  In this work we consider a class of nonlocal non-autonomous evolution
equations, which generalizes the model of neuronal activity that arises in
Amari (1979).
  Under suitable assumptions on the nonlinearity and on the parameters present
in the equation, we study, in an appropriated Banach space, the assimptotic
behavior of the evolution process generated by this equation. We prove results
on existence, uniqueness and smoothness of the solutions and on the existence
of pullback attracts for the evolution process associated to this equation. We
also prove a continuous dependence of the evolution process with respect to
external stimulus function present in the model. Furthermore, using the result
of continuous dependence of the evolution process, we also prove the upper
semicontinuity of pullback attracts with respect to stimulus function. We
conclude with a small discussion about the model and about a biological
interpretation of the result of continuous dependence of neuronal activity with
respect to the external stimulus function.
