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     

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.     

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.     

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     

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.     

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 Mathematical Model of the Human Circadian System and Its Application to Jet Lag

A mathematical model of the circadian system is described that is appropriate for application to jet lag. The core of the model is a van der Pol equation with an external force. Approximate solutions of this equation in which the external force is composed of a constant and an oscillating term are investigated. They lead to analytical expressions for the amplitude and period of free-running rhythms and for the frequency limits of the entrainment region. The free-running period increases quadratically with stiffness. Both period and amplitude depend on the value of the constant external force. The width of the range of entrainment is mostly determined by the external force, whereas the relative position of this range follows the intrinsic period of the oscillator. Experiments with forced and spontaneous internal desynchronization were evaluated using these analytical expressions, and estimates were obtained for the intrinsic period of the oscillator, its stiffness, and the external force. A knowledge of these model parameters is essential for predictions about circadian dynamics, and there are practical implications for the assessment of the adaptation after rapid transmeridian travel.     

A Mathematical Model of the Human Circadian System and Its Application to Jet Lag

A mathematical model of the circadian system is described that is appropriate for application to jet lag. The core of the model is a van der Pol equation with an external force. Approximate solutions of this equation in which the external force is composed of a constant and an oscillating term are investigated. They lead to analytical expressions for the amplitude and period of free-running rhythms and for the frequency limits of the entrainment region. The free-running period increases quadratically with stiffness. Both period and amplitude depend on the value of the constant external force. The width of the range of entrainment is mostly determined by the external force, whereas the relative position of this range follows the intrinsic period of the oscillator. Experiments with forced and spontaneous internal desynchronization were evaluated using these analytical expressions, and estimates were obtained for the intrinsic period of the oscillator, its stiffness, and the external force. A knowledge of these model parameters is essential for predictions about circadian dynamics, and there are practical implications for the assessment of the adaptation after rapid transmeridian travel.     

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.     

The Stability of Clathrate Hydrates: Temperature Dependence of Dissociation Pressure in Xe and Ar Hydrate

Abstract

The thermodynamic stability of clathrate hydrates I and II encaging xenon or argon has been investigated by examining the temperature dependence of the dissociation pressure. The evaluation of the stability is made based on the generalized van der Waals and Platteeuw theory developed by Tanaka and Kiyohara [J. Chem. Phys. 98, 4098 (1993)]. In the new treatment, the free energy of formation of hydrates in equilibrium with ice is calculated by taking the coupling of the host lattice vibrations with guests into consideration. The predicted dissociation pressures of Xe and Ar hydrates agree well with experiments in higher temperature range. A poor agreement between experiment and calculation for Ar clathrate hydrate at low temperature is improved by the use of a quantum mechanical partition function for a harmonic oscillator in evaluating the free energy difference between ice and empty hydrate.     

Zum Mechanismus der biologischen 24-Stunden-Periodik

Summary The level (=arithmetic average of all instantaneous values)of a self-sustained oscillation in general influences all properties of the oscillation, including period, amplitude and shape of the oscillation, and the rate of exchange of energy between the oscillator and its environment. Only when the non-linear damping factor does not depend on the instantaneous value of the oscillating function, but only on the amplitude of the oscillation, are the other properties independent of the average level. The differential equations describing self-sustained oscillations cannot be solved exactly, but methods of approximation are applicable. Numerical solutions to several different forms of the equations will be discussed.In the simplest case (van der Pol equation) all properties of the self-sustained oscillation (e.g. period, amplitude) are extreme when the level is zero. The oscillation continues only within a given range of levels (oscillating range); outside this range, the oscillation damps out. In other modifications of the equation, the oscillating function cannot assume a zero value. In all cases, the extent to which the average level influences the different properties depends on the factor , which describes the position of the oscillation within the range between harmonic and relaxation types of oscillation.In the elementary van der Pol equation, the correlation between level and frequency changes sign within the oscillating range; that is, the circadian rule, demanding an always positive correlation between level and frequency, cannot be fulfilled. Only with an additional non-linearity in the energy of recoil does the correlation remain unchanged in sign throughout the oscillating range. A stability condition demands a positive sign for this non-linearity, and hence, for the correlation (fulfilling the circadian rule); if the sign is negative (violating the circadian rule), the oscillation becomes unstable. With an additional term of the third order, the oscillation acquires a two-peaked shape typical of many circadian oscillations.A simple differential equation describing all general properties of the circadian periodicity must fulfil these conditions: the oscillation must be self-sustained and limited to positive values; and the energy of recoil must be non-linear with a positive coefficient to obtain the appropriate correlation between level and frequency. In the equations here developed the environment directly influences only one parameter of the oscillation, i.e. the level. In addition to the circadian periodicity, the differential equations here examined describe the behavior of several other biological oscillations.

---

---

Die benutzten mathematischen Begriffe folgen — soweit dort angeführt — den Benennungen des DIN-Blattes 1311; im Anhang I sind die wichtigsten Begriffe noch einmal zusammenfassend definiert.     

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.     

Molecular dynamics simulations of carbon nanotube oscillators deformed by encapsulated copper nanowires

Pure carbon nanotube (CNT) oscillators are compared to the corresponding CNT oscillators encapsulating copper nanowires (Cu@CNTs) by molecular dynamics simulations. The classical oscillation theory provides a fairly good estimate of the mass dependence of the operating frequency when the CNT surface is not deformed by the Cu nanowire. The structural deformations of the CNT induced by the encapsulated copper nanowire have a greater effect on the oscillation frequency than the mass of the copper nanowire. The excess forces of the Cu@CNT oscillator are slightly higher than those of the CNT oscillator and the excess van der Waals forces induced by the inter-wall interactions are 17 times higher than the excess forces induced by the Cu nanowire–CNT interactions.     

On the Role of Density Fluctuations in the Equation of State of a Simple Fluid

Abstract

We address the wellknown problems intorduced into the theory of fluids by density fluctuations in the form of van der Waals loops and nonclassical critical phenomena. A clean separation of long and short range density fluctuations is achieved by use of cell-constrained models which display well-defined van der Waals loops and classical behaviour around the critical point. For a pure Lennard-Jones fluid with occupancy restricted to 1 or 8 particles per cell, the phase diagram is determined by Monte Carlo simulation. By considering the deviations from the normal simulations without cell constraint, the effects of longer range density fluctuations are exposed. The system size dependence of the van der Waals loops present in all simulations of fluids is analyzed in terms of the GvdW free energy density functional theory, which is formuiated on the basis of the cell concept. The loops are found to gradually disappear either with greater cel occupancy or increasing total particle number in the simulation box.     

A Relaxation Oscillator Description of the Burst-Generating Mechanism in the Cardiac Ganglion of the Lobster,Homarus americanus

Properties of the neural mechanism responsible for generating the periodic burst of spike potentials in the nine ganglion neurons were investigated by applying brief, single shocks to the four small cells with extracellular electrodes placed near the trigger zones of the small cells. The shock elicited a burst if presented during the latter portion of the silent period, terminated a burst during the latter portion of the burst period, and was followed by a newly initiated burst during the early portion of the burst period. The resultant changes in burst and silent period durations were quantitatively described by a second-order non-linear differential equation similar to the van der Pol equation for a relaxation oscillator. The equation also qualitatively described changes in firing threshold of the small cells during the burst cycle. The first derivative of the solution to the equation is similar to slow transmembrane potentials in neurons that are involved in generation of burst activity in other crustacean cardiac ganglia.     

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.     

Monte Carlo Study of the Buckingham Exponential-six Fluid

Abstract

In this paper we report the results of extensive Monte Carlo simulations of a pure fluid of Buckingham modified exponential-six molecules. Data are presented for the configurational energy and pressure covering a wide range of temperatures and densities. These data are interpreted using the generalized van der Waals partition function with a novel separation into free volume and mean potential terms. We find, surprisingly, that the Buckingham fluid is described by a simple van der Waals-like equation of state provided that the b parameter is temperature dependent and chosen in a theoretically correct manner.     

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.     

The spike generation zone of the ampullary electroreceptor

Sensory input to the central nervous system begins with a transduction step, specialized to the sensory modality involved, resulting in the production of postsynaptic electrical input to the outermost branches of a dendritic tree. Spatiotemporal summation of this slow input as it converges upon the axon then initiates the production of or modulates the rate of ongoing production of fast neural spikes destined for the central nervous system. We present a novel circuit design consisting of an operational amplifier, a tunnel diode and linear passive components, intended to model the spike generation zone at which the transformation of neural input from slow to fast format takes place. Our circuit is shown to be a relaxation oscillator of the van der Pol type. Simulated postsynaptic current modulates the frequency of spike production by the relaxation oscillator model, producing a stimulus-response characteristic which can be compared with those observed in vivo. Stimulus-response data for our model match data available in the literature for the ampullary electroreceptor of elasmobranch fish.