Asymptotic behavior for a nonlocal model of neural fields

In this paper we consider the non local evolution problem $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u =- u + K(f\circ u ) \ \ in \ \ \Omega ,\\ u = 0 \ \ in \ \ \mathbb {R}^N\backslash \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\), \(f: \mathbb {R}\rightarrow \mathbb {R}\) and K is an integral operator with a symmetric kernel. We prove existence and some regularity properties of the global attractor. We also characterize the global attractor, using the properties of a Lyapunov functional for this model.