BioSystems 71 (2003) 61–69 A qualitative model of a motor circadian rhythm Beatriz Fuentes-Pardo a,∗ , Ana M. Guzmán-Gómez b , Miguel Lara-Aparicio b , Santiago López de Medrano c a Departamento de Fisiolog´ ıa, Facultad de Medicina, UNAM, Ciudad Universitaria, Apdo. Postal 70-250, Mexico D.F. CP 04510, Mexico b Departamento de Matemáticas, Facultad de Ciencias, UNAM, Mexico D.F. CP 04510, Mexico c Instituto de Matemáticas, UNAM, Mexico D.F. CP 04510, Mexico Received 2 November 2002; received in revised form 24 March 2003 To Arthur T. Winfree, in memoriam Abstract In Nature it is possible to observe diverse rhythms. Because of their adaptive characteristics, the circadian rhythms are of major importance and have been the subject of numerous experimental and theoretical studies. In this article, we give a presentation of the main results we have obtained about the motor circadian rhythm along some years of collaboration between biologists and mathematicians. We present a mathematical model simulating changes in frequency, synchronization and amplitude of the circadian oscillation during two developmental stages of the crayfish, namely, the juvenile and the adult stages. We report also some work in progress on the simulation of the phase response curve and on a simplified model of the rhythm. © 2003 Elsevier Ireland Ltd. All rights reserved. Keywords: Motor circadian rhythm; Mathematical model; Non-linear oscillators 1. General properties of circadian rhythms Among the established periodical processes driven by one (or more than one) biological clock are the circadian rhythms. The basic rhythmic element of cir- cadian rhythms is an endogenous and innate structure that in constant environmental conditions expresses a period of approximately 24 h, and may be affected by a number of external factors. Besides the presence of a pacemaker, some empir- ically derived generalizations have been proposed as ∗ Corresponding author. E-mail addresses: bfpardo@servidor.unam.mx (B. Fuentes-Pardo), amgg@hp.fciencias.unam.mx (A.M. Guzm´ an-G´ omez), aparicio@servidor.unam.mx (M. Lara-Aparicio), lmedrano@servidor.unam.mx (S. L´ opez de Medrano). descriptions of major features of circadian rhythms: a close dependence on light, in accordance to the well known Aschoff’s rule. Aschoff (1960) proposed that the circadian rhythms of diurnal organisms increase their frequency, amplitude and activity period when the external light increases and that, under this same situation, the opposite (diminution in all of these three parameters) occurs in nocturnal species. Circadian rhythms are readily phase-shifted by sin- gle perturbations in the light or temperature (among others) regimes. The character of the phase shift (phase-lead, phase-lag, and even the absence of change), depends not only on intensity and duration of the perturbing signal, but, especially, on the phase at which the circadian cycle was perturbed. This implies that the pacemaker of the circadian system possesses a differential sensitivity to the perturbing stimuli dur- ing the 24-h cycle. Immediately after the perturbing 0303-2647/$ – see front matter © 2003 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/S0303-2647(03)00110-2