Existence and Stability of Traveling Pulses in a Neural Field Equation with Synaptic Depression

2013 ◽  
Vol 12 (4) ◽  
pp. 2032-2067 ◽  
Author(s):  
Grégory Faye
Keyword(s):  
Field Equation ◽  
Synaptic Depression ◽  
Neural Field ◽  
Traveling Pulses ◽  
Existence And Stability ◽  
Neural Field Equation

Localized radial bumps of a neural field equation on the Euclidean plane and the Poincaré disc

Nonlinearity ◽  
2013 ◽  
Vol 26 (2) ◽  
pp. 437-478 ◽  
Author(s):  
Grégory Faye ◽  
James Rankin ◽  
David J B Lloyd
Keyword(s):  
Field Equation ◽  
Euclidean Plane ◽  
Neural Field ◽  
Neural Field Equation

scholarly journals Rich spectrum of neural field dynamics in the presence of short-term synaptic depression

2015 ◽  
Vol 92 (3) ◽  
Author(s):  
He Wang ◽  
Kin Lam ◽  
C. C. Alan Fung ◽  
K. Y. Michael Wong ◽  
Si Wu
Keyword(s):  
Synaptic Depression ◽  
Neural Field ◽  
Short Term ◽  
Rich Spectrum

Existence and Stability of Traveling Pulses in a Continuous Neuronal Network

2005 ◽  
Vol 4 (4) ◽  
pp. 954-984 ◽  
Author(s):  
David J. Pinto ◽  
Russell K. Jackson ◽  
C. Eugene Wayne
Keyword(s):  
Neuronal Network ◽  
Traveling Pulses ◽  
Existence And Stability

scholarly journals Existence and stability of periodic solutions in a neural field equation

2021 ◽  
pp. 271-295
Author(s):  
Karina Kolodina ◽  
Vadim Kostrykin ◽  
Anna Oleynik
Keyword(s):  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Activation Function ◽  
Step Function ◽  
Neural Field ◽  
Neural Field Model ◽  
Existence And Stability ◽  
Laurent Operator ◽  
The Stability ◽  
Necessary And Sufficient

We study the existence and linear stability of stationary periodic solutions to a neural field model, an intergo-differential equation of the Hammerstein type. Under the assumption that the activation function is a discontinuous step function and the kernel is decaying sufficiently fast, we formulate necessary and sufficient conditions for the existence of a special class of solutions that we call 1-bump periodic solutions. We then analyze the stability of these solutions by studying the spectrum of the Frechet derivative of the corresponding Hammerstein operator. We prove that the spectrum of this operator agrees up to zero with the spectrum of a block Laurent operator. We show that the non-zero spectrum consists of only eigenvalues and obtain an analytical expression for the eigenvalues and the eigenfunctions. The results are illustrated by multiple examples.

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