Identification of a free energy functional in an integro-differential equation model for neuronal network activity.
type: Item
instance of: scholarly article
title: Identification of a free energy functional in an integro-differential equation model for neuronal network activity.
zbMATH Open document ID: 1070.45008
published in: Applied Mathematics Letters
publication date: 2005-03-07T00:00:00Z
review text: The author deals with the integrodifferential convolution equation \[ D_tu(x,t)+u(x,t)=\int_{{\mathbb R}^n} w(x-y)P(u(y,s)-\theta)\,dy,\qquad (x,t)\in {\mathbb R}^n \times {\mathbb R}, \tag{1} \] where: (i) \(n=1,2,3\); (ii) \(\theta\) is a constant; (iii) \(w\) is an even function compactly supported or quickly decaying away from \(0\) such that \(\int_{{\mathbb R}^n} w(x)\,dx=1\); (iv) \(P\in C^1({\mathbb R})\) is a nonnegative increasing function. The author shows that any solution to (1) satisfies the energy equality \[ D_tE(u(\cdot,t))=-\int_{{\mathbb R}^n} P'(u(x,t)-\theta)[D_tu(x,t)]^2\,dx\leq 0, \tag{2} \] where \[ \begin{aligned} E(u(\cdot,t))= &{1\over 4}\int_{{\mathbb R}^n}\int_{{\mathbb R}^n}w(x-y)[P(u(x,t)-\theta)-P(u(y,t)-\theta)]^2\,dx\,dy\\ &+\int_{{\mathbb R}^n} dx\int_0^{u(x,t)}P(s-\theta)[s-P(s-\theta)]\,ds. \end{aligned}\tag{3} \] Moreover, setting \(v=P(u-\theta)\) and assuming \(P(-\theta)=0\), the A. shows that the energy formulated in terms of \(v\) is similar to that found by \textit{J. J. Hopfield} [Neurons with graded response have collective computational properties like those of two-state neurons, PNAS 81, 3088--3092 (1984)]. Finally, under the assumption \(P'(s-\theta)\neq 0\) for all \(s\in {\mathbb R}\), he proves that the critical points of \(E\) coincide with the solutions to equation (1).
reviewed by: Alfredo Lorenzi
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 92B20
Mathematics Subject Classification ID: 45Q05
zbMATH DE Number: 2140781
zbMATH Keywords: integrodifferential convolution equation
zbMATH Keywords: energy functionals
zbMATH Keywords: neuronal networks
zbMATH Keywords: inverse problem
zbMATH Keywords: critical points
author: Donald A. French
MaRDI profile type: Publication
full work available at URL: j.aml.2004.07.007
OpenAlex ID: W2060983374
Wikidata QID: Q115360994
cites work: Spatially structured activity in synaptically coupled neuronal networks. I: Traveling fronts and pulses
cites work: Multiple Bumps in a Neuronal Model of Working Memory
cites work: Spatially structured activity in synaptically coupled neuronal networks. II: Lateral inhibition and standing pulses
cites work: PDE Methods for Nonlocal Models
cites work: Stationary bumps in networks of spiking neurons
cites work: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity
cites work: Q3034912
cites work: The computational modeling of nonequilibrium thermodynamics of the martensitic transformations
cites work: Asymptotic behavior of solution to the Cahn-Hillard equation
cites work: An integrodifferential model for phase transitions: stationary solutions in higher space dimensions
cites work: Neurons with graded response have collective computational properties like those of two-state neurons.
DOI: 10.1016/J.AML.2004.07.007
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description: scientific article; zbMATH DE number 2140781