Text S1 - Modeling the Seasonal Adaptation of Circadian Clocks by Changes in the Network Structure of the Suprachiasmatic Nucleus

The supporting Text S1 provides several theoretical definitions and derivations generalizing the findings of the main study. In Section 1 the concept of phase response curves is introduced and it is shown how to compute them numerically. In Section 2 a general equation for the dynamics of the in-phase distribution of an arbitrary oscillator network with small heterogeneity between the oscillators is derived. The derivation is based on the well-known phase reduction method introduced by Kuramoto and a linearization around small phase differences. In Section 3 we derive the stationary phase-distribution and the locked frequency using the pseudoinverse of the matrix M of the linearized system. Moreover, we show that the stationary phase distribution is among other factors mostly influenced by near-zero singular/eigenvalues of the matrix M. In Section 4 we consider several special cases of oscillator coupling. In particular we derive that for a coupling function that is the same for all oscillators the networks Laplacian and consequently its near-zero singular/eigenvalues determine the stationary phase-distribution. In Section 5 we connect the findings from Section 2–4 to previous work from spectral graph theory, showing that the network structure, and in particular the occurrence of communities with nodes that are well connected within the community but weakly connected between communities, is tightly related to the singular/eigenvalue spectrum of the Laplacian. Moreover, we calculate and discuss the spectra for the winter and summer topology of our SCN network. In Section 6 we additionally analyze theoretically the entrainment to an external stimulus of a single amplitude-phase oscillator from our study. We derive novel entrainment bounds for several special cases of rigid and weak oscillators. These bounds are derived to compare them against the entrainment of the whole SCN network. Section 7 contains all supplementary figures along with figure captions of our study.
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