Optimal system size for complex dynamics in random neural networks near criticality.

Optimal system size for complex dynamics in random neural networks near criticality Gilles Wainrib and Luis Carlos García del Molino Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 23, 043134 (2013); doi: 10.1063/1.4841396 View online: http://dx.doi.org/10.1063/1.4841396 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/23/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Non-negative sparse autoencoder neural networks for the detection of overlapping, hierarchical communities in networked datasets Chaos 22, 043141 (2012); 10.1063/1.4771600 Optimal and suboptimal networks for efficient navigation measured by mean-first passage time of random walks Chaos 22, 043129 (2012); 10.1063/1.4768665 Coherent periodic activity in excitatory Erdös-Renyi neural networks: The role of network connectivity Chaos 22, 023133 (2012); 10.1063/1.4723839 Chaotic phase synchronization in a modular neuronal network of small-world subnetworks Chaos 21, 043125 (2011); 10.1063/1.3660327 The development of generalized synchronization on complex networks Chaos 19, 013130 (2009); 10.1063/1.3087531

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CHAOS 23, 043134 (2013)

Optimal system size for complex dynamics in random neural networks near criticality Gilles Wainrib1,a) and Luis Carlos Garcıa del Molino2,b) 1

Laboratoire Analyse G eom etrie et Applications, Universit e Paris XIII, Villetaneuse, France Institute Jacques Monod, Universit e Paris VII, Paris, France

2

(Received 3 June 2013; accepted 22 November 2013; published online 17 December 2013) In this article, we consider a model of dynamical agents coupled through a random connectivity matrix, as introduced by Sompolinsky et al. [Phys. Rev. Lett. 61(3), 259–262 (1988)] in the context of random neural networks. When system size is infinite, it is known that increasing the disorder parameter induces a phase transition leading to chaotic dynamics. We observe and investigate here a novel phenomenon in the sub-critical regime for finite size systems: the probability of observing complex dynamics is maximal for an intermediate system size when the disorder is close enough to criticality. We give a more general explanation of this type of system size resonance in the framework of extreme values theory for eigenvalues of random C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4841396] matrices. V

The human brain is composed of approximately one hundred billion neurons, and each neuron is typically connected through ten thousand synaptic connections. Understanding the dynamical behavior of such complex systems is a key challenge in neuroscience and beyond. The study of mathematical models of many interacting dynamical units appears to be an efficient approach to gain new insights into this problem. Due to the enormous amount of unknown parameters in these models, for instance, the synaptic connections’ strengths between each pair of neurons, a classical modeling strategy is to consider these parameters to be randomly drawn according to a given distribution, giving rise to disordered dynamical systems. In this framework, it is possible to study how the statistical properties of the disorder will affect the dynamical behavior of the system. In the context of neural networks, it is known that, at the limit of infinite number of neurons, increasing the level of disorder induces to a phase transition leading to complex chaotic dynamics. Interestingly, the critical regime displays many important features in terms of information processing capacity and has found applications in the field of machine learning. Although the behavior is rather clear in the limit of infinite system size, the situation is substantially different for finite size systems. In particular, various attractors (equilibria, limit cycles, chaos) may co-exist and contribute to a much richer dynamical repertoire, even in the sub-critical regime. We investigate in this article the probability of observing complex dynamics and report a novel phenomenon of system size resonance: close to criticality, there exists an intermediate system size for which the probability of chaos is maximal. We provide a theoretical explanation of this effect in the framework of random matrix theory. This property may have significant implications for the understanding of

dynamical systems of complex networks with a modular structure.

I. INTRODUCTION

Most complex systems involve a large number of interacting elements and are often modeled and studied within the framework dynamical systems on random networks. Typical examples range from condensed matter physics19 to biology17,26 and social sciences.24 In particular, the hundreds of billions of heterogeneous synaptic connections between neurons in the brain constitute a paradigmatic example of a disordered system of dynamical agents coupled through a random connectivity matrix. To understand the impact of this heterogeneity on brain dynamics, a random neural network model has been introduced in Refs. 1 and 26 for 1 i n;

(1)

where J ¼ ðJij Þ is a Ginibre9 random matrix with Gaussian i.i.d. entries such that E½Jij ¼ 0 and E½Jij2 ¼r2 =n and /ðÞ is a smooth odd sigmoid function with unit slope at the origin /0 ð0Þ ¼ 1: The variables xi represent the activity of a neuron i (or neuronal population) and Jij is the synaptic weight from neuron j to neuron i. Notice that zero is a trivial equilibrium of Eq. (1). It is known since26 (see also Refs. 2 and 28) that in the limit n ! 1 one can derive a mean field description of the above model, and an analysis of the mean field equations26 leads to the two following regimes: (i) (ii)

a)

Email: [email protected] Email: [email protected]

b)

1054-1500/2013/23(4)/043134/7/$30.00

n X dxi ¼ xi þ Jij /ðxj Þ; dt i¼1

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if r < 1, the only equilibrium point is zero and attracts all trajectories; if r > 1, there exist infinitely many equilibria and limit cycles but none of them is stable. The only stable attractors are chaotic in the sense of positive C 2013 AIP Publishing LLC V


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Lyapunov exponents. To be more precise, one must be aware that the study of the limit n ! 1 is in fact a difficult mathematical problem. A major breakthrough was obtained by Sompolinsky and collaborators26 using a mean-field method which has not been properly justified mathematically so far to our knowledge. However, when a white noise perturbation is added to the equations, a rigorous proof of the convergence is obtained in the work of Ben Arous and Guionnet2 (see also Ref. 21). This abrupt transition is also present in terms of linear stability analysis and can be recovered using a classical result of random matrix theory. Indeed, the circular law10,27 states that the empirical distribution of the complex eigenvalues for n n random matrices with i.i.d. centered coefficients of variance n1, converges to the uniform measure on the unit disk when n ! 1. Therefore, in the limit n ! 1, the trivial equilibrium zero becomes unstable as r crosses 1. Based on a combination of experimental and theoretical considerations, it has been argued in Refs. 4, 7, and 13 that living neuronal networks, as well as many other complex systems,3 may evolve in the vicinity of the phase transition also called the edge of chaos.15,16 Recently, this specific regime has been shown to be particularly important in the field of reservoir computing,12 a popular class of machine learning algorithms, because of the increased information processing capacity near criticality.6

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Therefore, it is of interest to study the behavior of this system close to the phase transition. Although the situation is rather clear in the infinite system size limit, as stressed in Ref. 26, the picture is different for finite size systems where, close to criticality, several stable equilibria, limit cycles, and chaotic attractors may co-exist (Fig. 1). A fine knowledge of the statistical properties of the eigenvalues distribution of random matrices is fundamental to characterize the behavior of various disordered systems, in particular, in terms of stability properties.18 In the critical region, the extreme eigenvalues of J play a crucial role in the stability in the finite size system, suggesting that its behavior will not be properly described by the mean-field equations. Indeed, in the case of finite size matrices, despite most eigenvalues being inside the unit circle, there is a nonzero probability that eigenvalues close to the border have a modulus slightly larger than 1, leading to unstable modes from the stability point of view. Our aim is to study the probability of observing spontaneous activity (limit cycle or chaotic attractor) in system (1) near the phase transition, specifically for values of disorder r slightly below criticality. As n ! 1,this probability shall converge to zero, but as we will show, this convergence is not monotonous, this probability is actually reaching a maximum for an intermediate value of n. It is instructive to mention that several phenomena of system-size resonance in interacting dynamical systems have been previously observed in the perspective of stochastic or coherence resonance,20,23 revealing that the response of such systems to

FIG. 1. Illustration of the various sample time-series of system (1) with a given choice of the disorder parameter r ¼ 1 and system size n ¼ 40. Each figure corresponds to a random choice of the connectivity matrix J and random initial conditions.


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external stimuli is optimal in some sense for an intermediate number of units. Furthermore, the sensitivity of near-critical systems to other parameters, such as noise or time-scales, have been investigated in Ref. 14 which also revealed resonance-type phenomena. Therefore, in this context, although the fact that the probability of observing complex dynamics does not change monotonically with system size n appears as a new phenomenon, it remains in line with the above-mentioned literature on resonance effects. The paper is organized as follows. We will first investigate the probability of spontaneous activity directly on system (1) numerically (Sec. II) and then provide a deeper theoretical explanation of the observed phenomenon of system size resonance through the study of extreme eigenvalues of random matrices (Sec. III). Finally, in Sec. IV, we discuss our results. II. PROBABILITY OF COMPLEX DYNAMICS

To determine whether the neural network is activated or not, we compute the maximal Lyapunov exponent K of the attractors. Depending on the value of K, one can distinguish three cases: (i) if K < 0, the attractor is a fixed point (might be other than 0), the network is not activated, (ii) if K ¼ 0, the attractor is a limit cycle, the network is activated and exhibits regular activity, and (iii) if K > 0, the attractor is chaotic, the network is activated and exhibits irregular activity. In terms of K, the probability of observing spontaneous activity, whether it is chaotic or periodic, is given by P[K 0]. In the numerical simulations, an estimation of the maxi is computed mal Lyapunov exponent of the attractors K using the variational equations as shown in Ref. 25. Starting from a random initial condition, we evolve a reference solution ðxi ðtÞÞ1in according to Eq. (1) and a normalized difference vector on the tangent plane ðdðx; tÞi Þ1in driven by the Jacobian of Eq. (1). At each time step, the modulus of d is normalized to a small quantity d0 so that ðxðtÞi þ dðx; tÞi Þ1in is always in the vicinity of the reference solution. After some iterations, the reference orbit will be close to one of the attractors of the system, and the difference vector will point in the direction corresponding to the maximal Lyapunov exponent of the reference trajectory. ¼ Dt PT=Dt log jdj, where T is the samThen, we estimate K i¼1 T d0 pling time and Dt is the time step. To circumvent approxima ¼0 tions error in the case of periodic attractors, we set K 29 when periodicity is detected directly on the trajectories. Numerical results are shown in Fig. 2. The main unexpected phenomenon is that the probability of observing spontaneous activity for r < 1, close enough to 1, displays a maximum for an intermediate value of the network size n (Fig. 2(a)). Surprisingly, this probability first increases with n until an optimal system size nðrÞ, and then decreases to zero as expected from Ref. 26. Moreover, as a gets closer to 1, both the optimal size nðrÞ and the probability of spontaneous activity at n ¼ nðrÞ increase. In the limit case r ¼ 1, we 0] tends to 1 as n do not observe a maximum and P[K becomes larger. The estimations of the probabilities corresponding to ¼ 0] (Fig. 2(b)) and chaotic oscillations limit cycles P[K

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> 0] (Fig. 2(c)) have a similar behavior. The main difP[K > 0] are reached at larger ference is that the maxima for P[K ¼ 0]. As shown in values of n than the ones for P[K Fig. 2(d), this difference means that the larger the system is, the more likely is that the spontaneous activity takes the form of chaotic attractors. Therefore, numerical simulations have revealed a novel type of system size resonance effect, showing an enhancement of the probability of observing complex dynamics for an intermediate system size. In Sec. III, we study this new phenomenon within the framework of random matrix theory. III. EXTREME EIGENVALUES OF RANDOM MATRICES

From numerical simulations, we have shown the existence of an intermediate system size which maximizes the probability of observing complex dynamics. This phenomenon has not been described previously, and we intend to provide a theoretical explanation through the lens of random matrix theory. Indeed, as mentioned in Ref. 26, the phase transition can be inferred from the study of the linearized problem around the trivial equilibrium: in the limit n ¼ 1, linear stability is lost when r > 1 as a direct consequence of the circular law. Moreover, if ðkk Þ1kn denotes the complex eigenvalues of J, and if one defines q1 ¼ max Reðkk Þ; 1kn

(2)

then q1 < 1 implies that the trivial equilibrium is globally attractive. Therefore, a natural quantity to study is the probability pn that the trivial equilibrium zero is linearly stable. Such a question has a direct translation in terms of random matrix theory pn ¼ P½q1 1 and still K < 0 due to the existence non trivial attracting equilibrium points. However, we expect both quantities to have similar properties. To study q1 , we need to build upon the extreme value theory for the eigenvalues random matrices. Contrary to classical extreme value theory, originally developed in the context of independent random variables (for some background reading on extreme value statistics, see Ref. 8), there is a strong dependence structure among the eigenvalues, which turns out to be well described within the framework of determinantal point processes. In fact, the theory of extremal eigenvalues of complex Gaussian matrices has been investigated recently in Ref. 5. Based on these results, we explain the optimal size phenomenon in the case of complex matrices. In the case of real


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FIG. 2. Numerical estimation of the probability of observing (a) spontaneous activity, (b) limit cycles, (c) chaotic trajectories, and (d) chaotic attractor given spontaneous activity, as a function of n and for different values of r 2{0.95,0.96,0.97,0.98,0.99,1}. For each matrix size and each value of r 1, 5.104 realizations of the random matrix have been analyzed and the sigmoid function used was tanh(). Standard deviations errors are smaller than the points, therefore not shown. Numerical estimations of the maximal Lyapunov exponent were performed after cutting a first period of transient dynamics.

Gaussian matrices, the situation is more delicate, and we only provide numerical results in Fig. 3 showing the same phenomenology. ðnÞ More precisely, for r > 0, we consider fkk ðrÞg1kn the complex eigenvalues of an n n random matrix J such that the coefficients Jij are i.i.d complex Gaussian random r2 I2 Þ: variables on C ¼ R2 with Jij N ð0; 2n We denote the maximum of the real parts of the eigenvalues of J by

and its complementary probability qn ðrÞ ¼ 1 pn ðrÞ that is the probability that there exists at least one eigenvalue with a real part larger than 1. Our main theoretical result is the following: Theorem 1. For any r 2 ð0; 1Þ; the integer n ðrÞ such that qn ðrÞ ¼ sup qn ðrÞ satisfies n2N

lim n ðrÞ ¼ þ1:

r!1

(6)

Moreover, qn(r) can be made arbitrary close to1 kðnÞ max ðrÞ :¼ max Reðkk Þ: 1kn

(4) lim qn ðrÞ ¼ 1:

We are interested in the probability that all the eigenvalues have a real part smaller than 1 pn ðrÞ :¼ P½knmax ðrÞ < 1

(5)

r!1

(7)

Heuristically, this property stems from a competition between (i) the fact that when the number of eigenvalues n increases, the maximal real part tends to be higher (the larger


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FIG. 3. Real Ginibre ensemble: numerical estimation of qn(r) and pn(r). A total of 5 105 Monte-Carlo runs were computed for each different matrix sizes and value of r.

a set of variables is, the larger its maximum) and (ii) the convergence of the spectral density to the unit disk that implies a concentration of kðnÞ max ðrÞ close to r < 1. The proof of theorem 1 is based on Ref. 5, where the asymptotic law for large n of the largest real part of eigenvalues for a class non-Hermitian random matrix is studied, developing extreme value theory for determinantal processes. Essentially, theorem 2.5 of Ref. 5 states that 1 t ðnÞ lim P kmax ð1Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi t þ cn ¼ ee ; (8) n!1 2 n log n for any t 2 R, with 1 cn ¼ pffiffiffi 2 2

rffiffiffiffiffiffiffiffiffiffi 1=4 5 lnðnÞ 4pffiffi2 lnðlnðnÞÞ lnð2 pÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : n n lnðnÞ

(9)

This result is in fact a in line with standard results of extreme value theory on the convergence to the Gumbel distribution for the law of the maximum of a set of random variables (Type I domain, see Ref. 8) and is therefore a rather natural result to expect. Proof. From Eq. (8), we claim that for any d 2 (0,1) lim P½knmax ð1Þ < 1 þ ð1 dÞcn ¼ 0:

n!1

(10)

Indeed, let d 2 (0,1) and > 0 fixed, and t < 0 such that t2 ee < . Then, from Eq. (8), there exists n0 1 such that for all n n0 t p ffiffiffiffiffiffiffiffiffiffiffiffi P kðnÞ ð1Þ < 1 þ c 1 þ < : (11) n max 2cn nlog n pffiffiffiffiffiffiffiffiffiffiffiffi when Moreover, as cn nlogn !p1 n ! 1 one can choose ffiffiffiffiffiffiffiffiffiffiffiffi n0 such that d < t = 2cn nlog n . Then, by monotonicity of the distribution function, we deduce

h i P kðnÞ ð1Þ < 1 þ c ð1 dÞ n max P kðnÞ max ð1Þ < 1 þ cn 1 þ

t pffiffiffiffiffiffiffiffiffiffiffiffiffi 2cn n log n

< ;

which proves claim (10). Now, we want to show that pn ðrÞ can be made arbitrarily small when r ! 1 . First, since the law of kðnÞ max ðrÞ is the same as the law of r kðnÞ max ð1Þ; we observe that h i ð1Þ < 1 þ ð1 dÞc P kðnÞ n max h i ðrÞ < 1 þ rð1 dÞc ð1 rÞ :¼ fn : ¼ P kðnÞ n max We know that fn ! 0 when n ! 1, so there exists n1 1 (independent of r) such that for all n n1 , fn < : Moreover, since cn ! 0, we can choose n~ðrÞ such that for all n n~ðrÞ; rð1 dÞcn ð1 rÞ > 0 say, we choose it such that ð1 dÞcn~ ðrÞ 2ð1 rÞ=r. As n~ðrÞ ! 1 when r ! 1 , there exist r0 2 ð0; 1Þ such that n~ðr0 Þ n0 . Then, by monotonicity of the distribution function, we deduce that pn~ðr0 Þ ðr0 Þ fn~ðr0 Þ < :

(12)

As a consequence, we deduce that n ðrÞ ! 1 when r ! 1 as stated in Eq. (6) and that qn ðrÞ ! 1 as stated in Eq. (7). As far as the real Ginibre ensemble is concerned (real coefficients), the above theoretical analysis does not apply directly. Indeed, the spectral joint probability density in this case lacks the determinantal structure, which is a key in the work of Ref. 5. However, very recently extreme value theory for the spectral radius of real matrices has been studied,22 suggesting that a similar result could be obtained in this context. In fact, numerical simulations confirm that the same phenomenon occurs. In Fig. 3, a numerical estimation of qn ðrÞ is computed and shows that for r close enough to 1, qn ðrÞ first increases then decreases as a function of n, similar


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to the maximum observed on the maximal Lyapunov exponent. IV. DISCUSSION

This phenomenon of system size resonance may appear in a wider class of problems, involving the maximum of a set n convergent random variables. It is instructive to discuss this idea on a toy example. Consider a sequence of families ðnÞ of real i.i.d. random variables ðXi Þ1in and denote ðnÞ

ðnÞ

Mn ¼ maxðXi ; 1 i nÞ:We assume that P½Xk x ¼ an ¼ 1 bn is increasing to 1 when n ! 1 for a given x 2 R. By the i.i.d. assumption, one writes P½Mn > x ¼ 1 ð1 bn Þn :

(13)

We deduce that if bn ¼ o(n1) then P½Mn > x ! 0 when n ! 1. Now, we can wonder whether this convergence is monotonic or if there exists an intermediate value of n for which P½Mn > x is maximal. This can be answered by differentiating Eq. (13) with respect to n (considered as a real variable)

Finally, although we have carried out the numerical investigation of the maximal Lyapunov exponent only on the random neural network model (1), the theoretical analysis of kðnÞ max using random matrix theory is much more general and suggests that the system resonance phenomenon may be observed in various complex systems, and in particular in disordered systems close to the phase transition. In this perspective, a promising direction would be to develop our approach towards the study of random genetic networks to unravel how their rich dynamical repertoire11 is affected by system size and disorder parameters. ACKNOWLEDGMENTS

G.W. wants to thank Amir Dembo and Ofer Zeitouni for their help in the theoretical part of this paper and Lenya Ryzhik for his support during year 2010–2011 at Stanford University, where the starting part of this work was done. The authors also want to thank Jonathan Touboul and Khashayar Pakdaman for helpful discussions.

1

@n ð1 ðan Þn Þ ¼ @n ð1 enlnðan Þ Þ ¼ enlnðan Þ ðlnðan Þ þ na0n =an Þ (14) and looking n such that an logðan Þ þ na0n ¼ 0:

(15)

We find, for instance, that if an ¼ 1 ejn with j < 1, then P½Mn > x is maximal for n around j1 . The real parts of the eigenvalues of J satisfy similar assumptions as the toy model, except the strongest one that is independence. This difficulty has been solved using the approach of determinantal processes,5 which was the starting point of our analysis. Moreover, one can view the results on the Lyapunov exponent as a consequence of this general principle. There is a competition between two opposite phenomena as the system size increases. On the one hand, consistent with an extreme value behavior, increasing system size increases the likelihood of obtaining a large maximal Lyapunov exponent, since one looks at the maximum on n variables: the maximal Lyapunov tends to increases as n increases. Of course, on the other hand, there is another competitive phenomenon at play. As n increases, self-averaging principle drives the system towards the mean-field behavior which converges to the trivial zero solution for r < 1. Therefore, as n increases to infinity, the maximal Lyapunov exponent should become negative. This competition results in the emergence of an intermediate system size which enhances the probability of observing spontaneous activity. In terms of perspectives, our result may shed a new light on the behavior of modular networks, composed by clusters of randomly connected agents and sparse connections between clusters. Indeed, each cluster size may be viewed as a parameter that controls the propensity to generate complex dynamics within each cluster.

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C. Quininao and J. Touboul, “Limits and dynamics of randomly connected neuronal networks,” preprint arXiv:1306.5175 (2013). 22 B. Rider and C. D. Sinclair, “Extremal laws for the real Ginibre ensemble,” preprint arXiv:1209.6085 (2012). 23 R. Toral, C. R. Miraso, and J. D. Gunton, “System size coherence resonance in coupled FitzHugh-Nagumo models,” Europhys. Lett. 61(2), 162 (2003). 24 J. Scott, Social Network Analysis: A Handbook (Sage Publications Limited, 2000). 25 C. Skokos, The Lyapunov Characteristic Exponents and Their Computation, Lecture Notes in Physics Vol. 790 (Springer, Berlin, Heidelberg, 2010), pp. 63–135. 26 H. Sompolinsky, A. Crisanti, and H. J. Sommers, “Chaos in random neural networks,” Phys. Rev. Lett. 61(3), 259–262 (1988). 27 T. Tao, V. Vu, and M. Krishnapur, “Random matrices: Universality of ESDs and the circular law,” Ann. Probab. 38(5), 2023–2065 (2010).

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29

G. Wainrib and J. Touboul, “Topological and dynamical complexity of random neural networks,” Phys. Rev. Lett. 110(11), 118101 (2013). The precision of this method is limited by two reasons asides from truncation error. First, the precision increases as T=Dt increases but for obvious reasons we have to keep this quantity finite. Second, the method computes the maximal Lyapunov exponent of the reference trajectory, not the one of the attractor itself. In general, we assume that after evolving the system for a while the reference trajectory is close enough to the attractor. This assumption is easy to check by direct examination of the trajectories when the attractor is a limit cycle or a fixed point, but it is impossible to distinguish a chaotic trajectory from a long transient regime. However, as we is clearly are only interested in the sign of K the error is irrelevant when K 0, the error becomes relevant to the positive or negative. Only when K analysis. Therefore, to detect periodicity, we rely instead on a direct analy ¼ 0 when periodicity is detected. sis of the trajectories, and we set K


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