Early in the study of Limulus CE, it was discovered that when the eye was stimulated with beams of light that could be focused on individual ommatidia, the firing frequency of a given E-cell axon in the optic nerve could be reduced by focusing a second beam of light on a neighboring ommatidium. The relation between the firing frequencies of the two separate, stimulated E-cells was found to be described by a pair of simultaneous, piecewise-linear algebraic equations (Ratliff et al., 1963):
where r1 and r2 are the steady-state firing frequencies of the E-cell axons from the two stimulated ommatidia; e1 and e2 are the frequencies at which the E-cells would fire if stimulated one at a time (the other ommatidium being dark); K12 and K21 are the reciprocal inhibition constants; <12 and <21 are threshold firing rates, above which inhibition takes place. For example, if (r2 - <12) < 0, then K12 = 0, etc. Also, if [e1 - K12(r2 - <12)] < 0, r1 = 0, etc. (Clearly, there are no negative frequencies in nature.) The process described by the two equations above was called lateral inhibition (LI), because the lateral plexus evidently carried information about retinula cell depolarization from a stimulated ommatidium to its neighbors, causing a reduction in their firing rates. Figure 5.3-4 shows a systems block diagram describing the two LI equations above. Note that the LI dyad is an overall, positive feedback system when operating in its linear range. Neglecting the dc offsets, the dc loop gain of the system is: Al = + K12 K21.
The linearity of the dyadic model was demonstrated by Ratliff (1964). Ratliff et al., (1963) were able to extend the dyadic relation above to a more general N ommatidium model:
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