Robustness and size-dependence of circadian rhythms in multiscale suprachiasmatic-nucleus networks

TL;DR

By increasing the average degree with network size, this work reproduces size-dependent rhythms and shows that they arise from network connectivity, whereas low-degree networks fragment and fail to sustain oscillations and provide a framework for linking multi-scale network structure to biological timekeeping.

Abstract

Understanding how multi-scale network structure influences circadian rhythms in the suprachiasmatic nucleus (SCN) is essential for uncovering the principles of rhythmic robustness and synchronization. Previous studies using synthetic SCN networks suggested a size-dependent phenomenon, in which rhythmic activity initially strengthens with network size and then saturates, but it remains unclear whether this occurs in real SCN networks. Here, we apply geometric branch growth (GBG) and geometric renormalization (GR) to generate self-similar scaled-up and scaled-down replicas from a single-scale functional mouse SCN network. Unlike synthetic models, these SCN replicas do not exhibit size-dependent rhythms: average period, amplitude, and synchronization remain stable across scales. By increasing the average degree with network size, we reproduce size-dependent rhythms and show that they arise from network connectivity, whereas low-degree networks fragment and fail to sustain oscillations. Disrupting clustering self-similarity slightly reduces synchronization, but circadian rhythms remain robust, indicating that average degree, rather than clustering, is the dominant structural driver. These results highlight the resilience of SCN rhythms to network scaling and provide a framework for linking multi-scale network structure to biological timekeeping.

Figures (14)

• Figure 1: Self-similarity of scaled-up and scaled-down SCN networks. (a) Schematic illustration of the GBG and GR models. Layer $l=0$ corresponds to the original SCN network. Different node colors indicate communities detected by the Louvain algorithm Blondel2008. Yellow and blue arrows represent the scaling-up and scaling-down processes of the multiscale unfolding under the GBG and GR models, respectively. For clarity, GR layers are relabeled as $l=-1,-2,\dots$ and GBG layers as $l=1,2,\dots$. In each layer, node size is proportional to $\log k$, where $k$ is the node degree, and node colors inherit those of their ancestors in the original network ($l=0$). (b-d) Topological properties of scaled-up and scaled-down SCN networks. (b) Complementary cumulative degree distribution. (c) Degree-dependent clustering coefficient. (d) Degree--degree correlations. (e) Connection probability $p(\chi_{ij}^{(l)})$ as a function of the effective distance $\chi_{ij}^{(l)}$ in layer $l$. The red curve indicates the theoretical prediction from Eq. \ref{['eq:con_pro_S1']} for the original SCN network at $l=0$.
• Figure 2: Circadian rhythms in scaled-up and scaled-down SCN replicas. (a) Time series of the average output $\overline{Y_1}$ across different layers $l$ for coupling strength $K = 0.8$. (b) Corresponding period distribution $P(T_i)$ of individual oscillators for $K = 0.8$. (c) Average period $T$, (d) SCN amplitude $A$, and (e) synchronization degree $R$ as functions of network size $N$ for different coupling strengths. Error bars represent one standard deviation from the mean. Results are obtained using the Becker--Weimann model under constant-darkness conditions and averaged over $50$ realizations with different initial conditions and independently sampled $\mu_i$.
• Figure 3: Effects of average degree on circadian rhythms. (a) Complementary cumulative degree distribution; inset: average degree as a function of network size $N$, showing the expected increase with $N$. (b) Degree-dependent clustering coefficient, (c) Degree--degree correlations. In (a-c), degrees are rescaled as $k_{\mathrm{res}} = k / \bar{k}$. (d) Average period $T$, (e) SCN amplitude $A$, and (f) synchronization degree $R$ as functions of $N$ in the perturbed networks from Null-$k$ model under varying coupling strengths. Error bars represent one standard deviation from the mean. The black dashed vertical lines indicate the original networks. Results are averaged over $50$ realizations with different initial conditions and independently sampled $\mu_i$.
• Figure 4: Evolution of neuronal oscillators in networks with different average degrees. (a, b) Time series of a representative neuronal oscillator $Y_{1,i}$ and the population-averaged output $\overline{Y_1}$ in layer $l\!=\!-2$ from the Null-$k$ model and the GR replica, respectively. (c) Corresponding network snapshot from the Null-$k$ model. (d, e) Time series of a representative neuronal oscillator $Y_{1,i}$ and the population-averaged output $\overline{Y_1}$ in layer $l\!=\!3$ from the Null-$k$ model and the GBG replica, respectively. (f) Corresponding network snapshot from the Null-$k$ model. The GR replica at layer $l=-2$ and the GBG replica at layer $l=3$ are used as references for comparison.
• Figure 5: Impact of disrupting self-similarity in the clustering coefficient on circadian rhythms. (a) Complementary cumulative degree distribution. (b) Degree-dependent clustering coefficient; inset: average clustering coefficient as a function of network size $N$, showing the expected increase with $N$. (c) Degree--degree correlations. In (a-c), degrees are rescaled as $k_{\mathrm{res}} = k / \bar{k}$. (d) Average period $T$, (e) SCN amplitude $A$, and (f) synchronization degree $R$ as functions of $N$ in the perturbed networks from Null-$c$ model under varying coupling strengths. Error bars represent one standard deviation from the mean. Results are averaged over $50$ realizations with different initial conditions.