Stroboscopic motion reversals in delay-coupled neural fields

TL;DR

The work shows that delay-induced interareal coupling in a pair of neural-field rings yields a discrete lattice of coexisting traveling bump speeds, enabling intrinsic motion representations and stroboscopic percepts without external sampling clocks. Using an interface reduction, the authors derive low-dimensional delay differential equations that capture speed selection, stability, and entrainment, revealing how continuous forcing yields veridical tracking while impulsive stroboscopic forcing drives transitions between speed states and even motion reversals. Spatially dependent delays further deform the speed lattice in a geometry-dependent manner but preserve the core multistability mechanism, signaling robust applicability to realistic conduction heterogeneity. Overall, the study provides a mechanistic, analytically tractable account of motion aliasing and wagon-wheel-like percepts as emergent properties of delayed neural interactions in distributed visual-processing circuits.

Abstract

Visual illusions provide a window into the mechanisms underlying visual processing, and dynamical neural circuit models offer a natural framework for proposing and testing theories of their emergence. We propose and analyze a delay-coupled neural field model that explains stroboscopic percepts arising from the subsampling of a moving, often rotating, stimulus, such as the wagon-wheel illusion. Motivated by the role of activity propagation delays in shaping visual percepts, we study neural fields with both uniform and spatially dependent delays, representing the finite time required for signals to travel along axonal projections. Each module is organized as a ring of neurons encoding angular preference, with instantaneous local coupling and delayed long-range coupling strongest between neurons with similar preference. We show that delays generate a family of coexisting traveling bump solutions with distinct, quantized propagation speeds. Using interface-based asymptotic methods, we reduce the neural field dynamics to a low-dimensional system of coupled delay differential equations, enabling a detailed analysis of speed selection, stability, entrainment, and state transitions. Regularly pulsed inputs induce transitions between distinct speed states, including motion opposite to the forcing direction, capturing key features of visual aliasing and stroboscopic motion reversal. These results demonstrate how delayed neural interactions organize perception into discrete dynamical states and provide a mechanistic explanation for stroboscopic visual illusions.

Figures (7)

• Figure 1: A. Schematic of the delay-coupled ring model \ref{['dubdelay']}. Two orientation rings receive rotating inputs $I_1$ and $I_2$ and evolve according to within-layer coupling $w(x)$ and delayed cross-layer coupling $w_c(x)$. This abstraction can be interpreted as orientation-selective populations in distinct visual areas (e.g., V1 and MT) exchanging delayed signals. B. Conceptual illustration of the wagon wheel illusion: although a stimulus rotates in one direction, stroboscopic sampling can generate the percept of motion in the opposite direction. In our model, this corresponds to a traveling bump solution propagating counter to the input. We introduce this figure here to foreshadow the main phenomenon analyzed in later sections: how delayed coupling in neural fields provides a minimal circuit mechanism for generating illusory percepts.
• Figure 2: Half-width and stability of symmetric stationary bumps. A. Eigenvalues determining stability of the wide bump occur at the zeros of the Evans function $E(\lambda)$ -- the intersection of the real and imaginary zero level sets. Here $\bar{w} = 1,$$\tau = 10$ and $\theta = 0.1$ and we study the eigenvalues of the associate wide bump ($a_w$). B. Narrow bump ($a_n$) has a set of eigenvalues with positive real part. C. Stable/wide and unstable/narrow branches annihilate in a saddle node bifurcation as the threshold $\theta$ is increased to a critical value $\theta_c = 1 + \bar{w}$ for $\bar{w} = 1,2,3$. D. Saddle-node occurs at $\bar{w}_c = -1 + \theta$ as the cross-coupling strength is decreased ($\theta = 0.1$).
• Figure 3: Families of metastable traveling bumps generated by delayed coupling.(A) Traveling bump speeds $c$ as functions of the interlayer delay $\tau$ for fixed coupling strength $\bar{w}$. Black curves correspond to input-free traveling bump solutions of the delay-coupled neural field equations, each containing a wide stable and narrow unstable bump. Red curves indicate extraneous branches on which a proper bump width cannot be defined. Delays generate infinitely many branches of admissible propagation speeds. Here $\theta = 0.1$ and $\bar{w} = 1.$(B) Dependence of traveling bump speeds $c$ on the interlayer coupling strength $\bar{w}$ for fixed delay $\tau = 20$. Stable and unstable branches coexist and exchange stability at saddle-node bifurcations. (C) Spatiotemporal evolution of neural activity for a stable traveling bump corresponding to the green dot in (A) and (B). Color indicates activity on the orientation ring as a function of time, with diagonal bands corresponding to coherent bump motion at a constant speed predicted by the interface reduction.
• Figure 4: Entrainment of traveling bumps to continuous motion.(A) Regions of entrainment in the $(c,I_0)$ plane. Dark gray indicates parameter values for which no entrained traveling bump exists. Light gray indicates existence of an entrained solution that is linearly unstable. White indicates existence of a linearly stable entrained solution. Colored markers denote parameter values used in panels (B)--(D). Panels (B)--(D) show the spatiotemporal evolution of the neural field $u_1(x,t)$ for the corresponding parameter values. (B) Below the entrainment threshold (red marker), showing failure of tracking and loss of coherent motion representation. (C) Unstable entrained solution (blue marker), illustrating transient tracking followed by drift or distortion of the bump profile. (D) Stable entrained solution (green marker), showing veridical motion perception: the bump remains coherent and rotates at the same speed and in the same direction as the stimulus, with a constant phase lag. Parameters: $\bar{w}=1$, $\theta=0.1$, $\tau=10$.
• Figure 5: Speed selection under impulsive forcing.(A) Contour plot of the Evans function \ref{['eq:evans']} associated with the autonomous inter-impulse dynamics, illustrating the stability of discrete traveling speeds. Here $r =2$ and $\bar{w} =1.$(B) Time series of the sliding-window speed $v(t)$ following a single impulse. A weak impulse (blue) does not cross a separatrix and relaxes back to the original speed, while a stronger impulse (red) induces a transition to a neighboring stable speed. The parameters are the same as panel A, with the impulse amplitudes at $0.5$ and $2$ respectively. (C) Asymptotic speed $\lim_{t\to\infty} v(t)$ under repeated impulses applied at fixed intervals $\Delta T$, showing convergence to a discrete set of quantized speeds. Positive asymptotic speeds slower than the stimulus correspond to aliasing, while negative speeds correspond to anti-aliasing (motion reversal). The impulse amplitude $A$ is set at $1,$ with $\tau = 10,$$\bar{w} = 1,$ and $\theta = 0.1.$(D) Asymptotic speed as a function of input speed $r$ and pulse period normalized by the delay, $\Delta T/\tau$, revealing banded regions of aliasing ($0<v<r$) and anti-aliasing ($v<0$) organized by impulse timing. Parameters $\theta = 0.1$, $\bar{w} = 1$; impulse amplitude $A=1$; period $\Delta T$, and input speed $r$ are varied.