Modulation analysis of forced non-linear oscillators for biological modelling.

The synchronization of self-sustained oscillations by a forcing oscillation is of interest in a number of biological models. It has been considered for circadian rhythm modelling, heart-rate variability studies and forced breathing experiments. Outside the range of synchronization, conditions of almost-entrainment occur in which changes in amplitude and/or frequency are apparent. It is shown in this paper that such conditions can be analysed as modulation phenomena using the analytical method of harmonic balance. The degree of non-linearity in the self-sustained oscillation affects the nature of modulation, in that increasing distortion gives a trend towards frequency rather than amplitude modulation. The analytical results compare favourably with spectral analysis of simulated oscillators.     

Synchronization of non-linear biochemical oscillators coupled by diffusion

The behaviour of similar coupled non-linear oscillators of the type \(\dot x\) =f(x, y, µ \(\dot y\) =g(x, y, µ is to be investigated. The oscillators are assumed to be coupled by diffusion gradients. If some conditions on the magnitude of the diffusion coefficients are satisfied, it is proved that: 1) if the oscillators have the same period (identical value of the parameter μ) and different phases before coupling, after coupling they tend to synchronize the phases; 2) if the periods of the oscillators are not too different (in terms of the values of the parameter μ) before coupling, after coupling they tend to oscillate with the same period. It is suggested the possible role of diffusion as a synchronizing mechanism in some biological phenomena.     

On partial contraction analysis for coupled nonlinear oscillators

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in flocks of oscillators or dynamic elements.     

A non-linear analysis for spatial structure in a reaction-diffusion model

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on the practical Thomas reaction-diffusion mechanism which exhibits bifurcation to non-uniform states. The analytical results compare favourably with the numerical solutions. The sequential patterns generated by this model by variations in a parameter related to the reaction-diffusion domain indicate its capacity to represent certain key morphogenetic features required in a recent model by Kauffman for pattern formation in theDrosophila embryo.     

The method of harmonic balance applied to coupled asymmetrical van der Pol oscillators for intestinal modelling

The spontaneous electrical rhythms recorded from the gastro-intestinal tract of humans and animals have been successfully modelled by an array of interconnected van der Pol oscillators. To account for asymmetry in the recorded waveforms (with particular reference to the human small intestine) an additional term in the van der Pol dynamics has been included. It is shown that the method of harmonic balance can be used to give analytical results for this asymmetrical condition. The non-linear algebraic equations are solved by hill-climbing to give values of d.c., fundamental and second harmonic amplitudes together with the entrained frequency. The results correlate well with actual measurements made on an analogue simulation by three different methods for waveshape factors of 0.1 and 1.0     

Dynamics of coupled stomatal oscillators

Stomata are microscopic openings in the leaves of green plants which permit gas exchange. Stomata exhibit oscillatory opening and closing behavior under certain environmental conditions in addition to a daily (diurnal) cycle. In order to explore the effects of coupling between neighboring stomata we present a mathematical model of the dynamics of a system of N coupled stomatal oscillators. An individual stomate is modeled to either remain closed, oscillate periodically, or remain open, depending on the local water potential. Coupling between neighboring stomata is accomplished in the model by taking into account the flow of water in the leaf as well as by oscillator phase coupling.Analysis of the model shows that under certain conditions it exhibits a stable spatially uniform synchronized behavior, referred to here as the in-phase mode. It is also shown that under non-uniform illumination the system may behave in a more complicated fashion.     

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.     

Mode analysis of physiological oscillators intercoupled via pure time delays

The study of bi-directionally coupled oscillators is relevant in biological modelling of such systems as gastro-intestinal electrical activity, cardiac pacemarkers, cardiovascular and respiratory interactions and circadian rhythms. Interconnecting pathways in biological systems often exhibit pure time-delay characteristics. In this paper the multiple-mode limit-cycle behaviour of such systems is analysed using the method of harmonic blance. It is shown that the coupling time delay radically affects the number, frequency and amplitudes of entrained limit-cycles.     

Synchronization of coupled nonidentical genetic oscillators

The study of the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using a control theory approach, we provide a theoretical method for analysing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.     

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.     

Modeling of a bipedal robot using mutually coupled Rayleigh oscillators

The objective of the work presented here was the modeling of a bipedal robot using a central pattern generator (CPG) formed by a set of mutually coupled Rayleigh oscillators. We analyzed a 2D model, with the three most important determinants of gait, that performs only motions parallel to the sagittal plane. Using oscillators with integer relation of frequency, we determined the transient motion and the stable limit cycles of the network formed by the three oscillators, showing the behavior of the knee angles and the hip angle. A comparison of the plotted graphs revealed that the system provided excellent results when compared to experimental analysis. Based on the results of the study, we come to the conclusion that the use of mutually coupled Rayleigh oscillators can represent an excellent method of signal generation, allowing their application for feedback control of a walking machine.Acknowledgements The authors would like to express their gratitude to CNPq and CAPES for the financial support provided during the course of this research.     

Formation of a regular dissipative structure: a bifurcation and non-linear analysis.

A non-linear reaction diffusion model of a negative feedback epigenetic control system is presented. The model involves synthesis of the mitotic inducing and inhibiting proteins, simultaneously with intercellular self-diffusion and cross-diffusion of the latter only. The importance of negative cross-diffusion for creating a regular dissipative structure is shown. A bifurcation analysis of the non-linear diffusive system has been performed and it is concluded that bifurcation is supercritical. Lastly, using Liapunov's direct method, it is shown that the pattern evolved by the system is globally asymptotically stable.     

The behavior of rings of coupled oscillators

Coupled oscillators in a ring are studied using perturbation and numerical methods. Stability of waves with nearest neighbor weak coupling is shown for a class of simple oscillators. Linkens' [23] model for colorectal activity is analyzed and several stable modes are found. Stability of waves with general (non nearest neighbor coupling) is determined and comparisons to the nearest neighbor case are made. Approximate solutions to a ring with inhomogeneities are compared with numerical simulations.Supported by a NSF Grant No. MCS8300885 and the Alfred Sloan Foundation     

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.     

Synchronization in a pool of mutually coupled oscillators with random frequencies

An exact solution to a model of mutually interacting sinusoidal oscillators is found. Limits on the variation of the native frequencies are determined in order for synchronization to occur. These limits are computed for different distributions of native frequencies.This research was supported by NSF Award No. MCS8300885 and the Alfred Sloan Foundation.     

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.     

Two coupled oscillators as a model for the coordinated finger tapping by both hands

Recently, it was found that rhythmic movements (e.g. locomotion, swimmeret beating) are controlled by mutually coupled endogeneous neural oscillators (Kennedy and Davis, 1977; Pearson and Iles, 1973; Stein, 1974; Shik and Orlovsky, 1976; Grillner and Zangger, 1979). Meanwhile, it has been found out that the phase resetting experiment is useful to investigate the interaction of neural oscillators (Perkel et al., 1963; Stein, 1974). In the preceding paper (Yamanishi et al., 1979), we studied the functional interaction between the neural oscillator which is assumed to control finger tapping and the neural networks which control some tasks. The tasks were imposed on the subject as the perturbation of the phase resetting experiment. In this paper, we investigate the control mechanism of the coordinated finger tapping by both hands. First, the subjects were instructed to coordinate the finger tapping by both hands so as to keep the phase difference between two hands constant. The performance was evaluated by a systematic error and a standard deviation of phase differences. Second, we propose two coupled neural oscillators as a model for the coordinated finger tapping. Dynamical behavior of the model system is analyzed by using phase transition curves which were measured on one hand finger tapping in the previous experiment (Yamanishi et al., 1979). Prediction by the model is in good agreement with the results of the experiments. Therefore, it is suggested that the neural mechanism which controls the coordinated finger tapping may be composed of a coupled system of two neural oscillators each of which controls the right and the left finger tapping respectively.     

n:m Phase-locking of weakly coupled oscillators

Two weakly coupled oscillators are studied and the existence of n:m phase-locked solutions is shown. With the use of a slow time scale, the problem is reduced to a two-dimensional system on an invariant attracting torus. This system is further reduced to a one-dimensional dynamical system. Fixed points of this system correspond to n:m phase-locked solutions. The method is applied to a forced oscillator, linearly coupled - systems, and a pair of integrate and fire neuron models.     

An unusual form of phase walk in a system of coupled oscillators

The authors describe an unusual form of phase walk (i.e., a progressive change in phase angle between coupled oscillators) using the 10-Hz rhythmic discharges of the inferior cardiac and vertebral postganglionic sympathetic nerves (CN and VN, respectively) in hypercapnic, baroreceptor-denervated, and vagotomized cats anesthetized with urethane. Unlike phase walk ascribable to weakened coupling (desynchronization of oscillators), the phase walk of VN 10- Hz activity relative to CN10-Hz activity 1) recurred on the time scale of the respiratory cycle, 2) was bidirectional with CN-VN phase angle increasing during expiration and decreasing during inspiration, and 3) occurred over a range equivalent to one-half the period of the 10-Hz rhythm rather than a full cycle. Moreover, this form of phase walk occurred during strong coupling of the 10-Hz oscillators, as reflected by CN-VN coherence values approaching 1.0. The authors propose that the bidirectional phase walk reflects a state of strong coupling of the 10-Hz oscillators controlling the CN and VN, the angle of which is reset from cycle to cycle by the continuously changing level of activity in their respiratory inputs. In addition, the data demonstrate that frequency and amplitude modulation of sympathetic nerve discharge can be independently regulated by respiratory inputs.     

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.