This 2D version lays a single row of ommatidia out flat and shows the correlator circuit working in real time: the moving stripe strip at top is the stimulus, the bar row below it is each ommatidium's instantaneous luminance L(t), and the scrolling strip at the bottom is a genuine spacetime plot of every neighbouring pair's correlator output over the last few seconds.
Delay line (low-pass filter of time constant τ):
dL_d/dt = (L − L_d) / τ
Correlator output for neighbours 1 (left) and 2 (right):
R(t) = L_d1(t)·L2(t) − L_d2(t)·L1(t)
R > 0 → motion left→right (toward neighbour 2)
R < 0 → motion right→left
R ≈ 0 → no coherent motion, or a spatial frequency the pair cannot resolve
Every ommatidium here integrates its own delayed copy L_d each frame from the real differential equation above — nothing in the readout, the bar colours or the waterfall is a lookup table; it is the literal output of that circuit running on canvas 2D each frame.
Spatial aliasing: two adjacent ommatidia are separated by a fixed interommatidial angle Δφ. This is the eye's sampling interval, so the Nyquist limit of ordinary signal sampling applies:
resolvable stripes: k · Δφ < π
aliased (may reverse): k · Δφ ≥ π
When the grating is finer than the eye can sample, or ω is cranked high enough that the stripe pattern outruns the correlator's delay line, the circuit can lock onto the wrong beat frequency and report motion in the opposite direction of the true stimulus. Watch the waterfall's diagonal streaks flip slope when this happens — the same sampling-theorem limit that makes a wagon wheel appear to spin backwards on film, but here it is a real, documented limit of insect optomotor responses, not an illusion of a shutter.
- Angular velocity ω — speed and direction of the moving stripe pattern across the eye's field of view.
- Stripe count k — spatial frequency of the grating; higher k means finer stripes and a shorter angular period.
- Delay τ — the correlator's low-pass time constant; response peaks when the stripe pattern takes about τ to cross one interommatidial angle.
Real-world relevance: this exact correlator (in parallel arrays, pooled over thousands of ommatidia) drives a fly's optomotor stabilisation reflex and a bee's odometry during foraging flight.