A multiscale model to investigate circadian rhythmicity of pacemaker neurons in the suprachiasmatic nucleus

A minimal model of peripheral clocks reveals differential circadian re-entrainment in aging

The mammalian circadian system comprises a network of endogenous oscillators, spanning from the central clock in the brain to peripheral clocks in other organs. These clocks are tightly coordinated to orchestrate rhythmic physiological and behavioral functions. Dysregulation of these rhythms is a hallmark of aging, yet it remains unclear how age-related changes lead to more easily disrupted circadian rhythms. Using a two-population model of coupled oscillators that integrates the central clock and the peripheral clocks, we derive simple mean-field equations that can capture many aspects of the rich behavior found in the mammalian circadian system. We focus on three age-associated effects that have been posited to contribute to circadian misalignment: attenuated input from the sympathetic pathway, reduced responsiveness to light, and a decline in the expression of neurotransmitters. We find that the first two factors can significantly impede re-entrainment of the clocks following perturbation, while a weaker coupling within the central clock does not affect the recovery rate. Moreover, using our minimal model, we demonstrate the potential of using the feed-fast cycle as an effective intervention to accelerate circadian re-entrainment. These results highlight the importance of peripheral clocks in regulating the circadian rhythm and provide fresh insights into the complex interplay between aging and the resilience of the circadian system.

Limits of entrainment of circadian neuronal networks

Circadian rhythmicity lies at the center of various important physiological and behavioral processes in mammals, such as sleep, metabolism, homeostasis, mood changes, and more. Misalignment of intrinsic neuronal oscillations with the external day-night cycle can disrupt such processes and lead to numerous disorders. In this work, we computationally determine the limits of circadian synchronization to external light signals of different frequency, duty cycle, and simulated amplitude. Instead of modeling circadian dynamics with generic oscillator models (e.g., Kuramoto-type), we use a detailed computational neuroscience model, which integrates biomolecular dynamics, neuronal electrophysiology, and network effects. This allows us to investigate the effect of small drug molecules, such as Longdaysin, and connect our results with experimental findings. To combat the high dimensionality of such a detailed model, we employ a matrix-free approach, while our entire algorithmic pipeline enables numerical continuation and construction of bifurcation diagrams using only direct simulation. We, thus, computationally explore the effect of heterogeneity in the circadian neuronal network, as well as the effect of the corrective therapeutic intervention of Longdaysin. Last, we employ unsupervised learning to construct a data-driven embedding space for representing neuronal heterogeneity.

A Phenomenological Mouse Circadian Pacemaker Model

Mathematical models have been used extensively in chronobiology to explore characteristics of biological clocks. In particular, for human circadian studies, the Kronauer model has been modified multiple times to describe rhythm production and responses to sensory input. This phenomenological model comprises a single set of parameters which can simulate circadian responses in humans under a variety of environmental conditions. However, corresponding models for nocturnal rodents commonly used in circadian rhythm studies are not available and may require new parameter values for different species and even strains. Moreover, due to a considerable variation in experimental data collected from mice of the same strain, within and across laboratories, a range of valid parameters is essential. This study develops a Kronauer-like model for mice by re-fitting relevant parameters to published phase response curve and period data using total least squares. Local parameter sensitivity analysis and parameter distributions determine the parameter ranges that give a near-identical model and data distribution of periods. However, the model required further parameter adjustments to match characteristics of other mouse strains, implying that the model itself detects changes in the core processes of rhythm generation and control. The model is a useful tool to understand and interpret future mouse circadian clock experiments.

Network rewiring and plasticity promotes synchronization of suprachiasmatic nucleus neurons

In mammals, circadian rhythms throughout the body are orchestrated by the master clock in the hypothalamic suprachiasmatic nucleus (SCN), where SCN neurons are coupled with neurotransmitters to generate a uniform circadian rhythm. How the SCN circadian rhythm is so robust and flexible is, however, unclear. In this paper, we propose a temporal SCN network model and investigate the effects of dynamical rewiring and flexible coupling due to synaptic plasticity on the synchronization of the neural network in SCN. In networks consisting of simple Poincaré oscillators and complex circadian clocks, we found that dynamical rewiring and coupling plasticity enhance the synchronization in inhomogeneous networks. We verified the effect of enhanced synchronization in different architectures of random, scale-free, and small-world networks. A simple mean-field analysis for synchronization in plastic networks is proposed. Intuitively, the synchronization is greatly enhanced because both the random rewiring and coupling plasticity in the heterogeneous network have effectively increased the coupling strength in the whole network. Our results suggest that a proper network model for the master SCN circadian rhythm needs to take into account the effects of dynamical changes in topology and plasticity in neuron interactions that could help the brain to generate a robust circadian rhythm for the whole body.

Modelling the functional roles of synaptic and extra-synaptic γ-aminobutyric acid receptor dynamics in circadian timekeeping

In the suprachiasmatic nucleus (SCN), γ-aminobutyric acid (GABA) is a primary neurotransmitter. GABA can signal through two types of GABAA receptor subunits, often referred to as synaptic GABAA (gamma subunit) and extra-synaptic GABAA (delta subunit). To test the functional roles of these distinct GABAA in regulating circadian rhythms, we developed a multicellular SCN model where we could separately compare the effects of manipulating GABA neurotransmitter or receptor dynamics. Our model predicted that blocking GABA signalling modestly increased synchrony among circadian cells, consistent with published SCN pharmacology. Conversely, the model predicted that lowering GABAA receptor density reduced firing rate, circadian cell fraction, amplitude and synchrony among individual neurons. When we tested these predictions, we found that the knockdown of delta GABAA reduced the amplitude and synchrony of clock gene expression among cells in SCN explants. The model further predicted that increasing gamma GABAA densities could enhance synchrony, as opposed to increasing delta GABAA densities. Overall, our model reveals how blocking GABAA receptors can modestly increase synchrony, while increasing the relative density of gamma over delta subunits can dramatically increase synchrony. We hypothesize that increased gamma GABAA density in the winter could underlie the tighter phase relationships among SCN cells.

Astrocytic Modulation of Neuronal Activity in the Suprachiasmatic Nucleus: Insights from Mathematical Modeling

The suprachiasmatic nucleus (SCN) of the hypothalamus consists of a highly heterogeneous neuronal population networked together to allow precise and robust circadian timekeeping in mammals. While the critical importance of SCN neurons in regulating circadian rhythms has been extensively studied, the roles of SCN astrocytes in circadian system function are not well understood. Recent experiments have demonstrated that SCN astrocytes are circadian oscillators with the same functional clock genes as SCN neurons. Astrocytes generate rhythmic outputs that are thought to modulate neuronal activity through pre- and postsynaptic interactions. In this study, we developed an in silico multicellular model of the SCN clock to investigate the impact of astrocytes in modulating neuronal activity and affecting key clock properties such as circadian rhythmicity, period, and synchronization. The model predicted that astrocytes could alter the rhythmic activity of neurons via bidirectional interactions at tripartite synapses. Specifically, astrocyte-regulated extracellular glutamate was predicted to increase neuropeptide signaling from neurons. Consistent with experimental results, we found that astrocytes could increase the circadian period and enhance neural synchronization according to their endogenous circadian period. The impact of astrocytic modulation of circadian rhythm amplitude, period, and synchronization was predicted to be strongest when astrocytes had periods between 0 and 2 h longer than neurons. Increasing the number of neurons coupled to the astrocyte also increased its impact on period modulation and synchrony. These computational results suggest that signals that modulate astrocytic rhythms or signaling (e.g., as a function of season, age, or treatment) could cause disruptions in circadian rhythm or serve as putative therapeutic targets.

A New Theory of Gender Dysphoria Incorporating the Distress, Social Behavioral, and Body-Ownership Networks

When postmortem studies related to transgender individuals were first published, little was known about the function of the various identified nuclei. Now, over 2 decades later, significant progress has been made associating function with specific brain regions, as well as in identifying networks associated with groups of behaviors. However, much of this progress has not been integrated into the general conceptualization of gender dysphoria in humans.

When postmortem studies related to transgender individuals were first published, little was known about the function of the various identified nuclei. Now, over 2 decades later, significant progress has been made associating function with specific brain regions, as well as in identifying networks associated with groups of behaviors. However, much of this progress has not been integrated into the general conceptualization of gender dysphoria in humans. I hypothesize that in individuals with gender dysphoria, the aspects of chronic distress, gender atypical behavior, and incongruence between perception of gender identity and external primary sex characteristics are all directly related to functional differences in associated brain networks. I evaluated previously published neuroscience data related to these aspects and the associated functional networks, along with other relevant information. I find that the brain networks that give individuals their ownership of body parts, that influence gender typical behavior, and that are involved in chronic distress are different in individuals with and without gender dysphoria, leading to a new theory—that gender dysphoria is a sensory perception condition, an alteration in the sense of gender influenced by the reflexive behavioral responses associated with each of these networks. This theory builds upon previous work that supports the relevance of the body-ownership network and that questions the relevance of cerebral sexual dimorphism in regard to gender dysphoria. However, my theory uses a hierarchical executive function model to incorporate multiple reflexive factors (body ownership, gender typical/atypical behavior, and chronic distress) with the cognitive, reflective process of gender identity.

Multi-scale modeling of the circadian modulation of learning and memory

We propose a multi-scale model to explain the time-of-day effects on learning and memory. We specifically model the circadian variation of hippocampus (HC) dependent long-term potentiation (LTP), depression (LTD), and the fear conditioning paradigm in amygdala. The model we built has both Goodwin type circadian gene regulatory network (GRN) and the conductance model of Morris-Lecar (ML) type to explain the spontaneous firing patterns (SFR) in suprachiasmatic nucleus (SCN). In the conductance model, we also include N-Methyl-D-aspartic acid receptor (NMDAR) to study the circadian dependent changes in LTP/LTD in hippocampus and include both NMDAR and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) dynamics to explain the circadian modulation of fear conditioning paradigm in memory acquisition, recall, and extinction as seen in amygdala. Our multi-scale model captures the essential dynamics seen in the experiments and strongly supports the circadian time-of-the-day effects on learning and memory.

Protein sequestration versus Hill-type repression in circadian clock models

Circadian (∼24 h) clocks are self-sustained endogenous oscillators with which organisms keep track of daily and seasonal time. Circadian clocks frequently rely on interlocked transcriptional-translational feedback loops to generate rhythms that are robust against intrinsic and extrinsic perturbations. To investigate the dynamics and mechanisms of the intracellular feedback loops in circadian clocks, a number of mathematical models have been developed. The majority of the models use Hill functions to describe transcriptional repression in a way that is similar to the Goodwin model. Recently, a new class of models with protein sequestration-based repression has been introduced. Here, the author discusses how this new class of models differs dramatically from those based on Hill-type repression in several fundamental aspects: conditions for rhythm generation, robust network designs and the periods of coupled oscillators. Consistently, these fundamental properties of circadian clocks also differ among Neurospora, Drosophila, and mammals depending on their key transcriptional repression mechanisms (Hill-type repression or protein sequestration). Based on both theoretical and experimental studies, this review highlights the importance of careful modelling of transcriptional repression mechanisms in molecular circadian clocks.

Neuronal oscillations on an ultra-slow timescale: daily rhythms in electrical activity and gene expression in the mammalian master circadian clockwork

Neuronal oscillations of the brain, such as those observed in the cortices and hippocampi of behaving animals and humans, span across wide frequency bands, from slow delta waves (0.1 Hz) to ultra-fast ripples (600 Hz). Here, we focus on ultra-slow neuronal oscillators in the hypothalamic suprachiasmatic nuclei (SCN), the master daily clock that operates on interlocking transcription-translation feedback loops to produce circadian rhythms in clock gene expression with a period of near 24 h (< 0.001 Hz). This intracellular molecular clock interacts with the cell's membrane through poorly understood mechanisms to drive the daily pattern in the electrical excitability of SCN neurons, exhibiting an up-state during the day and a down-state at night. In turn, the membrane activity feeds back to regulate the oscillatory activity of clock gene programs. In this review, we emphasise the circadian processes that drive daily electrical oscillations in SCN neurons, and highlight how mathematical modelling contributes to our increasing understanding of circadian rhythm generation, synchronisation and communication within this hypothalamic region and across other brain circuits.

Calcium Circadian Rhythmicity in the Suprachiasmatic Nucleus: Cell Autonomy and Network Modulation

Circadian rhythms of mammalian physiology and behavior are coordinated by the suprachiasmatic nucleus (SCN) in the hypothalamus. Within SCN neurons, various aspects of cell physiology exhibit circadian oscillations, including circadian clock gene expression, levels of intracellular Ca²⁺ ([Ca²⁺]_i), and neuronal firing rate. [Ca²⁺]_i oscillates in SCN neurons even in the absence of neuronal firing. To determine the causal relationship between circadian clock gene expression and [Ca²⁺]_i rhythms in the SCN, as well as the SCN neuronal network dependence of [Ca²⁺]_i rhythms, we introduced GCaMP3, a genetically encoded fluorescent Ca²⁺ indicator, into SCN neurons from PER2::LUC knock-in reporter mice. Then, PER2 and [Ca²⁺]_i were imaged in SCN dispersed and organotypic slice cultures. In dispersed cells, PER2 and [Ca²⁺]_i both exhibited cell autonomous circadian rhythms, but [Ca²⁺]_i rhythms were typically weaker than PER2 rhythms. This result matches the predictions of a detailed mathematical model in which clock gene rhythms drive [Ca²⁺]_i rhythms. As predicted by the model, PER2 and [Ca²⁺]_i rhythms were both stronger in SCN slices than in dispersed cells and were weakened by blocking neuronal firing in slices but not in dispersed cells. The phase relationship between [Ca²⁺]_i and PER2 rhythms was more variable in cells within slices than in dispersed cells. Both PER2 and [Ca²⁺]_i rhythms were abolished in SCN cells deficient in the essential clock gene Bmal1. These results suggest that the circadian rhythm of [Ca²⁺]_i in SCN neurons is cell autonomous and dependent on clock gene rhythms, but reinforced and modulated by a synchronized SCN neuronal network.

Inhibitory and excitatory networks balance cell coupling in the suprachiasmatic nucleus: A modeling approach

Neuronal coupling contributes to circadian rhythms formation in the suprachiasmatic nucleus (SCN). While the neurotransmitter vasoactive intestinal polypeptide (VIP) is considered essential for synchronizing the oscillations of individual neurons, γ-aminobutyric acid (GABA) does not have a clear functional role despite being highly concentrated in the SCN. While most studies have examined the role of either GABA or VIP, our mathematical modeling approach explored their interplay on networks of SCN neurons. Tuning the parameters that control the release of GABA and VIP enabled us to optimize network synchrony, which was achieved at a peak firing rate during the subjective day of about 7Hz. Furthermore, VIP and GABA modulation could adjust network rhythm amplitude and period without sacrificing synchrony. We also performed simulations of SCN networks to phase shifts during 12h:12h light-dark cycles and showed that GABA networks reduced the average time for the SCN model to re-synchronize. We hypothesized that VIP and GABA balance cell coupling in the SCN to promote synchronization of heterogeneous oscillators while allowing flexibility for adjustment to environmental changes.

Diurnal Expression of the Per2 Gene and Protein in the Lateral Habenular Nucleus

The suprachiasmatic nucleus plays an important role in generating circadian rhythms in mammals. The lateral habenular nucleus (LHb) is closely linked to this structure. Interestingly, the LHb shows a rhythmic firing rate in vivo and in vitro, and sustained oscillation of rhythmic genes in vitro. However, under the in vivo condition, whether rhythmic gene expression in the LHb has circadian rhythms remains unknown. In this study, we examined LHb tissue in rats to determine Period2 (Per2) gene and protein expression at six zeitgeber time points (ZT2, ZT6, ZT10, ZT14, ZT18, and ZT22) in a 12-h light and 12-h dark (LD) environment. We found that in the LD environment, Per2 gene expression and PER2 protein levels in the LHb were higher in the day and lower in the night, showing periodic oscillation, with a peak at ZT10 and a trough at ZT22 (Per2 mRNA) and ZT18 (PER2 protein). We conclude that Per2 expression and PER2 protein levels in the LHb have rhythmic oscillation in vivo. This study provides a basis for further study on the role of the LHb in the circadian rhythm system.

Distinct roles for GABA across multiple timescales in mammalian circadian timekeeping

The suprachiasmatic nuclei (SCN), the central circadian pacemakers in mammals, comprise a multiscale neuronal system that times daily events. We use recent advances in graphics processing unit computing to generate a multiscale model for the SCN that resolves cellular electrical activity down to the timescale of individual action potentials and the intracellular molecular events that generate circadian rhythms. We use the model to study the role of the neurotransmitter GABA in synchronizing circadian rhythms among individual SCN neurons, a topic of much debate in the circadian community. The model predicts that GABA signaling has two components: phasic (fast) and tonic (slow). Phasic GABA postsynaptic currents are released after action potentials, and can both increase or decrease firing rate, depending on their timing in the interspike interval, a modeling hypothesis we experimentally validate; this allows flexibility in the timing of circadian output signals. Phasic GABA, however, does not significantly affect molecular timekeeping. The tonic GABA signal is released when cells become very excited and depolarized; it changes the excitability of neurons in the network, can shift molecular rhythms, and affects SCN synchrony. We measure which neurons are excited or inhibited by GABA across the day and find GABA-excited neurons are synchronized by-and GABA-inhibited neurons repelled from-this tonic GABA signal, which modulates the synchrony in the SCN provided by other signaling molecules. Our mathematical model also provides an important tool for circadian research, and a model computational system for the many multiscale projects currently studying brain function.

A model for the fast synchronous oscillations of firing rate in rat suprachiasmatic nucleus neurons cultured in a multielectrode array dish

When dispersed and cultured in a multielectrode dish (MED), suprachiasmatic nucleus (SCN) neurons express fast oscillations of firing rate (FOFR; fast relative to the circadian cycle), with burst duration ∼10 min, and interburst interval varying from 20 to 60 min in different cells but remaining nevertheless rather regular in individual cells. In many cases, separate neurons in distant parts of the 1 mm recording area of a MED exhibited correlated FOFR. Neither the mechanism of FOFR nor the mechanism of their synchronization among neurons is known. Based on recent data implicating vasoactive intestinal polypeptide (VIP) as a key intercellular synchronizing agent, we built a model in which VIP acts as both a feedback regulator to generate FOFR in individual neurons, and a diffusible synchronizing agent to produce coherent electrical output of a neuronal network. In our model, VIP binding to its (VPAC₂) receptors acts through G_s G-proteins to activate adenylyl cyclase (AC), increase intracellular cAMP, and open cyclic-nucleotide-gated (CNG) cation channels, thus depolarizing the cell and generating neuronal firing to release VIP. In parallel, slowly developing homologous desensitization and internalization of VPAC₂ receptors terminates elevation of cAMP and thereby provides an interpulse silent interval. Through mathematical modeling, we show that this VIP/VPAC₂/AC/cAMP/CNG-channel mechanism is sufficient for generating reliable FOFR in single neurons. When our model for FOFR is combined with a published model of synchronization of circadian rhythms based on VIP/VPAC₂ and Per gene regulation synchronization of circadian rhythms is significantly accelerated. These results suggest that (a) auto/paracrine regulation by VIP/VPAC₂ and intracellular AC/cAMP/CNG-channels are sufficient to provide robust FOFR and synchrony among neurons in a heterogeneous network, and (b) this system may also participate in synchronization of circadian rhythms.

Brain clock driven by neuropeptides and second messengers

The master circadian pacemaker in mammals is localized in a small portion of the brain called the suprachiasmatic nucleus (SCN). It is unclear how the SCN produces circadian rhythms. A common interpretation is that the SCN produces oscillations through the coupling of genetic oscillators in the neurons. The coupling is effected by a network of neuropeptides and second messengers. This network is crucial for the correct function of the SCN. However, models that study a possible oscillatory behavior of the network itself have received little attention. Here we propose and analyze a model to examine this oscillatory potential. We show that an intercellular oscillator emerges in the SCN as a result of the neuropeptide and second messenger dynamics. We find that this intercellular clock can produce circadian rhythms by itself with and without genetic clocks. We also found that the model is robust to perturbation of parameters and can be entrained by light-dark cycles.

It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms

Biological rhythms, generated by feedback loops containing interacting genes, proteins and/or cells, time physiological processes in many organisms. While many of the components of the systems that generate biological rhythms have been identified, much less is known about the details of their interactions. Using examples from the circadian (daily) clock in three organisms, Neurospora, Drosophila and mouse, we show, with mathematical models of varying complexity, how interactions among (i) promoter sites, (ii) proteins forming complexes, and (iii) cells can have a drastic effect on timekeeping. Inspired by the identification of many transcription factors, for example as involved in the Neurospora circadian clock, that can both activate and repress, we show how these multiple actions can cause complex oscillatory patterns in a transcription–translation feedback loop (TTFL). Inspired by the timekeeping complex formed by the NMO–PER–TIM–SGG complex that regulates the negative TTFL in the Drosophila circadian clock, we show how the mechanism of complex formation can determine the prevalence of oscillations in a TTFL. Finally, we note that most mathematical models of intracellular clocks model a single cell, but compare with experimental data from collections of cells. We find that refitting the most detailed model of the mammalian circadian clock, so that the coupling between cells matches experimental data, yields different dynamics and makes an interesting prediction that also matches experimental data: individual cells are bistable, and network coupling removes this bistability and causes the network to be more robust to external perturbations. Taken together, we propose that the interactions between components in biological timekeeping systems are carefully tuned towards proper function. We also show how timekeeping can be controlled by novel mechanisms at different levels of organization.

Molecular modeling of ErbB4/HER4 kinase in the context of the HER4 signaling network helps rationalize the effects of clinically identified HER4 somatic mutations on the cell phenotype

In the ErbB/HER family of receptor tyrosine kinases, the deregulation of the EGFR/ErbB1/HER1, HER2/ErbB2, and HER3/ErbB3 kinases is associated with several cancers, while the HER4/ErbB4 kinase has been shown to play an anti-carcinogenic role in certain tumors. We present molecular and network models of HER4/ErbB4 activation and signaling in order to elucidate molecular mechanisms of activation and rationalize the effects of the clinically identified HER4 somatic mutants. Our molecular-scale simulations identify the important role played by the interactions within the juxtamembrane region during the activation process. Our results also support the hypothesis that the HER4 mutants may heterodimerize but not activate, resulting in blockage of the HER4-STAT5 differentiation pathway, in favor of the proliferative PI3K/AKT pathway. Translating our molecular simulation results into a cellular pathway model of wild type versus mutant HER4 signaling, we are able to recapitulate the major features of the PI3K/AKT and JAK/STAT activation downstream of HER4. Our model predicts that the signaling downstream of the wild type HER4 is enriched for the JAK-STAT pathway, whereas downstream of the mutant HER4 is enriched for the PI3K/AKT pathway. HER4 mutations may hence constitute a cellular shift from a program of differentiation to that of proliferation.

Causes and consequences of hyperexcitation in central clock neurons

Hyperexcited states, including depolarization block and depolarized low amplitude membrane oscillations (DLAMOs), have been observed in neurons of the suprachiasmatic nuclei (SCN), the site of the central mammalian circadian (~24-hour) clock. The causes and consequences of this hyperexcitation have not yet been determined. Here, we explore how individual ionic currents contribute to these hyperexcited states, and how hyperexcitation can then influence molecular circadian timekeeping within SCN neurons. We developed a mathematical model of the electrical activity of SCN neurons, and experimentally verified its prediction that DLAMOs depend on post-synaptic L-type calcium current. The model predicts that hyperexcited states cause high intracellular calcium concentrations, which could trigger transcription of clock genes. The model also predicts that circadian control of certain ionic currents can induce hyperexcited states. Putting it all together into an integrative model, we show how membrane potential and calcium concentration provide a fast feedback that can enhance rhythmicity of the intracellular circadian clock. This work puts forward a novel role for electrical activity in circadian timekeeping, and suggests that hyperexcited states provide a general mechanism for linking membrane electrical dynamics to transcription activation in the nucleus.

Modeling the seasonal adaptation of circadian clocks by changes in the network structure of the suprachiasmatic nucleus

The dynamics of circadian rhythms needs to be adapted to day length changes between summer and winter. It has been observed experimentally, however, that the dynamics of individual neurons of the suprachiasmatic nucleus (SCN) does not change as the seasons change. Rather, the seasonal adaptation of the circadian clock is hypothesized to be a consequence of changes in the intercellular dynamics, which leads to a phase distribution of electrical activity of SCN neurons that is narrower in winter and broader during summer. Yet to understand this complex intercellular dynamics, a more thorough understanding of the impact of the network structure formed by the SCN neurons is needed. To that effect, we propose a mathematical model for the dynamics of the SCN neuronal architecture in which the structure of the network plays a pivotal role. Using our model we show that the fraction of long-range cell-to-cell connections and the seasonal changes in the daily rhythms may be tightly related. In particular, simulations of the proposed mathematical model indicate that the fraction of long-range connections between the cells adjusts the phase distribution and consequently the length of the behavioral activity as follows: dense long-range connections during winter lead to a narrow activity phase, while rare long-range connections during summer lead to a broad activity phase. Our model is also able to account for the experimental observations indicating a larger light-induced phase-shift of the circadian clock during winter, which we show to be a consequence of higher synchronization between neurons. Our model thus provides evidence that the variations in the seasonal dynamics of circadian clocks can in part also be understood and regulated by the plasticity of the SCN network structure.

Circadian clocks drive the temporal coordination of internal biological processes, which in turn determine daily rhythms in physiology and behavior in the most diverse organisms. In mammals, the 24-hour timing clock resides in the suprachiasmatic nucleus (SCN) of the hypothalamus. The SCN is a network of interconnected neurons that serves as a robust self-sustained circadian pacemaker. The electrical activity of these neurons and their synchronization with the 24-hour cycle is established via the environmental day and night cycles. Apart from daily luminance changes, mammals are exposed to seasonal day length changes as well. Remarkably, it has been shown experimentally that the seasonal adaptations to different photoperiods are related to the modifications of the neuronal activity of the SCN due to the plasticity of the network. In our paper, by developing a mathematical model of the SCN architecture, we explore in depth the role of the structure of this important neuronal network. We show that the redistribution of the neuronal activity during winter and summer can in part be explained by structural changes of the network. Interestingly, the alterations of the electrical activity patterns can be related with small-world properties of our proposed SCN network.

Modeling the emergence of circadian rhythms in a clock neuron network

Circadian rhythms in pacemaker cells persist for weeks in constant darkness, while in other types of cells the molecular oscillations that underlie circadian rhythms damp rapidly under the same conditions. Although much progress has been made in understanding the biochemical and cellular basis of circadian rhythms, the mechanisms leading to damped or self-sustained oscillations remain largely unknown. There exist many mathematical models that reproduce the circadian rhythms in the case of a single cell of the Drosophila fly. However, not much is known about the mechanisms leading to coherent circadian oscillation in clock neuron networks. In this work we have implemented a model for a network of interacting clock neurons to describe the emergence (or damping) of circadian rhythms in Drosophila fly, in the absence of zeitgebers. Our model consists of an array of pacemakers that interact through the modulation of some parameters by a network feedback. The individual pacemakers are described by a well-known biochemical model for circadian oscillation, to which we have added degradation of PER protein by light and multiplicative noise. The network feedback is the PER protein level averaged over the whole network. In particular, we have investigated the effect of modulation of the parameters associated with (i) the control of net entrance of PER into the nucleus and (ii) the non-photic degradation of PER. Our results indicate that the modulation of PER entrance into the nucleus allows the synchronization of clock neurons, leading to coherent circadian oscillations under constant dark condition. On the other hand, the modulation of non-photic degradation cannot reset the phases of individual clocks subjected to intrinsic biochemical noise.

Circadian regulation of sleep-wake behaviour in nocturnal rats requires multiple signals from suprachiasmatic nucleus

The dynamics of sleep and wake are strongly linked to the circadian clock. Many models have accurately predicted behaviour resulting from dynamic interactions between these two systems without specifying physiological substrates for these interactions. By contrast, recent experimental work has identified much of the relevant physiology for circadian and sleep-wake regulation, but interaction dynamics are difficult to study experimentally. To bridge these approaches, we developed a neuronal population model for the dynamic, bidirectional, neurotransmitter-mediated interactions of the sleep-wake and circadian regulatory systems in nocturnal rats. This model proposes that the central circadian pacemaker, located within the suprachiasmatic nucleus (SCN) of the hypothalamus, promotes sleep through single neurotransmitter-mediated signalling to sleep-wake regulatory populations. Feedback projections from these populations to the SCN alter SCN firing patterns and fine-tune this modulation. Although this model reproduced circadian variation in sleep-wake dynamics in nocturnal rats, it failed to describe the sleep-wake dynamics observed in SCN-lesioned rats. We thus propose two alternative, physiologically based models in which neurotransmitter- and neuropeptide-mediated signalling from the SCN to sleep-wake populations introduces mechanisms to account for the behaviour of both the intact and SCN-lesioned rat. These models generate testable predictions and offer a new framework for modelling sleep-wake and circadian interactions.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

A multicellular model for differential regulation of circadian signals in the core and shell regions of the suprachiasmatic nucleus

We developed a multicellular model of the mammalian circadian clock characterized by a high degree of heterogeneity with respect to single cell periodicity and behavior (intrinsic and driven oscillators), neurotransmitter release (VIP, GABA and glutamate synthesis) and spatial organization (core and shell regions), mimicking structural patterns within the suprachiasmatic nucleus (SCN) associated with distinct circadian functions. We simulated the SCN core and shell separately utilizing experimentally derived connectivity schemes for the two subdivisions as observed within the rat SCN. The core was modeled via a small world network characterized by VIP and GABA co-localization, whereas the shell was simulated as a nearest neighbor network promoting local GABAergic connections. To study the function of the axonal plexus extending from the densely innervated ventrolateral region to distal areas across the dorsomedial SCN, directed long range links from the core to the shell were gradually introduced via a probability p(cs) that ranged from 0 to 1. A probability value of 0 excluded core-shell interactions, whereas p(cs)=1 achieved maximal connectivity between the two regions. Our model exhibited a threshold in the number of core-to-shell links required for sufficient cell-to-cell coordination to maintain periodicity and rhythmic behavior across the entire model network (including both shell and core populations) in constant darkness as well as 12:12h light-dark cycles. By contrast, constant light was shown to increase phase synchronization across the shell while core populations remained poorly synchronized, suggesting differential light response across the two SCN compartments. We further simulated increasing percentages of intrinsic oscillators and demonstrated a negative correlation between the number of intrinsic oscillators distributed across the SCN and the ability of the system to produce synchronized signals. Simulations that differed with respect to the placement of intrinsic oscillators supported the hypothesis that improved synchronization is achieved with networks characterized by localized intrinsic oscillators placed exclusively within the shell versus networks containing uniformly distributed intrinsic oscillators in both SCN compartments. This study has successfully reproduced a number of spatiotemporal and behavioral attributes of the SCN, providing a useful computational tool to correlate observed circadian phenotypes with distinct chemoarchitectural properties of spatially localized neural populations.

Multicellular model for intercellular synchronization in circadian neural networks

We developed a multicellular model characterized by a high degree of heterogeneity to investigate possible mechanisms that underlie circadian network synchronization and rhythmicity in the suprachiasmatic nucleus (SCN). We populated a two-dimensional grid with 400 model neurons coupled via γ-aminobutyric acid (GABA) and vasoactive intestinal polypeptide (VIP) neurotransmitters through a putative Ca(2+) mediated signaling cascade to investigate their roles in gene expression and electrical firing activity of cell populations. As observed experimentally, our model predicted that GABA would affect the amplitude of circadian oscillations but not synchrony among individual oscillators. Our model recapitulated experimental findings of decreased synchrony and average periods, loss of rhythmicity, and reduced circadian amplitudes as VIP signaling was eliminated. In addition, simulated increases of VIP reduced periodicity and synchrony. We therefore postulated a physiological range of VIP within which the system is able to produce sustained and synchronized oscillations. Our model recapitulated experimental findings of diminished amplitudes and periodicity with decreasing intracellular Ca(2+) concentrations, suggesting that such behavior could be due to simultaneous decrease of individual oscillation amplitudes and population synchrony. Simulated increases in Cl(-) levels resulted in increased Cl(-) influx into the cytosol, a decrease of inhibitory postsynaptic currents, and ultimately a shift of GABA-elicited responses from inhibitory to excitatory. The simultaneous reduction of IPSCs and increase in membrane resting potential produced GABA dose-dependent increases in firing rates across the population, as has been observed experimentally. By integrating circadian gene regulation and electrophysiology with intracellular and intercellular signaling, we were able to develop the first (to our knowledge) multicellular model that allows the effects of clock genes, electrical firing, Ca(2+), GABA, and VIP on circadian system behavior to be predicted.

A multiscale modeling approach to investigate molecular mechanisms of pseudokinase activation and drug resistance in the HER3/ErbB3 receptor tyrosine kinase signaling network

Multiscale modeling provides a powerful and quantitative platform for investigating the complexity inherent in intracellular signaling pathways and rationalizing the effects of molecular perturbations on downstream signaling events and ultimately, on the cell phenotype. Here we describe the application of a multiscale modeling scheme to the HER3/ErbB3 receptor tyrosine kinase (RTK) signaling network, which regulates critical cellular processes including proliferation, migration and differentiation. The HER3 kinase is a topic of current interest and investigation, as it has been implicated in mechanisms of resistance to tyrosine kinase inhibition (TKI) of EGFR and HER2 in the treatment of many human malignancies. Moreover, the commonly regarded status of HER3 as a catalytically inactive 'pseudokinase' has recently been challenged by our previous study, which demonstrated robust residual kinase activity for HER3. Through our multiscale model, we investigate the most significant molecular interactions that contribute to potential mechanisms of HER3 activity and the physiological relevance of this activity to mechanisms of drug resistance in an ErbB-driven tumor cell in silico. The results of our molecular-scale simulations support the characterization of HER3 as a weakly active kinase that, in contrast to its fully-active ErbB family members, depends upon a unique hydrophobic interface to coordinate the alignment of specific catalytic residues required for its activity. Translating our molecular simulation results of the uniquely active behavior of the HER3 kinase into a physiologically relevant environment, our HER3 signaling model demonstrates that even a weak level of HER3 activity may be sufficient to induce AKT signaling and TKI resistance in the context of an ErbB signaling-dependent tumor cell, and therefore therapeutic targeting of HER3 may represent a superior treatment strategy for specific ErbB-driven cancers.

Circadian regulation of cardiovascular function: a role for vasoactive intestinal peptide

The circadian system, driven by the suprachiasmatic nucleus (SCN), regulates properties of cardiovascular function. The dysfunction of this timing system can result in cardiac pathology. The neuropeptide vasoactive intestinal peptide (VIP) is crucial for circadian rhythms in a number of biological processes including SCN electrical activity and wheel running behavior. Anatomic evidence indicates that SCN neurons expressing VIP are well positioned to drive circadian regulation of cardiac function through interactions with the autonomic centers. In this study, we tested the hypothesis that loss of VIP would result in circadian deficits in heart rate (HR) and clock gene expression in cardiac tissue. We implanted radiotelemetry devices into VIP-deficient mice and wild-type (WT) controls and continuously recorded HR, body temperature, and cage activity in freely moving mice. Under light-dark conditions, VIP-deficient mice displayed weak rhythms in HR, body temperature, and cage activity, with onsets that were advanced in phase compared with WT mice. Similarly, clock gene expression in cardiac tissue was rhythmic but phase advanced in mutant mice. In constant darkness, the normal circadian rhythms in HR were lost in VIP-deficient mice; however, most mutant mice continued to exhibit circadian rhythms of body temperature with shortened free-running period. The loss of VIP altered, but did not abolish, autonomic regulation of HR. Analysis of the echocardiograms did not find any evidence for a loss of cardiac function in VIP-deficient mice, and the size of the hearts did not differ between genotypes. These results demonstrate that VIP is an important regulator of physiological circadian rhythmicity in the heart.