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.     

Oscillatory isozymes as the simplest model for coupled biochemical oscillators

We analyze a simple model for two autocatalytic reactions catalyzed by two distinct isozymes transforming, with different kinetic properties, a given substrate into the same product. This two-variable system can be viewed as the simplest model of chemically coupled biochemical oscillators. Phase-plane analysis indicates how the kinetic differences between the two enzymes give rise to complex oscillatory phenomena such as the coexistence of a stable steady state and a stable limit cycle, or the co-existence of two simultaneously stable oscillatory regimes (birhythmicity). The model allows one to verify a previously proposed conjecture for the origin of birhythmicity. In other conditions, the system admits multiple oscillatory domains as a function of a control parameter whose variation gives rise to markedly different types of oscillations. The latter behavior provides an explanation for the occurrence of multiple modes of oscillations in thalamic neurons.     

The heart as a system of coupled nonlinear oscillators

In various approximations the heart is considered as a system of coupled nonlinear oscillators which are characterized by their phase transition curve (PTC) only. The system exhibits many phenomena which are known from ECG recordings such as multi-level-Wenckebach periodicity, dissociation, induction or removal of a tachycardia by an extrasystole, pseudo-block and the gap phenomenon. All these can be discussed within the model on a quantitative level.     

The behaviour of coupled biochemical oscillators as a model of contact inhibition of cellular division

Intercellular communication of molecules between normal cells by tight junctions, and lack of this in some cancer cells (Loewenstein), can explain contact inhibition of cellular division in tissues. A general theory has been based on assuming the continual rise and fall (intrinsic oscillation) of a key substance x in each cell, with the period of the cell cycle. Periods are asynchronous in different cells, and x is exchanged between cells in contact by diffusion. A reduction in the resultant amplitude of fluctuation of x results, so that it does not reach the threshold x_t required for division to ensue; hence contact inhibition.The mathematical model is defined in its simplest form, and the sets of differential equations for arrays of cells are solved, from the isolated cell to the cell in an infinite sheet. The relative probability of division, P, is computed by numerical analysis from the area of resultant curves of x that lies above the threshold x_t. P depends on four dimensionless parameters, the order of coupling n (the number of cells directly communicating with a given cell), the total number of cells N in the aggregate, the communication constant K, and x_t, as a fraction of the amplitude of the intrinsic oscillation. The degree of synchrony, measured by the coefficient of variation σ of the periods, is important. If σ < ± 4%, contact inhibition is much reduced. The theory predicts that a paradoxical “contact-facilitation” is possible for very small aggregates of cells. For a cell in an infinite sheet, the amplitude of oscillation of x is reduced approximately by the factor $\text{1}\text{nK}$. For normal cells K is probably > 1, for cancer cells that lack communication, K is probably «< 1. However, two other basic causes for lack of regulation of tissue growth (cancer) could be excessive intrinsic oscillation of x, cf. x_t, and partial or complete synchronization of groups of cells by some unknown mechanism.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment.     

Extending the HKB model of coordinated movement to oscillators with different eigenfrequencies

 We study the dynamics of a system of coupled nonlinear oscillators that has been used to model coordinated human movement behavior. In contrast to earlier work we examine the case where the two component oscillators have different eigenfrequencies. Problems related to the decomposition of a time series (from an experiment) into amplitude and phase are discussed. We show that oscillations at multiples of the main frequency of the oscillator system may occur in the phase and amplitude due to the choice of a coordinate system and how these oscillations can be eliminated. We derive an explicit equation for the dynamics of the relative phase of the oscillator system in phase space that enables a direct comparison between theory and experiment. Received: 30 December 1994/Accepted in revised form: 27 June 1995     

The stability of entrainment conditions forRLC coupled van der pol oscillators used as a model for intestinal electrical rhythms

A commonly accepted mathematical model for the slow wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected relaxation oscillators. The method of harmonic balance is used here to obtain analytical results for the entrained frequencies and amplitudes of two oscillators coupled with a parallelRLC network. By perturbations and linearisation about these values the conditions for stable limit-cycles are found and regions in theRLC parameter space which give one or two stable limit-cycle conditions are derived. These analytical results are compared with simulated results and found to creelate well for a waveshape factor of ε=0.1 and fairly well for ε=1.0. The single limit-cycle region corresponds to the requirement for a single mode having a frequency higher than the uncoupled value in small-intestinal data, while the double limit-cycle region corresponds to the two rhythms found in human large-intestinal activity.     

The fundamental organization of cardiac mitochondria as a network of coupled oscillators

Mitochondria can behave as individual oscillators whose dynamics may obey collective, network properties. We have shown that cardiomyocytes exhibit high-amplitude, self-sustained, and synchronous oscillations of bioenergetic parameters when the mitochondrial network is stressed to a critical state. Computational studies suggested that additional low-amplitude, high-frequency oscillations were also possible. Herein, employing power spectral analysis, we show that the temporal behavior of mitochondrial membrane potential (DeltaPsi(m)) in cardiomyocytes under physiological conditions is oscillatory and characterized by a broad frequency distribution that obeys a homogeneous power law (1/f(beta)) with a spectral exponent, beta = 1.74. Additionally, relative dispersional analysis shows that mitochondrial oscillatory dynamics exhibits long-term memory, characterized by an inverse power law that scales with a fractal dimension (D(f)) of 1.008, distinct from random behavior (D(f) = 1.5), over at least three orders of magnitude. Analysis of a computational model of the mitochondrial oscillator suggests that the mechanistic origin of the power law behavior is based on the inverse dependence of amplitude versus frequency of oscillation related to the balance between reactive oxygen species production and scavenging. The results demonstrate that cardiac mitochondria behave as a network of coupled oscillators under both physiological and pathophysiological conditions.     

Two coupled oscillators: simulations of the circadian pacemaker in mammalian activity rhythms

In the activity rhythms of captive small mammals a variety of features, most notably “splitting”, suggest that two coupled oscillators may constitute the pacemaker system which underlies the rhythms. A phenomenological model proposed by Pittendrigh is developed and expanded here using an explicit quantitative structure. It is found that such a system can simulate several qualitative features in the experimental data: the inter-dependence of free-running period (gt) and activity time (α) with changing light intensity described in Aschoff's rule, after-effects on τ and α of prior conditions, and the occasional existence of two stable phase relationships, with different τ values for a given light intensity, as observed in “splitting”. It is hoped that the model will suggest experiments aimed at the elucidation of the physiological basis of these phenomena.     

Human sleep and circadian rhythms: a simple model based on two coupled oscillators

We propose a model of the human circadian system. The sleep-wake and body temperature rhythms are assumed to be driven by a pair of coupled nonlinear oscillators described by phase variables alone. The novel aspect of the model is that its equations may be solved analytically. Computer simulations are used to test the model against sleep-wake data pooled from 15 studies of subjects living for weeks in unscheduled, time-free environments. On these tests the model performs about as well as the existing models, although its mathematical structure is far simpler.Supported by NIGMS Grant No. 5-R01-GM-30719-03     

Nuclear envelope breakdown is coordinated by both Nup358/RanBP2 and Nup153, two nucleoporins with zinc finger modules

When higher eukaryotic cells transition into mitosis, the nuclear envelope, nuclear pore complexes, and nuclear lamina are coordinately disassembled. The COPI coatomer complex, which plays a major role in membrane remodeling at the Golgi, has been implicated in the process of nuclear envelope breakdown and requires interactions at the nuclear pore complex for recruitment to this new site of action at mitosis. Nup153, a resident of the nuclear pore basket, was found to be involved in COPI recruitment, but the molecular nature of the interface between COPI and the nuclear pore has not been fully elucidated. To better understand what occurs at the nuclear pore at this juncture, we have probed the role of the nucleoporin Nup358/RanBP2. Nup358 contains a repetitive zinc finger domain with overall organization similar to a region within Nup153 that is critical to COPI association, yet inspection of these two zinc finger domains reveals features that also clearly distinguish them. Here, we found that the Nup358 zinc finger domain, but not a zinc finger domain from an unrelated protein, binds to COPI and dominantly inhibits progression of nuclear envelope breakdown in an assay that robustly recapitulates this process in vitro. Moreover, the Nup358 zinc finger domain interferes with COPI recruitment to the nuclear rim. Consistent with a role for this pore protein in coordinating nuclear envelope breakdown, Nup358-specific antibodies impair nuclear disassembly. Significantly, targeting either Nup153 or Nup358 for inhibition perturbs nuclear envelope breakdown, supporting a model in which these nucleoporins play nonredundant roles, perhaps contributing to COPI recruitment platforms on both the nuclear and cytoplasmic faces of the pore. We found that an individual zinc finger is the minimal interface for COPI association, although tandem zinc fingers are optimal. These results provide new information about the critical components of nuclear membrane remodeling and lay the foundation for a better understanding of how this process is regulated.     

Proteomics as a tool for tapping potential of entomopathogens as microbial insecticides

Biopesticides are collective pest control harnessing the knowledge of the target pest and its natural enemies that minimize the risks of synthetic pesticides. A subset of biopesticides; bioinsecticides, are specifically used in controlling insect pests. Entomopathogens (EPMs) are micro‐organisms sought after as subject for bioinsecticide development. However, lack of understanding of EPM mechanism of toxicity and pathogenicity slowed the progress of bioinsecticide development. Proteomics is a useful tool in elucidating the interaction of entomopathogenic fungi, entomopathogenic bacteria, and entomopathogenic virus with their target host. Collectively, proteomics shed light onto insect host response to EPM infection, mechanism of action of EPM’s toxic proteins and secondary metabolites besides characterizing secreted and membrane‐bound proteins of EPM that more precisely describe relevant proteins for host recognition and mediating pathogenesis. However, proteomics requires optimized protein extraction methods to maximize the number of proteins for analysis and availability of organism's genome for a more precise protein identification.     

Mode analysis of a tubular structure of coupled non-linear oscillators for digestive-tract modelling

A commonly accepted mathematical model for the slow-wave electrical activity of the gastro-intestinal tract of humans and animals comprises a set of interconnected nonlinear oscillators. Using a van der Pol oscillator with third-power conductance characteristics as the unit oscillator a number of structures have been analysed using a matrix Krylov-Bogolioubov method linearisation. The mode analysis of one-dimensional chains and two-dimensional arrays has been reported. In this paper the method has been extended to consider a tubular structure which is relevant to modelling small-intestinal rhythms. It is shown that this structure is capable of producing stable single models, non-resonant double modes and degenerated modes. General expressions are obtained for anm×n structure and examples given of two special conditions of 3×4 (i.e. odd numbers of oscillators in a ring) and 4×3 cases. The analytical results obtained for these two cases have been vertified experimentally using an electronic implementation of coupled van der Pol oscillators. Results obtained using fifth-power non-linear oscillators are summarised.     

Generation of theta oscillations by weakly coupled neural oscillators in the presence of noise

Neuronal oscillations are a robust phenomenon occurring in a variety of brain regions despite considerable amounts of noise. In this article classical phase-response theory is generalized to the case of noisy weak-coupling regimes by deriving an iterated map for the asynchrony of spikes in an oscillation cycle. Two criteria are introduced to check the validity of our approximations: One criterion tests the assumption that all neurons fire exactly once per cycle, the other criterion tests for linearity. The framework is applied to stellate cells of the medial entorhinal cortex layer II. We find that rhythmogenesis is more robust in the case of excitatory noise as compared to inhibitory noise. It is shown that a network of stellate cells can also act as a generator of theta if the neurons are connected via a fast-oscillating network of inhibitory interneurons.     

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.     

A model for the coordinated stepping of cytoplasmic dynein

Cytoplasmic dynein play an important role in transporting various intracellular cargos by coupling their ATP hydrolysis cycle with their conformational changes. Recent experimental results showed that the cytoplasmic dynein had a highly variable stepping pattern including “hand-over-hand”, “inchworm” and “nonalternating-inchworm”. Here, we developed a model to describe the coordinated stepping patterns of cytoplasmic dynein, based on its working cycle, construction and the interaction between its leading head and tailing head. The kinetic model showed how change in the distance between the two heads influences the rate of cytoplasmic dynein under different stepping patterns. Numerical simulations of the distribution of step size and striding rate are in good quantitative agreement with experimental observations. Hence, our coordinated stepping model for cytoplasmic dynein successfully explained its diverse stepping patterns as a molecular motor. The cooperative mechanism carried out by the two heads of cytoplasmic dynein shed light on the strategies adopted by the cytoplasmic dynein in executing various functions.     

Adenosine 5''-triphosphate consumption by smooth muscle as predicted by the coupled four-state crossbridge model.

We have proposed a four-state crossbridge model to explain contraction and the latch state in arterial smooth muscle. Ca(2+)-dependent crossbridge phosphorylation was the only postulated regulatory mechanism and the latchbridge (a dephosphorylated, attached crossbridge) was the only novel element in the model. In this study, we used the model to predict rates of ATP consumption by crossbridge phosphorylation (JPhos) and cycling (JCycle) during isometric and isotonic contractions in arterial smooth muscle; then we compared model predictions with experimental data. The model predicted that JPhos and JCycle were similar in magnitude in isometric contractions, and both increased almost linearly with myosin phosphorylation. The predicted relationship between isometric stress and ATP consumption was quasihyperbolic, but approximately linear when myosin phosphorylation was below 35%, in agreement with most of the available data. Muscle shortening increased the predicted values of JCycle up to 3.7-fold depending on shortening velocity and the level of myosin phosphorylation. The predicted maximum work output per ATP was 7.4-7.8 kJ/mol ATP and was relatively insensitive to changes in myosin phosphorylation. The predicted increase in JCycle with shortening was in agreement with available data, but the model prediction that work output per ATP was insensitive to changes in myosin phosphorylation was unexpected and remains to be tested in future experiments.     

Further evidence that the circadian clock in Drosophila is a population of coupled ultradian oscillators

We hypothesize that ultradian oscillators are coupled to yield a composite circadian clock in Drosophila. In such a system, period would be a function of the tightness of coupling of these oscillators, increasing as coupling loosens. Ultradian oscillations would become apparent under weak coupling or in the absence of coupling. A new technique for calculating signal-to-noise ratio (SNR) for biological rhythms to characterize their precision has yielded support for this hypothesis. SNR of rhythms of the allelic series of mutations at the period (per) locus of Drosophila melanogaster were compared. Per(o) was the noisiest, grading through perL, per+, and pers, the least noisy. SNR decreases significantly with increasing period in pers, per+, and perL; per(o) typically has multiple ultradian oscillations and the lowest SNR. At least 70% of perL individuals also exhibit ultradian periodicities.