Mathematics – Dynamical Systems
Scientific paper
2011-08-11
Mathematics
Dynamical Systems
Scientific paper
The cortex is composed of large-scale cell assemblies sharing the same individual properties and receiving the same input, in charge of certain functions, and subject to noise. Such assemblies are characterized by specific space locations and space-dependent delayed interactions. The mean-field equations for such systems were rigorously derived in a recent paper for general models, under mild assumptions on the network, using probabilistic methods. We summarize and investigate general implications of this result. We then address the dynamics of these stochastic neural field equations in the case of firing-rate neurons. This is a unique case where the very complex stochastic mean-field equations exactly reduce to a set of delayed differential or integro-differential equations on the two first moments of the solutions, this reduction being possible due to the Gaussian nature of the solutions. The obtained equations differ from more customary approaches in that it incorporates intrinsic noise levels nonlinearly and make explicit the interaction between the mean activity and its correlations. We analyze the dynamics of these equations, with a particular focus on the influence of noise levels on shaping the collective response of neural assemblies and brain states. Cascades of Hopf bifurcations are observed as a function of noise amplitude, for noise levels small enough, and delays, in a finite-population system. The presence of spatially homogeneous solutions in law is discussed in different non-delayed neural fields and an instability, as noise amplitude is varied, of the homogeneous state, is found. In these regimes, very complex irregular and structured spatio-temporal patterns of activity are exhibited including in particular wave or bump splitting.
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