Dynamics of Stochastic Neuronal Networks and the Connections to Random Graph Theory

Abstract

Abstract. We analyze a stochastic neuronal network model which corresponds to an all-to-all network of discretized integrate-and-fire neurons where the synapses are failure-prone. This network exhibits different phases of behavior corresponding to synchrony and asynchrony, and we show that this is due to the limiting mean-field system possessing multiple attractors. We also show that this mean-field limit exhibits a first-order phase transition as a function of the connection strength -as the synapses are made more reliable, there is a sudden onset of synchronous behavior. A detailed understanding of the dynamics involves both a characterization of the size of the giant component in a certain random graph process, and control of the pathwise dynamics of the system by obtaining exponential bounds for the probabilities of events far from the mean.

Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

“…With a view towards an eventual understanding of the interactions between complex networks and complicated dynamics, we study the evolution of discrete stochastic neuronal dynamics on these networks and the propensity of these dynamics to synchronize. The authors and collaborators have studied these exact dynamics on simpler graphs in [29,30], and the current work can be thought of an extension of those papers.…”

“…The purpose of this paper is to understand more fully the precise dependence of the synchronization properties of neuronal dynamics on the underlying networks on which they are defined. It was shown in [29,30] that random neuronal dynamics defined on "all-to-all" networks have certain interesting synchronization properties -in particular, in certain limits there exist discontinuous phase transitions between different attractors. We perform a comprehensive numerical study below, and show that, in a certain sense to be made precise below, while certain random topologies (e.g.…”

Quantitative Neuroscience: From Chalk Board to Bedside

“…The first theme of this issue explores two lines of mathematical research that have been particularly fruitful: 1) neural synchronization [19,21], the fundamental process by which spatially distributed neural centers bind a sensory stimulus and coordinate their activities to respond to it, and 2) multistability [2,10,12], the concept that treatments may be possible by applying jolts of electricity to the right location in the brain at the right time [11,16]. Attention is drawn to the effects of time delays and random perturbations ("noise") on neural control and information processing.…”

“…Obviously the most promising mechanisms for neural processing are those which are robust in the presence of noise and delay. In other words, it is not only necessary to develop analytical expressions that describe neural synchronization regardless of the number of neurons [5], it is also necessary to understand the effects of noise of the dynamics of synchronizing neural populations [2]. Similarly the widespread occurrence of time-delayed feedback in neural pathways raises questions as to the role of time delays in information processing [10] and whether new effects arise from the interplay between noise…”

“…Brain-computer interfaces now make it possible to translate thought into action [6,7,8,15], replace lost limbs with robotic ones [9], prevent epileptic seizures [11,16] and even to alleviate the symptoms of neuro-degenerative diseases, such as Parkinson's [14]. Is it possible to do even better?The first theme of this issue explores two lines of mathematical research that have been particularly fruitful: 1) neural synchronization [19,21], the fundamental process by which spatially distributed neural centers bind a sensory stimulus and coordinate their activities to respond to it, and 2) multistability [2,10,12], the concept that treatments may be possible by applying jolts of electricity to the right location in the brain at the right time [11,16]. Attention is drawn to the effects of time delays and random perturbations ("noise") on neural control and information processing.…”