Dynamical mean-field 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 previously proposed by the author [Phys. Rev. E 67, 041903 (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 neurons with couplings whose average coordination number Z may change from local ( Z<<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 2N -dimensional stochastic differential equations are transformed to 13-dimensional 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 worse because of increased heterogeneous couplings, although the average network distance becomes shorter. Results calculated by our theory are in good agreement with those by direct simulations.