A gradient flow formulation for the stochastic Amari neural field model

Abstract

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.

Appendix A: Cylindrical Wiener processes in Hilbert spaces

Let $\{v_{i}\}_{i\in \mathbb {N}}\subset H$ be a complete orthonormal system for H and let $\{\beta _{t}^{i}\}_{i\in \mathbb {N}}$ be a collection of independent real-valued standard Brownian motions modeled on a filtered normal probability space $(\varOmega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\ge 0},\mathbb {P})$. Then the cylindrical Wiener process $\{W_{t}\}_{t\ge 0}$ with covariance $Q={\text {Id}}$ has the formal representation

$$\begin{aligned} W_{t}=\sum _{i=1}^{\infty }v_{i}\beta _{t}^{i},\quad t\ge 0, \end{aligned}$$

(A.1)

which is a standard Wiener process in a weaker separable Hilbert space U, such that there exists a Hilbert–Schmidt embedding $\iota :H\rightarrow U$, $\iota \in L_{2}(H,U)$ and $\{W_{t}\}_{t\ge 0}$ has the covariance operator $\iota \iota ^{*}$. By Da Prato and Zabczyk (2014, Proposition 4.7 and Proposition 4.8), then $(\iota \iota ^{*})^{\frac{1}{2}}(U)=H$, and one can always find U and $\iota$ with the above properties such that the representation (A.1) holds; see Da Prato and Zabczyk (2014, Chapter 4) for details. For our purposes, it is sufficient to set U equal to the abstract completion of $L^{2}({\mathcal {B}})$ with respect to the alternative scalar product $(u,v)_{U}:=\sum _{n=1}^{\infty }n^{-2}\hat{u}_{n}\hat{v}_{n}$, where $u,v\in H$ and $\{\hat{u}_{n}\},\{\hat{v}_{n}\}\in \ell ^{2}$ are such that $u=\sum _{n=1}^{\infty }\hat{u}_{n}v_{n}$ and $v=\sum _{n=1}^{\infty }\hat{v}_{n}v_{n}$. Then $\iota :={\text {Id}}$ is a Hilbert–Schmidt embedding from H into U.