Mean-field dynamics of a random neural network with noise

Abstract

We consider a network of randomly coupled rate-based neurons influenced by external and internal noise. We derive a second-order stochastic mean-field model for the network dynamics and use it to analyze the stability and bifurcations in the thermodynamic limit, as well as to study the fluctuations due to the finite-size effect. It is demonstrated that the two types of noise have substantially different impact on the network dynamics. While both sources of noise give rise to stochastic fluctuations in case of the finite-size network, only the external noise affects the stationary activity levels of the network in the thermodynamic limit. We compare the theoretical predictions with the direct simulation results and show that they agree for large enough network sizes and for parameter domains sufficiently away from bifurcations.

Interval stability for complex systems

“…When the coupling is introduced, the stationary levels persist although may shift due to the input from the peers. As shown in [44], the mean-field dynamics of a homogeneous neural network may be sometimes approximated by a one-dimensional equation similar to the one describing a single node of (10). Therefore the network (10) may be interpreted as a population of connected sub-networks, or an inhomogeneous neural network with incorporated clusters [45] (see also [46,47]).…”

Clustering promotes switching dynamics in networks of noisy neurons

“…We consider a network comprising N neurons arranged into clusters, such that intra-cluster connectivity is larger than the connectivity between neurons from different clusters. The local dynamics of a given neuron i from cluster X follows the rate model 30,35,39,40 dr…”

“…Our immediate goal is to derive a secondorder stochastic mean-field (macroscopic) model for an arbitrary cluster by appropriately averaging the local (microscopic) neuronal dynamics. To this end, we first introduce an Ansatz regarding the local variables, 30,35 which will ultimately allow us to treat the nonlinear threshold term Hðv Xi Þ.…”

“…To complete the model, it is necessary to specify the gain function H. In general, the gain function should meet the requirements that it is zero for sufficiently small input and that it saturates for large enough input, whereas for intermediate input values, H should just be smooth and monotonous. For convenience of analytical study, 30,35 we adopt the following form of H:…”

“…This presents generalization of our previous work, where we have considered bistability and slow fluctuations in a network with simple random connection topology. 30,35 Investigating the impact of clustered topology on collective dynamics is biologically plausible, given that neural networks with statistically inhomogeneous wiring are inherent to mammalian neocortex, 36,37 where the clustered structures with stronger synapses and increased connection probability make up the so-called cell assemblies. Earlier studies have indicated that clustered connectivity could give rise to bistability or multistability, 4,25,38 potentially allowing for switching dynamics between interacting populations, considered as a likely paradigm for decision-making processes during perception or cognition.…”