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Multiple Cusp Bifurcation

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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 Section 5.3.3 we showed that any weakly connected neural network of the form

$$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right) $$

at a multiple cusp bifurcation is governed by the canonical model

$$ x'_i = r_i + b_i x_i + \sigma _i x_i^3 + \sum\limits_{j = 1}^n {c_{ij} } x_{ij} ,{\text{ i = 1,}} \ldots {\text{,n,}} $$
((11.1))

where ′ = d/dτ, τ is slow time, x i , r i , b i c ij are real variables, and σ i = ±1. In this chapter we study some neurocomputational properties of this canonical model. In particular, we use Hirsch’s theorem to prove that the canonical model can work as a globally asymptotically stable neural network (GAS-type NN) for certain choices of the parameters b1,…, b n . We use Cohen and Grossberg’s convergence theorem to show that the canonical model can also operate as a multiple attractor neural network (MA-type NN).

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© 1997 Springer Science+Business Media New York

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Hoppensteadt, F.C., Izhikevich, E.M. (1997). Multiple Cusp Bifurcation. In: Weakly Connected Neural Networks. Applied Mathematical Sciences, vol 126. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1828-9_11

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  • DOI: https://doi.org/10.1007/978-1-4612-1828-9_11

  • 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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