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

By extending a semi-analytical, dynamical mean-field approximation (DMA) previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 41903 (2003)], we have developed a generalized theory which takes into account a wide range of couplings in small-world networks consisting of noisy N -unit FitzHugh-Nagumo (FN) neurons: the coordination number on the average Z is allowed to vary from local (Z ≪ N) to global couplings (Z = N − 1) while the concentration of random couplings p is allowed to vary from regular (p = 0) to completely random (p=1). The original 2N -dimensional stochastic differential equations are transformed to 13-dimensional deterministic differential equations expressed in terms of means, variances and covariances for three kinds of correlations: the on-site correlation, the correlation for a coupled pair and that for a pair without direct couplings. The firing-time precision and the synchronization ratio 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 worsen because of increased heterogeneous couplings, although the average network distance becomes shorter. Results calculated by DMA are in good agreement with those by direct simulations. PACS No. 87.10.+e 84.35.+i 05.45.-a 07.05.Mh Typeset using REVTEX E-mail: [email protected] 1