Predicting Perception of the Wagon Wheel Illusion

Stroboscopic illumination of a rapidly rotating disk with radial spokes leads to a range of different stationary and moving images as the angular rotation frequency of the disk and the strobe frequency are varied. We compare predictions from the standard correlation model of motion perception with a model based on phase locking observed during periodic stimulation of an integrate-and-fire nonlinear oscillator. The close agreement between theoretical predictions and experimental observations suggests the possibility that periodic forcing of nonlinear neural oscillations may play a role in motion perception.

Figure 1

Illustration of the wagon wheel illusion. The first three panels illustrate three successive positions of a disk with 4 spokes during a clockwise rotation of 84° per strobe flash. The right panel shows the superimposition of the three images, leading to the wagon wheel illusion with a counter clockwise rotation of 6° per strobe flash.

Figure 2

Predicted zones for the wagon wheel illusion based on the correlation function analysis with superimposed experimental data collected from one of the authors in a determination of the boundaries of the resonance zones. ${\tau }_s$ is the time between strobe flashes, and ${\tau }_r$ is the recurrence time. The experimentally determined borders for a rotating disk with 4 spokes are indicated by the superimposed symbols: $m=1$ (circles); $m=2$ (triangles); $m=3$ (squares); $m=4$ (diamonds); $m=6$ (star). For a disk with $N$ spokes, the colored zones have $mN$ spokes: $m=1$ (purple); $m=2$ (green); $m=3$ (blue); $m=4$ (red); $m=5$ (gray); $m=6$ (yellow). We assume $T=200\mathrm{ms}$.

Figure 3

Model of a neural oscillator that is periodically stimulated [13]. The stimulus induced by the strobe is modeled by red vertical bars from the baseline of length $d$ and with a (dimensionless) time interval between the stimuli of ${\tau }_s/{\tau }_r$. The firing of the neuron occurs when the activity reaches threshold, indicated by a blue vertical bar. (a) No stimulation; (b) $1:1$ entrainment with $d=0.3$ and ${\tau }_s/{\tau }_r=0.8$; (c) $3:2$ entrainment with $d=0.3$ and ${\tau }_s/{\tau }_r=0.7$.

Figure 4

Theoretically computed locking zones from the neural oscillator model in Fig. 3 with superimposed experimental data collected from one of the authors in a determination of the boundaries of the resonance zones. We assume $T=200\mathrm{ms}$. Symbols are as in Fig. 2.

Figure 5

Theoretically computed locking zones from the neural oscillator model in Fig. 3 and superimposed data points for clockwise rotating disks with 4, 8, or 16 spokes with rotation speed constant at 2 Hz and variable strobe frequency (see text and supplementary material [22]). The asterisks represent points of multistability. We assume $T=200\mathrm{ms}$. Symbols are as in Fig. 2.