Dynamical mean-filed approximation to small-world networks of spiking   neurons: From local to global, and/or from regular to random couplings
  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. 
Hideo Hasegawa (Tokyo Gakugei Univ.)