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Supplementary Figure 1 The product of two Hill functions (a-b) with sufficiently distant midpoints is equivalent to the Hill function with the largest midpoint (c). Namely: Hill(x, K 1) · Hill(x, K 2) ≈ Hill(x, K 2) where K 1<
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Supplementary Figure 2 Hill functions of Hill functions (a-b) can also be approximated by Hill functions. (c-d) Hill (Hill(x, K 2), K 1) = (1+K 1)−1. Hill(x, K 1 K 2(1+K 1)−1). In this case the midpoint of the resulting Hill function depends on the specific values of the midpoints of the original Hill functions considered in the composition of the Hill-of-Hill function. (e-h) Hill(x, K 1 · Hill (x, K 2)) = (x + K 2)/(x + K 1 + K 2) = Hill (x, (K 1 + K 2)) + f(x), where f(x) = K 2/(x + K 1 + K 2). Notably, f(x⟶0) = K 2/(K 1 + K 2) whereas f(x⟶∞) ≈ 0, so that the resulting Hill curve is essentially comprised within the interval [K 2/(K 1 + K 2),1). (GIF 50.6KB)

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Supplementary Figure 3 Bifurcation diagrams for a modified ChI model and prototypical sets of (a-c) AM-encoding and (d-f) FM-encoding L-R parameters. The bifurcation diagrams were computed after introduction into the ChI model of the rate of glutamate-dependent IP3 production, v glu , as a free bifurcation parameter, namely . This figure shows that the ChI model can still display oscillations in presence of an external non-specific bias of IP3 production. This is a first suggestion that the corresponding glutamate-dependent G-ChI model may also display oscillations. The parameters are taken from Table 1. (GIF 44.6KB)

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