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深层关联，重点聚焦于视觉麦凯效应的建模[Mackay, Nature 1957]。尽管分岔理论已成为处理神经科学问题的主流数学方法，尤其在描述因参数变化导致的V1自发性斑图形成方面贡献显著，但当涉及局部感觉输入场景时，该理论面临挑战。这一局限在Mackay心理物理实验中尤为明显：视觉刺激信息的冗余导致不规则形状产生，使得分岔理论与多尺度分析效能降低。为此，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。在该框架下，我们将感觉输入视为控制函数，通过视网膜-皮层映射获取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"麦凯射线"信息。从控制论角度，我们讨论了含线性与非线性响应函数的Amari型方程精确可控性性质。针对视觉麦凯效应建模，我们通过调节表征神经元内连接的参数，确保在无感觉输入时皮层活动指数级稳定至稳态。最后，通过定性与定量研究证明，该模型能完整复现Mackay所报告诱发光后像的所有关键特征。