Wilson–Cowan neural field equation

Example

A neural field describes the activity of a sheet of neurons, with a position variable $x \in \R^d$ and time $t$. Let $u_e(x, t)$ and $u_i(x, t)$ be the mean activities of the excitatory and the inhibitory populations, with time constants $\tau_e, \tau_i > 0$. Let $w_{ab} \colon \R^d \to [0, \infty)$ be integrable connectivity kernels, let $S_e, S_i$ be increasing sigmoid firing-rate functions, and let $h_e, h_i$ be external inputs. A commonly used form of the Wilson–Cowan neural field equations, with refractory terms neglected, is $$\tau_e \, \partial_t u_e = -u_e + S_e\big(w_{ee} * u_e - w_{ei} * u_i + h_e\big), \qquad \tau_i \, \partial_t u_i = -u_i + S_i\big(w_{ie} * u_e - w_{ii} * u_i + h_i\big),$$ where $(w * u)(x, t) = \int_{\R^d} w(x - y) \, u(y, t) \, dy$. Each equation has a time derivative of the unknowns and a space integral of them, so the system is a nonlinear integro-differential equation.

Check: for constant inputs, a spatially constant state $(\bar u_e, \bar u_i)$ is a steady state exactly when $$\bar u_e = S_e\big(W_{ee} \bar u_e - W_{ei} \bar u_i + h_e\big), \qquad \bar u_i = S_i\big(W_{ie} \bar u_e - W_{ii} \bar u_i + h_i\big),$$ where $W_{ab} = \int_{\R^d} w_{ab}(z) \, dz$, because $w * \bar u = W \bar u$ for a constant $\bar u$. These are the equations of the space-clamped Wilson–Cowan model.