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.

翻译：本文研究了视觉感知与初级视觉皮层（V1）神经活动数学建模之间的复杂联系，重点关注视觉麦凯效应的建模（Mackay，Nature 1957）。尽管分岔理论已成为解决神经科学问题的重要数学方法，尤其在描述V1中参数变化引起的自发斑图形成方面，但在局部感觉输入场景下仍面临挑战。例如，在Mackay的心理物理学实验中，视觉刺激信息的冗余性导致不规则形状，使得分岔理论和多尺度分析效果不佳。为此，我们基于Amari型神经场模型的输入-输出可控性提出数学视角。在该框架中，我们将感觉输入视为控制函数，通过视网膜-皮层映射获得视觉刺激的皮层表征，捕捉其独特特征，包括Mackay漏斗斑图中心高度局部化的信息（“麦凯射线”）。从控制论角度，讨论了Amari型方程在线性和非线性响应函数下的精确可控性性质。针对视觉麦凯效应建模，我们调整代表神经元内连接的参数，确保在无感觉输入时皮层活动指数稳定收敛至稳态。随后，通过定量和定性研究证明该模型能复现Mackay报告中后像的所有关键特征。