) + P(A) = F(A)

and we have again obtained a 1-dimensional map. We now make use of the condition P (0) -- P (T) -- 0 to note that A -- 0 is a fixed point of the map F. PRCs derived from real neurons as well as those derived numerically from conductance-based models satisfy this condition, which implies that there is no effect on spike timing if the input comes at the precise time the cell is spiking. (This is not true for the well-known integrate-and-fire model. Thus, the latter requires artificial "absorption" conditions (Mirollo and Strogatz [45]) that are not necessary for more biological PRCs.) Differentiating F and setting A = 0, we find that F (0) -- (1 +

P' (0)) 2

so that a necessary and sufficient condition for stability of synchrony is that - 2 < P' (0) < 0. Crook et al. [13] showed numerically that the presence of adaptation due to a lowthreshold, slow-potassium current causes the PRC to have a negative region after spiking,

Mechanisms of phase-locking and frequency control

41

with P' (0) < 0. As long as P' is not too negative (for example, if the interaction is not too strong), then the map calculation shows how this alteration of the PRC allows synchrony to be stabilized. In contrast, without adaptation, neurons tend to have strictly positive PRCs when coupled with excitation. The effects of the change on the PRC, and the effect of the latter on stability of synchrony, mirrors changes in the adjoint V*(t); indeed V*(t) is a kind of "infinitesimal PRC" (Hansel et al. [23]). (See Figure 2.1 and Section 2.2.) More recently, Ermentrout et al. [ 18] used such methods to investigate the effects of different kinds of adaptation currents on synchrony between excitatory neurons. They showed that low threshold adaptation currents and high threshold ones can have different mechanisms for creation of synchrony. The effects work in sample model neurons capturing behavior near creation of limit cycles via saddle-node bifurcations on an invariant circle [ 15, 251. The Ermentrout et al. [ 18] treatment is based on weak interactions, using a convolution with an infinitesimal PRC to create the PRC. For larger interactions, predictions from the infinitesimal PRC can be misleading. This was shown by Ackers et al. [ 1], who constructed the PRC numerically from the Hodgkin-Huxley equations for pairs of cells in the medial entorhinal cortex, a structure that gates the inputs and outputs to and from the hippocampus. The more direct construction reveals global properties of synchronization not apparent in the infinitesimal PRC: for large values of the conductance of the low current, which is associated with prominent subthreshold oscillations for the uncoupled cell, the stable antiphase solution predicted by the infinitesimal PRC approach disappears, and the synchronous solution becomes globally stable instead of bistable with antiphase. These global changes are seen to be associated with the Hopf bifurcation that produces subthreshold oscillations in the uncoupled cells (not found in the equations in (Ermentrout et al. [ 18])). Thus, subthreshold oscillations are seen to be relevant to the domain of stability of the synchronous solution. We note that the spike-timing maps associated with the PRCs can be globally defined. This is not true of the maps from the "hold-and-fire" systems, which cease to be valid sufficiently far from the synchronous solution. The reason is that a different set of spike times must be considered far from synchrony: in the maps discussed in Sections 4.1 and 4.3, the independent variable is t2 - tl, where the ti are nearby spike times of the two cells, not dependent on which of the cells is first, provided that the "hold" due to inhibition is larger than It2 - tl [. For larger initial differences in spike time, the input most relevant to the receiving oscillator changes, changing the map. The PRCs discussed above keep the order of the relevant spikes the same from cycle to cycle, and the map does not become undefined for large values of the initial difference in spike times.

5. Interactions of bursting neurons

As described in Rubin and Terman [52] and Wang and Rinzel [70], neurons can have complicated firing patterns known as bursts, consisting of a rapid sequence of spikes followed by a quiescent period. Some biophysical models for bursting are given in those references. In studying the interactions of bursting cells, we consider the envelope of the spikes (as do Rubin and Terman); a much-used simple example of such equations are the so-called

42

N. Kopell and G.B. Ermentrout

Morris-Lecar equations (Morris and Lecar [46]), originally designed to describe a calcium spike in muscle (not nerve) tissue. These equations were introduced as (2.8) in an earlier section. In these equations, the activation w of the potassium current is slow (i.e., 1/ r << 1). The equations can be understood metaphorically as envelope equations, or as equations for a calcium plateau on which the sodium spikes would ride if the spike currents were included. The Morris-Lecar equations have phase-plane geometry similar to those of the van der Pol oscillator or its variation, the FitzHugh-Nagumo equations. The nullcline dr~ dt = 0 is cubic-shaped, and the nullcline d w / d t = 0 is monotone-increasing. Depending on the relative positions of the nullclines, the equations can have a stable fixed point on either the left-hand branch or the right-hand branch of the cubic. With the intersection in the middle (and 1/r sufficiently small), the equations have a stable limit cycle. A variation of these equations uses different currents, but has a similar geometry (Wang and Rinzel [69]). The Morris-Lecar equations and the FitzHugh-Nagumo equations have a "fast-slow" structure, with the voltage equation significantly faster than the equation for the gating variable. Many results about networks of such cells use only this and other geometric structure, not the details of the biophysical descriptions. In this section, we discuss the interaction of two such neural oscillations, giving results complementary to those in Rubin and Terman [52,51], and using the fast-slow structure in a somewhat different way.

5.1. Fast threshold modulation (FTM) Models of interacting bursting neurons generally use different models of synaptic currents than models using spiking neurons. For the latter, the spikes are very quick or instantaneous, while the currents that result from the spikes have a finite lifetime. Therefore, it is necessary to describe the currents (as above) as explicit functions of time. For bursting neurons, the active state of the cell is a significant fraction of its period. If the time span of the synaptic current from a single spike is short compared to the time of the burst, we can approximate the current by considering it a function of the voltage of the presynaptic cell; the current is on when that voltage (averaged over spikes) is sufficiently high and off when that voltage falls below some threshold. An often-used model of a synaptic current has the same form as an intrinsic current, but the gating of the current is by the voltage of the pre-synaptic cell. Thus, a term of the form --gsyn(V)(V- Vsyn)

(5.1)

is added to the fight-hand side of (2.8a), where v is the voltage of the post-synaptic cell and is the voltage of the pre-synaptic cell. Note that this is the form of the calcium current in the Morris-Lecar equations; that current has one gate (an activation gate) that is instantaneous, so there is no extra differential equation associated with this current. The constant Vsyn is the reversal potential associated with the synapse. A low value of Vsyn corresponds to an inhibitory synapse, since the current (5.2) will then be an outward (hyperpolarizing) current; a value of Vsyn that is above the voltage values produced during activity gives an excitatory synapse, producing inward (depolarizing) current.

Mechanisms of phase-locking and frequency control

43

SYNCHRONOUS ORBIT

-,c

EXCITED

Fig. 5.1. Excitation raises the nullcline of the fast variable. For the synchronous solution of the coupled system, this changes the position of the trajectory in phase space.

The function gsyn(V) has the same form as the gating in the calcium current of the Morris-Lecar equations: it is a sigmoidal function. If this function saturates for low and high values of its argument, and if the voltages in the active and inactive states correspond to saturated tails of the sigmoid, a further simplification is possible: on the left branch, the synaptic current is zero, while on the right (active) branch, gsyn(13) is a constant gsyn independent of ~3. (A limiting version of this is the Heaviside function used in Rubin and Terman [52].) Thus, the voltage equation for a coupled cell does not have ~ explicitly in the equation; one need only keep track of which branch of the O-nullcline the partner cell is on. When the partner cell is active, the effect of the synaptic term is to raise or lower the v ' = 0 nullcline with a change in shape; an excitatory synapse raises the nullcline and an inhibitory one lowers the nullcline (see Figure 5.1). This kind of coupling was called by Kopell and Somers "fast threshold modulation" (FTM). The current in (5.1) corresponds to a simple form of synapse described in Rubin and Terman [52], ones that are "direct" and rapid compared to intrinsic time scales. That chapter concentrated on the important effects of time scales of the synapses; as shown there, inhibition can be synchronizing if the synapses decay sufficiently slowly compared to some intrinsic processes, and excitation can be desynchronizing if the synapse is fast and if the active portion of the burster is much shorter than the silent portion. The techniques used in that chapter focus on behavior along slow manifolds, and make computations of synchronization or desynchronization using distance metrics within the slow manifold. In our case, the lack of extra time-scales in the synapse brings into relief other aspects of the geometric structure that are better approached using a different kind of metric. 5.1.1. Synchronization via excitatory FTM. Identical cells coupled by excitatory FTM have a synchronous trajectory; this solution is not the same as that of an uncoupled cell. The construction of the synchronous solution uses the fast-slow structure of Equations (2.8). For the coupled system, the slow equation is (2.8b), with v = v(w) determined by setting the right-hand side of (2.8a) to zero. For the unexcited cell, the equation is given by (2.8a), and for a cell receiving excitation there is an extra term -gsyn(l) - 1)syn) added to the righthand side of (2.8a). The trajectory of the synchronous solution is shown in Figure 5.1.

44

N. Kopell and G.B. Ermentrout

wl

w, w0 v

u

Fig. 5.2. The jump from the inactive branch to the active branch leaves the variable w unchanged.

(Also see Rubin and Terman [52].) Note that the period of the synchronous solution is in general longer than that of the isolated cell, because the cells have a longer stretch of nullcline to traverse. Though the speed along the raised nullcline can be larger than that of the unexcited nullcline, in most examples (including Morris-Lecar), this effect is not enough to offset the effect of changing the length of the nullcline. We now show that excitatory coupling through fast threshold modulation leads to synchronization of the cells in the limit e --+ 0. This depends critically on both the fast synapse and the fact that the cells are bursters, with a non-zero active time. (Compare with Rubin and Terman [52] and the other sections of this chapter.) We first consider two identical cells, and then discuss the effects of heterogeneity. The central idea is to use the fast-slow structure of the equations, and the time to flow between points on the slow manifold as a distance metric. Suppose that two cells start on the inactive (left) branch at some non-zero distance. When the first jumps to its active state, it changes the nullcline that governs the lagging cell. If the latter is below the knee of the raised nullcline, it will also fire, synchronously in the limit e --+ 0. Right after the jumps, the two cells are on the right-hand branch of the raised nullcline, with the values of w the same as those they had before the jump. (See Figure 5.2.) We denote those values of w by w0 and Wl. The crucial hypothesis is that the time it takes a cell to move between w0 and w l is larger on the left-hand branch than on the right-hand one. This can be easily checked for the Morris-Lecar equations, and holds for many classes of equations with similar cubic nullclines. The speed along a branch is determined by (2.8b), and hence is proportional to the distance between w and w ~ ( v ) . Note that this distance is much greater on the righthand branch than on the left one for values of w corresponding to points near the knee of the lower cubic. Thus, there is a compression in the time metric across the jump. As the trajectories of the cells move up the right-hand side of the raised cubic, they become closer to one another in the Euclidean metric, but stay the same distance in the time metric. A similar mechanism can work on the jump down from the activated to inactive branch. Each jump further compresses the distance between the cells in the time metric, synchronizing the cells.

Mechanisms of phase-locking and frequency control

45

A related analysis in Rubin and Terman [52] shows synchronization using the Euclidean distance metric. One advantage of focusing on the time metric is that it shows clearly how the rate of synchronization can depend on geometric properties of the oscillations (Somers and Kopell [58]). For example, two coupled Morris-Lecar equations converge to synchrony considerably faster than two FitzHugh-Nagumo equations ( v ' = f ( v ) - w, w' = ~v; f ( v ) -- v(1 - v2)). The essential reason is that the periodic trajectory of the ML equations is slow near the left knee, since w ~ ( v ) - w is small in absolute value in that region; the latter is much larger on the right-hand branch of the cubic slow manifold, leading to a large compression in the time-metric across the jump to the active state. By contrast, the FN equation has a much lower compression across that jump because Ivl is not that different along the two slow branches of the cubic nullcline for the same value of w. Again, this analysis is also relevant to the jump down to the inactive state. Other papers that use a time metric to analyze stability of solutions involving bursting neurons are LoFaro and Kopell [38], Terman and Wang [60] and Rubin and Terman [51]. Suppose now that the cells are not identical; we continue to assume the interaction is via fast threshold modulation. The arguments sketched above show that if the differences are not too large (but still O(1) with respect to ~), there are solutions in which the jumps are synchronous. In order for this to happen, the cell lagging on each of the branches must jump before it reaches the knee of its own (excited or unexcited) nullcline. Thus the coupled pair responds to heterogeneity in the cells by changing the wave-form of at least one of the cells, while preserving the timing of the jumps (Somers and Kopell [59]). Asymmetry in the coupling has similar consequences: if the sizes of the synaptic conductances are different in the two directions, the two cells have different excited nullclines to follow. The mathematics is identical to that in which the coupling is the same but the intrinsic dynamics are different. (See Terman and Wang [60] for related work about heterogeneity.) Note that there is a dramatic contrast in the reaction to heterogeneity (intrinsic or coupling) between bursters coupled by FTM and oscillators coupled via phase differences. (Compare to Section 2.3 of this chapter.) There is also another contrast between phase difference and FTM coupling: long chains of oscillators take a large number of cycles to settle into the phase-locked state, while long chains of FTM oscillators can lock within a couple of cycles (Somers and Kopell [58]). Related work about fast synchronization in a network of integrate-and-fire neurons is in Campbell et al. [7]. Recently, Izhikevich [27] has derived a set of phase models that show some of the properties of relaxation oscillators interacting via FTM. He considers models of the form # x ' = f (x, y),

v ' = g(x, y),

where # is a small parameter and the system has a limit cycle with discontinuities analogous to those of relaxation oscillators. For these systems, he is able to compute the singular adjoint trajectory as # --+ 0 and show that there is stable synchronization in the presence of non-uniformities in frequencies. However, the method requires that the coupling strength e satisfy e << #, so the coupling must be extremely weak. 5.1.2. Antiphase via excitatory FTM. Suppose now that the time to traverse the righthand branch in the uncoupled system is less than the time to traverse the left-hand branch.

N. Kopell and G.B. Ermenovut

46

A

B ESCAPE

,..,B,TEO---- \

UNCOUPLED

Fig. 5.3. A. When the active cell reaches the upper knee, it releases the inactive cell from inhibition. B. When the inactive cell reaches the lower knee, it escapes from the inhibition imposed by the other cell.

(A precise hypothesis is found in Kopell and Somers [36].) In the coupled system, there is a set of initial conditions for which one cell stays on the inactive branch during the entire excursion of the other cell on the active branch (see Figure 5.3). With some further conditions, it can be shown that there is a stable antiphase solution with initial conditions in that set. The mechanism for the antiphase solution is very different and more subtle than that for the synchronous solution. During the active pulse of one cell, the other cell is displaced from the nullcline associated with the uncoupled cell to the (raised) nullcline associated with the active synapse (Figure 5.1). In a large class of equations that includes the MorrisLecar equations, the movement along the displaced nullcline is slower than along the original one. Thus, the cell that stays in the inactive state experiences a "virtual" delay (the difference between the distance it actually went during the excursion minus the distance it would have gone in that time in the absence of coupling). This is true even though the coupling is excitatory, and leads to advances of the cell for other initial conditions. The analysis of the behavior is done in terms of the delay as a function of position of one cell when the other begins its active excursion. From this, it is possible to get a Poincar6 map, taking the position of cell 1 when cell 2 begins its active phase to the position of cell 2 when cell 1 is at that spot; the fixed point of this is the antiphase solution. Again, time metrics greatly facilitate the analysis of the map. 5.1.3. FTM and Frequency control in "half-center" oscillators. As shown in Rubin and Terman [52], bursting cells that are mutually inhibitory, with fast inhibition, do not stably synchronize. In this case, the same mechanism that gives rise to synchronization for fast excitatory synapses leads to antiphase. An oscillating network with the two cells in antiphase is known as a "half-center" oscillator. Such behavior is frequently found as part of networks governing locomotion or other motor patterns, especially if there is antiphase symmetry in the motor outputs (Marder et al. [42]). In Rubin and Terman [52], it was shown that periodic solutions can exist for the coupled pair even if neither is an oscillator. Here we focus on some properties of the antiphase solutions, including the control of their frequency.

Mechanisms of phase-locking and frequency control

47

When cells are locked but not synchronous, there can be different ways to trigger a switch between active and inactive states of a cell. There are two kinds of distinctions we note: which of the two cells governs the switch (Wang and Rinzel [69]), and whether the mechanism is intrinsic or synaptic (Skinner et al. [56]). We start with the intrinsic mechanisms. For these, the synaptic threshold is between the active and inactive branches of both the original and inhibited nullcline. The exact value of the threshold voltage plays no role in the behavior of the coupled network. The first of these intrinsic mechanisms has been called "release" (Wang and Rinzel [69]); it is illustrated in Figure 5.3A. In this setup, the active cell reaches the end of its active plateau and jumps down. Because its jump traverses the synaptic threshold, this changes the governing nullcline for the other cell and releases it from inhibition. As in Figure 5.2, if the position of the inactive cell is below the new nullcline, it will jump to the active state. The release-induced jump is sometimes called "post-inhibitory rebound". Note that a periodic solution with this mechanism requires that the active state be long enough to allow the inactive cell to traverse enough of the left-hand nullcline to be able to jump up at the same time as the jump down of the active cell. Figure 5.3B illustrates the mechanism known as "escape": the inactive cell, which is on the left-hand branch of the inhibited nullcline, reaches the lower knee of that nullcline, escaping the inhibition from the other cell and jumping up. This lowers the governing nullcline of the other cell, allowing it to jump down if its trajectory is high enough on this nullcline. Note that a periodic solution of this type requires the inactive state to be sufficiently long. For each of these intrinsic mechanisms, the coupled state is an oscillation with the cells in antiphase and the active and inactive states of each identical in length. Neither mechanism requires that the uncoupled cells be capable of oscillations. There are, however, some constraints on the cells for the half-center oscillator to work. For example, in the release mechanism, there may or may not be a stable critical point for the uninhibited cell and/or the inhibited one; what is critical is that the inhibited cell is able to rebound when it is released. Further mechanisms of antiphase oscillations are obtained by relaxing the restriction that the synaptic threshold lie between the active and inactive branches. If the synaptic threshold is on one of the branches, then the signal given from one cell to another changes as the former traverses that branch, not during its jump between branches. This has a large effect on the wave-form and frequency of the resulting oscillation. (See Figure 5.4A.) If the threshold is in the low-voltage (inactive) region, the inactive cell escapes when it crosses the threshold, not when it reaches the lower knee; this is "synaptic escape". In this region, increasing the synaptic threshold increases the period of the coupled system. If the threshold is in the high-voltage (active) region, the release also occurs before the releasing cell reaches its own higher knee; this is "synaptic release" (Figure 5.4B). In that region, increasing the synaptic threshold decreases the period. If the threshold is between the branches, changing the former does not affect the period of the coupled system. It is possible to probe a real network to see what mechanism it is using to switch between states. For the network controlling heartbeat in the leech, this was done using a "dynamic clamp", an apparatus that replaces the real chemical synapses between the cells by ones that are computer-generated to have prescribed characteristics. By varying the synaptic

N. Kopell and G.B. Ermentrout

48

A

B SYNAPTIC ESCAPE

SYNAPTIC RELEASE

!

_f'M

!

! !

Fig. 5.4. A. The active cell releases the other cell from inhibition when it crosses the synaptic threshold and stops inhibiting the other cell. B. The inactive cell escapes before it reaches the knee: once it crosses the synaptic threshold, it inhibits the active cell, which then jumps down and stops inhibiting the first cell.

threshold and measuring the frequency of the resulting network, it was found that the period increased with threshold in the low voltage range and decreased with it in a higher voltage range, consistent with the use of synaptic escape mechanisms in the low voltage region and synaptic release in the higher voltage range (Sharp et al. [53]). 5.1.4. Variations: coupling between dissimilar cells. In this section, we have so far focused on synchronous and anti-synchronous behavior of two bursting neurons coupled by fast excitatory or inhibitory synapses. Here we briefly mention that such synapses between two dissimilar cells can lead to new behavior; these results underscore how flexible even a two-cell network can be. LoFaro and Kopell [38], considered a two-cell subnetwork from a well-studied invertebrate network. In this network, the cells are coupled by inhibition and one of the cells has a slow inward current that turns on when the voltage is low. They showed that modulation of the slow current could lead to behavior with one cell bursting N times for each burst of the slower cell, with N adjustable by varying the magnitude or voltage activation curve (moc(v)) of the slow current; changes of kinetics of the cell (rate of reset when voltage is high) could allow the same network to produce N : M ratios of bursting instead of just N : 1. The mathematical essence is the use of time-metric techniques to reduce the full (5-dimensional) equations to a one-dimensional map, from which the network behavior could be deduced. Working with other aspects of the same cells, Nadim et al. [47] showed that, if one of the synapses is more complicated, the network can display switches in control of frequency. The complication is "depression" in the synapses: instead of (5.1), in which there is just one voltage-dependent gate, the synapse is treated as also having an inactivation gate with a monotone-decreasing hoe(v), as in the Introduction. In this network, a small increase in the size of the maximal conductance of the depressing synapse can start a regenerative

Mechanisms of phase-locking and frequency control

49

mechanism, leading to a large increase in the synaptic current and a switch from a mechanism in which the intrinsic properties of the cells control the network frequency to one in which the kinetics of the synapse controls the frequency. Even if the membrane currents are very simple (no voltage-gated membrane currents, only a leak current), a network can produce oscillations using the nonlinearities in the synapses. Manor et al. [39] modeled another subnetwork of the same invertebrate system, producing a half-center oscillation from two cells, neither of which were modeled with voltage-gated currents. The cells are different in their degree of excitability, leading one to be on and the other off in the absence of synaptic input of a third excitatory cell; the latter, turned on and off by the first two, acts more slowly to change the relative excitability of the two primary cells.

5.2. Electrical coupling of cells or compartments In Sections 2.5 and 3.3 we used phase equations and the spike response method to show that weak electrical coupling could lead to antiphase solutions; in this section, we shall be concerned with unintuitive behavior of electrical coupling in the strong coupling regime. We shall illustrate this by discussing work concerning a single cell, but one extended in space, and (conceptually) discretized into different "compartments". As in networks of spiking cells, electrical synapses are modeled as discrete diffusion in the voltage equation only. The same mathematical formalism used for coupling between compartments is also used to describe electrical coupling between cells, and we give some examples in which such coupling does not have the effect of synchronizing cells. 5.2.1. Strong coupling within a dendrite. In recent work on dopamine-releasing cells, Wilson and Callaway [74] have been investigating the hypothesis that the firing patterns of the cells depend on the differences in dynamics along the dendrites of the cell; they also have evidence for oscillations in the cell, and interpret their data to suggest that each single cell can be thought of as a chain of oscillating compartments, electrically coupled, with a gradient in frequency along the chain. Their first model used a Morris-Lecar-like set of equations for each compartment. A simpler set of equations that reproduces the numerical behavior of that model uses FitzHugh-Nagumo oscillators coupled electrically (Medvedev and Kopell [44], Medvedev [43]). For each of two compartments, the equations are ev' = f (v) - u + d(f~ - v), U t --

(5.2)

(.OU,

where v is voltage of the compartment in question, u is here the concentration of calcium in that compartment, and ~3 is the voltage in the other compartment. The function f (v) is a cubic, co governs the period in this compartment, as does ~ in the other compartment. co and ~ need not be even approximately the same. The coupling d is very strong, i.e., d >> 1. The strength of the coupling implies that the voltage in the two compartments are

N. Kopell and G.B. Ermentrout

50

almost the same, even though the natural periods of the oscillators are not necessarily close. However, the values of u in the two coupled compartments need not be close. The system turns out to have an unexpected mathematical structure: in the limit of large coupling, there is an invariant cylinder foliated by periodic orbits, with an integral that is constant on each orbit. For d large but not infinite, this integral becomes a Lyapunov function, with slow drift between orbits that are periodic when d = oo (see Figure 5.4). The analysis uses two independent small parameters, e and 1/d. The critical idea (backed up by asymptotic expansions, and then proved rigorously) is that (fi - v) is O ( 1 / d ) , so the term d(~ - v) is O(1) in 1/d. Since ~ ~ v, one can think of d(~ - v) (to lowest order) as a function of v alone and the branch on which the two trajectories lie; we denote this function c(v). One works with singular orbits, combinations of solutions to the slow and fast equations with e --+ 0. For definitions of fast and slow equations see Rubin and Terman [52,51 ]. The slow equations for (5.2) are U l = f (v) + c(v),

(5.3)

u2 -- f (v + 6 c ( v ) ) - c(v), !

U 1 ~ 0)11)1,

=

+

Here ~ = 1/d and we are using indices 1 and 2 for the compartments, with v -----vl and V2 = v + 3c(v). Note that the coupled equations (5.3) can now be treated as uncoupled within each branch, as in fast threshold modulation. From (5.3) it is possible to get a differential equation for c(v), by differentiating the last two equations and using the first two. To lowest order, one gets 0) 1 - - 0)2

co(v) -- ~

(5.4)

f (v) + K ,

0)1 + 0 ) 2

where K is an arbitrary constant. This can be used in (5.3) to get u l and U2 as functions of v to lowest order in 6. It is to next order that one sees the mathematical structure. One can compute this using asymptotic expansions, but it is conceptually clearer to see that there is a Lyapunov function for the singular orbits, which is given explicitly as

L(ul,

1 u2) -- ~

ul Li

u2 t - A

,

0)2

where

- f f (v(t)) dt. A--T ~o1~o2 Each singular periodic orbit can be associated with a value of K in (5.4) and L = L ( K ) ; there is a unique value of K for which L = 0, and L (K) > 0 otherwise. Furthermore, over

Mechanisms o f phase-locking and frequency control

51

one full singular cycle, L decreases by O(6), so L is an integral for 6 = 0 and a Lyapunov function for 6 > 0. The fact that larger coupling leads to longer-lasting transients is not a property of all relaxation oscillators coupled through the fast system. By changing the nullcline of the slow (u) variable to v : Vu, where V = O(1) in e and 6, the rate of synchronization becomes almost independent of 6. There are also slow nullclines for which the coupled system has a 1-parameter family of periodic solutions for every value of d.

5.3. Electric coupling of heterogeneous cells Strong diffusion-like coupling between identical cells tends to synchronize the cells. If the elements being coupled are very different, the effect of strong coupling is less intuitive. Manor et al. [40] and Smolen et al. [57] have shown that oscillations can emerge from the electrical coupling of cells that are not oscillators, provided that some appropriate "average" of the cells can oscillate. Kopell et al. [32] considered electrical coupling of a relaxation oscillator and a simple (one-dimensional) bistable cell. They used a new geometrical technique to transform information about the geometry of the fast equation v' = f (v, w) of the oscillator to information about the jumps between high and low voltage when the cells are coupled. The technique shows how the electrical coupling can keep the oscillator pinned at a high or low voltage level longer than the uncoupled oscillator, and that larger coupling may actually be less effective at this pinning than coupling of some intermediate size.

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[12] S.M. Crook, G.B. Ermentrout and J.M. Bower, Dendritic and synaptic effects in systems of coupled cortical oscillators, J. Comp. Neurosci. 5 (1998), 315-329. [ 13] S.M. Crook, G.B. Ermentrout and J.M. Bower, Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators, Neural Comp. 10 (1998), 837-854. [14] G.B. Ermentrout and N. Kopell, Frequency plateaus in a chain of weakly coupled oscillators, I, SIAM J. Math. Anal. 15 (1984), 215-237. [15] G.B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math. 46 (1986), 233-253. [16] G.B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in coupled neural oscillators, J. Math. Biol. 29 (1991), 195-217. [17] G.B. Ermentrout and N. Kopell, Fine structure of neural spiking and synchronization in the presence of conduction delays, Proc. Nat. Acad. Sci. USA 95 (1998), 1259-1264. [ 18] G.B. Ermentrout, M. Pascal and B. Gutkin, The effects of spike frequency adaptation and negative feedback on the synchronization of neural oscillators, Neural Comp. 13 (2001), 1285-1310. [19] W. Gerstner, Time structure of the activity in neural network models, Phys. Rev. E 51 (1995), 738-758. [20] W. Gerstner, J.L. van Hemmen and J. Cowen, What matters in neuronal locking?, Neural Comp. 8 (1996), 1653-1676. [21] C.M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions, J. Comp. Neurosci. 1 (1994), 11-38. [22] S.K. Han, C. Kurrer and Y. Kuramoto, Dephasing and bursting in coupled neural oscillators, Phys. Rev. Lett. 75 (1995), 3190-3193. [23] D. Hansel, G. Mato and C. Meunier, Synchrony in excitatory neural networks, Neural Comp. 7 (1995), 307-337. [24] EC. Hoppensteadt and E.M. Izhikevich, Weakly Connected Neural Networks, Springer, New York (1997). [25] E.M. Izhikevich, Class 1 neural excitability, conventional synapses, weakly connected networks and mathematical foundations ofpulse-coupled models, IEEE Trans. Neur. Networks 10 (1999), 499-507. [26] E.M. Izhikevich, Neural excitability, spiking and bursting, Internat. J. Bifur. Chaos 10 (2000), 1171-1286. [27] E.M. Izhikevich, Phase equations for relaxation oscillations, SIAM J. Appl. Math. 60 (2000), 1789-1805. [28] S.R. Jones, D. Pinto, T. Kaper and N. Kopell, Alpha-frequency rhythms desynchronize over long cortical distances: a modeling study, J. Comp. Neurosci. 9 (2000), 271-291. [29] J. Karbowski and N. Kopell, Multispikes and synchronization in a large-scale neural network with delays, Neural Comp. 12 (2000), 1573-1606. [30] W.M. Kistler, W. Gerstner and L. van Hemmen, Reduction of the Hodgkin-Huxley equations to a singlevariable threshold model, Neural Comp. 9 (1997), 1015-1045. [31] N. Kopell, Toward a theory of modelling central pattern generators, Neural Control of Rhythmic Movements, A. Cohen, S. Grillner, S. Rossignol, eds, Wiley, New York (1987), 369-413. [32] N. Kopell, L. Abbott and C. Soto-Trevino, On the behavior of a neural oscillator electrically coupled to a bistable element, Phys. D 121 (1998), 367-395. [33] N. Kopell and G.B. Ermentrout, Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39 (1986), 623-660. [34] N. Kopell and G.B. Ermentrout, Phase transitions and other phenomena in chains of oscillators, SIAM J. Appl. Math. 50 (1990), 1014-1052. [35] N. Kopell, G.B. Ermentrout, M. Whittington and R.D. Traub, Gamma rhythms and beta rhythms have different synchronization properties, Proc. Nat. Acad. Sci. USA 97 (2000), 1867-1872. [36] N. Kopell and D. Somers, Anti-phase solutions in relaxation oscillators coupled through excitatory interactions, J. Math. Biol. 33 (1995), 261-280. [37] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer, New York (1984). [38] T. LoFaro and N. Kopell, Timing regulation in a network reduced from voltage-gated equations to a onedimensional map, J. Math. Biol. 38 (1999), 479-533. [39] Y. Manor, F. Nadim, J. Ritt, S. Epstein, E. Marder and N. Kopell, Network oscillations generated by balancing asymmetric inhibition in passive neurons, J. Neurosci. 19 (1999), 1765-2779. [40] Y. Manor, J. Rinzel. I. Segev and Y. Yarom, Low amplitude oscillations in the inferior olive: a model based on electrical coupling of neurons with heterogeneous channel densities, J. Neurophysiol. 77 (1997), 27362752.

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[41 ] E. Marder and R.L. Calabrese, Principles of rhythmic motor pattern generation, Physiological Rev. (1996), 687-717. [42] E. Marder, N. Kopell and K. Sigvardt, How computation aids in understanding biological networks, Neurons, Networks and Motor Behavior, Stein, Selverston, Grillner, eds, MIT Press, Cambridge, MA (1998), 139-150. [43] G. Medvedev, Some problems on pattern formation and oscillations in mathematical biology, Ph.D. Thesis, Boston University (1999). [44] G. Medvedev and N. Kopell, Synchronization and transient dynamics in chains of FitzHugh-Nagumo oscillators with strong electrical coupling, SIAM J. Appl. Math. (2001) (in press). [45] R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math. 50 (1990), 1645-1662. [46] C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical J. 35 (1981), 193-213. [47] E Nadim, Y. Manor, N. Kopell and E. Marder, Synaptic depression creates a switch that controls the frequency of an oscillator circuit, Proc. Nat. Acad. Sci. USA 96 (1999), 8206-8211. [48] D. Pinto, S.R. Jones and N. Kopell, Modulation of ionic currents changes rhythms expressed and synchronization properties (2001) (in preparation). [49] J. Rinzel and G.B. Ermentrout, Analysis of neural excitability and oscillations, Methods of Neuronal Modeling, C. Koch and I. Segev, eds, 2nd edn., MIT Press, Cambridge, MA (1998), 251-292. [50] P.R. Roelfsema, A.K. Engel, E K6nig and W. Singer, Visuomotor integration is associated with zero timelag synchronization among cortical areas, Nature 385 (1997), 157-161. [51] J. Rubin and D. Terman, Geometric analysis of population rhythms in synaptically coupled neuronal networks, Neural Comp. 12 (2000), 597-645. [52] J. Rubin and D. Terman, Geometric singular perturbation analysis of neuronal dynamics, Handbook of Dynamical Systems, Vol. 2, B. Fiedler, ed., Elsevier, Amsterdam (2002), 93-146. [53] A.A. Sharp, EK. Skinner and E. Marder, Mechanisms of oscillation in dynamic clamp constructed two cell half-center circuits, J. Neurophysiol. 76 (1996), 867-883. [54] A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Nat. Acad. Sci. USA 89 (1992), 2471-2474. [55] W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Ann. Rev. Physiol. 55 (1993), 349-374. [56] EK. Skinner, N. Kopell and E. Marder, Mechanisms for oscillation and frequency control in networks of mutually inhibitory relaxation oscillators, J. Comp. Neurosci. 1 (1994), 69-87. [57] E Smolen, J. Rinzel and A. Sherman, Why pancreatic islets burst but single ~ cells do not. The heterogeneity hypothesis, Biophys. J. 64 (1993), 1668-1680. [58] D. Somers and N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybern. 68 (1993), 393-407. [59] D. Somers and N. Kopell, Waves and synchrony in arrays of oscillators of relaxation and non-relaxation type, Phys. D 89 (1995), 169-183. [60] D. Terman and D.L. Wang, Global competition and local cooperation in a network of neural oscillators, Phys. D 81 (1995), 148-176. [61 ] R.D. Traub and R. Miles, Neuronal Networks of the Hippocampus, Cambridge Univ. Press, Cambridge, UK (1991). [62] R.D. Traub, M.A. Whittington, E.H. Buhl, J.G.R. Jeffreys and H.J. Faulkner, On the mechanism of the gamma-beta shift in neuronal oscillations induced in rat hippocampal slices by tetanic stimulation, J. Neurosci. 19 (1999), 1088-1105. [63] R.D. Traub, M.A. Whittington, I.M. Stanford and J. Jeffreys, A mechanism for generation of long-range synchronous fast oscillations in the cortex, Nature 328 (1996), 621-624. [64] R.D. Traub, J.G.R. Jeffreys and M. Whittington, Fast Oscillations in Cortical Circuits, MIT press, Cambridge, MA (1999). [65] C. van Vreeswijk, L.E Abbott and G.B. Ermentrout, When inhibition not excitation synchronizes neural firing, J. Comp. Neurosci. 1 (1994), 313-322. [66] G. de Vries, H.-R. Zhu and A. Sherman, Diffusively coupled bursters: Effects of heterogeneity, Bull. Math. Biol. 60 (1998), 1167-1200.

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[67] A. von Stein, C. Chiang and E K6nig, Top-down processing mediated by interareal synchronization, Proc. Nat. Acad. Sci. 97 (2000), 14748-14753. [68] X.-J. Wang and G. BuzsLki, Gamma oscillation by synaptic inhibition in an interneuronal network model, J. Neurosci. 16 (1996), 6402-6413. [69] X.-J. Wang and J. Rinzel, Alternating and synchronous rhythms in reciprocally inhibitory model neurons, Neural Comp. 4 (1992), 84-97. [70] X.-J. Wang and J. Rinzel, Oscillatory and bursting properties of neurons, The Handbook of Brain Theory and Neural Networks, M.A. Arbib, ed., MIT Press, Cambridge, MA (1995), 686-691. [71] J. White, C. Chow, J. Ritt, C. Soto-Trevino and N. Kopell, Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons, J. Comp. Neurosci. 5 (1998), 5-16. [72] M.A. Whittington, R.D. Traub and J. Jeffreys, Synchronized oscillations in interneuron networks driven by metabotropic glutamate receptor activation, Nature 373 (1995), 612-6 15. [73] T. Williams and G. Bowtell, The calculation of the frequency-shift function for chains of coupled oscillators, with application to a network model of the lamprey locomotor pattern generator, J. Comp. Neurosci. 4 (1997), 47-55. [74] C.J. Wilson and J.C. Callaway, A coupled oscillator model of the dopaminergic neuron of the substantia nigra, J. Neurophysiol. 83 (2000), 3084-3100. [75] A.T. Winfree, The Geometry of Biological Time, Springer, New York (1980).

CHAPTER

2

Invariant Manifolds and Lagrangian Dynamics in the Ocean and Atmosphere Christopher Jones *t and Sean Winkler* Division of Applied Mathematics, Brown University, Providence, R102912, USA

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2. Geometry of Lagrangian transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

2.1. Basics of transport theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fluid exchange in the Gulf Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. History of transport studies of the Gulf Stream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 60 61

3.1. Ring pinch-off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.2. Kinematic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Dynamically consistent models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A meandering jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64

4.1. An evolved jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. A periodic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64 67

4.3. Invariant manifolds

67

4.4. Lobe dynamics

............................................

..............................................

5. Finite-time transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Effective invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Transport calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. An adiabatic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. A theory for finite-time invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. An eddy-shedding event . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 70 71 72 74 74 75

7. Geometry, statistics, and the Antarctic polar vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Relative dispersion and coherent structure boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Application to the polar vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76 78 80

8. Vorticity and viscosity

85

.............................................

8.1. Potential vorticity fluxes

.........................................

8.2. Viscosity induced transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88

9. Conclusions and future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

References

90

.....................................................

*Partially supported by the Office of Naval Research under grant N00014-93-1-0691. tpartially supported by the National Science Foundation under grant DMS-9704906. H A N D B O O K OF D Y N A M I C A L SYSTEMS, VOL. 2 Edited by B. Fiedler 9 2002 Elsevier Science B.V. All rights reserved 55

56

C. Jones and S. Winkler

Abstract The ocean presents an exciting array of dynamical phenomena. The complex time dependence of the flow fields, the relevance of effects at a variety of scales, and the difficulty in collecting data all serve to make this a fertile area for the application of dynamical systems. We address here the question of Lagrangian transport in and around coherent features in the ocean. Examples of such features are the Gulf Stream and rings that detach from it. The focus is on the development of geometric techniques to track and quantify this transport. The key method is the isolation of invariant manifolds that act as boundaries of distinguished regions in the flow, such as eddies and recirculating cells, and orchestrate the transport associated with a given feature. In simple models, these are stable and unstable manifolds of fixed points. In flows with complex time dependence, such as occur in the ocean, such manifolds may not always make sense. We make the case here that these basic concepts of dynamical systems need to be adapted to accommodate such complex flows and that, moreover, this can be achieved with meaningful and significant results for the ocean and atmosphere.

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

57

1. Introduction

Despite the immensity of the ocean and the complexity of its dynamics, coherent medium and large-scale structures are clearly identifiable and persistent. The Gulf Stream in the North Atlantic and its analog in the Pacific, the Kuroshio Current, are striking examples. They can be easily distinguished from their surrounding waters through satellite data that records such properties as sea-surface height and temperature. In Figure 1, averaged sea surface temperature is shown and the sharp gradients reveal the Gulf Stream with considerable clarity. Its direction, spatial and temporal persistence, and meandering are all evident from such data. There are also eddies that detach from the Gulf Stream on a regular basis. These socalled Gulf Stream rings are oceanic versions of hurricanes that carry water through an ocean region with contrasting properties. They can survive for months while traveling

Fig. 1. Water surface temperature from AVHRRdata for a time interval of 6.76 days ending January 19, 1999. (Courtesy of the Space OceanographyGroup, Johns Hopkins UniversityApplied Physics Laboratory.)

58

C. Jones and S. Winkler

westward back toward the eastern United States. After a certain time, they are usually re-entrained into the Gulf Stream. These coherent features, currents such as the Gulf Stream and Gulf Stream rings alike, do not pass through regions of the ocean without leaving their mark. Indeed, they impact the waters through which they pass in a number of ways. One of the most evident is the straightforward process of fluid exchange between a given coherent feature and its ambient water. Indeed, such a feature does not live as exactly the same body of water but, while maintaining its overall identity, is involved in a continual swapping of water with the surrounding ocean. This process of "communication" between a coherent structure and the ambient waters has implications for the overall distribution of critical fluid properties, such as heat and salinity, as well as the distribution of biological nutrients. The consequences of changes in these processes may be far-reaching; see, for instance, the article by Calvin [ 15] for a discussion of implications for the climate. Dynamical systems techniques have recently been developed in the context of oceanic Lagrangian dynamics with the aim of elucidating this exchange process. The goal of this work has been to use these methods to track and quantify fluid redistribution in active regions, such as the Gulf Stream. The analysis promises also to expose the physical mechanisms underlying the fluid exchange processes and give an assessment of their influence on ambient waters. This article will give a description of these developments and point to some of the challenges this line of work has opened up. The key technique is the use of invariant manifolds to delineate regions of specific flow characteristics. The interactions between different manifolds then underpin the fluid exchange process. The theory of invariant manifolds is well-developed for simple flows that are steady or periodic in time, such as would occur in basic model flows. The challenge lies in the fact that dynamical systems concepts in general, and those underlying invariant manifolds in particular, rest on infinite time concepts, while in realistic flow fields the structures of interest may persist for only finite spans of time. It is not clear in this context whether the traditional concept of an invariant manifold makes sense.

2. Geometry of Lagrangian transport The motion of a fluid particle is a trajectory of a dynamical system whose vector field is the velocity field of the flow, found by solving the appropriate partial differential equations (such as the Navier-Stokes or Euler equations). The phase space is the physical space and coherent features can be understood as geometric structures familiar to researchers in dynamical systems. In the simplest models in which the flow field is periodic, these structures are invariant manifolds, stable and unstable, of saddle fixed points. The transport of fluid is effected by the transverse intersection of these stable and unstable manifolds. This mechanism of fluid transport was earlier invoked as an explanation of mixing in laboratory scale fluids, see Aref [1] and Ottino [35]. This work goes under the title of chaotic advection as it is the chaotic nature of the transport that makes the mixing so effective. For the ocean, we are not as interested in the "chaotic" nature of this effect, but rather its power to describe transport.

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

59

2.1. Basics o f transport theory A two-dimensional, incompressible flow admits a representation by a streamfunction 7z(x, y, t) in the following manner ~c = u ( x , y, t) =

O ~ ( x , y, t)

-- v(x, y, t) -- -

Oy OTt(x, y, t) Ox

where u and v are the velocities in the x and y directions respectively. From a dynamical systems point of view, this set of equations is a Hamiltonian system with the streamfunction 7t being the Hamiltonian. In a steady flow, the streamfunction is independent of time, and the dynamical system is therefore autonomous. Lagrangian particle paths will coincide with streamlines, allowing a complete description of the transport by studying level sets of the Hamiltonian. In general, the phase space will be divided into regions of recirculation around elliptic points separated by heteroclinic trajectories associated with hyperbolic fixed points. These separatrices are the invariant manifolds of the hyperbolic points and act as boundaries to fluid transport. Now, suppose the streamfunction is time-periodic with period T. To understand the Lagrangian motions of fluid particles in this case, one studies the period T Poincar6 map, denoted 79. When the time-periodic vector field results from a small periodic perturbation of a steady streamfunction, standard results from dynamical systems theory guarantee that hyperbolic fixed points of the steady system will perturb to hyperbolic fixed points of the associated Poincar6 map, and their attendant invariant manifolds perturb as well [21]. However, the perturbed invariant manifolds will, in general, intersect transversely. Such intersections are heteroclinic points, and the resulting complex manifold structure is known as a heteroclinic tangle. Suppose pl and p2 are distinct hyperbolic fixed points of the Poincar6 map, and that the unstable manifold W u (pl) intersects transversely the stable manifold W s (p2) (see Figure 2). Invariance of the manifolds and the time-periodicity of the flow imply that if the manifolds intersect once, they will in fact intersect infinitely many times. Let q be a homoclinic point, and let U [ p l , q] denote the segment of W u ( p l ) joining pl to q. Similarly, let S[p2, q] denote the segment of W s (p2) joining p2 to q. The point q is called a primary intersection point (pip) if U [ p l , q] and S[p2, q] intersect only at q. A lobe is a region bounded by the segments of W u (Pl) and W s (P2) that join an adjacent pair of pips. The invariance of the manifolds guarantees that lobes are mapped to lobes. For incompressible flows, the area of a lobe will remain constant under iteration of 79. By examining a typical heteroclinic tangle, a mechanism for transport via the dynamics of the lobes becomes apparent. Three distinct types of particle motion are evident in Figure 2. Motion in region 7~1 is left to right, or prograde, and unbounded. Similarly, motion in region 7~3 is right to left, or retrograde, and unbounded. Within the recirculation region 7~2, the fluid circulates counterclockwise. The phase space may be clearly partitioned into regions of distinct motion using the manifolds as boundaries. By choosing a pip q0, U [ p l , q0] U S[p2, q0] defines a boundary separating 7~1 from J'~2.

60

C. Jones and S. Winkler i

!

i

,

i

|

~3 -

WS(px)

i,

i 1

~w~(p~)

~

1 -

. 5 3~"~

$ 3 ..

~t~_o "',,~q-a

. . ~_g

/ f ; -" ]4_1

_

i -.~

BO x

~~....~=T~

i

ql ~ ~

~ ~

-

A1

WU(pl)~~o

-

I

t=2T

2

i

~

i

/

/

i

Fig. 2. Chaotic transport for a time-periodic flow. Transverse intersections of the stable manifold W s (P2) (dashed line) and unstable manifold W u (Pl) (solid line) result in large fluxes across the boundary separating 7~ 1 and 7~2.

Let 79k, k e Z, denote iterations of the Poincar6 map, and let qk =~ 79k(qo). Define the lobes A0 -- S[q0, q0] t_J U[q0, q0] and B0 -- S[q0, q-l] U U[q0, q-l], and Ak =- 79k(Ao), Bk -~ 79k(Bo). With these definitions, consider the action of the Poincar6 map on the lobes A0 and B0. A0 is in region 7~1, whereas A1 is in region 7~2, and B0 is in region 7~2 while B1 is in region Tr This exchange of fluid between T~l and 7~2 represents Lagrangian transport via the lobe dynamics mechanism. The lobes A0 and B0 are called turnstile lobes, since they are the lobes involved in fluid exchange from one region to the other in one iteration of the Poincar6 map. The area of Ao (respectively B0) represents the amount of fluid transported from 7~1 to 7~2 (respectively 7r to ~1) in one time period, and dividing this amount by the period defines an average flux across the boundary. This is the basic idea of lobe dynamics in time-periodic flows, with more detailed descriptions available in the works of MacKay et al. [30], Rom-Kedar et al. [42,43], Wiggins [47], Camassa and Wiggins [ 16]. Similar results may be formulated for quasiperiodic flows (see [4,5,47,19]) in terms of two-dimensional Poincar6 maps, and generalized for adiabatic flows as well (see [28,29]).

2.2. Fluid exchange in the Gulf Stream The problem of fluid exchange in the ocean can be viewed in terms of this transport theory. The first conclusion from the above considerations that is relevant here is that time dependence of the underlying flow field is critical in this process of fluid exchange. The Gulf Stream can be viewed as a snake emanating from the separation at Cape Hatteras on the East Coast of the United States (Figure 1). If the humps and troughs of the snake remained steady, at least in a moving frame, then the fluid in the jet of the Gulf Stream would flow inexorably with the current and the edges of the current would not see exchange of

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

61

Fig. 3. Cartoon of Lagrangian transport near the Gulf Stream via lobe dynamics.

fluid, only different fluid particles flowing at very different speeds. However, the humps and troughs are not steady but pulsating relative to one another (in any moving frame). A cartoon for this process can be developed by strategically placing some key dynamical features on the Gulf Stream jet. In Figure 3, fixed points of the flow are placed at the top of "hills" of the meandering jet. This is dynamically reasonable, except that the points cannot be truly fixed as the entire meander is moving to the northeast. However, by viewing the structure in a moving frame, the idea of a fixed point at each peak is more reasonable. Inside each meander will be a recirculation zone and this enforces a separation point at the peak. This stagnation point is the fixed point mentioned above. In this picture, there are now three distinct dynamic regimes: the core of the jet, the recirculation zone and the retrograde motion north of the jet (with analogous structures on the south side). The demarcation of the boundary between these three regimes is supplied by the stable and unstable manifolds of the separation point. Due to the complex time dependence and the resulting intersections of invariant manifolds, as depicted in Figure 3, water in the northern retrograde flow can be carried into the recirculation zone and, similarly, water from the recirculation zone can be carried into the jet. This cartoon of transport is definitely easiest to understand if the flow is periodic. However, any realistic fluid flow modeling the ocean will not be periodic, and there are some fundamental issues in extending these ideas to the general, non-periodic case.

3. History of transport studies of the Gulf Stream The realization that fluid trajectories in the Gulf Stream may not be regular came relatively recently. In the mid 1980s, Bower and Rossby completed the first comprehensive sub-

62

C. Jones and S. Winkler 75N

70H

.:,-...':~~ 9

. i

~

/)/ /

65H

60N

55N

40N

35N

'

35N

/

76H

70N

65H

60H

65H

S P A G H E T T I DIAGRAM OF RAFOS FLOATS 1984-1985 Fig. 4. Float trajectories from Bower and Rossby [8].

surface float study of the Gulf Stream [8]. The results were surprising in that float paths appeared to have little predictable structure. Indeed, apart from an initial period during which most floats more or less followed the Gulf Stream path, the floats tended to drift off either to the south or north. The overall result is what is known as a spaghetti plot, as the paths look as if somebody dropped a pile of spaghetti on a map of the North Atlantic (see Figure 4). This work challenged the prevailing view of the Gulf Stream as a channel that would funnel water inexorably along its northeasterly path.

3.1. Ring pinch-off Bower and Lozier [7] posed the question as to whether the complicated Lagrangian float trajectories could be explained by the pinch-off of rings from the Gulf Stream. Ring pinchoff is a mechanism by which two neighboring troughs (or peaks) of the meander come together and create a ring that detaches from the Gulf Stream. This mechanism effects the bulk transport of fluid from the South to the North (or vice-versa) and also involves the ejection of jet water as parts of the jet encircle the ring. This is however a comparatively rare event (about 10-20 per year) and would seem to be an implausible explanation of the likely ejection of any given fluid particle from the jet as indicated by the Bower-Rossby spaghetti plots.

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

63

3.2. Kinematic models From the dynamical systems perspective, it is not surprising that a flow with complex time dependence could preserve its essential character, while at the same time any individual particle path could belie this character, appearing to disobey the guide set forth by the flow. This avenue of explanation was first pursued by Bower herself, who developed a kinematic model for a meandering jet that exposed the different possible regimes of motion, including recirculation regions above the troughs and under the hills of the jet [6]. This could explain some of the swirling motions observed in the float data, but fell short of explaining the ejection of fluid parcels from the Gulf Stream as the underlying streamfunction was time-independent and thus did not allow crossing of separatrices. Samelson suggested that some simple time dependence be put into the coefficients of the streamfunction, inducing a periodic undulation of the meander [45]. The fluid exchange in such a model is effectively represented in Figure 2, which explains in a natural way the motion of fluid from one regime to another. While the analysis of kinematic models shows convincingly that time dependence of the velocity field can explain fluid exchange, it cannot offer insight into the underlying physical mechanisms for the time dependence and hence the exchange. Moreover, these models cannot be used to estimate the amount of fluid involved in the exchange as they are based on streamfunctions that are merely hypothesized and not derived from the governing equations. In other words, the only physics in these equations is the incompressibility of the fluid that leads to the existence of the streamfunction itself. The question then remained as to how much water is actually involved in the exchange process. Were it shown to be insignificant in comparison to the water transported by ring pinch-off, the oceanographers would be free to ignore it.

3.3. Dynamically consistent models A further step was taken by del-Castillo-Negrete and Morrison [ 17] and Pratt et al. [39]. In each of these papers, a dynamically consistent model was developed with qualitatively similar behavior to that of the Bower-Samelson model. The models are formed by superimposing on a base jet normal modes arising from linearizing the barotropic equations around the jet. The idea is to take two neutral modes and add them to the jet with coefficients of different magnitude. The first (larger amplitude) mode will endow the jet with a meander and the second (smaller amplitude) mode will make it time-dependent. The time dependence will be non-trivial provided the frequencies of the two modes are incommensurate. The result is behavior similar to that observed by Samelson [45], and the analysis gives further corroboration to the idea that the time-dependent variation of a meandering jet causes significant fluid exchange between different constituent parts of a jet. Since the models contain much more physics than the kinematic models, they provide a useful indicator that such effects are present. However, if one asks the question as to the quantities involved in such exchanges, their usefulness diminishes. This point is rather subtle, but the reason that they allow exchange of fluid between regions of ostensibly different motion is that they are actually solutions of a linearized equation. It follows from the observations of

C. Jones and S. Winkler

64

Brown and Samelson [14] that a comparable nonlinear solution would have no exchange. In other words, the amount of transport between regions will be dependent upon, even commensurate with, the extent to which the linearized problem differs from the full nonlinear problem. If the approximation improved then the amount of fluid involved in the transport would be decreased. If the question being asked is whether this effect of "chaotic advection" occurs then these models give a satisfactory, and affirmative, answer; but if the issue is how great an effect it is, then the model will not be useful. The most reasonable approach is then to work directly with numerical models. The next section describes an analysis of such a model. With the Gulf Stream in mind, it seemed natural to analyze a meandering jet and assess the transport between different regimes.

4. A meandering jet In the work of Flierl et al. [20], the barotropic fl-plane equations were solved numerically with the aim of determining initial data that would generate a meandering jet. Although not formulated as a serious model of any specific ocean region, the model captures the fundamental effects that generate a meandering jet such as the Gulf Stream. This section describes a precursor to the studies of this model by Miller et al. [32] and Rogerson et al. [41 ], which represent the first full Lagrangian transport analysis for a numerical model. The governing equation for the flow is the equation for the potential vorticity

Oq -+- J (7r, q) = D, Ot where 7t is the streamfunction, q = V2~ -~- fly is the potential vorticity,/3 is the variation of the Coriolis parameter with latitude,

is the Jacobian of (f, g), and D is a dissipation term. The dissipation term can be viewed in two ways: first, as an aid to avoid numerical instability due to the development of high wave number effects and, secondly, as a crude model of eddy diffusivity in the ocean. Eddy diffusivity can in turn be viewed as a regularization of small-scale (turbulent) effects. In this study, D - ~1 V 4 ~. Periodic boundary conditions in both the zonal and meridional directions are assumed, and the flow is approximated pseudospectrally in a square computational domain of non-dimensional length L D. The initial data are set as a weakly perturbed horizontal jet; in other words, the linear approximation is used as a guide to seed the equations with a meandering jet that subsequently saturates into a fully nonlinear mode with the characteristics of a meandering jet (see [20] for details on how this is achieved).

4.1. An evolved jet The evolution of the flow for the parameter setting (Re, fl, n0) = (103, 0.103, 3) is shown in Figure 5 in terms of the potential vorticity field. (The Reynolds number Re is related to

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

65

.__.__--

t=20

t=40

t=60

m

t=80 O

-4

_

I

I

4

I

I

8

I

I

12

l

I

16

I

l

20

1

l

24

Fig. 5a. Evolution of the numerically-simulated flow as represented by the potential vorticity field for (Re, j3, n0) = (103, 0.103, 3). The early nonlinear evolution of the unstable jet, t = 20, 40, 60, 80.

the dissipation term, and no sets the number of meanders in the spatial domain, see [20].) The flow is unstable and develops nonlinearly from the initial perturbed zonal jet to a finiteamplitude meandering configuration. The generation and evolution of coherent structures in the flow for times t = 0 to t ~ 100 is discussed in detail by Flierl et al. By t -- 200 (Figure 5b), the flow has saturated into a configuration characterized by a primary meander with nearly constant eastward propagation and nearly periodic meander amplitude pulsa-

66

C. Jones and S. Winkler

t=200

L=220

i

t=240

4t

I

i

/

i

I

i

I

i

I

I

I

I

t=260

0-4

I

0

I

4

I

I

8

I

I

12

I

I

16

I

i

20

I

I

24

Fig. 5b. The nearly time-periodic meandering jet flow, t = 200, 220,240,260.

tion. It is this late-stage regime, in which the large-scale flow is nearly-periodic, that is of interest in this analysis. In a broad sense, the flow is composed of three regions exhibiting qualitatively different types of motion. The jet core is the region represented by the open potential vorticity contours meandering between y ~ • in Figure 5b. To the north and south of the jet in between meanders, there are recirculation regions, and the retrograde region in the far field is characterized by a westward flow.

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

67

4.2. A periodic approximation When viewed in a stationary reference frame, the numerical solution exhibits time dependence associated with the propagation of the jet meanders. Using power spectra, two dominant time periods can be identified. The first, To, corresponds to a spatial easterly migration and the problem is cast in a moving frame that effectively brings this period to zero. The second dominant period Tl = 56.50 is used as an approximate period for the flow itself. In other words, the flow is treated as a time-periodic flow of period T1. This Tl-periodic flow agrees exactly with the full numerical solution for time t = to to t -- to + T1 - 6t but deviates from it for times t >~ to + T1. Enforcing periodicity in this manner allows constructions based on calculations as t --+ -+-~. Since stable and unstable manifolds are defined by asymptotic conditions, this infinite time construction is used in a crucial way. The manifolds indicate how the chaotic motion is geometrically constrained, so to better understand this transport mechanism in the numerically simulated meandering jet flow, an analysis is performed on the model data to construct such structures. Lagrangian trajectories are computed for the flow using a 4th-order Runge-Kutta time integration with At = 0.05 and linear interpolation to approximate the velocity at intermediate times. The time dependence in the flow is on a much longer time scale than the interval At, and using a higher-order interpolation appears to have little effect on the computed results. A spatial interpolation is also required to estimate the velocity at off-grid points, for which there are a number of approaches (see [2,32]). The method used here is a bicubic polynomial interpolation requiring values of 7t, ~Px, ~ , and 7txy on a local 2 • 2 grid. Smoothness through the first derivatives of ~(x, y, t) ensures that the interpolation preserves the Hamiltonian structure present in the 2-D incompressible flow. The necessary derivatives are precomputed spectrally and transformed back to a physical space for the interpolation. This approach is a compromise between accuracy and computational efficiency. The Hamiltonian structure of the two-dimensional flow is retained with full spectral accuracy at the grid points, and some computational efficiency is gained from employing a local interpolation scheme. The construction of Lagrangian trajectories from numericallygenerated velocity fields is described in detail in [32].

4.3. lnvariant manifolds For each parameter setting, fixed points and manifolds are calculated for a single cat's eye structure on the northern side of the jet. Two fixed points, denoted p l and p2, are located and their associated eigenvalues and eigenvectors are computed. For the Poincar6 map 79, stable eigenvalues are those satisfying 0 < IUI < 1, unstable eigenvalues satisfy I)~L'I > 1, and the incompressibility can be expressed in terms of the determinant of 79 as 1D79[ -- ,V',ks = 1. The one-dimensional stable manifold for the fixed point is approximated by the backward time iteration of a short segment of initial points aligned with the stable eigenvector. The unstable manifold is computed by iterating forwards in time a short line segment of initial points aligned with the unstable eigenvector. The invariant curves presented here are all computed to a resolution of 0.02; that is, the Euclidean distance between adjacent points on the curve does not exceed 0.02.

68

C. Jones and S. Winkler

The extreme stretching rates 0~u "~ 103) in the vicinity of the hyperbolic fixed points makes it difficult to numerically identify these points and to completely resolve their invariant manifolds. Approaches used to address these difficulties are discussed in [32]. However, as a consequence of using the bicubic spatial interpolation, the computed lobe areas remain constant to at least three decimal places over several iterations of the map, even with the strong stretching of the lobe boundaries. Moreover, the Jacobian computed at the hyperbolic fixed points is very close to the theoretical value of 1.

4.4. Lobe dynamics Figure 6 shows the stable and unstable manifolds, W s and W u, with the fixed points pl and p2 for the parameter setting (Re, fl, no) = (103, 0.103, 3); these fixed points are in the vicinity of meander crests. The transverse intersection of the stable manifold W s (pl), and the unstable manifold W u (p2), indicates that there is fluid exchange across the exterior boundary defined as the union U[p2, qe] t,3 S[pl, qe]. (Recall the notation U[a, b] to denote the segment of W u connecting points a and b, and similarly for S[a, b], a segment of W s .) There are four distinct sets of lobes that result from this transverse intersection, labeled A, B, C, and D and indexed consistently with forward iterations of the Poincar6 map 79, i.e., 7~

7~ P 7~ >Ao > AI > A2 >'", 7~ 7~> 7~> 7~ > Bo BI B2 > -".

Under forward iteration of the map, fluid in lobes A l and C1 leaves the recirculation region and enters the region of retrograde motion while fluid contained in lobes B1 and D1 enters the recirculation region from the retrograde region. That is, the exchange of fluid across the exterior boundary takes place when fluid passes from lobes A l . . . . . D1 to lobes A2 . . . . . D2. As mentioned above, the designation of turnstile lobes depends on how the boundary between regions is defined (determined by the choice of qe). However, the identification of which lobe sets are entering and leaving a particular region is not affected by the choice of qe, and therefore the resulting transport estimates (discussed below) are not affected by this choice. The transverse intersection of W u (pl) and W s (p2) indicates that there is fluid exchange across the interior boundary, defined as U[pl, qi] t,3 S[p2, qi] (Figure 6(b)). The turnstile lobes along the interior boundary are labeled E, F, G and H. Lobes Fl and Hi enter the recirculation region from the edge of the jet core while lobes El and G l are mapped from the recirculation region to a region near the edge of the jet core (lobe G2 is not shown completely in Figure 6(b)). Exchange takes place across the interior boundary when fluid is mapped from lobes E1 . . . . . HI to lobes E2 . . . . . H2. Lobe areas are calculated using Green's theorem and can be used to estimate the transport associated with the Lagrangian motion described above. The dimensional transport is simply, A

A

T -- D* A/T1,

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

69

Table 1 Summary of the transport estimates. The exterior boundary separates the recirculation region from the retrograde region. The interior boundary separates the recirculation region and the edge of the jet core. The cumulative nondimensional lobe areas are listed along with the associated dimensional transport in Sverdrups (10 6 m 3 s - l ) . The dimensional scales are described in the text. Fluid leaving vortex

Fluid entering vortex

(Re, r, no)

Bndry

Lobes

Area

Transport

Lobes

Area

Transport

(103,0.103,3)

exter inter

B,D F, H

3.031 0.724

4.69 Sv 1.12 Sv

A, C E, G

3.031 0.724

4.69 Sv 1.12 Sv

WS(pl)

_

_

w~(p~)

3

(~)

Y 2

_

A1 6

8

i0

.

12

.

.

14

q~~.

.

Pl ~ ~ ~ / / o

16

~_~

_

18

20

x

E1

5

"",,

P2

4

(b)

\\ -

\

/

\

/

\~, -|

|

1/ t

i

,

'

Fig. 6. Computed stable (dashed line) and unstable (solid line) manifolds for (Re, fl, n0) = (103, 0.103, 3) and the resulting lobes. The heavier lines define our choice of boundary as described in the text. (a) Along the exterior boundary, U[p2,qe ] U S[pl,qe], fluid in lobes A 1 and C1 moves from the recirculation region to the outer retrograde region while lobes B 1 and D 1 are mapped from the retrograde region to the recirculation region. (b) Along the interior boundary, U[p 1, qi] U S[p2, qi], lobes E 1 and G 1 transport fluid from the recirculation region to the edge of the jet core. Lobes F1 and H1 are mapped from the edge of the jet core to the recirculation region.

70

C. Jones and S. Winkler

where A" is the dimensional lobe area, ~ is the dimensional period of the flow, and D* is a characteristic length scale representing the depth over which the transport takes place. Table 1 shows transport estimates for D* -- 500 m as applicable to the Gulf Stream, along with the horizontal length scale L* - 105 m and velocity scale U* = 1.75 m s - l ( u * -0.87 m s- 1) for nondimensional/3 - 0.103. The resulting transport fluxes are significant, although those associated with the exterior boundary exceed those of the interior by a factor of at least four. In comparison with the flux of water across the Gulf Stream associated with ring detachment, about 3 Sverdrup, the fluxes along the exterior boundary are the same order of magnitude. They must thus be viewed as substantial and cannot be ignored in any overall mass flux budget, see [41] for further discussion.

5. Finite-time transport While the above calculations make a striking case for the significance of Lagrangian transport in the overall budget of ocean transport, the artificial reconstruction of a flow field defined for all time by enforcing periodicity is unrealistic. As mentioned a number of times above, many structures in an ocean flow will persist for only finite spans of time. Moreover, the entire character of a flow may change and thus the templates against which the transport is being calculated may cease to be valid. Despite the temporary nature of coherent structures, they are identifiable for long time spans and they play key organizing roles in the overall flow during their lifetime. For instance, the Gulf Stream rings carry water that is quite saline to the coastal region of the East Coast. The transport of water in and out of these structures is a significant feature in the overall fluid budget and therefore a theory is needed that allows us to quantify that transport in an effective manner. Flows spanning a finite time of observation thus form the appropriate context for which a realistic theory of transport needs to be developed. To dynamicists, the limitation to finite time is a challenge, as the fundamental constructions of dynamical systems are based upon asymptotic information. Stable and unstable manifolds are defined by the behavior of solutions as t - . -+-c~, respectively. A coherent feature such as an eddy could naturally be thought of as defined by a surrounding stable manifold, but if the feature lives in a highly time-dependent flow and persists for only finite time, a stable manifold cannot be defined. A theory of "finite-time" stable and unstable manifolds is clearly needed. In the following, we outline an operational procedure for generating "effective" invariant manifolds. These play the same role in organizing the flow and orchestrating the transport as do stable and unstable manifolds of fixed points in steady or periodic flows. The underlying idea is to proceed in exactly the same way as when computing (numerically) an invariant manifold in a periodic system, or one for which infinite time data is available. The basic numerical strategy is articulated well by Nusse and Yorke [34] who adopt the term "straddling" for this procedure. The operation is most easily visualized for a saddle fixed point in the plane: a segment of initial data is picked that is transverse to the stable direction; under the evolution of the flow, or the map, it will be stretched out in the unstable directions while part of the segment close to the fixed point will be anchored by the exponential contraction along stable direction. The idea in the following is to implement this

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

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procedure in other cases where some semblance of a saddle fixed point exists dynamically, but which is perhaps time dependent in a sufficiently complex manner so as to preclude the existence of an actual fixed point. For instance, the region may have the hyperbolic character of that near a saddle fixed point for only a finite time.

5.1. Effective invariant manifolds A theory was developed in the paper by Miller et al. [32] that gives a practical approach to generating these "effective invariant manifolds". The term "effective" has both its meanings here: they are effective in the sense that they are manifestations in these complex flows of material surfaces that play the role of stable and unstable manifolds in transport studies in simpler systems, and also they are effective in that they do effect the transport under consideration. The idea is as follows. The data available and useful for the meandering jet spans over a time interval of approximately 60 time units. Before that time, the jet is forming out of the (linear) initial data and, after that, the jet is decaying due to the drain of viscosity (note that no forcing is included in that model which might have preserved the jet by balancing viscosity). However, during that time span, the characteristic structures of a meandering jet are evident. These include the central jet itself and its attendant recirculation zones lying in the troughs of the meanders and under the hills. In the analysis described in the previous section, the enforcement of periodicity allowed us to play freely with time integrations without worry as to whether we would run out of data by exiting the time window of available data. The consequence was that fixed points, or more generally trajectories, could be isolated, their stable and unstable manifolds generated, and their role in governing the dynamics determined. The natural question arises as to whether that freedom is really needed for the computations of the stable and unstable manifolds. In other words, even when the periodic approximation is being made, only a finite time span of data is used in generating the invariant manifolds. Of course, this is inevitable as in any numerical computation only a finite span of the data set is used. To answer this question, it is useful to think how an invariant manifold calculation is carried out in practice. If a saddle type fixed point is known, then an initialization of wellchosen data points near that fixed point will be stretched out along the unstable manifold by the flow and trace out the unstable manifold. The initial points can be "well-chosen" in a number of ways. They might, for instance, be chosen as a small circle surrounding the fixed point. The flow will squash the circle onto the unstable directions, the force of the unstable part will then elongate the evolved circle, and a caricature of the unstable manifold will emerge. A more refined technique can be implemented if some information is readily available about the unstable directions. In this case, an initialization of a curve of points could be chosen along an unstable direction and iterated under the flow. Various refinements of this procedure have been used, but ultimately they all largely depend on the stretching of the flow in the unstable directions. Of course, it is important to pack the initial data set densely enough so as to avoid developing gaps in the sketched manifold. The greatest danger lies near the fixed point itself as the flow carries all trajectories away from this unstable point. This potential problem can be resolved by re-seeding the initial data set if neighboring points separate more than some predetermined tolerance.

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C. J o n e s a n d S. W i n k l e r

The main interest then is whether the finite-time data sets available from the numerical runs of the meandering jet suffice for carrying out this procedure. The physics gives us a hint. A given coherent structure, such as an eddy, appears to generate a hyperbolic trajectory as a key part of what defines it. Indeed, the boundary of such a structure is reasonably viewed in a Lagrangian manner as the stable manifold of a hyperbolic point. We have come to expect these regions of strong stretching and compressing as standard accompaniments of a coherent structure. It is not unreasonable to believe that if a structure is well-defined for a sufficient interval of time, then a data set will be available to generate the invariant manifolds. At the beginning, it was an article of faith that we would be able to obtain invariant manifolds in this way. However, it has been borne out in all the examples, and it is now quite sensible to conjecture that the coherence of a structure and an attendant region of sufficient hyperbolicity to generate the invariant manifolds are really the same thing.

5.2.

Transport

calculations

In Miller et al. [32], finite time invariant manifolds were computed for the meandering jet flow of Section 4 using these ideas. The parameters used in this study were ( R e , fl, n o ) (104, 0.207, 4), and computations were carried out for the time interval 0 <~ t ~< 122, where time has been shifted by 300 units to guarantee that the flow has fully evolved from the initial conditions. T ,~ 30.5 is the dominant time period of the flow, but in this case no periodic approximation is made. The resulting manifold calculations are shown in Figure 7, with solid lines representing unstable manifolds and dashed lines representing stable manifolds. For the time interval used in this analysis, the evolution of six lobes may be followed. In Figure 7 the lobes are labeled C - H and the iteration indices 0 . . . . . 4 correspond to the time slices, t = 0, t = 30.5, t = 61.0, t = 91.5 and t = 122.0, respectively. For the aperiodic flow the exterior boundary may be changing in time, but nonetheless it is quite clear which regions of phase space are going from retrograde motion to vortex motion and which regions of fluid are leaving the vortex. That is, these finite-time stable and unstable manifolds clearly delineate regions of phase space having

Table 2 Estimated transport for the aperiodic flow covering 0 <~t ~<122. The labels vort and retr refer to the vortex and retrograderegions, respectively. "Period" indicates the time interval in which the exchange takes place. The final column 79 is the lobe area from the Poincar6 map for the corresponding time period. Lobe

Exchange

Period

Area

79

C D E F G H

vort --+ retr retr --+ vort vort --+ retr retr ~ vort vort --+ retr retr --+ vort

0.0-30.5 30.5-61.0 30.5-6 1.0 61.0-91.5 61.0-91.5 91.5-122.0

1.112 0.982 1.055 0.955 1.004 0.889

1.137 1.037 1.037 0.992 0.992 0.931

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Fig. 7. Finite time stable and unstable manifolds for the aperiodic flow. Reprinted from Physica D, Volume 110, ED. Miller, C.K.R.T. Jones, A.M. Rogerson and L.J. Pratt, Quantifying transport in numerically generated velocity fields, 105-122, 1997, with permission from Elsevier Science. qualitatively different Lagrangian motion. For example, in going from t = 30.5 to t = 61.0 fluid in lobe E1 passes out of the vortex into the retrograde region (lobe E2) and continues in retrograde motion for at least another time period (lobe E3). At the same time, fluid in lobe D1 will exit from retrograde motion during the next time period, entering the vortex (lobe D2). Although the definition of the boundary may change from slice to slice, one

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can still identify turnstile lobes and quantify a time-averaged flux between the two regions. The lobe areas are summarized in Table 2, showing that the flux is slowly decreasing with time. For comparison, the Poincar6 map was computed using a periodic truncation on each of the four time intervals. The flux estimates computed from the maps for these truncated periodic flows are included in Table 2 along with the time-dependent fluxes computed from the full aperiodic data set. A key conclusion to be drawn from this table is that the analysis of the full aperiodic flow produces fluxes comparable to those deduced from the periodic approximation. This then serves to reinforce the conclusion drawn earlier that this transport mechanism is a significant factor in the overall budget of fluid exchange into and out of the jet.

6. An adiabatic approach The work of Miller et al. [32] demonstrated that infinite amounts of data were not necessary to carry out the construction of manifolds. Prompted by this observation, Haller and Poje investigated the idea of finite time transport further [23]. Their idea was to exploit the structure that could be concluded from frozen-time slices, an Eulerian perspective, when the flow field changed (relatively) slowly in time. The underlying assumption of their work holds when the Eulerian time scale is long compared with the Lagrangian time scale. The main result in [23] gives conditions under which the presence of a stagnation point in each Eulerian data slice during a finite time interval implies the existence of an underlying saddle-type Lagrangian trajectory. Additionally, this hyperbolic trajectory will possess locally invariant stable and unstable manifolds. Due to the finite extent of the data set, the hyperbolic trajectory and its manifolds will not be unique; however, they are determined up to exponentially small errors, and will thus appear unique in numerical simulations for data sets of sufficient length. The technical conditions of the theorem may be found in the original paper, but the basic idea is that the time dependence of the velocity field should not be too fast. Let p(t) denote the hyperbolic stagnation point that is assumed present in each Eulerian time slice. Then the conditions of the theorem impose bounds on the maximal Lagrangian velocity # -- maxt~[t-,t+] I/~(t)l of the stagnation point, the rate of change of the angle formed by the eigenvectors of the stagnation point, and on the strength of the nonlinearity of the vector field near p(t). In essence, one wants an Eulerian stagnation point to manifest an underlying Lagrangian trajectory that organizes the nearby particle flow in the familiar manner of a hyperbolic saddle. It turns out that this will occur provided the Eulerian stagnation point moves sufficiently slowly enough and that its (frozen time) eigenvectors do not rapidly "scissor" or rotate. All of this makes intuitive sense and is formulated precisely in [23].

6.1. A theory for finite-time invariant manifolds There have been two contexts in which theories of finite-time invariant manifolds have been developed. The first is the adiabatic theory due to Haller and Poje, described above, and the second is due to Sandstede et al. [44]. The context of this latter development is

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

75

of flow fields that have an infinite time span as a certain parameter tends to zero. In both cases, the construction of the manifolds follows Perron's method. The data is given over a certain time interval [t-, t+]. In contrast to the assumptions used in [23] described above, for the development in [44], t - -- t - (e) and t + = t + (e) with t - --+ - c ~ as e --+ 0 and t + --+ +cx~. The vector field is first extended to all t E R in a C ~ fashion via bump functions. The extended vector field will be smooth in the domain for all time and will coincide with the time-restricted vector field in the time interval [t-, t+]. A stable manifold W s is defined by fixing a box B surrounding the hyperbolic trajectory and requiring that W s contains the set of initial conditions which never leave B in forward time. Trajectories associated with these initial conditions are shown to satisfy an integral equation that guarantees exponential attraction to a Lagrangian hyperbolic trajectory F (t) in the flow. A contraction mapping argument then verifies that the integral equation has a unique C 1 solution, and it follows that W s is a C 1 manifold. A similar argument with time reversed establishes W u. In [23], The underlying Lagrangian hyperbolic trajectory F ( t ) is C 1 close to the Eulerian hyperbolic trajectory p ( t ) . Now, any given modification of the vector field outside of the time interval [t-, t +] will yield such results. Since an infinite choice of these modifications exists, F (t) and its associated stable and unstable manifolds will not be unique. However, one can argue that for sufficiently large finite time intervals, each of these different solutions will be exponentially close to "master" manifolds W s and W u (the unique manifolds that would exist if the data were given for infinite time). The conclusion is that in practical applications, numerical resolution will render it impossible to distinguish these exponentially small errors, and the calculations will yield the desired objects as accurately as numerically possible.

6.2. A n e d d y - s h e d d i n g event Once the conditions for the existence of Lagrangian saddle-type dynamics have been verified, one may confidently proceed with manifold calculations to understand the geometry of the flow and make transport estimates. Haller and Poje applied these ideas to study eddyjet interactions during detachment and entrainment [38]. The Eulerian velocity fields are given by a reduced gravity, single layer primitive equation model that solves the shallow water equations. This leads to a double gyre circulation pattern with the gyres separated by a strong eastward flowing jet. Eddy detachment events are observed in the time-dependent Eulerian fields (see Figure 8). A hyperbolic stagnation point between the eddy and the jet is evident in the height field. As the eddy is shed from the jet, this hyperbolic stagnation point eventually loses its saddle-type structure; hence, the finite-time formulation is well suited to this application. In the analysis of the eddy-shedding event, the first step is to locate the Eulerian hyperbolic stagnation point in each time slice during the interval of interest. The relevant quantities needed to test the conditions of the theorem are estimated numerically to determine the time interval over which the theorem is satisfied. Manifolds are calculated following the straddling approach of Miller et al., with initial segments chosen along the stable and unstable eigenvectors of the Eulerian stagnation points. The geometry of the eddy-shedding event is strikingly different from that of the meandering jet model. The stable and unstable manifolds do not appear to intersect at any point

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Fig. 8. Topology of the double-gyre circulation. Contours of instantaneous layer depth sometime during year 5 of the simulation. Depths less than 500 m are shown dashed. Note the detaching eddy and its attendant Eulerian stagnation point (near (1200, 1500)).

other than the hyperbolic trajectory. Upon consideration of the dynamics, this makes perfect sense. In incompressible flows, lobes maintain a constant area as they are transported by the flow. In this problem, the object of interest is an eddy that grows in area as it forms and slowly moves away from the jet. Thus, the manifolds exhibit an open structure that allows fluid to flow into the growing eddy. In fact, the manifold structure reveals a long mixing channel that entrains fluid from far downstream of the eddy itself and not just from the surrounding water (see Figure 9).

7. Geometry, statistics, and the Antarctic polar vortex The numerical computations of manifolds following the methods of the preceding sections depend on some correlation between the Eulerian and Lagrangian dynamics. Since the initial segments for the manifold calculations are chosen based on considerations of the Eulerian field, this approach is difficult to apply in situations where the Eulerian picture yields little information. For example, consider again the meandering jet flow. The Eulerian streamlines do not provide the requisite hyperbolic stagnation points necessary

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Fig. 9. Invariant manifolds of the cold core ring at t = 95 days (top), and t -- 110 days (bottom). Stable manifold, W s solid, unstable manifold, W" dashed. The stable manifold encloses a long channel that funnels fluid into the eddy.

for the initialization of a straddling calculation. Since the hyperbolic trajectories of interest propagate eastward with the jet, they only appear in the Eulerian data when an appropriate change of reference frame is made. A priori knowledge of the physics in the meandering jet flow suggests the shift to a moving reference frame, but in general, the existence of such a frame is not likely. This exact difficulty arises in the study of the springtime atmospheric circulation near the Antarctic polar vortex. The basic flow pattern is similar to the meandering jet model; in this case, a jet encircles the polar region in the southern hemisphere, creating the polar vortex. During the southern spring, atmospheric conditions and solar radiation facilitate the photochemical destruction of ozone within the vortex, forming the infamous ozone hole. Of interest is whether the jet acts as a barrier to transport, effectively trapping the ozone depleted air within the vortex. If significant transport across the jet does occur, then current estimates of ozone depletion may be inaccurate. Transport associated with this flow has been studied extensively (see [9-12,36] and references therein). The application of finite time transport theory to this problem was first proposed by Bowman [ 13]. The velocity field is taken from the United Kingdom Meteorological Office (UKMO)

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stratospheric assimilation for July 1, 1996 to June 30, 1997. Data is given on surfaces of constant potential temperature, on which the flow is approximately two-dimensional and incompressible. The velocity field and other relevant quantities are derived from observational data, and the complicated flow dynamics precludes any simplifying assumptions on the time dependence. The jet is composed of a near-stationary wave of wavenumber 1 and higher frequency nonstationary waves, with the largest moving component of the jet dominated by wavenumber 2. However, the erratic phase and amplitude of the wavenumber 2 component combined with the influences of higher modes defies any attempt to shift to a moving reference frame. As previously mentioned, such irregular time dependence means that any relevant hyperbolic trajectories are not likely to appear as Eulerian stagnation points, and the methods of Haller and Poje are not readily applicable to this problem.

7.1. Relative dispersion and coherent structure boundaries Bowman has proposed a method for identifying hyperbolic structures in this setting [ 13]. The idea is to infer the manifold geometry by considering the stretching associated with the hyperbolicity of these structures. Recall that segments that straddle a stable manifold will stretch in forward time in the direction of the unstable manifold. For some initial time to, consider a small segment in the phase space defined by the endpoints x(t0) and y(t0). After integrating the trajectories through these initial conditions for time T, the quantity Ix(T) - y(T)I

Ix(to)- y(to)l provides a measure of the length stretch of the segment (cf. [35] where length stretch is defined in the limit as the initial segment length goes to zero). In Bowman's finite strain method, a staggered grid of initial conditions is integrated and the resulting length stretch of each initial segment defined by this grid is assigned to the midpoint of the segment. This yields a scalar field, associated with both the initialization time to and the integration time T - to, which is then plotted against the grid of initial segment midpoints. Approximations to the stable manifolds should then appear as local maxima in the finite strain plot, while unstable manifolds will maximize the backward time finite strain calculation. The finite strain is a particular manifestation of a more general concept from Lagrangian statistics, namely relative dispersion. Given a pair of trajectories with initial conditions x(0) and y(0), the relative dispersion associated with the pair after time T is the quantity Ix(T) - y(T)I. Traditionally, this quantity is averaged over an ensemble of initial pairs to yield a single value R(T). This statistical approach is common in the study of turbulence (see [40] and the extensive references therein). However, plotting R ( T ) against the grid of initial conditions provides insight into the location of structures at time to that influence the dynamics of passive tracers in the flow. This approach has been taken by von Hardenberg et al. to study baroclinic vortices in the atmosphere [26]. In this case, the relative dispersion is defined for a grid of initial conditions by initializing eight "satellite" particles for each grid node. The relative dispersion at each node is the average square distance between the final position of the node and its eight satellites after time T. The authors also compute

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

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finite-time Lyapunov exponents to measure stretching along trajectories. This is a local calculation that has been used previously to study atmospheric mixing and transport [37]. Doerner et al. have explored a direct relationship between local maxima of the finite-time Lyapunov exponent field and stable manifolds of hyperbolic sets [ 18] (see also [22]). For short times, the relative dispersion and Lyapunov exponent plots are similar, as both calculate local stretching along trajectories. However, for longer times the relative dispersion measures global advective effects of the full nonlinear velocity field, whereas the Lyapunov exponents still measure local linear stretching along trajectories. Numerical experiments suggest that there is a correspondence between local maxima of these scalar fields and invariant manifolds. This provides a link between Lagrangian statistics and ideas from the theory of geometric dynamical systems. A key concept underlying both approaches is that of coherent structures. From the dynamical systems perspective, it is the invariant manifolds that define the boundaries of coherent structures such as cat's eye vortices, lobes, and eddies. However, the definition of a coherent structure may vary depending on the application of interest; Haller and Yuan review several approaches in the context of two-dimensional turbulence and propose a definition based on the stability of material lines [25]. In this approach, the authors measure the amount of time that a trajectory spends in a region of strong hyperbolicity, and this time is plotted against initial conditions to distinguish stable and unstable material surfaces. The common theme in all these approaches is that of measuring dynamically relevant scalar quantities associated with tracer trajectories and visualizing the spatial distribution of these scalars to determine the geometry of coherent structures. Relative dispersion is proposed here as an appropriate scalar for defining coherent structures and their boundaries. A formulation based on the integration of a single grid is used. Consider a two-dimensional grid at some initial time to with grid nodes Xi,j (to). The node's four neighbors, Xi+l,j, Xi-l,j, Xi,j+l, and Xi,j--l, a r e used to define relative dispersion: Ri,j (to, T) =

IXi+l,j(T) -- X i - l . j ( T ) ] IXi+l.j(tO) -- X i - l . j ( t o ) l

+

]xi,j+l ( T ) - xi,j_ l (T)] Ixi,j+l (to) - x i . j - I (t0)l

For each grid node, Ri.j(to, T) captures the horizontal and vertical stretching associated with the segments defined by the neighboring nodes. Using this "cross" at each gridpoint is natural for two-dimensional problems, as it accounts for stretching in any direction and associates this stretching with the intersection point of the cross (the grid node). The twopoint finite strain method is biased toward stretching normal to segments, and the eightpoint satellite method contains two superfluous segments. Local maxima of the relative dispersion field act as boundaries between regions with similar Lagrangian fates. For example, consider a vortex in a shear layer. The interior of the vortex is a coherent structure associated with a low value of R, as the relative dispersion for initial conditions inside the vortex is bounded by its width. Initial conditions on the vortex edge will exhibit large relative dispersion since neighboring interior tracers circulate in the vortex while neighboring exterior tracers move rapidly downstream in the shear flow. In general, the relative dispersion provides a hierarchy of coherent structures based on its time evolution properties. As an example, recall the geometry of the meandering jet model. Intuitively, the longest-lived coherent structures are the cat's eye vortices, whereas

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lobes are shorter-lived structures. For material volumes near the vortex core, the relative dispersion is low for long integration times, as the volume rotates en masse but does not stretch significantly. For lobes, Ri,j(to, T) is small for short times while the lobe maintains its general shape, but increases as the lobe nears a hyperbolic trajectory and elongates along the unstable manifold. As with the finite strain, considerations of stretching associated with hyperbolic invariant sets suggest that invariant manifolds manifest themselves as local maxima of the relative dispersion field. However, the relative dispersion may also detect non-hyperbolic features such as shear layers that can be useful for defining coherent structures as well. To demonstrate the kind of image this technique produces, consider the forced, damped Duffing oscillator. The structure of the invariant manifolds of the hyperbolic fixed point near the origin has been studied extensively for this system (see, for example, [21]). Figure 10(a) shows the results of a standard straddling calculation of the manifolds for a particular choice of parameters. Figure 10(b) shows the scalar field R = R ( 0 , - 1.5T) R (0, 1.5 T), the backward time relative dispersion minus the forward time relative dispersion with initial time 0 and final integration time 4-1.5 T. T is the period of the forcing, 2zr. Note the correspondence between the stable manifold in Figure 10(a) and large negative values of R (local maxima of the forward time relative dispersion calculation). A similar correspondence is evident between the unstable manifold and local maxima of the backward time relative dispersion calculation.

7.2. Application to the polar vortex Figure 11 shows the relative dispersion computation (again plotted as the difference of the backward and forward time computations) for several different initial times in the UKMO atmospheric data set. In each panel, the final integration time is -+-8 days. The cat's eye structure of the flow is apparent in these plots, with the wavenumber 2 dominance clear, though higher wavenumber effects are also apparent. Lobe-like structures similar to those found in the meandering jet problem are also visible. However, these lobes are quite large, as they are not the result of a small perturbation of a steady vector field. Filaments of high R in both forward and backward time are nearly parallel near the vortex edge, indicating the strong shear induced by the jet. The vortex edge is commonly defined as the region with the largest potential vorticity gradient [31]. In Figure 12, contour lines of the potential vorticity are overlaid on a plot of the forward and backward time relative dispersion calculation initialized on October 11, 1996. The potential vorticity is taken from the UKMO data set, and here the relative dispersion is shown with a logarithmic scaling to highlight the curve of local minima of R present at the jet core. The relative dispersion is minimized here since nearby initial conditions are translated together by the jet. A strong potential vorticity gradient is evident in the region of the jet core. Hence, the relative dispersion calculation provides dynamical support for the definition of the vortex edge as the region of the largest potential vorticity gradient. To gain further insight into the nature of the transport induced by this flow geometry, material lines may be calculated via straddling. The initial segment for the straddling

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

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Fig. 10. Forced, damped Duffing oscillator. (a) Invariant manifolds for the fixed point near the origin, computed with a standard straddling method. The red curve is the unstable manifold, and the blue curve is the stable manifold. (b) Relative dispersion. Pseudocolor plot of R = R ( 0 , - 1 . 5 T ) - R(0, 1.5T), where T = 2zr is the period of the forcing. Large positive values, in shades of red and yellow, indicate large stretching in backward time. Large negative values, in shades of dark blue, indicate large stretching in forward time.

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N

Fig. 11. Relative dispersion for UKMO atmospheric data. Pseudocolor plot of R = R(to,-8) - R(t0, 8) with to = (a) 26 September, (b) 1 October, (c) 6 October, (d) 11 Octoberof 1996. Coordinates are longitude, latitude.

computation is chosen by first plotting the R field and identifying the intersections of the forward and backward time relative dispersion calculations. Following ideas from the preceding sections, these intersections are treated as effective hyperbolic trajectories for the straddling computation. For example, a small segment is initialized at one such intersection near 100 ~ longitude, 45~ latitude on September 26 (see Figure 1 l(a)), chosen to lie along the red curve (maxima of the backward time relative dispersion) and straddling the blue curve (maxima of the forward time relative dispersion). This segment, referred to as an unstable material line, is then advected forward under the velocity field. Similarly, a stable material line is initialized on October 21 and advected in backward time. Figure 13 shows a time series of the evolution of these stable and unstable material lines for the intermediate time interval spanning October 1 to October 11, 1996. A comparison of Figure 13 to Figures 11 and 12 reveals that the advected material lines align with maxima of the relative dispersion throughout this time period. This consistency supports the use of relative dispersion to identify coherent structure boundaries. Five lobes, labeled A - E are identified and tracked in Figure 13 (compare to Figure 7). Transport of fluid from the edge of the jet to the surf zone occurs as long, thin lobes on the poleward side of the recirculation zone move counterclockwise, eventually reaching the boundary between the retrograde and recirculating regions (e.g., lobes C and E in

Invariant manifolds and Lagrangian dynamics in the ocean and atmosphere

83

Fig. 12. Potential vorticity contours (black) overlaid on (a) forward and (b) backward time relative dispersion calculations (reds are high values, blues are low values). A logarithmic scale is used for the plot to highlight the local minimum at the jet core, manifested by the blue curve with peaks near 60~ latitude.

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Fig. 13. Evolution of stable (green) and unstable (red) material lines between 1 October and 11 October 1996. Five lobes are identified and followed, and their areas are summarized in Table 3.

Fig. 14. Actual positions of stable (green) and unstable (red) material lines on 11 October 1996, superimposed on a map of the Southern hemisphere.

I n v a r i a n t m a n i f o l d s a n d L a g r a n g i a n d y n a m i c s in the o c e a n a n d a t m o s p h e r e

85

Table 3 Estimated transport by the lobes associated with one cat's eye structure. The labels vort and retr refer to the vortex and retrograde regions, respectively. Lobe A B C D E

Exchange vort retr vort retr vort

--+ retr --+ vort -+ retr --+ vort ~ retr

Area in km 2 ( x 106) 0.704 7.64 0.717 1.07 2.39

Figure 13). They are then entrained into the retrograde region and once again stretch and fold into thin filaments, this time in the surf zone (e.g., lobe A). At the same time, lobes upstream in the surf zone are pulled into the recirculation zone and eventually stretch out along the jet edge (e.g., lobes B and D). This is the familiar lobe dynamics mechanism, only in this case the flow is completely aperiodic, so the configuration of the lobes never repeats. Comparing this picture to the case of the meandering jet model reveals a strikingly similar qualitative picture of the Lagrangian transport mechanism. The areas of the lobes may be computed using Green's Theorem (taking into account the Earth's sphericity) to obtain estimates of transport between the retrograde and recirculation zones. These calculations are summarized in Table 3. The magnitude of the transport resulting from the lobe dynamics may be better appreciated by viewing the lobes relative to their actual positions over the Earth (see Figure 14). Material lines may be followed for longer times, revealing a very complicated geometry (Figure 15). Transport between the two cat's eye structures can be seen as some of the lobes beginning near the jet are ejected from the vortex into the recirculation zone, then become re-entrained into the next vortex. Additionally, the repeated stretching and folding of the material lines gives rise to small-scale structures, leading to enhanced mixing. The stretching effect is especially pronounced near the strong hyperbolic regions at the peaks of the wave in the jet, where filamented structures appear. Hence, the manifold calculations suggest that the breaking of waves in the surf zone provides an efficient means for mixing in midlatitude regions. Note also that the material lines accumulate on the vortex edge, but no lobes enter the vortex region. This suggests little isentropic transport across the vortex edge by the lobe dynamics mechanism.

8. Vorticity and viscosity Following idealized fluid particles as in the transport studies discussed so far has a potential pitfall: if the fluid properties that are measurable and of interest physically are not preserved by the fluid motion, then their redistribution under flow evolution is not captured by this approach. A key dynamical variable in oceanographic studies is potential vorticity. Potential vorticity is made up of the usual (local) fluid vorticity and the planetary vorticity due to the rotation of the Earth. In an ideal, inviscid fluid, potential vorticity is conserved along fluid trajectories, and so potential vorticity redistribution could be concluded from

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Fig. 15. Stable (green) and unstable (red) material lines on 11 October 1996. Repeated stretching and folding is evident, but the material lines to not penetrate the vortex core.

transport studies based on fluid particle motion. However, in the ocean itself, potential vorticity is not conserved. The dissipation of potential vorticity is effected by a number of mechanisms involving a variety of scales, many of which remain mysterious. In fact, dissipation remains largely resistant to modeling. Eddy diffusivity is a simplistic, but common way, of parameterizing small-scale effects that promote the dissipation of potential vorticity. The idea is that turbulent motion at small scales creates a drain on potential vorticity by transferring it to sub-grid scales that will not be reflected in the large-scale motion. Eddy diffusivity is simply the parameterization of this effect through a viscous term in the equations. In numerical models, such a viscous term is usually included for reasons of stability of the calculations. A natural correspondence is then made between this viscous term and eddy viscosity. In cases where potential vorticity is conserved, a constraint is imposed on the Lagrangian dynamics that has significant implications, see Brown and Samelson [ 14] for further discussion. In particular, the conservation of potential vorticity has implications for the transport itself. The dissipation of potential vorticity by viscosity, or other effects, therefore raises two interesting questions: (1) Can Lagrangian transport studies accurately reflect the redistribution of potential vorticity?

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(2) What role does viscosity play in promoting transport that may not be present in the inviscid limit?

8.1. Potential vorticity fluxes It is natural to suspect that in numerical models in which potential vorticity is dissipated, a characteristic of all numerical models discussed here, the relationship between potential

Fig. 16. Potential vorticity difference q ( x ( t f ) , y ( t f ) ) - q(x(to),y(to) ) plotted against initial conditions (x(to), y(to)). A stable manifold of the hyperbolic trajectory near the island boundary is shown in black.

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vorticity fluxes and material transport will be broken. As part of the work on recirculation past an island, Miller et al. [33] have exhibited a striking relationship between potential vorticity change and the effective invariant manifolds. Based on an idea in [24], Figure 16 shows the color-coded potential vorticity difference over two sets of 30 time units, with a corresponding stable manifold of an appropriate hyperbolic point on the island. The picture reveals an extraordinary correlation between boundaries of contrasting regions of potential vorticity differences and the stable manifold. The right hand-column shows an analogous picture over a backward time interval. The point to be concluded from these pictures is that the dominant potential vorticity changes are a result of the advective fate of a fluid particle. If a particle is dragged into the boundary layer near the island, for instance, then whether it moves north or south will have an enormous impact on the change in potential vorticity it experiences. In turn, whether it is a fluid particle destined to move north or south along the island boundary is information encoded in the lobe structures and revealed by the invariant manifolds. A key problem in oceanography is to measure potential vorticity fluxes in active regions. A critical element of any such analysis is choosing the boundaries through which the flux is measured. A standard approach is to use boundaries of coherent structures based on time averaged flow fields, see [27]. Such definitions may lead to deceptive results if the structure actually deviates from its time-averaged version significantly. A Lagrangian definition of the boundary, of the kind discussed here, is more appropriate. The picture discussed above shows the strong correspondence between the manifolds and the change in potential vorticity and suggests the use of effective invariant manifolds for measuring potential vorticity fluxes. A systematic study of the different ways this can be done for the island problem is given in [33 ].

8.2. Viscosity induced transport The second question raised by taking viscosity into account is whether the viscosity itself induces transport between the constituent parts of a given coherent structure and between the structure and ambient waters. This question was taken up by Balasuriya et al. [3]. A simple flow for an inviscid fluid can be imagined that is steady, possibly in a moving frame, and which exhibits a cat's eye or eddy-like structure in which a heteroclinic loop or a homoclinic orbit surround a recirculating region. Since potential vorticity is conserved in an inviscid flow field, such a steady flow is a natural model. An important question is then whether the addition of viscosity, which would lead to a perturbed flow field, would render a flow field with intersecting stable and unstable manifolds. This is a daunting question as the perturbed flow field is not known, only that it satisfies a certain partial differential equation. Nevertheless, a surprisingly strong result is shown in [3]. Indeed, a Melnikov calculation is carried out and an expression derived for it that depends only on the inviscid flow field. The overall conclusion is that viscosity can promote transport both into and out of a coherent structure and that the flux can be estimated through the Melnikov function, which affords an expression in terms of the inviscid flow field. One issue that arose through this work was whether the result could hold for perturbed flow fields that stayed close to the inviscid field for only finite times. A complete theory

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would require an affirmative answer to this question as there are no guarantees that the viscous field will stay close to its inviscid limit for infinite time. Prompted by this question, Sandstede et al. [44] developed the theory of finite-time invariant manifolds discussed above. The extraordinary result is that the modifications outside of the fixed time interval contribute only at higher order in the calculation of the distance function, and so the Melnikov function remains valid.

9. Conclusions and future challenges The pictures that usually reveal a coherent structure in the ocean are Eulerian in nature. Contours of temperature or sea surface height from satellite data depict the Gulf Stream clearly as in, for instance, Figure 1. In numerical models, streamfunction contours or contours of potential vorticity at frozen times are often shown to exhibit coherent structures, such as vortices, jets etc. While these are satisfying pictures of the effect under consideration, they contain little dynamic information as they depend purely on information at a fixed time. Additionally, viewing the contours at different fixed times is not illuminating since the contours do not stack up in general to form material surfaces. The concept being put forward here is that an improved understanding of these phenomena is obtained through depictions of Lagrangian structures that delineate the coherent features. In contrast to Eulerian characterizations that represent features in terms of distributions of scalars at frozen times, Lagrangian characterizations contain dynamic information about the fluid, such as which particles will be entrained into the feature and which will leave the feature. In this article and the papers to which it refers, it is shown that such a Lagrangian map of coherent features can be formed by generating what we have called "effective invariant manifolds". These are slices of material surfaces that are tied to regions of strong hyperbolicity (stretching and compressing). These regions themselves are attendant to the coherent features under consideration. An operational strategy is given by the work of Miller et al. [32], which depends on the idea of "straddling" the stable directions to pin the manifold down, and extending it under the expansion of the flow in the unstable directions. These manifolds lead to naturally defined boundaries of the coherent feature and also contain all the information concerning transport into and out of the feature, as well as between its constituent parts. It should be kept in mind that any curve generated this way will amount to a slice of some invariant surface in the three-dimensional phase space (two space dimensions + time), a so-called "material surface" in the terminology of a fluid dynamicist. This can be achieved by seeding any initial curve at a fixed time and applying the evolution of the flow. The point is, however, that the initial seeding should be chosen to generate a material surface that reveals information about the transport in the flow field. The key in any given problem is how to find the "hyperbolic" region to initially seed the material surface with a "straddling" segment. We have seen three different approaches to this above: (1) Periodic approximation: If the flow field is close to having a dominant time period, then an initial approximation can be made by enforcing periodicity and repeating the velocity data from one time interval, based on that dominant period. The hyperbolic fixed

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points, and their stable and unstable directions, of this periodic flow can be used as a first approximation. Judiciously chosen initial segments to be used in the Miller et al. routine can then easily be developed. These are then used to generate effective invariant manifolds for the full flow field. (2) Eulerian approximation: If the Eulerian time scale is long compared with the Lagrangian time scale, then it is reasonable to use the frozen-time Eulerian fields as first approximations. The idea is to use the saddle points of, say, streamfunction contours as initial guesses for the "hyperbolic" regions. This is the basis for the Haller-Poje approach and is shown by them to be very effective in the eddy-shedding events. (3) Relative dispersion: In cases of extremely complex time dependence, neither of the above methods will work well and a direct method is needed to locate the regions of strong hyperbolicity. Such a method is supplied by the dispersion computations, described above in the context of the Antarctic polar vortex. One of the main challenges is to extend these transport ideas to three-dimensional flows. Little is known in this context, even for tame steady or periodic flows. Effects of vertical flows are important, but a fully three-dimensional theory is not as important as might be initially thought. Indeed, the ocean is well stratified and even multi-layer computations trusted by oceanographers may involve only a small number of layers. The difficulties surrounding the collection of data in the ocean is part of what makes oceanography so challenging. For sub-surface data, only sparse Lagrangian data can be collected using floats that render information only about their own tracks. Since the floats are individually quite expensive and their placement a significant logistical operation, the amount of data available now, or in the future, is not great. It is thus a fundamental issue to design optimal experimental systems that maximize the information resulting from any given float experiment. The dynamical systems analysis discussed in this paper promises to delineate the regions in physical space with specific fates, such as entrainment into an eddy, faithful following of a jet, etc. It is natural therefore to incorporate this insight into an optimal design for a float experiment system. This issue has become the focus of much current research. There is a danger of a circular argument as the above shows how to glean dynamical information from observational data, but here it is being suggested that the observational data should be based on the results of the dynamical systems analysis. The work by Toner et al., discussed in [46], is aimed at breaking this circle through the use of reconstructions of approximate velocity fields and numerical models. This area of observational design is one where dynamical systems can have tremendous impact in a truly practical fashion. Its potential remains to be fulfilled, but the prospects are good and the results so far are promising.

References [1] H. Aref, Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Ann. Rev. Fluid Mech. 15 (1983), 345-389. [2] S. Balachandar and M.R. Maxey, Methods for evaluating fluid velocities in spectral simulations of turbulence, J. Comp. Phys. 83 (1989), 96-125. [3] S. Balasuriya, C.K.R.T. Jones and B. Sandstede, Viscous perturbations of vortici~-conserving flows and separatrix splitting, Nonlinearity 11 (1998), 47-77.

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[41 D. Beigie, A. Leonard and S. Wiggins, Chaotic transport in the homoclinic and heteroclinic tangle regions of quasi-periodically forced two-dimensional dynamical systems, Nonlinearity 4 (1991), 775-819. [5] D. Beigie, A. Leonard and S. Wiggins, The dynamics associated with the chaotic tangles of two-dimensional quasiperiodic vector fields: theoo, and applications, Nonlinear Phenomena in Atmospheric Oceanic Sciences, EG. Carnevale and R. Pierrehumbert, eds, IMA Vol. Math. Appl., Vol. 40, Springer, Berlin (1992), 47-138. [61 A.S. Bower, A simple kinematic mechanism for mixing fluid parcels across a meandering jet, J. Phys. Ocean. 21 (1991), 173-180. [7] A.S. Bower and M.S. Lozier, A closer look at particle exchange in the Gulf Stream, J. Phys. Ocean. 24 (1994), 1399-1418. [81 A.S. Bower and H.T. Rossby, Evidence of cross-frontal exchange processes in the Gulf Stream based isopycnal RAFOSfloat data, J. Phys. Ocean. 19 (1989), 1177-1190. [9] K.P. Bowman, Evolution of the total ozone field during the breakdown of the antarctic circumpolar vortex, J. Geophys. Res. 95 (1990), 16529-16543. [lO] K.E Bowman, Barotropic simulation of large-scale mixing in the antarctic polar vortex, J. Atmospheric Sci. 50 (1993), 2901-2914. [ll] K.P. Bowman and N.J. Mangus, Observations of deformation and mixing of the total ozone field in the antarctic polar vortex, J. Atmospheric Sci. 50 (1993), 2915-2921. [121 K.E Bowman, Rossby wave phase speeds and mixing barriers in the stratosphere. Part I: Observations, J. Atmospheric Sci. 53 (1996), 905-916. [13] K.P. Bowman, Manifold geometry, and mixing in observed atmospheric flows, Preprint (1999). [14] M.G. Brown and R.M. Samelson, Particle motion in vorticity-conserving, two-dimensional incompressible flows, Phys. Fluids 6 (1994), 2875-2876. [15] W.H. Calvin, The great climate flip-flop, Atlantic Monthly 281 (1) (1998), 47-64. [16] R. Camassa and S. Wiggins, Chaotic advection in a Rayleigh-Bdnardflow, Phys. Rev. A 43 (1991), 774797. [171 D. del-Castillo-Negrete and P.J. Morrison, Chaotic transport by Rossby waves in shear flow, Phys. Fluids A 5 (1993), 938-965. [18] R. Doerner, B. Htibinger, W. Martienssen, S. Grossman and S. Thomae, Stable manifolds and predictability' of dynamical systems, Chaos, Solitons, & Fractals 10 (1999), 1759-1782. [19] J.Q. Duan and S. Wiggins, Fluid exchange across a meandering jet with quasi-periodic variabili~, J. Phys. Ocean. 26 (1996), 1176-1188. [20] G.R. Flied, E Malanotte-Rizzoli and N.J. Zabusky, Nonlinear waves and coherent vortex structures in barotropic fl-plane jets, J. Phys. Ocean. 17 (1987), 1408-1438. [21] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983). [22] G. Haller, Distinguished material surfaces and coherent structures in 3D fluid flows, Preprint (2000). [23] G. Haller and A.C. Poje, Finite time transport in aperiodic flows, Phys. D 119 (1998), 352-380. [24] G. Haller and A.C. Poje, Lagrangian transport and diffusion in two-dimensional aperiodic flows, Preprint

(2OOO). [25] G. Hailer and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulence, Phys. D 147 (2000), 352-370. [26] J. von Hardenberg, K. Freadrich, F. Lunkeit and A. Provenzale, Transient chaotic mLving during a baroclinic life cycle, Chaos 10 (2000), 122-134. [27] W.R. Holland and EB. Rhines, Example of eddy-induced ocean circulation, J. Phys. Ocean. 10 (1980), 1010-1031. [281 T.J. Kaper and S. Wiggins, Lobe area in adiabatic Hamiltonian systems, Phys. D 51 (1991), 205-212. [291 T.J. Kaper and G. Kovarir, A geometric criterion for adiabatic chaos, J. Math. Phys. 35 (1994), 1202-1218. [301 R.S. MacKay, J.D. Meiss and I.C. Percival, Transport in Hamiltonian systems, Phys. D 13 (1984), 329-354. [311 M.E. McIntyre, On the Antarctic ozone hole, J. Atmospheric Terr. Phys. 51 (1989), 29-43. [32] P.D. Miller, C.K.R.T. Jones, A.M. Rogerson and L.J. Pratt, Quantifying transport in numerically generated velocity'fields, Phys. D 110 (1997), 105-122. [331 P.D. Miller, L.J. Pratt, K.R. Helfrich and C.K.R.T. Jones, Material transport and potential vorticity flux in a model of recirculation past an island, Preprint (2000).

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[34] H.E. Nusse and J. Yorke, Dynamics: Numerical Explorations, 2nd edn., Appl. Math. Sci., Vol. 101, Springer, Berlin (1998). [35] J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge Univ. Press, Cambridge (1989). [36] E Paparella, A. Babiano, C. Basdevant, A. Provenzale and P. Tanga, A Lagrangian study of the Antarctic polar vortex, J. Geophys. Res. 102 (1997), 6765-6773. [37] R.T. Pierrehumbert and H. Yang, Global chaotic mixing on isentropic surfaces, J. Atmospheric Sci. 50 (1993), 2462-2480. [38] A.C. Poje and G. Haller, Geometry of cross-stream mixing in a double-gyre ocean model, J. Phys. Ocean. 29 (1999), 1649-1665. [39] L.J. Pratt, M.S. Lozier and N. Beliakova, Parcel trajectories in barotropic jets: Neutral modes, J. Phys. Ocean. 25 (1995), 1451-1466. [40] A. Provenzale, Transport by coherent barotropic vortices, Ann. Rev. Fluid Mech. 31 (1999), 55-93. [41] A.M. Rogerson, P.D. Miller, L.J. Pratt and C.K.R.T. Jones, Lagrangian motion and fluid exchange in a barotropic meandering jet, J. Phys. Ocean. 29 (1999), 2635-2655. [42] V. Rom-Kedar, A. Leonard and S. Wiggins, An analytical study of transport, mixing, and chaos in an unsteady vortical flow, J. Fluid Mech. 214 (1990), 347-394. [43] V. Rom-Kedar and S. Wiggins, Transport in two-dimensional maps, Arch. Rational Mech. Anal. 109 (1990), 239-298. [44] B. Sandstede, S. Balasuriya, C.K.R.T. Jones and P. Miller, Melnikov theory for finite-time vector fields, Nonlinearity 13 (2000), 1357-1377. [45] R.M. Samelson, Fluid exchange across a meandering jet, J. Phys. Oceanography 22 (1992), 431-440. [46] M. Toner, A.C. Poje, A.D. Kirwan Jr., B. Lipphardt and C. Grosch, Reconstructing basin-scale Eulerian velocity fields from simulated drifter data, J. Phys. Ocean. (to appear). [47] S. Wiggins, Chaotic Transport in Dynamical Systems, Springer, New York (1992).

CHAPTER

3

Geometric Singular Perturbation Analysis of Neuronal Dynamics Jonathan E. Rubin Department of Mathematics, Universib' of Pittsburgh, Pittsburgh, PA 15260, USA

David Terman Department of Mathematics, The Ohio State Universi~, Columbus, OH 43210, USA

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract In this chapter, we consider recent models for neuronal activity. We review the sorts of oscillatory behavior which may arise from the models and then discuss how geometric singular perturbation methods have been used to analyze these rhythms. We begin by discussing models for single cells which display bursting oscillations. There are, in fact, several different classes of bursting solutions; these have been classified by the geometric properties of how solutions evolve in phase space. We describe several of the bursting classes and then review related rigorous mathematical analysis. We then discuss the dynamics of small networks of neurons. We are primarily interested in whether excitatory or inhibitory synaptic coupling leads to either synchronous or desynchronous rhythms. We demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. Finally, we discuss larger networks of neuronal oscillators involving two distinct cell populations. In particular, we demonstrate how dynamical systems methods can be used to analyze recent models for sleep rhythms and other oscillations generated in the thalamus. The analysis helps to explain the generation of the different thalamic rhythms and the transitions between them, in both of which inhibition plays a crucial role.

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1. Introduction Oscillations arise throughout the central nervous system [47,38,79,89,9]. These oscillations have been implicated in the generation of sleep rhythms, epilepsy, Parkinsonian tremors, sensory processing, and learning [30,39,76,79,40]. Oscillatory behavior also arises in such activities as respiration, movement, and secretion [ 13,10]. Models for the relevant neuronal networks often exhibit a rich structure of dynamic behavior. The behavior of even a single cell can be quite complicated [11,13,82,83,31]; an individual cell may, for example, fire repetitive action potentials or bursts of action potentials that are followed by a silent phase of near quiescent behavior [57,58,96]. The bursting behavior may wax and wane on a slower time scale [18,3,19]. Examples of population rhythms include synchronous behavior, in which every cell in the network fires at the same time, and clustering [27,28,44], in which the entire population of cells breaks up into subpopulations or blocks; every cell within a single block fires synchronously and different blocks are desynchronized from each other. Of course, much more complicated population rhythms are also possible [89,86,44]. The activity can also propagate through the network in a wave-like manner [41,19,88,29,61 ]. In this article, we review numerous recent results in which geometric singular perturbation methods have been used to analyze the dynamics of neuronal networks. Much of the analysis presented is done in the singular limit. Nonetheless, the results are extremely useful for elucidating the roles of network parameters and components in generating various activity patterns. We begin in Section 2 by considering models for single cells which exhibit bursting oscillations. There are, in fact, several different types of bursting oscillations, and there has been considerable effort in trying to classify the underlying mathematical mechanisms responsible for these oscillations [58,4,37]. Models for bursting oscillations contain multiple time scales and this often leads to very interesting issues related to the theory of singular perturbations [46]. Even the simplest bursting models may exhibit exotic behavior, including chaotic dynamics, for certain parameter ranges [ 11,83]. In Section 3, we consider models for a pair of mutually coupled cells. The results in this section are complementary to related work in [43], Chapter 1 in this Handbook. Each cell will be modeled as a relaxation oscillator; one can view this as a simple depiction of a bursting neuron in which the active phase corresponds to the envelope of a burst's rapid spikes. We only consider a form of coupling between cells that represents chemical synapses, since this is the primary means by which neurons in the central nervous system communicate with each other. There are, however, many forms of synaptic coupling [40]. It may be excitatory or inhibitory and it may exhibit either fast or slow dynamics. We will be primarily interested in whether excitatory or inhibitory coupling leads to either synchronous or desynchronous rhythms. We will demonstrate that all four combinations are possible, depending on the details of the intrinsic and synaptic properties of the cells. In Section 4, we consider a larger network of relaxation oscillators with inhibitory and excitatory synaptic coupling, referred to as a globally inhibitory network [65,63]. This network may exhibit both synchronous and clustered solutions and we give sufficient conditions for when each of these rhythms arises. These results help to clarify how the intrinsic properties of individual cells interact with the synaptic properties to produce the emergent

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population rhythm. Inhibition, for example, may play multiple roles in producing different rhythms. Which rhythms arise depends on how inhibition interacts with intrinsic properties of the neurons; the nature of these interactions depends on the underlying architecture of the network. Finally, in Section 5, we consider a detailed model for certain sleep rhythms produced in the thalamus. Many of the results in this paper, as well as the particular form of the globally inhibitory network, are motivated by experimental, modeling, and computational studies of these rhythms (see [28,22], which provides a review of related works). After describing the rhythms, we discuss in some detail how one models these networks. The model for each cell is based on the Hodgkin-Huxley formalism [33]. We then discuss how the geometric analysis helps to explain the generation of different thalamic rhythms and the transitions between them.

2. Bursting oscillations 2.1. Introduction Certain neurons and other excitable cells exhibit bursting oscillations; this behavior is characterized by a silent phase of near steady state resting behavior alternating with an active phase of rapid, spike-like oscillations, as shown in Figure 1. Examples of biological systems which display bursting oscillations include the Aplysia R-15 neuron, insulin secreting pancreatic beta cells, and neurons in the hippocampus, cortex and thalamus. For reviews, see [96,67,37]. Figure 1 shows several different types of bursting oscillations. Figure 1A displays an example of square-wave bursting. This is characterized by abrupt periodic switching between the quiescent, or silent, phase and the active phase of repetitive firing. Note that the frequency of spikes decreases at the end of the active phase. Figure 1B illustrates elliptic bursting. Small amplitude oscillations occur during the silent phase and the amplitude of spikes gradually waxes and wanes. Finally, Figure 1C displays parabolic bursting. The spike rate first increases and then decreases in a parabolic manner. The mathematical mechanisms responsible for each class of bursting oscillation are described in terms of geometric properties of the corresponding phase space dynamics. The model for each involves multiple time scales and can be written as !

x =f(x,y), y' = eg(x, y).

(2.1)

Here, x 6 R n represents fast variables, while y 6 IR'n represents slow variables; e is a small singular perturbation parameter. We refer to the first equation in (2.1), with y considered as constant, as the fast subsystem (FS). The silent phase of the bursting solution corresponds to the passage of a trajectory of (2.1) near a manifold of fixed points of (FS). The active phase of repetitive spikes corresponds to the passage of the trajectory near a manifold of periodic solutions of (FS). It is the slow processes which modulate the fast dynamics between these two phases. Different classes of bursting oscillations are distinguished by

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the mechanisms by which the bursting trajectories switch between the silent and active phases. This is closely related to the global bifurcation structure of the fast subsystem with the slow variables treated as parameters. We note that models for bursting oscillations may exhibit other types of periodic solutions, as well as more exotic behavior including chaotic dynamics [ 11,82,83]. The models contain multiple time scales and this often leads to very interesting issues related to the theory of singular perturbations. For example, homoclinic orbits usually play an important role in the generation of these rhythms; the active phase of rapid oscillations may either begin or end (or both) as the bursting trajectory crosses a homoclinic point. At these points, standard singular perturbation methods may break down, so more delicate analysis is required. Moreover, the homoclinic orbits are often directly responsible for the generation of chaotic dynamics [82,83]. In the next section, we describe the mathematical mechanisms responsible for the three classes of bursting oscillations shown in Figure 1. The description is quite heuristic; however, the formal constructions are extremely useful in understanding what geometric ingredients are needed to generate a particular class of bursters and how the dynamics changes with respect to parameters.

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Rigorous results related to square-wave bursting are presented in Section 2.3. These results demonstrate that even the simplest models for bursting oscillations lead to very interesting problems associated with the geometric theory of singular perturbations. It need not be true, for example, that the bursting solution is uniquely determined for all e sufficiently small. Moreover, there are multiple ways in which chaotic dynamics may arise as parameters in the model are varied.

2.2. Bursting mechanisms We now describe the mathematical mechanisms responsible for the three classes of bursting oscillations shown in Figure 1. This description follows closely the pioneering work of Rinzel [58]. We only consider the simplest, lowest dimensional models which generate these solutions. As we shall see, square-wave and elliptic bursting can arise in threedimensional systems, while parabolic bursting requires at least four dimensions. In each case, we consider a system of the form (2.1) and make geometric assumptions concerning the set of fixed points and periodic solutions of (FS). Some assumptions are then needed on the equations governing the slow variables. These assumptions can usually be verified for a specific system by using numerical software. 2.2.1. Square-wave bursting.

Consider a three-dimensional version of (2.1) of the form

v ' = f (v, w, y), w ' = g(v, w, y),

(2.2)

y' = eh(v, w, y, ~). A concrete system which exhibits square-wave bursting is given in Remark 2.6. The fast subsystem (FS) now consists of the first two equations in (2.2) with the slow variable y considered as constant. In the third equation, )~ represents a fixed parameter. Later, we discuss complex bifurcations that arise when )~ is varied. The primary assumptions on (2.2) concern the bifurcation structure of the fast subsystem with y treated as a parameter. This structure appears in Figure 2, which also shows the projection of the square-wave bursting solution onto the corresponding bifurcation diagram. The set of fixed points of (FS) is assumed to be a Z-shaped curve in the (v, w, y) phase space. We denote this curve by S; only a portion of this Z-shaped curve is shown in Figure 2. The fixed points along the lower branch of S are stable solutions of (FS), while the fixed points on the middle branch of S are saddles. Fixed points along the upper branch of S may be stable or unstable. We also assume that there exists a one-parameter family of periodic solutions of (FS), denoted by P. These limit cycles originate at a (subcritical) Hopf bifurcation along the upper branch of S and terminate along a solution of (FS) that is homoclinic to one of the fixed points on the middle branch of S. Assumptions are also needed about the slow dynamics. We assume that the y-nullsurface {h = 0} defines a two-dimensional manifold that intersects S at a single point. This point lies on the middle branch of S between the homoclinic point and the left knee of S. Finally, h > 0 above {h = 0} and h < 0 below {h = 0}.

Geometric singular perturbation analysis of neuronal dynamics 20

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We now give a heuristic explanation for why this system generates a square-wave bursting solution. Suppose that e > 0 is small and consider a solution that begins close to the lower branch. Because this branch consists of stable fixed points of (FS), the trajectory quickly approaches a small neighborhood of the lower branch. The trajectory tracks leftward along the lower branch according to the slow dynamics, until it passes the left knee. This portion of the solution corresponds to the silent phase. Once past the left knee, the trajectory is attracted to near P, the branch of periodic solutions of (FS). This generates the fast repetitive spikes of the bursting solutions. The trajectory passes near P, with increasing y, until it reaches a neighborhood of the homoclinic orbit of (FS). Once it passes the homoclinic orbit, the fast dynamics eventually forces the trajectory back to near the lower branch of S and this completes one cycle of the bursting solution. This description is formal. It is not at all clear that if the system (2.2) satisfies the above assumptions, then, for all s sufficiently small, there exists a unique periodic solution corresponding to a bursting oscillation. In Section 2.3, we show, in fact, that such a result cannot

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be true, in general. We describe a result, proved in [46], that the bursting solution will be uniquely determined for all e sufficiently small, except for those e that lie in a certain very small set. REMARK 2.1. A crucial ingredient for square-wave bursting is bistability. This allows for a hysteresis loop between a lower branch of stable fixed points and an upper branch of stable limit cycles. It is also very important that the slow nullsurface {h = 0} lies between these two branches. If this last condition is not satisfied, then the system may exhibit other types of solutions. For example, suppose that {h = 0} intersects the lower branch of S. This point of intersection will then be a globally stable fixed point of (2.2). If, on the other hand, {h = 0} intersects S along its middle branch above the homoclinic point, then (2.2) may give rise to a stable limit cycle which always remains in the active phase near P. This type of solution is referred to as continuous spiking. Rigorous results concerning the existence of continuous spiking are presented in [82,83]. REMARK 2.2. Square-wave bursting arises in models for electrical activity in pancreatic /3-cells. It is believed that this activity plays an important role in the release of insulin from the cells. The first mathematical model for this bursting was due to Chay and Keizer [ 10]. There have been numerous related models, based on experimental data, since then. A review of these models, along with a detailed description of the more biological issues, is given in [67]. Square wave bursting also arises in recent models for respiratory CPG neurons [7] and models for pattern generation based on synaptic depression [81 ]. REMARK 2.3. It is still not clear what underlying biological mechanisms cause bursting oscillations in pancreatic/3-cells [ 15]. In fact, it is possible that bursting may not be generated by the intrinsic properties of a single cell; rather, the bursting may be a network property generated by the coupling between cells. Computational studies in [69] have demonstrated that a population of cells with electrical, or gap-junction, coupling can exhibit bursting oscillations although the parameters of each cell are set so that the cells do not oscillate without any coupling. Detailed analysis of two mutually coupled square-wave bursters is given in [66]. REMARK 2.4. Very complicated (global) bifurcations can take place as the parameters e or )~ are varied in (2.2). The singular perturbation parameter e controls the rate at which a bursting trajectory passes through the silent and active phases. In particular, the number of spikes per burst is O(1/e) and becomes unbounded as e --+ 0. It is demonstrated in [82] that Smale horseshoe chaotic dynamics can arise during the transition of adding a spike. For example, the solution shown in Figure 3B exhibits a somewhat random variation between 3 and 4 spikes per burst. In Section 2.3, we describe how spikes are added to the bursting solution as e decreases. Perhaps even more interesting is the bifurcation structure of (2.2) as )~ is varied. In the /3-cell models, ~, is related to the glucose concentration. As the glucose level gradually increases, the cells exhibit resting behavior, then bursting oscillations, and then continuous spiking. This is consistent with behavior exhibited by the model. As )~ increases, the

Geometric singular perturbation analysis of neuronal dynamics 0.2

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Fig. 3. Chaotic dynamics from the square-wave bursting model. Note that v represents a rescaled voltage. A. (top) Chaotic solution generated by a bifurcation in X. B. (bottom) Chaotic solution generated by a bifurcation in e.

y-nullsurface {h = 0} intersects the lower branch of S, then the middle branch of S below the homoclinic point, and then the middle branch of S above the homoclinic point. Numerical studies [11] and rigorous analysis [83] have shown that as )~ varies between the bursting and continuous spiking regimes, the bifurcation structure of solutions must be very complicated. A Poincar6 return map defined by the flow from a section transverse to the homoclinic orbit of (FS) will exhibit Smale-horseshoe dynamics for a robust range of parameter values. This leads to solutions in which the number of spikes per burst varies considerably, as shown in Figure 3A. REMARK 2.5. Some phenomenological, polynomial models for square-wave bursting have been proposed. See, for example, [32,54,16]. Analysis of models with two slow variables which exhibit square-wave bursting is given in [70]. REMARK 2.6. A system of equations which give rise to square-wave bursting is [59]: v

!

-

-(g.,m~(v)(v

-

vc~) + g ~ w ( v - v~)

+ g l ( v - vl) + g ~ c a Z ( y ) ( v -- V~)) + I,

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,,_/=

,,,)/,-(,,),

y' = 2 0 e ( - l Z g c a m ~ ( v ) ( v -

Vca) - Y),

where, gca - 4.0, gk = 8.0, gl -- 2.0, Vk -- --84, Vl -- --60, l)ca -- 120.0, I -- 45, gkca = 0.25, r = 0.23, e -- 0.005, and/z -- 0.02. The nonlinear functions are given by moo (v) 0 . 5 ( 1 . 0 + t a n h ( ( v + 1.2)/18)), w c ~ ( v ) = 0 . 5 ( 1 . 0 + t a n h ( ( v - 12)/17.4)), z ( y ) = y / ( 1 + y ) and r ( v ) - c o s h ( ( v - 12.0)/34.8).

2.2.2. E l l i p t i c bursting. Elliptic bursting can also arise in a system of the form (2.2) in which there are two fast variables and one slow variable. The bifurcation diagram of (FS) for an elliptic bursting scenario, with the slow variable treated as a parameter, is shown in Figure 4, which includes the projection of the elliptic bursting solution. Bistability is crucially important for the generation of elliptic bursting, just as it is for square-wave bursting. An important difference, however, is that for elliptic bursting, the

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Geometric singular perturbation analysis of neuronal dynamics

103

curve S of fixed points of (FS) need not be Z-shaped; there may be only one fixed point of (FS) for each value of y. The branch of periodic solutions P now originates at a subcritical Hopf bifurcation along S. To obtain bursting, we hypothesize that the slow variable decreases along S; that is, h < 0 near S. During the silent phase, the bursting solution evolves near the stable portion of S until it passes the Hopf point, beyond which the fixed points of S are no longer stable. Note, however, that the trajectory does not jump up to the active phase immediately after crossing the Hopf point. The slow variable y may traverse a distance that is O(1) with respect to e past the Hopf point before jumping up. This type of delayed behavior or slow-passage past a Hopf point has been studied extensively in the singular perturbation literature [53,2]. We must also hypothesize that there is a net increase in the slow variable as the bursting trajectory passes near P. More precisely, let (Vy(t), Wy(t)) denote the periodic solutions along the outer branch of P and suppose that their periods are T (y). Let

h- ( y ) -

r ( 1y )

~0 T(y) h(vy(t),

Wy(t), y ) d t

denote the average of h along these limit cycles. We assume that/t(y) > 0 for each y. Thus during the active phase, y increases until the bursting solution passes the turning point, or knee, of P. The fast dynamics then forces the trajectory back towards S and a new silent phase begins. REMARK 2.7. Both square-wave and elliptic bursting depend on bistability and hysteresis. An important difference between these two classes of bursting is how the active phase terminates. Square-wave bursting ends at a homoclinic bifurcation; the period of oscillation, therefore, increases at the end of each burst. For elliptic bursting, the active phase ends at a saddle node of periodic orbits. There is no pattern for the spike frequencies in general. REMARK 2.8. Elliptic bursting arises in models for thalamic neurons [21], rodent trigeminal neurons [52], and 40 Hz oscillations [48,92]. REMARK 2.9. Here we assumed that the branch of periodic orbits of (FS) originate at a subcritical Hopf bifurcation. Hoppensteadt and Izhikevich [34] show that elliptic bursting is also possible even if the Hopf bifurcation is supercritical. REMARK 2.10. A system of equations which give rise to elliptic bursting is [59]:

v' - -(gcamc~(v)(v - Vca) + gkw(v -- l)k) + gl(v - Vl) + gkcaZ(y)(v - vk)) + I, y' -- e ( - # g c a m ~ ( v ) ( v -

Y e a ) - Y),

where, gca -= 4.4, gk = 8.0, gt = 2.0, Vk = --84, vt = --60, V c a = 120.0, I = 120, gkca "- 0.75, ~b -- 1.2, e -- 0.04, and # = 0.016667. The nonlinear functions

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are given by mc~(v) = 0.5(1.0 + tanh((v + 1.2)/18)), w ~ ( v ) = 0.5(1.0 + tanh((v 2.0)/30.0)), z(y) = y/(1 + y) and r(v) = cosh((v - 2.0)/60.0). 2.2.3. Parabolic bursting. Both square-wave and elliptic bursting can be achieved in a system with only one slow variable. Moreover, both depend on bistability of the fast dynamics. Parabolic bursting, on the other hand, requires at least two slow variables and does not arise from a hysteresis phenomenon. A geometric model for parabolic bursting is the following. Consider a system of the form (2.1) where x 6 ~2 and y - (yl, Y2) E ~2. There are now two slow variables, namely yl and y2. We first describe the bifurcation structure of the fast subsystem with both slow variables considered as parameters. This is illustrated in Figure 5, where we plot one component of the fast variable, corresponding to the membrane potential, along with both slow variables. We also show the projection of a parabolic bursting solution onto this figure. We hypothesize that the set of fixed points of (FS) forms a Z-shaped surface. The fixed points along the lower branch of this surface are assumed to be stable fixed points of (FS). There is a curve along the upper branch of fixed points where Hopf bifurcations occur. Periodic solutions arise at these Hopf points and terminate along orbits homoclinic to the

Fig. 5. Bifurcation structure for parabolic bursting. Here, x denotes one component of the fast variable. The meshed surface represents the maximumand minimumx-values of periodic orbits of (FS) while the solid surface consists of fixed points of (FS). The solid trajectory is the projection of a parabolic bursting solution.

Geometric singular perturbation analysis of neuronal dynamics

105

fixed points along the lower fold of the fixed point surface. In Figure 5, we show the maximum and minimum values of the fast variable along each of these periodic solutions. The existence of a parabolic bursting solution also requires hypotheses on the slow dynamics. To write reduced equations which determine the evolution of the slow variables in the silent phase, denote the lower branch of fixed points by x = q~(y). The silent phase evolution of the slow variables is then given by the reduced equations

y - g(~(y), y),

(2.3)

where differentiation is now with respect to r = et. One obtains reduced equations for the evolution of the slow variables in the active phase using averaging. Suppose that y = (yl, y2) lies in the region where there exists a stable limit cycle of (FS). Let Xy(t) be the corresponding periodic solution of (FS) with period T (y) and consider the averaged quantity

~'(Y) -

r ( y1)

f0 r(y) g(xy(t),

y) dt.

The active phase evolution of the slow variables is then given by the averaged equations y -- ~(y).

(2.4)

Parabolic bursting corresponds to the existence of a closed curve in the slow (yl, y2) phase plane which passes through both the region of stable fixed points of (FS), where it satisfies (2.3), and the region of stable limit cycles of (FS), where it satisfies (2.4). This is illustrated in Figure 6. Note that the active phase of the bursting solution both begins and ends along a curve of homoclinic bifurcations. Since the periods of the limit cycles tend to infinity at the homoclinic bifurcations, the interspike interval is longer at both the beginning and end of each burst. This accounts for the parabolic nature of the period of fast oscillations. REMARK 2.1 1. Parabolic bursting is found in the Aplysia R-15 neuron [ 1]. Rinzel and Lee [60] considered a model due to Plant [56] and were the first to give a detailed analysis of parabolic bursting by dissecting the equations into fast and slow subsystems as described here. REMARK 2.12. Rigorous results related to parabolic bursting are given in [42,73,34]. REMARK 2.13. A system of equations which give rise to parabolic bursting is [59]:

v' - --(i~a(V) + i~(v) + i~(v) + i~ca(V, c) + icas(V, s)) + I,

!

J.E. Rubin and D. Terman

106

active phase

30-.. 20-. 10~ 0-, - 1 0 ...

homoclinic bifurcations - 2 0 -. - 3 0 -. -40 >

0.2

" ~

0.7

o.~5

0.1

Y2 Yl

0

0.5

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where, ica(V) = g c a m ~ ( v ) ( v - Yea), ik(V) = gkW(V-- Vk), il(V) = g l ( v - - Vl), ikca(V, C) = gkcaZ(C)(V -- Vk), and icas(V,S) = gcasS(V - Yea). The constants are given by gca = 4.0, gk - 8.0, gl = 2.0, Vk = --84, Vt -- --60, Yea -- 120.0, I = 65, gkca - 1.0, ~b -1.333, e = 0.002, /z = 0.025, rs -- 0.05, and gcas = 1.0. The nonlinear functions are given by m ~ ( v ) = 0.5(1.0 + tanh((v + 1.2)/18)), w ~ ( v ) = 0.5(1.0 + tanh((v - 12.0)/17.0)), rw(V) = cosh((v - 12.0)/34.0), z(c) = c/(1 + c), and s o = 0.5(1 + tanh((v - 12)/24)).

2.3. Rigorous results f o r square-wave bursting solutions Here we present a theorem that addresses the issue of when, and in what sense, the heuristic description of a square-wave bursting trajectory given in Section 2.2.1 can be rigorously justified. The theorem implies that there may exist a small range of values of e for which the bursting trajectory is not uniquely determined; moreover, for exactly these values, the bursting solution does not closely follow the heuristically defined orbit. The reason why such values of e exist is related to the mechanism by which spikes are added as e decreases, as we describe here. A more complete discussion is given in [46]. For the following theorem, we consider (2.2) and assume that the geometric assumptions described in Section 2.2.1 are satisfied. We also need some technical assumptions concerning the bounded solutions of the fast subsystem (FS). We assume that each of the stable fixed points along the lower branch of S is hyperbolic, the manifold P of periodic solutions is normally hyperbolic, the right knee of S is nondegenerate, and the homoclinic

Geometric singular perturbation analysis of neuronal dynamics

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orbit arises from the transverse intersection of stable and unstable manifolds. Finally, we assume that S, P, and orbits heteroclinic to S and P represent all the bounded solutions of (FS). THEOREM 2.1. The periodic bursting solution is uniquely determined and asymptotically stable f o r all values o f e > 0 sufficiently small except f o r those in a set o f the f o r m O. Moreover, U i % 1(ei -- 3i, ei + 6i) 9 The F~i and 3i can be chosen so that limi__+e~ei 6i - - 6 i + 1 >

C182

and

~i ~ C2 e -k/ei

f o r some positive constants C1, C2 and k.

The proof of Theorem 2.1 is given in [46]. An important step in the proof is to explain how the bursting solution adds spikes as e varies. It is during these transitions that the heuristic construction is not justified. Here we give a geometric description of how these transitions take place. The number of spikes is determined by how many times the bursting trajectory spirals around in phase space near P. The active phase terminates when the bursting trajectory passes near the homoclinic orbit of (FS) and jumps down to the lower branch of S. The key to understanding how spikes are added, therefore, is to understand what determines when the trajectory jumps down to the silent phase. As we describe below, this crucially depends on the center-unstable and center-stable manifolds of the fixed points along the middle branch of S. The center-stable manifolds serve to separate those trajectories which continue to spiral in the active phase and those which jump down to the silent phase. Note that, when e = 0, there are two trajectories in the unstable manifold of each of the fixed points along the middle branch of S. The union of these trajectories forms the centerunstable manifold of the middle branch. One of these trajectories evolves towards the active phase, looping around the upper branch. Suppose that the homoclinic orbit of (FS) is at Y = Yh. If y < Yh, then this trajectory approaches one of the periodic solutions along P, while if y > Yh, then it ultimately approaches a stable fixed point along the lower branch. The other unstable trajectory evolves directly towards the silent phase and approaches the stable fixed point along the lower branch. Now the stable manifolds to the fixed points along the middle branch separate the two branches of unstable trajectories. Hence, if a trajectory lies close to the middle branch, it will either give rise to a spike or jump down to the silent phase depending on which side of the appropriate stable manifold it lies on. What we have described so far holds for e = 0, however, this all carries over for small e > 0. To make this more precise, let W~ and W~ be the union of all the stable and unstable manifolds to the fixed points along the middle branch when e = 0. (We exclude small neighborhoods of the left and right knee