Skip to main content

Local Analysis of Singularly Perturbed WCNNs

  • Chapter
Weakly Connected Neural Networks

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

  • 1094 Accesses


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

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

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 9.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations


Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Science+Business Media New York

About this chapter

Cite this chapter

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.

Download citation

  • DOI:

  • 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

Publish with us

Policies and ethics