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.

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。重点在于对视觉MacKay效应[D. M. MacKay, Nature, 180 (1957), pp. 849--850]进行建模。尽管分岔理论一直是解决神经科学问题的重要数学方法，特别是在描述V1中因参数变化导致的自发模式形成方面，但在存在局部感觉输入的场景中，该方法面临挑战。这在MacKay的心理物理实验中尤为明显，其中视觉刺激信息的冗余导致不规则形状的产生，使得分岔理论和多尺度分析效果有限。为解决此问题，我们采用基于Amari型神经场模型输入-输出可控性的数学视角。在此框架下，我们将感觉输入视为控制函数，即通过视觉刺激的视网膜-皮层映射获得的皮层表征，该映射捕捉了刺激的显著特征。这包括MacKay漏斗图案“MacKay射线”中心高度局部化的信息。从控制理论的角度，我们讨论了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MacKay效应建模，我们调整代表神经元内连接性的参数，以确保在缺乏感觉输入时皮层活动能指数稳定至静止状态。随后，我们进行了定量和定性研究，以证明该模型捕捉了MacKay所报告诱导后像的所有基本特征。