A mathematical model of the visual MacKay effect

TL;DR

This work reframes the MacKay visual illusion within an Amari-type neural field model for V1 by treating the retinal stimulus as a distributed control and analyzing the input–output map Ψ(I) that links sensory input to cortical steady states. It proves well-posedness and exponential stabilization of the network below the pattern-formation threshold μ0, develops exact controllability results for both linear and nonlinear regimes, and demonstrates that the MacKay effect can be replicated when symmetry-breaking, highly localized input is introduced. The key theoretical finding is that the MacKay after-image is essentially a linear phenomenon in this framework, with nonlinear saturation damping high-frequency components, supported by numerical simulations that reproduce MacKay rays and targets in both linear and nonlinear settings. Overall, the paper provides a controllability-based, geometrically informed approach to replicating complex visual illusions and suggests applicability to a broader class of psychophysical phenomena.

Abstract

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.

Figures (9)

• Figure 1: The MacKay effect mackay1957, showcasing the illusion induced by the stimulus on the left, referred to as "MacKay rays". This stimulus leads to an illusory perception of concentric rings superimposed in its background, as illustrated on the right. To see the illusory contours, look at the centre of the black circle in the image on the left. The adaptation of this figure is based on the original representation from mackay1957 and zeki1993.
• Figure 2: The stimuli used in mackay1957. Left: "MacKay-rays". Right: "MacKay-target".
• Figure 3: Possible response functions on the left where $\operatorname{erf}$ is the Gauss error function, and on the right a $2$D DoG kernel $\omega$. Here, $\kappa=2$, $\sigma_1=2$, and $\sigma_2=4$.
• Figure 4: Funnel pattern on the left (respectively in the retina and V1). Tunnel pattern on the right (respectively in the retina and V1).
• Figure 5: MacKay effect (right) on the "MacKay rays" (left). We use the linear response function $f(s)=s$. The sensory input is chosen as $I(x) = \cos(5\pi x_2)+\varepsilon H(2-x_1)$, $\varepsilon = 0.025$, where $H$ is the Heaviside step function.

Theorems & Definitions (64)

• Definition 3.1: Stationary state
• Lemma 3.2
• Theorem 3.3
• proof
• Definition 3.4
• Proposition 3.5
• proof
• Definition 4.1: Exact controllability
• Proposition 4.2
• proof