Analysis of limit-cycle conditions in intercoupled relay oscillators with reference to gastrointestinal modelling

This paper presents an exact analysis of a mutually coupled relay oscillator based on a method orignated by Tsypkin. Limit-cycle frequencies and phases can be determined exactly using this method, unlike other approximate methods based on describing functions and harmonic balance techniques. A new method of exact determination of limit-cycle stability is also shown to give excellent agreement with simulation studies. Different types of intercoupling are shown to give different stability conditions, and these are discussed in relation to gastrointestinal (GI) smooth muscle modelling. GI tract electrical activity has previously been modelled using bidirectionally coupled nonlinear oscillators, and the results of the present analysis of relay oscillators is compared with other studies using van der Pol dynamics.     

A statistical model of the human core-temperature circadian rhythm

We formulate a statistical model of the human core-temperature circadian rhythm in which the circadian signal is modeled as a van der Pol oscillator, the thermoregulatory response is represented as a first-order autoregressive process, and the evoked effect of activity is modeled with a function specific for each circadian protocol. The new model directly links differential equation-based simulation models and harmonic regression analysis methods and permits statistical analysis of both static and dynamical properties of the circadian pacemaker from experimental data. We estimate the model parameters by using numerically efficient maximum likelihood algorithms and analyze human core-temperature data from forced desynchrony, free-run, and constant-routine protocols. By representing explicitly the dynamical effects of ambient light input to the human circadian pacemaker, the new model can estimate with high precision the correct intrinsic period of this oscillator ( approximately 24 h) from both free-run and forced desynchrony studies. Although the van der Pol model approximates well the dynamical features of the circadian pacemaker, the optimal dynamical model of the human biological clock may have a harmonic structure different from that of the van der Pol oscillator.     

A novel approach to the van der pol oscillator: Natural frequency and entrainment

Abstract

The physics of the van der Pol oscillator as realized by the Meissner circuit is discussed by analogy to the beat phenomenon and by a consequent analysis of current balance. The current balance method leads to a new, very accurate equation for the dependence of the oscillator frequency on the feedback parameter. Several aspects of entrainment (existence, limited frequency range, dependence on parameters, phase shift) can be explained, too. Numerical results are presented which have been obtained by solving the homogeneous and inhomogeneous van der Pol equation with a Runge‐Kutta method.     

A simpler model of the human circadian pacemaker

Numerous studies have used the classic van der Pol oscillator, which contains a cubic nonlinearity, to model the effect of light on the human circadian pacemaker. Jewett and Kronauer demonstrated that Aschoff's rule could be incorporated into van der Pol type models and used a van der Pol type oscillator with higher order nonlinearities. Kronauer, Forger, and Jewett have proposed a model for light preprocessing, Process L, representing a biochemical process that converts a light signal into an effective drive on the circadian pacemaker. In the paper presented here, the authors use the classic van der Pol oscillator with Process L and Jewett and Kronauer's model of Aschoff's rule to model the human circadian pacemaker. This simpler cubic model predicts the results of a three-pulse human phase response curve experiment and a two-pulse amplitude reduction study with as much, or more, accuracy as the models of Jewett and Kronauer and Kronauer, Forger, and Jewett, which both employ a nonlinearity of degree 7. This suggests that this simpler cubic model should be considered as a potential alternative to other models of the human circadian system currently available.     

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.     

On the complex dynamics of intracellular ganglion cell light responses in the cat retina

We recorded intracellular responses from cat retinal ganglion cells to sinusoidal flickering lights, and compared the response dynamics with a theoretical model based on coupled nonlinear oscillators. Flicker responses for several different spot sizes were separated in a smooth generator (G) potential and corresponding spike trains. We have previously shown that the G-potential reveals complex, stimulus-dependent, oscillatory behavior in response to sinusoidally flickering lights. Such behavior could be simulated by a modified van der Pol oscillator. In this paper, we extend the model to account for spike generation as well, by including extended Hodgkin-Huxley equations describing local membrane properties. We quantified spike responses by several parameters describing the mean and standard deviation of spike burst duration, timing (phase shift) of bursts, and the number of spikes in a burst. The dependence of these response parameters on stimulus frequency and spot size could be reproduced in great detail by coupling the van der Pol oscillator and Hodgkin-Huxley equations. The model mimics many experimentally observed response patterns, including non-phase-locked irregular oscillations. Our findings suggest that the information in the ganglion cell spike train reflects both intraretinal processing, simulated by the van der Pol oscillator, and local membrane properties described by Hodgkin-Huxley equations. The interplay between these complex processes can be simulated by changing the coupling coefficients between the two oscillators. Our simulations therefore show that irregularities in spike trains, which normally are considered to be noise, may be interpreted as complex oscillations that might carry information.To the memory of Prof. Otto-Joachim Grusser     

A van der Pol coupled-oscillator model as a basis for developing a system for suppressing MHD instabilities in a tokamak

The main parameters of tokamak discharges are known to be limited by large-scale MHD instabilities. Sometimes, the instabilities lead to a rapid (on time scales of tens of microseconds) disruption of the discharge current and to the release of all the energy stored in the plasma column at the discharge chamber wall. This process, which is called the disruptive instability, may have irreversible catastrophic consequences for the operation of a fusion reactor. In the present paper, a study is made of the dynamics of self-oscillations in systems of two and six van der Pol coupled oscillators. A van der Pol coupled-oscillator model is used to develop a multivariable feedback controller based on the combined principle of compensating for internal cross feedbacks within the object and introducing damping feedbacks in each control channel. By using mathematical simulation methods, it is shown that the controller designed guarantees the suppression of self-oscillations in a system of van der Pol oscillators over a fairly broad range of parameters of the object under control (and thereby provides the structural stability of the object). The nonlinear control system model makes it possible to suppress coupled MHD perturbations developing in a tokamak plasma.     

Integer-ratio entrainment in mutually-coupled non-linear oscillators

In recent years entrainment conditions for mutually-coupled, non-linear oscillators have been studied for a number of biomedical applications and using different analytical methods. The emphasis has been on entrainment between oscillators of similar frequencies. In this paper entrainment conditions are considered for oscillators having intrinsic frequency ratio of about 3:1 and which exhibit integer-ratio synchronization. This condition has application in the study of blood pressure regulation particularly in relation to respiratory effects. Coupling has been observed between respiration and the vasomotor activity associated with the baroreceptor reflex, which has an intrinsic 0·1 Hz component. At normal breathing frequencies the frequency ratio of the respiratory and vasomotor components is in the region of 3:1 hence integer-ratio entrainment is feasible. Using a coupled van der Pol model the entrainment zones for different parameters are described. The parameters considered allow for varying amounts of output, output rate and delay in the intercoupling structure. In particular, it is shown that the entrainment regions are strongly affected by the nature of the coupling. Within these zones the harmonic balance method is developed to provide an analytical solution to frequency, amplitudes and phase conditions. The assumed solution is valid only for certain regions of the stability zones and the reason for this is demonstrated and the means whereby this can be overcome are indicated.     

Computing van der Waals energies in the context of the rotamer approximation

The rotamer approximation states that protein side-chain conformations can be described well using a finite set of rotational isomers. This approximation is often applied in the context of computational protein design and structure prediction to reduce the complexity of structural sampling. It is an effective way of reducing the structure space to the most relevant conformations. However, the appropriateness of rotamers for sampling structure space does not imply that a rotamer-based energy landscape preserves any of the properties of the true continuous energy landscape. Specifically, because the energy of a van der Waals interaction can be very sensitive to small changes in atomic separation, meaningful van der Waals energies are particularly difficult to calculate from rotamer-based structures. This presents a problem for computational protein design, where the total energy of a given structure is often represented as a sum of precalculated rigid rotamer self and pair contributions. A common way of addressing this issue is to modify the van der Waals function to reduce its sensitivity to atomic position, but excessive modification may result in a strongly nonphysical potential. Although many different van der Waals modifications have been used in protein design, little is known about which performs best, and why. In this paper, we study 10 ways of computing van der Waals energies under the rotamer approximation, representing four general classes, and compare their performance using a variety of metrics relevant to protein design and native-sequence repacking calculations. Scaling van der Waals radii by anywhere from 85 to 95% gives the best performance. Linearizing and capping the repulsive portion of the potential can give additional improvement, which comes primarily from getting rid of unrealistically large clash energies. On the other hand, continuously minimizing individual rotamer pairs prior to evaluating their interaction works acceptably in native-sequence repacking, but fails in protein design. Additionally, we show that the problem of predicting relevant van der Waals energies from rotamer-based structures is strongly nonpairwise decomposable and hence further modifications of the potential are unlikely to give significant improvement.     

Dynalets: A new method for modelling and compressing biological signals. Applications to physiological and molecular signals

The biological information coming from electrophysiologic sensors like ECG, pulse sensor or from molecular signal devices like NMR spectrometry has to be visualized and manipulated in a compressed way for an efficient medical use by clinicians, if stored in scientific data bases or in personalized patient records repositories. Here, we define a new transform called Dynalet based on Liénard ordinary differential equations susceptible to model the mechanism at the source of the studied signal, and we propose to apply this new technique first to the modelling and compression of real biological periodic signals like ECG and pulse rhythm. We consider that the cardiovascular activity results from the summation of cellular oscillators located in the cardiac sinus node and we show that, as a result, the van der Pol oscillator (a particular Liénard system) fits well the ECG signal and the pulse signal. The reconstruction of the original signal (pulse or ECG) using Dynalet transform is then compared with that of Fourier, counting the number of parameters to be set for obtaining an expected signal-to-noise ratio. Then, we apply the Dynalet transform to the modelling and compression of molecular spectra obtained by protein NMR spectroscopy. The reconstruction of the original signal (peak) using Dynalet transform is again compared with that of Fourier. After reconstructing visually the peak, we propose to periodize the signal and give it to hear, the whole process being called the protein “stethoscope”.     

Transition Between Advance and Delay Responses to Eastbound Transmeridian Flights

In response to eastbound transmeridian flights, which result in zeitgeber phase advance shifts, adaptation of the circadian system to the new time zone by phase delays and advances are observed. The delay response to an advance zeitgeber shift has been called an antidromic response. For the shift at which the transition from an advance to an antidromic response occurs, the term critical shift is introduced.

For the study of critical shifts, a flight experiment across nine time zones and numerical simulations of a van der Pol equation have been evaluated. The interest is focussed on the determination of a range for critical abrupt shifts. An abrupt shift means that the ensemble of zeitgebers including geophysical zeitgebers and the rest-activity cycle is shifted immediately in the new time zone. The range of critical advance shifts has been estimated to reach from + 7 to + 10 hr. In the literature, results were reported which would imply a much wider range. The discussion of these observations shows that the actual shifts were presumably not abrupt in the quoted experiments.

The consequences of critical shifts for jet lag symptoms are investigated. If reduced circadian amplitudes and long times taken for the resynchronization contribute to the feeling of jet lag, the symptoms will be worst for shifts close to the critical one, as numerical simulations revealed. Manipulations of such shifts with the aim to alleviate jet lag are discussed.     

Solitary wave and spatially locked solitary pattern in a chemical reaction system

We investigate the behavior of a one-dimensional two component dynamical system. The dynamical equations are obtained by extracting an essence out of equations which describe the behavior of a biochemical reaction catalyzed by an allosteric protein. The obtained dynamical equations are similar to van der Pol equations. The dynamical equations are solved numerically. In the continuous system, a solitary wave is found to occur in certain ranges of the parameter space. The condition of occurrence of the solitary wave is investigated. The solitary wave can be induced by various initial perturbations, including rectangular ones with space-wise length longer than a certain critical value. The property of the solitary wave is similar to that of the impulses in nervous systems. In the discrete system, a spatially locked solitary pattern is found to occur in certain ranges of the parameter space.     

Dynamical correlation of non-stationary signals in time domain—A comparative study

In a thorough study, the multitaper (MTM) and the extended continuous wavelet-transform (CWT) coherence-analysis methods were compared in terms of there application in determining the dynamics from the electroencephalogram (EEG) and electromyogram (EMG) signals of patients with Parkinsonian tremor. The main aim of the study in a biological point of view is to analyze whether the basic tremor frequency and its “first harmonic” frequency of Parkinsonian tremor are really harmonically related or are in fact distinct processes.The extension of the CWT is achieved by using a Morlet wavelet as the analysis window with an adjustable relative bandwidth which gives the flexibility in setting a desired frequency resolution. In order to obtain a perspective view of the two methods, they were applied to two different model signals to determine their actual threshold in detecting short-lived changes in the analysis of non-stationary signals and to determine their noise thresholds by adding external noise to the signals to test the reduction in coherence to be not merely due to the random fluctuations in stochastic signals. Beyond applying an autoregressive 2nd-order and a coupled van der Pol model system, however, also true EEG and EMG data from five Parkinson patients were used. The results were compared in terms of the time and frequency resolutions of these two methods, and it was determined that the multitaper method was able to detect reduction in power and coherence as short as 1 s. The extended CWT analysis only revealed gaps that were longer than 3 s.The time gaps in the coherence indicate the loss of connection between the cortex and muscle during the respective time intervals. This more accurate analysis of the MTM was also seen in the dynamical EEG–EMG coherence at the tremor frequency and its “first harmonic” of Parkinsonian patients.In terms of our “biological” aim, this shows distinct prevalence of the corticomuscular coupling at those frequencies over time. Applying this method to biological data reveals important aspects about their dynamics, e.g., the distinct dynamics between basic frequency and “first harmonic” frequency over time in Parkinsonian tremor.     

Coupled oscillators utilised as gait rhythm generators of a two-legged walking machine

The gait of current two-legged walking machines differs from that of humans, although the kinematic structures of these machines' legs frequently imitate human limbs. This paper presents a method of generating the trajectories of hip and knee joint angles resulting in a gait pattern similar to that of a human. For this purpose the solutions of coupled van der Pol oscillator equations are utilised. There is much evidence that these equations can be treated as a good model of the central pattern generator generating functional (also locomotional) rhythms in living creatures. The oscillator equations are solved by numerical integration. The method of changing the type of gait by changing appropriate parameter values in the oscillator equations is presented (change of velocity and trajectory of leg-ends). The results obtained enable enhanced control of twolegged walking systems by including gait pattern generators which will assume a similar role to that of biological generators.     

Vespidae (Hymenoptera) of Vietnam 3: Synoptic key to Vietnamese species of the polistine genus Ropalidia, with notes on taxonomy and distribution

Nineteen species of the paper wasp genus Ropalidia are recorded from Vietnam, of which three are new records for Vietnam. A key to all of the species, distribution records and taxonomic remarks are given. New synonymies proposed are: Ropalidia bicolorata parvula van der Vecht, 1962, under R. bicolorata van der Vecht, 1962; R. hongkongensis juncta van der Vecht, 1941, under R. hongkongensis (de Saussure, 1854); Icaria jucunda Cameron, 1898, and R. marginata sundaica van der Vecht, 1941, under R. marginata (Lepeletier, 1836); R. rufocollaris atrata van der Vecht, 1941, under R. rufocollaris (Cameron, 1900); and R. stigma rufa van der Vecht, 1941 and R. stigma nigrolineata van der Vecht, 1962, under R. stigma (Smith, 1858).     

Noise-induced neural impulses

The firing pattern of neural pulses often show the following features: the shapes of individual pulses are nearly identical and frequency independent; the firing frequency can vary over a broad range; the time period between pulses shows a stochastic scatter. This behaviour cannot be understood on the basis of a deterministic non-linear dynamic process, e.g. the Bonhoeffer-van der Pol model. We demonstrate in this paper that a noise term added to the Bonhoeffer-van der Pol model can reproduce the firing patterns of neurons very well. For this purpose we have considered the Fokker-Planck equation corresponding to the stochastic Bonhoeffer-van der Pol model. This equation has been solved by a new Monte Carlo algorithm. We demonstrate that the ensuing distribution functions represent only the global characteristics of the underlying force field: lines of zero slope which attract nearby trajectories prove to be the regions of phase space where the distributions concentrate their amplitude. Since there are two such lines the distributions are bimodal representing repeated fluctuations between two lines of zero slope. Even in cases where the deterministic Bonhoeffer-van der Pol model does not show limit cycle behaviour the stochastic system produces a limit cycle. This cycle can be identified with the firing of neural pulses.     

A model for the speed of memory retrieval

The memory retrieval process of number problems with external noise is studied with the use of the Bonhoeffer–van der Pol oscillator model. Three cell assembly responses are simulated, coding one true number and two neighboring erroneous. The time of a correct response, T _c, was averaged over statistical assemblies of numerous trials. It is demonstrated that T _c takes a minimum value for a certain noise intensity. This result correlates well with experimental data by Usher and Feingold (2000). The location of the minimum as a function of the time delay between two consecutive simulation trials is investigated.