Figure 4.

Aschoff’s rule: a strong permanent external stimulus causes arrhythmic behavior as the oscillations are dampened off. (a) Time curves where the superscript u indicates that these time curves stem from the extended model consisting of (1) to (5) where (2) is replaced by (8) for (10) with $\alpha =1$ and $c=1$. (b) Time curve of M calculated from the model (1) to (5) and of $M^u$ calculated from (1) to (5) where (2) is replaced by (8) for (10) with $\alpha =1$ and $c=1$. In (c) and (d) a weak permanent external stimulus causes dampening of the amplitude of the oscillations, however they are still measurable. The experiment is as in (a) and (b) with $c=0.5$. Units and dimensions for (a)–(d): abscissa t: time in hours, ordinate U: amount of substance (M, $M^u$ mRNA, $P_i^u$, $\left\{0,,,1,,,2,,,N\right\},$ protein, u intensity of illumination). In (e) we have on the abscissa the illuminance c and on the ordinate we have the period length T in hours. The periods of the endogenous clocks of the fly and the mammalian model under a constant external stimulus. The blue graph is from the fly model and the red one is from the mammalian model. The data points for the fly model are $\left(0,,,23.7\right)$, $\left(0.1,,,23.5\right)$, $\left(0.2,,,23.4\right)$, $\left(0.3,,,23.7\right)$, $\left(0.4,,,24.5\right)$, $\left(0.5,,,25.4\right)$, $\left(0.6,,,26.4\right)$, $\left(0.7,,,27.4\right)$, $\left(0.8,,,28.5\right)$, $\left(0.9,,,29.6\right)$. The data points for the mammalian model are $\left(0,,,23.7\right)$, $\left(0.1,,,22.8\right)$, $\left(0.2,,,22.2\right)$, $\left(0.3,,,21.7\right)$, $\left(0.4,,,21.5\right)$, $\left(0.5,,,21.5\right)$, $\left(0.6,,,21.8\right)$, $\left(0.7,,,22.3\right)$, $\left(0.8,,,22.9\right)$, $\left(0.9,,,23.7\right)$.