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Local Analysis of Singularly Perturbed WCNNs

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Weakly Connected Neural Networks

Part of the book series: Applied Mathematical Sciences ((AMS,volume 126))

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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 ∈ ℝk Y i ∈ ℝm 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.

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© 1997 Springer Science+Business Media New York

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Hoppensteadt, F.C., Izhikevich, E.M. (1997). Local Analysis of Singularly Perturbed WCNNs. In: Weakly Connected Neural Networks. Applied Mathematical Sciences, vol 126. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1828-9_6

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  • DOI: https://doi.org/10.1007/978-1-4612-1828-9_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7302-8

  • Online ISBN: 978-1-4612-1828-9

  • eBook Packages: Springer Book Archive

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