Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
- 作者：Jonathan Touboul ; jonathan.touboul@college-de-france.fr
- 关键词：Noise ; Neural fields ; Collective dynamics ; Bifurcations ; Turing instabilities
- 刊名：Physica D
- 出版年：2012
- 期刊代码：55_01672789
- 类别：ph
- 出版时间：1 August, 2012
- 卷：241
- 期：15
- 页码：1223-1244
- 文件大小：2994 K
摘要
In this manuscript we analyze the collective behavior of mean-field limits of large-scale, spatially extended stochastic neuronal networks with delays. Rigorously, the asymptotic regime of such systems is characterized by a very intricate stochastic delayed integro-differential McKean-Vlasov equation that remain impenetrable, leaving the stochastic collective dynamics of such networks poorly understood. In order to study these macroscopic dynamics, we analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics and sigmoidal interactions. In that case, we prove that the solution of the mean-field equation is Gaussian, hence characterized by its two first moments, and that these two quantities satisfy a set of coupled delayed integro-differential equations. These equations are similar to usual neural field equations, and incorporate noise levels as a parameter, allowing analysis of noise-induced transitions. We identify through bifurcation analysis several qualitative transitions due to noise in the mean-field limit. In particular, stabilization of spatially homogeneous solutions, synchronized oscillations, bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow further exploring the role of noise in the nervous system.