Identification of a free energy functional in an integro-differential equation model for neuronal network activity.
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).
integrodifferential convolution equation
scientific article; zbMATH DE number 2140781
scientific article