Stochastic Modeling of System Function in a Network of Biological Oscillators

Many living organisms have evolved to anticipate daily circadian cycles and changing seasons of their environment. In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus, a brain region of about 20,000 neurons, serves as the master circadian clock coordinating timing throughout the body and entraining to daily external light cycles. The remarkable precision of the SCN clock relies on intercellular signaling. In its absence, each SCN neuron and the SCN as a whole have significantly less stable oscillations. Though there are candidate signaling neuropeptides and anatomical surveys of the SCN, it is still unknown how the SCN as a whole responds to changes in the environment and regulates function in the body. We model the unstable oscillations in individual cells by developing a stochastic model based on the cell clock's gene regulatory network, then investigate the intercellular signaling properties of the SCN to understand its behavior as a whole. Though many existing deterministic models contain details of the gene regulation in the cell, their output has been compared to the behavior of the SCN as a whole, rather than to individual cells. Characterizing properties of individual cells such as period, phase, and synchronization is challenging due to their non-linear and unstable oscillations. We developed a wavelet analysis method to characterize cell behavior in biological experiments and compare with stochastic cell models. This analysis led to an examination of how period distributions could be influenced by stochastic fluctuations in a nonlinear cell oscillator model, and a hypothesis that the poor or strong oscillators observed in biological experiments could be a stochastic oscillator operating near a bifurcation point, between non-oscillatory and oscillatory conditions.It was observed in SCN tissue and in the SCN stochastic model that the oscillator is less likely to shift phase in response to a vasoactive intestinal polypeptide (VIP) dose at circadian time (CT4) than at other times. A reexamination of the behavior of the SCN as a whole, when modeled as linked stochastic oscillators, led to the hypothesis that the cells of the SCN synchronize to each other using a ``phase tumbling'' process. Our hypothesis is that the SCN synchronizes by its cells shifting with a wide phase distribution when they are perturbed at phases not near CT4. Rather than shifting in a deterministic manner, where all the cells stay synchronized and shift together to a new light schedule, they instead temporarily desynchronize then reorganize aligned to the new light/dark cycle. Within a few cycles the system can rapidly shift to a new light schedule. This rapid re-entrainment to both new light/dark and temperature schedules was confirmed in mice by first desynchronizing the SCN using a neuropeptide that has been considered a synchronizing agent, vasoactive intestinal polypeptide, or by a brief bright light exposure before exposing the animals to a new shifted schedule.Finally, since the behavior of the SCN as a whole may depend on the network topology of its intercellular connections, we applied an information theoretic measure to infer pairwise functional connections between neurons in the SCN. We first validated the method on several model networks. After inferring connection networks of three SCN's, we modeled those networks in our stochastic SCN model and confirmed that we could re-infer the bio-inspired networks. We found that the SCN, at least for these experimental samples, appears to have a small-world network topology and is scale-free. We hope that our results have helped to illuminate how stochastic fluctuations in the SCN system contribute to its behavior

Stochastic Modeling of System Function in a Network of Biological Oscillators

Many living organisms have evolved to anticipate daily circadian cycles and changing seasons of their environment. In mammals, the suprachiasmatic nucleus (SCN) of the hypothalamus, a brain region of about 20,000 neurons, serves as the master circadian clock coordinating timing throughout the body and entraining to daily external light cycles. The remarkable precision of the SCN clock relies on intercellular signaling. In its absence, each SCN neuron and the SCN as a whole have significantly less stable oscillations. Though there are candidate signaling neuropeptides and anatomical surveys of the SCN, it is still unknown how the SCN as a whole responds to changes in the environment and regulates function in the body. We model the unstable oscillations in individual cells by developing a stochastic model based on the cell clock's gene regulatory network, then investigate the intercellular signaling properties of the SCN to understand its behavior as a whole. Though many existing deterministic models contain details of the gene regulation in the cell, their output has been compared to the behavior of the SCN as a whole, rather than to individual cells. Characterizing properties of individual cells such as period, phase, and synchronization is challenging due to their non-linear and unstable oscillations. We developed a wavelet analysis method to characterize cell behavior in biological experiments and compare with stochastic cell models. This analysis led to an examination of how period distributions could be influenced by stochastic fluctuations in a nonlinear cell oscillator model, and a hypothesis that the poor or strong oscillators observed in biological experiments could be a stochastic oscillator operating near a bifurcation point, between non-oscillatory and oscillatory conditions.It was observed in SCN tissue and in the SCN stochastic model that the oscillator is less likely to shift phase in response to a vasoactive intestinal polypeptide (VIP) dose at circadian time (CT4) than at other times. A reexamination of the behavior of the SCN as a whole, when modeled as linked stochastic oscillators, led to the hypothesis that the cells of the SCN synchronize to each other using a ``phase tumbling'' process. Our hypothesis is that the SCN synchronizes by its cells shifting with a wide phase distribution when they are perturbed at phases not near CT4. Rather than shifting in a deterministic manner, where all the cells stay synchronized and shift together to a new light schedule, they instead temporarily desynchronize then reorganize aligned to the new light/dark cycle. Within a few cycles the system can rapidly shift to a new light schedule. This rapid re-entrainment to both new light/dark and temperature schedules was confirmed in mice by first desynchronizing the SCN using a neuropeptide that has been considered a synchronizing agent, vasoactive intestinal polypeptide, or by a brief bright light exposure before exposing the animals to a new shifted schedule.Finally, since the behavior of the SCN as a whole may depend on the network topology of its intercellular connections, we applied an information theoretic measure to infer pairwise functional connections between neurons in the SCN. We first validated the method on several model networks. After inferring connection networks of three SCN's, we modeled those networks in our stochastic SCN model and confirmed that we could re-infer the bio-inspired networks. We found that the SCN, at least for these experimental samples, appears to have a small-world network topology and is scale-free. We hope that our results have helped to illuminate how stochastic fluctuations in the SCN system contribute to its behavior

Synchrony and entrainment properties of robust circadian oscillators

Systems theoretic tools (i.e. mathematical modelling, control, and feedback design) advance the understanding of robust performance in complex biological networks. We highlight phase entrainment as a key performance measure used to investigate dynamics of a single deterministic circadian oscillator for the purpose of generating insight into the behaviour of a population of (synchronized) oscillators. More specifically, the analysis of phase characteristics may facilitate the identification of appropriate coupling mechanisms for the ensemble of noisy (stochastic) circadian clocks. Phase also serves as a critical control objective to correct mismatch between the biological clock and its environment. Thus, we introduce methods of investigating synchrony and entrainment in both stochastic and deterministic frameworks, and as a property of a single oscillator or population of coupled oscillators

A neuropeptide speeds circadian entrainment by reducing intercellular synchrony.

Shift work or transmeridian travel can desynchronize the body's circadian rhythms from local light-dark cycles. The mammalian suprachiasmatic nucleus (SCN) generates and entrains daily rhythms in physiology and behavior. Paradoxically, we found that vasoactive intestinal polypeptide (VIP), a neuropeptide implicated in synchrony among SCN cells, can also desynchronize them. The degree and duration of desynchronization among SCN neurons depended on both the phase and the dose of VIP. A model of the SCN consisting of coupled stochastic cells predicted both the phase- and the dose-dependent response to VIP and that the transient phase desynchronization, or "phase tumbling", could arise from intrinsic, stochastic noise in small populations of key molecules (notably, Period mRNA near its daily minimum). The model also predicted that phase tumbling following brief VIP treatment would accelerate entrainment to shifted environmental cycles. We tested this using a prepulse of VIP during the day before a shift in either a light cycle in vivo or a temperature cycle in vitro. Although VIP during the day does not shift circadian rhythms, the VIP pretreatment approximately halved the time required for mice to reentrain to an 8-h shifted light schedule and for SCN cultures to reentrain to a 10-h shifted temperature cycle. We conclude that VIP below 100 nM synchronizes SCN cells and above 100 nM reduces synchrony in the SCN. We show that exploiting these mechanisms that transiently reduce cellular synchrony before a large shift in the schedule of daily environmental cues has the potential to reduce jet lag

A neuropeptide speeds circadian entrainment by reducing intercellular synchrony.

Shift work or transmeridian travel can desynchronize the body's circadian rhythms from local light-dark cycles. The mammalian suprachiasmatic nucleus (SCN) generates and entrains daily rhythms in physiology and behavior. Paradoxically, we found that vasoactive intestinal polypeptide (VIP), a neuropeptide implicated in synchrony among SCN cells, can also desynchronize them. The degree and duration of desynchronization among SCN neurons depended on both the phase and the dose of VIP. A model of the SCN consisting of coupled stochastic cells predicted both the phase- and the dose-dependent response to VIP and that the transient phase desynchronization, or "phase tumbling", could arise from intrinsic, stochastic noise in small populations of key molecules (notably, Period mRNA near its daily minimum). The model also predicted that phase tumbling following brief VIP treatment would accelerate entrainment to shifted environmental cycles. We tested this using a prepulse of VIP during the day before a shift in either a light cycle in vivo or a temperature cycle in vitro. Although VIP during the day does not shift circadian rhythms, the VIP pretreatment approximately halved the time required for mice to reentrain to an 8-h shifted light schedule and for SCN cultures to reentrain to a 10-h shifted temperature cycle. We conclude that VIP below 100 nM synchronizes SCN cells and above 100 nM reduces synchrony in the SCN. We show that exploiting these mechanisms that transiently reduce cellular synchrony before a large shift in the schedule of daily environmental cues has the potential to reduce jet lag

Functional network inference of the suprachiasmatic nucleus

In the mammalian suprachiasmatic nucleus (SCN), noisy cellular oscillators communicate within a neuronal network to generate precise system-wide circadian rhythms. Although the intracellular genetic oscillator and intercellular biochemical coupling mechanisms have been examined previously, the network topology driving synchronization of the SCN has not been elucidated. This network has been particularly challenging to probe, due to its oscillatory components and slow coupling timescale. In this work, we investigated the SCN network at a single-cell resolution through a chemically induced desynchronization. We then inferred functional connections in the SCN by applying the maximal information coefficient statistic to bioluminescence reporter data from individual neurons while they resynchronized their circadian cycling. Our results demonstrate that the functional network of circadian cells associated with resynchronization has small-world characteristics, with a node degree distribution that is exponential. We show that hubs of this small-world network are preferentially located in the central SCN, with sparsely connected shells surrounding these cores. Finally, we used two computational models of circadian neurons to validate our predictions of network structure