Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

动态神经网络中的同步振荡

目前有一种假设认为同一视觉对象是由一群神经元的同步振荡活动来表征的。这一神经元发放活动的时间特性,是解决视觉信息处理中“结合问题（Ｂｉｎｄｉｎｇｐｒｏｂｌｅｍ）”的可能机制。本文用我们所提出的一种简化现实性神经网络模型［１］所构造的时滞非线性振子网络［２］,模拟生物神经网络的同步振荡活动。并考虑了振子各参数的设置与振荡活动的关系,以及网络振子间耦联对同步活动的影响．     

Noise Induces Oscillation and Synchronization of the Circadian Neurons

The principle clock of mammals, named suprachiasmatic nucleus (SCN), coordinates the circadian rhythms of behavioral and physiological activity to the external 24 h light-dark cycle. In the absence of the daily cycle, the SCN acts as an endogenous clock that regulates the ~24h rhythm of activity. Experimental and theoretical studies usually take the light-dark cycle as a main external influence, and often ignore light pollution as an external influence. However, in modern society, the light pollution such as induced by electrical lighting influences the circadian clock. In the present study, we examined the effect of external noise (light pollution) on the collective behavior of coupled circadian oscillators under constant darkness using a Goodwin model. We found that the external noise plays distinct roles in the network behavior of neurons for weak or strong coupling between the neurons. In the case of strong coupling, the noise reduces the synchronization and the period of the SCN network. Interestingly, in the case of weak coupling, the noise induces a circadian rhythm in the SCN network which is absent in noise-free condition. In addition, the noise increases the synchronization and decreases the period of the SCN network. Our findings may shed new light on the impact of the external noise on the collective behavior of SCN neurons.     

Forced desynchronization of dual circadian oscillators within the rat suprachiasmatic nucleus

The circadian clock in the suprachiasmatic nucleus of the hypothalamus (SCN) contains multiple autonomous single-cell circadian oscillators and their basic intracellular oscillatory mechanism is beginning to be identified. Less well understood is how individual SCN cells create an integrated tissue pacemaker that produces a coherent read-out to the rest of the organism. Intercellular coupling mechanisms must coordinate individual cellular periods to generate the averaged, genotype-specific circadian period of whole animals. To noninvasively dissociate this circadian oscillatory network in vivo, we (T.C. and A.D.-N.) have developed an experimental paradigm that exposes animals to exotic light-dark (LD) cycles with periods close to the limits of circadian entrainment. If individual oscillators with different periods are loosely coupled within the network, perhaps some of them would be synchronized to the external cycle while others remain unentrained. In fact, rats exposed to an artificially short 22 hr LD cycle express two stable circadian motor activity rhythms with different period lengths in individual animals. Our analysis of SCN gene expression under such conditions suggests that these two motor activity rhythms reflect the separate activities of two oscillators in the anatomically defined ventrolateral and dorsomedial SCN subdivisions. Our "forced desychronization" protocol has allowed the first stable separation of these two regional oscillators in vivo, correlating their activities to distinct behavioral outputs, and providing a powerful approach for understanding SCN tissue organization and signaling mechanisms in behaving animals.     

Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus

The suprachiasmatic nuclei (SCN) host a robust, self-sustained circadian pacemaker that coordinates physiological rhythms with the daily changes in the environment. Neuronal clocks within the SCN form a heterogeneous network that must synchronize to maintain timekeeping activity. Coherent circadian output of the SCN tissue is established by intercellular signaling factors, such as vasointestinal polypeptide. It was recently shown that besides coordinating cells, the synchronization factors play a crucial role in the sustenance of intrinsic cellular rhythmicity. Disruption of intercellular signaling abolishes sustained rhythmicity in a majority of neurons and desynchronizes the remaining rhythmic neurons. Based on these observations, the authors propose a model for the synchronization of circadian oscillators that combines intracellular and intercellular dynamics at the single-cell level. The model is a heterogeneous network of circadian neuronal oscillators where individual oscillators are damped rather than self-sustained. The authors simulated different experimental conditions and found that: (1) in normal, constant conditions, coupled circadian oscillators quickly synchronize and produce a coherent output; (2) in large populations, such oscillators either synchronize or gradually lose rhythmicity, but do not run out of phase, demonstrating that rhythmicity and synchrony are codependent; (3) the number of oscillators and connectivity are important for these synchronization properties; (4) slow oscillators have a higher impact on the period in mixed populations; and (5) coupled circadian oscillators can be efficiently entrained by light–dark cycles. Based on these results, it is predicted that: (1) a majority of SCN neurons needs periodic synchronization signal to be rhythmic; (2) a small number of neurons or a low connectivity results in desynchrony; and (3) amplitudes and phases of neurons are negatively correlated. The authors conclude that to understand the orchestration of timekeeping in the SCN, intracellular circadian clocks cannot be isolated from their intercellular communication components.     

Spontaneous synchronization of coupled circadian oscillators

In mammals, the circadian pacemaker, which controls daily rhythms, is located in the suprachiasmatic nucleus (SCN). Circadian oscillations are generated in individual SCN neurons by a molecular regulatory network. Cells oscillate with periods ranging from 20 to 28 h, but at the tissue level, SCN neurons display significant synchrony, suggesting a robust intercellular coupling in which neurotransmitters are assumed to play a crucial role. We present a dynamical model for the coupling of a population of circadian oscillators in the SCN. The cellular oscillator, a three-variable model, describes the core negative feedback loop of the circadian clock. The coupling mechanism is incorporated through the global level of neurotransmitter concentration. Global coupling is efficient to synchronize a population of 10,000 cells. Synchronized cells can be entrained by a 24-h light-dark cycle. Simulations of the interaction between two populations representing two regions of the SCN show that the driven population can be phase-leading. Experimentally testable predictions are: 1), phases of individual cells are governed by their intrinsic periods; and 2), efficient synchronization is achieved when the average neurotransmitter concentration would dampen individual oscillators. However, due to the global neurotransmitter oscillation, cells are effectively synchronized.     

Intercellular coupling confers robustness against mutations in the SCN circadian clock network

Molecular mechanisms of the mammalian circadian clock have been studied primarily by genetic perturbation and behavioral analysis. Here, we used bioluminescence imaging to monitor Per2 gene expression in tissues and cells from clock mutant mice. We discovered that Per1 and Cry1 are required for sustained rhythms in peripheral tissues and cells, and in neurons dissociated from the suprachiasmatic nuclei (SCN). Per2 is also required for sustained rhythms, whereas Cry2 and Per3 deficiencies cause only period length defects. However, oscillator network interactions in the SCN can compensate for Per1 or Cry1 deficiency, preserving sustained rhythmicity in mutant SCN slices and behavior. Thus, behavior does not necessarily reflect cell-autonomous clock phenotypes. Our studies reveal previously unappreciated requirements for Per1, Per2, and Cry1 in sustaining cellular circadian rhythmicity and demonstrate that SCN intercellular coupling is essential not only to synchronize component cellular oscillators but also for robustness against genetic perturbations.     

A network of (autonomic) clock outputs

The circadian clock in the suprachiasmatic nuclei (SCN) is composed of thousands of oscillator neurons, each of which is dependent on the cell-autonomous action of a defined set of circadian clock genes. A major question is still how these individual oscillators are organized into a biological clock producing a coherent output that is able to time all the different daily changes in behavior and physiology. We investigated which anatomical connections and neurotransmitters are used by the biological clock to control the daily release pattern of a number of hormones. The picture that emerged shows projections contacting target neurons in the medial hypothalamus surrounding the SCN. The activity of these pre-autonomic and neuro-endocrine target neurons is controlled by differentially timed waves of, among others, vasopressin, GABA, and glutamate release from SCN terminals. Together our data indicate that, with regard to the timing of their main release period within the light-dark (LD) cycle, at least 4 subpopulations of SCN neurons should be discerned. The different subgroups do not necessarily follow the phenotypic differences among SCN neurons. Thus, different subgroups can be found within neuron populations containing the same neurotransmitter. Remarkably, a similar distinction of 4 differentially timed subpopulations of SCN neurons was recently also discovered in experiments determining the temporal patterns of rhythmicity in individual SCN neurons by way of the electrophysiology or clock gene expression. Moreover, the specialization of the SCN may go as far as a single body structure; i.e., the SCN seems to contain neurons that specifically target the liver, pineal, and adrenal.     

A Network of (Autonomic) Clock Outputs

The circadian clock in the suprachiasmatic nuclei (SCN) is composed of thousands of oscillator neurons, each of which is dependent on the cell‐autonomous action of a defined set of circadian clock genes. A major question is still how these individual oscillators are organized into a biological clock producing a coherent output that is able to time all the different daily changes in behavior and physiology. We investigated which anatomical connections and neurotransmitters are used by the biological clock to control the daily release pattern of a number of hormones. The picture that emerged shows projections contacting target neurons in the medial hypothalamus surrounding the SCN. The activity of these pre‐autonomic and neuro‐endocrine target neurons is controlled by differentially timed waves of, among others, vasopressin, GABA, and glutamate release from SCN terminals. Together our data indicate that, with regard to the timing of their main release period within the light‐dark (LD) cycle, at least 4 subpopulations of SCN neurons should be discerned. The different subgroups do not necessarily follow the phenotypic differences among SCN neurons. Thus, different subgroups can be found within neuron populations containing the same neurotransmitter. Remarkably, a similar distinction of 4 differentially timed subpopulations of SCN neurons was recently also discovered in experiments determining the temporal patterns of rhythmicity in individual SCN neurons by way of the electrophysiology or clock gene expression. Moreover, the specialization of the SCN may go as far as a single body structure; i.e., the SCN seems to contain neurons that specifically target the liver, pineal, and adrenal.     

Photic desynchronization of two subgroups of circadian oscillators in a network model of the suprachiasmatic nucleus with dispersed coupling strengths

The suprachiasmatic nucleus (SCN) is the master circadian clock in mammals and is composed of thousands of neuronal oscillators expressing different intrinsic periods. These oscillators form a coupled network with a free-running period around 24 h in constant darkness and entrainable to the external light-dark cycle (T cycle). Coupling plays an important role in setting the period of the network and its range of entrainment. Experiments in rats have shown that two subgroups of oscillators within the SCN, a ventrolateral (VL) subgroup that receives photic input and a dorsomedial (DM) subgroup that is coupled to VL, can be desynchronized under a short (22-h) T cycle, with VL entrained to the cycle and DM free-running. We use a modified Goodwin model to understand how entrainment of the subgroups to short (22-h) and long (26-h) T cycles is influenced by light intensity, the proportion of neurons that receives photic input, and coupling heterogeneity. We find that the model's critical value for the proportion of photically-sensitive neurons is in accord with actual experimental estimates, while the model's inclusion of dispersed coupling can account for the experimental observation that VL and DM desynchronize more readily under the 22-h than under the 26-h T cycle. Heterogeneous intercellular coupling within the SCN is likely central to the generation of complex behavioral patterns.     

A network of (autonomic) clock outputs

The circadian clock in the suprachiasmatic nuclei (SCN) is composed of thousands of oscillator neurons, each dependent on the cell-autonomous action of a defined set of circadian clock genes. A major question is still how these individual oscillators are organized into a biological clock that produces a coherent output capable of timing all the different daily changes in behavior and physiology. We investigated which anatomical connections and neurotransmitters are used by the biological clock to control the daily release pattern of a number of hormones. The picture that emerged shows projections contacting target neurons in the medial hypothalamus surrounding the SCN. The activity of these pre-autonomic and neuro-endocrine target neurons is controlled by differentially timed waves of vasopressin, GABA, and glutamate release from SCN terminals, among other factors. Together our data indicate that, with regard to the timing of their main release period within the LD cycle, at least four subpopulations of SCN neurons should be discernible. The different subgroups do not necessarily follow the phenotypic differences among SCN neurons. Thus, different subgroups can be found within neuron populations containing the same neurotransmitter. Remarkably, a similar distinction of four differentially timed subpopulations of SCN neurons was recently also discovered in experiments determining the temporal patterns of rhythmicity in individual SCN neurons by way of the electrophysiology or clock gene expression. Moreover, the specialization of the SCN may go as far as a single body structure, i.e., the SCN seems to contain neurons that specifically target the liver, pineal gland, and adrenal gland.     

A Network of (Autonomic) Clock Outputs

The circadian clock in the suprachiasmatic nuclei (SCN) is composed of thousands of oscillator neurons, each dependent on the cell‐autonomous action of a defined set of circadian clock genes. A major question is still how these individual oscillators are organized into a biological clock that produces a coherent output capable of timing all the different daily changes in behavior and physiology. We investigated which anatomical connections and neurotransmitters are used by the biological clock to control the daily release pattern of a number of hormones. The picture that emerged shows projections contacting target neurons in the medial hypothalamus surrounding the SCN. The activity of these pre‐autonomic and neuro‐endocrine target neurons is controlled by differentially timed waves of vasopressin, GABA, and glutamate release from SCN terminals, among other factors. Together our data indicate that, with regard to the timing of their main release period within the LD cycle, at least four subpopulations of SCN neurons should be discernible. The different subgroups do not necessarily follow the phenotypic differences among SCN neurons. Thus, different subgroups can be found within neuron populations containing the same neurotransmitter. Remarkably, a similar distinction of four differentially timed subpopulations of SCN neurons was recently also discovered in experiments determining the temporal patterns of rhythmicity in individual SCN neurons by way of the electrophysiology or clock gene expression. Moreover, the specialization of the SCN may go as far as a single body structure, i.e., the SCN seems to contain neurons that specifically target the liver, pineal gland, and adrenal gland.     

The mammalian circadian timing system: from gene expression to physiology

Many physiological processes in organisms from bacteria to man are rhythmic, and some of these are controlled by self-sustained oscillators that persist in the absence of external time cues. Circadian clocks are perhaps the best characterized biological oscillators and they exist in virtually all light-sensitive organisms. In mammals, they influence nearly all aspects of physiology and behavior, including sleep-wake cycles, cardiovascular activity, endocrinology, body temperature, renal activity, physiology of the gastro-intestinal tract, and hepatic metabolism. The master pacemaker is located in the suprachiasmatic nuclei, two small groups of neurons in the ventral part of the hypothalamus. However, most peripheral body cells contain self-sustained circadian oscillators with a molecular makeup similar to that of SCN (suprachiasmatic nucleus) neurons. This organization implies that the SCN must synchronize countless subsidiary oscillators in peripheral tissues, in order to coordinate cyclic physiology. In this review, we will discuss some recent studies on the structure and putative functions of the mammalian circadian timing system, but we will also point out some apparent inconsistencies in the currently publicized model for rhythm generation.     

Cortical Resonance Frequencies Emerge from Network Size and Connectivity

Neural oscillations occur within a wide frequency range with different brain regions exhibiting resonance-like characteristics at specific points in the spectrum. At the microscopic scale, single neurons possess intrinsic oscillatory properties, such that is not yet known whether cortical resonance is consequential to neural oscillations or an emergent property of the networks that interconnect them. Using a network model of loosely-coupled Wilson-Cowan oscillators to simulate a patch of cortical sheet, we demonstrate that the size of the activated network is inversely related to its resonance frequency. Further analysis of the parameter space indicated that the number of excitatory and inhibitory connections, as well as the average transmission delay between units, determined the resonance frequency. The model predicted that if an activated network within the visual cortex increased in size, the resonance frequency of the network would decrease. We tested this prediction experimentally using the steady-state visual evoked potential where we stimulated the visual cortex with different size stimuli at a range of driving frequencies. We demonstrate that the frequency corresponding to peak steady-state response inversely correlated with the size of the network. We conclude that although individual neurons possess resonance properties, oscillatory activity at the macroscopic level is strongly influenced by network interactions, and that the steady-state response can be used to investigate functional networks.     

A cellular oscillator model for periodic pattern formation

In this paper, we present a model for pattern formation in developing organisms that is based on cellular oscillators (CO). An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell autonomous or non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. We apply a CO model to vertebrate somitogenesis and show that we can reproduce the dynamics of periodic gene expression patterns observed in the pre-somitic mesoderm. We also show how varying somite lengths can be generated with the CO model. We then discuss the model in view of experimental evidence and its relevance to other instances of biological pattern formation, showing its versatility as a pattern generator.     

Coupling-induced synchronization in multicellular circadian oscillators of mammals

In mammals, circadian rhythms are controlled by the neurons located in the suprachiasmatic nucleus (SCN) of the hypothalamus. Each neuron in the SCN contains an autonomous molecular clock. The fundamental question is how the individual cellular oscillators, expressing a wide range of periods, interact and assemble to achieve phase synchronization. Most of the studies carried out so far emphasize the crucial role of the periodicity imposed by the light-dark cycle in neuronal synchronization. However, in natural conditions, the interaction between the SCN neurons is non-negligible and coupling between cells in the SCN is achieved partly by neurotransmitters. In this paper, we use a model of nonidentical, globally coupled cellular clocks considered as Goodwin oscillators. We mainly study the synchronization induced by coupling from an analytical way. Our results show that the role of the coupling is to enhance the synchronization to the external forcing. The conclusion of this paper can help us better understand the mechanism of circadian rhythm.     

Quantitative analysis of phase wave of gene expression in the mammalian central circadian clock network

Background

The suprachiasmatic nucleus (SCN), the master circadian clock, is a heterogeneous oscillator network, yet displays a robust synchronization dynamics. Recent single-cell bioluminescent imaging revealed temporal gradients in circadian clock gene expression in the SCN ex vivo. However, due to technical difficulty in biological approaches to elucidate the entire network structure of the SCN, characteristics of the gradient, which we refer to as phase wave, remain unknown.

Methodology/Principal Findings

We implemented new approaches, i.e., quantitative analysis and model simulation to characterize the phase waves in Per2::Luciferase clock reporter gene expression of the rat SCN slice. Our quantitative study demonstrated not only a high degree of synchronization between the neurons and regular occurrence of the phase wave propagation, but also a significant amount of phase fluctuations contained in the wave. In addition, our simulations based on local coupling model suggest that the intercellular coupling strength estimated by the model simulations is significantly higher than the critical value for generating the phase waves. Model simulations also suggest that heterogeneity of the SCN neurons is one of the main factors causing the phase wave fluctuations. Furthermore, robustness of the SCN network against dynamical noise and variation of the natural frequencies inherent in these neurons was quantitatively assessed.

Conclusions/Significance

To our knowledge, this is the first quantitative evaluation of the phase wave and further characterization of the SCN neuronal network features generating the wave i.e., intercellular synchrony, phase fluctuation, strong local coupling, heterogeneous periodicity and robustness. Our present study provides an approach, which will lead to a comprehensive understanding of mechanistic and/or biological significance of the phase wave in the central circadian oscillatory system.     

Evaluation of the Oscillatory Interference Model of Grid Cell Firing through Analysis and Measured Period Variance of Some Biological Oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ ³ /(4πσ) ² seconds where µ is the mean period of an oscillator in seconds and σ² its variance in seconds². We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.