- cos(2n;x/ X) sin[(2n/X)(x - 8)]) sin(rae) + cos(2nV X) cos[(2n/X))x - 8)]) cos(rae)}

The DDC output is taken as Ay = y' - y . Combining the terms, using trigonometric identities, and noting that ra = 2nv/X, finally yields

Thus, the DDC output to a moving sinewave object shows the first positive peak for v = X/4e mm/s (positive v for object motion from left to right). The first negative peak occurs for object motion from right to left at v = -X/4e. Ay also increases as the ratio 8/X increases.

As shown above, the DDC response to a moving, spatial, sinewave object demonstrates that the DDC works as a model motion detector. How can the basic DDC operation be realized in the optic lobes of insects? The angular separation between ommatidia gives the parameter, 8. The temporal delay, e, might be associated with the propagation time for an analog signal along a thin dendrite or axon. The multiplier is harder to conceptualize. It might involve a linear summation of two logarithmic signals at a nerve membrane, then an exponential operation in the spike generation process to give the antilog product in terms of spike frequency. (Such a y = ln(A) + ln(B) operation, followed by f Y K exp(y) spike generation was proposed by the author in 1975 for a model of signal-to-noise improvement in insect vision.) The outputs of many such dyadic machines would have to be combined so that the whole compound eye displays directional sensitivity, as in nature. The interested reader wishing to pursue this topic further should consult the papers by Martin Egelhaaf (1985a, b, c) on theoretical neuronal models for motion detection circuits in the fly's optic lobes, as well as the many papers authored or co-authored by Werner Reichardt.

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