Compound Eye Motion Detector: The Reichardt Correlator

Stimulus

Angular velocity ω 40°/s →

Stripe count (spatial freq. k) 10

Correlator delay τ 35 ms

Elementary Motion Detector

EMD response R̄ (correlator output)

0.000 —

Interommatidial Δφ

–

Sampling k·Δφ / π

–

How it works

Each facet (ommatidium) of a compound eye reports the light intensity L(t) arriving from one small angular direction. Insects estimate motion not with a lens-and-retina image, but with a local circuit compares two neighbouring ommatidia: the Hassenstein–Reichardt correlator, first identified from beetle optomotor behaviour and now confirmed as the T4/T5 circuit in the fly lobula.

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 facet here integrates its own delayed copy L_d each frame from the real differential equation above — the correlator sign and strength in the readout is the actual output of that circuit, not a lookup.

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, the correlator can lock onto the wrong beat frequency and report motion in the opposite direction of the true stimulus — the same aliasing 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.

Ommatidium luminance Rightward-tuned pair Leftward-tuned pair

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