Synchronization in a network of model neurons

We study the spatiotemporal dynamics of a network of coupled chaotic maps modelling neuronal activity, under variation of coupling strength $ϵ$ and degree of randomness in coupling $p$. We find that at high coupling strengths $(ϵg{ϵ}_{\text{fixed}})$ the unstable saddle point solution of the local chaotic maps gets stabilized. The range of coupling where this spatiotemporal fixed point gains stability is unchanged in the presence of randomness in the connections, namely ${ϵ}_{\text{fixed}}$ is invariant under changes in $p$. As coupling gets weaker $(ϵl{ϵ}_{\text{fixed}})$, the spatiotemporal fixed point loses stability, and one obtains chaos. In this regime, when the coupling connections are completely regular $(p=0)$, the network becomes spatiotemporally chaotic. Interestingly however, in the presence of random links $(pg0)$ one obtains spatial synchronization in the network. We find that this range of synchronized chaos increases exponentially with the fraction of random links in the network. Further, in the space of fixed coupling strengths, the synchronization transition occurs at a finite value of $p$, a scenario quite distinct from the many examples of synchronization transitions at $p\ensuremath{\rightarrow}0$. Further we show that the synchronization here is robust in the presence of parametric noise, namely in a network of nonidentical neuronal maps. Finally we check the generality of our observations in networks of neurons displaying both spiking and bursting dynamics.