Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

Abstract

We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (Erdős-Rényi) and "small-world" networks give rise to synchronization phenomena similar to that in "all-to-all" networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in "scale-free" networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In marked contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the "hubs" in the network.

Spreading dynamics on spatially constrained complex brain networks

“…We note that in, e.g. [9,10,11] (and in the present work), network nodes are to be interpreted as 'neural units' comprising many neurons, and representing a single cortical column, say. Detailed consideration of synaptic signalling models is therefore not appropriate.…”

“…Whilst actual estimates for mean degree in the rat cortex are not available, such connectivity is typical of that employed in the literature (see, e.g. [9,10,11] and references therein), particularly when one accounts for the large spread of observed degrees (e.g., for r = 0.35 the degree ranges from 8 to 48 whilst for r = 0.5 it lies between 20 and 87), and so serves to illustrate our methodology. For brevity, in the figures that follow, we illustrate the differences in spreading dynamics obtained in each network for the choices r = 0.35 and r = 0.5.…”

Synchrony in stochastically driven neuronal networks with complex topologies

“…Particular attention was frequently paid to which features of the phase response curve are conducive to synchrony [27]. We will be concerned with developing an inherently statistical approach applicable to all-excitatory networks [23,24,[29][30][31][32] that have stochastic driving and only a statistical description of their connectivity architecture.…”

“…A central advance presented in the present work therefore addresses the interplay between the network topology and the time-evolving ensemble of the neuronal membrane potentials that enables cascading total firing events to still take place with high probability in the appropriate parameter regimes. We are thereby attempting to contribute insight from another perspective to the broad question of how the statistical properties of the network are reflected in the statistical properties of the network dynamics [10,[32][33][34][35][36][37]. Moreover, we are here building a connection between our previous results concerning the statistical description of cascading total firing events in all-to-all coupled IF networks [30,31] with those addressing steady-state statistics of firing rates and membrane-potential correlations in the asynchronous operating regime of IF networks with specific nontrivial architectural connectivity [33].…”

Cascades on a stochastic pulse-coupled network

“…To model systems in which transient cascades of two distinct and opposing influences can form, we extend the stochastic pulse-coupled neural network model of DeVille and Peskin192021; hereafter referred to as the DP model. First, by allowing each integrate-and-fire oscillator2223 to produce both positive and negative pulses that compel coupled oscillators to move closer to an upper or lower boundary (represented by distinct firing states), respectively.…”