Neural Field Equations with Random Data

Abstract. We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view these spatially extended, nonlocal evolution equations as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution for uncertainty quantification (UQ) and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyze in abstract form neural fields that have been spatially discretized, setting the foundations for analyzing UQ schemes.