Figure 1. Step and multipulse adaptation.

(A) The basic model (1) consists of the two variables y and x linked by a negative feedback loop (of gains k₁ and k₂), an external input u, and two first order degradation terms (of rates δ_x and δ_y). (B) Various levels of adaptation for the model (1), according to the ratio δ_x/δ_y. The upper and lower parts of the panel show the response to the two main input protocols described in the text, steps and multiple pulse pairs (here a series of 4 pulse pairs in which the second pulse is progressively delayed with respect the first; the 4 double pulse responses are shown all simultaneously, and the 4 first pulses of each pair are all identical and overlapping). Notice that the step responses resembles those of Figure 5 of Koshland et al.¹¹. The ratio δ_x/δ_y determines the amount of the two forms of adaptation mentioned in the text, step adaptation and multipulse adaptation. In particular perfect step adaptation requires exact integral feedback (i.e., δ_x = 0) and corresponds to no recovery in multipulse adaptation (leftmost plots). Moving from left to right of the panel as we increase the ratio δ_x/δ_y, step adaptation decreases and the recovery in multipulse adaptation becomes faster. See Supplementary Figure S1 for blown-up plots of the various cases.