Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis
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.
KeywordsLocalized state Neural field equation Reversible Hopf-bifurcation Normal form Orbital stability Numerical continuation
Mathematics Subject Classification37G05 34D20 37M05 92B20
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- Brezis N (1983) Analyse fonctionnelle. Théorie et applications. MassonGoogle Scholar
- Burke J, Knobloch E (2007) Normal form for spatial dynamics in the Swift–Hohenberg equation. Discret Continuous Dyn Syst Ser S, pp 170–180Google Scholar
- Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X (1997) AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont)Google Scholar
- Faye G, Chossat P (2011) Bifurcation diagrams and heteroclinic networks of octagonal h-planforms. J Nonlinear Sci (accepted for publication)Google Scholar
- Faye G, Chossat P, Faugeras O (2011) Analysis of a hyperbolic geometric model for visual texture perception. J Math Neurosci 1(4)Google Scholar
- Haragus M, Iooss G (2010) Local bifurcations, center manifolds, and normal forms in infinite dimensional systems. EDP Sci. Springer Verlag UTX seriesGoogle Scholar
- Kozyreff G, Chapman SJ (2006) Asymptotics of large bound states of localized structures. Phys Rev Lett 97:044502, 1–4Google Scholar
- Veltz R, Faugeras O (2010) Illusions in the ring model of visual orientation selectivity. Technical report, arXiv (submitted to PLoS Comp Biol)Google Scholar