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.     

Are circadian oscillators structurally unstable?

The common belief is that all biological oscillations are of limit cycle type. It is shown in this article that the phase response curves simulated on a two-species Lotka-Volterra linear (i.e. non-limit cycle type) oscillator, do look similar to those obtained by experimental methods by different workers. The form of the phase response curves, the existence of singularities and the mirror-image symmetry of opposite perturbations are modelled on the Lotka-Volterra system. The study, which is strongly indicative of the possibility that the underlying oscillator (or oscillators) is (are) not structurally stable, also indicates the necessity of designing critical experiments, capable of distinguishing between limit cycle and non-limit cycle oscillators, since the single-pulse phase resetting does nothing to distinguish between them.     

Modeling Temperature Compensation in Chemical and Biological Oscillators

All physicochemical and biological oscillators maintain a balance between destabilizing reactions (as, for example, intrinsic autocatalytic or amplifying reactions) and stabilizing processes. These two groups of processes tend to influence the period in opposite directions and may lead to temperature compensation whenever their overall influence balances. This principle of “antagonistic balance” has been tested for several chemical and biological oscillators. The Goodwin negative feedback oscillator appears of particular interest for modeling the circadian clocks in Neurospora and Drosophila and their temperature compensation. Remarkably, the Goodwin oscillator not only gives qualitative, correct phase response curves for temperature steps and temperature pulses, but also simulates the temperature behavior of Neurospora frq and Drosophila per mutants almost quantitatively. The Goodwin oscillator predicts that circadian periods are strongly dependent on the turnover of the clock mRNA or clock protein. A more rapid turnover of clock mRNA or clock protein results, in short, a slower turnover in longer period lengths. (Chronobiology International, 14(5), 499–510, 1997)     

How to Achieve Fast Entrainment? The Timescale to Synchronization

Entrainment, where oscillators synchronize to an external signal, is ubiquitous in nature. The transient time leading to entrainment plays a major role in many biological processes. Our goal is to unveil the specific dynamics that leads to fast entrainment. By studying a generic model, we characterize the transient time to entrainment and show how it is governed by two basic properties of an oscillator: the radial relaxation time and the phase velocity distribution around the limit cycle. Those two basic properties are inherent in every oscillator. This concept can be applied to many biological systems to predict the average transient time to entrainment or to infer properties of the underlying oscillator from the observed transients. We found that both a sinusoidal oscillator with fast radial relaxation and a spike-like oscillator with slow radial relaxation give rise to fast entrainment. As an example, we discuss the jet-lag experiments in the mammalian circadian pacemaker.     

Biological oscillators can be stopped—Topological study of a phase response curve

Many biological oscillators are stable against noise and perturbation (e.g. circadian rhythms, biochemical oscillators, pacemaker neurons, bursting neurons and neural networks with periodic outputs). The experiment of phase shifts resulting from discrete perturbation of stable biological rhythms was developed by Perkel and coworkers (Perkel et al., 1964). By these methods, they could get important insights into the entrainment behaviors of biological rhythms. Phase response curves, which are measured in these experiments, can be classified into two types. The one is the curve with one mapping degree (Type 1), and the other is that with zero mapping degree (Type 0) (Winfree, 1970). We define the phase response curve mathematically, and explain the difference between these two types by the homotopy theory. Moreover, we prove that, if a Type 0 curve is obtained at a certain magnitude of perturbation, there exists at least one lower magnitude for which the phase response curve cannot be measured. Some applications of these theoretical results are presented.     

The topology of phase response curves induced by single and paired stimuli in spontaneously oscillating chick heart cell aggregates.

The topological properties of the phase resetting of biological oscillators by an isolated stimulus delivered at various phases of the cycle depend on whether the stimulus is "weak" or "strong." When multiple stimuli are delivered to the oscillator, the response to stimulation also depends on the time between the stimuli, and the rate at which the oscillator returns to an underlying limit cycle attractor. If the time between two consecutive "weak" stimuli is sufficiently short, the effects produced by the pair of stimuli may be characteristic of a single "strong" stimulus. These results are demonstrated in a model experimental system, spontaneously beating aggregates of cells derived from embryonic chick heart, and are illustrated by consideration of a simple theoretical model of nonlinear oscillators, the Poincaré oscillator.     

Two coupled neural oscillators as a model of the circadian pacemaker

Pittendrigh first found that the circadian rhythm of locomotor activity in nocturnal rodents split into two components. Hoffman then reported that the splitting phenomenon was even more reproducible in the small diurnal primate Tupaia. These “splitting” experiments and many other experiments suggest that two coupled oscillators may constitute the circadian pacemaker system. Pittendrigh proposed a phenomenological two-oscillator model. Daan and Berde developed a quantitative model assuming that the interaction between the two constituent oscillators is by instantaneous resets. Their model system can simulate several qualitative features in the experimental data. As the assumption of instantaneous resets seems to be unnatural, we study two limit cycle oscillators, which are coupled continuously to each other, as a model of the circadian pacemaker. We assume the following points, (i) One oscillator in a resting state does not affect another oscillator, (ii) Two oscillators are identical, (iii) The coupling is symmetrical. By the theory of Hopf bifurcation it is found that the general two-oscillator system has two stable periodic solutions. One is the in-phase solution where the two constituent oscillators oscillate in phase synchrony. Another is the anti-phase solution where the two oscillators oscillate 180 ° out of phase. The former corresponds to a single pattern of locomotor activity and the latter corresponds to a splitting pattern. Furthermore, we study specific two-neural oscillators, which are linearly coupled to each other. By the method of secondary bifurcation we find that the model shows simultaneous stability of the two alternative phase relationships and the hysteresis phenomena found in Tupaia. A natural period of the uncoupled constituent oscillator is longer than that of the in-phase solution but it is shorter than that of the anti-phase solution. This is in agreement with the data of Tupaia.     

Modeling experimental time series with ordinary differential equations

Recently some methods have been presented to extract ordinary differential equations (ODE) directly from an experimental time series. Here, we introduce a new method to find an ODE which models both the short time and the long time dynamics. The experimental data are represented in a state space and the corresponding flow vectors are approximated by polynomials of the state vector components. We apply these methods both to simulated data and experimental data from human limb movements, which like many other biological systems can exhibit limit cycle dynamics. In systems with only one oscillator there is excellent agreement between the limit cycling displayed by the experimental system and the reconstructed model, even if the data are very noisy. Furthermore, we study systems of two coupled limit cycle oscillators. There, a reconstruction was only successful for data with a sufficiently long transient trajectory and relatively low noise level.     

Control of multistability in ring circuits of oscillators

The essential dynamics of some biological central pattern generators (CPGs) can be captured by a model consisting of N neurons connected in a ring. These circuits, like many oscillatory nonlinear circuits of sufficient complexity, are capable of multistability, that is, of generating different firing patterns distinguished by the phasic relationships between the firing in each circuit element (neuron). Moreover, a shift in firing pattern can be induced by a transient perturbation. A systematic approach, based on phase-response curve (PRC) theory, was used to determine the optimum timing for perturbations that induce a shift in the firing pattern. The first step was to visualize the solution space of the ring circuit, including the attractive basins for each stable firing pattern; this was possible using the relative phase of N−1 oscillators, with respect to an arbitrarily selected reference oscillator, as coordinate axes. The trajectories in this phase space were determined using an iterative mapping based only on the PRCs of the uncoupled component oscillators; this algorithm was called a circuit emulator. For an accurate mapping of the attractive basin of each pattern exhibited by the ring circuit, the emulator had to take into account the effect of a perturbation or input on the timing of two bursts following the onset of the perturbation, rather than just one. The visualization of the attractive basins for rings of two, three, and four oscillators enabled the accurate prediction of the amounts of phase resetting applied to up to N−1 oscillators within a cycle that would induce a transition from any pattern to any another pattern. Finally, the timing and synaptic characterization of an input called the switch signal was adjusted to produce the desired amount of phase resetting. Received: 29 May 1998 / Accepted in revised form: 18 September 1998     

Phase response curves for social entrainment

Summary Phase shifts in free-running activity rhythms of male golden hamsters,Mesocricetus auratus, often occur when they establish a new territory and home after a cage change. Similar shifts also often occur after pairs of animals interact with each other for half an hour. When these events take place during the middle of the hamsters' subjective day, they produce phase advances: when late in the subjective night, they produce phase delays. Repeated social interactions at the same time of day can entrain activity rhythms in a way consistent with the shape of the phase response curves. Not all individuals become entrained, as is predictable from the modest amplitude of the phase response curve. The effects of social interactions and of other disturbances may be mediated through an oscillator phased by general arousal. The present findings have implications for the interpretation of drug-induced changes in biological rhythms.     

PRC Bisection Tests

This communication presents a new method for evaluating phase response curves (PRCs). A PRC describes the phase shifts produced in an oscillator by stimuli applied at different initial phase‐states of that oscillator. In the PRC bisection tests, we repeatedly cut in half the circular distribution of the initial phase‐states of the oscillator when stimuli are given. Empirically, we locate that optimal diameter which best bisects the circular distribution of phase responses into arcs of relative phase advance and phase delay. We compute a D score reflecting the success of the best bisection. The null hypothesis of a random distribution of phase responses by initial phase is tested with a Monte Carlo procedure, which computes D_r scores from random combinations of phase shifts with initial phases, thus determining the probability, given the null hypothesis, that the observed D score was from a random distribution. The bisection procedure can be extended to examine whether stronger phase shifts are produced in one phase response curve than in contrasting curves. Also, the bisection procedure yields an estimate of the inflection point of the phase response curve. A method is given to estimate the power of the PRC bisection test.     

Multiple pulse interactions and averaging in systems of coupled neural oscillators

Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University     

PRC bisection tests

This communication presents a new method for evaluating phase response curves (PRCs). A PRC describes the phase shifts produced in an oscillator by stimuli applied at different initial phase-states of that oscillator. In the PRC bisection tests, we repeatedly cut in half the circular distribution of the initial phase-states of the oscillator when stimuli are given. Empirically, we locate that optimal diameter which best bisects the circular distribution of phase responses into arcs of relative phase advance and phase delay. We compute a D score reflecting the success of the best bisection. The null hypothesis of a random distribution of phase responses by initial phase is tested with a Monte Carlo procedure, which computes D_r scores from random combinations of phase shifts with initial phases, thus determining the probability, given the null hypothesis, that the observed D score was from a random distribution. The bisection procedure can be extended to examine whether stronger phase shifts are produced in one phase response curve than in contrasting curves. Also, the bisection procedure yields an estimate of the inflection point of the phase response curve. A method is given to estimate the power of the PRC bisection test.     

A phase response curve for circannual rhythm in the varied carpet beetle <Emphasis Type="Italic">Anthrenus verbasci</Emphasis>

We know that entrainment, a stable phase relationship with an environmental cycle, must be established for a biological clock to function properly. Phase response curves (PRCs), which are plots of phase shifts that result as a function of the phase of a stimulus, have been created to examine the mode of entrainment. In circadian rhythms, single-light pulse PRCs have been obtained by giving a light pulse to various phases of a free-running rhythm under continuous darkness. This successfully explains the entrainment to light-dark cycles. Some organisms show circannual rhythms. In some of these, changes in photoperiod entrain the circannual rhythms. However, no single-pulse PRCs have been created. Here we show the PRC to a long-day pulse superimposed for 4 weeks over constant short days in the circannual pupation rhythm in the varied carpet beetle Anthrenus verbasci. Because the shape of that PRC closely resembles that of the Type 0 PRC with large phase shifts in circadian rhythms, we suggest that an oscillator having a common feature in the phase response with the circadian clock, produces a circannual rhythm.     

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.     

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.     

Engineering entrainment and adaptation in limit cycle systems

Periodic behavior is key to life and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modeling or generating periodic behavior is done by using oscillators, i.e., dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is twofold: (1) to provide a framework for characterizing and designing oscillators and (2) to review how classes of well-known oscillators can be understood and related to this framework. The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators that might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phase-locking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely offline methods and online methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular, the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modeling of physical or biological phenomena.     

Circadian phase response curves for dark pulses in the hamster

Summary Hamsters maintained under constant illumination were exposed to 2- or 6-h pulses of darkness at various phases of their circadian activity rhythms. When presented around the time of activity onset, the pulses resulted in phase advances, and when presented toward the end of daily activity, they resulted in phase delays. Since others have shown that light pulses presented at the same phases in constant darkness cause phase shifts in the opposite directions, these results indicate that phase response curves for light and dark pulses are mirror images.Dark pulses also caused phase-dependent changes, both transient and long-lasting, in the period of the free-running rhythms, and a few pulses were immediately followed by splitting of the activity rhythms into two components. Such effects may reflect a differential responsiveness of two coupled oscillators to dark pulses.Abbreviations CT circadian time - DD constant dark - LD lightdark - LL constant light - PRC phase response curve - SD subjective day - SN subjective night - period of a circadian rhythm Supported by grants from the NSERC of Canada to B. Rusak and to G.V. Goddard. We are grateful to Dr. Goddard for his support and encouragement     

Unfolding an electronic integrate-and-fire circuit

Many physical and biological phenomena involve accumulation and discharge processes that can occur on significantly different time scales. Models of these processes have contributed to understand excitability self-sustained oscillations and synchronization in arrays of oscillators. Integrate-and-fire (I+F) models are popular minimal fill-and-flush mathematical models. They are used in neuroscience to study spiking and phase locking in single neuron membranes, large scale neural networks, and in a variety of applications in physics and electrical engineering. We show here how the classical first-order I+F model fits into the theory of nonlinear oscillators of van der Pol type by demonstrating that a particular second-order oscillator having small parameters converges in a singular perturbation limit to the I+F model. In this sense, our study provides a novel unfolding of such models and it identifies a constructible electronic circuit that is closely related to I+F.     

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.