Stochastic Wilson–Cowan models of neuronal network dynamics with memory and delay

Abstract

We consider a simple Markovian class of the stochastic Wilson-Cowan type models of neuronal network dynamics, which incorporates stochastic delay caused by the existence of a refractory period of neurons. From the point of view of the dynamics of the individual elements, we are dealing with a network of non-Markovian stochastic two-state oscillators with memory, which are coupled globally in a mean-field fashion. This interrelation of a higher-dimensional Markovian and lower-dimensional non-Markovian dynamics is discussed in its relevance to the general problem of the network dynamics of complex elements possessing memory. The simplest model of this class is provided by a three-state Markovian neuron with one refractory state, which causes firing delay with an exponentially decaying memory within the two-state reduced model. This basic model is used to study critical avalanche dynamics (the noise sustained criticality) in a balanced feedforward network consisting of the excitatory and inhibitory neurons. Such avalanches emerge due to the network size dependent noise (mesoscopic noise). Numerical simulations reveal an intermediate power law in the distribution of avalanche sizes with the critical exponent around −1.16. We show that this power law is robust upon a variation of the refractory time over several orders of magnitude. However, the avalanche time distribution is biexponential. It does not reflect any genuine power law dependence. belongs to statistical physics or system biophysics. Physical models such as SOC are especially important and helpful here.The Wilson and Cowan model [12] presents one of the well-established models of neuronal network dynamics [13]. It incorporates individual elements in a simplest possible fashion as two-state stochastic oscillators with one quiescent state and one excited state, and random transitions between these two states which are influenced by the mutual coupling among the network elements. The model has been introduced in the deterministic limit of huge amounts of coupled elements in complete neglect of the intrinsic mesoscopic noise and became immensely popular with the years [13], being used e.g. to describe neuronal oscillations in visual cortex within a mean-field approximation [1,14]. Recently, the previously neglected mesoscopic noise effects were incorporated in this model for a finite-size network [15,16]. Such a noise has been shown to be very important, in particular, for the occurrence of the critical avalanche dynamics [15] absent in the deterministic Wilson-Cowan model and also for the emergence of oscillatory noisy dynamics [16]. At first glance, such a noisy dynamics can look like a chaotic deterministic one. Deterministic chaos influenced by the noise can also be a natural feature of a higher-dimensional dynamics, beyond the original two-dimensional (2D) Wilson-Cowan mean field model. Indeed, deterministic chaos has been found in the brain dynamics some time ago [1,17]. However, it cannot be described within the memoryless Wilson-Cowan model because the...

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