In this study, we investigate the intricate connection between visual perception and the mathematical modelling of neural activity in the primary visual cortex (V1), focusing on replicating the 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 localised 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. This framework views the sensory input as a control function, cortical representation via the retino-cortical map of the visual stimulus that captures the distinct features of the stimulus, specifically the central redundancy in MacKay's funnel pattern ``MacKay rays''. From a control theory point of view, the exact controllability property of the Amari-type equation is discussed both for linear and nonlinear response functions. Then, applied to the MacKay effect replication, we adjust the parameter representing intra-neuron connectivity to ensure that, in the absence of sensory input, cortical activity exponentially stabilises to the stationary state that we perform quantitative and qualitative studies to show that it captures all the essential features of the induced after-image reported by MacKay

翻译：本研究探究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关联，重点关注MacKay效应[Mackay, Nature 1957]的复现。尽管分岔理论作为解决神经科学问题的主流数学方法，尤其在描述V1中因参数变化引发的自发模式形成方面成效显著，但在局部感觉输入场景下仍面临挑战。这一局限在Mackay的心理物理学实验中尤为明显——视觉刺激信息的冗余性导致不规则形状产生，使得分岔理论与多尺度分析的有效性降低。为突破此困境，我们基于Amari型神经场模型的输入-输出可控性框架，采用数学视角进行探究。该体系将感觉输入视作控制函数，通过视觉刺激的视网膜-皮层映射捕捉刺激的显著特征（特别是MacKay漏斗模式"MacKay射线"的中心冗余性）。从控制论角度，分别探讨了线性和非线性响应函数下Amari型方程的精确可控性特征。继而应用于MacKay效应复现时，通过调节表征神经元间连接性的参数，确保在无感觉输入条件下皮层活动指数级稳定至平稳态。我们进行的定性与定量研究证实，该模型完整复现了MacKay所报告的诱发电后像的所有本质特征。