Existence and Stability of Persistent States in Large Neuronal Networks

VOLUME 86, NUMBER 18

PHYSICAL REVIEW LETTERS

30 APRIL 2001

Existence and Stability of Persistent States in Large Neuronal Networks D. Hansel Laboratoire de Neurophysique et de Physiologie du Système Moteur, EP 1848 CNRS, Université René Descartes, 45 rue des Saints Pères, 75270 Paris Cedex 06, France

G. Mato Comisión Nacional de Energía Atómica and CONICET, Centro Atómico Bariloche and Instituto Balseiro (CNEA and UNC), San Carlos de Bariloche, R. N., Argentina (Received 11 August 2000) We study the existence and stability of persistent states in large networks of quadratic integrate-and-fire neurons. The networks consist of two populations, one excitatory and one inhibitory. The stability of the asynchronous state is studied analytically. Our study demonstrates the role of recurrent inhibition and inhibitory-inhibitory interactions in stable persistent activity in large neuronal networks. DOI: 10.1103/PhysRevLett.86.4175

PACS numbers: 87.18.Sn, 05.45.– a, 87.10. +e, 87.19.La

A fundamental issue in brain research is how information about the external world is stored and represented in the central nervous system. It has been proposed by Hebb [1], that the representation of an object in short term memory consists of the set of cortical cells that this object activates when it is presented. According to this hypothesis, the reverberation of excitation across this set is able to maintain activity even when the stimulus has been withdrawn, thus keeping a memory of the object. The Hopfield model of associative memory [2] and the model of Seung et al. [3] for the control of eye position rely on this idea. Similar ideas have been proposed to explain the persistence of neural activity in the delay period in match-to-sample tasks [4]. For the activity to persist, a sufficiently strong excitatory feedback is required. This poses the problem of the control of the neural activity at firing rates in the range 10– 30 spikes兾sec, as observed in experiments. It has been proposed recently that this can be solved if the excitatory synapses are slow and saturating [5]. However, the existence of such properties for excitatory synapses is still unclear [6]. Another possibility is that inhibition controls the firing rate [7]. However, inhibition can also induce synchrony [8–10] and this can be incompatible with low rate persistent activity. Indeed, if the neurons fire synchronously in some short time window the fast excitatory feedback will decay before the neurons fire their next spike. In this paper we investigate analytically the role played by inhibition in the existence and the stability of an asynchronous persistent state (APS) with physiological level of activity. The network model we study consists of two large populations of neurons, with NE excitatory 共E兲 neurons and NI inhibitory 共I兲 neurons. Neuron 共i, a兲 (i 苷 1, . . . , Na , a 苷 E, I) is characterized by a dynamical variable Xia which satisfies [11] t

dXia syn th stim 苷 X 2 2 Iia 1 Iia 共t兲 1 Iia . dt

(1)

0031-9007兾01兾86(18)兾4175(4)$15.00

This equation has to be supplemented with a condition for firing: if Xia reaches a threshold value Xth . 0 at time tspike , it fires an action potential and Xia is instantaneously 1 兲 苷 Xr . The external stimuli reset to Xr , 0, i.e., X共tspike and the synaptic interactions between neurons are represyn stim and Iia , respectively. sented by Iia In the absence of synaptic interactions, neuron 共ia兲 is active only if the external stimulus is suprathreshold stim th 共Iia . Iia 兲. In that case the firing rate of the neuron is th stim 兲, 2 Iia nia 苷 F共Iia

(2)

p where for px . 0, F共x兲 苷 px G共x兲兾t with: G共x兲 苷 21 tan21 共Xth 兾 x 兲 2 tan In particular, for q 共Xr 兾 x 兲. th stim th stim th stim , nia ~ Iia 2 Iia Iia ! Iia and ni,a ~ Iia 2 Iia stim for large Iia [12]. We assume that the thresholds are distributed according to

Qa 共I th 兲 苷 A exp关2共I th 2 I¯ath 兲2 兾2s 2 兴

(3)

for jI th 2 I¯ath j , 2s, and Qa 共I th 兲 苷 0 otherwise. The constant A ensures normalization. The synaptic coupling between the cells gives rise to the current X syn syn Iia 苷 Iij,ab 共t兲 , (4) j,b

where the contribution of each synapse is modeled as X syn b Iij,ab 苷 gij,ab fb 共t 2 tspike,j 兲 . (5) spikes

Here, summation is performed over all the spikes emitted prior to time t by neuron 共 j, b兲. The constant gij,ab measures the strength of the synapses that neuron 共 j, b兲 makes on neuron 共i, a兲. Excitatory (inhibitory) interactions correspond to gij,ab . 0 (gij,ab , 0). We assume that the network has no spatial structure (except for the segregation © 2001 The American Physical Society

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into two populations) and that gij,ab 苷 gab 苷 Gab 兾Nb for all i, j, where Gab are independent of Nb . The couplings are normalized to ensure that the total synaptic input to a neuron remains finite in the thermodynamic limit, Nb ! `. The function fb 共t兲 is [13] ∑ µ ∂ µ ∂∏ 1 t t fb 共t兲 苷 exp 2 2 exp 2 Q共t兲 , t1b 2 t2b t1b t2b (6) where Q共t兲 is the Heaviside function. For simplicity the rise and decay times of a synapse depend only on its type, excitatory or inhibitory. The asynchronous state (AS) in a highly connected network is particularly simple. Because each cell is affected by many uncorrelated synaptic inputs (within its integration time window), the total synaptic current it receives is time independent up to small temporal fluctuations of p the order of 1兾 Nb . This is because in the AS the spatial summation of the synaptic inputs is equivalent to their time average. In our model, the total synaptic current on a neuron is then X syn Iia 苷 Gab nb (7) b苷E,I

independently on i. The average firing rate of the neurons P in population b has been denoted by nb 苷 1兾Nb i苷1,Nb nib . The self-consistent relationship between the firing rates and the external stimuli,

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∂ µ X stim th nia 苷 F Iia , 1 Gab nb 2 Iia

completes the description of the network state. stim We now assume Iia 苷 0. In the following we discuss the stability properties of a state with given average firing rates when the coupling parameters are varied. For each set of coupling strengths the suitable average threshold has to be determined. Two situations occur when this is possible: (1) part of the support of threshold distribution is below 0, and (2) all the thresholds are above 0. In the second case the state coexists with a state where all the neurons are quiescent [14]. We are mostly interested in the second case, which we call the persistent state. To analyze the stability of the AS in our two-population heterogeneous network model, we have generalized the method introduced by Abbott and van Vreeswijk [15] to study homogeneous one-population integrate-and-fire networks. One writes the dynamical equation satisfied by the probability distribution of the “voltage” variables Xia . Linearizing this equation around the AS, one looks for solutions with time dependence exp共lt兲. A spectral equation determines the growth rates l. The AS loses stability continuously when some parameter is changed if one of the eigenvalues crosses the imaginary axis. At this point, l 苷 im. Details of the derivation will be given elsewhere [16]. The spectral equation reads

Y ∑ 共im 1 g1a 兲 共im 1 g2a 兲 2 g1a g2a Uaa ∏ 苷 UEI UIE , g1a g2a a苷E,I where g1a 苷 1兾t1a , g2a 苷 1兾t2a , µ ∂ A1b p m 1 A2b 2 i A3b , Uab 苷 Gab 2 m 2 A1b 苷

Z

`

0

A2b

A3b

Z ` Pb 共n兲n 1 dn Hb 共n兲 , F 21 共n兲 0 ∂µ ∂ µ ` X m 2 m 苷 Hb , 2pn 2pn n苷1 dn

m µ ∂ y Hb 共 y12np 兲 苷 dy cot 2 共 y 1 2np兲2 n苷1 2p m µ ∂ Z p y Hb 共 y 兲 1 dy cot , 2 y2 0 ` Z X

(10)

(11)

(12)

(13)

an

exp共2im兾n兲 关cos共f兲 1 sin共f兲 im 兴 an

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GEE 苷 1兾F 0 共GEE nmin 2 IEth 兲

(14)

nmin 苷 F共GEE nmin 2 IEth 兲 ,

(15)

and

an µ Pb 共n兲n cos共a 1 f兲 1 sin共a 1 f兲 im Hb 共n兲 苷 an 2F 21 共n兲 关1 1 共 im 兲2 兴

关1 1 共 im 兲2 兴 p and a 苷 2G关F 21 共n兲兴, f 苷 2 tan21 关Xr 兾 F 21 共n兲兴.

(9)

An eigenvalue with m 苷 0 corresponds to an instability to perturbations in the firing rate of the neurons. Modes with m fi 0 correspond to an instability in which neurons synchronize. If m is of the order of the firing rates, this synchrony is on the time scale of the spikes. On the other hand, if m is much smaller, synchronized bursting arises [10,16]. In the absence of inhibition, and for a given coupling GEE , the firing rate of the APS cannot be arbitrarily small. Indeed, if the firing rate is smaller than some value nmin , given by the solution of the equations

p

with

2

(8)

b苷E,I

∂

the APS is unstable with m 苷 0 (for simplicity we assume here that s 苷 0) [17]. In order to get a low rate, the coupling and the threshold have to be small, namely, GEE ~ nmin and 0 , IEth ~ GEE . To show how inhibition stabilizes APS even at low rates we assume first that t1E 苷 t1I and t2E 苷 t2I and that the average firing rates of the two populations are the same (this imposes a constraint on the average thresholds).

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Under these assumptions, the spectral equation depends only on the trace T and the determinant D of coupling matrix Gab . Remarkably, one sees that, if T 苷 D 苷 0, the real solutions of Eq. (9) are 2g1 and 2g2 . Therefore, in this case, the AS is stable with respect to rate instability, at any firing rate. Moreover, if s 苷 0, all the complex solutions of Eq. (9) are purely imaginary. Hence, the AS is marginally stable with respect to instability to synchrony. Introducing small heterogeneities in the neuron thresholds makes the AS stable. If, in addition, the excitatory-toexcitatory interactions 共GEE 兲 and the excitatory-toinhibitory interactions 共GIE 兲 are strong enough, it can be easily proved that the AS is persistent. Therefore there is no limitation in the firing rate of stable APS, provided the I-to-I connections and the recurrent inhibition (characterized by GEI GIE ) are appropriately chosen. We now consider the more general case in which the time constants of excitation and inhibition differ. As in the previous case, rate instability of the APS is prevented even if GEE is large, provided GII and the recurrent inhibition are strong enough. However, the APS can lose stability due to synchrony. This depends critically on the values of the synaptic time constant. For instance, for a threshold distribution with variance s 苷 0.05, GEE 苷 2.95, GEI 苷 21, GIE 苷 6.72, t1E 苷 t1I 苷 1 msec, and GII 苷 0, an APS with nE 苷 nI 苷 20 Hz exists independently of t2E and t2I . The stability lines of this state are shown in Fig. 1a as a function of g2E and g2I . For fast inhibition, an excitation slower than 100 msec is required to have a stable APS. For very slow inhibition, the required time constant scales like t2I . At intermediate values of t2I , excitation is always slower than 30 msec. Increasing jGII j allows one to take faster excitation. This is shown in Fig. 1b for GII 苷 22.75, where one sees that for inhibition, with a time constant t2I 苷 6 msec (which is typical for GABAA synapses), an excitation faster than t2E 苷 1.8 msec (within the AMPA range) is compatible with a stable APS. More generally, we find that the AS is stable in an intermediate range of values of g2E bounded by two instability lines. On the upper (plain) line, m decreases continuously when gI goes to zero (slow inhibition). In this limit, m becomes much smaller than the firing rate, therefore the instability leads to burst synchrony. On the other hand, when gI increases, m becomes comparable to the average firing rate of the neurons, signaling an instability to spike synchrony. The lower (dashed) line corresponds also to spike synchrony. It terminates at a finite value of g2I , when g2E 苷 0. At this point the effect of the E population on the I population is an excitatory drive constant in time. Therefore, at this point, synchrony is the result of the I-I interactions. More generally, the lower transition line corresponds to the emergence of spike synchrony due to I-I interactions [8–10]. These results show that introducing I-I interactions reduces the minimal t2E compatible with a stable APS. Note also that, when GII is large and t2I is small, excitation cannot be too slow because this lead to spike synchrony.

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Stable APS

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FIG. 1. Phase diagram for the stability of the APS for nE 苷 nI 苷 20 Hz (other parameters in the text): (a) GII 苷 0; (b) GII 苷 22.75. Inset: distribution of firing rates for both populations in the APS.

In reality, even in the “quiescent” state the neurons can display spontaneous very low rate random firing. This can be obtained in our model if an additional noise is introduced in the right-hand side of Eq. (1). In that case the existence and stability of the AS cannot be studied analytically with the techniques used here. However, one expects that, if the noise is not too strong, the above results remain qualitatively valid. This is shown in Fig. 2, where the average instantaneous firing rate of the two populations is plotted as a function of time when the system starts in the spontaneous firing state and is stimulated by transient excitation. The parameters of the network are the same as in Fig. 1b and the synaptic time constants t2E 苷 3 msec and t2I 苷 6 msec. The noise level has been chosen so that, in the spontaneous state, the average rate is about 1.5 Hz. One sees that the network is bistable. One can switch from the spontaneous state to the persistent state with a transient excitation and back with a transient inhibition. Our mean-field approach relies on an all-to-all connectivity, or more generally on a connectivity which is proportional to the network size in the thermodynamic limit. If this is not the case, the noise induced in the dynamics 4177

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discussions with N. Brunel, B. Gutkin, C. Chow, C. van Vreeswijk, and K. Martin are acknowledged. We thank D. Golomb and C. Meunier for careful reading of the manuscript. 0

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FIG. 2. Switching between the spontaneously active state and the APS in a simulation of a network with NE 苷 NI 苷 800 neurons. Parameters are the same as in Fig. 1a. A depolarizing (hyperpolarizing) stimulus is applied at 500 msec during 100 msec (2000 msec during 50 msec). Solid line: E population. Dotted line: I population. Inset: stimulus as a function of time.

by the spatial fluctuations does not vanish in the thermodynamic limit [18]. If this effective noise is too strong it can make the APS unstable. However, we have checked that, for the parameters of Fig. 2, the APS remains stable for connectivity as small as K 苷 50 synapses per neuron (25 E and 25 I synapses) which is largely compatible with available anatomical data. The method we have used is general and can be applied to study the stability of the AS in heterogeneous networks, whether it is persistent or not [10]. In particular, it can be shown that, in a two-population network, I-I interactions can suppress nonpersistent synchronous oscillations [16]. This role of I-I interactions has been noted previously by Tsodyks et al. in the framework of a two-population rate model [19]. This result can be extended to the more general case of spiking neurons by using our approach. The main result of this paper, derived analytically for the quadratic integrate-and-fire model, is that recurrent inhibition as well as I-I interactions can play a crucial role in the existence and stability of APS. We have checked (using simulations) that this result also holds for conductance-based models that are more faithful to biophysical reality. In a recent paper [5] it has been argued, relying mainly on numerical simulations, that slow excitation of the NMDA type was required to achieve such states. However, I-I interactions were not included in this model. Recent anatomical and physiological studies have revealed that I-I connections are numerous and strong in the cortex [20]. Our conclusions fit nicely with these experimental results. This work was partially supported by Grant No. PICT97 03-00000-00131 from ANPCyT for G. M. and by Grant No. ECOS-SeCyT A99E01 for D. H. and G. M. Fruitful 4178

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[1] D. Hebb, The Organization of Behavior (Wiley, New York, 1949). [2] J. J. Hopfield, Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982); D. J. Amit, H. Gutfreund, and H. Sompolinsky, Phys. Rev. Lett. 55, 1530 (1985); D. J. Amit, H. Gutfreund, and H. Sompolinsky, Phys. Rev. A 35, 1007 (1987). [3] H. S. Seung et al., Neuron 26, 259 (2000). [4] J. M. Fuster, The Prefrontal Cortex (Raven, New York, 1988); A Compte et al., Cerebral Cortex (to be published); N. Brunel (to be published). [5] X.-J. Wang, J. Neurosci. 19, 9587 (1999). [6] A. K. McAllister and C. F. Stevens, Proc. Natl. Acad. Sci. U.S.A. 97, 6173 (2000). [7] N. Rubin and H. Sompolinsky, Europhys. Lett. 10, 465 (1989); P. E. Latham et al., J. Neurophysiol. 83, 808 (2000). [8] D. Hansel, G. Mato, and C. Meunier, Neural Comput. 7, 307 (1995); C. van Vreeswijk, L. F. Abbott, and G. B. Ermentrout, J. Comput. Neurosci. 1, 313 (1994); W. Gerstner and L. van Hemmen, Phys. Rev. Lett. 76, 1755 (1996); P. C. Bressloff and S. Coombes, Neural Comput. 12, 91 (2000). [9] X.-J. Wang and G. Buzsáki, J. Neurosci. 16, 6402 (1996). [10] D. Golomb, D. Hansel, and G. Mato, in “Neuroinformatics”, Handbook of Biological Physics Vol. 4, edited by F. Moss and S. Gielen (Elsevier, New York, 2000). [11] This quadratic integrate-and-fire model corresponds to the normal form of a conductance-based neuron near a saddlenode bifurcation. It can be derived exactly in the limit of low firing rates. This is in contrast with the (linear) integrate-and-fire model [L. Lapicque, J. Physiol. (Paris) 9, 620 (1907)] which is frequently used. The variable X is adimensional. In a normal form analysis near the bifurcation it can be related to a linear combination of the voltage and gating variables. [12] The parameters we use in this paper are Xt 苷 2Xr 苷 1.3, t 苷 8.33 msec to recover the f-I curve of the WangBuszaki model [9]. [13] W. Rall, J. Neurophysiol. 30, 1138 (1967). [14] In the first situation, bistability between two active states can be possible. [15] L. F. Abbott and C. van Vreeswijk, Phys. Rev. E 48, 1483 (1993). [16] D. Hansel and G. Mato (to be published). [17] Depending on t1E , t2E synchrony can occur before this rate instability. In general, nmin is a lower bound for the allowed firing rate of the AS. [18] D. Golomb and D. Hansel, Neural Comput. 12, 1095 (2000). [19] M. V. Tsodyks et al., J. Neurosci. 17, 4382 (1997). [20] A. Sik et al., J. Neurosci. 15, 6651 (1995); K. TarczyHornoch et al., J. Physiol. (London) 508.2, 351 (1998); A. Gupta, Y. Wang, and H. Markram, Science 287, 273 (2000).