Molecular dynamics study of carbon nanotube oscillator on gold surface

We investigated the substrate effect of carbon nanotube (CNT) oscillators using classical molecular dynamics simulations. Double-walled CNT oscillators on {100} gold surface were considered. The nanotube–gold interactions induced the compressive deformations of the outer nanotube and affected the transitional velocity and the energy dissipation of the nanotube oscillator. When the inner nanotube was extruded from the outer nanotube, the central regions of the outer nanotube were compressed by the nanotube–gold interactions and then, these compressive forces pushed out the inner nanotube and finally, the transitional velocity of the inner nanotube was slightly increased at the edges regions. Since the energy dissipation of the nanotube oscillator on gold surface was higher than that in vapor, the decrease of the transitional velocity for the nanotube oscillator on gold surface was greater than that for the nanotube oscillator in vapor.     

The Calculation of Frequency-Shift Functions for Chains of Coupled Oscillators, with Application to a Network Model of the Lamprey Locomotor Pattern Generator

Chains of coupled limit-cycle oscillators are considered, in which the coupling is assumed to be weak and only between adjacent oscillators. For such a system the change in frequency of an oscillator due to the coupling can be expressed, up to first order in thecoupling strength, by functions that depend only on the phase difference between the coupled oscillators. In this article a numerical algorithm is developed for the evaluation of these functions (the H-functions) in terms of a single oscillator and the interactions between coupled oscillators. The technique is applied to a connectionist model for the locomotor pattern generator in the lamprey spinal cord.An H-function so derived is compared to a function derived empirically(the C-function) from simulations of the same system. The phase lagsthat develop between adjacent oscillators in a simulated chain are compared with those predicted theoretically, and it is shown that coupling thatis functionally strong is nonetheless weak enough to behave as predicted.     

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.     

Interaction of coupled nonlinear oscillators having different intrinsic resting levels

The interaction among coupled oscillators is governed by oscillator properties (intrinsic frequency and amplitude) and coupling mechanisms. This study considers another oscillator property, the intrinsic resting level, and evaluates its role in governing oscillator interactions. The results of computer experiments on a chain of either three or five bidirectionally coupled nonlinear oscillators, suggest that an intrinsic resting level gradient, if present, is one of the factors governing the interaction between coupled oscillators. If there is no intrinsic frequency gradient, then an intrinsic resting level gradient is sufficient to produce many features of interaction among coupled oscillators. If both intrinsic frequency and intrinsic resting level gradients are present, then both of them determine the manner in which the coupled oscillators interact with each other.     

An oscillator network exhibiting a long-lasting response to an external signal

This study proposes an oscillator network to model the long-lasting responses observed in neural circuits. The responses of the proposed network model are represented by the temporal synchronization of the oscillators. The response duration does not depend on the natural frequency of the oscillators, which allows the responses to last much longer than the oscillation period of the oscillators. We can control the response duration by tuning the connection strengths between the oscillators and the external signal that triggers the responses. It is possible to break and restart the responses regardless of the way in which the oscillators are connected.     

Spatial patterns in coupled biochemical oscillators

Summary The effects of diffusion on the dynamics of biochemical oscillators are investigated for general kinetic mechanisms and for a simplified model of glycolysis. When diffusion is sufficiently rapid a population of oscillators relaxes to a globally-synchronized oscillation, but when diffusion of one or more species is slow enough, the synchronized oscillation can be unstable and a nonuniform steady state or an asynchronous oscillation can arise. The significance of these results vis-a-vis models of contact inhibition and zonation patterns is discussed.     

The Role of Axonal Delay in the Synchronization of Networks of Coupled Cortical Oscillators

Coupled oscillator models use a single phase variable to approximate the voltage oscillation of each neuron during repetitivefiring where the behavior of the model depends on the connectivityand the interaction function chosen to describe the coupling. Weintroduce a network model consisting of a continuum of theseoscillators that includes the effects of spatially decaying coupling and axonal delay. We derive equations for determining the stability of solutions and analyze the network behavior for two different interaction functions. The first is a sine function, and the second is derived from a compartmental model of a pyramidal cell.In both cases, the system of coupled neural oscillators can undergo a bifurcation from synchronous oscillations to waves.The change in qualitative behavior is due to the axonal delay,which causes distant connections to encourage a phase shift between cells. We suggest that this mechanism could contribute to the behavior observed in several neurobiological systems.     

TEMPORAL PATTERN OF LH SECRETION: REGULATION BY MULTIPLE ULTRADIAN OSCILLATORS VERSUS A SINGLE CIRCADIAN OSCILLATOR

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399–412, 2001)     

Temporal pattern of LH secretion: regulation by multiple ultradian oscillators versus a single circadian oscillator

The possibility that the 24h rhythm output is the composite expression of ultradian oscillators of varying periodicities was examined by assessing the effect of external continuously or pulsed (20-minute) Gonadotropinreleasing hormone (GnRH) infusions on in vitro luteinizing hormone (LH) release patterns from female mouse pituitaries during 38h study spans. Applying stepwise analyses (spectral, cosine fit, best-fit curve, and peak detection analyses) revealed the waveform shape of LH release output patterns over time is composed of several ultradian oscillations of different periods. The results further substantiated previous observations indicating the pituitary functions as an autonomous clock. The GnRH oscillator functions as a pulse generator and amplitude regulator, but it is not the oscillator that drives the ultradian LH release rhythms. At different stages of the estrus cycle, the effect of GnRH on the expression of ultradian periodicities varies, resulting in the modification of their amplitudes but not their periods. The functional output from the system of ultradian oscillators may superimpose a “circadian or infradian phenotype” on the observed secretion pattern. An “amplitude control” hypothesis is proposed: The temporal pattern of LH release is governed by several oscillators that function in conjunction with one another and are regulated by an amplitude-controlled mechanism. Simulated models show that such a mechanism results in better adaptive response to environmental requirements than does a single circadian oscillator. (Chronobiology International, 18(3), 399-412, 2001)     

Control of interaction strength in a network of the true slime mold by a microfabricated structure

The plasmodium of the true slime mold, Physarum polycephalum, which shows various nonlinear oscillatory phenomena, for example, in its thickness, protoplasmic streaming and concentration of intracellular chemicals, can be regarded as a collective of nonlinear oscillators. The plasmodial oscillators are interconnected by microscale tubes whose dimensions can be closely related to the strength of interaction between the oscillators. Investigation of the collective behavior of the oscillators under the conditions in which the interaction strength can be systematically controlled gives significant information on the characteristics of the system. In this study, we proposed a living model system of a coupled oscillator system in the Physarum plasmodium. We patterned the geometry and dimensions of the microscale tube structure in the plasmodium by a microfabricated structure (microstructure). As the first step, we constructed a two-oscillator system for the plasmodium that has two wells (oscillator part) and a channel (coupling part). We investigated the oscillation behavior by monitoring the thickness oscillation of the plasmodium in the microstructure with various channel widths. It was found that the oscillation behavior of two oscillators dynamically changed depending on the channel width. Based on the results of measurements of the tube dimensions and the velocity of the protoplasmic streaming in the tube, we discuss how the channel width relates to the interaction strength of the coupled oscillator system.     

Resetting biological oscillators—A stochastic approach

We present a stochastic approach to phase-resetting of an ensemble of oscillators. In order to describe stimulation-induced dynamical phenomena we develop a stochastic model which consists of an ensemble of phase oscillators interacting via random forces. Every single oscillator is submitted to a phase stimulus. The ensemble's dynamics is determined by a Fokker-Planck equation. The stationary states are calculated explicitly, whereas the transients are analysed numerically. If the stimulus of a given (non-vanishing) intensity is administered at a critical initial cluster phase for a critical duration T _crit the ensemble's synchronized oscillation is annihilated. A transition from type 1 resetting to type 0 resetting occurs when the stimulation duration exceeds T _crit. Stimulation causes a shift of the mean frequency of every single oscillator. This frequency shift is explicitly calculated by deriving the mean first passage time. The model shows that there is a subcritical intensity which is connected with an enhanced vulnerability to stimulation. The desynchronized states, the so-called black holes, are typically associated with a double peak in the ensemble's phase distribution. This is important for analysing experimental data because simple peak-detection algorithms are not able to extract the underlying dynamics.Our results are discussed from the experimentator's point of view so that the insights derived from our model can improve data analysis and design of stimulation experiments.     

Systems and synthetic biology approaches in understanding biological oscillators

Background: Self-sustained oscillations are a ubiquitous and vital phenomenon in living systems. From primitive single-cellular bacteria to the most sophisticated organisms, periodicities have been observed in a broad spectrum of biological processes such as neuron firing, heart beats, cell cycles, circadian rhythms, etc. Defects in these oscillators can cause diseases from insomnia to cancer. Elucidating their fundamental mechanisms is of great significance to diseases, and yet challenging, due to the complexity and diversity of these oscillators. Results: Approaches in quantitative systems biology and synthetic biology have been most effective by simplifying the systems to contain only the most essential regulators. Here, we will review major progress that has been made in understanding biological oscillators using these approaches. The quantitative systems biology approach allows for identification of the essential components of an oscillator in an endogenous system. The synthetic biology approach makes use of the knowledge to design the simplest, de novo oscillators in both live cells and cell-free systems. These synthetic oscillators are tractable to further detailed analysis and manipulations. Conclusion: With the recent development of biological and computational tools, both approaches have made significant achievements.     

Phase plane description of crayfish swimmeret oscillator

Crayfish swimmeret system shows rhythmic, coordinated behavior when the command fibers are stimulated chronically by electrical pulses, and the oscillating frequency becomes faster with increasing stimulus frequency. This behavior is organized by the distributed neural oscillators in the abdominal ganglia. We investigated the dynamics of the neural oscillators which are controlled by command fibers. Phase resetting experiment technique was used for this purpose; a temporary cessation of commanding pulses, which was regarded as suppressive perturbation for the neural oscillator, was applied to the chronically stimulated oscillator, and phase transition curves (PTCs) were measured. For the short cessation of command pulses, type 1 PTCs were obtained. With increasing cessation length, the PTC shifted downward, and finally changed into type 0. We also measured PTCs for temporarily increased stimulus frequency, which was an excitatory perturbation for the neural oscillator and increased the frequency of the oscillation transiently. For the short excitatory perturbation, the PTCs were also type 1 and shifted upward. PTCs changed their shapes from type 1 into type 0, as increasing the perturbation length. These shapes of the PTCs contain important information about the properties of the neural oscillator. Analyzing these PTCs, we present a phase plane diagram which describes the character of the command control of the neural oscillator.     

Transient and steady state phase response curves of limit cycle oscillators

The experiment of phase shifts resulting from discrete perturbations of stable biological rhythms has been carried out to study entrainment behavior of oscillators. There are two kinds of phase response curves, which are measured in experiments, according to as one measures the phase shifts immediately or long after the perturbation. The former is the first transient phase response curve and the latter is the steady state phase response curve. We redefine both curves within the framework of dynamical system theory and homotopy theory. Several topological properties of both curves are clarified. Consequently, it is shown that we must compare the shapes of both two phase response curves to investigate the inner structures of biological oscillators. Moreover, we prove that a single limit cycle oscillator involving only two variables cannot simulate transient resetting behavior reported by Pittendrigh and Minis (1964). In other words, the circadian oscillator of Drosophila pseudoobscura does not consist of a single oscillator of two variables. Finally we show that a model which consists of two limit cycle oscillators is able to simulate qualitatively the phase response curves of Drosophila.     

Frequency modulation of large oscillatory neural networks

Dynamical systems which generate periodic signals are of interest as models of biological central pattern generators and in a number of robotic applications. A basic functionality that is required in both biological modelling and robotics is frequency modulation. This leads to the question of whether there are generic mechanisms to control the frequency of neural oscillators. Here we describe why this objective is of a different nature, and more difficult to achieve, than modulating other oscillation characteristics (like amplitude, offset, signal shape). We propose a generic way to solve this task which makes use of a simple linear controller. It rests on the insight that there is a bidirectional dependency between the frequency of an oscillation and geometric properties of the neural oscillator’s phase portrait. By controlling the geometry of the neural state orbits, it is possible to control the frequency on the condition that the state space can be shaped such that it can be pushed easily to any frequency.     

Zum Mechanismus der biologischen 24-Stunden-Periodik

Summary The biological 24-hour-periodicity is based upon an endogenous (self-sustained) oscillation which is synchronized with the earth's rotation by periodically changing factors of the environment, primarily by the alternation of light and dark. These external Zeitgebers affect the phase of the endogenous oscillation. Theoretically, there are four different simple types of phase-control; all complicated types are combinations of these four types. In model experiments the behaviour of an oscillation in each of the four cases of phase-control is clearly demonstrated.The comparison of model experiments and biological experiments suggests that in organisms a specific combination-type of phase-control occurs. In this combination-type, a change in frequency is always positively correlated with a change in average level of the oscillation. Both parameters of the oscillation increase in light-active organisms and decrease in dark-active organisms with increasing light-intensity (circadian rule). In organisms both parameters are coupled by means of non-linear elements.The differential equation describing the 24-hour-periodcity is characterized by certain non-linearities. One of these makes the oscillation self-sustained and simultaneously couples the frequency of the oscillation to the average level, in the sense postulated by the circadian rule. The magnitude of the non-linearity is such that the resulting oscillation is intermediated between a harmonic and a relaxation type of oscillation, but has more characteristics of a harmonic oscillation. A second non-linearity which also couples frequency and level positively concerns the energy of recoil.All general properties of the biological 24-hour-periodicity can be reproduced by the described oscillator model. Some special properties (e.g. pattern) are more easily understood by the assumption of two coupled oscillators; the second oscillator, following the same general laws described above, is controlled by the first one. The oscillator hypothesis can be applied to biological periodicities with other frequencies; in general, the higher the frequency of a system the more the oscillation tends towards a relaxation type of oscillation.     

Oscillator decomposition of infant fNIRS data

The functional near-infrared spectroscopy (fNIRS) can detect hemodynamic responses in the brain and the data consist of bivariate time series of oxygenated hemoglobin (oxy-Hb) and deoxygenated hemoglobin (deoxy-Hb) on each channel. In this study, we investigate oscillatory changes in infant fNIRS signals by using the oscillator decompisition method (OSC-DECOMP), which is a statistical method for extracting oscillators from time series data based on Gaussian linear state space models. OSC-DECOMP provides a natural decomposition of fNIRS data into oscillation components in a data-driven manner and does not require the arbitrary selection of band-pass filters. We analyzed 18-ch fNIRS data (3 minutes) acquired from 21 sleeping 3-month-old infants. Five to seven oscillators were extracted on most channels, and their frequency distribution had three peaks in the vicinity of 0.01-0.1 Hz, 1.6-2.4 Hz and 3.6-4.4 Hz. The first peak was considered to reflect hemodynamic changes in response to the brain activity, and the phase difference between oxy-Hb and deoxy-Hb for the associated oscillators was at approximately 230 degrees. The second peak was attributed to cardiac pulse waves and mirroring noise. Although these oscillators have close frequencies, OSC-DECOMP can separate them through estimating their different projection patterns on oxy-Hb and deoxy-Hb. The third peak was regarded as the harmonic of the second peak. By comparing the Akaike Information Criterion (AIC) of two state space models, we determined that the time series of oxy-Hb and deoxy-Hb on each channel originate from common oscillatory activity. We also utilized the result of OSC-DECOMP to investigate the frequency-specific functional connectivity. Whereas the brain oscillator exhibited functional connectivity, the pulse waves and mirroring noise oscillators showed spatially homogeneous and independent changes. OSC-DECOMP is a promising tool for data-driven extraction of oscillation components from biological time series data.     

Frequency tuning in animal locomotion

In locomotion that involves repetitive motion of propulsive structures (arms, legs, fins, wings) there are resonant frequencies f(*) at which the energy consumption is a minimum. As animals need to change their speed, they can maintain this energy minimum by tuning their body resonances. We discuss the physical principles of frequency tuning, and how it relates to forces, damping, and oscillation amplitude. The resonant frequency of pendulum-type oscillators (e.g. swinging arms and legs) may be changed by varying the mass moment of inertia, or the vertical acceleration of the pendulum pivot. The frequency of elastic vibrations (e.g. the bell of a jellyfish) can be tuned with a non-linear modulus of elasticity: soft for low deflection amplitudes (low resonant frequency), and stiff for large displacements (high resonant frequency). Tuning of elastic oscillations can also be achieved by changing the effective length or cross-sectional area of the elastic members, or by allowing springs in parallel or in series to become active. We propose that swimming and flying animals generate oscillating propulsive forces from precisely placed shed vortices and that these tuned motions can only occur when vortex shedding and the simple harmonic motion of the elastic elements of the propulsive structures are in resonance.     

Synchronization in a neural network of phase oscillators with the central element

A neural network model is considered which is designed as a system of phase oscillators and contains the central oscillator and peripheral oscillators which interact via the central oscillator. The regime of partial synchronization was studied when current frequencies of the central oscillator and one group of peripheral oscillators are near to each other while current frequencies of other peripheral oscillators are far from being synchronized with the central oscillator. Approximation formulas for the average frequency of the central oscillator in the regime of partial synchronization are derived, and results of computation experiments are presented which characterize the accuracy of the approximation.