  We study solutions to a recently proposed neural field model in which
dendrites are modelled as a continuum of vertical fibres stemming from a
somatic layer. Since voltage propagates along the dendritic direction via a
cable equation with nonlocal sources, the model features an anisotropic
diffusion operator, as well as an integral term for synaptic coupling. The
corresponding Cauchy problem is thus markedly different from classical neural
field equations. We prove that the weak formulation of the problem admits a
unique solution, with embedding estimates similar to the ones of nonlinear
local reaction-diffusion equations. Our analysis relies on perturbing weak
solutions to the diffusion-less problem, that is, a standard neural field, for
which weak problems have not been studied to date. We find rigorous asymptotic
estimates for the problem with and without diffusion, and prove that the
solutions of the two models stay close, in a suitable norm, on finite time
intervals. We provide numerical evidence of our perturbative results.
