A Mathematical Model of the Visual MacKay Effect

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 3, Page 2138-2178, September 2024.
Abstract.This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849–850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay’s psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay’s funnel pattern “MacKay rays.” From a control theory point of view, the Amari-type equation’s exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.