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Our approach involves solving the integral of the partial integro differential equation directly using collocation techniques, alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods we study localised activity patterns in two different neural field models; namely, the Amari equation, for which we consider stationary bump solutions, and an extended version of the Amari equation that admits both stationary and travelling bump solutions. We solve both equations on a variety of domains, including a flat periodic domain, the curved surface of the torus and the folded surface of the rat cortex.

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