The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network

Transcription

1 SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 4, No. 4, pp c 25 Society for Industrial and Applied Matematics Te Dynamic Range of Bursting in a Model Respiratory Pacemaker Network Janet Best, Alla Borisyuk, Jonatan Rubin, David Terman, and Martin Wecselberger Abstract. A network of excitatory neurons witin te pre-bötzinger complex (pre-bötc) of te mammalian brain stem as been found experimentally to generate robust, syncronized population bursts of activity. An experimentally calibrated model for pre-bötc cells yields typical square-wave bursting beavior in te absence of coupling, over a certain parameter range, wit quiescence or tonic spiking outside of tis range. Previous simulations of tis model sowed tat te introduction of synaptic coupling extends te bursting parameter range significantly and induces complex effects on burst caracteristics. In tis paper, we use geometric dynamical systems tecniques, predominantly a fast/slow decomposition and bifurcation analysis approac, to explain tese effects in a two-cell model network. Our analysis yields te novel finding tat, over a broad range of synaptic coupling strengts, te network can support two qualitatively distinct forms of syncronized bursting, wic we call symmetric and asymmetric bursting, as well as bot symmetric and asymmetric tonic spiking. By elucidating te dynamical mecanisms underlying te transitions between tese states, we also gain insigt into ow relevant parameters influence burst duration and interburst intervals. We find tat, in te two-cell network wit synaptic coupling, te stable family of periodic orbits for te fast subsystem features spike asyncrony witin oterwise syncronized bursts and terminates in a saddle-node bifurcation, rater tan in a omoclinic bifurcation, over a wide parameter range. As a result, square-wave bursting is replaced by wat we call top at bursting (also known as fold/fold cycle bursting), at least for a broad range of parameter values. Furter, spike asyncrony is a key ingredient in saping te dynamic range of bursting, leading to a significant enancement in te parameter range over wic bursting occurs and an abrupt increase in burst duration as an appropriate parameter is varied. Key words. square-wave bursting, fast/slow decomposition, synaptic coupling, averaged equations, bifurcation analysis, respiratory pacemaker AMS subject classifications. 34C15, 34C29, 37G15, 37N25, 92C2 DOI / Introduction. Te inspiratory pase of te respiratory rytm is believed to originate in a group of neurons in a region of te brain stem referred to as te pre-bötzinger complex (pre-bötc) [28]. Witin te pre-bötc, wen coupling among cells is removed, tere are silent cells, cells tat spike continuously, and intrinsically bursting cells tat generate groups of spikes separated by pauses [28, 12, 14]. Cells in all of tese classes seem capable of reg- Received by te editors February 28, 25; accepted for publication (in revised form) by M. Golubitsky July 29, 25; publised electronically November 18, 25. Tis researc was supported by te National Science Foundation, troug awards DMS to JR, DMS to DT, and DMS-1125 to te Matematical Biosciences Institute. ttp:// Te Matematical Biosciences Institute, Te Oio State University, Columbus, OH 4321 (jbest@mbi.oio-state. edu, borisyuk@mbi.oio-state.edu, wm@mbi.oio-state.edu). Department of Matematics and Center for te Neural Basis of Cognition, University of Pittsburg, Pittsburg, PA 1526 (rubin@mat.pitt.edu). Department of Matematics, Te Oio State University, Columbus, OH 4321 (terman@mat.oio-state.edu). 117

2 118 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER ular oscillatory bursting, if provided wit appropriate inputs experimentally, and tus tese cells are sometimes called pacemaker cells. Experiments in brain slices ave sown tat a synaptically coupled network of pre-bötc pacemaker cells can display syncronous bursting oscillations [28, 18]. In two papers, Butera and collaborators presented experimentally constrained conductancebased models for individual pacemaker cells in te pre-bötc as well as for a network of tese cells [1, 2]. In te network, bot excitatory synaptic coupling between cells and a depolarizing input current from a tonically firing population were included, wereas te persistence of respiratory rytms in pre-bötc under experimental blockage of synaptic inibition justified its omission [14]. For te most part, eac cell was coupled to all oter cells, altoug similar results were found wit less complete connectivities. Following Butera, Rinzel, and Smit, let te parameter g syn e denote te maximal conductance of an excitatory synaptic input from one cell to anoter, and let g tonic e denote te conductance of te tonic depolarizing current, wic is taken to be identical for all cells. A focal point of te Butera network study was te caracterization of te dynamic range of bursting in te model network. Te dynamic range ere refers bot to te range of g tonic e over wic te network displays bursting beavior, for a fixed g syn e, and to te corresponding range of burst frequencies produced. Uncoupled model pre-bötc cells are square-wave bursters, over a range of g tonic e. In teir simulations, Butera, Rinzel, and Smit found tat introducing synaptic coupling among identical model cells, by increasing g syn e from zero to a nonzero level, increased te range of g tonic e over wic syncronized bursting oscillations occurred, relative to te bursting range for a single cell [2]. More precisely, te coupled network would burst syncronously for te same g tonic e values tat led to single cell bursting, as well as for an interval of g tonic e tat would cause continuous firing in a single cell. Tis effect was nonmonotonic, suc tat as g syn e was increased, te bursting range of g tonic e would reac a maximum and ten would begin to srink back toward tat observed for g syn e =. Butera, Rinzel, and Smit also used simulations to map out te canges in burst frequency and oter burst caracteristics wit canges in g syn e and g tonic e. In particular, tey found tat wile te bursting range of g tonic e increased as g syn e increased from zero, network bursts wit at least some nonzero g syn e values acieved a more limited range of burst frequencies tan acieved wit g syn e =. Te primary goal of tis work is to provide a toroug matematical analysis of te mecanisms underlying most of tese findings. We employ a fast/slow decomposition [2, 22] to focus on ow canges in g syn e and g tonic e affect te bifurcation structure of te Butera pacemaker cell model. Tis approac allows us to elucidate te nature of te transitions from quiescence to bursting and from bursting to spiking in te network, as g syn e and g tonic e are separately varied. We note tat wile bot g syn e and g tonic e are conductances for inward, excitatory currents, increasing tese parameters may ave very different effects on network dynamics. In particular, increasing g syn e may transform te network from spiking to bursting and ten back to spiking; owever, increasing g tonic e can never transform te network from spiking to bursting. Importantly, our analysis raises te distinction tat bursting and tonic spiking in a coupled pair of cells can be symmetric, in tat te trajectories converge to, and oscillate regularly about, an axis of symmetry, or asymmetric, depending on features tat we derive from te network dynamics. In addition to explaining ow tese different activity patterns arise, our results include an analysis of transitions between tem. In te bursting

3 THE DYNAMIC RANGE OF BURSTING 119 regime, tis leads to an understanding of ow synaptic coupling and excitatory inputs combine to influence te silent and active pase durations, and ence te period, of bursting. Because te Butera, Rinzel, and Smit pacemaker cell model is a square-wave burster under appropriate parameter coices, te results presented ere advance te current matematical understanding of transitions between activity modes in general networks of cells capable of square-wave bursting [2, 3, 31, 11, 22]. Te analysis also demonstrates ow coupling cells tat exibit one type of beavior, namely, spiking, can lead to a different firing pattern, namely, bursting. Furtermore, our results, wile matematical in caracter, are relevant to te study of te biology of respiration in tat tey elucidate dynamical mecanisms tat can lead to various activity patterns, wic may be experimentally distinguisable in te pre-bötc, along wit te implications of tese mecanisms for quantitative aspects of network activity. In section 2 of te paper, we introduce te full Butera model and te details of te fast/slow decomposition tat we employ, including te key matematical features tat combine to govern bot te network dynamics in te model and te influence of g tonic e,g syn e on network beavior. Tis analysis, in te case g syn e =, explains te transition from quiescence to bursting to tonic spiking in a single uncoupled cell. Next, in section 3, we start wit a brief discussion of ow te transition from quiescence to bursting seen witout synaptic coupling carries over directly to coupled cells. Following tis, we turn to te muc more complex transition from bursting to spiking in te presence of synaptic coupling. We progress troug several levels of analysis of te associated penomena. First, we consider te special case of a single self-coupled cell. Second, we consider a pair of coupled cells under a strong syncrony assumption. Finally, we consider a pair of coupled cells wit no restrictions imposed on teir evolution. Tis progression demonstrates ow eac aspect of te dynamics of te freely evolving coupled cell pair contributes to te overall transition landscape. In particular, our analysis illustrates ow te asyncrony of spikes during te active pases of bursts can extend te dynamic range of bursting in a synaptically coupled pair of cells. In section 4, we explain ow variations in g syn e and g tonic e lead to canges in burst duration and interburst intervals, based on te bifurcation structures elucidated in te earlier sections. Finally, certain aspects of te qualitatively different transition mecanisms tat we find underlying te switc between bursting and tonic spiking in different parameter regimes lead to different experimental implications, wic we describe as part of te discussion in section Model and basic fast/slow decomposition Te Butera model. Te results of Butera, Rinzel, and Smit sow tat single-cell bursting, matcing experimentally observed properties of pre-bötc cells, can be initiated by te fast activation of a persistent sodium current, I NaP, and terminated by te slow inactivation of tis same current [1]. Tus, using te Hodgkin Huxley formalism, te membrane potential dynamics of eac pre-bötc cell witin a coupled network can be modeled by te equation (1) v i =( I NaP I Na I K I L I tonic e I syn e )/C, were eac term on te rigt-and side denotes an ionic current troug te cell membrane and te derivative is wit respect to time t. Specifically, we ave I NaP =ḡ NaP m P, (v i ) i (v i

4 111 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER E Na ),I Na = ḡ Na m 3 (v i )(1 n i )(v i E Na ),I K = ḡ K n 4 i (v i E K ),I L = ḡ L (v i E L ), and I tonic e = g tonic e (v i E syn e ). Te functions and parameters in tese currents are identical to tose presented in [1] and are listed in te appendix for completeness. Units for all variables are also given in te appendix. Tese units are used for all simulations, figures, and analysis in tis work, and we omit explicit mention of tem trougout te rest of te paper. Te dynamic auxiliary variables i,n i satisfy (2) i = ɛ( (v i ) i )/τ (v i ), (3) n i =(n (v i ) n i )/τ n (v i ) wit functions (v i ),τ (v i ),n (v i ),τ n (v i ) also specified in [1] and given in te appendix. We ave introduced te parameter ɛ in (2) to empasize tat te i will be considered as slow variables in te upcoming analysis. Te arcitecture of synaptic connections in te network contributes to te form of te synaptic current I syn e. We will consider a single self-coupled cell and a pair of coupled cells. In bot cases, let I syn e = g syn e s i (v i E syn e ) were (4) s i = α s (1 s i )s (v j ) s i /τ s, wit te function s (v) and te constants α s,τ s specified in te appendix. In te self-coupled cell case, i = j = 1, wile wit a pair of coupled cells, i, j {1, 2} wit j =3 i Fast/slow decomposition and bifurcation structure for a single cell. For a single cell, let us omit te subscripts i = j = 1 on te dependent variables in te model. In system (2) (3), ɛ/τ (v) 1/τ n (v) for all relevant v; furter, te evolution of is muc slower tan tat of v, as given by (1). Tus, it is natural to treat as a parameter and to consider te bifurcation structure of te fast subsystem (1), (3), and (4) as varies, a standard approac described, for example, in [2, 22]. Of course, in te full model, does evolve, and te position of te -nullcline determines te sign of te cange in at eac location in pase space. Tus, te position of te -nullcline relative to te bifurcation structures of te fast subsystem will contribute crucially to te dynamics of te network. An example of te relevant bifurcation structures, for (g tonic e,g syn e )=(.2, ), appears in Figure 1. For eac fixed, te fast subsystem, wic we now take as (1), (3) since g syn e =, as 1, 2, or 3 critical points. Te collection of all suc points forms a curve in (, v, n)-space, wic we call te fast nullcline and denote by S. Te solid/dased, S-saped curve in Figure 1 is te projection of S to (, v)-space. Note tat tis nullcline as 3 brances over an intermediate range of values. At an value near.8, te middle and lower brances come togeter in a saddle-node bifurcation; we refer to te coalescence point as te lower knee of S. Similarly, at an value near 1.5, te middle and upper brances coalesce in a saddle-node bifurcation at te upper knee of S. Te lower branc consists of stable critical points for (1), (3), wile points on te middle branc are unstable saddles. Points on te upper branc are unstable for small. As increases, a subcritical Hopf bifurcation occurs along te upper branc of critical points, above wic te critical points are stable. A family of unstable periodic orbits emanates from tis bifurcation. Tis family meets wit a second,

5 THE DYNAMIC RANGE OF BURSTING 1111 v nullcline P HB 3 6 S 1 1 Figure 1. Te bifurcation diagram for te fast subsystem (1), (3) wit (g tonic e,g syn e) =(.2, ), projected into (, v)-space, along wit te -nullcline. Te solid (dased) black curve is te curve S of stable (unstable) critical points of (1), (3) wit fixed at te levels indicated on te abscissa. A family of unstable periodic orbits, wit maxima and minima labeled by open circles, emanates from S in a Hopf bifurcation at te point marked HB. Tis family coalesces wit te family of stable periodic orbits P, wit maxima and minima labeled by dark, tick curves, in a saddle-node bifurcation at near 1.3. Te tick grey curve sows te -nullcline, namely, = (v), were =. outer family of periodics in a saddle-node bifurcation at a larger value tan te Hopf point. Te outer periodics are stable and terminate in a omoclinic bifurcation as decreases from te saddle-node. We will denote tis outer family by P. Finally, te -nullcline, or slow nullcline, intersects S in tree places, wic are critical points of te full system (1) (3) (wit g syn e = ). Te only stable critical point occurs on te fast nullcline s lower branc and is attracting. As g tonic e increases, wit g syn e =, it as tree effects on te bifurcation diagram for te fast subsystem. Increasing g tonic e causes te lower part of S to move to smaller values, causes P to move to smaller values, and causes te omoclinic point to move toward te lower knee of S. Tese effects can be seen in te left column of Figure 2. Tese canges will ave significant implications for te dynamics of te model cell. For comparison wit te case of a coupled pair of cells, to be considered in section 3, it is important to note tat we ave numerically computed te saddle quantity [15] of te omoclinic point on te middle branc of S for g syn e = and a range of values of g tonic e. Te saddle quantity remains negative over all relevant g tonic e, wic implies tat it is indeed a stable family of periodic orbits tat emanates from eac omoclinic point, and te saddle quantity decreases as g tonic e increases, corresponding to te fact tat te omoclinic point approaces te left knee of S as g tonic e increases (Figure 2). Examples of voltage traces derived from te evolution of (1), (2), and (3), wit g syn e =,

6 1112 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER v g tonic e P g syn e P 3 6 S S 1.5 LK HB g tonic e 1 1 HB.8 LK g syn e Figure 2. Dependence of te bifurcation structure of (1), (3), and (4) on g tonic e and g syn e. Te upper plots sow curves of critical points and families of periodic orbits for varying values of g tonic e and g syn e. Left: g tonic e =,.4,.7,g syn e =. Rigt: g tonic e =.2,g syn e =, 4, 8. Larger values correspond to more leftward structures. Te bottom plots sow ow te positions of te lower knee (LK) and Hopf bifurcation point (HB) vary wit g tonic e (wit g syn e =) and g syn e (wit g tonic e =.2), respectively. Note tat te lower knee and indeed te entire curve of critical points are approximately invariant under canges in g syn e. corresponding to a single uncoupled cell, are sown in Figure 3. Observe tat as g tonic e increases, te cell switces from quiescence to bursting to tonic spiking, as also sown in [2, 24]. In te quiescent case in Figure 3A, te trajectory is attracted to a stable critical point. In te bursting solution sown in Figure 3B, te trajectory spends some time on te lower branc of S, were it is below te slow nullcline, suc tat slowly increases. Tis is referred to as te silent pase of te solution. Altoug te two nullclines intersect very close to te lower knee, and it is difficult to discern in te figure, te intersection now occurs on te middle branc of S. Tus, te trajectory can reac te lower knee and jump up to P, and oscillations ensue, yielding te active pase of te solution. P lies above te slow nullcline, so decreases during te active pase. Finally, te trajectory approaces te omoclinic bifurcation were P terminates, and it falls back to te lower branc. Tis form of bursting is called square-wave bursting and as been analyzed extensively in previous work [3, 2, 3, 31, 16]. Note tat in te bottom panel of Figure 3B, tere is an interval of -values, extending on bot sides of =.6, for wic te dynamics of te fast subsystem are bistable. Specifically, for eac in tis range, tere are a stable critical point on te lower branc of S and a stable

7 THE DYNAMIC RANGE OF BURSTING 1113 v A B C D 4 t v Figure 3. Voltage traces (top row) and bifurcation diagrams wit superimposed trajectories and (grey) - nullclines (bottom row). In all panels, g syn e =. Te parameter g tonic e takes values.2 (A - quiescence; stable critical point on te lower branc of S denoted by ),.3 (B - bursting),.4 (C - bursting), and.7 (D - tonic spiking). In te top row, te scale bar corresponds to 2 seconds. Note te different -axis scales in eac panel in te bottom row. periodic orbit from P. Te case in Figure 3C again represents square-wave bursting, but te range of bistable values is muc smaller tan in Figure 3B. In tis case, tis leads to sort bursts relative to Figure 3B. Finally, in Figure 3D, tere is no region of bistability, and te trajectory is pinned in te vicinity of P, suc tat tonic spiking results. Note from te bottom part of Figure 3D tat te trajectory extends bot above and below te slow nullcline (grey curve). Wile it is above (below) te slow nullcline, decreases (increases). In te attracting state for te network, te net drift in is zero, leading to te pinning and continuous spiking seen ere [31]. Rater tan varying g tonic e, we can keep g tonic e fixed and consider te effect of varying g syn e on te bifurcation structure of te fast subsystem, now including (4), wit i = j =1, corresponding to a single self-coupled cell. Because of te influence of (4), canges in g syn e are not equivalent to canges in g tonic e. In particular, s along all brances of te fast nullcline S, were v does not become muc larger tan 3, due to te form and parameters of s (v), as given in te appendix. Tus, increasing g syn e leaves te projection of S to (, v)-space largely uncanged, as seen in te rigt column of Figure 2. Increasing g syn e from does cause P to move to smaller values, owever, since s can become significant at te larger v values reaced along P. Tis sift widens te range of values for wic bistability occurs in te fast subsystem. Furter, wit its leftward motion, a greater part of tis family lies below te slow nullcline, resulting in a decrease in te leftward drift during te

8 1114 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER active pase of a burst. Eventually, tis effect can cause pinning, corresponding to a transition from bursting to tonic spiking. We explore te transitions between activity modes more systematically in te subsequent sections of te paper. 3. Analysis of transitions between modes of activity Te transition from quiescence to bursting. As noted in section 2, a cell or network of identical cells is quiescent wen te fast and slow nullclines ave an intersection on te lower branc of S. Given a network in te quiescent state, bursting can be induced by increasing g tonic e. Indeed, as also observed in [2] (see Figure 2 of [2], also reproduced in Figure 18 below), te value of g tonic e at wic te switc from quiescence to bursting occurs, namely, g tonic e.26, depends only very weakly on te value of g syn e. Te mecanism underlying te switc from quiescence to bursting is tat as g tonic e increases, S, and in particular its lower knee, moves leftward in te (, v)-plane, to smaller -values, as mentioned in section 2 (Figure 2). Since te slow nullcline is independent of g tonic e, tis trend causes te lowest v intersection of te nullclines (call it p) to transition from lying on te lowest branc of S to lying on te middle branc of S, by passing troug te lower knee of S. In tis transition, an eigenvalue of te linearization of (1) (4) about p crosses from te negative real axis to te positive real axis, suc tat on te middle branc, p is an unstable critical point of (1) (4). Trajectories starting in te silent pase now flow past te lower knee of S and are attracted to te family of periodic orbits P. Bursting, rater tan tonic spiking, results from te transition for te parameter values of interest due to a combination of two factors seen in Figure 3B; tere is always bistability between te lower branc and P wen tis transition occurs, and tere is a net leftward drift in during te active pase. Finally, te transition is relatively independent of g syn e because, as noted in section 2 (e.g., Figure 2, rigt top panel), g syn e as little impact on te position of S and ence on te position of te critical point p Te transition from bursting to tonic spiking in a self-coupled cell. Te transition from bursting to tonic spiking is muc more complex tan tat from quiescence to bursting. In fact, tere are several different mecanisms underlying te transition from bursting to tonic spiking, depending on parameter values. Here we will briefly return to te simplest case of a single self-coupled cell, as considered in subsection 2.2; note tat tis case is also equivalent to a pair of coupled cells tat are completely syncronized. As we sall discuss in te subsequent subsections, te completely syncronized solution is generally unstable wit respect to te full system, and coupled cells fire spikes tat are out of pase in te stable bursting and tonic spiking solutions. However, te progression in analysis presented in tis and subsequent subsections will illustrate te precise way in wic asyncrony between cells witin te spiking pase can fundamentally alter te fast/slow bifurcation structure and be a significant ingredient in determining te model s dynamic range of bursting. Consider a single, self-coupled cell, wic satisfies (1) (4) wit (v i, i,n i,s i ) replaced by (v,, n, s). As in subsection 3.1, we analyze tis system using fast/slow analysis wit as te slow variable, and representative bifurcation diagrams are sown in Figure 2. Define te -nullsurface G = {(v,, n, s) : = (v)} and let p = G S as in subsection 3.1. Note

9 THE DYNAMIC RANGE OF BURSTING 1115 tat (1) (4) exibits square-wave bursting if tere is an interval of -values were te fast subsystem exibits bistability and p lies between te lower knee of S and te omoclinic point P S. Alternatively, if p lies at a smaller -value tan tat of te omoclinic point on te middle branc of S, ten, in te limit ɛ in(2), system (1) (4) will exibit tonic spiking. Tus, as demonstrated in [31], te transition from square-wave bursting to tonic spiking, in te limit ɛ, occurs wen te omoclinic point on te middle branc of fixed points crosses G p g syn e = g syn e = g tonic e Figure 4. Te curve of fixed points p of (1) (4), togeter wit te curves of omoclinic points P S of te fast subsystem (1), (3), and (4) for g syn e =and g syn e =2, as a function of g tonic e. Te intersections of tese curves yield te values of g tonic e at wic te transition from bursting to tonic spiking is predicted to occur, based on a fast/slow decomposition of te single self-coupled cell. Note tat altoug p switces from te lower branc of S to te middle branc at g tonic e.26, te -value of p is a monotonically decreasing function of g tonic e, because all of S moves toward smaller values as g tonic e increases. As illustrated in te examples in Figure 2 and particularly in Figure 4, te omoclinic point for te self-coupled cell lies at smaller tan tat for te uncoupled cell. Tus, g tonic e must be increased more for G to cross te curve of omoclinic points in te self-coupled case, and te transition to tonic spiking occurs at a iger value of g tonic e tan for te uncoupled cell; tat is, a self-coupled cell as a larger dynamic range of bursting oscillations tan an uncoupled cell as. To finis tis analysis, we use XPPAUT [9] to follow te curve in (g tonic e,g syn e ) parameter space were G intersects te omoclinic point P S. Tis generates a transition curve, sown in Figure 5, wit a sape tat qualitatively matces tat in Figure 2 of [2] (see Figure 18 below). Tere is a significant quantitative difference between te two results, owever, wit te curve in Figure 5 substantially underestimating te extent of te bursting region. Tus, we conclude tat te dynamics of a single self-coupled cell, wile interesting in teir own rigt, do not capture te complexity of te bursting and spiking beaviors in te

10 1116 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER pre-bötc model wit multiple, synaptically coupled cells TONIC SPIKING g syn e BURSTING g tonic e Figure 5. Te transition curve (G P S) between bursting (to te left) and tonic spiking (to te rigt) predicted by analysis of a single self-coupled cell. Tis curve significantly underestimates te extent of te bursting region Te transition from bursting to spiking in coupled cells wit 1 = 2. In te previous section, we assumed tat te cells were completely syncronized and concluded tat tis does not accurately predict te full increase in dynamic range for te coupled system. Figure 6 illustrates wy tis sould not be surprising. Here we sow te voltage traces of te two cells for (g tonic e,g syn e )=(.5, 8). Note tat wile te cells appear to burst togeter, teir spikes fire out-of-pase. We must, terefore, extend te fast/slow analysis to te case in wic we consider asyncronous spiking. Tis will be done in two steps. In tis section, we assume tat te slow variables 1 and 2 are equal; we can ten perform te fast/slow analysis wit a single slow variable, = 1 = 2. As we sall see, tis assumption leads to an accurate prediction for te transition to tonic spiking for large values of g syn e (see Figure 7). Moreover, te resulting bifurcation structure as some rater novel features not seen in te analysis of te self-coupled cell. For moderate and low values of g syn e, we can no longer assume tat 1 = 2 ; tese must be considered as separate slow variables (Figure 7). Te two-slow-variable analysis will be carried out in te next subsection. Denote te system of eigt equations, consisting of (1) (4) taken wit bot i = 1 and i =2,by(1) i (4) i. Figure 8 sows an example of te bifurcation diagram generated by te fast subsystem consisting of te six equations (1) i,(3) i,(4) i wit 1 = 2 = as te single bifurcation parameter. Tis diagram is projected onto te (, v 1 )-plane. Note tat two families of periodic orbits emanate from te single curve of equilibria S in distinct subcritical Hopf bifurcations. As we move from rigt to left along te -axis, starting above bot Hopf points, te critical points on S are stable. Tey lose stability in te first Hopf bifurcation, wic gives rise to an unstable family of periodic orbits, labeled as IP in Figure 8 and consisting

11 THE DYNAMIC RANGE OF BURSTING 1117 v 1 v v time time Figure 6. Bursting solutions of te full model (1) (4). Here, (g tonic e,g syn e) =(.5, 8). Te left panel sows tat te bursts appear to be syncronized. Te rigt panel sows tat te spikes actually occur in antipase..4.3 g ton e =.5 g syn e = g ton e =.75 g syn e = time Figure 7. Plots of 1 (red) and 2 (blue) as functions of time during a single burst cycle. If g syn e is large (top), ten 1 2, wile for moderate or low values of g syn e (bottom), we cannot assume tat 1 2.

12 1118 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER of in-pase oscillations, as is decreased. Bot brances of periodics in IP, wic merge at a saddle-node bifurcation, are unstable wit respect to te fast subsystem, except possibly for te outer branc in some relatively small neigborood of te saddle-node bifurcation. Te second family of periodics (call it AP) occurs at lower and corresponds to antipase oscillations. Tis family consists of tree brances. Te branc tat emanates from te subcritical Hopf consists of unstable limit cycles. Tis branc terminates at a saddle-node of periodic orbits, at = R in Figure 8, were it coalesces wit a second branc of periodic orbits. Tis second branc is stable, at least away from a relatively small neigborood of te saddle-node bifurcation. It will be very important in te analysis and we label it as AP S. Tis branc terminates in anoter saddle-node bifurcation of periodic orbits, at = L in Figure 8, were it coalesces wit a tird branc of unstable periodics. Te tird branc terminates in an orbit omoclinic to te middle branc of S. (Note tat te upper branc corresponding to tis family lies very close to tat of AP S, and ence cannot be distinguised at te scale sown in Figure 8.) A similar emergence of antipase and in-pase periodic orbit families is also seen wen diffusive coupling is introduced between square-wave bursters derived from a model for bursting in pancreatic β-cells [25]. v IP AP S L R _ 1 _ 2 _ 3 _ 4 _ 5 _ Figure 8. Bifurcation structure of te fast subsystem for (g tonic e,g syn e) =(.5, 8). Here we assume tat = 1 = 2 is te bifurcation parameter. Te branc of fixed points S is sown in blue. Tere are two brances of periodic orbits; in-pase solutions (IP) are sown in red, wile antipase solutions are sown in green. Te stable portion of te antipase branc is denoted as AP S and exists on te interval [ L, R]. Te projection of a bursting solution (purple) onto tis bifurcation diagram is sown in te rigt panel. Note tat te active pase ends at a saddle-node of periodic solutions of te fast subsystem. Remark 3.1. For values below bot Hopf bifurcations, linearization of te 6-dimensional fast subsystem around eac critical point on te upper branc of S yields four eigenvalues wit positive real parts. As S is followed around te upper knee, altoug all four unstable eigenvalues become real, two of tese cross troug te origin, by symmetry. Similarly, te oter two unstable eigenvalues stabilize at te lower knee, suc tat te critical points on te lower branc of S are indeed stable. Remark 3.2. Numerical calculations suggest tat wen te fast subsystem is linearized about te omoclinic point at wic te tird branc of AP terminates, wic lies on te middle branc of S, te unstable pair of eigenvalues as larger magnitude tan tat of te leading stable eigenvalues. Because te multiplicity of tese eigenvalues comes from symmetry

13 THE DYNAMIC RANGE OF BURSTING 1119 and not degeneracy, te saddle quantity [15] is relevant, and based on tis, te periodic orbits on tis tird branc are unstable, as we observe in our bifurcation diagrams and direct numerical simulations. Tis differs from te standard square-wave bursting scenario, seen in te case of te single, self-coupled cell in subsection 2.2, in wic te leading stable eigenvalue as larger magnitude tan te unstable eigenvalue and stable periodic orbits emerge from te omoclinic point as is increased. Figure 8 also sows te projection of te bursting solution sown in Figure 6 onto te fast subsystem bifurcation diagram. As usual, te silent pase lies along te lower branc of S and te active spiking pase begins wen te trajectory reaces te lower knee of S. During te active pase, te trajectory lies close to AP S and te active pase ends wen te trajectory reaces te saddle-node of periodics. Note tat tis bifurcation structure no longer corresponds to square-wave bursting, were spiking ends at a omoclinic orbit, but rater represents a different bursting class (see also [11, 27, 4, 26]). We ave, terefore, demonstrated tat synaptic coupling of cells leads to a cange in te class of bursting activity tat occurs. As we demonstrate below, tis will contribute to te fact tat te coupled system as an increased dynamic range. A 3-dimensional caricature of tis induced form of bursting is illustrated in Figure 9. We sall refer to tis bursting class, wic is called fold/fold cycle bursting in [11], as top at bursting. fast 1 fast 2 Figure 9. A scematic illustration of a top at burster. A similar top at structure would arise from a system wit two fast variables and one slow variable or from a projection of a iger-dimensional system, suc as we consider, onto two fast dimensions and one slow dimension.

14 112 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER Top at bursters ave several important features tat distinguis tem from square-wave bursters. Te active pase of a square-wave burster ends at a omoclinic orbit. For tis reason, te spike frequency becomes small at te end of eac burst. For top at bursters, te active pase ends at a saddle-node of limit cycles. Hence, te spike frequency approaces some fixed value, bounded away from zero, at te termination of burst activity. A second difference between square-wave and top at bursting is related to te transition to tonic spiking as a parameter, suc as g tonic e, is varied. Recall tat for a square-wave burster, tis transition takes place as te omoclinic point crosses te slow nullsurface, denoted by G earlier. For top at bursters, tis transition arises from a very different mecanism. To understand tis new mecanism, we use singular perturbation metods to reduce te full system of (1) i (4) i to a reduced system for just te slow variables. Since we are now assuming tat 1 = 2, tis will lead to a reduction of te full model to a single equation. Te reduction is carried out separately for te silent and active pases. Wile in te silent pase, te solution lies close to te lower branc of S and we invoke a steady state approximation. Tat is, introduce te slow time variable τ = ɛt in (1) i (4) i and ten set ɛ =. Te rigt and sides of (1) i,(3) i, and (4) i ten become zero and we may solve for fast variables (v i,n i,s i ), i =1, 2, in terms of. Wile tere are multiple possible solutions, we coose tat wit te smallest v, corresponding to te silent pase. As a result, since we use te same for i = 1 and i = 2, we obtain (v 1,n 1,s 1 )=(v 2,n 2,s 2 ) in te silent pase. After substituting v 1 = v 2 into (2), we obtain a single equation for te evolution of in te silent pase. For te active pase, we use te metod of averaging. Suppose tat AP S exists for L R (Figure 8). For L R, let (v i (t, ),n i (t, ),s i (t, )), i =1, 2, be te corresponding antipase periodic orbit of te fast subsystem and assume tat its period is T (). Ten, in te limit ɛ, te evolution of during te active pase is governed by te averaged equation (5) ḣ = 1 T () T () ( (v i (t, )) )/τ (v i (t, ))dt a(). Here, differentiation is wit respect to τ. We may use v 1 or v 2 in (5), since we are assuming tat 1 = 2 and we are terefore integrating over a common periodic orbit, belonging to te stable family AP S, for i = 1 and i = 2, altoug te cells may be out of pase along te orbit. Now te system exibits bursting if a() < for all ( L, R ). In tis case, te solution drifts to te left wile oscillating along AP S. Te onset of tonic spiking occurs at te minimal value of g tonic e for wic tere exists a stable fixed point of (5) in[ L, R ] tat as te lower knee of S in its basin of attraction. In teory, suc a fixed point could arise at te saddle-node of periodic orbits at L (Figure 8), yielding a unique tonic spiking solution, or it could first appear via a double zero of a() in( L, R ), leading to a saddle-node bifurcation of tonic spiking solutions, one stable and one unstable, as g tonic e increases [27, 4, 26]. (Recall tat we only evaluate (5) along periodic orbits in AP S, ignoring possible unstable tonic spiking solutions corresponding to te unstable branc of periodic orbits tat is also born at = L.) Our simulations sow tat a() is a monotone decreasing function on [ L, R ] for eac fixed g tonic e. Tus, bursting occurs, wit a() < on[ L, R ], for sufficiently small g tonic e,

15 THE DYNAMIC RANGE OF BURSTING 1121 and te transition from bursting to tonic spiking appens at te minimal g tonic e suc tat a( L ) =. An example is given in Figure 1. a() 5 x 1 3 g tonic e =.5 g tonic e =.61 g tonic e =.62 g tonic e = L Figure 1. Te function a() plotted over [ L, LK] for g syn e =8and several g tonic e values, were LK is te -value for te lower knee of S. For all g tonic e, a() remains monotone decreasing. As g tonic e increases from.61 to.62, a zero of a() occurs at = L, and tis zero moves away from L toward larger -values as g tonic e increases furter. Note tat te actual values of L, LK depend on g tonic e, and ence we omit numerical lables from te -axis; all of te curves sown ave been aligned according to teir respective L values for comparison. Te criterion a( L ) = gives an accurate prediction for te value of g tonic e at wic te transition from bursting to tonic spiking occurs for large values of g syn e. For small and moderate values of g syn e, tis curve does not matc te actual transition; it severely underestimates te increase in dynamic range of bursting activity. Te reason for tis discrepancy is tat, for small and moderate values of g syn e, te beavior of te full system is inconsistent wit te assumption tat 1 = 2. We must, terefore, extend our fast/slow analysis to te case of two slow variables Te transition from bursting to tonic spiking in te full model for two coupled cells Using slow averaged dynamics in te oscillation region to analyze activity states. Te previous subsections demonstrate tat to capture te full picture of te dynamic range of bursting for two coupled pre-bötc cells, it is necessary to consider te full four-equation model (1) (4) for eac cell. Again, tere is a natural fast/slow decomposition, acieved by taking 1, 2 as slow variables; below, we refer to te fast subsystem to mean te oter six equations wit 1, 2 frozen. Rater tan visualizing fast subsystem bifurcation structures, we will now consider dynamics projected to te ( 1, 2 )-plane. To start, fix (g tonic e,g syn e ) and note tat for some pairs ( 1, 2 ), te fast subsystem will

16 1122 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER support regular, stable tonic spiking, wile for oters, suc sustained oscillations will not exist. We can use direct simulation of te fast subsystem (e.g., fixing 2, varying 1 systematically, and ten repeating for a different 2 ), to estimate a boundary curve for te oscillation region O in ( 1, 2 )-space, suc tat for ( 1, 2 )-values below tis curve, regular, stable oscillations do not exist for te fast subsystem. In wat follows, we denote tis boundary curve as B. Remark 3.3. We use te term regular oscillations to refer specifically to periodic solutions in wic te two cells fire in alternation, wit constant interspike intervals. We will return to te issue of regular versus irregular oscillations of te fast subsystem later in tis subsection. We use averaging to reduce te full system to a system of two equations for just te slow variables. For g(v, ) ( (v) )/τ (v), te reduced system can be written as (6) 1 T (1, 1 = 2 ) T ( 1, 2 ) 2 = 1 T ( 1, 2 ) g(v 1p ( 1, 2 ; t), 1 ) dt a 1 ( 1, 2 ), T (1, 2 ) g(v 2p ( 1, 2 ; t), 2 ) dt a 2 ( 1, 2 ), were ( 1, 2 ) O, T ( 1, 2 ) is te period of te fast subsystem periodic orbit for tis coice of ( 1, 2 ), and v 1p,v 2p are te time courses of v 1,v 2 around te orbit, wic bot depend on bot 1 and 2, since te orbit itself does. Note tat tonic spiking corresponds to a stable fixed point of (6). In fact, as we now demonstrate, te complete transition from bursting to tonic spiking for te full system can be understood by analyzing te pase planes generated by (6). Figure 11 illustrates pase planes of (6) wit g syn e = 3 and four values of g tonic e. Note tat for tis value of g syn e, te analysis in te preceding section, in wic we assumed tat 1 = 2, does not give an accurate prediction for wen te transition from bursting to spiking takes place for te full system. In eac panel of Figure 11, te black curve represents B, te boundary of te oscillation region. Wen a bursting solution crosses B, it falls back to te silent pase (not sown in te figure), and spiking activity stops until a subsequent burst cycle begins. Te red and blue curves in Figure 11 are numerically computed averaged nullclines, namely, A 1 = {( 1, 2 ):a 1 ( 1, 2 )=} and A 2 = {( 1, 2 ):a 2 ( 1, 2 )=}. Fixed points of (6) are given by te intersections of tese nullclines, and one can usually determine te stability of te fixed points by considering te nullcline configuration. Note tat to estimate te positions of A 1 and A 2, we simulate te fast subsystem (1) i,(3) i,(4) i. Tis consists of fixing 2 and systematically varying 1 to identify locations were eiter a 1 ( 1, 2 )or a 2 ( 1, 2 ) is sufficiently close to zero, repeating te process for eac 2 on a partition of te relevant 2 range, wic corresponds to te interior of te region O as determined by te location of B. In Figure 11A, g tonic e =.57. Note tat A 1 and A 2 are not present in O, and terefore bot 1 and 2 remain negative along every trajectory in O. Hence, every solution of te averaged slow equations (6) must eventually leave O troug B and te full system (1) i (4) i exibits bursting. Te bursting is symmetric in te sense tat trajectories of (6) converge to te line L {( 1, 2 ): 1 = 2 } over successive burst cycles and oscillate symmetrically about it wile in O, and ence we refer to tis as symmetric bursting. In general, tis is a top at burster and can be analyzed using te one slow variable analysis described in te preceding section.

17 THE DYNAMIC RANGE OF BURSTING 1123 A B p C D q B q B p.15 p q A Figure 11. Averaged pase planes, corresponding to (6), wit superimposed trajectories of (1) i (4) i, for g syn e =3. Trougout tis figure, te jump-down curve B is solid black, te nullclines A 1, A 2 are red and blue, respectively, te symmetry axis L is dased black, and trajectories are green. (A) For g tonic e =.57, ḣ 1, ḣ2 are negative everywere in te oscillatory region O. Tus, every solution of te averaged equations leaves te oscillatory region O troug B and te system exibits symmetric bursting. Te trajectories ere correspond to te flow of (1) i (4) i during te active pases of several bursts only, wit te silent pases omitted. (B) For g tonic e =.83, tere is an unstable fixed point p in O were te averaged nullclines intersect. Te system exibits asymmetric bursting. Again, te trajectory sown is from te active pase of bursting. (C) For g tonic e =.87, te averaged nullclines intersect at tree fixed points in O, namely, p, wic is still unstable, and q A,q B, wic are stable. Te system exibits asymmetric tonic spiking. An asymmetric tonic spiking solution is sown in green; similar solutions exist near q A. (D) For g tonic e =.91, p is a stable fixed point and te system exibits symmetric spiking. Here B is not visible since it lies at smaller ( 1, 2) values tan tose sown in tis plot. In Figure 11B, g tonic e =.83 and te full system still exibits bursting. However, te slow system (6) now as a fixed point, denoted by p in Figure 11B, inside of O. Indeed, tis fixed point enters O troug te intersection point of B wit te line L {( 1, 2 ): 1 = 2 } as g tonic e increases. Using te fact tat te slope of te 2 -nullcline at p is more negative tan te slope of te 1 -nullcline tere and te reflection symmetry of (6), we ave n b ḣ1/ 2 = ḣ2/ 1 <n s ḣ1/ 1 = ḣ2/ 2 <. Tus, te eigenvalues n s ± n b of te linearization of (6) about p ave opposite signs, suc tat p is an unstable saddle of (6). Te stable manifold of p lies along te line L, wile eac solution of (6) tat does not begin along L must eventually leave te oscillation region troug B, after crossing troug A 1 and A 2 and experiencing a cange in te sign of ḣ1 and ḣ2, respectively. As a result, te full system generically exibits bursting oscillations. We sall refer to tis as asymmetric bursting since 1 2 along te solution. We note tat it is essential ere to consider two-slow-variable

18 1124 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER analysis. If, as in te preceding section, we assumed tat 1 = 2, ten we would incorrectly predict tat te full system exibits tonic spiking as soon as p enters O, wic occurs at significantly smaller g tonic e tan te actual spiking onset. Tat is, p is stable wit respect to solutions of (6) tat lie along te stable manifold L. Tis explains wy te analysis in te preceding section does not accurately predict te full dynamic range of rytmic bursting oscillations. Furter, te fact tat no saddle-node bifurcation gives rise to critical points of (6) along L, away from B, asg tonic e increases corroborates our earlier claim tat no saddle-node bifurcation occurs in te fixed points of (5). For Figure 11C, we set g tonic e =.87. Wile tere is still an unstable fixed point p O, te averaged nullclines A 1 and A 2 now intersect at two new fixed points, labeled as q A and q B, in te oscillatory region O. Tese fixed points are stable, as can be seen from te configuration of te nullclines, and tey represent tonic spiking of te full system. We say tat tis is asymmetric tonic spiking because 1 2 at q A and q B ; tat is, te stable fixed points do not lie along te axis of symmetry L. Finally, suppose tat g tonic e =.91. In tis case, as sown in Figure 11D, p is a stable fixed point of (6) and te full system exibits symmetric tonic spiking. Te configuration of te nullclines A 1 and A 2 at p as now switced from te previous cases. Tat is, as we increase g tonic e from.87 to.91, a pitcfork bifurcation occurs. In tis bifurcation, te stable fixed points q A and q B come togeter at p, and p switces from being a saddle to being a stable node. Figure 11D also sows an example of ow a tonic spiking trajectory oscillates symmetrically about p. It is important to note tat, even in tese symmetric tonic spiking solutions, we expect v 1 and v 2 to be antipase. Tis can be cecked for small g syn e by calculating te H-function [13, 1]. Te functions H(φ) and H odd (φ) (H(φ) H( φ))/2 for g tonic e =1.5 and g syn e =1 appear in Figure 12. A zero of H odd (φ) represents a paselocked, periodic solution of te full system, wic is stable (unstable) if te derivative of H odd is positive (negative) tere. Since te pase sift in a solution is given by te value of φ at wic te corresponding zero of H odd (φ) occurs, Figure 12 predicts tat v 1,v 2 will be exactly antipase for tis (g tonic e,g syn e ) (see also Figure 6). 2 H(φ) φ 1.5 H (φ) odd φ 1 Figure 12. H-function and its odd part H odd for g tonic e =1.5 and g syn e =1. Since H odd (.5)=and H odd(.5) >, te antipase symmetric spiking solution is predicted to be stable. Remark 3.4. We ave also numerically computed te H-function for symmetric bursting

19 THE DYNAMIC RANGE OF BURSTING 1125 for particular values of g tonic e,g syn e. Te results agree wit our analysis, sowing tat spikes are out of pase witin te stable solution. Te results also suggest tat a completely antipase solution, in wic te cells take turns bursting, sould also be stable. However, it is important to note tat tis calculation is relevant in te weak coupling limit. Our simulations sow tat suc antipase bursting solutions indeed may stably exist, but only for extremely small g syn e. Furter consideration of antipase bursting solutions is outside of te scope of tis work. Figure 13 sows regions in (g tonic e,g syn e ) parameter space were te full coupled system (1) i (4) i is predicted to exibit symmetric bursting (SB), asymmetric bursting (AB), asymmetric spiking (AS), and symmetric spiking (SS). As seen above, te SB region corresponds to te absence of fixed points in O, and te symmetry expected ere refers to an approximate equality of 1 and 2. We ave not yet justified wy solutions in SB sould, in general, ave 1 2, owever, and tis is discussed in subsection in te context of syncronization of bursts. Te blue curve corresponds to wen te fixed point p first appears in O as g tonic e is varied, representing te transition from SB to AB. Tis is were te one-slow variable analysis described in te previous section predicts tat tere sould be te transition from bursting to tonic spiking. Te green curve corresponds to te transition from AB to AS. Recall tat tis occurs wen te additional intersections of te averaged nullclines A 1 and A 2, namely, te stable fixed points q A and q B, appear in O. Te red curve corresponds to te transition to SS. Tis corresponds to te occurrence of a pitcfork bifurcation for te slow averaged equations (6) g syn e 5 4 symmetric spiking (SS) silent symmetric bursting (SB) asymmetric bursting (AB) asymmetric spiking (AS) g tonic e? Figure 13. A summary of ow te activity of a pair of coupled pre-bötc cells depends on te parameters g tonic e,g syn e. Eac solid curve represents a boundary between regions in (g tonic e,g syn e)-space corresponding to different activity patterns. Te question mark indicates tat for very weak coupling g syn e, numerical difficulties prevent us from distinguising precisely were te AS SS transition occurs. See te text for a full discussion of te regions and transitions specified in tis figure.

20 1126 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER Canges in te transition patway as g syn e is increased. In Figure 13, te region between te black line and te green curve were it exists, or te blue curve were te green curve does not exist, gives te set of parameter values for wic bursting is predicted. Tis gives excellent quantitative agreement wit te simulation results from [2]. Note from Figure 13 tat qualitatively different transitions troug activity states occur for g syn e above or below a tresold of approximately 7.5. Figure 14 sows examples of SB and SS solutions for g syn e =8. As g syn e is increased to larger values, te AB and AS regions in (g tonic e,g syn e ) space srink, as sown in Figure 13. For all g syn e < 7.5, te AS region persists, altoug it becomes so narrow tat it can ardly be distinguised from AB on te scale used in Figure 13. Note tat in fact tere cannot be a direct transition from SB to AB to SS. Tat is, in te AB state, te unstable symmetric fixed point p of (6) lies in O, and in te SS state, tis fixed point is stable. Te stabilization occurs troug a pitcfork bifurcation as g tonic e is increased, wic requires te existence of te two stable equilibria q A,q B in O for g tonic e sufficiently close to, but below, te onset of SS. For suc g tonic e values, AS will occur. A B g syn e =8 g ton e = g syn e =8 g ton e = Figure 14. Averaged pase planes from (6), wit superimposed trajectories of te full system (1) i (4) i, for g syn e =8, corresponding to a direct transition from SB to SS. Te labels ere are as in Figure 11. (A) Symmetric bursting solution for g tonic e =.6. Te green trajectory sown travels first from te upper rigt part of te region to te lower left, were it its te black boundary curve B. At tis point, te cells enter te silent pase and 1, 2 bot increase. Correspondingly, te trajectory ere moves back from lower left to upper rigt, altoug te cells are not spiking and te dynamics of (6) are irrelevant. Te jump up to te active pase for te next burst cycle corresponds to te trajectory turning around and eading back toward B. Note tat 1 and 2 become closer during te silent pase and jump up, suc tat te trajectory subsequently travels close to L (black dased line). (B) Symmetric spiking solution for g tonic e =.63. Te red and blue curves are te nullclines A 1 and A 2, respectively, of (6). Te inset sows ow te sample trajectory sown approaces te fixed point p were A 1 A 2 occurs. In teory, tere could be a direct transition from SB to AS, if q A,q B were to enter O before p as g tonic e were increased. However, our numerical simulations indicate tat bot te AB and te AS regions terminate togeter, at g tonic e 7.5. Te scematic diagram in Figure 15 illustrates te transition from SB AB AS SS to SB SS tat occurs as g syn e is raised troug 7.5. As noted above, Figure 14 gives examples of te dynamics in O

21 THE DYNAMIC RANGE OF BURSTING 1127 q B p p q A SB AB AS SS SB SS g tonic e g tonic e g syn e < 7.5 g syn e > 7.5 Figure 15. A scematic diagram sowing ow te cange occurs in te bifurcation diagram for te dynamics of (6) inside O as g syn e crosses troug 7.5. Te orizontal black lines indicate te presence of te fixed point p in O, wile te black curves denote te fixed points q A and q B; stable fixed points are given by solid curves, wile unstable ones are indicated by dased curves. As g syn e increases toward 7.5, te AB and AS regions become narrower, until tey cease to exist togeter at g syn e 7.5. on bot sides of te SB SS transition for g syn e = Details of activity patterns witin regions. Te analysis illustrated in Figure 11 caracterizes a pat in (g tonic e,g syn e ) space along wic all four activity states occur as g tonic e increases. Wile tis same set of transitions arises for an interval of g syn e values, subtle differences in activity witin te same state may emerge for different (g tonic e,g syn e ) values, based on wat appens wen trajectories leave O. We next consider a mecanism underlying tese differences, and ten we briefly return to te issue of syncronization of bursting solutions. To understand ow differences in te details of asymmetric bursting can arise, note tat te cells are only coupled troug te variables s i, eac of wic depends on v j. For eac i, we can consider te (v i, i ) bifurcation diagram generated by te dynamics of (v i,n i ) wit i as a bifurcation parameter and wit s i also frozen. Tis will yield a picture similar to tose in Figure 2, wit te value of s i (for fixed g syn e ) selecting te relative positions of P and of te omoclinic orbit tat terminates P; for s i treated as a fixed constant in tis way, canges in s i also affect te position of S, unlike in te rigt panel of Figure 2. In reality, te s i ave fast dynamics, so one can tink of te (v i, i ) bifurcation diagram as jumping around rapidly, driven by canges in s i, but at eac instant in time, tere exists an appropriate diagram.

22 1128 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER Based on tis collection of structures, for eac fixed (g tonic e,g syn e ), tere exists a curve H of (, s)-values suc tat at eac value on te curve, te (v, n)-system as a omoclinic orbit. Note tat ds/d is negative on H (cf. Figure 2); an example appears in Figure 16A. For a trajectory of (6) to exit O troug te boundary curve B, it is necessary but not sufficient tat te (, s) coordinates for one cell sould move to te nonoscillatory side of H, were no periodic oscillations are supported by te (v i,n i ) dynamics. If one cell, say, cell 1, does cross H, ten te input from te oter cell, via s 1, may pull it back across, causing regular network oscillations to continue. If tis does not appen, ten te trajectory of cell 1 will be attracted to te lower branc of S, causing s 2 to drop. One possibility is tat tis loss of synaptic input will pull cell 2 across H as well, terminating te pair s oscillations. Tis is exactly te case in wic exit from O troug variation of one or bot of te parameters 1, 2 yields an abrupt transition from tonic spiking to quiescence in te fast subsystem (1) i,(3) i, (4) i, as illustrated in Figure 16B. For g syn e = 3 and oter intermediate values of g syn e,at least away from a small neigborood of te AB AS transition, tis possibility is realized. Correspondingly, wen trajectories of te slow averaged equations (6) tat start from initial conditions in O leave O, te fast variables stop oscillating altogeter and te subsequent silent pase dynamics of te full system (1) i (4) i causes ( 1, 2 ) to grow. Eventually, oscillations return, wit ( 1, 2 ) somewere in O, and te dynamics of (6) becomes relevant again. A B 1 omoclinic curve cell 1 cell s v t Figure 16. Exit from O for an asymmetric bursting solution wit g syn e =3,g tonic e =.8. (A) Wen cell 1 crosses te omoclinic curve (H in te text) from above to below, it pulls cell 2 down wit it, resulting in a cessation of oscillations, wit s 1,s 2 wile 1, 2 increase (in te silent pase). Te arrows sow te direction of time evolution for cell 1, as it transitions from its final oscillation (down arrow) to te silent pase (orizontal arrow) to its return to te active pase (up arrow). Te evolution for cell 2 is similar. (B) Correspondingly, te transition across B yields an abrupt switc from oscillations to quiescence in te dynamics of te fast subsystem (1) i,(3) i, and (4) i. Here, a crossing of B was implemented by decreasing 1 from.169 to.168, at time 999, wit 2 =.18. Te v time course is only sown for one cell; it was qualitatively similar for te oter cell. An alternative scenario, wic arises most prominently for small g syn e, is tat even for s 2 =, cell 2 can continue to oscillate. In tis case, it is possible tat successive oscillations of cell 2 can cause cell 1 to resume oscillating after cell 1 crosses H, even toug a single

23 THE DYNAMIC RANGE OF BURSTING 1129 A.2 B cell 1 cell s.1 omoclinic curve 1 v t C 1 D v t Figure 17. Te sustained oscillations of one cell can rescue te oscillations of anoter cell to wic it is coupled, as sown ere for g syn e =2,g tonic e =.88. (A) Even wen cell 1 crosses from above to below te curve of omoclinic orbits (H in te text), cell 2 continues to oscillate. Te coupling from cell 2 pulls cell 1 back across H, were it resumes oscillations. (B) Te transition across B yields a switc from tonic spiking to irregular sustained oscillations in te dynamics of te fast subsystem (1) i,(3) i, and (4) i. Here, a crossing of B was implemented by decreasing 1 from.1693 to.1692 at time 999, wit 2 =.25. Te v time course is only sown for one cell; it was qualitatively similar for te oter cell. (C) Te dynamics of te full system sows asymmetric bursting wit sort interburst intervals, wit a cange in burst cycle occurring wen cell 2 (blue) fires two consecutive spikes. (D) Tis asymmetric bursting solution (green) remains very close to B (black) in te ( 1, 2)-plane; te red and blue curves sow te nullclines A 1, A 2 of (6) as tey terminate on B. oscillation of cell 2 does not; in particular, as seen in Figure 17A, cell 2 never crosses H. Wen tis form of rescued oscillation arises in te fast subsystem (1) i,(3) i,(4) i wit 1, 2 fixed, as sown in Figure 17B, tis does not qualify as regular tonic spiking, and tus by our definition ( 1, 2 ) do not lie in O. Furter, tis effect yields bursting solutions of te full system (1) i (4) i featuring a very small interburst interval, in wic one cell never spends

24 113 J. BEST, A. BORISYUK, J. RUBIN, D. TERMAN, AND M. WECHSELBERGER time in te silent pase; see Figure 17C. Figure 17D sows a corresponding example of an asymmetric bursting solution wit g syn e = 2, projected onto te ( 1, 2 )-plane, wic differs from tat sown in Figure 11B for g syn e = 3 in tat te projection of te burst trajectory onto ( 1, 2 ) stays very close to B for all time. If te net drift in ( 1, 2 ) during suc a solution were actually zero, ten tere could exist a bursting solution of te full system (1) i (4) i tat never enters O. In summary, te transition across B corresponds to different fast subsystem dynamics for different (g tonic e,g syn e ) values, leading to differences in te details of te asymmetric bursting tat results. We empasize tat te existence of suc possibilities does not affect te validity of our analysis of transitions between bursting and tonic spiking; as long as tere is no stable fixed point of (6) ino, regular tonic spiking of te full system will not occur. Finally, from te idea of considering canges in bifurcation structure as bot s and vary, it becomes clear tat te syncronization of te cells in bursting solutions relates in part to a form of fast tresold modulation (FTM) [29, 33]. In teory, FTM can act at eiter or bot of te jump down to te silent pase and te jump up to te active pase. Based on our simulations, most of te compression toward syncrony occurs in te silent pase and in te jump up to te active pase of eac burst (e.g., bottom panel of Figure 7). Wen one cell, say, cell 1, reaces te lower knee of its corresponding critical point curve S 1 and begins to oscillate, te coupling from cell 1 to cell 2 sifts S 2 to te left, advancing te jump-up time of cell 2. Tis can allow compression in te -coordinates of te cells relative to te uncoupled case, in wic 2 would ave ad to evolve to larger values before jumping up. During tis additional evolution in te uncoupled case, 1 would ave been decreasing, leading to an approximately constant magnitude of 2 1 before and after jump-up. Tere is also compression in te silent pase, wic in teory could be analyzed using te slow dynamics [32, 21]. In te AB case, after tis compression and FTM bring trajectories toward syncrony, tey are pused away from te axis of symmetry L in te active pase by te flow of (6) ino. In te SB case, no suc instability occurs to counteract syncronization. It remains to explore te full details of syncronization of bursts in te SB region in te full 8-dimensional system (1) i (4) i. 4. Burst duration and interburst interval of coupled pre-bötc cells. Our analysis in te previous section explained te dynamic range of bursting of coupled pre-bötc cells. We next give an explanation for te numerically observed canges in burst duration (active pase) and interburst interval (silent pase) under variations of (g syn e,g tonic e ), as sown in Figure 18. Te features of te different bursting regimes, symmetric (SB) and asymmetric (AB), are critical for understanding ow te burst duration is determined Te symmetric bursting regime. Te onset of bursting is described in section 3.1 and is due to te crossing of te -nullsurface G from te stable lower branc to te unstable middle branc of S. Recall tat tis crossing is almost independent of g syn e, because te position of te lower knee of S depends only very weakly on g syn e, and appens at g tonic e.26. After te onset of bursting, we are in te symmetric (or top at) bursting regime, wic was analyzed in section 3.3. If we fix g tonic e in tis SB regime and increase g syn e, ten te burst duration as well as te interburst interval increase. Te reason is te following: as g syn e increases, te Hopf

25 THE DYNAMIC RANGE OF BURSTING g syn e (ns) 6 4 silent 2 symmetric bursting (SB) asymmetric bursting (AB) g tonic e (ns) Figure 18. Simulated burst durations and (inter)burst intervals from Butera, Rinzel, and Smit [2]. (A) Te color-coded plot sows ow burst duration in a pair of coupled pre-bötc cells canges wit g tonic e and g syn e. Te transition curves tat we ave computed for te onset and offset of symmetric and asymmetric bursting, from Figure 13, are sown for comparison, illustrating in particular tat te transition from symmetric to asymmetric bursting is responsible for te abrupt increase in burst duration wit g tonic e. (B) Interburst interval increases wit g syn e and decreases as g tonic e increases. Te color-coded plots of burst duration and interburst interval sown ere appeared in [2] and are used wit permission of te American Pysiological Society. point as well as te stable branc of periodic orbits AP S corresponding to te top at burster move to te left, wile te lower knee of S is fixed, increasing te bistable region of te top at burster; an example appears in Figure 19A. Tus, solutions stay longer in bot te active pase and te silent pase for increased g syn e. If we fix g syn e in te SB regime and increase g tonic e, ten te lower knee of S moves to te left. Te Hopf point and te stable branc of periodic orbits AP S associated to te top at burster move to te left as well, but tey do so more slowly, as seen in Figure 19B and analogously to wat is sown in te bottom left panel of Figure 2. Tis causes a net decrease in te size of te bistable region. Furter, tis smaller bistable region is moved to te left, wit AP S becoming closer to te -nullsurface G and te lower branc of S becoming farter from G (see, e.g., te bottom row of Figure 3). Tese canges cause bot a slower drift in te active pase and a faster drift in te silent pase. It follows immediately tat for increased g tonic e te interburst interval decreases, because te bistable region gets smaller and te drift