Waves in Spatially-disordered Neural Fields: a Case Study in Uncertainty Quantification

Neural field models have been used for many years to model a variety of macroscopic spatiotemporal patterns in the cortex. Most authors have considered homogeneous domains, resulting in equations that are translationally invariant. However, there is an obvious need to better understand the dynamics of such neural field models on heterogeneous domains. One way to include heterogeneity is through the introduction of randomly-chosen “frozen” spatial noise to the system. In this chapter we investigate the effects of including such noise on the speed of a moving “bump” of activity in a particular neural field model. The spatial noise is parameterised by a large but finite number of random variables, and the effects of including it can be determined in a computationally-efficient way using ideas from the field of Uncertainty Quantification. To determine the average speed of a bump in this type of heterogeneous domain involves evaluating a high-dimensional integral, and a variety of methods are compared for doing this. We find that including heterogeneity of this form in a variety of ways always slows down the moving bump.