  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.
