Journal of Mathematical Biology

, Volume 66, Issue 6, pp 1303–1338 | Cite as

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

Article

Abstract

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.

Keywords

Localized state Neural field equation Reversible Hopf-bifurcation Normal form Orbital stability Numerical continuation 

Mathematics Subject Classification

37G05 34D20 37M05 92B20 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.NeuroMathComp LaboratoryINRIA, ENS ParisSophia-AntipolisFrance
  2. 2.J-A Dieudonné LaboratoryCNRS and University of Nice Sophia-AntipolisNice Cedex 02France

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