Local Analysis of Singularly Perturbed WCNNs

Abstract

In this chapter we study the local dynamics of singularly perturbed weakly connected neural networks of the form

$$\left\{ {\begin{array}{*{20}{c}} \hfill {\mu {{X}_{i}}^{\prime } = Fi\left( {{{X}_{i}}{{Y}_{i}},\lambda ,\mu } \right) + \varepsilon {{P}_{i}}\left( {X,Y,\lambda ,\rho ,\mu ,\varepsilon } \right)} \\ \hfill {{{Y}_{i}}^{\prime } = {{G}_{i}}\left( {{{X}_{i}},{{Y}_{i}}\lambda ,\mu } \right) + \varepsilon {{Q}_{i}}\left( {X,Y,\lambda ,\rho ,\mu ,\varepsilon } \right)} \\ \end{array} ,\varepsilon ,\mu \ll 1} \right.$$

(6.1)

where X _i ∈ ℝ^k Y _i ∈ ℝ^m are fast and slow variables, respectively; τ is a slow time; and ′ = d/dτ. The parameters ε and μ are small, representing the strength of synaptic connections and ratio of time scales, respectively. The parameters λ ∈ Λ and ρ ∈ R. have the same meaning as in the previous chapter: They represent a multidimensional bifurcation parameter and external input from sensor organs, respectively. As before, we assume that all functions F _i , G _i , which represent the dynamics of each neuron, and all P _i and Q _i which represent connections between the neurons, are as smooth as necessary for our computations.