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

Grégory Faye 1 James Rankin 1, * Pascal Chossat 1, 2
* Auteur correspondant
1 NEUROMATHCOMP - Mathematical and Computational Neuroscience
CRISAM - Inria Sophia Antipolis - Méditerranée , JAD - Laboratoire Jean Alexandre Dieudonné : UMR6621
Abstract : The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize work- ing (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 equiva- lent 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.
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Journal of Mathematical Biology, Springer Verlag (Germany), 2013, 66 (6), pp.1303-1338. 〈10.1007/s00285-012-0532-y〉
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Grégory Faye, James Rankin, Pascal Chossat. Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis. Journal of Mathematical Biology, Springer Verlag (Germany), 2013, 66 (6), pp.1303-1338. 〈10.1007/s00285-012-0532-y〉. 〈hal-00807366〉

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