Benjamin Pfeuty et al- Electrical Synapses and Synchrony: The Role of Intrinsic Currents

8/3/2019 Benjamin Pfeuty et al- Electrical Synapses and Synchrony: The Role of Intrinsic Currents

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Behavioral/Systems/Cognitive

Electrical Synapses and Synchrony: The Role of

Intrinsic Currents

Benjamin Pfeuty,1 German Mato,2 David Golomb,3,4 and David Hansel1,51Laboratoire de Neurophysique et Physiologie du Systeme Moteur, Centre National de la Recherche ScientifiqueUnite Mixte de Recherche 8119, Universite

Rene Descartes, 75270 Paris Cedex 06, France, 2Comision Nacional de Energia Atomica and Consejo Nacional de Investigaciones Cientificas y Tecnicas,

Centro Atomico Bariloche and Instituto Balsiero, Universidad Nacional de Cordoba, 8400 San Carlos de Bariloche, Argentina, 3Department of Physiology

and Zlotowski Center for Neuroscience, Faculty of Health Sciences, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel, 4Mathematical

Biosciences Institute, Ohio State University, Columbus, Ohio 43210, and 5Interdisciplinary Center for Neural Computation, The Hebrew University,

Jerusalem 91904, Israel

Electrical synapses are ubiquitous in the mammalian CNS. Particularly in the neocortex, electrical synapses have beenshownto connectlow-threshold spiking (LTS) as well as fast spiking (FS) interneurons. Experiments have highlighted therolesof electrical synapses in the

dynamics of neuronal networks. Here we investigate theoretically how intrinsic cell properties affect the synchronization of neurons

interactingbyelectricalsynapses.Numericalsimulationsofanetworkofconductance-basedneuronsrandomlyconnectedwithelectrical

synapses show that potassium currents promote synchrony, whereas the persistentsodiumcurrent impedes it. Furthermore, synchronyvaries with the firing rate in qualitatively different ways depending on the intrinsic currents. We also study analytically a network of

quadratic integrate-and-fire neurons. We relate the stability of the asynchronous state of this network to the phase-response function

(PRF),whichcharacterizes theeffect of small perturbations on thefiring timingof theneurons.In particular, we show that thegreater the

skew of thePRF towardthe firsthalf of theperiod, themorestable theasynchronous state.Combining oursimulationswithour analyticalresults, we establish general rules to predict the dynamic state of large networks of neurons coupled with electrical synapses. Our work

provides a natural explanation for surprising experimental observations that blocking electrical synapses may increase the synchronyof

neuronal activity. It also suggests different synchronization properties for LTS and FS cells. Finally, we propose to further test our

predictions in experiments using dynamic clamp techniques.

Key words: electrical synapses; conductance-based model; synchrony; neuronal network model; intrinsic currents; neocortex

IntroductionElectrical synapses are sites at which gap-junctions bridge themembranes of two neurons. They have long been known to existin invertebrates (Watanabe, 1958; Furshpan and Potter, 1959),but only recently has evidence of their ubiquity been unequivo-cally found in the mammalian brain. Electrical synapses arepresent in the inferior olive (Llinas and Yarom, 1986), the hip-pocampus (Draghun et al., 1998; Venance et al., 2000), the cere-

bellum (Mann-Metzer and Yarom, 1999), the locus coereleus(Christieet al., 1989;Alvarez et al., 2002), thestriatum(Kitaet al.,1990), the neocortex (Galarreta and Hestrin, 1999; Gibson et al.,

1999), the reticular thalamic nucleus (Landisman et al., 2002),and between motoneurons (Kiehn and Tresch, 2002).

Experiments have revealed that electrical synapses are in-volved in synchronizing neural activity (Draghun et al., 1998;Mann-Metzer and Yarom, 1999; Beierlein et al., 2000; Perez-Velazquezand Carlen, 2000; Tamas et al., 2000; Deanset al., 2001;Hormuzdi et al., 2001). In contrast to the results of these studies,it has been reported recently that inspiratory motoneurons may

display more strongly synchronized activity in presence of car-benoxolone (CBX), a blocker of electrical synapses, than in thecontrol situation (Bou-Flores and Berger, 2001). Therefore, inthis case, electrical synapses desynchronize neural activity.

Thedynamicsof networks of neurons interacting viachemicalsynapses have been studied extensively (Golomb et al., 2001).However, only a few theoretical studies have addressed the dy-namics of networks in which neurons are coupled by electricalsynapses. Models for pattern generation in the lobster pyloricsystem have been investigated (Kepler et al., 1990; Abbott et al.,1991; Meunier, 1992). Stable antiphase locking and transitionsbetween in-phase and antiphase locking were demonstrated inpairs of model neurons coupled by electrical synapses, and itsimpact on rhythmogenesis was investigated (Sherman and Rin-zel, 1992; Cymbalyuk et al., 1994; Han et al., 1995). More re-

Received March 13, 2003; revised April 21, 2003; accepted April 22, 2003.

ThisworkwassupportedbyNorthAtlanticTreatyOrganizationPhysicalandEngineeringScienceand Technology

CollaborativeLinkage Grant977683, Les Programmes Internationauxde Cooperation ScientifiqueCentreNational

de la Recherche Scientifique (number 837), LAction Concertee Incitative Neurosciences integratives et computa-

tionnelles (Ministere de la Recherche, France), and project A99E01 from the Comision Asesora Cient fica de la

CooperacionArgentino-Francesa.Researchby D.G.wassupportedbyIsraelScienceFoundationGrant657/01and by

the National Science Foundation, under Agreement 0112050. We thank Y. Loewenstein, C. van Vreeswijk, B. Con-

nors,andY.YaromforstimulatingdiscussionsandB.Connors,Y.Loewenstein,andC.Meunierforcarefulandcritical

reading of this manuscript.

Correspondence should be addressed to Dr. David Hansel, Laboratoire de Neurophysique et Physiologie du

Systeme Moteur, 45 Rue des Saints- Peres, 75270 Paris Cedex 06, France. E-mail: dav [email protected]

paris5.fr.

Copyright 2003 Society for Neuroscience 0270-6474/03/236280-15$15.00/0

6280 The Journal of Neuroscience, July 16, 2003 23(15):62806294


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(2003). For Ired Icred, the solution of Equation 6 diverges in finite time.

This corresponds to the firing of an action potential. If one supplementsEquation 6 with the condition that the variable vred is reset to Vr V*immediately after vred reaches some threshold, VT, from below, this

yields a reduced model that accurately describes the dynamics of theneuron in the limit Iext

red3 Ic

red. In particular, the currentfrequencyrelationship( IFcurve) of this model andthe neuronbehave similarly inthis limit, for any value of the parameters Vr and VT. When Iext

red Icred is

not small, the reduced model is no longer an exact description of theneuronal dynamics. However, one can fit the parameters so that the re-duced model provides a good approximation of the IF curve of theneuron. The difference, VT Vr, will be called the reset depth.

It is convenient to rewrite the subthreshold dynamics of the reducedmodel in terms of the dimensionless variables vred and I

red defined by:

vred A

Cred0vred V* (7)

Ired A

Cred2 I

red02, (8)

where 0 has the dimension of a time. This yields:

0dv red

dt v red

2 Iext

red Ic

red. (9)

The variable vred isresetto V r A(Vr V*)0 /Cred whenever it reachesthe threshold value: V T A(VT V*)0 /Cred. Note that V r 0. Asimilar model has been used to study networks of excitatory and inhibi-tory neurons (Latham et al., 2000; Hansel and Mato, 2003).

The subthreshold dynamics (Eq. 9) need to be supplemented with amodel for the suprathreshold part of the membrane potential timecourse. Assuming that the width of the action potentials is much smallerthan the interspikes, we represent their time course by a -function ateach time a spike is fired. Therefore, the (dimensionless) reduced mem-brane potential of the neuron can be written:

V redt vredt

spikest tspike, (10)

where measures the integral over time of the suprathreshold part of anaction potential.

The model definedby Equations9 and 10willbe called theQIFmodel.To simplify notations of the reduced model, we will drop the index(red) and the tildes.

Phase reduction in the weak coupling limit. Let us consider a neuronfiring action potentials periodically with an interspike T. A small andinstantaneous perturbation applied at time t after a spike induces asmall change in the timing of the subsequent spikes. This change, whichdepends on t, or equivalently on 2t/T, can be characterized bya function, Z(), which measures the delay or the advance induced inthe firing times after the perturbation. A positive value ofZ() indicatesthat theperturbation advances thesubsequent spikes. A negative value of

Z() indicates a delay. The effect of noninstantaneous weak perturba-tions (such as spikelets) can be estimated by convolving the phase-response function (PRF), Z, with the perturbation. It can be shown thatthe dynamic behavior of a network of weakly interacting neurons can becompletelydescribedin terms of the response function in the frameworkof the phase reduction approach (Kuramoto, 1984; Ermentrout and Ko-pell, 1986;Hanselet al., 1993, 1995; Golomband Hansel, 2000; Neltner etal., 2000; Lewis and Rinzel, 2003). In this approach, a phase variable isassociated witheach neuronin thenetwork. This variable,i(i 1, . . . ,

N), measures the time elapsed since the last action potential fired byneuron i. For a network of identical neurons coupled with electricalsynapses, one can show (see Appendix) that the phase variables follow aset ofn first-order coupled differential equations:

di

dt v

j1

n

Jiji j it, i 1, . . . , N (11)

(i j ) the phase coupling between neurons i and j given by:

i j g

20

2

Zu iVu j Vu idu .

(12)

Jij is the connectivity matrix, and i(t) is a white noise with zero meanand variance

2

20

2

Zu 2du

with , the SD of the noise of the nonreduced dynamics.Numerical integration. In the simulations of the conductance-

based model, the differential equations were integrated using thesecond-order Runge-Kutta scheme with fixed-time step: t 0.01 msec. Averaged quantities (firing rate, CV, ) were com-puted over a time period of 1 sec after discarding a transient of500 msec.

ResultsSingle neuron and coupling properties of theconductance-based modelFiring properties of single neuronsIn the absence of persistent sodium and slow potassium currents(the control model), our conductance-based model neuronfires tonically for large enough external currents, Iext Ic 0.16A/cm 2. The frequency of the discharge in response to a step ofcurrent is plotted as a function of the step amplitude (IFcurve)in Figure 1A. For Iext larger than but close to Ic, the minimumvalue of the external current required for the neuron to fire, the

firing frequency can be arbitrarily small, because action poten-tials appear at Ic through a saddle-node bifurcation [for a defini-tion of a saddle-node bifurcation, see Strogatz (1994) and Rinzel

Figure 1. IFcurves of the conductance-based model neuron. Each panel corresponds to adifferentsetofintrinsicconductancevalues.Thefiringratewascomputedfromthesteady-state

response of the neuronto stepsof constant external current of different amplitudes.A, Controlcase: gK 9 mS/cm

2, gKs gNaP 0. B, gK 2.5 mS/cm2, gKs gNaP 0. Note that the

scale of the x-axis is one-half the scale used in A. The main effect on the IFcurve of thereduction ofgK is multiplicative (change in the gain). C, gK 9 mS/cm

2, gNaP 0.2 mS/cm2,

gKs 0.Themaineffectonthe IFcurveofthepersistentsodiumissubtractive(changeintherheobase). D, gK 2.5 mS/cm

2, gKs 0.2 mS/cm2, gNaP 0. The slow potassium current

reducesthegainoftheneuronandpreventslow-frequencyfiring.Italsolinearizesthe IFcurve(compare with B).

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and Ermentrout (1998)]. Decreasing gK does not significantly

modify the resting potential and the rheobase of the neuron, butit increases the gain of the IF curve (Fig. 1 B). The slow potas-sium (IKs) and persistent sodium (INaP) currents modify the on-setof firing, because these currents areactivated at rest. As shouldbe expected, the persistent sodium current increases the excit-ability of the neuron and reduces its rheobase, Ic (Fig. 1C). For

gNaP 0.08 mS/cm2, the neuron is spontaneously active. As

shown in Figure 1 D, the main effects of adding IKs on the IFcurve are a translation to the right, a linearization, and, if gKs islarge enough, a suppression of the ability of the neuron to fire atlow rates (because for large enough gKs, a subcritical Hopf bifur-cationoccurs at the onset of firing[for a definition of a subcriticalHopf bifurcation, seeStrogatz (1994)and Rinzeland Ermentrout

(1998)].The response of the neuron to a step of external current,

whose value is adjusted to give a firing frequency of 50 Hz, isplotted in Figure 2 for different combinations of the intrinsiccurrents. The action potentials are shown with a higher temporalresolution in Figure 3C. In the control case (Fig. 2A; and Fig. 3C,solid line), action potentials are narrow and are followed by astrong afterhyperpolarization (AHP) that brings the membranepotential 15 mV below its resting value. When gK is decreased(Fig. 2 B), theresting potentialof theneuron does notchange, butthe AHP is reduced and the membrane potential always remainsabove rest. This conductance also controls the width of the spikeand the depth of the AHP [Fig. 3C, compare the solid line (gK

9 mS/cm

2

) with the dotted line (gK 2.5 mS/cm

2

)].The persistent sodium and the slow potassium currents sig-nificantly modify the resting potential and the external current

required to adjust the firing rate. In contrast, they only slightlyaffect the depth of the repolarization, measured from threshold,after the action potential (Fig. 2C,D). The size and width of theaction potentials remain almost unchanged when the conduc-tances of these currents are varied [Fig. 3C, compare the dottedline with the dashed-dotted line (effect ofgKs) and the solid linewith the dashed line (effect ofgNaP)].

Properties of the spikeletsWhen a neuron fires an action potential, it induces a depolariza-tion, called a spikelet, in the neurons that are connected to it. Thespikelets generated by a neuron firing one isolated spike in re-sponse to a very brief but strong pulse of current (Fig. 3A) orfiring tonically at 50 Hz (Fig. 3C), are shown in Figure 3, B and D,respectively, for different parameters of the ionic currents.

In all four cases displayed, the width of the spikelets is muchlarger than the width of the presynaptic spikes that generate

them. This is because the spikelets are a filtered version of thepresynaptic membrane potential profile.Thesize of thespikeletis practically unaffected by thepresence

Figure 2. Theresponseoftheneurontoastepofcurrent.Astepofcurrentof200msecwasinjected into theneuron.No noisewas included. Themembranepotential beforeand after the

currentinjectionisindicatedtotheleftofeachpanel.Inallcases,theexternalcurrentduringthe

stepwaschosentoobtainadischargerateof50Hz.A, Control case.Externalcurrent,Iext, variesfrom0to1.10A/cm 2.Restingpotentialis 63mV.B,gK 2.5mS/cm

2,gKsgNaP 0. Iextvariesfrom0to0.48 A/cm 2. Note that theconcavityof themembrane potentialtimecourseisalwaysupwardincontrasttoA.C,gK 9mS/cm

2,gNaP 0.2mS/cm2,gKs 0.Theneuron

is hyperpolarized by the injection of a negative current, Iext 1.55 A/cm2, before the

current step, to prevent spontaneous firing. The step amplitude is 1.0 A/cm 2. D, gK 2.5mS/cm2, gKs 0.2 mS/cm

2, gNaP 0. The external current varies from 0 to 4.88 A/cm2.

The resting membrane potential is more hyperpolarized than in B, because of the slow potas-sium current that is activated at rest. The (shifted) time course of the membrane potentialduringthe dischargeis very similar tothe tracein B. Notethe transienthyperpolarizationright

after the switching off of the current, which is caused by the slow relaxation ofIKs back to itsresting value.

Figure3. Thespikeletsinducedfor differentcombinations of ioniccurrents. Solidline,gK 9 mS/cm 2, gKs gNaP 0 (control case). Dotted line, gK 2.5 mS/cm

2, gKs gNaP 0.Dashed line, gK 9 mS/cm

2, gKs 0, gNaP 0.2 mS/cm2. Dashed-dotted line, gK 2.5

mS/cm 2,gKs 0.2mS/cm2,gNaP 0.A, Anaction potentialgenerated by a shortand strong

current pulse. The duration and the amplitude A of the pulse are the same for all four traces: 1 msec and A 50 A/cm 2. The neuron was hyperpolarized to prevent firing whenpersistent sodium was present and to obtain the same initial membrane potential for all four

parametersets.Thesolidlineand thedashedline overlap. B, Spikeletinduced bya presynapticneuronfiringanactionpotentialasin A. Thesynapticconductanceisg 0.005mS/cm2,whichcorrespondstoacouplingcoefficient(Amitaietal.2002)CC 5%.Thepostsynapticneuronhasthesameintrinsicpropertiesasthepresynapticneuron.C,Aconstantexternalcurrentisinjected

tomake theneuron fire at 50 Hz.The external current is Iext 1.10A/cm

2

forthe solid line,Iext 0.48A/cm2 for the dotted line, Iext 4.88A/cm

2 for the dashed-dotted line, andIext 0.55A/cm

2 for the dashed line. The subthreshold voltage of the neuron when thepersistent sodium is added (dashed line) is below the control (solid line), although the persis-tentsodiumcurrentacceleratesthedepolarizationoftheneuron,becauseherewecomparethefiring patterns for the same firing rate. The external current is therefore smaller when thepersistent sodium is present than in the control case. D, Spikelets induced by a presynaptic

neuron firing tonically, as in C. The postsynaptic and the presynaptic neurons have the sameintrinsic properties.

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of the persistent sodium (Fig. 3B,D, com-pare the solid line with the dashed line), be-cause this current does not greatly affect theshape of the action potential. The only no-ticeable effect of the persistent sodium onthe spikelet is to slow down its decay timecourse. This is because it reduces the input

conductance of the postsynaptic neuron (by50% at 76 mV).

The modulation of the spikelets by thedelayed rectifier potassium current is pri-marily caused by changes in the presynapticneuron voltage time course. When gK is re-duced, the action potential is broader, andthis induces an increase in the amplitudeand the width of the spikelets. Although theslow potassium current does not greatly af-fect the width of the action potential, it con-tributes to bringing the membrane of thepresynaptic neuron transiently below itsholding potential. This explains why in Fig-ure 3B the spikelet is narrower and displaysa faster and deeper repolarization in thepresence ofIKs.

The spikelets generated by tonically fir-ing neurons at low firing rates are similar tothose in Figure 3B (data not shown). Forhigh-enough firing rates, the slow potas-sium saturates, affecting the dynamics pri-marily as would a constant hyperpolarizingcurrent. Therefore, its effect on the voltage traces and on thespikelets becomes less pronounced (Fig. 3, compare B and D).

Synchrony in the conductance-based network modelThe goal of this section is to show how synchrony properties ofour conductance-based network model depend on the intrinsiccurrents of the neurons. Further understanding of the effectsdescribed in this section will be provided in the next section byinvestigating analytically a simplified and more abstract model.

Synchrony is modified when intrinsic conductances are changed Apair of conductance-based neuronsThe effect of intrinsic currents on neuronal network dynamicscan first be demonstrated by studying the dynamics of a pair ofneurons. Here, as an example, we show that a persistent sodiumcurrent tends to promote antiphase locking, whereas a potassium

current promotes in-phase locking. In Figure 4A, the traces oftwo identical neurons (gK 9 mS/cm2; gKs gNaP 0) are

plotted over a time interval of 100 sec. Both neurons receive anoisy input that makes them fire at 50 Hz. These traces suggestthat the neurons fire action potentials in synchrony. This is con-firmed by the cross-correlation of the traces computed over alonger time interval (100 sec), which displays a strong peak cen-tered around t 0 msec, as shown in Figure 4 B. In contrast,when thepersistent sodiumconductanceis large enough(gNaP0.4 mS/cm2), neurons tend to fire in antiphase (Fig. 4C), and thepeak of the cross-correlogram is shifted to 10 msec (Fig. 4 D). If aslow potassium current is added (gKs 0.15 mS/cm

2) whilekeeping the persistent sodium conductance the same, antiphase

locking becomes unstable and neurons tend to fire in-phase (Fig.4 E, F). Therefore, a slow potassium current is a promoter ofsynchronous activity.

Large networks of conductance-based neuronsIn Figure 5 the synchrony measure, , is plotted against the con-ductances gK, gKs, and gNaP. Examples of raster plots of networkactivity are also displayed in this figure, for particular values of

the intrinsic conductances (Fig. 5, insets). In each of the panels,the external input (average deviation and SD) is kept constant.Therefore, when the intrinsic conductances vary, the average fir-ing rate of the neurons and the CV of their interspike intervalschange. In particular, the firing rate decreases (respectively in-creases) when the potassium (respectively the persistent sodium)conductances increase (top right figures in each of the panels inFigure 5).

In Figure 5A, the effect ofgK on the synchrony level is shown(gKs gNaP 0). When gK is small enough (gK 4.5 mS/cm

2),is very close to zero. A detailed study shows that in this region,is on the order of 1/N, where Nis the number of neurons inthe network, and vanishes in the limit of a very large network

(data not shown): non-zero values ofare attributable to finitesize effects, and the network is asynchronous. For gK of 4.5mS/cm2, increases rapidly up to 0.35. For example, in theraster plot shown in the right inset of Figure 5A (control situa-tion, gK 9 mS/cm

2), 0.34. This corresponds to a state inwhich the neurons fire together within time windows of approx-imately one-third of the period.

Figure 5B shows the effect of the slow potassium current forgNaP 0 and gK 2.5 mS/cm

2. For gKs 0, the neurons fireasynchronously. At gKs 0.06 mS/cm

2, starts to increase. Forlarge gKs, saturates at a value, 0.55, that is substantiallylargerthan in thecontrol situation. This value ofcorrespondstoa tight synchrony of the action potentials fired by the neurons

(Fig. 5B inset on the right).In Figure 5C, is plotted against gNaP for gK and gKs at theircontrol values (gK 9 mS/cm

2, gKs 0). Clearly, is a decreas-

Figure 4. Effect of persistent sodium and potassium currents on the synchronization of a pair of neurons: phase versus

antiphasestability.Resultsfrom numericalsimulations. The synaptic conductance isg 0.005mS/cm2

.TheSDofthenoiseis 0.3 mV/msec 1/2. In all of the simulations, neurons were firing in antiphase at the beginning of the simulation (data notshown).Spiketrainswerecomputedwithatimebinof1msec.Thespikecross-correlograms( B, D, F)werecalculatedfromspiketrainsrecorded over 100sec(after discarding a transientof 5 sec)and were normalized tothe expectednumber ofspikes duringthis time interval. Neurons fire at an average firing rate of 50 Hz. A, B, Traces of the voltage of the two neurons and thecross-correlograms of the spike trains for the control parameters and Iext 1.08A/cm

2. Neurons tend to fire in-phase. C, D,

Persistentsodiumcurrentstabilizesantiphaselocking.ParametersaregK9mS/cm2,gNaP 0.4mS/cm

2,andgKs0. Iext1.38A/cm 2. E, F,Slowpotassiumconductancedestabilizesantiphaselocking.ParametersaregK 9mS/cm

2,gNaP 0.4mS/cm2, and gKs 0.15 mS/cm

2. Iext 0.80A/cm2. Neurons tend to fire in-phase.



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ing function of gNaP. When gNaP is large enough (gNaP 0.1mS/cm2), the neurons fire asynchronously.

In the results presented in Figure 5, the firing rate of the neu-rons is not controlled. Therefore, the origin of the variation ofwith the intrinsic current conductances is not clear. It may beattributable to the fact that, in general, synchrony depends on the

firing rate for given intrinsic currents. It might also be a result ofthe fact that dynamic properties of neurons with different intrin-siccurrentsmay be differenteven if they have thesame firingrate.

Dependence of synchrony on intrinsic currents at fixed firing ratesTo clarify further how potassium and sodium currents affect thesynchrony of the network, we performed another set of numeri-cal simulations in which the external input and the noise levelwere tuned to keep constant both the average firing rate of theneurons and the CV of their spike trains when the conductances

of the intrinsic currents were changed. The results are shown inFigure 6 (three panels on the left). In all of the simulations, theaverage firing rate of the network was 50 Hz and the CV 0.12(Fig. 6, panels on the right).

The level of synchrony increases with the potassium conduc-tances. For small-enough gK or gKs, the activity of the network isweakly correlated. A detailed study of the dependence ofwiththe network size reveals that for gK (respectively gKs) smaller than

g*K 4 mS/cm2 (respectivelygKs smaller thang*Ks 0.075 mS/cm

2),ison the order of1/N(data not shown). Therefore in this rangeof conductances, the network is in theasynchronous state.Atg*K and

g*Ks, a transition to a synchronous state occurs, and beyond thesevalues the network settles into a synchronous state. This occurs al-

though the level of noise has been increased.In contrast to the potassium conductances, which promotesynchrony, the persistent sodium current impedes it, as shown in

Figure 5. Effectofintrinsiccurrentsonthesynchronizationofalargenetwork.Resultsfromnumerical simulations. Details on the simulation and the network parameters are given in

Materials and Methods. The synaptic conductance is g 0.005 mS/cm2

. In all of the simula-tions, the SD of the noise was set at 0.6 mV/msec 1/2. The synchrony level, (left), theaveragefrequency,andtheaverageCVoftheneuronaldischargesareplottedasafunctionofanintrinsicconductancethatisvaried.A,Theconductance,gK,varies;gKsgNaP 0.Theexternalcurrent is Iext 0.8A/cm

2. For gK g*K,increases and saturates. Inset, Raster plot repre-senting the spiking times of 100 neurons for gK 9 mS/cm

2 (neurons are arrayed vertically,and the x-axis corresponds to time where the scale denotes 25 msec). Panels on the right

indicate that the average frequency of the neurons decreases when gK increases, because thegain of the IFcurve decreases. The variability of the neuronal discharge varies slightly andnonmonotonically in the investigated range ofgK. B, The conductance, gKs, varies; gK 2.5mS/cm2,gNaP 0.The externalcurrent is Iext 2A/cm

2.Inset,RasterplotsareforgKs 0andforgKs 0.1mS/cm

2.AtlargevaluesofgKs,aslightdecreaseoccurs,buttheactivityofthenetworkremainsstronglysynchronized.PanelsontherightindicatethattheaveragefiringrateoftheneuronsisstronglyreducedwhengKsvaries,primarilybecauseIKshasastrongsubtractiveeffect on the IFcurve of the neurons (the rheobase increases with gKs). C, The conductance,

gNaP,varies;gK 9mS/cm 2,gKs 0.Theexternalcurrentis Iext0.8A/cm 2.Thesynchronyis reduced when gNaP increases. Inset, Raster plot for gNaP 0.2A/cm

2. Panels on the rightindicatethattheaveragefiringrateoftheneuronsincreaseswith gNaP,primarilybecauseofthe

subtractive effect ofINaP on the IFcurve of the neurons (the rheobase is reduced when gNaPincreases).

Figure 6. The synchrony level depends on the intrinsic currents when the frequency is keptconstant.Results fromnumericalsimulations. Details on the parametersare givenin Materials

and Methods. The synaptic conductance is g 0.005 mS/cm2

. The external current, Iext, andtheSDofthenoise,, werechangedto control theaveragefiringrate andtheaverageCVof theneurons.In allof thesimulations,theaveragefiring rate is50 Hz 10%,andtheCV 0.12.Ineach of the panels, the corresponding values ofIext andare plotted versus the conductance

that is varied (panels to the right). A, The conductance, gK, varies; gKs gNaP 0. B, Theconductance,gKs, varies;gK 2.5mS/cm

2,gNaP 0. C, The conductance,gNaP, varies;gK 9mS/cm 2, gKs 0.



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Figure 6C. For large-enoughgNaP, synchrony is even destroyed. Asharp transition to the asynchronous state occurs for gNaP 0.15mS/cm2.

Relying on the way intrinsic currents affect spikelet modula-

tion (Fig. 3), one might naively expect that potassium currentsthat reducethe size of the spikelets shouldalso reducesynchrony,and that persistent sodium current, which increases this size,should promote it. As a matter of fact, we found a trend that wasexactly the opposite.

Firing rate affects synchrony differently when different intrinsiccurrents are involvedWe also investigated in what ways the network dynamic statedepends on the firing rate of the neurons. The firing rate waschanged by varying the average external input, and was plottedas a function of the frequency of the neurons for several combi-nations of potassium and sodium conductances.

Figure 7 plots the synchrony measure, , versus the average

firing rate of the neurons for four sets of conductance parametersandtwo levelsof noise.The qualitative behavior ofas a functionof the firing rate depends on the intrinsic currents involved.

As expected, is always larger for the smaller level of noise( 0.3 mV/msec1/2) (Fig. 7,circles) than for the largerone (0.6 mV/msec1/2) (Fig. 7, squares). In particular, for 0.6 mV/msec1/2, the noise is strong enough to prevent synchrony in theentire range of firing frequencies for gK 3.5 mS/cm

2 (Fig. 7B)and for gNaP 0.15 mS/cm

2 (Fig. 7C). This level of noise issufficient to destroy synchrony as well in the control situation,but only at 20 Hz (Fig. 7A).

In thecontrol case,increases monotonously with thefiring rate(except for a small decrease followedby a slight increase between 50

and100Hzfor 0.3mV/msec

1/2

). In contrast, forsmallergK (e.g.,gK 3.5 mS/cm2), we find a nonmonotonous variation of syn-

chrony: increases at low firing rates but decreases again for firing

rates of20 Hz.Thenetworkstate becomesasynchronousforfiringrates of60 Hz (Fig. 7B, squares). If the slow potassium current isadded (gKs 0.15 mS/cm

2), again varies monotonously with thefiring rate (Fig. 7D). Finally, with enough persistent sodium currentand not overly strong noise (gK 9 mS/cm

2, with gNaP 0.15mS/cm2, gKs 0) (Fig. 7C, squares), varies nonmonotonouslywith the firing rate, as in the reducedgK case (Fig. 7B).

Synchrony in networks of quadratic integrate-and-fireneurons depends on the phase response functionThePRF,which characterizes how neurons respondto small pertur-bations, is a key concept to understand the relationship betweenintrinsic properties of neurons and their collective dynamics(Kuramoto, 1984; Hansel et al., 1993; van Vreeswijk et al., 1994;Kopell and Ermentrout, 2002). This function depends on the excit-ability properties of the neurons and therefore is determined by theintrinsic currents involved in their dynamics. One expects that thedifferences in the dynamic behavior of our conductance-based net-work model fordifferent sets of parameters reflect thechanges in theexcitability properties of the neurons. However, relying on numeri-calsimulations alone, itwould verydifficult toestablish generalprin-ciplestorelatetheexcitability properties andtheneuronal PRFto thesynchronization properties. Below, we consider a network of QIFneurons (Eqs. 9, 10) fully connected by electrical synapses. As weshow below, the qualitative properties of the excitability of the QIFneurons crucially depend on the parameters (the threshold and thereset voltages and the external current). Thanks to its relative sim-plicity, this model can be investigated using analytical techniques. Itreveals how, in the framework of this model, one can relate thestability of antiphase locking of a pair of neurons and thestability ofthe asynchronous state of a large network to the shape of the PRF oftheneurons.Subsequently, we showthat similarrules canbe appliedto conductance-based models, and that they provide a unifiedframework explainingthediversity of behaviorfoundin ournumer-

ical simulations of the conductance-based model.

The membrane potential and the PRF of QIF neuronsIn the Appendix, it is shown that between two successive spikesthe subthreshold membrane potential of a QIF neuron reads:

v(t)Iext Ic tantIext Ic0 tan1Vr

Iext Ic, (13)

where tmeasures the time elapsed since the first spike. The firingperiod, T, is determined by the condition: v(T) VT. Therefore:

T 0

tan1 VT

Iext

Ic

tan1 Vr

Iext

Ic

Iext Ic

. (14)

In the weak coupling limit,the dynamics of the network are com-pletely determined by the phase-response function, Z() (seeMaterials and Methods), which depends on three parameters,namely, the firing frequency of the neurons (or equivalently theexternal current Iext), thethresholdVT, and thereset potential Vr.The response function of the QIF model can be derived analyti-cally as shown in the Appendix. One finds:

Z 0

vT22

Iext Ic

. (15)

Changing the ratio Vr/VT for a fixed reset depth, VT Vrhas a strong influence on the shape of the trajectory of the phase-

Figure 7. Synchrony varies with the firing rate in a way that depends on the intrinsic cur-rents. Results from numerical simulations. Details on the simulation parameters are given inMaterialsandMethods.Thesynapticconductanceisg 0.005mS/cm2.Theaveragefiringrateof the neurons is varied by changing the external current. In each of the panels, the results aredisplayedfortwonoiselevels.Circles, 0.6mV/(msec)1/2.Squares, 0.3mV/(msec)1/2.

A, gK 9 mS/cm2, gKs gNaP 0. The external current was varied between 0.15 and 2.7

A/cm 2 (100 Hz).B,gK 3.5 mS/cm2,gKs gNaP 0. The external current, Iext,wasvaried

between 0.11 and 1.4 A/cm 2. C, gK 9 mS/cm2, gKs 0, gNaP 0.15 mS/cm

2. Iext wasvaried between 1.17and0.8A/cm 2. D,gK 3.5mS/cm

2,gKs 0.15mS/cm2,gNaP 0.

Iext was varied between 2.3 and 7.2 A/cm2. The firing rate cannot be reduced to 20 Hz

because of the presence of the slow potassium cur rent (see also Fig. 1 D).



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response function of the neuron, as shown in Figure 8. In Figure8A, the membrane potential of a QIF neuron firing at 50 Hz isplotted for three values of the ratio Vr/VTand a fixed reset depth:VT Vr 3. For Vr/VT 0.1, the concavity of the subthresh-old trajectory is always directed upward. In contrast, for Vr/VT 10, it is always downward. For Vr/VT 1, the concavitychanges from upward to downward.

It is easy to see that Z() 0 for all , and that it is with onemaximum. Thelocationof themaximum ofZdepends on Vr, VT,and Iext. This is depicted in Figure 8, where the voltage trace andthe response function are plotted for different values of theseparameters. For Vr/VT 1, Z() is symmetric and its maxi-mum is always at . For Vr/VT 1, the maximum ofZislocated in the first half of the firingperiod, whereas forVr/VT

1, it is located in the second half. The value of Z at the maximumalso depends on Vr, VT. It has smaller values when Vr/VT 1.The larger the firing rate, the stronger the dependence of Z onVr/VT(Fig. 8, compare B, for 50 Hz, and C, for 10 Hz).

A pair of identical QIF neurons without noiseWhen, in the absence of noise, two identical neurons interact in asymmetric way, they reach, at large time, a phase-locked state inwhich the two neurons fire action potentials with a fixed phaseshift, . In the Appendix, it is shown that is a solution to theequation:

0, (16)

wherethe function

() 1/2(() ()),and ()istheeffective phase coupling (see Materials and Methods). Because ofthe symmetryof the systemand the 2periodicity of the function

, there are always at least two solutions tothis equation: 0 and . The firstsolution corresponds to the state in whichthe two neurons fire in-phase (in-phaselocking), the second one to the state inwhich they fire in antiphase (antiphaselocking). Other solutions with 0

may also exist. However, only solutions thatare stable can be reached at large time.Therefore, out of all of the solutions toEquation 16, only those that satisfy the con-dition (see Appendix):

d

d 0 (17)

correspond to phase-locked states that thetwo neurons can eventually reach startingfrom appropriate initial conditions. In thefollowing, we focus on the stability of theantiphase state of a pair of QIF neurons.

Stability conditions of antiphase lockingThe voltage trajectory consists of threeparts: (1) a subthreshold part, during whichthe membrane potential increases with timefrom a value Vr to a threshold value VT; (2)a spike, modeled by a -function of ampli-tude, and (3)a resetting, where potentialisinstantaneously brought back to Vr. Then,as explained in the Appendix, antiphaselocking of a pair of QIF neurons is stable if:

d

d gSsp Sr Ssub 0

(18)

with:

Ssp vdZ

d (19)

Sr VT Vr)Z() (20)

Ssub 1

20

2

Zdv

d d. (21)

Thefirstterm,Ssp, is thecontribution of thepresynapticspikesto the destabilization of antiphase locking. The second term, Sr, isthe contribution of the instantaneous reset of the membrane po-tential. The last term, Ssub, corresponds to the effect of the cou-pling between the two neurons when the presynaptic neuron issubthreshold. These three terms are plotted in Figure 9A as afunction of the ratio Vr/VT, for 1, VT Vr 3, and afrequencyv 50 Hz.

The terms Sr and Ssub (Fig. 9, dotted line and dashed-dottedline, respectively) are invariant under the transformationVr/VT 3 VT/Vr. That is why the corresponding curves inFigure 8B are symmetric around Vr/VT 1 (note the logarith-mic scale of the x-axis). Sr is always positive and therefore desta-

bilizes antiphase locking. In contrast, Ssub is always negative andleads to stabilized antiphase locking. The sum of the two terms isalso plotted in Figure 9A (double-dotted-dashed line) to show

Figure 8. Voltage traces and phase-response functions of the QIF neuron. The voltage traces and the phase-response func-tions were computed using Equations 13 and 15, respectively. 0 10 msec. In A and B, the external current is such that thefiring rate is 50 Hz. A, The voltage trace of the neuron in response to a constant injected current for three values of the ratioVr/VT. The reset depth is the same in all three cases: VT Vr 3. Note the changes in the concavity of the voltage tracebetweenthespikeswhenVr/VTincreases.TheexternalcurrentisIext0.97for Vr/VT 1andfor Vr/VT10,and Iext0.66 forVr/VT 0.1. B, The phase-response function,Z,forthesamecasesasinA. C,Sameasin B,butforafiringrateof10Hz. The phase-response changes less than in B when Vr/VTvaries.



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that their overall contribution is destabilizing except for verysmall or very large values ofVr/VT.

Because the maximum ofZ moves from the first part of theperiod to the second part of the period when Vr/VT crosses 1(Fig. 8), the sign of the derivative ofZatalso changes from

negative to positive, and the term Ssp (dashed line) tends to de-stabilize antiphase locking for Vr/VT 1. The sum of all threeterms ofthe right sideof Equation18 isplotted inFigure 9A (solidline). Antiphase locking is stable only if the phase at which Zreaches its maximum is small enough.

The phase diagram for the stability of antiphase locking as afunction of Vr/VT and the firing frequencies is displayed inFigure 9C. The stability of antiphase locking requires thatVr/VT be small enough. Moreover, the range of values ofVr/VT where it is stable increases with the frequency of theneurons. This is because the contribution of the term Ssp in theright side of Equation 18 grows with the firing rate and tends todominate the contribution of the two other terms. In contrast, at

low frequencies, the response function weakly depends onVr/VT(Fig. 8), and Ssp, which is proportional to the frequency,is negligible. Moreover, Sr Ssub is positive except for very small

or very large values ofVr/VT. Therefore, for small frequencies,the domain of stability of antiphase locking is very narrow.

This analysis shows that for a pair of neurons, the stability ofantiphase locking depends critically on the shape of the phase-response function and in particular on the sign of the derivativeof the response function at the mid-period. The more skewedtoward the right (the second part of the firing period) the re-

sponse function of the neurons, the more unstable antiphaselocking becomes.

Large networks of weakly coupled QIF neuronsThe stability analysis of the asynchronous state of a large networkof neurons coupled all-to-all receiving a noisy input is also sim-plified if one assumes weak coupling. Using the phase-reductionapproach, it can be shown (see Appendix) that the asynchronousstate is stable if:

n ngv ImZn n g ImZnvn) n22Z0

2 0.

(22)

For all integers, n 0, with Zn and vn the nth-Fourier compo-

nents of the function Z() and v(), respectively (Eqs. 13, 15)defined byZn 1/20

2Z()e 2ind and a similar equationfor vn. Iffor some n,n is negative (respectively positive), it is saidthat the mode of order n is stable (respectively unstable). For theasynchronous state to be stable, the modes at any order must bestable. It is unstable when at least one mode is unstable.

The first two terms in Equation 22 correspond to the effect ofthe interaction. The first term represents the effect of the spikesand can be written:

n nq

2 0

2

sinnZd. (23)

The second term, n ng Im(nZnvn), corresponds to the com-bined effect of the reset of the membrane potential and its subse-quent subthreshold evolution.

The sign of their sum depends on the parameters, Vr, VTandon the frequency of the neurons, v. The last term in Equation 22corresponds to the effect of the noise. It is always positive. There-fore, as should be expected, noise increases the stability of theasynchronous state (because of the negative sign in front of thisterm in Eq. 22). The stability of the asynchronous state dependson the competition between the first two terms and the last term.In particular, the sign ofn depends on the ratio

2/g. Note thatbecause of the factor n2 in the last term of Equation 22, thestability of the modes increases rapidly with their order.

We first consider the case of a noiseless network (i.e., with0). In Figure 9B, we plotted 1 (dashed line) and 1 (dashed-dotted line) against Vr/VT. The qualitative behavior of thesequantities is similar to those ofSsp and Ssub Sr, respectively, fora pair of neurons. In particular 1 is symmetric (in a semiloga-rithmic scale) around Vr/VT 1. Moreover, 1 increasesmonotonically from negative values to positive values whenVr/VTincreases, and it changes sign at Vr/VT 1.This can beeasily explained. Indeed, for Vr/VT 1 (respectively, Vr/VT 1), the functionZ() is skewed toward(respectively)where sin() 0 [respectively sin() 0]; therefore, for n 1, theintegral in Equation 23 is negative (respectively positive) and1 0(respectively1 0). In particular,becausefor Vr/VT 1,Z() is

symmetric around , the integral in Equation 23 vanishes forn 1 [becausesin() sin()]. The signof1 (solid line) isprimarily determined by the sign of 1. It is negative for small

Figure 9. Stability of the asynchronous state of QIF networks at weak coupling in the ab-senceofnoise.ResultsfromtheanalyticalstudyoftheQIFmodel.Timeconstant 0 10msec.The size of the spikes is 1. A, A pair of identical neurons. The quantity () is plotted(solidline). Thethree contributions tothis quantity (Eq. 21)are also plotted:Ssub (dotted line),

Sr (dashed-dotted line), and Ssp (dashed line). The double-dotted-dashed line corresponds toSr Ssub. All of these quantities were computed from Equations 21, 13, and 15. Note thelogarithmicscaleonthex-axis.Theneuronsfireat50Hz. B,Alargenetworkofall-to-allweaklycoupledidenticalneurons:thequantity1.Theasynchronousstateisstablewithrespecttothemode n 1 if1 0. The quantities1 (dashed line) and1 (double-dotted-dashed line)are also plotted. The firing rate of the neurons is 50 Hz. C, The phase diagram of antiphaselocking fora pair of identicalneuronsin theweakcouplinglimit.Notethe logarithmic scale onthe x-axis. The solid lines separate the region in which antiphase locking is unstable (in the

middle) from the regions in which it is stable (on the left and in the small region on the right).TheparametersVr and VTarevaried,whereastheresetdepthiskeptconstant: VT Vr 3.Theboundary linesare defined by thecondition: () 0. D,Thephasediagramforthestabilityof the asynchronous state in a large network of all-to-all coupled neurons. The solid line is theboundary between the domains in which the asynchronous state is stable (to the left) andunstable (to the right).



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Vr/VTand becomes positivefor Vr/VT 1. Thereforethemoden 1isstableonlyifVr/VTis small enough.Similaranalysis canbeperformed for the modes n 1. It reveals that ifVr/VT is not toolarge, the mode n 1 determines the stability of the asynchronousstate.

The phase diagram for the stability of the asynchronous stateas a function of Vr/VT and the firing frequency is plotted inFigure 9D. It is very similar to the phase diagram for the stability

of the antiphase state for a pair of neurons (Fig. 9C). The onlyqualitative difference between these two phase diagrams is thatfor a pair of neurons, antiphase locking is stable at low firing ratesand very large Vr/VT, whereas for a large network, the asyn-chronous state is unstable in that limit. Therefore, we can con-clude that neurons coupled with electrical synapses will be moreeasily synchronized if their phase-response functions are skewedto the right than to the left.

The phase diagram for the stability of the asynchronous stateis plotted in Figure 10 for different levels of noise. For large-enough firing rates, the instability lines for 2/g 0 and 2/g0.01 are very close to each other. The distance between the twolines increases when the frequency decreases. At some value,

*, the two lines separate completely. At *, the line for 2

/g0.01 changes direction and continues toward the right of thephase diagram (large Vr/VT). In contrast, the line for 0continues toward the left of the diagram. In particular, for2/g0.01, the asynchronous state is stable for any value ofVr/VTforsmall-enough firing rates. As a matter of fact, one can show ana-lytically that the asynchronous state is stable for any value ofVr/VT if the firing rate is smaller than a critical value, *

(2/g),which vanishes with 2/g. The size of the region in which theasynchronous state is stable increases with the noise level, as ex-pected. In particular,

*, and the frequency range in which the

asynchronous state is stable for all Vr/VT, increases with 2/g.

How synchrony depends on the ratio Vr/VT at fixed fre-

quencies can be deduced from the phase diagram in Figure 10.For low and moderate noise levels (e.g., 2/g 0.01 and 2/g0.05), we find three generic behaviors as a function of Vr/VT:

(1) At low firing rates, the asynchronous state is stable for allVr/VT. (2) At high firing rates, the asynchronous state is stablefor small Vr/VTand becomes unstable when Vr/VTincreases.(3) In some intermediate range of firing rates, the stability of the

asynchronous statevaries nonmonotonously with Vr/VT: whenVr/VT increases from 0, the asynchronous state is first stable,then unstable, and stable again for large Vr/VT. The size of theintermediate region in which the asynchronous state is unstabledecreases when the noise level increases.

Figure 10 also allows us to predict how the stability of theasynchronous state depends on the frequency for fixed Vr/VT.For small-enough Vr/VT, the asynchronous state is stable at allfiring rates. The range of Vr/VT where this happens is largerwhen the noise is stronger. For large-enough Vr/VT, the asyn-chronous state is stable at low firing rates and becomes unstablewhen the firing rate is large enough. If the noise is not too strong,the stability of the asynchronous state varies nonmonotonously

in some intermediate range ofVr/VT 1. In this domain, theasynchronous state is stable at low rates, loses stability when therate increases, but becomes stable again at high firing rates.

Interpretation of the simulation results of theconductance-based networkOur analysis of the synchrony properties of QIF networks showsthat the stability of the asynchronous states depends crucially onthe shape of the phase-response function,Z(), and in particularon the location of its maximum. This suggests that the desyn-chronization effect of the persistent sodium and the synchroni-zation effect of the potassium currents revealed by our numericalsimulations may be related to the way these currents shape the

phase-response functions of the neurons.The function, Z(), for our conductance-based neuronmodel is plotted in Figure 11 for different values of gK, gKs, and

Figure 10. Stability of the asynchronous (AS) state in a large network of all-to-all weaklycoupledQIFneuronsinthepresenceofnoise.Timeconstant 010msec.Thesizeofthespikesis 1. Thesolidlineseparates theregionin whichthe asynchronousstateis stablefromthe

regioninwhichitisunstablefor2

/g 0.05.Thedashedlineisfor2

/g 0.01.Forcompar-ison,theinstabilityboundaryisalsoplottedforthenoiselesscase(dottedline).TheparametersVr and VTare varied, but the reset depth is kept constant: VT Vr 3. Note the logarithmicscale on the x-axis.

Figure 11. The phase-response functions depend on the intrinsic currents. The

conductance-based model is defined in Materials and Methods. The phase-response functionwas computed for the same four sets of intrinsic conductances as in Figure 1 using XPPAUTsoftware (Ermentrout, 2002). For each set of parameters, the external current is such that thefiring frequency of the neuron is 50 Hz. A, Control cases: gK 9 mS/cm

2, gNaP gKs 0.Externalcurrentis Iext 1.10A/cm

2. B,TheconductanceofthedelayedrectifierisreducedtogK 2.5 mS/cm

2 (gNaP 0, gKs 0 as in the control case). External current is Iext 0.48A/cm 2. C, gNaP 0.2 mS/cm

2 [other currents as in the control (gK

9 mS/cm 2, gKs

0)].Externalcurrentis Iext 0.55A/cm

2. D,TheparametersareasinB,butgKs 0.2mS/cm2.

External current is Iext 4.88A/cm2.



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gNaP. In Figure 11AC, Z() 0 for all , except in a very tinyregion after the spike peak. In Figure 11 D, the region of negative

Z() is larger. However, in all four cases, negative values ofZ()are much smaller than positive values. This means that most ofthe time, a small and brief depolarizing perturbation advancessubsequent firing of action potentials.

In allof thecases presentedin Figure11, the functionZ()has

only one maximum, whose location depends on the intrinsiccurrents. Comparison of Figure 11A with Figure 11 B shows thatthe maximum ofZis shifted to the right when gK increases, andthat the global shape of the phase-response function is moreskewed to the right (respectively to the left) for large (respectivelysmall) values of gK. The slow potassium current has an effectsimilar to that seen for the delayed rectifier current. It distorts thephase-response function toward the second half of the firing pe-riod. Therefore, increasing the potassium conductances has asimilar effect on the phase-response functionof the conductance-based neuron as increasing Vr/VTdoes for the QIF neuron. Incontrast, the persistent sodium current shifts the peak of the re-sponse to the left and skews the shape of the phase-responsefunction in that direction, as seen by comparing Figure 11A withFigure 11C. Hence, increasing gNaP in the conductance-basedmodel and decreasing Vr/VT in the QIF model have the samequalitative effect.

We are now able to understand the results of our numericalsimulations. In Figure 6, A and B, the network state is asynchro-nous for small potassium conductances (both delay-rectifier andslow-potassium) and becomes synchronous when these conduc-tances are increased. A similar behavior is found in the QIF net-work forv v

*(): theasynchronous state is stablewhen Vr/VT

is small, and the asynchronous state is destabilized when Vr/VTincreases enough. This stems from the fact that the qualitativechanges in the phase-response function of the conductance-based neuron when gK increases and the QIF neuron when

Vr/VTincreases are qualitativelysimilar. Conversely, increasingthe conductance of the persistent sodium current has the sameeffect on the phase-response function of the neuron as a decreaseofVr/VT. This explains the results of Figure 6C.

The phase diagram of Figure 10 also explains the differentbehaviors we found in our simulations for fixed-current conduc-tances when the frequencyvaries (Fig. 7) in presence of noise. Forinstance, in the control case, the response function is skewed tothe right. This is equivalent to Vr/VT 1 for the QIF model.This explains the monotonous transition from the asynchronousstate at low firing rates to synchrony when the firing rate in-creases, as shown in Figure 7, A and D. The nonmonotonousbehavior in Figure7, B and C, can also beexplained. In Figure 7C,

gK is smaller than in the control model. The corresponding effec-tive value ofVr/VTis in the intermediate range, where the asyn-chronous state changes stability nonmonotonously with the fre-quency. A similar argument can account for the results in Figure7B, but here the change in the phase-response function is attrib-utable to the persistent sodium current.

DiscussionHow intrinsic currents affect synchrony in networks ofneurons connected by electrical synapsesIn this work, we have derived explicit rules for the stability of theasynchronous state in networks of neurons interacting via elec-trical synapses. We have shown analytically how the shape of the

PRF, which depends on the model parameters, determines thestability of the asynchronous state in QIF networks. We haveextended these results to provide a unified framework explaining

the diversity of behavior found in our numerical simulationswhen potassium and sodium conductances are changed.

One can understand intuitively how potassium and sodiumcurrents shape the PRF. The current, IK, increases the refractori-ness of the neuron; therefore, it reduces its responsiveness tosmall depolarization after a spike. A similar effect, but strongerand more lasting, occurs with IKs. This intuitively explains why

we found that largerconductances of these currents skew thePRFtoward the second half of the firing period. Similar effects havebeen found by Ermentrout et al. (2001) for M current and AHPpotassium current. The current INaP is an inward current. It isalready activated near rest, and depolarizing perturbations am-plify this activation. Hence INaP increases the responsiveness ofthe neuron after a spike and shifts the maximum of the responsefunction towardthe first half of theperiod, as we have found here.

We predict that potassium currents, like IK and IKs, tend topromote synchrony of neurons coupled via electrical synapsesand that, in contrast, INaP tends to oppose it. Our other predic-tions concern the dependency of the synchrony level with thefiring rate. We expect that these conclusions do not depend onthe particular models we have chosen for the intrinsic currents.

Assumption of weak couplingThe stability of the asynchronous state of a large network of all-to-allconnected QIFneurons can be calculated analyticallyin theabsence of noise for any coupling strength using the populationdensity method (Abbott and van Vreeswijk, 1993; Hansel andMato, 2003). An alternative approach is the phase-reductionmethod. Although it is only mathematically justified for weakinteractions, it leads to nontrivial results that remain valid in areasonable range of interaction strengths. Moreover, it isstraightforward to study the effect of noise in this framework.These were themotivations to usethe phase-reductionmethod inthe framework of which all of the analytical results of this paper

were derived.At finite coupling strength, an additional instability of the

asynchronous state appears at low firing rates and small Vr/VT.It corresponds to the impossibility of controlling low firing rateswhen the coupling is too strong and the AHP is too small. This isbecause for a small AHP, the recurrent electrical synapse interac-tions tend to depolarize the neurons on the average, and thisincreases their firing rates. This instability is similar to the rateinstability in networks of excitatory neurons (Hansel and Mato,2003).

A detailed analysis of the QIF network at finite couplingstrength shows that in a broad range of coupling strengths, thephase diagrams for two coupled neurons and for large networks

are similar to those derived here, except for the additional rateinstability line. A full report of this analysis will be publishedelsewhere.

Simulation results were performed here for a coupling con-ductance,g 0.005 mS/cm2. This corresponds to coupling coef-ficients (Amitai et al., 2002) CC 5% and to a spikelet size of0.5 mV (Fig. 3). We have verified that similar results have beenfound for conductances four times larger (data not shown).Therefore the conclusions of the present work are relevant forelectrical synapse conductances in the physiological range (Ga-larreta and Hestrin, 2001, 2002; Amitai et al., 2002).

Related works

Stable antiphase locking has been demonstrated in previous stud-ies for a pair of neurons coupled via electrical synapses. Shermanand Rinzel (1992) used numerical simulations to show that this



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nonintuitive phenomenonoccurs in a simple conductance-basedmodel. However, they did not relate it to the intrinsic propertiesof their model. Antiphase locking was found by Han et al. (1995)in simulations of coupled MorrisLecar oscillators. For thismodel, they computed the effective phase interaction, , andshowed that their simulation results could be predicted from it.Ourwork further clarifies theconditions of stableantiphase lock-

ing in that it gives general qualitative criteria that the PRF mustsatisfy, which allows us to predict how it depends on intrinsiccurrents.

Phase locking of neurons connected by electrical synapses hasbeen also investigated analytically (Chowand Kopell, 2000; Lewisand Rinzel, 2003). In both studies, neurons were modeled in thesubthreshold range as leaky integrators, but the spikes were de-scribed in different ways. Chow and Kopell (2000) used a phe-nomenological model to describe the time course of spikes,whereas Lewis and Rinzel (2003), like us, modeled them with a-function. Both studies found stable antiphase locking at lowfiring rates and destabilization for large-enough firing rates. Thiscan be understood using Equations 1821 as follows. For a pas-sive integrator,Z(),Z() 1/[I

extT] exp(T/2

m),whereT is

the firing rate, Iext is the external current, and m is the neuronalmembrane time constant (Kuramoto, 1991; Hansel et al., 1995).The value ofZdiverges exponentially with the period at all , asdo Ssub and Sr. Detailed analysis reveals that the most divergent ofthese terms is Ssub 0. Hence, for low-enough firing rates,() 0. The term Ssp is positive (Z() 0) and decreaseswith T, whereas Ssub and Sr increase. Hence, for a large-enoughfrequency, () changes sign. Therefore for passive integrators,antiphase locking is stable at low rates and loses stability as thefiring rate increases. In contrast, we have found that the QIFmodel behaves differently than the LIF except in the limit of verylarge Vr/VT. Therefore our results show that the predictions ofChow and Kopell (2000) and Lewis and Rinzel (2003) may be

relevant only for neurons with strong potassium currents.

Relationship to experimentsElectrical synapses connect fast spiking (FS) as well as low-threshold spiking (LTS) interneurons (Kawaguchi and Kubota,1997; Gibson et al., 1999). Recently, the synchrony level of pairsof LTS and FS cells (characterized by the amplitude of the peak oftheir spikes cross-correlation) has been studied as a function ofthe firing rate v(Mancilla et al., 2002). It has been found that forFS cells, synchrony increases with v but decreases for LTS cells.Our results may explain this difference if one assumes that INaP isstronger in LTS than in FS cells and/or that potassium conduc-tances are stronger in FS than in LTS cells. This assumption is

compatible with thefact that FS andLTS cells have differentfiringpatterns: in response to current injection, the action potentials ofFS but not LTS cells are followed by a pronounced AHP, whichmay reflect stronger potassium conductances in FS than in LTScells. A test of this explanation would be to show that the PRFs ofthese LTS and FS cells have different shapes.

The dynamic clamp technique (Sharp et al., 1993) makes itpossible to artificially manipulate the intrinsic currents of theneurons and to couple two neurons via artificial electrical syn-apses. It provides a nice way of further testing our predictions.Therefore, it enables systematic verification of the effects of so-dium and potassium currents on the synchronization propertiesof a pair of neurons (Fig. 4) and how they correlate with their

PRF.Several studies in the mouse and the rat have reported thepresence of electrical synapses in hypoglossal and other inspira-

tory brainstem and phrenic motoneurons (Mazza et al., 1992;Rekling and Feldman, 1998). Recently, it has been found thatinspiratory motoneuron synchrony is modulated by electricalsynapses: blocking these synapses increases their level of syn-chrony (Bou-Flores and Berger, 2001). An explanation of thesurprising findings of Bou-Flores and Berger (2001) is suggestedby ourwork. Indeed, since strong INaP (Powers and Binder, 2003)

is present in these neurons, electrical synapses can reduce thelevel of synchrony of their activity compared with the case inwhich electrical synapses are blocked. Synchrony in the lattersituation would be attributable, for example, to GABAergic syn-aptic interactions between the neurons or to a spatiallycorrelatedtime-dependent external input. As a matter of fact, we have ver-ified that this effect occurs in numerical simulations of our net-work model with GABAergic synapses added. A detailed study ofthis phenomenon will be reported elsewhere.

AppendixThe IFcurve, the membrane potential, and the phase-response function of the QIF neuronThe subthreshold membrane potential, v(t), of a QIF neuronsatisfies the dynamic equation:

0dv

dt v2 Iext Ic. (24)

Integrating this differential equation, one finds that its generalsolution is for Iext Ic:

vt Iext Ic tan tIext Ic0 , (25)where is a constant of integration. The condition that at t 0the membrane potential of the neuron is at its reset value, Vr,

determines . One finds:

tan1 VrIext Ic. (26)The function v(t) increases monotonously to infinity. Therefore,after some time T, v(t) reaches the threshold, VT. Atthattime, theneuron fires an action potential, and v(t) is instantaneously resetto Vr. Subsequently, v(t) starts again to increase until it againreaches the thresholdafter another time T. Therefore, the neuronis firing periodically. The condition: v(T) VT determines thefiring period. One finds:

T 0tan

1

VT

Iext Ic

tan1

Vr

Iext IcIext Ic . (27)The IFcurve of the QIF neuron is given byv 1/T(Iext). Whenthe subthreshold membrane potential, v(t) reaches VT, the neu-ron fires an action potential represented by a -function:

Vt vt spikes

(t tspike), (28)

where measures the integral over time of the suprathresholdpart of an action potential.

The response function Z is the phase resetting curve of the

neuron (Kuramoto, 1984; Hansel et al., 1993; Rinzel and Ermen-trout, 1998) in the limit of vanishing small perturbations of themembrane potential. For the case of one-dimensional models, it



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can be written as dvd 1

(Kuramoto, 1991; Hansel et al.,

1995). Using Equation 24, one finds for the QIF model:

Z 0

vT22

Iext Ic

. (29)

Phase interaction for electrical synapsesThe phase coupling function, (), between two identical neu-rons, i andj, interacting synaptically depends on the phase differ-ence, i j , of the two neurons and is given by (Kuramoto,1984; Hansel et al., 1993, 1995; Kopell and Ermentrout, 2002):

i j 1

20

2

Zu iIsyn(u i, u j)du,

(30)

where Z is the phase-response function of the neurons andIsyn(i , j) is the synaptic current coming from the presynapticneuron j to the postsynapticneuroni. For electrical synapses, thiscurrent is:

Isyn(i, j) gVi Vj. (31)

Phase locking of a pair of weakly interacting neuronsWhen, in the absence of noise, two identical neurons interactweakly in a symmetric way, they reach a phase-locked state atlarge time. This is easilyshownin thephase-reduction framework(Hansel et al., 1993, 1995; Van Vreeswick et al., 1994; Kopell andErmentrout, 2002). The phases, 1 and 2 , of the two neurons

satisfy the two coupled equations:

d1

dt v 1 2 (32)

d2

dt v 2 1. (33)

Subtracting these equations, one finds:

d

dt )2(), (34)

where is thephase shift between thetwo neurons, 12 ,

and

()

1

2(() ()). At large time, the phase shift, ,reaches a fixed point (i.e., it is such that d/dt 0) and the twoneurons become locked with a phase shift that satisfies thecondition:

0. (35)

Because of the symmetry of the system and the 2periodicity ofthe function , there are always at least two solutions to thisequation.One is the in-phase solution, 0,and the other is theantiphase solution: . Other solutions with 0 mayalso exist. However, only those that are stable can be reached atlarge time. Linearizing Equation 34 around these solutions, onefinds that the stability condition is:

d

d 0, (36)

where combining Equations 30 and 31:

g

20

2

ZuVu Vudu. (37)

Differentiating this function with respect to , one finds:

g

20

2

ZuVu du. (38)

The membrane potential V can be expanded in its components(spike, reset, and subthreshold component) as:

V u v u Vr VTu vu, (39)

which allows us to rewrite ():

g

20

2

Zuu

Vr VT0

2

Zuu 0

2

Zuvu du. (40)Integrating by parts the first term and using the properties of the-function, we finally obtain:

g v dZd (VT Vr)Z()

1

2

0

2

Zuvu du

. (41)

n neurons: stability of the asynchronous state for the QIFnetwork with noiseThe stability analysis of the asynchronous state can be investi-gated for a network of phase oscillators in the presence of whitenoise (zero mean and variance D). For a general phase-couplinginteraction, (), one can show (Kuramoto, 1984) that the asyn-chronous state is stable if for all n 0:

Real(n) 0, (42)

where we have defined:

n in0 n n2D, (43)

with n , the nth-Fourier component of the 2periodic function():

n 1

20

2

e2in d. (44)

For a network of QIF neurons interacting with electrical syn-apses (Eqs. 24, 28) the nth-Fourier component of phase interac-tion has the form:

n gZnvn v, (45)

whereZn and vn are the nth-Fourier components of the functionsZ() and v() defined in Equations 29 and 25. The variance of



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the noise, D, depends on the average of the phase-response func-tion,Z0 1/20

2Z()d andisgivenbyD 2Z02 (Kuramoto,

1984), where 2 is the variance of the white noise in the full QIFmodel. This yields the stability condition for the asynchronousstate:

ng v ImZn ng ImZnvn n2Z0

2 0 (46)

Note that Z0 depends on the parameters Vr, VTand on the firingfrequency of the neurons. Therefore, the desynchronizing effectof the noise depends on these parameters. In particular, because

Z0 decreases when the firing rate increases, the noise is moreefficient at stabilizing the asynchronous state at low firing ratesthan at high firing rates.

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