Coarse-­‐Graining the Dynamics of Coupled Oscillators to Model Neurons in the Suprachiasmatic Nucleus

Abstract

The circadian rhythm is the oscillation of certain genes and proteins with approximate periods of 24 hours in many organisms, including humans. In mammals, the circadian rhythm is controlled by a part of the hypothalamus called the suprachiasmatic nucleus (SCN). Our model of the SCN, described by twenty-­‐one ordinary differential equations, makes two parameters heterogeneous, an intrinsic one and a structural one. The intrinsic heterogeneity is included by selecting the value of one of the parameters in the ODEs from a Gaussian distribution for each neuron; therefore, each neuron would have a different value for the parameter. We constructed small-­‐world networks to model the connectivity of neurons, and the structural heterogeneity we considered was “degree,” the number of other neurons in the network a neuron was connected to. We looked for regimes in which the variation is explained by both the structural and intrinsic heterogeneity, not just the intrinsic one. We also propose a way to simulate the SCN using coarse projective integration, which would allow for the simulation of more neurons over a longer period of time.