A mathematical model of the visual MacKay effect
arxiv(2023)
摘要
This paper investigates the intricate connection between visual perception
and the mathematical modelling of neural activity in the primary visual cortex
(V1). The focus is on modelling the visual 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 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 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. 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
modelling, 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.
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