Dynamical mean-filed approximation to small-world networks of spiking neurons: From local to global, and/or from regular to random couplings

Abstract

By extending a dynamical mean-field approximation (DMA) previously proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 41903 (2003)], we have developed a semianalytical theory which takes into account a wide range of couplings in a small-world network. Our network consists of noisy \(N\)-unit FitzHugh-Nagumo (FN) neurons with couplings whose average coordination number \(Z\) may change from local (\(Z \ll N \)) to global couplings (\(Z=N-1\)) and/or whose concentration of random couplings \(p\) is allowed to vary from regular (\(p=0\)) to completely random (p=1). We have taken into account three kinds of spatial correlations: the on-site correlation, the correlation for a coupled pair and that for a pair without direct couplings. The original \(2 N\)-dimensional {\it stochastic} differential equations are transformed to 13-dimensional {\it deterministic} differential equations expressed in terms of means, variances and covariances of state variables. The synchronization ratio and the firing-time precision for an applied single spike have been discussed as functions of \(Z\) and \(p\). Our calculations have shown that with increasing \(p\), the synchronization is {\it worse} because of increased heterogeneous couplings, although the average network distance becomes shorter. Results calculated by out theory are in good agreement with those by direct simulations.