Collective dynamics and self-organisation of stochastic neuronal systems influenced by synaptic time delay

Abstract

The focus in the present thesis lies with the synchronization mediated self-organization phenomena in populations of globally coupled stochastic excitable or bursting units subjected to interaction delays. Excitable local dynamics follows the Fitzhugh- Nagumo model, canonical for type II excitability, whereas the bursting units are represented by the Hindmarsh-Rose model. The study comprises two complementary lines of research. One is aimed at extending the analogy regarding the complex forms of collective behavior exhibited by the assemblies of coupled nonlinear autonomous oscillators and those made up of excitable units. Within the second line of research, our main contribution consists in developing the mean-field based models for the collective dynamics of the assemblies of excitable or bursting units, whose microscopic dynamics is governed by large sets of coupled stochastic delay-differential equations. This is instigated by the notion that any population displaying a collective mode may be treated as macroscopic oscillator. While the framework itself rests on implementing the cumulant approach complemented by the Gaussian approximation, one of the principal gains presents the ability to recast the problem of (stochastic) bifurcations affecting the stability of the stationary state of the exact system in terms of flows containing only several deterministic delaydifferential equations, where noise intensity may act as a bifurcation parameter...