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：本文研究视觉感知与初级视皮层（V1）神经活动数学建模之间的深层联系，重点是对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因参数变化而产生的自发模式形成方面，但在处理局部感觉输入场景时面临挑战。这在Mackay的心理物理实验中尤为明显——视觉刺激信息冗余导致不规则形状，使得分岔理论和多尺度分析效力降低。为应对这一难题，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-皮层映射得到的视觉刺激皮层表征捕捉其独特特征，包括MacKay漏斗模式中心高度局域化的信息（"麦凯射线"）。从控制论角度，我们讨论了具有线性和非线性响应函数的Amari型方程的精确可控性。在视觉麦凯效应建模中，通过调整表征神经元间连接性的参数，确保无感觉输入时皮层活动指数级稳定至稳态。随后进行的定量与定性研究证明，该模型能捕捉MacKay报告的所有诱发光后像本质特征。