On the mathematical replication of the MacKay effect from redundant stimulation

In this study, we investigate the intricate connection between visual perception and the mathematical modelling of neural activity in the primary visual cortex (V1), focusing on replicating the MacKay effect [Mackay, Nature 1957]. 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 localised 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 multi-scale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. This framework views the sensory input as a control function, cortical representation via the retino-cortical map of the visual stimulus that captures the distinct features of the stimulus, specifically the central redundancy in MacKay's funnel pattern ``MacKay rays''. From a control theory point of view, the exact controllability property of the Amari-type equation is discussed both for linear and nonlinear response functions. Then, applied to the MacKay effect replication, we adjust the parameter representing intra-neuron connectivity to ensure that, in the absence of sensory input, cortical activity exponentially stabilises to the stationary state that we perform quantitative and qualitative studies to show that it captures all the essential features of the induced after-image reported by MacKay