Chaotic Balanced State in a Model of Cortical Circuits

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1 ARTICLE Communicated by Misha Tsodyks Chaotic Balanced State in a Model of Cortical Circuits C. van Vreeswijk H. Somolinsky Racah Institute of Physics and Center for Neural Comutation, Hebrew University, Jerusalem, Israel The nature and origin of the temoral irregularity in the electrical activity of cortical neurons in vivo are not well understood. We consider the hyothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells. We study a network model with excitatory and inhibitory oulations of simle binary units. The internal feedback is mediated by relatively large synatic strengths, so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold. The connectivity is random and sarse. The mean number of connections er unit is large, though small comared to the total number of cells in the network. The network also receives a large, temorally regular inut from external sources. We resent an analytical solution of the mean- eld theory of this model, which is exact in the limit of large network size. This theory reveals a new cooerative stationary state of large networks, which we term a balanced state. In this state, a balance between the excitatory and inhibitory inuts emerges dynamically for a wide range of arameters, resulting in a net inut whose temoral uctuations are of the same order as its mean. The internal synatic inuts act as a strong negative feedback, which linearizes the oulation resonses to the external drive desite the strong nonlinearity of the individual cells. This feedback also greatly stabilizes the system s state and enables it to track a time-deendent inut on time scales much shorter than the time constant of a single cell. The satiotemoral statistics of the balanced state are calculated. It is shown that the autocorrelations decay on a short time scale, yielding an aroximate Poissonian temoral statistics. The activity levels of single cells are broadly distributed, and their distribution exhibits a skewed shae with a long ower-law tail. The chaotic nature of the balanced state is revealed by showing that the evolution of the microscoic state of the network is extremely sensitive to small deviations in its initial conditions. The balanced state generated by the sarse, strong connections is an asynchronous chaotic state. It is accomanied by weak satial cross-correlations, the strength of which vanishes in the limit of large network size. This is in contrast to the synchronized chaotic states exhibited by more conventional network models with high connectivity of weak synases. Neural Comutation 10, (1998) c 1998 Massachusetts Institute of Technology

2 1322 C. van Vreeswijk and H. Somolinsky 1 Introduction The ring atterns of neurons in the cortex of intact animals often exhibit a strong degree of temoral irregularity. This can be seen by the broad intersike interval histograms (ISI) of cortical neurons, which are tyically close to those generated by a Poisson rocess with a short refractory eriod (Abeles, 1991; Bair, Koch, Newsome, & Britten, 1994; Burns & Webb, 1976; Douglas, Martin, & Whitteridge, 1991; Softky & Koch, 1993). The irregular neuronal dynamics is also manifested in intracellular recordings of the membrane otential, which exhibit strong temoral uctuations. One of the long-standing roblems in cortical dynamics is understanding the origin of this irregularity and its comutational imlications (Douglas & Martin, 1991; Ferster & Jagadeesh, 1992). In vitro exeriments show that cortical neurons re in a relatively regular fashion when they are injected with a constant current. Thus, the irregularity of the in vivo neuronal activity must be due to uctuations in their synatic inut (Holt, Softky, Koch, & Douglas, 1996; Mainen & Sejnowski, 1995). These uctuations may be due to variations in the intensity of the sensory stimuli or may result from the stochastic action of synases. However, since cortical cells have thousands of synatic contacts, one would exect that the summation of the synatic inuts at the soma averages out most of the uctuations in the synatic inut and yields a membrane otential with only a small residual uctuation. This is a articularly dif cult issue in conditions where the cortex is vigorously active so that the cell receives many synatic inuts within a single integration time constant (Holt et al., 1996; Softky & Koch, 1993). One ossible resolution of this roblem is to assume that the uctuating synatic inuts are substantially correlated and therefore are not averaged out. Indeed, the sike trains of airs of neurons in cortex and in thalamus are often correlated in a relatively narrow time scale (of the order of 10 msec) (Abeles, 1991; Gray & Singer, 1989; Perkel, Gerstein, & Moore, 1967a, 1967b; Vaadia et al., 1995). However, the observed size of these correlations indicates that in general, only a small fraction of the neuronal activity is tightly correlated. Another ossibility, addressed in this article, is that although the inuts to a cell are only weakly correlated, the cell is sensitive to the residual correlations in the somatic otential. Several mechanisms that generate enhanced sensitivity of a cell to small uctuations in its otential have been exlored (Bell, Mainen, Tsodyks, & Sejnowski, 1994; Ermentrout & Gutkin, in ress; Gerstein & Mandelbrot, 1964; Shadlen & Newsome, 1994, 1995; Softky, 1995; Troyer & Miller, 1997). One ossibility is that the excitatory and inhibitory inuts to a cortical cell are balanced in a way that leaves the cell s average otential close to threshold, and its ring attern is therefore suscetible to small uctuations. An interesting question is what might be the mechanism that leads to this balance. An interesting recent study (Tsodyks & Sejnowski, 1995) exlored the

3 Chaotic Balanced State in a Model of Cortical Circuits 1323 ossible involvement of local cortical dynamics in balancing excitation and inhibition. This numerical study invoked a strong stochasticity in the synatic action in the form of a large failure robability. In a related study (Amit & Brunel, 1997a, 1997b), the variability in the network activity is at least artially due to uctuating external inuts to the local network. In addition, neither study roerly addresses imortant issues concerning the behavior of the models and the robustness of their variability as the network size is scaled u. In this article, we investigate the hyothesis that the intrinsic deterministic dynamics of local cortical networks is suf cient to generate strong variability in the neuronal ring atterns. Neuronal dynamics is highly nonlinear; hence it may seem natural to exect that neuronal networks with deterministic dynamics will exhibit chaotic behavior. However, studies of simle models of large networks with a high degree of connectivity (Abbott & van Vreeswijk, 1993; Gerstner & van Hemmen, 1993; Grannan, Kleinfeld, & Somolinsky, 1992; Hansel, Mato, & Meunier, 1995; van Vreeswijk, 1996; Wilson & Cowan, 1972; Tsodyks, Mitkov, & Somolinsky, 1993) reveal that in the absence of external sources of strong stochastic noise, they tend to settle into temorally ordered states of tonic ring or oscillations. Recent extensive numerical study (Bush & Douglas, 1991; Hansel & Somolinsky, 1992, 1996) of a model of local circuits in visual cortex with realistic conductance-based dynamics has shown the existence of arameter regimes in which these networks exhibit strongly irregular states, denoted as synchronized chaotic states. These chaotic states are generated by the emergence of strong synchrony in the uctuating activity of different neurons, which consistently generates a strong uctuating feedback to each cell. Thus, this is a network realization of the scenario of correlated synatic inuts. The resulting atterns of activity show strongly synchronized bursting atterns, tightly timed by the common inhibitory feedback. Although bursting atterns are sometimes observed in cortical networks, these synchronized chaotic states are hard to reconcile with the Poisson-like weakly correlated ring atterns, commonly observed in cortex. In this work, we exlore the ossibility that local networks with intrinsic dynamics evolve toward states that are characterized by strong chaos in conjunction with weak cross-correlations, through the mechanism of balancing between excitation and inhibition. This ossibility raises several questions: What are the conditions under which a network evolves to a state in which the excitatory and inhibitory inuts are balanced? What are the characteristics of this balanced state? Does the balanced state reresent a cooerative state that is qualitatively distinct from the synchronized chaotic state? What are the functional advantages of the balanced state?

4 1324 C. van Vreeswijk and H. Somolinsky We study these questions using a network model with the simli ed dynamics of binary elements. The architecture consists of excitatory and inhibitory oulations connected by sarse random connections. An essential ingredient of our model is the introduction of strong connections among the units. A cell is connected, on the average, to K other cells, and K is large. However, the ga between the threshold of the cell and its resting otential is only of the order of K excitatory inuts. Thus, the network will saturate unless a dynamically develoed balance between the excitatory inuts and the inhibitory inuts to a cell emerges. Indeed, our analytical solution of the model in the limit of large network size shows that in a broad range of arameters, the network settles into a stable balanced state. An interesting feature of our theory is that it goes far beyond calculating the roerties of the macroscoic order arameters. The theory yields a comlete statistical characterization of the balanced state. It shows that the balanced state is associated with a strong Poisson-like ring attern and also with a broad inhomogeneity in the average rates of individual neurons. Finally, we address the ossible functional imlications of the balanced state by showing that the network is caable of fast tracking of temoral changes in the external inut to the network. In section 2, we resent the model s dynamics and architecture. Section 3 resents the mean- eld dynamic equations of the evolution in time of the two macroscoic order arameters, which are the rates of activity of the two suboulations. The mean- eld theory is exact in the limit of large network size, N, and 1 K N. In section 4 the behavior of the oulation rates in the balanced state is studied. Section 5 is devoted to the satial and temoral distribution of activity within the network. Section 6 addresses the stability of the balanced state. It shows that there is a comfortable arameter regime where the balanced state is stable. We also discuss what haens to the network when the balanced xed oint is unstable. Section 7 considers the effect of inhomogeneity in the local thresholds. We show that in the resence of inhomogeneity, the distribution of rates acquires a characteristic skewed shae with a long tail, qualitatively similar to the observed distribution of rates in oulations of neurons in the cortex of behaving monkeys. In section 8, we evaluate the sensitivity of the temoral uctuations in the local instantaneous activities to a small change in the initial condition. We conclude that a small change in the initial condition leads raidly to a comlete loss of memory of the unerturbed initial conditions. Thus, our network shows the main characteristics of chaotic systems. Section 9 studies the dynamical resonse of the system to dynamic changes in the external inut and shows the fast tracking caabilities of the network. In section 10 we discuss the results and some oen issues. Details of the theory are outlined in aendixes A and B. A reliminary reort on this work was ublished in van Vreeswijk and Somolinsky (1996).

5 Chaotic Balanced State in a Model of Cortical Circuits 1325 E E Excit. J EE J EI J IE E I External Inhib. J II Figure 1: A schematic reresentation of the network architecture. Excitatory connections are shown as oen circles; inhibitory ones as lled circles. 2 The Model We consider a network of N 1 excitatory and N 2 inhibitory neurons. The network also receives inut from excitatory neurons outside it (see Figure 1). We will use either the subscrit 1 or E to denote the excitatory oulation and 2 or I for the inhibitory one. The attern of connections is random but xed in time. The connection between the ith ostsynatic neuron of the kth oulation and the jth resynatic neuron of the lth oulation, denoted J ij kl, is J kl / K with robability K/N k and zero otherwise. Here k, l D 1, 2. The synatic constants J k 1 are ositive and J k 2 negative. Thus, on average, K excitatory and K inhibitory neurons roject to each neuron. We will call K the connectivity index. The state of each neuron is described by a binary variable s. The value s D 0 (s D 1) corresonds to a quiescent (active) state. The network has an asynchronous dynamics where only one neuron udates its state at any given time. The udated state of the udating neuron at time t is sk i(t) D H(ui k (t)), (2.1) where H(x) is the Heaviside function, H(x) D 0 for x 0 and H(x) D 1 for x > 0. The total synatic inut, u i k to the neuron, relative to the threshold, h k, at time t is u i k (t) D 2X ld1 N l X jd1 J ij kl s j l (t) C u0 k h k, (2.2) where u 0 k denotes the constant external inut to the kth oulation. As exlained in aendix B, the recise de nition of the order of udates is not essential. One model is a stochastic model in which each neuron udates its

6 1326 C. van Vreeswijk and H. Somolinsky state at time intervals that have Poisson statistics. This model is the simlest to analyze. However, it has the drawback that it introduces a stochastic element (the random choice of the udating neuron). An alternative model is a fully deterministic one in which each neuron udates its state at equally saced times where the time between udates is different for each neuron. We show in aendix B that the two models have the same mean- eld equations. In both cases, the mean interval between consecutive udates of a neuron of the kth oulation is t k. We will use time units such that t E D 1 so that the only indeendent time arameter is t t I. To corresond to oint rocesses, we de ne a sike as the transition from the assive (0) to the active (1) state. Note that the ring rate, r i k, of neuron i in oulation k is different from the average value, m i k (t), of si k because before the cell can sike, it has to udate to the assive state. However, if neuron i of the kth oulation udates to the active state in two consecutive udates, the synases rojecting from this cell stay active after the second udate, even though no new sike is emitted. However, if m i k, which we will call the activity rate, is small, the robability of two consecutive udates to the active state is low, and thus for small m i k, the activity rate and the ring rate are nearly equal. Indeed, if we assume that at each udate the robability of being in the active state is m i k (very nearly true in this model for low rates, as shown in section 5.3), the ring rate is given by r i k D mi k (1 mi k ) /t k. A central ingredient of our model is the assumtion that the total excitatory feedback current and the total inhibitory current into a cell are large comared to the neuronal threshold. We model this by choosing thresholds h k that are of order 1 and by assuming that the strength of individual synases is of order 1/ K, that is, the coef cients J kl are of order unity. Furthermore, as will be seen later, it is crucial that the excitatory inuts from the external sources too are large comared to the threshold. This is modeled by denoting these inuts as u 0 k D E k m 0 K k D 1, 2, (2.3) where E k is of order unity and 0 < m 0 < 1 reresents the mean activity of the external neurons. We will use the notation E 1 D E (2.4) for the external inut to the excitatory oulation and E 2 D I (2.5) for the external inut to the inhibitory neurons. We assume that the external inut is temorally regular. Since the model neurons are threshold elements, the absolute scale of u k i is irrelevant. We therefore set J EE D J IE D 1, (2.6)

7 Chaotic Balanced State in a Model of Cortical Circuits 1327 so that the only connection arameters from the network are the inhibitory and external ones. We will denote the former as J E J EI I J I J II, (2.7) where J I, J E > 0. 3 Mean-Field Equations for Poulation Rates The dynamics of our model can be described by mean- eld theory, which is exact in the limit of large N k. To de ne this limit, we assume that N I /N E is held xed as the network size N D N E C N I grows. The nature of the mean- eld theory deends on the assumed relationshi between the network size and the connectivity index. Conventional mean- eld theory assumes that the networks are fully connected, de ned here to mean that K/N is xed as N! 1. Here we assume sarse connectivity de ned by assuming that K is xed as N grows. We are rimarily interested in temoral variability that is resent in highly connected networks, which are either fully connected or sarsely connected with large connectivity index K. Therefore we will focus on the case of large K. Technically, we will rst take the limit N! 1 and then the limit K! 1. In reality, networks have a large xed size and connectivity, so that the distinction between full and sarse connectivity may be roblematic. Nevertheless, the sarse limit is aroriate as long as 1 K N k, k D 1, 2. (3.1) The mean- eld theory of our model for arbitrary xed K is resented in aendix A. Taking the large K limit rovides a substantial simli cation of the mean- eld equations. In this limit, most of the roerties of the system can be exressed in terms of the rst and second moments of the neuronal activity levels, as will be shown here and in the following sections. We rst consider the oulation-averaged ring rates of the excitatory and inhibitory cells as m k (t) D [sk i(t)] D 1 XN k N k id1 sk i (t), k D 1, 2, (3.2) where [...] denotes a oulation average. In aendix A we show that the average activities satisfy in the large K limit t k d dt m k(t) D m k (t) C H Here H is the comlementary error function, H(z) Z 1 z uk ak. (3.3) dx 2 e x2 /2, (3.4)

8 1328 C. van Vreeswijk and H. Somolinsky H(x) Figure 2: Comlementary error function H(x). The error function varies sigmoidally from 1 for x! 1 to 0 at x! 1. x shown in Figure 2. The quantities u k (t) and a k (t) are! 2X u k (t) D K J ( kl m l (t) C E k m 0 h k (3.5) and ld1 a k (t) D 2X (J kl ) 2 m l (t), (3.6) ld1 resectively. Equation 3.5 denotes the oulation average of the total inut to a neuron in the kth oulation, relative to threshold. Equation 3.6 denotes the variance of this inut. Note that the external oulation does not contribute to variance because we assumed that the inut is the same for all the neurons in a oulation. In the case of a constant external inut, m 0 (t) D m 0, the network settles into a state in which the average activities are constant, m k (t) D m k, given by the stable xed oints of equation 3.3, m k D H uk ak, (3.7)

9 Chaotic Balanced State in a Model of Cortical Circuits 1329 where the mean inuts are u E D (Em 0 C m E J E m I ) K h E, (3.8) u I D (Im 0 C m E J I m I ) K h I. (3.9) The variance of the inuts is a k D m E C J 2 k m I. (3.10) Equation 3.3 re ects the fact that the instantaneous inut to each neuron u k i (t) uctuates across the oulation of neurons, and these uctuations obey a gaussian statistics in the large K limit. The exressions for the mean and variance of the inut to a cell can be derived in the large K limit by the following arguments. The oulation average inuts is u k (t) D [u i k (t)] D 2X XN l [ J ij kl ] [ s j l (t) ] C u0 k h k. (3.11) ld1 jd1 The oulation average [J ij kl ] is equivalent to a quenched average over the random connectivity and is therefore equal to J kl K /N, yielding equation 3.5. Note that on the right-hand side (r.h.s.) of equation 3.11 we have neglected the correlations between the random uctuations in the activity of a cell and the articular realization of its outut connectivity. This is justi ed since such correlations are weak in the large N limit. Similarly, the variance a k of the inut is a k (t) D [(du i k (t))2 ] D 2X N l X l,l 0 D1 j, j 0 D1 [ (d(j ij kl s j l (t)) )2 ], (3.12) where dx X [X]. Observing that [(J ij kl s j l (t))2 ] D Jkl 2 m l /N, whereas [(J ij kl s j l (t))]2 D Jkl 2 m2 l K /N 2, which is negligible, one obtains equation Poulation Rates in the Balanced State In a balanced state, the temoral uctuations in the inuts are of the same order as the distance between the mean inut relative to threshold (even when K is large). To show this, we need to robe the network temoral roerties. Here we study the necessary consequences of the balanced state on the behavior of the oulation rates. A necessary condition for a balanced state is that both the excitatory and the inhibitory oulations do not re at their maximum rate, or are comletely silent, when we take the limit

10 1330 C. van Vreeswijk and H. Somolinsky K! 1. In other words we look for solutions with 0 < m k < 1 in the large K limit. To have equilibrium rates with m k 6D 0, 1 in the large K limit, both u E and u I have to be nite in this limit. This means that the r.h.s. of equations 3.8 through 3.9 vanish to leading order. This leads to the following equations: Em 0 C m E J E m I D O(1/ K). (4.1) Im 0 C m E J I m I D O(1/ K). (4.2) Thus, in the large K limit we obtain m E D J IE J E I J E J I m 0 A E m 0. (4.3) m I D E I J E J I m 0 A I m 0. (4.4) Since both A E and A I have to be ositive, the couling strengths have to satisfy E I > J E J I > 1 (4.5) or E I < J E J I < 1. (4.6) Besides this balanced solution, we should also examine the ossibility of unbalanced solutions in which either m k D 0 and u k is of order K and negative, or m k D 1 and u k is of order K and ositive. Equation 4.6 admits an unbalanced solution in which m E D 0. In this solution, m I is to leading order in k given by m I D Im 0 /J I (since the leading order in u I should vanish) so that u E D K(E JE I/J I )m 0 < 0. (4.7) Furthermore if J E < 1 and J I < 1, there exists a solution with m E D m I D 1 even for m 0 D 0. In this solution, u k satis es to leading order u k D K(1 Jk ) (k D E, I), (4.8) so u k is of order K and ositive.

11 Chaotic Balanced State in a Model of Cortical Circuits 1331 Thus if we require that there be no stationary solutions with m E D 0, 1 or m I D 0, 1 for small m 0, the following constraints have to be satis ed: E I > J E J I > 1. (4.9) J E > 1. (4.10) It is straightforward to show that these constraints eliminate all ossible unbalanced states. Throughout the article, we will assume that equations 4.9 and 4.10 are satis ed and that m 0 is small enough so that A k m 0 < 1. Equations 4.3 and 4.4 imly that the network activity rates grow linearly with the external rate, m k D A k m 0, even though microscoic dynamics is highly nonlinear. This is because the network dynamically nds an oerating oint at which the net inut in both oulations is balanced. Thus, the linearity in the network rates re ects the linearity of the synatic summation underlying our model. 4.1 The Net Inut. Equations 4.3 and 4.4 determine the average rates of the oulations, but they must be consistent also with the general equilibrium results of equation 3.7. According to equations 4.3 and 4.4, the leading O( K) contributions to u k cancel each other. Thus, the net value of u k is determined by subleading contributions, such as corrections of order 1/ K to m k. In fact, equations 3.7 should be viewed as equations that determine the net synatic inuts u k given the mean activity rates m k, equations 4.3 and 4.4. It is useful to denote by h(m) the scaled inut of m, de ned as the solution of the equation m D H( h). (4.11) Thus, equation 3.7 reduces to u k D a k h(m k ). (4.12) The activity of neurons in cortex is usually much less than the saturation rate. It is therefore useful to consider the limit where m 0 1. In this regime, m k 1, and we can use the aroximation H(x) ¼ ex( x2 /2) 2 x (4.13) to obtain q h(m) ¼ 2 log m. (4.14)

12 1332 C. van Vreeswijk and H. Somolinsky Substituting this result in equation 4.12 yields q u k ¼ 2 a k log m k. (4.15) This relation between m k and u k will be needed to calculate the rate distribution (see section 5.1). 4.2 Finite K Corrections. For nite K the residuals of order 1/ K in the rates are not negligible, so that equations 4.3 and 4.4 no longer hold exactly. For nite K, the equilibrium activities satisfy m k D F k (m E, m I ), with F k given by equation A.5. However, as long as m k À K 1, (4.16) the gaussian assumtion of the inut statistics is a good aroximation; hence, equations 3.7 still hold. Thus, the leading nite K corrections can be incororated by resorting to the full mean- eld equations: equations 3.3 through In articular, the nite K equations for the xed oint are Em 0 C m E J E m I D (h E C a E h(m E ))/ K. (4.17) Im 0 C m E J I m I D (h I C a I h(m I ))/ K. (4.18) As long as m 0 is not small, the r.h.s. of these equations are small for large K; hence the corrections to the linear solution, equations 4.3 and 4.4, are small. When m 0 becomes suf ciently small (i.e., of order 1/ K or less), the strong nonlinearity in the single neuron dynamics reveals itself in a strong nonlinearity in the oulation resonse. In articular, the effect of the single neuron threshold h k becomes imortant. This is seen in Figure 3, where the oulation rates are evaluated by the nite K equations equations 4.17 and 4.18 with K D For comarison, we also show the straight lines redicted by the large K limit. Since the steady-state rates in cortical networks are usually low, it is sometimes useful to incororate the leading nite K corrections. Whenever we refer in subsequent gures to exlicit values for K, we use equations 4.17 and 4.18, unless otherwise stated. Excet for thresholding the oulation rates, the nite K corrections affect only the quantitative results, not the qualitative redictions of the simle large K theory. 5 Satial and Temoral Variability So far we have been concerned only with the oulation average rates m k. However, the fact that the oulation averages are not saturated does not necessarily imly that the system s state exhibits strong temoral variations.

13 Chaotic Balanced State in a Model of Cortical Circuits Network rates External rate Figure 3: The mean activity of the excitatory oulation (thick solid line) and the inhibitory oulation (thick dashed line) as a function of the inut activity. For a network in which cells receive inut from, on average, 1000 cells in each oulation. For comarison, the activities in the large K limit are also shown (thin solid line for the excitatory and thin dashed line for the inhibitory oulation, resectively). Parameter values: E D 1, I D 0.8, J E D 2, J I D 1.8, h E D 1, and h I D 0.7. Seci cally, a oulation average excitatory rate, m k, may be the outcome of a uctuating state where all the cells in the kth oulation re a fraction m k of the time. However, it can also be achieved by a frozen state in which a fraction m k of the cells re every time these cells are udated, while all other cells never re. In other words, the oulation average does not distinguish between temoral and satial uctuations of activity levels. Fortunately, the mean- eld theory fully characterizes the statistics of both the satial and the temoral uctuations in the activities in the balanced state. The statistics can be exressed by writing the instantaneous activity of a cell as a threshold function of two random variables x i and y i (t), sk ± u i(t) D H ² k C bk x i C ak b k y i (t). (5.1) The means u k are given by equations 3.8 and 3.9. The arameter b k is given

14 1334 C. van Vreeswijk and H. Somolinsky by b k D q E C J 2 k q I. (5.2) The order arameter q k is de ned as q k D 1 XN k (m i k N )2, (5.3) k id1 where m i k is the time-averaged activity rate of the ith cell, m i k hsi k (t)i. (5.4) The symbol h...i denotes average over long time. Both x i and y i (t) are indeendent gaussian variables with zero mean and unit variance. Quenched uctuations of synatic inuts. The term roortional to x i reresents a quenched random comonent of the synatic inut received by different cells and thus reresents a satial inhomogeneity in the rates. The origin of this inhomogeneity is twofold. Since the connectivity is random in our model, cells may differ in the number of synatic inuts they have. This comonent is given by d 1 hu i k i D 2X ld1 N l X jd1 dj ij kl [mj l ]. (5.5) Here, dj ij kl D Jij kl [Jij kl ]. In addition, different neurons are connected to different cells so that even if all the cells would have received the same number of inuts, the system would evolve into a state with a self-consistently develoed satial inhomogeneity. The second comonent can be written as d 2 hu i k i D 2X ld1 N l X jd1 J ij kl dmj l, (5.6) where dm i l mi l m l. Adding the two contributions yields [ (dhu i k (t)i )2 ] D 2X Jkl 2 q l D q E C Jk 2 q I D b k. (5.7) ld1 Thus, this variance reresents the uctuation in both the number and the identity of inut cells to the different cells.

15 Chaotic Balanced State in a Model of Cortical Circuits 1335 Temoral uctuations of synatic inuts. The term in equation 5.1 that is roortional to y i (t) reresents the stochastic comonent of the inuts to a cell a temorally uctuating comonent with a short-time correlation. This can be written as u i k (t) hui k i D 2X ld1 N l X jd1 J ij kl (s j k (t) mj ), k (5.8) from which one obtains [(u i k (t) hui k i2 ] D 2X Jkl 2 (m l q l ) D m E q E C Jk 2(m I q I ) ld1 D a k b k (5.9) Note that the variance of the temoral uctuations in the inuts deends on m k q k, which measures the temoral variability of the state. 5.1 Distribution of Time-Averaged Rates. The distribution of rates in the kth oulation is de ned as r k (m) N 1 k N k X id1 d(m m k i ). (5.10) The statistics of the time-averaged local rates can be derived by averaging equation 5.1 over y i (t) (which is equivalent to average over time), m i k D m k(x i ) D H uk C b k x i ak b k. (5.11) Thus, the distribution of m i k is fully determined by its rst two moments. Averaging this equation over x i yields equation 3.7. Similarly, squaring equation 5.11 and averaging over x i yields q k D Z Dx µ H uk C b k x ak b k 2. (5.12) Here we have used the gaussian measure, Dx dx ex( x 2 /2)/ 2. In general, q k satis es (m k ) 2 q k m k. The smaller q k is, the more homogeneous is the rate distribution. In a frozen state in which a fraction m k of the cells are active every time they are udated, while all other cells are always quiescent, q k is given by q k D m k. On the other hand, if all cells in the oulation have a robability m k of being active each time they are udated,

16 1336 C. van Vreeswijk and H. Somolinsky m k i D m k, q k D (m k ) 2. Equations 5.12 have two solutions: an unstable solution with q k D m k, corresonding to a frozen state, and a stable solution, (m k ) 2 < q k < m k, which corresonds to a temorally uctuating state. Although the frozen solution is unstable, its existence highlights the fact that the temoral variability in our system is urely of deterministic origin and is not induced by external stochastic sources. Generalizing equation 5.12, we can write, Z r k (m) D Dxd(m m k (x)). (5.13) In section A.1 we analyze the roerties of this distribution. A numerical evaluation ofr k (m) is shown in Figure 4, which dislays the rate distribution of the excitatory activity for different values of m E. The distribution is lotted against m/ m E. The synatic coulings were ket constant, while the mean rates were varied by adjusting the external rate m 0. For high mean activity levels, the distribution has a ronounced skewed shae. Note, however, that according to equation 5.11 the distribution of the time-averaged inuts u k i to the cells is gaussian for all values of m k. In the low rate limit, m 0 1, equation 5.12 can be solved using equations 4.13 through 4.15, yielding to leading order, q k D m 2 k C O(m3 k log m k ). (5.14) Thus, if the network evolves to a state with low average activity levels, m k 1, q k is slightly larger than m 2 k. The fact that q k m k imlies that the balanced state is characterized by strong temoral uctuations in the activity of the individual cells. On the other hand, the fact that q k is not exactly equal to m 2 k re ects the satial inhomogeneity in the time-averaged rates within a oulation. Equation 5.14 imlies that when the mean activity m k decreases, the width of the distribution is roortional to (m k ) 3/2 ; it decreases faster than the mean m k. Thus, for low mean activity, r k (m) becomes narrowly eaked at m D m k, as shown in Figure 4. The reason for the narrow eak is that in our model, the uctuations in the inut are related to the uctuations in the feedback from the network, hence their variance becomes small as the activity in the network decreases (see equation 5.2.) 5.2 Time-Delayed Autocorrelations. In order to comlete the statistical characterization of the balanced state, we have to determine the statistics of the temoral uctuations in the activities of single cells or, equivalently, the temoral uctuations in their inut. We have already stated that the temoral uctuations in u k i (t) obey gaussian statistics, with variance given by a k b k. To characterize its statistics fully, we have to evaluate its autocorrelations. Using arguments similar to those already outlined, it is straightforward to show that the autocorrelation of the inut is linearly related to

17 Chaotic Balanced State in a Model of Cortical Circuits Density Rate/mean rate Figure 4: Distribution of the activity rates of the excitatory oulation for two different values of the average rate in the large K limit: m E D 0.01 (solid line) and m E D 0.1 (dashed line). The distributions are shown as a function of the local rate divided by the mean rate. Parameter values as in Figure 3. the autocorrelations in the local activities, b k (t ) D [hdu k i (t)duk i (t C t )i] D q E(t ) C J 2 k q I(t ), (5.15) where q k (t ) is the time-delayed autocorrelations of the local activities, q k (t ) D N 1 k N k X hsi k(t)sk i (t C t )i, (5.16) id1 and as before h...i denotes average over t. Note that q k (0) D m k, whereas q k (t! 1) D q k. Likewise, b k (0) D a k, whereas b k (t! 1) D b k. Using this relation, the following self-consistent equation for q k (t ) (with t 0) is obtained: t k dq k (t ) dt D q k (t ) C Z 1 0 dt t k Z ex( t/t k ) µ Dx H uk b k (t C t ) x ak b k (t C t ) 2. (5.17)

18 1338 C. van Vreeswijk and H. Somolinsky 0.2 Autocorrelation t Figure 5: Poulation-averaged autocorrelation for the excitatory oulation in the large K limit (solid line). The dashed line shows the autocorrelation for a oulation of cells with the same rate distribution but Poissonian udating. Parameter values: t D 0.9, m 0 D 0.1, and other arameters as in Figure 3. Note that the integral over t in equation 5.17 results from averaging over the distribution of udate time intervals. The solution of this integral equation yields a function q k (t), which decays to its equilibrium value with a time constant of the order of t k. A numerical solution of equation 5.17 for q k (t ) is shown in Figure 5. As can be seen, the autocorrelations are larger than those redicted by Poisson statistics. This enhancement of short-time correlations re ect the refractoriness in the activities of the cells that roject the cell. 5.3 Numerical Realization of Synatic Inuts to a Cell. In order to demonstrate the nature of the uctuating synatic inuts to a single excitatory cell in the balanced state, we have numerically generated samles of stochastic gaussian rocesses, which simulate the uctuations of the synatic inuts to a single excitatory cell. In order to show exlicitly the effect of balancing, we have simulated searately the total excitatory and inhibitory comonents of u i E (t). The time average of the total excitatory (inhibitory) comonent is itself samled from a gaussian distribution with a mean K(m E C Em 0 ) ( KJ E m I ) and a variance q E (J E q I ). The time-deendent uctuations of the total excitatory (inhibitory) inut have a time-delayed au-

19 Chaotic Balanced State in a Model of Cortical Circuits Inut 0-4 Sikes t Figure 6: Temoral structure of the inut to an excitatory cell. The uer anel shows the total excitatory inut, consisting of the external inut and the excitatory feedback (uer trace), the total inhibitory inut (lower trace), as well as the net inut (middle trace). They are calculated by samling from the timecorrelated gaussian statistics redicted by the theory. The times when the cell switched to the active states are indicated. Parameter values: m 0 D 0.04 and other arameters as in Figure 5. K D 1000 was used to calculate the average inut. tocorrelation equal to q E (t ) q E (J 2 E (q I(t ) q I )). The results are shown in Figure 6, where we have used K D They demonstrate that the total excitatory and inhibitory inuts are large comared to the threshold and have uctuations that are small comared to their mean. Because the network is in the balanced state, the net inut is of the same order as the threshold, and the uctuations bring the inut above threshold at irregular intervals. In the lower art of the gure, we show the outut state of the cell. This is evaluated by generating the sequence of udate times and thresholding the net inut at these times. Because of the udate rule, the cell does not switch from assive to active every time the net inut crosses the threshold. In Figure 7 we resent the ISI histogram of the cell. Because the interval between sikes is a convolution of two random events rst, a transition from 1 to 0, and then a transition from 0 to 1 the ISI vanishes at small intervals. Thus, the above de nition of a sike to some extent catures the refractoriness of real sikes. In fact, if we ignore the short time correlations in the activities, the ISI of the ith (say, excitatory) cell with an average rate

20 1340 C. van Vreeswijk and H. Somolinsky P(t) t Figure 7: Intersike interval distribution for the cell shown in Figure 6 (solid line). The distribution was determined by measuring the time between consecutive switches from the inactive to the active state until intervals had been accumulated. The dashed line shows the intersike interval distribution for Poissonian udating with m i D m i can be shown to be simly I i (t) D m i(1 m i ) t E (1 2m i ) (ex( m it/t E ) ex( (1 m i )t/t E )), t 0. (5.18) This function rises linearly from zero and eaks at t / t E. For intervals of the order of t E /m i or longer, I(t) decays urely exonentially with a decay constant m i /t E as in the ISI of a single Poisson rocess with a rate m i /t E. Comarison with Figure 7 shows that this is indeed a very good aroximation of the ISI of our model. Finally, it should be noted that because of the sarsity of the connectivity, different cells receive inut from different suboulations, so that the uctuations in their inut will be only very weakly correlated. As a result, the correlations in their activity will be very small.

21 Chaotic Balanced State in a Model of Cortical Circuits Stability of the Balanced State To determine the stability of the balanced state, we have to study the resonse of the system to small erturbations in the oulation activity rates. However, because of the nature of the balanced state, we have to distinguish two scales of erturbations: local erturbations, in which the deviations in the rates are small comared to 1/ K, and global erturbations, in which these deviations are large comared to 1/ K. 6.1 Local Stability. Local stability of the balanced state requires that a suf ciently small erturbation in the oulations rates will decay to zero. In our case, a suf ciently small erturbation means that it initially causes only a small disrution of the balanced state. This means that the erturbations are small not only comared to m k but also comared to 1/ K, so that the erturbations of the inuts to the cells are initially small. We therefore consider a solution of equations 3.3 with an initial condition m k (0) D m k Cdm k (0) with a small dm k (0) where dm k (0) 1/ K. In this case, the erturbation of the total mean inut u k is also small; hence we can linearize the dynamic equations 3.3 around their xed oint. Thus, dm k (t) D m k (t) m k satisfy a linear equation of the form d t k dt dm X k(t) D dm k (t) C K f kl dm l (t). (6.1) ld1,2 Calculating f kl by artial differentiation of the r.h.s. of equation 3.3 yields f kl D ex( u2 k /2a k )J kl 2 ak. (6.2) Solving equations 6.1 one obtains dm k (t) D dm k,1 ex(l 1 t) C dm k,2 ex(l 2 t) where the eigenvalues l 1 and l 2 of the 2 by 2 equations (see equations 6.1) are both of order K. Requiring that their real art be negative yields a condition on t of the form t < t L, (6.3) where t L is of order 1; its recise value deends on the system arameters. Since both l 1 and l 2 are of order K, if t < t L, small erturbations will decay with an extremely short time constant of order 1/ K. This is due to the strong negative feedback, of order K, generated by the strong synatic coulings. 6.2 Global Stability. The local stability condition in equation 6.3 guarantees that a erturbation smaller than O(1/ K) will die out. It is therefore

22 1342 C. van Vreeswijk and H. Somolinsky imortant to ask whether the balanced state is stable also to erturbations that are large comared to this order. However, such erturbations will generate a large disrution in the inuts u k, of order K; hence, linearization of the dynamic equations is inadequate. We therefore have to consider the nonlinear evolution of erturbations in the rates under equations 3.3. In fact, since the erturbation destroys the balance between excitation and inhibition, H( u k / a k ) of equation 3.3 can be aroximated by H(u k ); hence the evolution of the erturbations is described by t k d dt dm k(t) D dm k (t) C H(dm E J k dm I ) m k. (6.4) These equations are iecewise linear and therefore can be solved exlicitly. One nds that the solution of these equations decays to zero rovided that the inhibitory time constant satis es t < t G, (6.5) where ( s ) J I m I (1 m I ) t G D J E min J E m E (1 m E ), m I, 1 m I. (6.6) m E 1 m E In conclusion, the global stability condition guarantees that starting from arbitrary initial values m k (0), the oulation rates eventually will aroach the balanced regime characterized by local elds u k, which are of order 1 and not K. In other words, the rates will deviate from the values of the balanced xed oint by at most O(1/ K) quantities. Whether they will actually aroach this xed oint or will converge to a limit cycle around it deends on the local stability condition equation 6.3. Deending on the system arameters t G may be greater or smaller than t L. Figure 8 shows the evolution of m E and m I in a network with K D 1000, for t D 1.3, when the network starts far away from the balanced state. The initial evolution is similar to the global dynamics. It converges to the neighborhood of the balanced xed oint in an oscillatory manner characteristic of the dynamics of equation 6.4. In Figure 8B, we show the late ortion of the dynamics, which corresonds to the local dynamics (see equations 6.1). For the arameters used in this gure, the large K critical inhibitory time constants are t L D 1.61 and t G D To illustrate the region of stability of the balanced state, we have calculated the hase diagram of the network in terms of two arameters: (1) the inhibitory time constant t and (2) the ratio between the external inut into the inhibitory oulation and the external inut into the excitatory oulation. We have chosen to scale m 0 so that the excitatory oulation rate is held xed. The results are shown in Figure 9, where both the local and global

23 Chaotic Balanced State in a Model of Cortical Circuits 1343 A B m I m I m E m E Figure 8: Evolution to the stable xed oint. The average inhibitory rate is lotted against the average excitatory rate. (A) The evolution of the rates when the rates are initialized far from their steady-state values. (B) A close-u view of the aroach to the xed oint. Parameters: t D 1.3. The other arameter values as in Figure 3. stability lines are resented. For these arameters, t L is always smaller than t G. 6.3 Regimes of Instability. Stability of the balanced state requires that t be smaller than both t L and t G. It is of interest to consider what haens if this condition is not ful lled. 1. Unbalanced limit cycle. t > maxft L, t G g. In this case equations 3.3 ossess a stable unbalanced limit cycle a stable oscillatory solution with u k (t) of order K. This is shown in Figure 10A. 2. Balanced limit cycle. t L < t < t G. In this case, erturbations that are of order 1 will decrease until they are of order 1/ K, while erturbations that are small comared to 1/ K will increase until they are of order 1/ K. Since there are no xed oints with dm k of order 1/ K, this means that there has to be a stable limit cycle with an amlitude of order 1/ K. Thus, in this regime, the system converges to a limit cycle that maintains the aroximate balance between excitation and inhibition. This is described schematically in Figure 10B. 3. Balanced xed oint with shrinking basin: t G < t < t L. In this case, erturbations of order 1 go to a global limit cycle, while erturbations much smaller than 1/ K evolve to the xed oint. There must be an unstable limit cycle with amlitude of order 1/ K that searates erturbations that go to the global limit cycle and erturbations that go to the xed oint.

24 t 1344 C. van Vreeswijk and H. Somolinsky Unstable 1 Stable I/E Figure 9: Critical time constants t G (solid line) and t L (dashed line) as a function of I/E. The external rate was adjusted to kee the excitatory activity level constant at m E D 0.1. I/E was varied from 0 to (J I m E )/(J E m E ). For this range of I/E, m I varies from m E /J I to 1. Parameters: E D 1, J E D 2.0, and J I D Inhomogeneous Thresholds So far we have considered networks of identical neurons, excet for their connectivity. Real neuronal systems exhibit a substantial inhomogeneity in single neuron roerties. It is therefore imortant to consider how such inhomogeneities affect the behavior of our system. We will model the inhomogeneity by a variability in the thresholds of the neurons. Inhomogeneities in the local thresholds may have a articularly strong effect in a balanced state with low mean activity. The reason is that the intrinsic uctuations are all generated by feedback from the network activity. Hence, they decrease in amlitude as the mean activity in the network dros. In articular, under these conditions, the intrinsic temoral uctuations may not be of suf ciently large amlitude to overcome the quenched disersion of local thresholds. Therefore, the imortant issue we address here is whether the balanced state remains temorally uctuating in the limit of low mean activity in the resence of inhomogeneous thresholds, or whether it becomes a frozen state. We will show that the answer to these questions deends not only on the width of the threshold distribution but also on the form of its

25 Chaotic Balanced State in a Model of Cortical Circuits A 0.8 B m I m I m E m E Figure 10: Different scenarios when the xed oint is unstable. (A) A case where t is larger than both t G and t L. Here the rates evolve to the global limit cycle where the amlitude of the oscillations is of order 1. The solid line shows the evolution when the network is initiated outside the limit cycle; the dashed line corresonds to the trajectory for initial rates inside the limit cycle. Parameters as in Figure 8, excet t D 1.8. (B) Shows schematically a case where t L < t < t G. The network evolves to a limit cycle with the amlitude of order K 1/2. The gure shows the evolution of the rates with initial conditions far from the xed oint. The insert shows an exanded view of the area around the xed oint, with the trajectory of rates starting outside the limit cycle (solid line) and the trajectory of a network that was initiated close to the xed oint. tail. We denote the local threshold of a neuron by hi k C h k, where h k is the oulation-averaged threshold and hi k is a quenched random variable with zero mean. We will call hi k the local threshold. The mean activity rate of neurons in the kth oulation that have a local threshold h is m k (h) D H h uk ak, (7.1) and hence the oulation-averaged rate is Z h uk m k D dhp(h) H, (7.2) ak where P(h ) denotes the quenched distribution of h, and u k and a k are given as before by equations 3.8, 3.9, and Note that we have absorbed the mean threshold, h k in the de nition of u k (see equations 3.8 and 3.9). 7.1 Distribution of Thresholds with Long Tails. We rst consider a distribution with a long tail of low thresholds. A concrete examle is 1 P(h ) D D 2 ex 1. 2 (h /D)2 (7.3)

26 1346 C. van Vreeswijk and H. Somolinsky In this case, the satial uctuations in the inuts (relative to thresholds) consist of two gaussian terms. One is induced by the random connectivity and has a variance a k, and the other is induced by the thresholds and has a variance D. The balance conditions that determine the oulation rates (equations 4.3 and 4.4) still hold. In addition,! u k, (7.4) ak C D 2 m k D H( which determines u k, and q k D Z Dx µ H uk D C b k x ak b k 2. (7.5) Now let us consider the limit of low mean rates, which is achieved by assuming that m 0 is small. For xed D, if the mean rates become suf ciently low so that m k D, the intrinsic variances a k and b k can be neglected comared with D; hence one obtains uk m k ¼ q k ¼ H. (7.6) D The fact that q k ¼ m k imlies that the state is essentially frozen, namely, m k (h) ¼ H(u k h ), (7.7) and, consequently, the distribution of mean rates has a distinct bimodal shae, r k (m) ¼ (1 m k )d(m) C m k d(m 1), m k 1, (7.8) as shown in Figure 11A. Thus, an unbounded threshold distribution has a relatively strong qualitative effect on the balance state, in the limit of low mean rate. 7.2 Bounded Distribution. We next consider the case of a bounded distribution of thresholds. As an examle, we take a distribution of h that is uniform between D/2 and CD/2, and zero otherwise. In this case, equation 7.2 yields m k D 1 D Z D/2 D/2 dh H uk C h ak. (7.9) To assess the effect of D, we analyze equation 7.9 in the low m k limit. In this case, the solution for u k is u k C D/2 D O( m k ). (7.10)