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

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### 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.

### Author and article information

###### Journal
16 March 2004
2004-09-08
cond-mat/0403415
10.1103/PhysRevE.70.066107