Qualitative similarities between the behavior of coupled oscillators and circadian rhythms

Entrainment of the suprachiasmatic nucleus network by a light-dark cycle

The synchronization of biological activity with the alternation of day and night (circadian rhythm) is performed in the brain by a group of neurons, constituting the suprachiasmatic nucleus (SCN). The SCN is divided into two subgroups of oscillating cells: the ventrolateral (VL) neurons, which are exposed to light (photic signal), and the dorsomedial (DM) neurons, which are coupled to the VL cells. When the coupling between these neurons is strong enough, the system synchronizes with the photic period. Upon increasing the cell coupling, the entrainment of the DM cells has been recently shown to occur via a very sharp (jumping) transition when the period of the photic input is larger than the intrinsic period of the cells. Here, we characterize this transition with a simple realistic model. We show that two bifurcations possibly lead to the disappearance of the endogenous mode. Using a mean-field model, we show that the jumping transition results from a supercritical Hopf-like bifurcation. This finding implies that both the period and strength of the stimulating photic signal, and the relative fraction of cells in the VL and DM compartments, are crucial in determining the synchronization of the system.

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.

Mechanism of phase splitting in two coupled groups of suprachiasmatic-nucleus neurons

The phase-splitting behavior of coupled suprachiasmatic-nucleus neurons has been observed in many mammals, and its mechanism is still not completely understood. Based on our previous work [C. Gu, J. Wang, and Z. Liu, Phys. Rev. E 80, 030904(R) (2009)] on the free-running periods of neurons in the suprachiasmatic nucleus, we present here a modified Goodwin oscillator model to explain the mechanism of phase splitting. In contrast to the previous phase model, the modified Goodwin oscillator model contains the information on both the phase and amplitude and, thus, can show more features than the purely phase model, including all three behaviors of synchronization, phase splitting, and amplitude death and the distributed periodicity in the regions of synchronization and phase splitting, etc. An analytic phase model is extracted from the modified Goodwin oscillator model to explain the dependence of periodicity on the parameters. Moreover, both the modified Goodwin oscillator model and the analytic phase model show that the ensemble frequency can be enhanced or reduced by the time delay.

Redox rhythmicity: clocks at the core of temporal coherence

Ultradian rhythms are those that cycle many times in a day and are therefore measured in hours, minutes, seconds or even fractions of a second. In yeasts and protists, a temperature-compensated clock with a period of about an hour (30-90 minutes) provides the time base upon which all central processes are synchronized. A 40-minute clock in yeast times metabolic, respiratory and transcriptional processes, and controls cell division cycle progression. This system has at its core a redox cycle involving NAD(P)H and dithiol-disulfide interconversions. It provides an archetype for biological time keeping on longer time scales (e.g. the daily cycles driven by circadian clocks) and underpins these rhythms, which cannot be understood in isolation. Ultradian rhythms are the foundation upon which the coherent functioning of the organism depends.

A model for "splitting" of running-wheel activity in hamsters

Splitting of locomotor activity rhythm in hamsters occurs when the animals are exposed for several weeks to constant light. The authors propose a mathematical model that explains splitting in terms of a switch in the sign of coupling of two oscillators, from positive to negative, due to long-term exposure to constant light. The model assumes that the two oscillators are not identical and that the negative coupling strengths achieved by each individual animal are variable. With these assumptions, the model provides a unified picture of all different splitting patterns presented by the hamsters, provides an explanation for why the two activity components cross each other during many patterns, and explains why the phase difference achieved by the split components is often near 180 degrees.

Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.

Molecular circadian oscillators: an alternative hypothesis

Results from experiments in different organisms have shown that elements of input pathways can themselves be under circadian control and that outputs might feed back into the oscillator. In addition, it has become clear that there might be redundancies in the generation of circadian rhythmicity, even within single cells. In view of these results, it is worth reevaluating our current working hypotheses about the pacemaker's molecular mechanisms and the involvement of single autoregulatory genes. On one hand, redundancies in the generation of circadian rhythmicity might make the approach of defining a discrete circadian oscillator with the help of single gene mutations extremely difficult. On the other hand, many examples show that components of signal transduction pathways can indeed be encoded by single genes. The authors have constructed a model placing an autoregulatory gene and its products on an input pathway feeding into a separate oscillator. The behavior of this model can explain the majority of results of molecular circadian biology published to date. In addition, it shows that different qualities of the circadian system might be associated with different cellular functions that can exist independently and, only if put together, will lead to the known circadian phenotype.

The circadian locomotor rhythm of Hemideina thoracica (Orthoptera; Stenopelmatidae): a population of weakly coupled feedback oscillators as a model of the underlying pacemaker

A control systems model consisting of a population of weakly-coupled feedback oscillators has been developed to simulate the circadian locomotor rhythm of the insect, Hemideina thoracica (Orthoptera; Stenopelmatidae). The model is an extension of a previously published single oscillator feedback model (Gander and Lewis, 1979) which successfully simulates entrainment, phase response curves, temperature compensation and Aschoff's Rule for Hemideina activity rhythms. The population model described here has the additional properties of predicting some of the free-run period lability (Pavlidis, 1978a, b) observed in the Hemideina rhythm (Christensen and Lewis, 1982) which is unexplained by single oscillator systems. Model behaviour is compared with the experimental data derived from the insect activity rhythms.

Entrainment of a bursting neuron--I. typical activity

Highly regular RPA-1 oscillator neurons of the land snail Helix pomatia L. are tested for entrainment by rhythmical stimulation. Both with orthodromic and direct hyperpolarizing pulses the bursting activity could be entrained to frequencies higher or lower than, and equal to, the spontaneous one, in this order of difficulty. Depolarizing pulses give mainly entrainment to higher frequencies, but synchronization to frequencies lower than the spontaneous one was also demonstrable. Each of these different driving relations implies the consecutive stimuli to become locked to a peculiar range of phases and not to a single phase of resonance inside the cycle.

Circadian rhythms in vasopressin deficient rats

Circadian rhythms in wheel running and drinking behavior were investigated using heterozygous an homozygous (diabetes insipidus) female Brattleboro rats Despite the lack of vasopressin in the suprachiasmatic nuclei of the diabetic rats, they showed coherent rhythms, both in cyclic light and in constant light. However, the periods of the free-running rhythms were longer for the diabetic rats, they were less active, and, of course, were severely polydipsic. Replacement treatment with systemic infusions of vasopressin reversed the polydipsia but did not affect the other measures.

Regularly firing neurones in the rat suprachiasmatic nucleus

The spontaneous discharge of some suprachiasmatic neurones in vivo and in vitro was found to exhibit a very constant interspike interval. In vivo these cells were comparatively rare and appeared to be mutually coupled. The findings are discussed in relation to coupled oscillator theories of circadian rhythm generation.