J Comput NeurosciDOI 10.1007/s10827-007-0026-x

Capturing the bursting dynamics of a two-cellinhibitory network using a one-dimensional map

Victor Matveev · Amitabha Bose · Farzan Nadim

Received: 13 November 2006 / Revised: 4 February 2007 / Accepted: 13 February 2007© Springer Science + Business Media, LLC 2007

Abstract Out-of-phase bursting is a functionally im-portant behavior displayed by central pattern gener-ators and other neural circuits. Understanding thiscomplex activity requires the knowledge of the in-terplay between the intrinsic cell properties and theproperties of synaptic coupling between the cells. Herewe describe a simple method that allows us to investi-gate the existence and stability of anti-phase burstingsolutions in a network of two spiking neurons, eachpossessing a T-type calcium current and coupled byreciprocal inhibition. We derive a one-dimensional mapwhich fully characterizes the genesis and regulation ofanti-phase bursting arising from the interaction of theT-current properties with the properties of synap-tic inhibition. This map is the burst length returnmap formed as the composition of two distinct one-dimensional maps that are each regulated by a differentset of model parameters. Although each map is con-structed using the properties of a single isolated modelneuron, the composition of the two maps accuratelycaptures the behavior of the full network. We analyzethe parameter sensitivity of these maps to determinethe influence of both the intrinsic cell properties and the

Action Editor: John Rinzel

V. Matveev (B) · A. Bose · F. NadimDepartment of Mathematical Sciences, New Jersey Instituteof Technology, Cullimore Hall, University Heights,Newark, NJ 07102-1982, USAe-mail: matveev@njit.edu

F. NadimDepartment of Biological Sciences, Rutgers University,Newark, NJ 07102, USA

synaptic properties on the burst length, and to find theconditions under which multistability of several burst-ing solutions is achieved. Although the derivation of themap relies on a number of simplifying assumptions, wediscuss how the principle features of this dimensionalreduction method could be extended to more realisticmodel networks.

Keywords Half-center bursting ·T-type calcium current · Poincaré return map ·Multistability · Dimensional reduction

1 Introduction

Electrical bursting activity is a widely observed phe-nomenon in neurons and hormone secreting cells(Selverston and Moulins 1986; Bertram and Sherman2000; Sohal and Huguenard 2001; Llinas and Steriade2006). Considerable effort has been made to clas-sify different types of bursting activity resulting inmany detailed mathematical models (Izhikevich andHoppensteadt 2004; Coombes and Bressloff 2005).Bursting activity often depends on properties of thenetworks in which these neurons lie. For exam-ple, bursting in reciprocally coupled inhibitory net-works can arise from ionic currents that producepost-inhibitory rebound, such as the low-thresholdtransient calcium current, known as the T-current(Perkel and Mulloney 1974; Wang and Rinzel 1994;Huguenard 1996).

Reciprocally inhibitory networks are common circuitelements in many neuronal systems, from the mam-malian neocortex and hippocampus to the invertebrate

J Comput Neurosci

central pattern generators, and play a crucial role inrhythmogenesis (Traub et al. 1996; Wang and Buzsaki1996). Central pattern generators, in particular, makeuse of reciprocal inhibition between pairs of neurons orpopulations of neurons as a general mechanism for pro-ducing out-of-phase oscillations (Satterlie 1985; Marderand Calabrese 1996). Consequently, reciprocal inhibi-tion has been the subject of many theoretical studiesthat have demonstrated the existence of a multitudeof possible network behaviors arising through distinctmechanisms (Skinner et al. 1994; Van Vreeswijk et al.1994; Wang and Rinzel 1994). In many cases, recip-rocally inhibitory networks demonstrate dynamicallycomplex outputs such as irregular oscillations or mul-tistability of distinct modes of activity (Terman et al.1998).

When considering networks or even pairs of recip-rocally coupled neurons, the high dimensionality of theensuing set of equations is often an obstacle in analyz-ing the dynamics of the system. Various methods suchas averaging and singular perturbation theory haveproved useful for reducing the dimensionality of largersystems of equations in a variety of network models(Butera 1998; Lee and Terman 1999; Medvedev 2005).These approaches involve tracking the behavior of asmaller number of variables in a phase space of lowerdimension than the original system. An alternative ap-proach has been to ignore certain state variables of thesystem, and instead track quantities that are experimen-tally and mathematically measurable. The inter-spikeinterval (ISI), the time between successive spikes ofa neuron, is one such quantity and Ermentrout andKopell (1998), for instance, use this approach to derivea one-dimensional map based on the ISI that theyuse to prove the existence and stability of synchronousspiking solutions in a hippocampal network.

We are interested in exploring the anti-phase burst-ing activity arising from the interplay between the low-threshold calcium T-current and synaptic inhibition ina reciprocally coupled network of two inhibitory neu-rons. In such a model, we observe that several stablebursting states may exist for the same set of para-meters (Fig. 1). Each of these states has a differentnumber of spikes per burst and thus different cycleperiods. In this study, we provide an analytically andnumerically tractable method for proving the existenceand stability of these anti-phase bursts. This methodexploits time-scale separations of variables to constructa one-dimensional Poincaré map whose fixed pointscorrespond to periodic anti-phase solutions. The map isthe burst length return map which tracks the length of aburst from one cycle to another, and, as such, does not

directly track the state variables of the governing setof equations. Interestingly, the construction of this mapdoes not even require the coupled network. Instead, itis constructed as the composition of two different maps,each of which can be derived by studying the propertiesof a single uncoupled cell. By restricting the two mapsto parameter ranges that are consistent with anti-phasebursts, we can use these single-cell maps to characterizethe solutions of the coupled two-cell network.

A primary advantage of our approach is that weare able to pinpoint how certain key parameters ofthe model affect each of the single-cell maps which,in turn, affect both the existence and the stability ofanti-phase solutions. In situations where this solutionis unstable, the map is used to construct higher orderperiodic (or possibly chaotic) solutions. Thus, we showthat a wide range of dynamic outputs of the reciprocallycoupled network can be examined by analyzing theone-dimensional Poincaré map.

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Fig. 1 A two-cell network of reciprocally inhibitory Morris–Lecar neurons with a T-current exhibits multiple stable periodicbursting solutions characterized by different number of spikesper burst for a single parameter set

J Comput Neurosci

2 Model

2.1 Single-cell dynamics

We first describe the dynamics of a single neuron. Thetwo cells are assumed to be identical, each modeledas a two-variable Morris–Lecar oscillator (Morris andLecar 1981), which we have modified to include a low-threshold Ca2+ current (T-current: IT), described indetail further below. The spiking of the model cell re-sults from the interplay between the dynamics of mem-brane potential v, and the recovery variable w, whichdescribes the activation of the potassium current:⎧⎪⎪⎪⎨

⎪⎪⎪⎩

Cm v′ = Iapp − gL [v − EL] − gCa m∞(v)

[v − ECa] − gK w [v − EK] − IT

w′ = φw∞(v) − wK

τw(v)

(1)

where Iapp is a constant applied current, gL, gCa andgK are conductances and EL, ECa and EK are reversalpotentials for the leak, fast high-threshold calcium andpotassium currents, respectively (Fig. 2). Both m∞ and

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Fig. 2 Single-cell dynamics: the Morris–Lecar oscillator (Eq. (1)with IT = 0). (a) In the absence of the T-current each uncoupledcell produces slow tonic spiking activity. (b) The phase-planediagram demonstrates the periodic spiking arising from the in-terplay between the dynamics of the membrane potential v andthe K+ current activation variable w. The model parameters arechosen to produce Type-I excitability. The dotted lines shows theT-current inactivation threshold vh

w∞ are monotonically increasing sigmoidal functionsof cell potential v. The functions and parameters aredescribed in detail in Appendix 1.

The T-current in Eq. (1) models the low-thresholdCa2+current with both activation and inactivation gat-ing. Since the T-current activation is faster than its in-activation kinetics (Huguenard and McCormick 1992),we make the simplifying assumption that the activa-tion variable is instantaneous and is described as asigmoidal function of the membrane potential a = a(v).The inactivation variable of the T-current is given bythe dynamic variable h. For the mathematical analysiswe make the further simplifying assumption that theactivation function and the steady-state inactivationfunction are Heaviside functions of the membrane volt-age. We discuss the implications of relaxing this as-sumption in the Discussion section. Also, the activationand steady-state inactivation curves are smoothed outto sigmoidal form in the simulations, as described inAppendix 1. The T-current is therefore described as

IT = gTah[v − ECa] (2)

a = H(v − vh) (3)

h′ ={

(1 − h)/τlo v < vh

−h/τhi v > vh(4)

where H(v) is the Heaviside function. The inactivationvariable h decreases to zero if the cell is depolarizedabove vh (inactivation), and increases to 1 if a cell is hy-perpolarized below vh (de-inactivation). We assume forsimplicity that the T-current activation and inactivationthresholds are equal.

Our choice of model parameters corresponds toType-I excitability (in the absence of synaptic couplingand the T-current): each cell spikes at a low baselinefrequency, as shown in Fig. 2(a). The phase portrait ofeach cell in the absence of the T-current is illustratedin Fig. 2(b). The nullclines for Eq. (1) are obtained bysetting the right-hand side equal to zero. When IT = 0the v-nullcline is cubic shaped, while the w-nullcline issigmoidal (see Fig. 2(b).) Note that the T-current inac-tivation threshold vh is chosen to be below the minimalvalue of v of the spiking trajectory of an uncoupledcell (Fig. 2). Therefore, the T-current is completelyinactivated during tonic spiking (h = 0), and plays norole in the dynamics of the uncoupled cell.

The main effect of IT is to produce a rebound burstof spikes in response to hyperpolarization, as shownin Fig. 3. If a hyperpolarizing current pulse lowers thecell potential below vh (horizontal bar in Fig. 3(a)),then a gradual de-inactivation of the T-current occurs,

J Comput Neurosci

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Fig. 3 Post-inhibitory rebound induced by the T-current. (a)Time course of cell potential (top panel), IT inactivation h(middle panel), and IT activation a (bottom panel), in responseto a hyperpolarizing pulse (horizontal bar) applied during t ∈[50, 80] ms. Hyperpolarization pushes v below IT inactivationthreshold, vh (dotted line), allowing IT to de-inactivate (h grows).When hyperpolarization is removed, IT is activated (bottompanel). The decrease in spike frequency during the reboundburst is caused by the decay of h (inactivation). (b) Effect of ahyperpolarizing pulse in the v-w phase plane. The positions onthe v-nullcline labeled as 1, 2, 3 and 4 correspond to the accord-ingly labeled time points in (a). The hyperpolarizing pulse lowersthe v-nullcline (2), creating a stable hyperpolarized equilibrium(circle). Note that the equilibrium lies to the left of vh, allowing ITto de-inactivate (h grows). When hyperpolarization is relieved,the v-nullcline shifts up (3), leading to a burst of action potentials(spike frequency is proportional to the elevation of the left kneeof the v-nullcline). During the burst IT gradually inactivates, andthe nullcline goes back to its unperturbed location (1, 4). Thetrajectory of the rebound burst sweeps the gray area

i.e. h increases. When such external hyperpolarizationis removed, the cell will depolarize above vh, causingthe activation of the T-current (a grows), which resultsin a burst of action potentials. In phase space, theactivation of the T-current lifts the v-nullcline aboveits resting position, increasing the spike frequency dueto larger distance of the periodic trajectory from thelower branch of the w-nullcline (Fig. 3(b)). During therebound burst, the T-current gradually inactivates (h

decreases) with time constant of inactivation equal toτhi (see Eq. (4)). This inactivation manifests itself in thedecrease of spike frequency during the burst. In phasespace, inactivation corresponds to the gradual loweringof the v-nullcline (Fig. 3(b)), from the elevated position(labeled 3) to the rest position (labeled 1, 4).

2.2 Effect of synaptic inhibition

The network we consider includes two neurons de-scribed by Eqs. (1–4), reciprocally coupled by synap-tic inhibition. The model for the synaptic current isadapted from Bose et al. (2001), and is given by (i, j =1, 2, i �= j):

Isyn = −gsyn si [v j − Einh]

s′i =

{(1 − si)/τg vi > vθ

−si/τsyn vi < vθ(5)

where vθ is the spike threshold that intersects themiddle branch of the v-nullcline and si is the gatingvariable describing the synaptic input from neuron i toneuron j. The synaptic growth (onset) time constant isshort compared to the spike width, thus allowing si toreach its maximal value with each action potential. Weconsider the case of intermediate to slow synaptic decaytime, with τsyn exceeding the spike width (τsyn > τw).This results in the following full set of model equations(i, j = 1, 2, i �= j):

Cm v′j = Iapp − IL(v j) − ICa(v j)

−IK(v j, w j) − IT(v j, a j, h j) − Isyn(si, v j)

w′j = φ

w∞(v j) − wK

τw(v j)

h′j =

{(1 − h j)/τlo v j < vh

−h j/τhi v j > vh(6)

s′j =

{(1 − s j)/τg v j > vθ

−s j/τsyn v j < vθ

a j = H(v j − vh)

The influence of inhibitory synaptic input to a cell isopposite to that of the T-current. In the v − w phaseplane, the effect of inhibition from cell i to j is to lowerthe cubic shaped v-nullcline, which causes a decreasein the frequency of spiking. If inhibition is sufficientlyweak (small gsyn), both cells will continue spiking, andtheir spiking trajectories are only slightly perturbedby the coupling. In this case the inhibitory input isnot enough to hyperpolarize the partner cell below

J Comput Neurosci

vh, and therefore the T-current remains inactive andplays no role in cell dynamics. For stronger synapticcoupling strength gsyn, however, a spike in one cell maybe sufficient to hyperpolarize the other cell below itsT-current de-inactivation threshold. In this case, theT-current will play a role in the network dynamics and,in particular, will make anti-phase bursting possible.

A quantity of central interest to us is the inter-spikeinterval (ISI). The ISI measures the time betweensuccessive spikes of the same neuron. We will useISItonic to denote the inter-spike interval of the tonicspiking uncoupled cell. We choose parameters so thatthis value is relatively large, as compared to the shorterISIs generated by the bursting mechanism describedbelow.

3 Results

3.1 Half-center (anti-phase) bursting

The IT -induced post-inhibitory rebound burst mech-anism can work synergistically or cooperatively withthe synaptic inhibition to entrain the network into a

periodic anti-phase bursting state (Fig. 4; see Perkeland Mulloney 1974, Huguenard 1996, and Destexheand Sejnowski 2003, for related work). This periodicbursting is a half-center oscillation where the two cellsare active out of phase with one another (see Fig. 4(a)).The half-center oscillation requires a sufficiently strongsynaptic coupling gsyn, so that a burst of one cell (say,cell 1) provides enough synaptic current to hyperpolar-ize the postsynaptic cell (cell 2) below vh, causing itsT-current to gradually de-inactivate (h2 grows). At thesame time, the T-current of cell 1 gradually inactivates(h1 decreases), and its ISIs grow larger, approachingthe uncoupled cell’s intrinsic spiking period, ISItonic.This increase in ISI eventually allows the inactive cell2 to escape from inhibition. This happens when theinter-spike interval increases beyond a certain criti-cal value, ISI (discussed in detail below). Once thesuppressed cell escapes from inhibition, its potentialincreases beyond vh. Since h2 is now non-zero, the IT

current rapidly activates, producing a burst of spikes,terminating the burst of cell 1, and hyperpolarizing itbelow vh. The process then repeats.

The escape mechanism underlying the burst termi-nation can be understood by considering the phase-

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Fig. 4 Periodic anti-phase bursting of the coupled system ofequations. (a) Time traces of the membrane potentials (toppanel), IT inactivation, h j (middle panel), and the synapticstrength variables of the two cells, s j (bottom panel). (b) Phaseplane dynamics of the network activity. The v-nullcline of thebursting cell gradually moves down as h decreases (inactivation).

The inhibitory synaptic input from the bursting cell lowers thev-nullcline of the postsynaptic cell, trapping the cell at an equilib-rium (filled circle). Double arrows indicate the oscillatory move-ment of the v-nullcline and the equilibrium point with each spikeof the bursting cell. The open circle on the v = vh line marks theescape point of the suppressed cell (IT activation threshold)

J Comput Neurosci

plane dynamics shown in Fig. 4(b). The v-nullclineof the bursting cell (thin black curve) is elevated,but gradually descends as IT inactivates (h1 decreases,black curve in middle panel of Fig. 4(a)), resulting ina gradual increase of the inter-spike interval for thiscell (black curve v1 in top panel of Fig. 4(a)). Theinhibitory synaptic input from the bursting cell keepsthe v-nullcline of the suppressed cell in a low position(thick black curve in (b)). Because of our assumptionthat τsyn > τw(v), the trajectory of the suppressed celllies in a neighborhood of the intersection of the v andw-nullclines (filled circle in (b)). However, this inter-section point moves left and right (double horizontalarrow in (b)) as the v-nullcline of the suppressed cellmoves up and down with each spike of the burstingcell (double vertical arrow). The nullcline moves downquickly which each spike, and moves up somewhatslowly following the dynamics of the synaptic variable,s1(t), shown in the bottom panel of Fig. 4(a) (blackcurve). Accordingly, the membrane potential v2 of thesuppressed cell oscillates up and down (top panel in(a)), following this nullcline movement. As the ISIof the bursting cell increases, s1(t) decays to smallerand smaller values during the inter-spike interval, andthe potential of suppressed cell moves closer to theT-current activation threshold (dotted vertical line,open circle in (b)). When the ISI is large enough toallow v2 to reach vh, the nullcline of the suppressed cellshifts up abruptly, due to the activation of its T-current,and the cell escapes from inhibition. The resulting burstof the previously suppressed cell terminates the burstof the active cell. The process then repeats with the twocells switching their roles.

The condition for escape of cell 2 can be describedanalytically by finding when it reaches the T-currentactivation threshold vh. This occurs for a specific valueof s, denoted s, that satisfies:

Itot(vh, w, s) = 0w = w∞(vh)

where Itot = Iapp− IL(v)− ICa(v)− IK(v, w)− Isyn(v, s).Geometrically, this condition can be interpreted as thevalue of s at which the v- and w-nullclines intersect onthe line v = vh (Fig. 4(b)). Solving for s and using thefact that w∞(vh) ≈ 0, we find

s = Iapp − gL[vh − EL] − gCam∞(vh)[vh − ECa]gsyn[vh − Einh] (7)

This value s is closely related to the ISI. Namely, thereexists a value ISI such that if s(0) = 1, then s(ISI) = s.Using Eq. (5), ISI satisfies:

ISI = −τsyn ln s (8)

Thus, ISI corresponds to the amount of time it takesa cell starting on the “s = 1” v-nullcline to reach theactivation threshold vh. Note that ISI is completelyindependent of the T-current properties since the sup-pressed cell feels no influence of IT (a = 0 for v < vh).

We now quantify the definition of bursting givenearlier. We say that a cell is in the bursting regime ifit exhibits a series of spikes for which the ISI < ISI.Thus the term ISI allows us to use aspects of thenullcline geometry of the Morris–Lecar model to definethe length of the burst.

3.2 The one-dimensional Poincaré map

The set of equations describing the dynamics of the two-cell network (Eq. (6)) forms a system of eight first orderdifferential equations and two algebraic equations (fora1 and a2). Despite the high-dimensional phase space,it is possible to construct a one-dimensional Poincarémap which can predict the existence and stability ofanti-phase burst states depicted in Fig. 1. The Poincarémap is the return map for the lengths of successive burstsand also determines the number of spikes per burst.It is constructed as the composition of two differentsingle-cell maps. The first map, denoted L = F(h�), de-termines the burst length of a cell as a function of h�,the level of T-current de-inactivation (h) at the begin-ning of the burst. The second map, denoted h� = G(L)

determines the amount of T-current de-inactivation asa function of the inter-burst length. We then define thereturn map P(L) by the relationship:

P(L) = F(G(L))

We will show that a fixed point of this map correspondsto an anti-phase bursting solution for the coupled net-work given by Eq. (6). Both of the maps F(h�) andG(L) are constructed using information obtained fromthe dynamics of a single cell, coupled with restrictionsimposed by the geometry of our model and by the con-dition of existence of a periodic half-center oscillation.

3.2.1 Dependence of burst length on the h� value:L = F(h�)

To reconstruct the relationship L = F(h�) betweenthe burst length and h�, the level of T-current de-inactivation at the beginning of the burst, we study asingle uncoupled cell. Suppose that at t = 0 the cellbegins a burst with h(0) = h�. We will find the durationof each inter-spike interval ISIn for a given value of h�,and then sum them up to obtain the burst length. Notethat the length of each ISIn is uniquely determined by

J Comput Neurosci

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Fig. 5 Dependence of burst length on h, the level of T-currentde-inactivation. (a) The relationship T(h) between ISI and hin a single cell. Filled circles mark the values of h and thecorresponding ISIs for a burst characterized by the initial con-dition h(0) ≡ h� = 0.12, and shown in panel (b). (b) Membranepotential (top panel) and T-current inactivation level h (bottompanel) for a numerically-generated burst with h(0) = 0.12 shownfor illustrative purposes. Filled circles label the value of h at thebeginning of each of the 10 inter-spike intervals (cf. (a)). (c)Dependence of each inter-spike interval on h�, the value of h at

the start of the burst. Each curve was generated using the T(h)

curve in panel (a) and Eqs. (9–10). Note that ISI1 is the curveT(h) shown in panel (a). The horizontal dotted line denotes ISI.The highlighted section of each curve indicates the last ISI ata given value of h�. The vertical dotted line corresponds to theburst in panel (b). (d) The dependence of burst length on h�

is obtained by adding all ISIn in panel (c) at each value of h�,plus ISI. In (c), numeric labels indicate the index of last ISI inthe burst, while in (d), the integers indicate the number of spikesper burst

the value of h(t) right before the corresponding inter-val, hn. This relationship is the same for all ISIn’s, andis given by a monotone decreasing function ISI = T(h)

as shown in Fig. 5(a). This curve may be thought of as aspecial version of the f -I curve of the cell, capturing thedependence of spiking period ISI = 1/ f on the levelof depolarization provided by the T-current, which isproportional to h. The value ISI1 = T(h�) yields thelength of the first inter-spike interval. The value of h atthe beginning of the second interval, h2, is determined

by the inactivation kinetics of h (see Eq. (4)): h2 =h� exp{−ISI1/τhi)}. Since h2 is smaller than h1 ≡ h�,the second inter-spike interval, ISI2 = T(h2), is greaterthan the first. Subsequent ISIs are obtained similarly.Thus:

ISIn = T(hn)

hn = hn−1 exp{−ISIn−1/τhi}h1 ≡ h� (9)

J Comput Neurosci

The pairs (hn, ISIn) are labeled along the curve T(h) inFig. 5(a), for the particular case h� = 0.12, correspond-ing to the numerical simulation shown in Fig. 5(b).

It is useful to represent the duration of each ISIn

within a burst as a function of h at the very beginning ofthe burst, h�. As Eq. (9) illustrates, the dependence ofeach of the ISIn intervals on h� is governed by the sameT(h�) curve that determines ISI1. Therefore for n > 1,

ISIn(h�) = T(hn) = T(h� exp

{ − [ISI1(h�) + ISI2(h�)

+... + ISIn−1(h�)]/τhi})

(10)

The ISIn(h�) curves are plotted in Fig. 5(c). Thesection of each curve highlighted in bold shows the lastinter-spike interval of the burst for a given value of h�

(in Fig. 5(b), the last ISIn is ISI10). The vertical dottedline corresponds to the burst shown in panel (b). Notethat each curve in the panel is obtained by stretchingthe curve T(h) in panel (a) to the right, as described byEq. (10).

To determine the burst length for any particularvalue of h� we first sum all ISIn values that lie below thecritical value ISI (horizontal dotted line in Fig. 5(c)).Recall that the critical value ISI is determined by Eq.(8) and is used to indicate the termination of a burst. Tocomplete the definition of F(h�), we use the fact that, inhalf-center oscillations, exactly one cell is active at anytime. Thus, ISI needs to be added to the sum of theISIs to calculate the time from the last spike of one cellto the first spike of the other (see Fig. 5(b)). Therefore,

L= F(h�)=n∑

j=1

ISI j(h�)+ ISI, ISI j < ISI, j=1..n (11)

The function F(h�), shown in Fig. 5(d), has twoimportant features. First, it is piecewise continuous,and second, it is decreasing on the subintervals whereit is continuous. The latter observation is easy to un-derstand since the original ISI vs. h� curve is itselfmonotone decreasing. The function F(h�) is discontin-uous because as h� is increased, new spikes are addedto the burst at certain values of h� denoted hcr j . In fact,hcr j are the values of h at which the ISI j curves intersectthe ISI line. As h� increases through each value hcr j ,the inter-spike interval ISI j satisfies ISI j < ISI, result-ing in the addition of another spike. For example, inFig. 5(c), the vertical line at h� = 0.12 shows that theassociated burst has 11 spikes and 10 ISIs. Thus thelength of the burst in this case is given by

L =10∑

j=1

ISI j + ISI.

If we were to increase h� to the value 0.15, then thevertical line h� = 0.15 would intersect the curve ISI11 ata value smaller than ISI. Therefore a new spike wouldbe added to the burst since ISI11 < ISI. The new burstlength would be

L =11∑

j=1

ISI j + ISI (12)

The size of the jump discontinuity in the burst lengthat each hcr j equals the length of the added inter-spikeinterval, ISI j(hcr j) = ISI.

Since the monotonic decay of the continuousstretches of F(h�) reflects the monotonic increase inspike frequency with increasing h�, the rate of thisdecay is proportional to the value of conductance gT ,which multiplies h in the expression for the IT (Eq. (2)).In fact, the effect of the variation of the gT parametercan be obtained by re-scaling the h� axis in Fig. 5(c,d)(see Fig. 7(a) and discussion below). The value ofthe inactivation time constant τhi affects the slope andthe magnitude of F(h�) in a similar manner. Increas-ing τhi slows down the T-current inactivation, decreas-ing the difference between successive ISIs shown inFig. 5(a–b), and thereby increasing the number ofISIs that fit below the ISI burst termination limitfor any given value of h�. This corresponds to theleftward “squeeze” of the ISIn(h�) curves in Fig. 5(c)toward the ISI1(h�) curve, thereby increasing the mag-nitude of F(h�) by increasing the number of spikesper burst.

In contrast, as discussed above, ISI is only sensitiveto the fast spike kinetics and the synaptic couplingparameters, and does not depend on the characteristicsof the T-current. However, an increase in ISI wouldhave a similar effect to an increase in τhi. Namely, itwould increase the number of spikes per burst, andalso increase the slope of the continuous stretches ofF(h�). In turn, an increase in gsyn or the membrane timeconstant would also influence F(h�) through the effectof these parameters on ISI.

Note that our method for constructing the F(h�)

easily generalizes to any model of single-cell dynam-ics, as long as each neuron has a well-defined f − Icurve, and all intrinsic currents except the T-current arefast compared to the time scales of synaptic inhibitionand the T-current inactivation and de-inactivation. Thespecifics of the single-cell model would only affect theT(h) curve in Fig. 5(a,c), while the value of ISI canalways be determined numerically even if the analyticalapproximation given by Eqs. (7, 8) does not apply.

J Comput Neurosci

0 100 200 300 4000.1

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Fig. 6 The function h� = G(L) obtained from Eq. (15) (a) andthe fixed points of the Poincaré return map (b). (a) The de-pendence of h�, the IT inactivation level at burst initiation, onthe length of the preceding inter-burst interval, L, with τlo =200 ms and τhi = 20 ms. The gray box corresponds to the axislimits in panel (b). (b) The stable periodic bursting solutionsare obtained as the intersections of the piece-wise continuous

L = F(h�) map see Fig. 5(d), and the inverse of the map shownin panel (a), L = G−1(h�) (dashed curve). The intersection pointsof the two curves correspond to the two stable periodic solutions,with 19 and 20 spikes per burst, respectively. Note that panel (b)implicitly describes the Poincaré return map, which is given byP(L) = F(G(L))

3.2.2 Dependence of h� on the burst length:h� = G(L)

The second part of Poincaré map, the dependence h� =G(L) (see Fig. 6(a)) is easier to construct. It measureshow the T-current of an uncoupled cell de-inactivateswhen the cell is silent. It follows straightforwardly fromthe first-order kinetics of the T-current de-inactivationgiven by Eq. (4):

dhdt

= 1 − hτlo

when v < vh

Solving this equation with h(0) = ho yields:

h(t, ho) = 1 − (1 − ho)e−t/τlo . (13)

The value ho is the minimum value of h obtained atthe end of the burst (see Fig. 4). However, in thecoupled network, there is no a priori way to know thevalue of ho. Nonetheless, this value will be boundedfrom above by h, defined as the value of h� at theintersection of ISI1 and ISI, and bounded from belowby h exp(−ISI/τhi). For definiteness, let ho = h. Evalu-ating Eq. (13) at t = L, we obtain:

h� = G(L) ≡ h(L, h) = 1 − (1 − h) exp (−L/τlo) (14)

We note that for a periodic solution, we can replaceh with its equilibrium value, h = h� exp(−L/τhi). Thisyields an equilibrium condition for Eq. (14):

h� = G(L) = 1 − exp(−L/τlo)

1 − exp(−L/τlo − L/τhi)(15)

The map h� = G(L) is plotted in Fig. 6(a) usingEq. (15). Note that this map is only a function of theinactivation and de-inactivation time constants of theT-current, τhi and τlo, with a much greater sensitivityto τlo than to τhi (assuming τhi << τlo). Thus, the twomaps h� = G(L) and L = F(h�) are controlled by twodistinct sets of model parameters. As will be shown be-low, this fact greatly simplifies the understanding of theparameter control of the network’s bursting dynamics.

3.2.3 Fixed points of the Poincaré map:P = F(G(L))

Periodic solutions of Eq. (6) correspond to the fixedpoints of the Poincaré map Leq = P(Leq) ≡ F(G(Leq)).Introducing an equilibrium value of h�, h�

eq = G(Leq),this equilibrium condition can also be written as Leq =F(h�

eq) = G−1(h�eq). Therefore, geometrically the peri-

odic solutions correspond to the intersections of the

J Comput Neurosci

graph of L = F(h�), given in Fig. 5(d), and the inverseof the map G(L) given by Eq. (15) and shown inFig. 6(a). In Fig. 6(b) we superimpose these two mapsand show their intersection points. Interestingly, thetwo curves share more than one intersection point,demonstrating that multiple bursting solutions can existfor the same parameter values. Indeed, below we ver-ify numerically the existence of such multiple burstingsolutions.

Apart from providing the information about the pe-riodic network activity, the Poincaré map simplifies theparameter sensitivity analysis of the model, illustratingthe control of the burst duration by various modelparameters. This is achieved by examining the influenceof model parameters on the two individual parts of thePoincaré map, the curves F(h�) and G(L). Consider forexample the effect of a variation of gT , the maximal IT

conductance. As discussed above, G(L) (Eq. (15)) isdependent on τhi and τlo, but is not sensitive to gT . Thedependence on gT of F(h�) is straightforward: namely,an increase in gT “squeezes” the L = F(h�) curve tothe left (h� axis scales down with gT), as shown inFig. 7(a). As a result, as gT is increased, the Poincarémap intersections shown in Fig. 6(b) would lie alongthe segments of F(h�) with larger number of spikesper burst.

In contrast, varying τlo affects only G(L) and notF(h�) (Fig. 7(b)). The change in G(L) occurs becausean increase in τlo effectively moves the G curve up(the L axis scales up with τlo). Thus, by changingdifferent model parameters, different components ofthe Poincaré map are affected, leading to multiple de-grees of control of the existence of periodic burstingsolutions.

3.2.4 Stability of periodic bursting solutions

The stability of the periodic bursting solutions can bedetermined by checking the slopes of the graphs L =F(h�) and L = G−1(h�) at a point of intersection. Aburst solution will be stable if |dP/dL|Leq < 1. Thisderivative satisfies

dPdL

∣∣∣∣

Leq

= dFdh�

∣∣∣∣h�

eq

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∣∣∣∣

Leq

(16)

The derivative dG/dL is easy to calculate fromEq. (14):

dGdL

= (1 − h) exp(−L/τlo)

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Fig. 7 Control of burst properties by parameters gT and τlo.(a) An increase in the value of gT from 1.0 to 1.08 mS/cm2 isequivalent to a horizontal contraction of the F(h�) map, but doesnot affect the G(L) map. Filled circles indicate the stable periodicsolutions for the parameter values corresponding to Fig. 6; opencircles label the stable periodic solutions after the value of gT is

increased from 1.0 to 1.08 mS/cm2. Note that the bursting statewith 19 spikes per burst disappears and is replaced by a solutionwith 21 spikes per burst. (b) An increase in the value of τlo from200 to 220 ms causes a vertical stretch of the G(L) map, but hasno effect on the F(h�) map. Note that the bursting solution with20 spikes is replaced with a solution with 18 spikes per burst

J Comput Neurosci

which clearly shows dependence on the parameter τlo.It is also indirectly affected by other parameters suchas τsyn since this synaptic time constant affects ISI andhence h. The derivative dF/dh� is harder to explicitlyquantify. Note, however, that since each inter-spikeinterval in the burst is inversely proportional to h,each spike contributes to the slope of the F(h�) curve.Therefore, the relationship between L and h� becomesprogressively more steep at higher values of τhi. Like-wise, the derivative dF/dh� grows with increasing gT

since this parameter increases the number of spikes perburst as well.

It is straightforward to numerically calculate thevalue of dF/dh� at a point of intersection. This is whatwe did, for example, in Fig. 6(b) to conclude the sta-bility of the obtained solutions. We further tested howwell the Poincaré map, based on single cell dynamics,correctly determines the dynamics of the coupled net-work by numerically solving the latter. Figure 8 showsthe numerically reconstructed phase diagram of themodel network, indicating the stable periodic activitystates as a function of the IT conductance, gT (seeAppendix 2 for details of the numerical crawl algorithmused to construct the phase diagram). Note the overlapbetween the regions of stability of the various burstingsolutions, in agreement with the multistability exhibitedby the Poincaré map in Fig. 6(b). The phase diagramalso validates the dependence of burst length on thevalue of gT inferred above by analyzing the Poincaré

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Fig. 8 Comparison of the bifurcation diagrams for the burstingstates as a function of T-current conductance, obtained fromthe Poincaré map (black) and by the numerical crawl algorithm(gray). The dashed lines and circles mark the values gT = 1.0 andgT = 1.08 mS/cm2 and the corresponding stationary states shownin Fig. 7(a)

return map (Fig. 7). Namely, it shows that an increasein gT increases the number of spikes per burst. Notethat a more dramatic increase in gT would also reducethe number of co-stable bursting solutions (cf. Fig. 11).

The analysis above suggests that the parameters τhi,τlo, gT and τsyn affect both the existence and the sta-bility of periodic solutions. For example, the parametervariations shown in Fig. 7 changed the number of spikesper burst but preserved the stability of the equilibriumpoints, as determined from Eq. (16). However, a moredrastic increase in gT or a decrease in τlo can changethe slopes of the corresponding parts of the Poincarémap to the extent that the equilibrium points becomeunstable.

The loss of stability can be also achieved by in-creasing the value of τsyn, as is shown in Fig. 9(a).According to Eq. (8), increasing τsyn increases ISI.With our choice of parameters, the value ISI intersectsthe curve T(h) along a very steep portion of this curve(Fig. 5). Thus an increase in ISI can change the burstlength quite significantly, but not change the corre-sponding h value much. Thus the slope of the functionF(h�) is quite sensitive to changes in ISI. Figure 9(b,c)demonstrates this fact and shows the dynamics ofthe network for τsyn = 10 ms for which no stable peri-odic solution exists.

3.2.5 Relaxing some assumptions leads to greatermultistability

The analysis presented above relies on a number ofassumptions. Notably, we assumed that synaptic decayis faster than spiking dynamics, which allowed us toformulate the escape condition in terms of the criticalvalue of ISI (Eq. (8)). Further, we assumed that vh

lies just below the spiking threshold of each cell. Thisensures that the bursting cell falls below the T-current(de-)inactivation threshold immediately after its burstis terminated by the escaping cell, leading to exact anti-phase solutions demonstrated in Figs. 2–7.

Figure 10 shows that the above constraints may berelaxed without loss of the qualitative features of thePoincaré map. In this figure the value of vh has beenlowered from −47.5 to −52 mV, and the synaptic decaytime is 1 ms. The time trace in panel (a) shows thatthere is a significant lag (labeled φ) between the escapetime of one cell and the time that the inhibition pushesthe other cell below the T-current threshold. The mainresult of this lag is a shift in the value of h at the transi-tion to the silent state, i.e. when v drops below vh. Thuswe can no longer utilize Eqs. (14) or (15) to calculate

J Comput Neurosci

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Fig. 9 Dynamics of the network near unstable equilibria. (a)Bifurcation diagram as a function of the synaptic decay timeconstant τsyn obtained from the single-cell map L = F(h�). Blackcurves label the stable periodic solutions, while the gray curveslabel the unstable solutions. Note that the stability of periodicbursting is lost near τsyn = 4.8 ms. The dashed lines mark thevalue τsyn = 4 ms corresponding to Figs. 4–8 and the value

τsyn = 10 ms corresponding to panel (b). (b) The dynamics ofthe network for τsyn = 10 ms, shown as a cobweb and obtainedby parsing the numerically integrated solution to the networkequations, shown in panel (c). Note the irregular pattern of thepeak value of T-current de-inactivation (h∗) apparent in thetime trace shown in (c), corresponding to the (possibly) chaoticdynamics of the discrete Poincaré map shown in (b)

the map h� = G(L). Instead, we numerically calculatethe map using a feed-forward network and the steady-state approximation. Namely, we consider a networkin which cell 1 inhibits cell 2 with cell 1 starting in thebursting regime with h = h� and cell 2 in the silent statewith h = h� exp(−L/τhi) ≡ hL. For different values ofL for cell 1, we calculate the time φ(h�) before v2 fallsbelow vh. This is done using simulations of the feed-forward network. We then use this information andthe steady-state approximation to numerically and it-eratively solve for the function h� = G(L). The inverseof the function h� = G(L) is plotted in Fig. 10(b). Notethat this map has a number of discontinuities that ariseat values of L below which an extra spike is requiredto inhibit the suppressed cell below vh (see Appendix 3for more details). Interestingly, in this case, the discon-tinuity increases the number of stable periodic solutions

that exist. As is shown, as many as six stable periodicbursting solutions exist, corresponding to the six pointsof intersection of the two maps, for the same valuesof parameters. Several of these solutions are shownin Fig. 1. Figure 11 shows the numerically-generatedphase diagram obtained by using the crawler method(see Appendix 2) on the coupled network for the newparameter values. Note that multistability is achievedfor a range of gT values, which provides evidence thatthe Poincaré map formalism (this time partially basedon a feed-forward map as opposed to a single-cell map)is valid. A qualitatively different case of multistabilityof periodic bursting states has been previously ana-lyzed in the context of a single-cell bursting model ofCanavier et al. (1991, 1994) by Butera (1998), using anapproach which is very similar to the one adopted inthis work (see below).

J Comput Neurosci

(a)

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Fig. 10 Poincaré map for the case of faster synaptic decay (τsyn =1 ms) and lower T-current activation/inactivation threshold (vh =−52 mV). (a) shows the potential and T-current inactivation ofthe two cells for the nine spikes per burst solution (shown alsoin Fig. 1). Note that the T-current dynamics of the two cells arenot completely anti-phase, due to the lag between cell escapetime and the time of suppression of the other cell below vh (φ,double arrow). (b) The Poincaré map. Note that the G(L) map isdiscontinuous; each discontinuity corresponds to the value of Lbelow which an extra spike is required to inhibit the suppressedcell below vh (see Appendix 3). Dashed line corresponds to theG(L) curve obtained under assumption of half-center bursting.Four of the six stable periodic bursting solutions are shown inFig. 1(b)

4 Discussion

4.1 Summary

There have been countless biophysical models of neu-rons and networks in the past two decades that havesuccessfully reproduced outputs of their biologicalcounterparts and made numerous useful predictions(Hines et al. 2004). In almost every case the modelsthemselves produce quite complex outputs which, dueto the large number of parameters and variables in-

volved, are quite difficult to analyze. In the currentstudy, we take one such model, a two-cell anti-phasehalf center oscillator producing bursting activity, andanalyze it by focusing on outputs of interest and re-ducing the original system of differential equations toa one-dimensional map. We then show that the analy-sis of the one-dimensional map provides informationabout both existence and stability of solutions to theoriginal high-dimensional system.

We studied a network of two identical neuronscoupled with reciprocally inhibitory synapses. Eachneuron was constructed as a Morris–Lecar model toproduce the spiking activity together with a low-threshold calcium (T-type) current. The synapses areaction-potential mediated with fast rise and interme-diate decay time constants. We described an analyticmethod that produces a Poincaré return map formed asthe composition of two one-dimensional maps (F andG) of an interval. Each of these maps is constructed byusing the properties of a single isolated neuron: the mapG yields the level of inactivation/de-inactivation (h�) ofthe T-current at the transition to bursting as a functionof the time interval during which the cell is inactive; themap F produces the burst duration L as a function ofh�. These two maps depend on different sets of modelparameters.

4.2 Significance

There is a large body of literature on reciprocally cou-pled inhibitory networks. These networks arise natu-rally in central pattern generating circuits that controlrhythmic motor activity (Perkel and Mulloney 1974;Marder and Calabrese 1996; Grillner et al. 2005). Insuch networks, anti-phase bursting outputs of neuronsis ubiquitous and results in the out-of-phase activity ofopposing groups of muscles such as flexors and exten-sors. Inhibitory networks have also been studied in thecontext of synchronization, especially in the presence ofslowly decaying synapses (Wang and Rinzel 1992; VanVreeswijk et al. 1994; Terman et al. 1998). Our analysisonly deals with anti-phase solutions and specificallyomits the analysis of synchrony.

Post-inhibitory rebound in networks with T-currentshas also been studied quite extensively (Wang andRinzel 1994; Huguenard 1996; Destexhe and Sejnowski2003). While our study fits in this context, our primaryreason for including the T-current in our model was notto achieve post-inhibitory rebound—the model cellsspike in the absence of input and thus do not need therebound to fire—but rather to work cooperatively withthe synapse to control the burst duration. The presenceof the T-current enables the model to produce bursting

J Comput Neurosci

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Fig. 11 Numerically reconstructed bifurcation diagram as a func-tion of T-current conductance, gT , for the case of fast synapticdecay and low vh. Dotted line corresponds to the value of gT inFig. (10)

activity with a large range of burst periods dependingon the T-current parameters. In fact, the presence ofthe T-current is essential for multistability of solutionswith different burst durations. However, our analysiswould still apply if the bursting was due to the slowly-activating potassium current rather than the T-current.

The primary advantage of our method is that it pro-vides a simple tool for the identification of parametersthat control the existence and stability of the burstingsolutions. For example, the map h� = G(L) was shownto depend only on the T-current inactivation and de-inactivation time constants, while the map L = F(h�)

depended both on the T-current properties and theproperties of the synapse. This analysis also enabledus to show how parameters of the synapse and of theT-current interact to determine the burst length andthe number of spikes per burst. A second advantage ofour method is that it reveals the circumstances underwhich single-cell or feed-forward maps—constructedby tracking a network quantity through a feed-forwardsubnetwork—can be used to characterize the dynamicsof the coupled (feedback) network. For example, boththe symmetric anti-phase bursting solutions and themore complex chaotic anti-phase bursting (see Fig. 9)are found as fixed points and chaotic trajectories of theone-dimensional map, respectively. Note that the mapsconstructed in our study are of lower dimension, andthus easier to analyze, than standard Poincaré returnmaps that track the state variables. Yet, we demon-strated that the one-dimensional map yields excellentqualitative and quantitative agreement with solutions

of the full network obtained by numerically solvingthe full set of equations (Figs. 8, 9(b); cf. Figs. 10(b)and 11).

Reduction to lower-dimensional maps to prove ex-istence and stability of solutions has been used ina wide variety of contexts (Butera 1998; Lofaro andKopell 1999; Lee and Terman 1999; Medvedev 2005;Terman 1994). Most of these studies involve trackingstate-variables in a lower-dimensional phase space. Forexample, Terman et. al. have used geometric singularperturbation theory to derive low dimensional mapsto find oscillatory solutions in a variety of networks(Rubin and Terman 2000; Terman 1994; Terman et al.1998). Butera (1998) used a Poincaré return map toanalyze the multistability of bursting solutions andchaotic bursting displayed by the aplysia bursting neu-ron model of Canavier et al. (1991, 1994). In each ofthese studies, a map, defined on a lower dimensionalslow manifold, tracks a subset of the state variables ofthe full system. Medvedev has shown how to constructone-dimensional maps using singular perturbation the-ory and averaging to understand the complicated dy-namics of a bursting neuron model of Chay (Chay andRinzel 1985; Medvedev 2005). This method involvesderiving a return map for the single slow variable ofthe model and understanding its bifurcation structure.A notable exception to tracking state variables can befound in the work of Ermentrout and Kopell (1998)in which the inter-spike interval (ISI) is tracked toconstruct a one-dimensional map associated with a hip-pocampal network. This map is formed as a combina-tion of two feed-forward maps, similar in spirit to thetwo single-cell maps in our study. The map constructedby Ermentrout and Kopell tracks spikes of differentcells in a four-cell network and shows that synchro-nization can be closely tied to the existence of a spikedoublet from a specific cell in the network.

Our approach bears close resemblance to the above-mentioned work of Butera (1998), who obtained the re-turn map capturing the multistable bursting of a modelcell as a composition of two separate components rep-resenting the spiking and the silent states of the burst-ing cell, respectively, In both studies, the componentrelated to the active phase is a piece-wise continuousmap, with each branch corresponding to a continuumof states with an identical number of action potentials.Finally, as in our case, one of the crucial parameters oftheir model exerted its influence on only one of the twocomponents of the map, allowing a simple descriptionof the burst length control by this parameter.

One significant consequence of the analysis pre-sented in this work is the demonstration of multiplefixed points corresponding to bursting solutions with

J Comput Neurosci

different numbers of spikes per burst. As shown inFigs. 6–11, the stability of multiple such fixed pointscan give rise to multistability of bursting activity (seeFig. 1). To our knowledge, this is the first modelingstudy that examines the possibility of emergence ofsuch network multistability from the interplay betweenthe synaptic interaction and the intrinsic cell dynamics.As discussed above, the existence of multiple stablebursting solutions has been previously considered in thecontext of intrinsic cell bursting (Canavier et al. 1991,1994; Butera 1998); however, it has not been carefullyexamined experimentally. On the other hand, there areknown examples of bistability (Lechner et al. 1996;Manor and Nadim 2001). It is intriguing to speculatewhether multiple stable periodic states could coexistin biological networks, enabling the system to quicklyswitch between distinct activity states. However, thebasin of attraction of the distinct bursting solutions maybe too narrow for these states to be reliably sustained.In biological systems, multistability of bursting activityis much more likely to manifest itself as cycle to cyclevariability in the number of spikes per burst due tonoise and other extrinsic influences on the network.Such variability is in fact observed in many half-centeroscillatory networks (Bartos et al. 1999; Masino andCalabrese 2002).

4.3 Relaxing the simplifying assumptions

In order to implement our map-based approach, wemade a few simplifying assumptions. First, we assumedthat the synaptic and recovery variables evolved slowlyin the silent state compared to the voltage variable.This allowed us to slave the v variable to the dynamicsof the w and s variables thus enabling us to deter-mine the value s (see Eq. (7)). This multiple-time-scale assumption is critical for using our single-cell mapapproach, but is less important if we wish to derivethe map by analyzing the feed-forward network. Forexample, when we relaxed the time-scale separation bychanging τsyn from 4 ms to 1 ms, thus allowing s toevolve more quickly in the silent state, we were still ableto use a feed-forward map to determine h� = G(L) (seeFig. 10 and Appendix 3), which enabled us to accuratelypredict the behavior of the full model.

In our analysis we assumed that the T-current ac-tivation threshold (va) and inactivation threshold (vh)are equal. If we consider the case va > vh, then the T-current would begin to inactivate while the neuron isstill in the silent state. If the inactivation rate is toofast, the effect of the T-current may be lost before itis activated. However, even if the inactivation rate is

not too fast, constructing a single-cell map for h� =G(L) would not be possible since we would need toknow how the silent cell responds to individual synapticevents. However, a feed-forward approach to constructthe map similar to the fast τsyn case would still bepossible, provided that v lies below va in the silent state.In this case s would be computed from Eq. (7) using thevalue of va instead of vh. Further, we used a Heavisidefunction to demarcate the inactivation and activationthresholds. These functions are smoothed out whenperforming numerical simulations. They could also besmoothed out in the analysis and would mostly affecthow we calculate the map h� = G(L). We would nolonger be able to explicitly calculate h, but would in-stead have to estimate it.

In this paper, the post-synaptic cell escapes frominhibition when it reaches the T-current activationthreshold. With a change of parameters, it is possiblefor this cell to instead be released from inhibition. Forexample, suppose each cell has a stable rest state at apotential larger than the T-current activation thresholdvh, and that the synaptic decay constant is large relativeto the intrinsic time constant τw. The difference inscales of the time constants would imply that the post-synaptic cell could not reach vh while the pre-synapticcell is still active. Once the pre-synaptic cell stopped fir-ing, its synaptic input would decay sufficiently to allowthe post-synaptic cell to fire. In this scenario, the firingpatterns of each cell would have a much wider spreadof ISIs. In order to calculate L = F(h∗) in this case, wehave to keep track of the time required for the nullclineof the bursting cell to cross the lower branch of the w-nullcline (i.e. time to the saddle-node bifurcation). Inaddition, we would then have to calculate the extra timethat the post-synaptic neuron needs to reach vh.

Another simplifying assumption of our model wasthat individual synaptic inputs are sufficient to forcethe inhibited cell below vh. In many cases, however,summation of multiple IPSPs is necessary to produceeffective inhibition of the postsynaptic neuron. We ex-amined one such example in Fig. 10 by lowering vh andusing faster synaptic decay. The result was the appear-ance of discontinuities in the map G. A full analysis ofthis case requires keeping track of synaptic summationto find the time interval between the first spike in theburst and the time that the opposite inhibited neuronfalls below vh. Such an analysis, as mentioned earlier,would require construction of G as a feed-forwardmap so that the effect inter-spike intervals have onthe summation of IPSPs can be taken into account.Although we did not address this issue in detail, themap G constructed for Fig. 10 shows the feasibility ofconstructing the one-dimensional map in this case.

J Comput Neurosci

A similar technique can be used to keep track ofany influence of short-term synaptic dynamics such asfacilitation and depression on the shape of the map G.Thus, the effect of short-term synaptic dynamics can beaccounted for by the one-dimensional maps so long asthese effects are short lived on the time scale of a burstand do not accumulate from burst to burst. Note thatthe presence of synaptic facilitation would possibly actsynergistically with the T-current in producing burstingoscillations by increasing the duration and effectivenessof inhibition and therefore allowing for additional de-inactivation of the T-current. However, the analysis ofthis synergistic interaction requires only modificationsin the map G because the map F is not strongly influ-enced by modifications of synaptic dynamics.

Note finally that we have only considered the caseof identical cells. However, the method readily gener-alizes to the heterogeneous case, in which case the mapcan be written as P(L) = F2(G2(F1(G1(L)))), wherethe subscript indicates the cell number correspondingto each of the single-cell maps. The analysis of thedynamics of a heterogeneous network is beyond thescope of the current work and will be explored in futurestudies.

4.4 Analysis of more complicated dynamicsand complex systems

The one-dimensional map based approach can be ap-plied to understand some of the more complicateddynamics of the two-cell network. For example, simplecobwebbing using the map (Fig. 9) revealed the exis-tence of chaotic solutions which were then numericallyobtained. The existence of higher order periodics canalso be explored in this way. In a network consistingof a larger number of cells, the formulation of a one-dimensional map may be considerably more difficult.However, if the ISI remains a quantity of interest, thenpresumably the map can be built as the compositionof several (more than two) single-cell maps, each ofwhich measures a quantity of specific interest over someportion of the trajectory.

4.5 Conclusions

We have described a simple analytical tool, a one-dimensional map constructed from single-cell prop-erties, for the analysis of the output of a relativelycomplex network model. This one-dimensional mapaccurately predicted the existence and stability of so-lutions to the network model as well as the presenceof chaotic solutions and the simultaneous stability ofmultiple solutions. The one-dimensional map was built

by focusing on a subset of solutions of interest, i.e.anti-phase bursts, and using some basic simplifyingassumptions. We emphasize that the usefulness of theone-dimensional map is primarily as an aid to under-standing the behavior of the full model network. Assuch, the map is not meant as a substitute for the fullmodel in producing and predicting outputs of the bio-logical network of interest. Rather, a thorough under-standing of biological neural networks would requirea combination of numerical simulations using detailedbiophysical models and the mathematical analysis ofthese high-dimensional models using reduction tech-niques such as the one described in this study.

Acknowledgements This work was supported, in part, by grantsfrom the National Science Foundation DMS 0417416 (VM),DMS 0615168 (AB) and National Institutes of Health MH-60605(FN).

Appendix

1 Model equations and parameters

The Morris–Lecar parameters in Eq. (6) are adaptedfrom Keener and Sneyd (1998):

Iapp = 14μA/cm2, Cm = 2μF/cm2, φ = 2/3, EK =−84, ECa = 120, EL = −60, gCa = 4, gK = 8, gL = 2.Here the voltage units are mV, and conductance unitsare mS/cm2.

The functions m∞, w∞ and τw(v) are given by

m∞(v) = 1 + tanh[(v + 12)/18]2

,

w∞(v) = 1 + tanh[(v + 8)/6]2

,

τw(v) = 1cosh[(v + 8)/12] (17)

The discontinuous dynamics of h j(t) and s j(t) inEqs. (3), (4) and (6) are smoothed out in the numericalsimulation using:

a j = σ(v j − vh)

h′j = σ(vh − v j)

1 − h j

τlo− σ(v j − vh)

h j

τhi

s′j = σ(v j − vθ )

1 − s j

τγ

− σ(vθ − v j)s j

τsyn

where σ(v) is the sigmoidal function given by

σ(v) = 12[1 + tanh(4v)]

J Comput Neurosci

The T-current inactivation time is set to τhi = 20ms, while the synaptic growth time is τγ = 0.2 ms. Thesynaptic reversal potential is set to Einh = −80 mV.

Except for the figure panels exploring the parametersensitivity of the network dynamics, the remaining pa-rameter values in Figs. 2–9 are: vh = −47.5 mV, τlo =200 ms, gsyn = 0.6 mS/cm2, gT = 1 mS/cm2, vθ = −35mV, and τsyn = 4 ms.

In Fig. 1, 10 and 11, these parameters are changedto: vh = −52 mV, τlo = 100 ms, gsyn = 1.1 mS/cm2, gT =1.4 mS/cm2, vθ = −3 mV, and τsyn = 1 ms.

The values of τhi and τlo given above are close to thevalues measured by Huguenard and McCormick (1992)of about 30 ms and 300 ms, respectively.

MATLAB code implementing the model can befound at http://web.njit.edu/~matveev/Burst.

Note that in principle the precise value of vθ is notcritical to the model dynamics, provided that vθ is lessthan the maximal potential of the spiking limit cycle.However, a more hyperpolarized level of vθ used inFigs. 2–9 allows for a more reliable inhibition by the cellinitiating the burst, ensuring that the potential of thecell terminating the burst quickly drops below vh. Onthe other hand, the use of a higher spiking thresholdenhances the situation whereby a single burst spikeis insufficient to suppress the potential of the partnercell below vh, a situation which holds in the case ofmultistability of solutions (Figs. 1, 10 and 11). However,the same effect could be achieved by varying gsyn alone.

Finally, in the computation of the single-cell mapL = F(h�), we ignored a minor contribution to theburst length of the duration between cell escape timeand the peak of first spike. This correction is easy to in-clude, and is incorporated in all numerical calculationsinvolving this map. Denoting the time to first spike as�t(h�), the sums in Eqs. (10–12) should be extendedto include �t(h�), and in Eq. (9), h1 should be set toh� exp[−�t(h�)/τhi].

2 Numerical reconstruction of burstbifurcation diagram

The phase diagrams shown in Fig. 8 (gray curve) and in11 are reconstructed numerically, by “crawling” alongthe relevant parameter direction. The algorithm is im-plemented in MATLAB, and involves two main parts:(1) the periodic state detector, and (2) the parameter“crawler”/continuer.

1. The detector of periodic solutions is implementedin the most straightforward way. Namely, modelequations are continuously integrated while check-ing for a pattern in the sequence of burst intervals.

A burst interval is defined as the time period be-tween two consecutive spikes of one cell containingat least one spike of the second cell. If a sequenceof such intervals is found to repeat itself a sufficientnumber of times with a given degree of accuracy,the corresponding state is assumed periodic, andthe number of spikes, the duration of the stateand its state vector are returned. For a symmetricanti-phase periodic bursting solution, the burstinterval sequence would normally represent thelengths of consecutive bursts of the two cells, suchas in { 35.4, 35.4, 35.4, ...}. However, in some casesthe synchronization of the last and first spikes inthe burst may occur, which may lead to a periodicsequence of length 4, such as { 25.2, 2.1, 1.4, 25.2,2.1, 1.4, 25.2, 2.1, 1.4 }. The latter sequence corre-sponds to the situation where the last spike of onecell occurs after the first spike of the partner cell.In this example 1.4 represents the first intra-burstinter-spike interval (containing one last spike ofthe partner cell), and 2.1 is the duration of the lastinter-spike interval of the partner cell (containingone first spike of cell 1). However, this complicationdoes not arise in the simulations shown in this work.Empirically, tracking the lengths of burst intervalsdefined above is more robust than tracking thesequence of the inter-spike intervals. Also, it au-tomatically takes into account the identity of thespikes as coming from a particular cell.

2. Once a periodic solution is detected, a variation ofa standard predictor-corrector step is performed.Namely, the relevant parameter is increased by acertain increment (or decreased, depending on thedirection of the crawl), and the periodic detectionis repeated, using the periodic state from the pre-ceding parameter point as the initial condition. Ifthe new periodic solution has the same numberof spikes (burst sequence signature) as for a pre-vious parameter value, a bigger parameter step isundertaken (parameter increment is stretched). Ifthe new periodic state is different from the old one,the parameter increment is decreased, and the newstate sequence is put into a queue along with thecorresponding parameter value and the full statevector. The process continues until the parameterstep drops below a certain minimal value, whichwill occur close to the boundary (bifurcation point)of the parameter interval supporting the corre-sponding periodic state.Once one boundary of the relevant parameter in-terval is detected, the crawl is re-started from theinitial parameter point, but in the opposite direc-tion (i.e. the parameter is decreased). When the

J Comput Neurosci

second parameter bound/bifurcation is detected,the parameter basin of the stable state has been de-termined. The process is then repeated for the nextperiodic state in the parameter queue. Recall thatthe queue is composed of activity states that thealgorithm encounters when the parameter value isincreased beyond the bifurcation point. Therefore,the algorithm “hops” from one state to another onewhich lies close to the first one in the parameterspace.

Note that the completeness of the reconstructed phasediagram is by no means guaranteed. For instance, if theparameter support of one state lies completely withinthe parameter state of the other cell, that state maynever be detected, unless it lies close to the bifurcationpoint of some third stable periodic state.

3 Case of discontinuous h� = g(L) map

Here we describe the reconstruction of the discon-tinuous h� = g(L) map shown in Fig. 10(b). The dis-continuity is directly related to the amount of timeneeded for the cell potential to drop below the vh

threshold after burst termination. This time duration islabeled φ and marked by a double arrow in Fig. 10(a).If the T-current threshold vh is close to the spikingthreshold of the cell, as is the case for the simulationsin Figs. 2–9, the potential of each cell will reach vh

almost instantaneously on burst termination, and φ isapproximately zero (φ is ignored in Eqs. (13–15)). Ifhowever vh is significantly lower than the potential ofthe cell upon burst termination, it may take severalinhibitory synaptic inputs to lower the potential belowvh.

φ depends on the level of T-current inactivation ofboth cells, φ = φ(h�, hL), where hL is the T-currentinactivation of the bursting cell at the end of theburst (see Fig. 10(a)). The larger hL, the higher is theV-nullcline of the cell on termination of its burst, andthe longer it takes for the potential of the cell to dropbelow the vh threshold. φ is also a function of h�, sinceh� determines the initial burst spiking frequency, andhence, it determines how quickly the inhibitory synapticinput can lower the potential of the suppressed cellbelow vh.

Thus, the de-inactivation of the T-current isdescribed by

h∗ = 1 + (ho − 1) exp[

− L − φ(h�, hL)

τlo

]

where ho (the minimal value of h(t)) and hL are deter-mined using the steady-state approximation, i.e. under

the assumption of periodicity in burst characteristics h�,φ and L. Examination of Fig. 10(a) leads to:

ho = h∗ exp[

− L + φ(h�, hL)

τhi

]

hL = h∗ exp[− L

τhi

]Thus, φ is a function of both h�

and L:

φ(h�, hL) = φ(h�, h∗ exp[−L/τhi]) ≡ (h�, L)

Therefore, we have

h� = 1 +(

h∗ exp[

− L + (h�, L)

τhi

]

− 1)

exp[

− L − (h�, L)

τlo

]

Solving for h� yields

h� =1 − exp

[− L−(h�,L)

τlo

]

1 − exp[− L+(h�,L)

τhi− L−(h�,hL)

τlo

] ≡ H(h�, L)

(18)

In the above equation, the function (h�, L) is de-fined numerically. Namely, the potential of one cell(bursting cell) is set above the spiking threshold, whilethe potential of the partner cell (the suppressed cell)is set below the excitation threshold, and their h val-ues are set respectively to h� and hL = h∗ exp[−L/τhi].Given these initial conditions, the feedforward networkequations are then integrated until the potential of thesuppressed cell reaches vh. The corresponding passagetime determines the value φ(h�, hL).

Since the right-hand side of Eq. (18) depends on h�

through (h�,L), this equation defines the h�=G(L)

map implicitly. It is solved iteratively for each valueof L:

h�n+1 = H(h�

n, L), n = 1, 2, ...

where the initial condition h�1 is obtained under the

approximation ho = 0, φ = 0:

h�1 = 1 − exp

[−L/τlo]

The iterative solution approaches the thin curve shownin Fig. 10(a). The discontinuity in this curve is the resultof the discontinuity of the function φ(h�, hL): as h�

decreases or hL increases past some critical value, itwill take one more inhibitory input to suppress the cellbelow vh, resulting in a jump increase in φ by one inter-spike interval.

J Comput Neurosci

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