Local Analysis of Singularly Perturbed WCNNs

  • Frank C. Hoppensteadt
  • Eugene M. Izhikevich
Part of the Applied Mathematical Sciences book series (AMS, volume 126)

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.

Keywords

Manifold Librium 

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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Frank C. Hoppensteadt
    • 1
  • Eugene M. Izhikevich
    • 1
  1. 1.Center for Systems Science and EngineeringArizona State UniversityTempe

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