Mutual synchronization in ensembles of globally coupled neural networks

D. H. Zanette, A. S. Mikhailov
1998 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
The collective dynamics in globally coupled ensembles of identical neural networks with random asymmetric synaptic connections is investigated. We find that this system shows a spontaneous synchronization transition, i.e., networks with synchronous activity patterns appear in the ensemble when the coupling intensity exceeds a threshold. Under further increase of the coupling intensity, the entire ensemble breaks down into a number of coherent clusters, until complete mutual synchronization is
more » ... ynchronization is eventually established. Synchronization phenomena in populations of globally coupled dynamical systems are a subject of intensive theoretical research. Since the pioneering study by Winfree ͓1͔, much attention has been attracted to investigations of large oscillator populations ͑see ͓2-4͔͒. However, it is getting increasingly clear that synchronization does not represent a special feature of oscillator systems. For instance, recent studies have shown that similar behavior is observed in systems formed by globally coupled Hamiltonian ͓5͔ or bistable ͓6͔ elements. Moreover, it is also known that mutual synchronization is possible ͓7͔ in populations of coupled chaotic dynamical systems, such as logistic maps ͓8͔ or Rössler oscillators ͓9͔. A detailed study of the synchronization transition in large populations of stochastic globally coupled systems has recently been performed ͓10͔. The importance of synchronization for functioning of biological systems has been extensively discussed ͓1,3,8͔. It has been emphasized that these effects should play a significant role in operation of the brain ͑see, e.g., ͓11͔͒. Indeed, theoretical investigations show that mutual synchronization easily develops in populations of globally coupled individual neurons ͓11-13͔. Examining the brain functions, one can, however, note that besides this strong kind of synchronization, resulting in identical states of all neurons in a uniform population, more subtle forms of synchronization should be present. The brain is essentially a system of interacting neural networks and the activity patterns of different networks may perhaps become synchronized while retaining their complex spatiotemporal dynamics. This puts forward a general theoretical problem of mutual synchronization in ensembles of coupled neural networks ͓11͔. In the present paper this problem is addressed by studying a simple model system where the neurons are represented by dynamical McCulloch-Pitts elements ͓14͔. A network is formed by such elements linked through activatory or inhibitory connections of varying weights. When asymmetric connection weights are chosen, such a network would generally exhibit complex spatiotemporal oscillations. We take an ensemble of identical networks that are linked together by in-troducing additional global cross-network interactions between neurons occupying equivalent positions in different networks of the ensemble. The simulations reveal that the ensemble can easily undergo a spontaneous synchronization transition. In the fully synchronous regime, all networks are characterized by the same complex spatiotemporal activity pattern of neurons. At lower intensities of the cross-network coupling, the ensemble breaks into several coherent clusters. We consider ensembles made of N identical neural networks each consisting of K neurons. The collective dynamics of an ensemble is described by the following algorithm: At time tϩ1, the activity x k i of a neuron kϭ1, . . . ,K belonging to a network iϭ1, . . . ,N is where h k i ϭ ͚ lϭ1 K J kl x l i (t) is the signal arriving at this neuron at time t from all other elements of the same network, J kl are the connection weights ͑the same for all networks͒, and ⌰(z) is a sigmoidal function. The two terms on the right-hand side of Eq. ͑1͒ have a clear interpretation. The first of them represents the individual response of a neuron to the total signal received from other elements in its own network. The second term depends on the global signal obtained by summation of individual signals received by neurons occupying the same positions in all networks of the ensemble ͑and hence it corresponds to global cross-network interactions͒. The parameter specifies the strength of global coupling. When global coupling is absent (ϭ0), the networks forming the ensemble are independent. On the other hand, at ϭ1 the first term vanishes and the states of respective neurons in all networks must be identical, since they are determined by the same global signal. For 0ϽϽ1, the ensemble dynamics is governed by an interplay between local coupling inside the networks and global coupling across them. Our analysis is based on numerical investigations. As the first step, we set up the connection weights between neurons in the individual network. Each of the connection weights J kl between neurons is chosen at random with equal probability from the interval between Ϫ1 to 1. The weights of forward and reverse connections are independently selected, and *Permanent address: Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche and Instituto
doi:10.1103/physreve.58.872 fatcat:veemz5tmsvgjtgitnyvcpvhqgu