Author’s copy Coupling and noise in the circadian clock synchronization Alessio Franci 1 , Marco Arieli Herrera-Valdez 1 Miguel Lara-Aparicio 1 , Pablo Padilla-Longoria 2 , 1 Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico. 2 Instituto de Investigación en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Mexico. afranci@ciencias.unam.mx, marcoh@ciencias.unam.mx, mla@ciencias.unam.mx, pablo@mym.iimas.unam.mx Keywords: Circadian rhythm, synchronization, nonlinear oscillators, Fokker-Planck, Hopf bifurcation. Abstract The general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength and noise play in the synchronization of a system of nonlinear, linearly coupled oscillators. First, we study a deterministic version of the model to find plausible regions in the parameter space for which synchronization is observed. Second, we focus on studying how noise and coupling interact in determining the synchronized behavior. To do so, we leverage the Fokker-Planck equation associated with the system. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used as a guide to further study coupled oscillations in biophysical nonlinear models. 1 Introduction The study of circadian rhythms has been a subject of great interest for a long time. However, the majority of the first studies were mainly based on observations in plants [1], and mathematical modeling of bio- logical rhythms was then at a very early stage [2–4]. There are two milestones in the study of circadian rhythms from a mathematical perspective, namely, the association of circadian rhythms to the existence of a limit cycle [5] and Arthur Winfree’s master book entitled "The Geometry of Biological Time" [6]. Related to the work of Kalmus, several mathematical models of circadian rhythm were presented in a Symposium on Biological Rhythms carried out at Cold Spring Harbor in the United States of America in 1960 [7–9]. Among these works, one of the most important entitled "Shock excited system as models for biological rhythms" was presented by Kalmus, a biologist, and Wigglesworth, a mathematician [5]. In this paper the authors associated a limit cycle to a circadian rhythm, using a hydraulic system as analogy. Lots of other important works on circadian rhythms were presented in this symposium, but the work of Kalmus and Wigglesworth was the crucial one in establishing a better mathematical formalism for the study of circadian rhythms. Although many researchers followed the theoretical path proposed by Kalmus and Wigglesworth, the mathematical study of circadian rhythms was finally established by Arthur Winfree (biologist and mathematician), who introduced topology for the description of several aspects of circadian rhythms. Lara-Aparicio et al. have worked on both mathematical modeling and experimental characterizations of different properties of circadian rhythms since 1993, when they first published an article on the ontogeny of circadian rhythm in the crayfish [10]. In their work, they built a mathematical model that phenomenologically captures a series of experimental results involving the synchronization of electro-retinogram activity in crayfish [11–13]. The model was constructed by coupling two van der Pol oscillators [14] represented by state vectors x =(x1,x2,x3) T and y =(y1,y2,y3) T . The coupled system from [10] is given by ˙ x = F (x; k, c, r) , ˙ y = F (y; l, C, R) (1) The parameters k and l represent the frequency of the oscillator; r and R are functions that represent the radii of the limit cycles, and c and C are also functions that represent the first coordinate of the center 1 PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.26447v1 | CC BY 4.0 Open Access | rec: 19 Jan 2018, publ: 19 Jan 2018