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.
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This work was partially funded by the ERC advanced grant NerVi number 227747.
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Faye, G., Rankin, J. & Chossat, P. Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis. J. Math. Biol. 66, 1303–1338 (2013). https://doi.org/10.1007/s00285-012-0532-y
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DOI: https://doi.org/10.1007/s00285-012-0532-y
Keywords
- Localized state
- Neural field equation
- Reversible Hopf-bifurcation
- Normal form
- Orbital stability
- Numerical continuation