Small-world network models of intercellular coupling predict enhanced synchronization in the suprachiasmatic nucleus

Abstract

The suprachiasmatic nucleus (SCN) of the hypothalamus is a multioscillator system that drives daily rhythms in mammalian behavior and physiology. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as the key intercellular synchronizing agent, we developed a multicellular SCN model to investigate the effects of cellular heterogeneity and intercellular connectivity on circadian behavior. A 2-dimensional grid was populated with 400 model cells that were heterogeneous with respect to their uncoupled rhythmic behavior (intrinsic and damped pacemakers with a range of oscillation periods) and VIP release characteristics (VIP producers and nonproducers). We constructed small-world network architectures in which local connections between VIP producing cells and their 4 nearest neighbors were augmented with random connections, resulting in long-range coupling across the grid. With only 10% of the total possible connections, the small-world network model was able to produce similar phase synchronization indices as a mean-field model with VIP producing cells connected to all other cells. Partial removal of random connections decreased the synchrony among neurons, the amplitude of VIP and cAMP response element binding protein oscillations, the mean period of intrinsic periods across the population, and the percentage of oscillating cells. These results indicate that small-world connectivity provides the optimal compromise between the number of connections and control of circadian amplitude and synchrony. This model predicts that small decreases in long-range VIP connections in the SCN could have dramatic effects on period and amplitude of daily rhythms, features commonly described with aging.

Figure 1

A schematic representation of the procedure used to construct different cellular networks on a 2-dimensional grid. (A) A nearest neighbor network with each neuron connected to its 4 nearest neighbors and vasoactive intestinal polypeptide (VIP) expressed by all cells within the network. (B) Small-world network with additional shortcut connections added to the nearest neighbor network according to a probability p and VIP expressed by all cells within the network. (C) Mean-field network with each neuron connected to every other neuron and VIP expressed by all cells within the network. (D) The small-world network that results when VIP production is randomly eliminated from a fixed percentage of neurons, yielding a heterogeneous network with both reciprocal and nonreciprocal connections.

Figure 2

Synchronization behavior of cellular networks on a 2-dimensional grid when vasoactive intestinal polypeptide (VIP) is introduced at t = 150 h. (A) Per mRNA time profiles of 10 randomly selected cells for the small-world network model (p = 0.05). All cells are rhythmic and the population exhibits a high degree of phase synchrony. (B) Per mRNA time profiles of 10 randomly selected cells with the nearest neighbor model (p = 0). Some cells are arrhythmic and the population exhibits a low degree of phase synchrony. (C) Synchronization index (SI) versus time for 4 values of the probability p, where circadian cycle 1 represents the time of VIP introduction. Despite having only 10% of the long-range connections, the small-world network model (p = 0.05) synchronizes as rapidly and achieves nearly the same final SI value as the mean-field model (p = 1). A small-world network model with fewer long-range connections (p = 0.01) yields lower SI values, and the nearest neighbor model (p = 0) produces poor synchronization. (D) Final SI values computed at the end of 8 cycle simulations and R values computed over the last 8 cycles of these simulations as a function of the probability p. For each p value, mean SI and R values (circles) and their standard deviations (error bars) were computed from the 10 network realizations. The large mean values and small standard deviations obtained for the small-world network model (p = 0.05, dashed line) show that additional shortcut connections (p > 0.05) do not enhance synchronization. (E) Final SI values obtained for small-world network models (p = 0.05) with different numbers of cells placed on the grid. For each cell number N, the mean SI value (circle) and its standard deviations (error bars) were computed from the 10 network realizations. The largest mean values and smallest standard deviations were obtained for N ≥ 225, demonstrating that the combination of large cell numbers and long-range connections enhances synchronization of heterogeneous cell populations.

Figure 3

The relationship between the network architecture and the resulting fractions of rhythmic cells and their period distributions. For each probability value, the mean (circle) and standard deviation (error bars) were computed from 10 network realizations. (A) The fraction of nonrhythmic cells as a function of the probability p. The population consisted of approximately 30% nonrhythmic cells for small p values, whereas the nonrhythmic population was completely eliminated in the small-world region (p = 0.05) where synchronization was enhanced. (B) The mean period reached a near maximal value of 24.2 h for p = 0.05 (dashed line) and decreased uniformly to 22.0 h for p = 0, suggesting that long-range connectivity improves the accuracy of circadian timekeeping. (C) The standard deviation of the period distribution reached a maximum of 2.5 h at p = 0 and decreased uniformly until a near minimal value of 0.8 h was reached at p = 0.05 (dashed line), suggesting that long-range connectivity enhances the precision of circadian timekeeping.

Figure 4

Percent increases in vasoactive intestinal polypeptide (VIP) and cAMP response element binding protein (CREB) oscillation amplitudes as a function of the probability p. The VIP (solid line) and CREB (dotted line) amplitudes obtained for the nearest neighbor model (p = 0) were used to scale the results. For each probability value, the mean (circle) and standard deviation (error bars) were computed from 10 network realizations. The mean amplitudes reached a maximum in the small-world region (p = 0.05, dashed line) and decreased uniformly with decreasing p.

Figure 5

The effect of different light schedules and the probability p on the synchronization index SI. For each light schedule and p value, the mean (circle) and standard deviation (error bars) were computed from 10 network realizations. A large increase in the mean SI value for constant darkness (solid line) was observed in the small-world region with a near maximal value obtained for p = 0.05 (dashed line). Constant light (dotted line) produced poor synchronization for all p values despite complete elimination of the nonrhythmic cell population present for constant darkness with p < 0.05, demonstrating a lack of synchronization among oscillating cells rather than a lack of rhythmicity at the single cell level. Schedules with alternating 12 h of light and darkness (dashed line) produced strong synchronization for all p values, suggesting that long-range connectivity may not be essential when light entrains the cell population.