Brunel 2000 Dynamics

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Journal of Computational Neuroscience 8, 183208 (2000)c 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Dynamics of Sparsely Connected Networks of Excitatoryand Inhibitory Spiking Neurons

NICOLAS BRUNEL LPS, Ecole Normale Sup erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

[email protected]

Received December 23, 1998; Revised June 9, 1999; Accepted June 9, 1999

Action Editor: G. Bard Ermentrout

Abstract. The dynamics of networks of sparsely connected excitatory and inhibitory integrate-and-re neuronsare studied analytically. The analysis reveals a rich repertoire of states, including synchronous states in whichneurons re regularly; asynchronous states with stationary global activity and very irregular individual cell activity;and states in which the global activity oscillates but individual cells re irregularly, typically at rates lower thanthe global oscillation frequency. The network can switch between these states, provided the external frequency, orthe balance between excitation and inhibition, is varied. Two types of network oscillations are observed. In the fastoscillatory state, the network frequency is almost fully controlled by the synaptic time scale. In the slow oscillatorystate, the network frequency depends mostly on the membrane time constant. Finite size effects in the asynchronousstate are also discussed.

Keywords: recurrent network, synchronization

1. Introduction

In recent years, many studies have been devoted tomodels of networks of simple spiking neurons, usingeither numerical simulations (e.g., Amit et al., 1990;Usher et al., 1994; Tsodyks and Sejnowski, 1995)or analytical methods in models of either fully con-

nected (e.g., Mirollo and Strogatz, 1990; Abbott andvan Vreeswijk, 1993; Hansel et al., 1995; Gerstner,1995) or locally coupled systems (e.g., Hopeld andHerz, 1995; Terman and Wang, 1995). Sparsely con-nected networks of binary excitatory and inhibitoryneurons have recently been studied by van Vreeswijk and Sompolinsky (1996, 1998). On the other hand,a theory fully describing the dynamical properties of a network of randomly interconnected excitatory andinhibitory spiking neurons is still lacking. To tacklethis problem, it seems natural to consider networks

of simple leaky integrate-and-re (IF) neurons. Thesenetworks are often used in simulation studies. Theyare simple enough to give some hope for an analyti-cal treatment; And last, single IF neurons have beenshown in many cases to provide a good approximationto the dynamics of more complex model neurons (e.g.,Bernander et al., 1991).

A rst step has been undertaken by Amit and Brunel(1997b). Using a self-consistent analysis, the averagering rates in the stationary states of a network of randomly connected excitatory and inhibitory neuronswere calculated as a function of the parameters of thesystem. In this process, it was shown that a consistenttheory of a network of cells that have irregular ringat low rate (a common situation in a living cortex ora living hippocampus) needs to take into account theuctuations in the synaptic inputs of a cell, since inthis regime it is those uctuations that drive neuronal


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184 Brunel

ring. In this study, a mean-eld approach has been in-

troduced, in which not only mean inputs but also theirvarianceare taken into account in a self-consistentway.The outcome of the analysis is the ring rates in sta-tionary states of the system, both in an unstructurednetwork and in a network structured by Hebbian learn-ing of external stimuli. This technique has also beenrecently applied to the simpler case of linear (withoutthe leak term) IF neurons (Fusi and Mattia, 1999).

Subsequently, a system composed exclusively of inhibitory neurons has been studied in more detail(Brunel and Hakim, 1999). A Hopf bifurcation line hasbeen found that separates regions with stationary andoscillatory global activity. Close to the bifurcation line,a simple reduced equation that describes the dynamicsof the global activity of the system has been derived.Thending of oscillatory activityin a network ofpurelyinhibitoryneurons agrees with previousmodeling stud-ies(e.g.,van Vreeswijket al., 1994; Wangand Buzs aki,1996;Traubetal.,1996;Whiteetal.,1998)andwithev-idence from both in vivo (McLeod and Laurent, 1996)and in vitro Whittington et al., 1995; Traub et al., 1996;Fisahn et al., 1998) experiments suggesting inhibitionplays an important role in the generation of network oscillations. Furthermore, the frequency of the oscilla-tion was found to depend mostly on the synaptic time

constants, in agreement with in vitro experimental data(Whittington et al., 1995), and other modeling stud-ies (Whittington et al., 1995; Traub et al., 1996; Whiteet al., 1998).

Neocortical and hippocampal networks in vivo arecomposed of a mixture of excitatory and inhibitoryneurons. It is therefore natural to ask the question of how thepresence of excitation in such a network affectsthe presenceand the characteristicsof the synchronizedoscillation. The interest of the model proposed in Amitand Brunel (1997b) andof the techniques introduced inBrunel and Hakim (1999) is that they set the stage foran analytical study of the problem. Such a study has themore general interest of providing for the rst time ananalytical picture of a system of randomly connectedexcitatory and inhibitory spiking neurons.

In the following we determine analytically

The characteristics (ring rates, coefcient of varia-tion of the neuronal interspike intervals) and the re-gion of stabilityof theasynchronous stationary statesof the system;

The frequency of the oscillations that appear on thevarious Hopf bifurcation lines that form the bound-ary of the region of stability;

The phase lag between excitatory and inhibitory

populations on the Hopf bifurcation line; and The autocorrelation function, or equivalently thepower spectrum, of the global activity in a nite net-work, in the asynchronous region.

The behavior of the system beyond the various Hopf bifurcation lines is studied both through numerical in-tegration of coupled nonlinear partial differential equa-tions and through numerical simulations of the model.

This allows us to characterize the phase diagramsof networks of sparsely connected excitatory and in-hibitory cells. They show a very rich behavior. De-pending on the values of the external frequency, thebalance between excitation and inhibition, and the tem-poral characteristics of synaptic processing, we nd

Synchronous regular (SR) states, where neurons arealmost fully synchronized in a few clusters and be-have as oscillators when excitation dominates in-hibition and synaptic time distributions are sharplypeaked;

Asynchronous regular (AR) states, with stationaryglobal activity and quasi-regular individual neu-ron ring when excitation dominates inhibition andsynaptic time distributions are broadly peaked;

Asynchronous irregular (AI) states, with stationaryglobal activity but strongly irregular individual ringat low rates when inhibition dominates excitation inan intermediate range of external frequencies;

Synchronous irregular (SI) states, with oscillatoryglobal activity but strongly irregular individual r-ing at low (compared to the global oscillation fre-quency) ring rates, when inhibition dominatesexcitation and either low external frequencies (slowoscillations) or high external frequencies (fast oscil-lations). When the average synaptic time constant ishigh enough, these two regions merge together.

We also discuss the implications of these results forthe interpretation of neurophysiological data.

2. The Model

We analyze the dynamics of a network composed of N integrate-and-re (IF) neurons, from which N E areexcitatory and N I inhibitory. Each neuron receives C randomly chosen connections from other neurons inthe network, from which C E = N E from excitatoryneurons and C I = N I from inhibitory neurons. It also


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Sparsely Connected Networks of Spiking Neurons 185

receives C ext connections from excitatory neurons out-

side the network. We consider a sparsely connectednetwork with =C E / N E =C I / N I 1.The depolarization V i ( t ) of neuron i (i =1, . . . , N )at its soma obeys the equation

V i ( t ) = V i ( t ) + R I i ( t ), (1)where I i ( t ) are the synaptic currents arriving at thesoma. These synaptic currents are the sum of the con-tributions of spikes arriving at different synapses (bothlocal and external). These spike contributions are mod-eled as delta functions in our basic IF model:

R I i ( t ) = j J i j k t t k j D , (2)

where the rst sum on the r.h.s is a sum on differ-ent synapses ( j =1, . . . , C +C ext ), with postsynapticpotential (PSP) amplitude (or efcacy) J i j , while thesecond sum represents a sum on different spikes ar-riving at synapse j , at time t = t k j + D , where t k j isthe emission time of k th spike at neuron j , and D isthe transmission delay. Note that in this model a singlesynaptic time D is present. For simplicity, we take PSPamplitudesequal at eachsynapsethat is, J i j = J > 0for excitatory external synapses, J for excitatory recur-rent synapses (note the strength of external synapses istaken to be equal to the recurrent ones), and g J forinhibitory ones. External synapses are activated by in-dependent Poisson processes with rate ext . When V i ( t )reaches the ring threshold , an action potential isemitted by neuron i , and the depolarization is reset tothe reset potential V r after a refractory period r p dur-ingwhichthe potentialis insensitiveto stimulation.Theexternal frequency ext will be compared in the follow-ing to the frequency that is needed for a neuron to reachthreshold in absence of feedback, thr =/( J C E ) .We rst study the case in which inhibitory and exci-tatory neurons have identical characteristics, as in themodel described above. This situation is referred to asmodel A . Then, taking into consideration physiologicaldata, we consider the case in which inhibitory and ex-citatory neurons have different characteristics. In thiscase the membrane time constants are denoted by E and I ; the synaptic efcacies are J EE J E (excita-tory to excitatory); J EI = g E J E (inhibitory to excita-tory); J IE J I (excitatory to inhibitory); J II = g I J I (inhibitory to inhibitory). Excitatory external synapsesare equal to recurrent excitatory synapsesthat is, J E for excitatory, and J I for inhibitory neurons. External

frequencies are denoted by E , ext , I ,ext . Last, delays

are Dab for synapses connecting population b to pop-ulation a , for a , b = E , I . This case is referred to asmodel B .

The parameter space remains large, even for suchsimple model neurons. In the following, using anatom-ical estimates for neocortex, we choose N E = 0.8 N , N I =0.2 N (80% of excitatory neurons). This impliesC E =4C I . We rewrite C I = C E that is, =0.25.The number of connections from outside the network is taken to be equal to the number of recurrent excita-tory ones, C ext = C E . We also choose E = 20 ms; =20 mV; V r =10 mV; r p =2 ms.The remaining parameters are, for model A, g, therelative strength of inhibitory synapses; ext , the fre-quency of the external input; J , the EPSP amplitude;C E , the number of recurrent excitatory connections;and D , the transmission delay. This makes a total of ve parameters.

For model B, there are additional parameters: the in-hibitory integration time constant I ; two EPSP ampli-tudes, J E and J I , for excitatory and inhibitory neuronstwo IPSP amplitudes (relative to the EPSP ones), g E and g I ; the frequencies of the external inputs are now E ,ext and I ,ext ; and four delays. This makes a total of 12 parameters.

Thus, we still face a huge parameter space. Theanalytical study that follows demonstrates that it isnonetheless possible to achieve a comprehensive un-derstanding of the possible behaviors of the system.

3. Formalism

The analysis proceeds along the lines of Amit andBrunel (1997b) and Brunel andHakim (1998). We startwith model A (identical excitatory and inhibitory neu-rons). We consider a regime in which individual neu-rons receive a large number of inputs per integrationtime , and each input makes a small contribution com-pared to the ring threshold ( J ). In this situation,the synaptic current of a neuron can be approximatedby an average part plus a uctuating gaussian part. Thesynaptic current at the soma of a neuron (neuron i ) canthus be written as

R I i ( t ) =( t ) + i ( t ), (3)

in which the average part ( t ) is related to the ringrate at time t D and is a sum of local and external


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186 Brunel

inputs

( t ) = l ( t ) + ext with l ( t ) =C E J (1 g )( t D), (4)

ext =C E J ext .The uctuating part, i ( t ) , is given by the uctua-tionin thesumof internalexcitatory, internal inhibitory,and external poissonian inputs of rates C E , C E andC E ext . Its magnitude is given by

2( t ) = 2l (t ) + 2ext with l ( t ) = J C

E (1 + g2)( t D) , (5) ext = J C E ext .

i ( t ) is a gaussian white noise, with i ( t ) = 0 andi ( t ) i ( t ) =( t t ) .

As a consequence of thenetwork sparserandom con-nectivity ( C N ), two neurons share a small num-ber of common inputs. Thus, the correlations of theuctuating part of the synaptic inputs of differentneurons are neglected in the limit C / N 0thatis, i ( t ) j ( t ) = 0 for i = j . The spike trains of all neurons in the network can be self-consistently de-scribed by random point processes that are correlatedonly because they share a common instantaneous r-ing rate ( t ) . In other words, between t and t +dt , aspike emission hasa probability ( t )dt of occurringforeach neuron, but these events occur statistically inde-pendently in different neurons. Note that this does notmean that the neurons have uncorrelated spike trainsand are thus not synchronized. In fact, they are uncor-related only when the global ring frequency is con-stant.If theinstantaneous ring rate varies in time, thespike trains will have some degree of synchrony. Thus,in the following, network states for which is con-

stant in time will be termed asynchronous , while thosefor which varies in time will be termed synchronous .On the other hand, the calculations done in this articledo not apply when signicant correlations appear be-yond those induced by a common time-varying ringrate ( t ) .

When correlations between the uctuating parts of the synaptic inputs are neglected, the system can bedescribed by the distribution of the neuron depolar-ization P ( V , t )that is, the probability of nding thedepolarization of a randomly chosen neuron at V at

time t , together with the probability that a neuron is

refractory at time t , p r ( t ) . This distribution is the (nor-malized) histogram of the depolarization of all neuronsat time t in the large N limit N . The stochasticequations ((1) and (3)) for the dynamics of a neuron de-polarization can be transformed into a Fokker-Planck equation describing the evolution of their probabilitydistribution (see, e.g., Risken, 1984)

P (V , t )

t = 2( t )

2 2 P (V , t )

V 2

+ V

[( V ( t )) P ( V , t )]. (6)The two terms in the r.h.s. of (6) correspond respec-tively to a diffusion term coming from the synapticcurrent uctuations and a drift term coming from theaverage part of the synaptic input. ( t ) and ( t ) arerelated to ( t D) , the probability per unit time of spike emission at time t D , by Eqs. (4) and (5).Equation (6) can be rewritten as the continuityequation

P ( V , t ) t =

S (V , t ) V

, (7)

where S is the probability current through V at time t (Risken, 1984):

S (V , t ) = 2 ( t )

2 P (V , t )

V (V ( t ))

P ( V , t ).

(8)

In addition to Eq. (6), we need to specify the bound-ary conditions at , the reset potential V r , and thethreshold . Theprobability current through gives theinstantaneous ring rate at t , ( t ) = S ( , t ) . To obtaina nite instantaneous ring rate, we need the absorbingboundary condition P ( , t ) =0. This is due to the factthat by denition of the IF neuron, the potential cannotbe above threshold, and hence P (V , t ) =0 for V > .Thus, a nite probability at the ring threshold wouldimply a discontinuity at . Because of thediffusivetermin Eq. (8), this would imply an innite probability cur-rent at and thus an innite ring probability. InsertingP (, t ) =0 in Eq. (8) gives the boundary condition forthe derivative of P at :

P V

( , t ) = 2( t ) 2 ( t )

. (9)

Similarly, at the reset potential V = V r , P ( V , t )must be continuous, and there is an additional


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probability current in V r due to neurons that just n-

ished their refractory period: what comes out at timet at the threshold must come back at time t + r p atthe reset potential. Thus, the difference between theprobability currents above and below the reset po-tential at time t must be proportional to the frac-tion of cells ring at t r p . This is expressed byS (V +r , t ) S (V r , t ) = ( t r p ) , which yields thefollowing derivative discontinuity,

P V

( V +r , t ) P V

(V r , t ) = 2( t r p )

2 ( t ). (10)

The natural boundary condition at V

= is that

P should tend sufciently quickly toward zero to beintegrablethat is,

limV

P (V , t ) =0 limV V P (V , t ) =0. (11)

Last, P ( V , t ) is a probability distribution and shouldsatisfy the normalization condition

P (V , t ) dV + p r ( t ) =1, (12)

in which

pr ( t ) = t

t t rp( u ) du

is the probability of the neuron being refractory attime t .

When excitatory and inhibitory cells have differentcharacteristics, we need to study the statistical prop-erties of both populations separately. For example, theaverage synaptic input a= E , I ( t ) of a cell in populationa = E , I is related to the ring rate of excitatory cellsat time t D E and of inhibitory cells at time t D I and is a sum of local and external inputs

a =C E J a a [a , ext + E ( t Da , E ) ga I ( t Da I )] (13)

and the uctuating part of these inputs have their mag-nitude, given by

2a = J 2a C E a a , ext + E ( t DaE )+ g2a I ( t Da I ) . (14)

The system is now described by the distributions of the

neuron depolarization Pa ( V , t )that is, the probabil-ity of nding the depolarization of a randomly chosenneuron of population a = E , I at V at time t , togetherwith the probability that a neuron is refractory at timet , pra ( t ) . These distributions obey

a Pa ( V , t )

t = 2a ( t )

22 Pa (V , t )

V 2

+ V

[(V a ( t )) Pa (V , t )],a = E , I . (15)

The partial differential equations governing the distri-butions of pyramidal cells and interneurons are cou-pled together through both a ( t ) and a ( t ) , which arerelated to the a ( t Da ) by Eqs. (13) and (14).Theboundaryconditionsaresimilar to Eqs. (9), (10),and (11):

Pa V

( , t ) =2a ( t ) a

2a ( t )(16)

Pa V

(V +r , t ) Pa V

(V r , t ) =2a ( t r p )

2a ( t )(17)

limV

Pa (V , t ) =0(18)

limV

VPa (V , t )

=0.

Last, the normalization conditions hold for both distri-butions.

4. Stationary States

4.1. Model A

Stationary states of the system have been rst stud-ied by Amit and Brunel (1997b). We give here a moredetailed account of their properties. In a stationary so-lution, P (V , t ) = P0(V ) , pr ( t ) = pr ,0 . Time indepen-dent solutions of Eq. (6) satisfying the boundaryconditions (9), (10), and (11) are given by

P0( V ) =20 0

exp (V 0)2

20

0

0

V 0 0

( u V r )eu2du , pr ,0 =0 r p ,

(19)

in which ( x) denotes the Heaviside function, ( x)

=1 for x > 0 and ( x) =0 otherwise, and 0 and 0


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188 Brunel

are given by

0 =C E J [ext +0(1 g ) ], (20) 20 =C E J 2 [ext +0 (1 +g2 ) ].

The normalization condition (12) provides the self-consistent condition, which determines 0 :

10 = r p +2

0 0

V r 0 0

due u2

u

dvev2

= r p +

0 0

V r 0 0

due u2(1 +erf (u )), (21)

where erf is the error function (see, e.g., Abramowitzand Stegun, 1970). Note that the analytical expres-sion for the mean rst passage time of an IF neu-ron with random Gaussian inputs (Ricciardi, 1977;Amit and Tsodyks, 1991) is recovered, as it should.Equations (20) and (21) as a self-consistent descrip-tion of the system were obtained by Amit and Brunel(1997b). The bonus of the present approach is that italso provides the stationary distribution of membranepotentials. In the regime ( 0) 0 (low ringrates), Eq. (21) becomes

0 ( 0)

0 exp ( 0 )2

20. (22)

To probe the nature and the number of stationarystates that can be found in the plane (g , ext ) , Eqs. (20)and(21) were solved numerically. Theresults for C E =4000, J =0.2 mV (100 simultaneousexcitatory spikesneeded to reach threshold from resting potential) areshown in Fig. 1. Differentvaluesof C E and J show verysimilar gures (differences are discussed in Section 6).ext is expressed in units of

thr = C E J ,

the external frequency needed for the mean input toreach threshold in absence of feedback. For the param-eters chosen in the gure, thr =1.25 Hz. The verticalline g = 4 is where feedback excitation exactly bal-ances inhibition. In the high ext , g < 4 region, only avery high-frequency (near saturation) stationary statecan be found (H state). In the high ext , g > 4 region,only a low activity state can be found (L state). Thetransition between H and L is smooth (see Fig. 1A) but

becomes more and more abrupt as C E increases. In the

low ext , g < 4 region, there are three stationary states,state H, an almost quiescent (Q) state ( 0 essentiallyzero), and an intermediate (I), unstable state. In thelow ext , g > 4 region, there is rst a region in whichonly the Q state is present; then increasing ext anotherregion, in which L, I, and Q states coexist. The sad-dle node bifurcation lines, on which two xed pointsmerge and disappear (the I and Q on the upper line, theI and L on the lower line), separate these regions. Sim-ple stability considerations (Amit and Brunel, 1997b)indicate that the I state is always unstable.

The stationary frequencies as g is varied are shownin the same gure. It shows the abrupt decrease in fre-quency for g near the balanced value g = 4. In theinhibition-dominated regime, g > 4, the frequency de-pends quasi-linearly on the external frequencies.

To understand better the difference between highand low activity states, the coefcient of variation(CV) of the interspike interval has been calculated (seeAppendix A.1). It is shown in Fig. 1C. In the highactivity region, the CV is essentially zero, since the in-terspike intervals are all close to the refractory period r p . When g 4, the CV abruptly jumps to a valuelarger than 2, indicating highly variable interspike in-tervals, contemporaneously with the abrupt decrease in

ring rates. It then decreases slowly as g increases fur-ther. When ring rates become very small the CV goesasymptotically to 1: spike trains become very close torealizations of Poisson processes. In the whole low-activity region, the CV is close to or higher than one,indicating highly irregular individual cell activity.

4.1.1. Simple Estimates of the Stationary FiringRates for Large C E . From Eqs. (21) and (20), the leadingorder term in an expansion of the stationary frequencyin 1 / C in the high-activity regime, g < 4 gives (seeAppendix A.2 for details):

0 = 1 r p 1 V r C E J (1 g )

. (23)

It gives a very good approximation of the frequenciesobtained by Eqs. (21) and (20). In this regime, the sta-tionary frequencyis almost independent on theexternalfrequency ext .

In the low-activity regime, g > 4, an approximateexpression for the stationary frequency can also bedetermined. It can be done by noticing that, whenext > thr and C E is large, the only consistent wayof solving Eqs. (21) and (20) is for the mean recurrent


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Figure 1 . Characteristics of the stationary (asynchronous) state. Parameters: C E =4000, J =0.2 mV. A: Bifurcation diagram of the system,in which saddle node bifurcations only are drawn (full lines). The vertical dotted line at g = 4 corresponds to the balanced network in whichfeedback excitation and inhibition exactly cancel. B: Left : Full lines : Frequencies in the H ( g < 4) and L ( g > 4) states as a function of g ,for ext / thr =0, 1, 2, 4 (from bottom to top). The curve corresponding to the positive frequency for ext = 0 is present only for g < 4 sincethe H state disappears near g =4, where it merges with the intermediate, unstable xed point (indicated by the dashed line). Right : Full lines :Firing rates in the L ( g > 4) states as a function of ext / thr , for g =4.5, 5, 6, and 8 (from top to bottom). Note the almost linear trend, witha gain roughly equal to 1 /( g 1) as predicted by the simple Eq. (24). Frequencies in the H region are close to the saturation frequency andare almost independent on the external frequency. Dashed lines : the unstable I state. C: Coefcient of variation (CV) of ISI. Note the sharp risefrom CV

0 for g < 4 to values greater than one for g > 4.


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190 Brunel

inputs to balance exactly the excess of the mean ex-

ternal currents above threshold. This leads to a lineardependence on the external frequency

0 = ext thr

g 1. (24)

The gain in this relationship between ext and 0 in-creases as g approaches the balanced state, g = 4.This explains the linear trend shown on the right of Fig. 1B.

4.2. Stationary States for Model B

We considerstationarysolutionsfor the two probabilitydistributions Pa (V , t ) = Pa 0(V ) , pra ( t ) = pra 0 , a = E , I . Time-independent solutions of Eq. (6) satisfyingthe boundary conditions (9), (10), and (11) are givenby

Pa 0(V ) =2a 0 a a 0

exp (V a 0)2

2a 0

a 0

a 0

V a 0 a 0

( u V r )eu2du , pr ,0 =a 0 r p

(25)

with

a 0 =C E J a a [ext + E 0 g I 0], (26) 2a 0 =C E J 2a a [ext + E 0 +g2 I 0].

The normalization condition (12) provides the self-consistent condition that determines both excitatoryand inhibitory frequencies

1

a 0 = r p

+2 a

a 0

a 0

V r a 0 a 0due u

2

u

dvev2 . (27)

The qualitative behavior of the stationary states isshown for a few values of J I on the left of Fig. 6.Qualitatively, the main difference with model A is thatthe transition between high (H) and low (L) activitystates is no more continuous but occurs now throughtwo saddle-node bifurcations that occur in the regiong < 4. Between these two lines H and L states coexist.We also see that the value of J I has a strong effect onthebifurcationdiagram already at the level of stationarystates: for low values of J I (compared to J E E / I ), the

transition from Q to L statesoccurs through twosaddle-

node bifurcations, as in model A, for any value of g .As J I increases toward J E E / I , inhibition becomesstronger, and it smoothens the transition between Land Q states that become continuous for large enoughvalues of g . Last, when inhibition becomes very strong,the coexistence region between L and Q states vanishes(see lower left Fig. 6). There is no longer a well-denedtransition between L and Q states. For such values of J I , the ring rates of inhibitory cells are much higher(typically 5 to 10 times higher) than those of excitatorycells in the low-activity regime.

5. Linear Stability of the Stationary Statesand Transitions Toward Synchrony

We now investigate in which parameterregion thetime-independent solution is stable. To simplify the studyof the Fokker-Planck equation (6), it is convenient torescale P , V and by

P = 2 0

0Q , y =

V 0 0

, =0(1 +n ( t )).(28)

y is thedifference between the membrane potentialandthe average input in the stationary state, in units of thestandard deviation of the input in the stationary state.n ( t ) corresponds to the relativevariation of theinstanta-neous frequencyaround the stationaryfrequency. Afterthese rescalings, Eq. (6) becomes

Q t =

12

2 Q y2 +

y

( y Q)

+n ( t D) G Q y +

H 2

2 Q y2

, (29)

G = C E J 0(g

1)

0 = 0, l 0 ,

(30)

H = C E J 2 0 (1 +g2 )

20 = 20, l 20

.

G is the ratio between the mean local inputs and 0(with a change of sign such that it is positive withpredominantly inhibitory interactions), and H is theratio between the variance of the local inputs and thetotal variance (local plus external). These parametersare a measure of the relative strength of the recurrentinteractions.


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Equation (29) holds in the two intervals < y< yr and yr < y < y . The boundary conditions forQ are now imposed at

y = 0

0and yr =

V r 0 0

.

Those on the derivatives of Q read

Q y

( y , t ) = 1 +n ( t )

1 + Hn ( t D),

(31) Q y

y+r , t Q y

yr , t = 1 +n ( t r p )1 + Hn ( t D)

.

The linear stability of the stationary solution of thepurely inhibitory system has been studied in detail byBrunel and Hakim (1998). The differences with thesituation described here are (1) the refractory periodis taken into account, and (2) the values of G , H , y , and yr may be very different from the case of thepurely inhibitory network, due to the presence of ex-citation. The stability analysis is done in a standardway (see, e.g., Hirsch and Smale, 1974) by expandingQ = Q0 + Q1 + and n = n1 + around thestationary solution (details are given in Appendix A).The linear equation obtained at rst order has solu-tions that are exponential in time, Q1

= exp(w t )

Q 1 ,

n1 exp(w t ) n1 , where w is a solution of the eigen-value equation (46). The stationary solution becomesunstable when the real part of one of these solutions be-comes positive. The outcome of the analysis are Hopf bifurcation lines on which the stationary state destabi-lizes due to an oscillatory instability. These Hopf bi-furcation lines areshown together with thesaddle-nodebifurcations in Fig. 2, for several values of the delay D .As for the case of saddle-node bifurcations, the preciselocation of theHopf bifurcations is only weakly depen-dent on the value of C E and J (provided C E is large and J small compared to ). There are three qualitatively

different branches that form the boundary of the regionof stability of the stationary solution.

5.1. Three Branches of the Boundaryof the Region of Stability

5.1.1. Transition to a Fast Oscillation of the Global Activity for Strong Inhibition and High External Fre-quencies. The branch that appears in the high g, highext region is similar to the one occurring in the purelyinhibitory network (Brunel and Hakim, 1998). The

qualitative picture of this instability is the following: an

increase in the global activity provokes a correspond-ing decrease in the activity after a time proportional to D if the feedback is inhibitory enough. This decreasewill itself provoke an increase at a time proportionalto 2 D , for essentially the same reason. More quantita-tively, the frequency of the oscillation is smaller than1/( 2 D) . In the present case, H , the ratio between thelocal and the total variances, is close to 1, and the os-cillatory instability appears whenever the mean recur-rent inhibitory feedack G / D . Its frequency is f 1/ 4 D (about 167 Hz for D = 1.5 ms, 125 Hzfor D = 2 ms, 83 Hz for D = 3 ms), as shown inFig. 3.

5.1.2. Transition to a Slow Oscillation of the Global Activity, for ext thr . The branch in the g > 4region when ext thr has a different meaning. Thisinstability is due to the fact that when the external fre-quency is of the order of magnitude of thr the network is again particularly sensitive to any uctuation in theglobal activity. In fact, a uctuation in this region eas-ily brings the network to a quiescent state. The network then needs a time that depends on the membrane timeconstant to recover. The network thus oscillates be-tween an essentially quiescent state and an active state.Its frequency is in the range 20 to 60 Hz for externalfrequencies around threshold.

There is a large gap in frequencies between bothoscillation regimes for small delays D. On the otherhand, for high enough D, the two branches merge atsomeintermediate external frequency. The correspond-ing frequencies on the instability line are shown inFig. 3.

Note that the lowest branch of the saddle-node bi-furcation at which the L stationary state appears hasnot been indicated in this picture since it does not cor-respond to a qualitative change in the behavior of the

network (only unstable xed pointsappearor disappearon this line).

5.1.3. Transition to the Synchronous Regular State Near the Balanced Point g = 4. When g becomesclose to the balanced value g =4, very high-frequencyoscillatory instabilities appear. These instabilities areessentially controlled by D and r p , with frequencies

k / D where k is a positiveinteger. Theyusuallyappear

for ext > 1, g 4, except when D / r p is an integeror very close to it. In this case the instability lines are


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192 Brunel

Figure 2 . Phase diagrams of the system, for differentvalues of the delay, indicatedon thecorresponding picture. In each case theasynchronous,stationary state is stable in the regions marked AR (regular ring, g < 4) or AI (irregular ring, g > 4), bounded by the Hopf bifurcation curves,indicated by full lines. On the Hopf bifurcation curves, an oscillatory instability appears, and one enters either a SR state (synchronous regularring) when excitation dominates, or SI states (synchronous irregular ring) when inhibition dominates. The short-dashed lines indicate thebifurcation curve on which the almost quiescent stationary state destabilizes. It is stable below the line. In the small triangular region betweenSR and SI states, the network settles in a quasi-periodic state in which the global activity exhibits a fast oscillation on top of a slow oscillation,and individual neurons re irregularly.

pushed toward lower values of g (see Fig. 2, D = r p=2 ms).

5.2. Wide Distribution of Delays

The analysis of the previous section can be generalizedto the case in which delays are no longer xed butare rather drawn randomly and independently at eachsynaptic site from a distribution Pr ( D) . It is describedin Appendix A.4. The results are illustrated in Fig. 4 for

a particular example. The effect of a wide distributionof delays on the three Hopf bifurcation branches are asfollows:

The high g, high ext branch progressively moves to-ward higher values. The frequency on the instabilityline is only weakly modied.

The ext thr branch remains qualitatively un-

changed. This is because this type of global oscil-lation is relatively independent on the delay D .


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Figure 3 . Frequencyof theglobaloscillation(imaginary partof thesolution of Eq. (46) with the largest real part, in the regions in whichthe real part is positive) as a function of the external inputs ext , forseveral values of the delay and g . Full lines : D = 1.5 ms. Long-dashed lines : D = 2 ms. Short-dashed lines : D = 3 ms. For eachvalue of D, three curves are shown, corresponding to g = 8, 6, 5(from top to bottom). Note that the frequencies for high ext areclose to 1 / 4 D (167, 125, and 83 Hz, respectively), while all curvesgather in the low ext region at around 20 Hz. In this region thefrequency essentially depends on the membrane time constant. Forshort delays there is a large gap between slow and fast frequencyranges, since these regions are well separated by the asynchronousregion. For large delays and g large enough, the frequency increasescontinuously from the slow regime to the fast regime.

Even a very small increase in the width of the dis-tribution stabilizes the stationary state in the wholelow g region. Thus, in this whole region, the network settles in a AR (asynchronous regular) state.

Figure 4 . Effect of a wide distribution of delays. A: All delays equal to 1.5 ms. B: Delays uniformly distributed between 0 and 3 ms.

Thus a wide distribution of delays has a rather dras-

tic effect on the phase diagram, by greatly expandingthe region of stability of the asynchronous stationarystate. The other regions that survive are qualitativelyunmodied.

5.3. Phase Diagrams Depend Only Weaklyon External Noise

One might ask the question whether the irregularityof the network dynamics is caused by the externalnoise or by the intrinsic stochasticity generated by the

quenched random structure of the network. To check that the irregularity in the AI and SI states is not aneffect of the external noise generated by the randomarrival of spikes through external synapses, the stabil-ity region of the asynchronous state has been obtainedwhen the external input has no noise component (bysetting ext = 0). The comparison between noisy andnoiseless external inputs is shown in Fig. 5. It showsthat the region of stability of the AI (asynchronous ir-regular) state is only weakly modied by the absenceof external noise. The CV in the AI region is alsoonly weakly modied by the removal of the external

noise. Numerical simulations conrm that in both AIand SI regions single neurons show highly irregularring. Thus, the irregularity of spike trains in this re-gion is an intrinsic effect of the random wiring of thenetwork.


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194 Brunel

Figure 5 . Effect of external noise on the stability of the asyn-chronous irregular state. Full lines : External noise is due to the Pois-son process of frequency ext coming on external synapses. Dashed lines : External noise is suppressed; theexternalinputis constant. Thedynamics of the system becomes deterministic, but in the inhibition-dominated regime, the intrinsic stochasticity generated by the ran-dom structure of recurrent collaterals (proportional to l ) gives riseto the irregular ring behavior.

5.4. Linear Stability and Synchronized States for Model B

The linear stability of stationary states can be studiedmuch as in model A. The analysis is described in somedetail in Appendix B. We show the results of such anal-ysis in the right part of Fig. 6, in the case of randomlydistributed delays, with a uniform distribution between0 and 4 ms. It shows the Hopfbifurcation lines thatformthe boundary of stability of the low-activity AI state.Qualitatively, these diagrams bear similarity to the di-agrams obtained for model A. In the large g , large ext an oscillatory instability with a frequency controlledby the delay is present, while for ext thr , a sloweroscillation appears, with frequency controlled by themembranetime constant.This line iscomposed,for low

J I , of two branches (see upper right Fig. 6): the near-horizontal branch close to ext = thr corresponds toan oscillation with frequencies between 60 and 90 Hz,while the near-vertical branch corresponds to an oscil-lation with frequencies between 10 and 15 Hz. In thesmall region with a triangular shape around g = 2,ext = thr , the network settles in quasi-periodic statewith a 60 Hz oscillation on top of a 15 Hz slower oscil-lation. As J I increases, the region in which the globalactivity exhibits a fast oscillation ( > 100 Hz) increases,while theregion with a slow oscillation breaksup in twoareas, one at high g with the intermediate-frequency

ranges (around 60 Hz), the second at low g with low-

frequency ranges (around 10 Hz). Last, for high J I , theregion with the slower oscillation disappears(see lowerright Fig. 6).

5.5. Phase Lag Between Excitatoryand Inhibitory Neurons

When excitatory and inhibitory cells have differentcharacteristics, the two populations will have in gen-eral a non-zero phase lag. This phase lag is calculatedin Appendix B as a function of the system parameters.In the fast oscillation regime, we show that the main

parameters controlling this phase lag are the synap-tic delays of inhibitory synapses, D EI and D II . In fact,in the case g I = g E , E ,ext = I , ext , and when the fre-quencyof the oscillation is large, the phase lagbetweeninhibitory and excitatory cells is well approximated by( D EI D II ) , where is the imaginary part of thesolution of Eq. (65) with positive real part. This resultcan be readily explained by the fact that in this regime,oscillations are caused by feedback inhibitory connec-tions. As a result, the phase lag will be negligible (lessthan a few degrees) when D EI = D II .Differences between the strength of inhibitoryconnections ( g

E = g

I ) or in the external inputs (

E ,ext

= I , ext ) also give rise to phase lags between bothpopulations.

6. Comparison Between Theory and Simulations

To perform a comparison between simulations andanalysis we need to choose a parameter set for whichthe number of connections per cell is not too large. Wechoose model A, with J =0.1 mV, C E =1000, and D =1.5 ms. The corresponding phase diagram in theplane g-ext predicted by the theory is shown in Fig. 7.Note that the main qualitative difference between thisdiagram and the one of a network with higher connec-tivity, like the one shown in Fig. 2 is that the regioncorresponding to the SI state with slow oscillation hassplit in two very small regionsone around g 4.5,the other for g > 7 .5. This is the main effect of vary-ing parameters C E and J on the bifurcation diagram.Note also that the line separating the AI from the SRstate has slightly moved toward the left, while the fast-oscillation region remains essentially unaffected.

In this gure, four points (shown by diamonds on thegure) are selected. They represent the four different


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Figure 6 . Bifurcation diagrams for model B. Parameters as in Fig. 1. Values of J I are indicated on the corresponding gure. Left : Saddle-nodebifurcations only. In each region of the diagram, stationary states are indicated by their corresponding initials: H (high activity), L (low activity),Q (quiescent state). Right : The Hopf bifurcation lines that dene the boundary of the AI low activity state are indicated together with thesaddle-node bifurcation lines.


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196 Brunel

Figure7 . Phasediagramof thenetworkcharacterizedby theparam-eters of Fig. 8 ( C E =1000, C I =250, J =0.1 mV, D =1.5 ms).Diamonds indicate the parameter sets chosen for the simulationsshown in Fig. 8.

phases that can be found in the system. Then, numeri-cal simulations are performed separately for these fourdifferent points, using a network of N E =10,000 ex-citatory neurons and N I = 2,500 inhibitory neurons(a connection probability of =0.1). Figure 8 showsboth single-cell behavior (rasters) and global activityfor each of these parameter sets. The analysis predictsqualitatively the behavior of the system in each case:

1. The network settles in a strongly synchronized statewith regularly ring neurons (upper left), when ex-citation dominates inhibition;

2. It settles in a state in which the global activity (LFP)exhibits a fast oscillation, close to 180 Hz in thiscase, while individual neurons re irregularly at acomparatively lower rate, about 60 Hz (upper right),when inhibition dominates, and the external fre-quency is high;

3. It settles in a state in which the global activity ex-

hibit strongly damped oscillations, and neurons reirregularly, when inhibition dominates, and the ex-ternal frequency is moderate (above threshold, butnot too high);

4. And last, it is in a slow oscillation regime, with afrequency close to 20 Hz, with very low individualneuron ring rates (about 5 Hz), when inhibitiondominates, and the external frequency is below butclose to threshold.

Note that in the strongly synchronized state A, cor-relations are present beyond the ones induced by a time

Table 1 . Comparison between simulations and theory in theinhibition-dominated irregular regimes: Average ring rates andglobal oscillation frequency.

Fir ing rate Global frequency

Simulation Theory Simulation Theory

B. SI, fast 60.7 Hz 55.8 Hz 180 Hz 190 Hz

C. AI 37.7 Hz 38.0 Hz

D. SI, slow 5.5 Hz 6.5 Hz 22 Hz 29 Hz

varying ring rate ( t ) . Thus, the analysis developedin the present article is not adequate to describe suchstates. However, the analysis does predict the transition

toward such synchronized states as soon as the excita-tion starts to dominate over inhibition.Both individual ring rate and global oscillation fre-

quency obtained in the simulation are compared withthe results of the analysis in Table 1. We see that inthe inhibition-dominated regimes, the analysis predictsquite well the ring frequency, even in the synchronousregions in which the stationary solution is unstable. Inthe asynchronous region, the agreement is very good.The frequencies of the global oscillation in the vari-ous synchronous regions are also well predicted by thecomplex part of the eigenvalue with largest real part of Eq. (46).

On the other hand some discrepancies with the an-alytical picture obtained so far can be observed. Theglobal activity is not quite stationary in the AI stateas predicted by the theory, and the global oscillationsexhibit some degree of irregularity in the SI states. Toaccount for these effects, a description of nite sizeeffects is needed.

6.1. Analytical Description of Finite Size Effects

Finite size effects have been studied analytically in thepurely inhibitory system by Brunel and Hakim (1999),

when the system is stationary or deviations to station-arity are small. When the dynamics is stochastic, sharptransitions can occur only in the limit N . Theyare smoothed by nite size effects, as can be seen bythe simulations in the AI state, which show some weak oscillatory behavior. In the sparse connectivity limit,the uctuations in the input of a given neuron i can beseen as the result of the randomness of two differentprocesses: the rst is the spike emission process S ( t )of the whole network; and the second, for each spikeemitted by the network, is the presence or absence of asynapse between the neuron that emitted the spike and


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Figure 8 . Simulation of a network of 10,000 pyramidal cells and 2,500 interneurons, with connection probability 0.1 and J E = 0.1 mV. Foreach of the four examples are indicated the temporal evolution of the global activity of the system (instantaneous ring frequency computed inbins of 0.1 ms), together with the ring times (rasters) of 50 randomly chosen neurons. The instantaneous global activity is compared in eachcase with its temporal average (dashed line). A: Almost fully synchronized network, neurons ring regularly at high rates ( g =3, ext / thr =2).B: Fast oscillation of the global activity, neurons ring irregularly at a rate that is lower than the global frequency ( g = 6, ext / thr = 4).C: Stationary global activity (see text), irregularly ring neurons ( g

=5, ext / thr

=2). D: Slow oscillation of the global activity, neurons ring

irregularly at very low rates ( g =4.5, ext / thr =0.9).


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198 Brunel

the considered neuron: if a spike is emitted at time t ,

i ( t ) =1 with probability C / N , and 0 otherwise. Theinput to the network is then R I i ( t ) = J i ( t ) S ( t D).

Both processes canbe decomposed between their meanand their uctuation,

i ( t ) = C N + i ( t ), S ( t ) = N ( t ) +S ( t ).

Thus the input becomes

R I i ( t ) =( t ) J N ( t ) i ( t ) J C

N S ( t ),

in which ( t ) is given by Eq. (4). The input is thesum of a constant part and of two distinct ran-dom processes superimposed on : the rst is uncor-related from neuron to neuron, and we have alreadyseen in Section 3 that it can be described by N uncor-related Gaussianwhitenoises i ( t ) , i =1, . . . , N where i ( t ) j ( t ) =i j ( t t ) . The second part isindependent of i : it comes from the intrinsic uctua-tions in the spike train of the whole network, whichare seen by all neurons. This part becomes negligi-ble when

= C / N

0 but can play a role as we

will see when C / N is nite. The global activity inthe network is essentially a Poisson process with in-stantaneous frequency N ( t ) . Such a Poisson processhas mean N ( t ) , which is taken into account in ,and a uctuating part that can be approximated by N 0( t ) , where (t ) is a Gaussian white noise thatsatises ( t ) = 0 and ( t )( t ) =( t t ) . Notethat for simplicity we take the variance of this noiseto be independent of time, which is the case when thedeviations to stationarity are small. These uctuationsare global and perceived by all neurons in the net-work. The idea of approximating the global activity,

a Poisson process, by a continuous Gaussian process is

P () = 2 H

( 1 2 H cos ( D) + H 2) +2 G (cos ( D) sin ( D) H ) +2G 2. (33)

justied by the large network size. It is similar to theapproximation at the single neuron level, of the synap-tic input by a Gaussian process. This allows the studyof nite size effects using such a continuous approxi-mation. Thus, the mean synaptic input received by the

neurons becomes

CJ ( t ) + J C 0 ( t ) + ext .Inserting this mean synaptic input in the drift term of the Fokker-Planck equation, we can rewrite Eq. (29)as

Q t =

12

2 Q y2 +

y

( y Q) +n ( t D)

G Q y +

H 2

2 Q y2 + H ( t )

Q y

,

(32)

As shown by Brunel and Hakim (1998) in the purelyinhibitory network, the effects of the small stochasticterm are the following:

In the stationary AI regime, a strongly damped os-cillatory component appears in the autocorrelationof the global activity that vanishes in the 0( N ) limit.

In the oscillatory SI regimes, it creates a phase dif-fusion of the global oscillation. The autocorrelationfunction of the global activity becomes a dampedos-cillatory function. In the limit 0, the dampingtime constant tends to innity.These two effects are now studied separately. This

allows a direct comparison of the simulation resultsshown in Fig. 8 with the theory.

6.2. Autocorrelation of the Global Activityin the Stationary Regime

The autocorrelation of the global activity in the station-ary regime is calculated in Appendix A.5. Itsamplitudeis proportional to . The power spectrum of the globalactivity is given by Eq. (55). When both and G are

large, the power spectrum can be well approximated by

The left part Fig. 9 shows the comparison of the powerspectrum of the global activity in the AI region ob-tained in the simulation with the analytical expressionEq. (55). It shows the location of the peaks in the powerspectrum are very well predicted by theory. The peak


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Figure 9 . Comparison between simulations and analysis taking into account nite size effects. A: Power spectrum of the global activityobtained in the simulation in the AI regime, g =5, ext / thr =2 (dashed line) is compared with the theoretical expression, Eq. (55) (full line).B: Amplitude of the autocorrelation of the global activity at zero lag vs connection probability. Straight line: theoretical prediction. +: results of the simulation.

near 100 Hz corresponds to the eigenvalue that is re-sponsible for the SI oscillatory instabilities, while thesmaller peak at 700 Hz corresponds to the eigenvaluethat is responsible for the SR instability (it is close to1/ D

=667 Hz). Note that the analysis slightly overes-

timates the size of the peaks. In the right part of Fig. 9,we compare the dependence of the amplitude of theuctuations of the global activity on the connectionprobability in both simulations and theory. In the sim-ulations, the connectionprobability wasvaried increas-ing or decreasing the network size keeping C =1000, J = 0.1 mV, and D =1.5 ms. The resulting autocor-relation of the relative uctuation of the global activityaround its mean value at zero lag is plotted asa functionof and shows a good agreement with the linear curvepredicted by the theory.

6.3. Phase Diffusion of the Global Oscillationin the Oscillatory Regime

Beyond the Hopf bifurcation lines, a global oscillationdevelops. Brunel and Hakim (1999) have shown howto describe analytically the oscillation close to the bi-furcation line using a weakly nonlinear analysis. Theoutcome of the analysis is a nonlinear evolution equa-tion for the deviation n1 of the instantanous ring ratefrom its stationary value. Finite size effects can alsobe incorporated in the picture using the stochastic term

of Eq. (32). This stochastic term gives rise to a phasediffusion of the global oscillation, and the AC of theglobal activity becomes a damped cosine function,whose coherence time (the characteristic time constantof the damping term in the AC function) depends lin-early on N / C = 1/ (for 1 C N and xed C ).Thus, thecoherencetime goes to innity as thenetwork size increases, at a xed number of connections perneuron. Simulations were again performed at variousconnection probabilities varying network size. Theyshow that for parameter corresponding to both casesB and D in Fig. 8, the coherence time increases as

decreases.

7. Discussion

The present study shows for the rst time a compre-

hensive analytical picture of the dynamics of randomlyinterconnected excitatory and inhibitory spiking neu-rons. In such a network, many types of states, char-acterized by synchronous or asynchronous global ac-tivity and regular or irregular single neuron activity,can be observed depending on the balance between in-hibition and excitation, and the magnitude of externalinputs. The analytical results include ring frequenciesand coefcient of variation of the interspike intervalsin both populations; region of stability of the variousasynchronous states; frequency of the global oscilla-tion, and phase lag between excitatory and inhibitory


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200 Brunel

populations, on the instability lines; and autocorrela-

tion of the global activity in a nite network, in theasynchronous region. Numerical analysis of the partialdifferential equations, together with numerical simula-tions, indicate that the qualitative featuresof the varioussynchronous states can be predicted by the knowl-edge of the various instability lines (that is, no qualita-tively new phenomenon appear far from the bifurcationlines).

Theform of sparseconnectivity chosenin thepresentarticle is such that each neuron receives exactly thesame number of inputs. This was done here for thesake of simplicity. Another, more realistic form of sparseness is the one in which a synapse is presentwith probability C / N at each pair of cells, indepen-dently from pair to pair, and therefore the number of inputs varies from cell to cell. This last form of sparse-ness has been considered in previous studies. Amit andBrunel (1997a) showed, both by simulations and bymean-eld analysis, that the main effect of variationsof the number of inputs from cell to cell in the station-ary, inhibition-dominated regime is to give rise to verywidespatial distributionsof ringratesamong neurons.Brunel and Hakim (1999) have compared both formsof sparse connectivities and observed only small dif-ferences between the synchronization properties of a

network of purely inhibitory neurons, again using bothsimulation and analysis. Thus, we expect that allow-ing a variability in the number of inputs from neuronto neuron will have only a mild effect on the proper-ties of the various synchronized states observed in theirregular, inhibition-dominated regimes.

Thesynapsesof themodel areoversimplied. Only asingle time scale is present, the transmission delay D,while experimentally measured postsynaptic currentsshow two additional time scales, a rise time (typicallyvery short for AMPA and GABA synapses), and a de-cay time. Previous analysis of two mutually coupledneurons, or of networks operating in the regular ringmode, have shown that the oscillatory properties, andin particular the frequency of the global oscillation,depend strongly on the decay time of inhibitory post-synaptic currents (see, e.g., van Vreeswijk et al., 1994;Hansel et al., 1995; Terman et al., 1998; Chow, 1998).Our model clearly outlines the importance of the trans-mission delay for the generation of fast oscillationsin networks operating in the irregular ring regime,but more work is necessary to understand the relativeroles of these different synaptic time constants. Thiscould be done incorporating more detailed synaptic

responses into the model but would increase signi-

cantly the complexity of the analysis.The asynchronous irregular (stationary global activ-ity, irregular individual ring) state was rst describedinAmit andBrunel (1997b), where itwas called sponta-neous activity .Asimilarstateisalsoobtainedforawiderange of parameters in the model of van Vreeswijk andSompolinsky (1996, 1998), in which there is no synap-tictime scale.Wehaveshown that such a state is generi-cally obtained in the inhibition-dominated regime, forawide range of external inputs. There hasbeen recently asurge of interest in the computational relevance of suchasynchronous states for the speed of information pro-cessing (Treves, 1993; Tsodyks and Sejnowski, 1995;Amit and Brunel, 1997a, 1997b; van Vreeswijk andSompolinsky, 1996). In fact, the reaction time to tran-sient inputs in such states is typically proportional tothe faster synaptic time scale rather than to the mem-brane integration time scale, which allows for a fastpopulation response, in contrast to single cells, whichtypically re zero or one spike in a time comparable tothis population response.

States in which neurons behave as oscillators aregenerically obtained in many models of fully con-nected spiking neurons (e.g., Mirollo and Strogatz,1990; Tsodyks et al., 1993; van Vreeswijk et al.,

1994; Gerstner, 1995; Gerstner et al., 1996). Wang andBuzs aki, (1996) observed such states in a purely in-hibitory randomly connected network. In the presentmodel, these states can be observed only when excita-tiondominatesinhibition.These states are synchronouswhen delays are sharply distributed and asynchronousas soon as the distribution of delays becomes wide.Note that inmost cases in which thenetworkis synchro-nized, these states correspond to the clustering phe-nomenon that has been discussed previously by, for ex-ample, Golomb and Rinzel (1994) and van Vreeswijk (1996).

Regimes similar to the synchronous irregular stateshave been obtained in some simulations of biophysi-cally detailed neurons (e.g., Traub et al., 1989). Theywere described analytically in (Brunel and Hakim,1999) in a network composed of inhibitory neuronsonly.

7.1. Relationships with Neurophysiological Data

Recordings from neocortical or hippocampal networksin vivo often do not show prominent eld oscillations,


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together with highly irregular individual cell ring

with low frequency (sometimes called spontaneousactivity ): a state similar to the asynchronous irregularstate described here (stationary global activity, irregu-lar single-cell ring).

However, prominent oscillations of the LFP havebeen described in many systems in vivo , including therat hippocampus (see, e.g., Buzsaki et al., 1983; Braginet al., 1995), the visual cortex of mammals, the olfac-torycortex, the somatosensorycortex, and the thalamus(for a review, see Gray, 1994). Such oscillations havebeen also recorded in slices of the rat hippocampus(Whittington et al., 1995; Traub et al., 1996; Fisahnet al., 1998; Draguhn et al., 1998). In the rat hippocam-pus, single-neuron recordings made in parallel withLFP recordings show in some cases irregular individ-ualneuron ring at a comparatively lowerrate (Buzsakiet al., 1992; Csicsvari et al., 1998; Fisahn et al., 1998).Thus, oscillatory states observed in this system seemfunctionally similar to the synchronous irregular statesdescribed in the present article. It seems interesting tonote that in the present model, the network typicallyswitches from the stationary to the oscillatory statethrough changes of the external inputs only. Further-more, two distinct frequency bands can be observed inthe model: a fast frequency range for high external in-

puts anda slow frequency range for low externalinputs.In the rat hippocampus, different frequency ranges areobserved: a 200 Hz oscillation is seen during sharpwaves, while 40 Hz oscillations are associated withtheta (

8 Hz) waves during exploratory behavior. Thesharp increase in ring rates of both excitatory and in-hibitory neurons during sharpwavesare consistent witha fast oscillation induced by a sharp increase in the ex-ternal input, as observed in our model. In fact, usingneurons with different characteristics (as in model B),much of the phenomenology of such an oscillation canbe accounted for (ring frequencies of excitatory andinhibitory neurons, frequency of the global oscillation,phase lag between excitatory and inhibitory popula-tions). It remains to be seen whether the introductionof more realistic postsynaptic currents preserves thisfast oscillation. Recently, Traub et al. (1999) proposedthat 200 Hz oscillations are caused by axo-axonal cou-pling of pyramidal cells through gap junctions. Moreexperimental and theoretical work is needed to clarifywhether chemical or electricalsynapsesare responsiblefor this fast oscillation.

Appendix A: Model A

A.1. Stationary Properties: Firing Frequencies, CV

In the asynchronous states, the input of each neuron of the network can be described as a stationary Gaussianprocess of mean and variance . The moments k of the interspike intervals, as a function of the reset po-tential x =V r , can then be computed by the recurrencerelations (see, e.g., Tuckwell, 1988):

2

2d 2 k dx 2 +( x)

d k dx = k k 1 .

Computation of the rst moment yields 1 = 1/ 0 ,where 0 is given by Eq. (21). The computation of thesecond moment gives

2 = 21 +2 y

yr e x

2dx

x

e y

2(1 +erf y)2 dy .

The coefcient of variation of the ISI is simply thevariance divided by the square mean ISI, and thus

CV =2 20 y

yr e x

2dx

x

e y

2(1 +erf y)2 dy .

A.2. Firing Frequency in the Excitation- Dominated Regime

We start from the equations giving the ring rate,

10 = r p +

0 0

V r 0 0

due u2(1 +erf (u )) (34)

0 =C E J [ext +0(1 g ) ], (35) 20 =C E J 2 [ext +0(1 +g2 ) ].

When the excitation dominates, the only solution tothese equations is a solution for which the ring fre-quencyis close to thesaturation frequency, 1 / r p . Sincetypically C E J / r p , the mean synaptic inputs 0are much larger than , V r and 0 . Thus, both boundsof the integral in Eq. (34) are negative and very large.For u , we have

eu2 (1 +erf (u )) 1u


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202 Brunel

and thus

10

r p [ln u ] 0 0V r 0

0

r p + ln

0 0 V r

r p +

V r 0

.

We now use 0 ext together with Eq. (35), to obtain

V r

0 10

V r C E J (1

g )

and nally

0 = 1 r p

1 V r

C E J (1 g ).

A.3. Linear Stability of the Stationary Solution

The aim of this section is to study the stability of thestationary solution

Q 0( y) =exp ( y

2

) y

y du exp(u2

) y > yr

exp ( y2) y

yr du exp(u 2) y < yr

(36)

of Eqs. (29) and (31). In the following, the linear oper-ator L is dened as

L [Q ] = 12

2 Q y2 +

y

( y Q ),

and the square bracket [ f ] y+ y denotes the discontinu-

ity of the function at ynamely, lim 0{ f ( y + ) f ( y )}. Note that r p , which had been neglectedin Brunel and Hakim (1999), has been reintroduced inall calculations.

The function Q can be expanded around the steadystate solution Q0( y) as Q( y) = Q0 ( y) + Q 1( y, t ) + , n ( t ) =n1( t ) + . At rst order, one obtains thelinear equation

Q 1 t =L [Q 1] +n1( t D) G

d Q 0dy +

H 2

d 2 Q 0dy2

,

(37)

where G and H are dened in Eq. (30), together with

the boundary conditions

Q 1( y , t ) =0, Q 1 y

( y ) = n1 ( t ) + Hn 1( t D)(38)

and

[Q 1] y+r yr = 0,

Q 1 y

y+r

yr = n1 ( t r p ) + Hn 1( t D). (39)

Eigenmodesof (37)have a simple exponential behavior

in time:

Q 1( y, t ) =exp ( t ) n1() Q 1( y,),n 1( t ) =exp ( t ), n1(),

and obey an ordinary differential equation in y:

Q 1 ( y, )

=L [ Q 1]( y, ) +e D Gd Q 0dy +

H 2

d 2 Q 0dy2

(40)

together with the boundary conditions

Q 1( y , ) =0, Q 1 y

( y ) = 1 + H exp ( D),Q 1( y , ) =0, Q 1 y

( y ) = exp ( r p ) + H exp ().

The general solution of Eq. (40) can be written as alinear superposition of two independent solutions 1,2of the homogeneous equation 1 / 2 + y +(1 )=0 plus a particular solution Q

p1 ( y, ) ,

Q 1 ( y, ) =

+1 () 1( y, ) + +1 () 2( y, )+ Q

p1 ( y, ) y > yr

1 () 1( y, ) + 1 () 2( y, )+ Q

p1 ( y, ) y < yr

(41)

with

Q p1 ( y, ) = e D G

1 + d Q 0( y)

dy

+ H

2(2 + )d 2 Q 0( y)

dy 2. (42)


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Solutions of the homogeneous equation 1 / 2 + y +(1 ) = 0 can be obtained as series ex-pansions around y = 0. They are linear combina-tions of two functions: the conuent hypergeometricfunction

1( y, ) = M [(1 )/ 2, 1/ 2, y2] (43)(see, e.g., Abramovicz and Stegun, 1970) and a com-bination of conuent hypergeometric functions

2 ( y, ) = 1+ 2

M 1

2,

12

, y2

+

2

2 y M 1

2 ,32 , y

2

. (44)

For more details and asymptotic expansions of thesefunctions, see (Brunel and Hakim, 1998). When y , 1 | y| 1 , while 2 | y| exp ( y2 ) . Thusfor Q 1( y, t ) to be integrable on [ , y ] we need torequire 1 =0 in (41).The Wronskian Wr of 1 and 2 has the simpleexpression

Wr ( 1 , 2) 12 12 = 2

(/ 2)exp ( y2).

(45)

The four boundary conditions (38) and (39) give alinear system of four equations for the four remain-ing unknowns +1 , 1 , +1 , and 1 . The condition 1 =0, needed to obtain an integrable Q 1( y, t ) , givesthe eigenfrequencies of the linear Eq. (37). For conve-nience we dene and W by

= 2Wr

, W ( y ) =Q p1 2 Q

p1 2

Wr( y ),

W ( yr ) = Q p1 2 Q

p1 2

Wr

y+r

yr

.

The equation for the eigenfrequencies of (37) is givenin terms of these functions by

( y )( 1 He D ) ( yr )( e rp He D )= W ( y ) W ( yr ). (46)

On a Hopf bifurcation line = i c . Eq. (46) can berewritten as1 Hei c D R( c ) ei c rp Hei c D

=S ( y , c ) R( c ) S ( yr , c ), (47)

where

( y, ) =2 ( y, )2 ( y, )

R() =( yr , )

( y , ) =exp y2r y2 +

yr

y ( y, ) dy

S ( y, ) =ei D G ( y, ) 2 y

1 +i

+ H y( y, ) 2(1 y2)

2 +i .

When y and (or) are large,

( y, ) y + y2 +2i +O (1/ y, 1/ ) (48) y + | y| for | y| (49) y +(1 +i ) for | y|. (50)

Thus in the large frequency limit c y2r , y2 , weobtain ( y, c ) (1 +i ) c , R becomes exponen-tially small, and Eq. (47) becomes

ei c H =( i 1)G H y c . (51)

Since y is nite and H < 1, we note that for Eq. (46)to have a such a root, we need G cthat is,

G = c sin( c D) H =sin ( c D) +cos ( c D).

A rst solution to these equations can be found for

c 2 D

, G 2 D , H 1.It corresponds to the branch in the upper right part of

the phase diagrams. Since in this region

0 ext thr

g 1,

we can nd an approximate expression for this insta-bility line in the plane ( ext / thr , g ):

ext thr =1 +

4K

g(1 +g ) g 1

+ 216 K 2 g(1 +g ) g 12

+ 2K

,


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204 Brunel

where

K = DC E thr = D J .Another solution can be found for any k > 0,

c 2 k

D, G (1 H ) 2 k D .

This solution can be found when g 4. As soon as theexcitation gets stronger than the inhibition, G becomesnegative and thuscrosses the high-frequency instabilityline.

A.4. Linear Stability for Randomly Distributed Delays

As discussed in Brunel and Hakim (1999) a wide dis-tribution of synaptic times P ( D) can be easily incor-porated in the calculation: all we need is replace e Din Eq. (46) by

P ( D)e D d D . A.5. Autocorrelation of the Global Activity

in the Stationary Regime

The autocorrelation of the global activity can be cal-culated, to rst order in the connection probability ,using techniques similar to those used in Section A.We start from Eq. (32) and linearize it. Now, in-stead of writing Q 1 and n1 as exponentials, we Fourier

Z () =W r ( y ) W r ( yr )

( y )( 1 He D ) ( yr )( e rp He D ) W ( y ) + W ( yr ),

transform them and obtain an ordinary differential

equation in y

i Q 1( y, ) =L [ Q 1]( y, ) +ei D n1()

Gd Q 0dy +

H 2

d 2 Q 0dy2

+ H ()d Q 0dy

(52)

Z () = H

1 +i ( y , ) 2 y R()( ( yr , ) 2 yr )

1 Hei D R() [ei rp Hei D ] S ( y , ) + R() S ( yr , ).

that is satised by the F.T. Q 1( y,), n1() of Q 1 , n1 ,together with the boundary conditionsQ 1 ( y , ) =0, Q 1 y

( y ) = n1() [1 + H exp (i D)],Q 1 ( y , ) =0, Q 1 y

( y ) = n1() [exp (i r p ) + H exp (c) ].The solution of Eq. (52) is

Q 1( y, )

= +1 1( y, i ) + +1 2 ( y, i )

+ n1() Q p1 ( y, i ) + R

p1 ( y, i ) y > yr

1 1( y, i ) + 1 2 ( y, i )+ n1() Q

p1 ( y, i ) + R

p1 ( y, i ) y < yr

(53)

where 1,2 and Q p1 are dened in Section A, +,1 and +,1 will be given by the boundary conditions, and R p1is and

R p1 ( y, i ) = H

1 +i d Q 0( y)

dy. (54)

The condition 1 = 0 then gives a linear relationshipbetween the F.T. of the global activity n 1 and the F.T.of the nite-size noise term :n1() = Z () (),

in which the linear-response term Z is given by

in which

W r ( y ) = R p1 2 R p1 2

Wr( y ),

W r ( yr ) = R p1 2 R

p1 2

Wr

y+r

yr .

Z () can be rewritten as


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The power spectrum of the global activity P () can

then readily be estimated

P () = Z () Z (). (55)The autocorrelation of the global activity is the FT of the power spectrum P () . It is linearly proportional to

as announced.A simpler expression for the power spectrum can be

obtained for large . In this limit, when G , wegetP () =

2 H ( 1 2 H cos ( D) + H 2 ) +2 G (cos ( D) sin ( D) H ) +2G 2

. (56)

Appendix B: Model B

We dene

Pa = 2 a a 0

a 0Q a , a = E , I ,

G aE = C E a E 0

a ,ext + E 0 +g2a I 0,

G a I = ga C E a I 0

a ,ext

+ E 0

+g2a I 0

, a = E , I ,

H aE = E 0

a ,ext + E 0 +g2a I 0,

H a I = g2a E 0

a ,ext + E 0 +g2a I 0, a = E , I ,

ya = V a 0

a 0, ya =

a 0 a 0

,

yar = V r a 0

a 0, a =a 0(1 +n a ( t )), a = E , I .

Equation (15) becomes

a Q

a t =L [Q a ] +b= E , I

b ( t Dab )

sb G ab Q a ya +

H ab2

2 Q a y2a

, (57)

where s E = 1, s I = 1. The boundary conditions at yar and ya become at ya

Q a ( ya , t ) =0, Q a ya

( ya , t ) = 1 +n a ( t )

1 + H aE n E (t DaE ) + H aI n I (t DaI )(58)

and at yr

[Q a ] y+ar yar =0,

Q a ya

y+ar

yar =

1 +n a ( t rp )1 + H aE n E (t DaE ) + H aI n I ( t DaI )

.

(59)

Moreover, Q a ( ya , t ) should vanish sufciently fast at ya = to be integrable.The steady-state solution is similar to the case E = I , except that indices a have to be added everywhere.

To nd out whether this solution is stable, we lin-earize around the stationary solution and obtain at rstorder

a Q a 1

t =L [Q a 1] +b=

E , I

n a 1 ( t Dab )

sb G abd Q a 0dya +

H ab2

d 2 Q a 0dy 2a

, (60)


Q a 1( ya , t ) =0, Q a 1 ya

( ya ) = n a 1( t ) +b= E , I

H ab nb1( t Dab )(61)

and

[Q a 1] y+ar yar =0,

Q a 1 ya

y+ar

yar = n a 1( t r p )

+b= E , I

H ab nb1( t Dab ). (62)Eigenmodesof (60)have a simple exponential behaviorin time

Q a 1( ya , t ) =exp ( t ) Q a ( ya ,),n a 1( t ) =exp ( t ) n a ()


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206 Brunel

and obey an ordinary differential equation in y

a Q a ( y, ) =L [ Q a ]( ya , ) +b

e Db

nb sb G b ad Q a 0dya +

H b2

d 2 Q a 0dy 2a

,

(63)


Q a ( ya , t ) =0, Q a ya

( ya ) = n a +b

H b nb exp( Db ),

Q a ( ya , t ) =0, Q a ya

( ya ) = n a exp ( r p )

+b

H ab nb exp ( Dab ).

The general solution of Eq. (40) is written as in theprevious section

Q a ( ya , )

=

+a () 1( ya , )

+ +a () 2( ya , )

+ Q pa ( ya , ) ya > yar a () 1( ya , ) + a () 2( ya , )

+ Q pa ( ya , ) ya < yar

(64)

with

Q pa ( ya , ) =b

Q pab ( ya , ) nb

Q pab ( ya , ) = e Dab sb G ab1 + a

d Q a 0 ( ya )dya

+ H ab

2(2 + a )d 2 Q a 0 ( ya )

dy 2a

and 1, 2 are given by Eqs. (43) and (44).The eight boundary conditions (61) and (62) give a

linear system of eight equations for the four remainingunknowns +a , a , +a , and a . The conditions a =0 needed to obtain integrable Q a ( ya , t ) give the twoequations

A EE n E + A EI n I =0 A IE n E + A II n I =0,

where

Aaa = ( ya ) ( yar )e rp

H aa e Daa [( ya ) ( yar )] W aa ( ya ) + W aa ( ya r ),

Aab = H ab e Dab [( ya ) ( yar )] W ab ( ya ) + W ab ( ya r ), a =b,

and

W ab ( y )

=Q pab 2 Q

pab 2

Wr( y ),

W ( yr ) = Q pab 2 Q

pab 2

Wr

y+r

yr .

The eigenvalues need to satisfy

det ( A) = A EE A II A IE A EI =0. (65)On a Hopf bifurcation line an eigenvalue becomesimaginary, = i c .On the bifurcation line

n I = A EE A EI n E .

This relationship gives both the ratio of the amplitudesof the global oscillation in the interneuron/pyramidalcell populations, together with the phase lag betweenboth populations.

There are two situations in which simplicationsoccur. The rst one corresponds to g I = G E G , E ,ext = I ,ext ext , D EE + D II = D EI + D IE , and tothe large-frequency limit c y2r , y2 . In this situationEq. (47) becomes

1 a

H aa ei c Daa = ( i 1)aa

eic Daa sa G aa c(66)

(compare Eq. (51)), and

n I =exp[ i c ( D EI D II )]n E .When D II = D EI , interneurons and pyramidal cells arecompletely locked together, and near the bifurcationline their global activities have the same amplitude.


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208 Brunel

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