Well posedness and stationary solutions of a neural field equation with synaptic plasticity

2017/02/02 by Juan Cordero Ceballos, Ceballos, Juan Cordero, Alejandro Jiménez Rodríguez +2
Computer Science · Mathematics · Neuroscience · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Neural Networks and Applications #Neural dynamics and brain function #math.AP #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.1702.00688

arxiv created 2017/02/02 · openalex publication_date 2017/02/02 · arxiv updated 2017/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the initial value problem associated to the neural field equation of Amari type with plasticity u_t(x,t)=-u(x,t)+∫_Ωw(x,y)[1+γg( u(x,t) - u(y,t) )] f(u(y,t)) dy, (x,t) ∈ Ω× (0, ∞), where Ω⊂ℝ^m, f and g are bounded and continuously differentiable functions with bounded derivative, and γ≥0 is the plasticity synaptic coefficient. We show that the problem is well posed in C_b(ℝ^m) and L¹(Ω) with Ω compact. The proof follows from a classical fixed point argument when we consider the equation's flow. Strong convergence of solutions in the no plasticity limit (γ→0) to solutions of Amari's equation is analysed. Finally, we prove existence of stationary solutions in a general way. As a particular case, we show that the Amari's model, after learning, leads to the stationary Schrödinger equation for a type of gain modulation.

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