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International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440013 (9 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127414400136

When Transitions Between Bursting ModesInduce Neural Synchrony

Reimbay Reimbayev and Igor Belykh∗Department of Mathematics and Statistics,

and Neuroscience Institute,Georgia State University, 30 Pryor Street,

Atlanta, GA 30303, USA∗ibelykh@gsu.edu

Received February 2, 2014

We study the synchronization of bursting cells that are coupled through both excitatory andinhibitory connections. We extend our recent results on networks of Hindmarsh–Rose burstingneurons [Belykh et al., 2014] to coupled Sherman β-cell models and show that the additionof repulsive inhibition to an excitatory network can induce synchronization. We discuss themechanism of this purely synergenic phenomenon and demonstrate that the inhibition leads tothe disappearance of a homoclinic bifurcation that governs the type of synchronized bursting. Asa result, the inhibition causes the transition from square-wave to easier-to-synchronize plateaubursting, so that weaker excitation is sufficient to induce bursting synchrony. We dedicate thispaper to the memory of Leonid P. Shilnikov, the pioneer of homoclinic bifurcation theory,and emphasize the importance of homoclinic bifurcations for understanding the emergence ofsynchronized rhythms in bursting networks.

Keywords : Bursting; synchronization; inhibition; excitation; neural networks.

1. Introduction

Neurons can generate complex patterns of burst-ing activity. Over the last thirty years, much workhas been dedicated to the classification of burstingrhythms [Rinzel, 1987; Rinzel & Ermentrout, 1989;Bertram et al., 1995; Izhikevich, 2000; Golubit-sky et al., 2001] and the bifurcation transitionsbetween them (see [Terman, 1991, 1992; Wang,1993; Belykh et al., 2000; Shilnikov et al., 2005;Frohlich & Bazhenov, 2006] and references therein).This includes chaotic bursting [Guckenheimer &Oliva, 2002] and transitions between tonic spikingand bursting, caused by homoclinic bifurcations ofa saddle-node periodic orbit and the blue sky catas-trophe [Turaev & Shilnikov, 1995; Shilnikov & Cym-baluyk, 2005].

When coupled in a network, bursting neuronscan attain different forms of synchrony: burst syn-chronization when only the envelopes of the spikessynchronize, complete synchrony, anti-phase burst-ing, and other forms of phase-locking. The emer-gence of a specific cooperative rhythm dependson the intrinsic properties of the coupled neu-rons, a type and strength of synaptic coupling,and network circuitry [Wang & Rinzel, 1992; vanVreeswijk et al., 1994; Kopell & Ermentrout, 2002;Elson et al., 2002; Somers & Kopell, 1993; Sher-man, 1994; Canavier et al., 1999; Rubin & Terman,2002; Kopell & Ermentrout, 2004; Belykh et al.,2005; Izhikevich, 2001; Belykh & Shilnikov, 2008;Shilnikov et al., 2008; Wojcik et al., 2011; Wojciket al., 2014]. Fast excitatory synapses are known to

∗Author for correspondence

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R. Reimbayev & I. Belykh

facilitate bursting synchrony (see [Kopell & Ermen-trout, 2004; Izhikevich, 2001; Belykh et al., 2005]).However, fast inhibition typically leads to anti-phase or asynchronous bursting [Wang & Rinzel,1992; Rubin & Terman, 2002; Belykh & Shilnikov,2008], unless the inhibitory connections are weakand the initial conditions are chosen close enoughtogether within the spiking phase of bursting [Jalilet al., 2012]. Making excitatory and inhibitorysynapses slower reverses their roles such that inhi-bition, not excitation leads to bursting synchrony[van Vreeswijk et al., 1994; Elson et al., 2002].

In a recent paper [Belykh et al., 2014], westudied networks of phenomenological Hindmarsh–Rose neuron models [Hindmarsh & Rose, 1984]and discovered a counterintuitive effect that fast,desynchronizing inhibition can induce synchronizedbursting when added to an excitatory network ofsquare-wave bursters. This nontrivial phenomenonappears as a result of a synergetic interactionbetween synchronizing excitation and desynchroniz-ing inhibition. Its key component is the ability ofinhibition to effectively change the type of burst-ing in the network, switching from square-wave toplateau-type bursting. Consequently, significantlyless excitation is necessary to trigger networksynchronization. In this paper, we demonstratethat a similar effect is observed in a network ofphysiologically-based Hodgkin–Huxley-type modelssuch as the pancreatic β-cell Sherman model [Sher-man, 1994], exhibiting square-wave and plateau-type bursting. Square-wave bursting [Rinzel, 1987]was named after the shape of the voltage traceduring a burst; it resembles a square wave due tofast transitions between the quiescent state andfast repetitive spiking (see Fig. 1). Square-wavebursters are notorious for their resistance to syn-chronization and require strong excitatory couplingto synchronize them. In the Izhikevich classification[Izhikevich, 2000], square-wave bursters, observed inthe Sherman model, correspond to an Andronov–Hopf/homoclinic burster. Its main signature is thepresence of a homoclinic bifurcation of a saddlein the 2-D fast subsystem. In the following, wewill show that the addition of inhibition inducessynchrony by making this homoclinic bifurcationdisappear and generating plateau-type bursting(Andronov–Hopf/Andronov–Hopf bursting, accord-ing to the Izhikevich classification).

Homoclinic bifurcation theory was primarilydeveloped by Leonid Pavlovich Shilnikov. By this

Fig. 1. Square-wave burster of the uncoupled Shermanmodels (1). The fast system displays a supercriticalAndronov–Hopf bifurcation at S = SAH1 and a homoclinicbifurcation (loop) at S = SHB. The spiking manifold is com-posed of limit cycles in the fast system and terminates atthe homoclinic bifurcation HB. The intersection of the fast(h(V )) and slow (S∞(V )) nullclines indicates a unique sad-dle point O of the full system. The red dotted curve showsthe route for bursting in the full system. The plane V = Θs

displays the synaptic threshold above which the presynapticcell can influence the postsynaptic one.

paper, we pay tribute to his pioneering contribu-tions [Shilnikov et al., 1998; Shilnikov et al., 2001]and argue that homoclinic bifurcations continue tobe a source of unexpected phenomena in both singleand coupled bursting cell models.

The layout of this paper is as follows. First,in Sec. 2, we introduce the network model and theSherman cell model as its individual unit. We showthat the uncoupled cell model exhibits square-wavebursting and discuss the generation mechanism.Then, in Sec. 3, we introduce the self-coupled sys-tem that governs the type of synchronous bursting.We show that the self-coupled system switches fromsquare-wave to plateau bursting with an increasein the excitatory and/or inhibitory couplings. Thisproperty is then used in Sec. 4 to analyze the vari-ational equations for the transverse stability of thesynchronous bursting solution, defined through theself-coupled system. Several stability arguments aregiven to explain the synergetic, synchronizing effectof combined excitation and inhibition. In Sec. 5,similar transitions to synchronized plateau-burstingare shown in a network with a varying reversalpotential. Finally, in Sec. 6, a brief discussion ofthe obtained results is given.

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When Transitions Between Bursting Modes Induce Neural Synchrony

2. The Network Model

We consider the simplest network of two cou-pled Sherman models with both excitatory andinhibitory connections:

τdVi

dt= F (Vi, ni, Si) + gexc(Eexc − Vi)Γ(Vj)

+ ginh(Einh − Vi)Γ(Vj),

τdni

dt= G(Vi, ni) ≡ n∞(Vi) − ni,

τsdSi

dt= H(Vi, Si) ≡ S∞(Vi) − Si, i, j = 1, 2.

(1)

Here, Vi represents the membrane potential ofthe ith cell. Function F (Vi, ni, Si) = −[ICa(Vi) +IK(Vi, ni) + IS(Vi, Si)] defines three intrinsic cur-rents: fast calcium, ICa, potassium, IK, and slowpotassium, IS, currents:

ICa = gCam∞(Vi)(Vi − ECa),

IK = gKni(Vi − EK),

IS = gSS1(Vi − EK).

The gating variables for ni and Si are the openingprobabilities of the fast and slow potassium cur-rents, respectively, and

m∞(Vi) =[1 + exp

(−20 − Vi

12

)]−1

n∞(Vi) =[1 + exp

(−16 − Vi

5.6

)]−1

S∞(Vi) =[1 + exp

(−35.245 − Vi

10

)]−1

.

Other intrinsic parameters are τ = 20, τS = 10000,gCa = 3.6, ECa = 25 mV, gK = 10, EK = −75 mV,gS = 4.

The cells are identical and the symmetricalsynaptic connections are fast and instantaneous.The parameters gexc and ginh are the excitatory andinhibitory coupling strengths. The reversal poten-tials Eexc = 10 mV and Einh = −75 mV make thesynapses excitatory and inhibitory, respectively asEexc > Vi (Einh < Vi) for all values of Vi(t). Thesynaptic coupling function is modeled by the sig-moidal function Γ(Vj) = 1/[1+exp−10(Vj −Θs)].The synaptic threshold Θsyn = −40 mV is set to

ensure that every spike in the single cell burst canreach the threshold (see Fig. 1). As a result, aspike arriving from a presynaptic cell j activatesthe synapse current (through Γ(Vj) switching from0 to 1) entering the postsynaptic cell i. Unlessnoticed otherwise, we will keep the above parame-ters fixed and only vary the synaptic strengths gexc

and ginh.The presence of the large parameter τS = 10000

on the left-hand side of the S-equation makes thesystem (1) slow–fast such that the (Vi, ni)-equationsrepresent the 2-D fast “spiking” subsystem for theith cell; the Si-equation corresponds to the slow1-D “bursting” system. Therefore, we use the stan-dard decomposition into fast and slow subsystems;the types of bursting that can exist in the uncou-pled cell systems (1) with gexc = 0 and ginh = 0are defined by the S-parameter sequences of phaseportraits of the 2-D fast system. This analysishas been performed for a similar pancreatic cell[Tsaneva-Atanasova et al., 2010] and revealed differ-ent types of bursting such as square-wave, plateauand pseudo-plateau bursting [Stern et al., 2008].Figure 1 illustrates the standard sequence of phaseportraits in the uncoupled systems (1) with gexc = 0and ginh = 0, giving rise to square-wave bursting.

The equilibrium point on the upper branch ofthe nullcline h(V ) in the 2-D fast subsystem under-goes a supercritical Andronov–Hopf bifurcation forS = SAH1, softly giving rise to a stable limit cyclethat encircles the unstable point and forms the spik-ing manifold for SAH1 < S < SHB. Its upper edgeis defined by a homoclinic bifurcation at S = SHB.Here, the stable limit merges into a stable homo-clinic loop and disappears. For the given locationof the slow nullcline S∞(V ), the trajectories jumpdown to the lower branch of the fast nullcline, creat-ing square-wave (fold/homoclinic) bursting. A moredetailed analysis of the phase portraits’ sequencesand bifurcations leading to square-wave burstingin other cell models such as the Hindmarsh–Rosemodel can be found in [Shilnikov & Kolomiets, 2008;Belykh & Hasler, 2011].

3. Self-Coupled System andIts Burster

Each cell in the network (1) receives one inhibitoryand one exciatory input from the other cell, there-fore the network system (1) has an invariant man-ifold D = V1 = V2 = V (t), n1 = n2 = n(t),

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R. Reimbayev & I. Belykh

S1 = S2 = S(t), that defines complete synchro-nization between the cells.

Synchronous dynamics on the manifold D isdefined by the self-coupled system:

τdV

dt= F (V, n, S) + gexc(Eexc − V )Γ(V )

+ ginh(Einh − V )Γ(V ),

τdn

dt= G(V, n)

τsdS

dt= H(V, S).

(2)

Note that the synchronous dynamics differs fromthat of the uncoupled cell as the former is governedby a system with extra coupling terms. Moreover,the synchronous dynamics and the type of burst-ing depend on the coupling strengths gexc and ginh.There are critical coupling strengths gexc and ginh atwhich square-wave bursting in the self-coupled sys-tem (2) turns into plateau-type bursting, depictedin Fig. 2. This happens through the disappearanceof the homoclinic bifurcation in the self-coupledsystem (2) due to increased coupling strengths.

Fig. 2. Plateau-type burster of the self-coupled model (2),governing the synchronous network dynamics. Note the dis-appearance of a homoclinic bifurcation in the fast system dueto the synaptic coupling. The stable limit cycle of the fastsystem disappears through a reverse Andronov–Hopf bifur-cation at S = SAH2, ending the spiking manifold. The reddotted curve shows the route for plateau-type bursting. Thenonsmooth part of the fast nullcline at V = Θs is due to thesynaptic coupling, turning on when the trajectory jumps upto the spiking manifold and crosses the threshold Θs. Thecoupling strengths gexc = 0.14 and ginh = 0.06 correspond tothe point b in the 2-D diagram of Fig. 3.

In the following, we will show that the disappear-ance of the homoclinic bifurcation (HBD) practi-cally coincides with the onset of stable synchronyin the system (1). While excitation alone is ableto transform square-wave into plateau-type burst-ing at some high values of gexc, inhibition does somore effectively and its addition lowers the com-bined coupling strength gexc + ginh.

4. Stability of Synchronization:A Synergetic Effect of Inhibitionand Excitation

The stability of synchronization in the network (1)is equivalent to the stability of the invariant mani-fold D. The variational equations for its infinitesi-mal transverse perturbations ∆V = V1 − V2, ∆n =n1−n2, and ∆S = S1−S2 read [Belykh et al., 2005;Belykh et al., 2014]:

τd

dt∆V = FV (V, n, S)∆V + Fn(V, n, S)∆n

+ FS(V, n, S)∆S − Ω(V )∆V

τd

dt∆n = GV (V, n)∆V + Gn(V, n)∆n

τd

dt∆S = HV (V, S)∆V + HS(V, S)∆S,

(3)

where

Ω(V ) = (gexc + ginh)Γ(V ) + (gexc(Eexc − V )

+ ginh(Einh − V ))ΓV (V ). (4)

Here, the partial derivatives are calculated at thepoint ∆V = 0,∆n = 0,∆S = 0; and V (t), n(t),S(t) is the synchronous bursting solution definedthrough the system (2). Note that the synapticcoupling function Γ(V ) along with its derivativeΓV (V ) = λ exp−λ(V −Θs)

(1+exp−λ(V −Θs))2 is non-negative. Hence,the contribution of the first term in (4), −(gexc +ginh)Γ(V )∆V is stabilizing for the zero fixedpoint of the variational system (3), correspond-ing to synchronous bursting. On the other hand,the contribution of the second coupling termin (4) can be destabilizing when ginh(Einh − V )exceeds gexc(Einh − V ), making the second termoverall negative. Note that increasing the inhibitorycoupling ginh makes this term more negativeand, therefore, promoting desynchronization as onewould expect. However, Fig. 3(a) indicates thatthe addition of inhibition to an excitatory network

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When Transitions Between Bursting Modes Induce Neural Synchrony

(a) (b)

Fig. 3. (a) The stability diagram for synchronization in the two-cell network (1). The color bar indicates the maximumvoltage difference V1 − V2, calculated over the last two bursts in the established rhythm. Blue (dark) zone corresponds to thezero voltage difference and shows the synchronization region. Note the unexpected effect when an increase of the inhibitorycoupling from 0 to 0.07 significantly lowers the synchronization threshold from about 0.18 to 0.07. Notice that the inhibitiondesynchronizes the cells in the absence of excitation (gexc = 0). The red dashed curve indicates the disappearance of thehomoclinic bifurcation (HBD) in the 2-D fast subsystem; it corresponds to the transition from square-wave to plateau burstingand practically coincides with the stability boundary between asynchronized and synchronized bursting. (b) (Top) Typicalout-of-phase voltage traces, corresponding to the red (“out-of-phase”) zone. (Bottom) Synchronization of plateau bursting inthe blue (“sync”) parameter region.

induces synchronization in a fairly wide range ofthe inhibitory strength ginh. Note that increasingginh first lowers the synchronization threshold andweaker excitation synchronizes the cells (e.g. fromgexc = 0.18 in the absence of inhibition to gexc =0.07 for ginh = 0.07). At the same time, the inhibi-tion cannot induce robust synchronization by itself(see the x-axis in Fig. 3(a), which corresponds to thedesynchronizing role of inhibition in the absence ofexcitation).

What is the cause of this highly unexpectedphenomenon? It is worth noticing that when com-plete (spike) synchronization occurs in the purelyexcitatory network (1) with ginh = 0, square-wavebursting, observed in the unsynchronized net-work (1) at lower values of gexc, turns into plateaubursting [see Fig. 3(b)]. This happens when theexcitatory coupling is strong enough to change thetype of bursting via the disappearance of the homo-clinic bifurcation in the fast subsystem of (2). Theimportant ingredient of the inhibition-induced syn-chronization in the network is that the inhibitionchanges the type of bursting much more effectivelythan the excitation [see Fig. 3(a)]. Consequently, a

much smaller amount of the combined force (gexc +ginh) is necessary for synchronization.

Why is plateau bursting easier to synchronize?Why does inhibition not desynchronize plateauoscillations as it seems to have an apparent desta-bilizing effect due to the second term in (4)?

These questions have been answered for net-works of Hindmarsh–Rose models in our recentpaper [Belykh et al., 2014]; here, we give additionaldetails and adapt the main arguments to the net-work of Sherman models (1), representing realisticHodgkin–Huxley-type cell models. Figure 4 answersthese questions by revealing the dynamics and sta-bility of synchronous bursting via the variationalsystem (3) and self-coupled system (2), describingthe synchronous solution V (t), whether stable orunstable. Figure 4(a)(top) shows the synchronoussolution of the self-coupled system (2), which isunstable as the coupling is not sufficiently strong[see the corresponding point a in Fig. 3(a)]. Thecontribution of the coupling term Ω(V ) to the sta-bilization of the synchronous solution is depicted inFig. 4(a)(bottom). When the voltage is above thesynaptic threshold Θs, only the first (stabilizing)

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R. Reimbayev & I. Belykh

(a) (b) (c)

Fig. 4. The role of inhibition in the stability and type of synchronous bursting. Cases (a), (b), and (c) correspond to pointsa, b, and c in Fig. 3. (a) (Top) The unstable synchronous solution and the fast nullcline h(V ) of the self-coupled system (4).(Bottom) The coupling term Ω is not strong enough to stabilize the synchronous solution, especially the subthreshold part ofthe spikes where the coupling is absent. (b) Increasing the inhibition shifts the part of the nullcline h(V ) above the thresholdcloser to the threshold V = Θs. However, it makes the amplitude of spikes smaller and leaves the spikes in the region above thethreshold where the coupling effectively synchronizes the spikes. Notice the transition from square bursting to plateau bursting.(c) Excessively strong inhibition dominating over excitation has a desynchronizing effect. It still forms plateau bursting inthe self-coupled system, governing synchronous bursting, but it creates a vertical part of the nullcline h(v) at V = Θs. Thebursting trajectory has to follow this part of the nullcline for a while when the cells experience a strong desynchronizing impact(notice the negative peak of Ω at V = Θs) and get desynchronized.

term (gexc + ginh)Γ(V ) matters as Γ(V ) becomesclose to 1. The second term is only essential forthe values of V, close to the threshold Θs as thederivative ΓV (V ) is close to the delta function atV = Θs. There is practically no coupling betweenthe cells when V is below the synaptic threshold Θs.The previous analysis of synchronization in exci-tatory networks of square-wave bursting cells bymeans of Lyapunov functions [Belykh et al., 2005;Belykh & Hasler, 2011] suggests that the stabiliza-tion of spikes via the coupling Ω(V ) amounts to sta-bilizing the entire synchronous trajectory, includingits subthreshold part.

Notice that the lower part of the spikes liesbelow the synaptic threshold Θs [to the left from Θs

in Fig. 4(a)(top)] where the synchronizing impact ofΩ(V ) is insignificant. Therefore, there is no synchro-nization for these values of gexc and ginh. Figure 4(b)corresponds to synchronized bursting, induced bystronger inhibition [point b in Fig. 3(a)]. In fact,increasing the inhibition has a two-fold effect. Itlowers the peak of Ω at V = Θs [Fig. 4(b)], makingthe contribution of the coupling smaller. However,at the same time it is capable of changing the typeof bursting such that the spikes of plateau burstingalmost entirely lie in the region above the threshold

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When Transitions Between Bursting Modes Induce Neural Synchrony

where the coupling term Ω stabilizes the (mostunstable) spiking part of the trajectory. Hence, theinhibition-induced transition to plateau burstingmakes synchronization stable. When the inhibitionbecomes stronger than the excitation, synchroniza-tion loses its stability [see, for example, point c inFig. 3(a)]. Note that the coupling Ω no longer favorsthe stability of synchronization at V = Θs as it hasa negative peak [Fig. 4(c)]. Moreover, the excessiveinhibition pushes the right branch of the nullclineh(V ) to the left so that the synchronous trajectoryexperiences this negative, desynchronizing impactfor a long time while crawling along the nullclineup to the right upper knee [see the nonsmooth partof the nullcline in Fig. 4(c)]. This results in desyn-chronization and the onset of asynchronous burstingin the network.

5. The Role of the ReversalPotential

In this section, we show that similar transitionsfrom square-wave to plateau bursting and windowsof induced synchronization also can be observedin the network (1) where the excitatory and

Fig. 5. The stability diagram for synchronization in thenetwork (1) with synaptic connections Ii

syn = gsyn(Esyn −Vi)Γ(Vj), i, j = 1, 2. The color bar indicates the maxi-mum voltage difference V1 − V2, calculated over the last twobursts in the established rhythm. Similar to Fig. 3(a), thered dashed curve displays the transition from square-waveto plateau bursting via the disappearance of the homoclinicbifurcation in the fast subsystem. This transition bound-ary coincides remarkably well with the onset of synchro-nized bursting. Notice the drop in the coupling strength gsyn,necessary for inducing synchronization in a window aroundEsyn = −40mV. This window corresponds to the window ofinduced synchronization in Fig. 3(a).

inhibitory connections Iisyn = gexc(Eexc−Vi)Γ(Vj)+

ginh(Einh−Vi)Γ(Vj), i, j = 1, 2 are replaced with twosynaptic connections Ii

syn = gsyn(Esyn − Vi)Γ(Vj),i, j = 1, 2. Depending on the value of the rever-sal potential Esyn, these synaptic connections canbe excitatory or inhibitory or be of a mixed typewhen Esyn lies somewhere in between the twoextremes 10 mV and −75mV. To some extent,decreasing Esyn from 10 mV amounts to increas-ing the inhibitory connections in the original net-work (1). Figure 5 shows the dependence of thesynchronization threshold coupling gsyn on thereversal potential Esyn. There is an optimal range ofEsyn close to the synaptic threshold Θsyn = −40 mVand corresponding to significantly improved syn-chronizability of the network. The vertical bound-ary of this synchronization region around Θsyn =−40 mV corresponds to the 45 line in Fig. 3(a).In fact, the coupling becomes purely inhibitoryfor the values Vi > Esyn = Θsyn as the factor(Esyn − Vi) < 0, and therefore the spikes cannotbe robustly synchronized.

6. Conclusions

Different types of bursting have significantly differ-ent synchronization properties. While square-wavebursters are known for their high resistance to spikesynchronization, elliptic and plateau-like burstersare much easier to synchronize and require a weakercoupling strength. Typically, fast nondelayed exci-tation promotes synchronization of bursters whilefast nondelayed inhibition desynchronizes them.Although, counterexamples of synchronizing fastnondelayed inhibition in the weak coupling casehave been reported [Jalil et al., 2012]. In this paper,we have shown that the onset of spike synchroniza-tion in a network of bursting cells is accompaniedby transitions from square-wave to plateau burst-ing. These transitions can be effectively enhancedby the addition of inhibition to a bursting networkwith excitatory connections. As a result, the inhi-bition, that desynchronizes the cells in the absenceof excitation, plays a synergetic role and helps theexcitation to make synchronization stable. In ourstudy, we have chosen the pancreatic β-cell Sher-man model, which exhibits various types of burst-ing and is capable of switching between them as abuilding block for the two-cell network. Our pre-liminary study shows that the reported synergeticeffect of combined excitation and inhibition is alsopresent in larger networks of bursting Sherman

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models with network topologies admitting completesynchronization. The role of network topology onsynchronization of other bursting cell models suchas the Hindmarsh–Rose models has been previ-ously studied for excitatory networks [Belykh et al.,2005; Belykh & Hasler, 2011] and for excitatory–inhibitory networks [Belykh et al., 2014], indicat-ing that the number of incoming excitatory andinhibitory connections is often the crucial quantity.A detailed stability analysis of complete synchro-nization and other phase-locked rhythms in largeexcitatory–inhibitory networks of Sherman modelsremains a subject of future work.

Acknowledgments

This work was supported by the National ScienceFoundation under Grant No. DMS-1009744 andGSU Brains & Behavior program. R. Reimbayevacknowledges support as a GSU Brains & BehaviorFellow.

References

Belykh, V. N., Belykh, I. V., Colding-Jørgensen, M. &Mosekilde, E. [2000] “Homoclinic bifurcations leadingto bursting oscillations in cell models,” Eur. Phys. J.E 3, 205–219.

Belykh, I., de Lange, E. & Hasler, M. [2005] “Synchro-nization of bursting neurons: What matters in the net-work topology,” Phys. Rev. Lett. 94, 188101.

Belykh, I. & Shilnikov, A. [2008] “When weak inhibi-tion synchronizes strongly desynchronizing networksof bursting neurons,” Phys. Rev. Lett. 101, 078102.

Belykh, I. & Hasler, M. [2011] “Mesoscale and clustersof synchrony in networks of bursting neurons,” Chaos21, 016106.

Belykh, I., Reimbayev, R. & Zhao, K. [2014] “Repulsiveinhibition promotes synchrony in excitatory networks:Help from the enemy,” (submitted), preprint availableat http://www2.gsu.edu/∼matixb/research.html

Bertram, R., Butte, M. J., Kiemel, T. & Sherman, A.[1995] “Topological and phenomenologial classicationof bursting oscillations,” Bull. Math. Biol. 57, 413.

Canavier, C. C., Baxter, D. A., Clark, J. W. & Byrne,J. H. [1999] “Control of multistability in ring circuitsof oscillators,” Biol. Cybern. 80, 87–102.

Elson, R. C., Selverston, A. I., Abarbanel, H. D. I. &Rabinovich, M. I. [2002] “Inhibitory synchronizationof bursting in biological neurons: Dependence onsynaptic time constant,” J. Neurophysiol. 88, 1166.

Frohlich, F. & Bazhenov, M. [2006] “Coexistence of tonicfiring and bursting in cortical neurons,” Phys. Rev. E74, 031922.

Golubitsky, M., Josic, K. & Kaper, T. [2001] “An unfold-ing theory approach to bursting in fast–slow systems,”Festschrift Dedicated to Floris Takens, Global Analy-sis of Dynamical Systems, pp. 277–308.

Guckenheimer, J. & Oliva, R. A. [2002] “Chaos in theHodgkin–Huxley model,” SIAM J. Appl. Dyn. Syst.1, 105–114.

Hindmarsh, J. L. & Rose, R. M. [1984] “A model ofneuronal bursting using three coupled first order dif-ferential equations,” Proc. Roy. Soc. London B 221,87.

Izhikevich, E. M. [2000] “Neural excitability, spiking andbursting,” Int. J. Bifurcation and Chaos 10, 1171–1266.

Izhikevich, E. M. [2001] “Synchronization of ellipticbursters,” SIAM Rev. 43, 315.

Jalil, S., Belykh, I. & Shilnikov, A. [2012] “Spikes mat-ter for phase-locked bursting in inhibitory neurons,”Phys. Rev. E 85, 036214.

Kopell, N. & Ermentrout, G. B. [2002] “Mechanisms ofphase-locking and frequency control in pairs of cou-pled neural oscillators,” Handbook of Dynamical Sys-tems, Vol. 2, ed. Fiedler, B. (Elsevier, Amsterdam),pp. 3–54.

Kopell, N. & Ermentrout, G. B. [2004] “Chemical andelectrical synapses perform complementary roles inthe synchronization of interneuronal networks,” Proc.Natl. Acad. Sci. USA 101, 15482.

Rinzel, J. [1987] “A formal classification of burstingmechanisms in excitable systems,” Mathematical Top-ics in Population Biology, Morphogenesis, and Neuro-sciences, Lecture Notes in Biomathematics, Vol. 71,eds. Teramoto, E. & Yamaguti, M. (Springer-Verlag,Berlin), pp. 267–281.

Rinzel, J. & Ermentrout, B. [1989] “Analysis of neu-ral excitability and oscillations,” Methods in NeuronalModeling, eds. Koch, C. & Segev, I. (MIT Press, Cam-bridge, MA), pp. 251–291.

Rubin, J. & Terman, D. [2002] “Synchronized activityand loss of synchrony among heterogeneous condi-tional oscillators,” SIAM J. Appl. Dyn. Syst. 1, 146.

Sherman, A. [1994] “Anti-phase, asymmetric, andaperiodic oscillations in excitable cells I. Coupledbursters,” Bull. Math. Biol. 56, 811–835.

Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua,L. O. [1998] Methods of Qualitative Theory in Non-linear Dynamics (Part I ), World Scientific Series onNonlinear Science, Ser. A, Vol. 4.

Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua,L. O. [2001] Methods of Qualitative Theory in Non-linear Dynamics (Part II ), World Scientific Series onNonlinear Science, Ser. A, Vol. 5.

Shilnikov, A. L., Calabrese, R. & Cymbaluyk, G. [2005]“Mechanism of bi-stability: Tonic spiking and burst-ing in a neuron model,” Phys. Rev. E 71, 205.

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August 19, 2014 10:27 WSPC/S0218-1274 1440013

When Transitions Between Bursting Modes Induce Neural Synchrony

Shilnikov, A. L. & Cymbaluyk, G. [2005] “Transitionbetween tonic-spiking and bursting in a neuron modelvia the blue-sky catastrophe,” Phys. Rev. Lett. 94,048101.

Shilnikov, A., Gordon, R. & Belykh, I. [2008]“Polyrhythmic synchronization in bursting networkmotifs,” Chaos 18, 037120.

Shilnikov, A. & Kolomiets, M. [2008] “Methods of thequalitative theory for the Hindmarsh–Rose model: Acase study. A tutorial,” Int. J. Bifurcation and Chaos10, 2141–2168.

Stern, J., Osinga, H., LeBeau, A. & Sherman, A. [2008]“Resetting behavior in a model of bursting in secre-tory pituitary cells: Distinguishing plateaus frompseudo-plateaus,” Bull. Math. Biol. 70, 68–88.

Somers, D. & Kopell, N. [1993] “Rapid synchronizationthrough fast threshold modulation,” Biol. Cybern. 68,393.

Terman, D. [1991] “Chaotic spikes arising from a modelof bursting in excitable membranes,” SIAM J. Appl.Math. 51, 1418–1450.

Terman, D. [1992] “The transition from bursting to con-tinuous spiking in excitable membrane models,” J.Nonlin. Sci. 2, 133–182.

Turaev, D. V. & Shilnikov, L. P. [1995] “On a blue skycatastrophe,” Soviet Math. Dokl. 342, 596–599.

Tsaneva-Atanasova, K., Osinga, H., Riel, T. & Sher-man, A. [2010] “Full system bifurcation analysis ofendocrine bursting models,” J. Theor. Biol. 264,1133–1146.

van Vreeswijk, C., Abbott, L. F. & Ermentrout, G. B.[1994] “When inhibition not excitation synchronizesneural firing,” J. Comput. Neurosci. 1, 313.

Wang, X.-J. & Rinzel, J. [1992] “Alternating and syn-chronous rhythms in reciprocally inhibitory modelneurons,” Neural Comput. 4, 84.

Wang, X.-J. [1993] “Genesis of bursting oscillations inthe Hindmarsh–Rose model and homoclinicity to achaotic saddle,” Physica D 62, 263–274.

Wojcik, J., Clewley, R. & Shilnikov, A. [2011] “Orderparameter for bursting polyrhythms in multifunc-tional central pattern generators,” Phys. Rev. E 83,056209.

Wojcik, J., Clewley, R., Schwabedal, J. & Shilnikov, A.[2014] “Key bifurcations of bursting polyrhythmsin 3-cell central pattern generators,” PLoS ONE 9,e92918.

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