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

We consider the initial value problem associated to the neural field equation of Amari type with plasticity $$u_t(x,t)=-u(x,t)+\int_{\Omega}w(x,y)[1+\gamma g( u(x,t) - u(y,t) )] f(u(y,t))\; dy, \;(x,t) \in \Omega \times (0, \infty),$$ where $\Omega\subset\mathbb{R}^m$, $f$ and $g$ are bounded and continuously differentiable functions with bounded derivative, and $\gamma\ge0$ is the plasticity synaptic coefficient. We show that the problem is well posed in $C_b(\mathbb{R}^m)$ and $L^1(\Omega)$ with $\Omega$ 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 ($\gamma\to0$) 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 Schr\"odinger equation for a type of gain modulation.