Abstract
We study complex networks of stochastic two-state units. Our aim is to model discrete stochastic excitable dynamics with a rest and an excited state. Both states are assumed to possess different waiting time distributions. The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution. The activation rate of any single unit is determined by its neighbors according to a random complex network structure. In order to treat this problem in an analytical way, we use a heterogeneous mean-field approximation yielding a set of equations generally valid for uncorrelated random networks. Based on this derivation we focus on random binary networks where the network is solely comprised of nodes with either of two degrees. The ratio between the two degrees is shown to be a crucial parameter. Dependent on the composition of the network the steady states show the usual transition from disorder to homogeneously ordered bistability as well as new scenarios that include inhomogeneous ordered and disordered bistability as well as tristability. The various steady states differ in their spiking activity expressed by a state dependent spiking rate. Numerical simulations agree with analytic results of the heterogeneous mean-field approximation.
Highlights
Discrete-state stochastic models can be used to describe discrete processes such as the orientation of a spin or the blinking of quantum dots [1]
The rest state is treated as an activation process with an exponentially distributed life time, whereas the latter in the excited state shall have a constant mean which may originate from any distribution
The activation rate of any single unit is determined by its neighbors according to a random complex network structure
Summary
Discrete-state stochastic models can be used to describe discrete processes such as the orientation of a spin or the blinking of quantum dots [1]. J. B (2017) 90: 14 any other parameters such as the noise intensity, the size and structure or possible dynamical states of the network. Apart from the expected homogeneous ordered bistability that is known from globally coupled units [7,12], inhomogeneous ordered and disordered states as well as tristable states are uncovered. These findings are confirmed via microscopic simulations using the original network structure. Given that the excited state possess an exponentially distributed waiting time, the spike trains are always nearly Poissonian. For a sharp-peaked waiting time density with no variance, the spike trains become highly coherent
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