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Weakly Connected Oscillators

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Weakly Connected Neural Networks

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

Abstract

In this chapter we study weakly connected networks

$$ \dot X_i = F_i \left( {X_i ,\lambda } \right) + \varepsilon G_i \left( {X,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n,}} $$
((9.1))

of oscillatory neurons. Our basic assumption is that there is a value of λ ∈ Λ such that every equation in the uncoupled system (ε = 0)

$$ \dot X_i = F_i \left( {X_i ,\lambda } \right),{\text{ X}}_i \in \mathbb{R}^m , $$
((9.2))

has a hyperbolic stable limit cycle attractor γ ⊂ ℝm The activity on the limit cycle can be described in terms of its phase \( \theta \in \mathbb{S}^1 \) of oscillation

$$ \dot \theta _i = \Omega _i \left( \lambda \right), $$

where Ωi(λ) is the natural frequency of oscillations. The dynamics of the oscillatory weakly connected system (9.1) can also be described in terms of phase variables:

$$ \dot \theta _i = \Omega _i \left( \lambda \right), + \varepsilon g_i \left( {\theta ,\lambda ,\rho ,\varepsilon } \right),{\text{ i = 1,}} \ldots {\text{,n}}{\text{.}} $$

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© 1997 Springer Science+Business Media New York

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Hoppensteadt, F.C., Izhikevich, E.M. (1997). Weakly Connected Oscillators. In: Weakly Connected Neural Networks. Applied Mathematical Sciences, vol 126. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1828-9_9

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  • DOI: https://doi.org/10.1007/978-1-4612-1828-9_9

  • 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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