Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

Entrainment in pacemakers characterized by a V-shaped PRC

The behaviour of a class of pacemakers characterized by a V-shaped PRC has been determined, for all possible frequencies and amplitudes of stimulation. The analytical study of the phase transition equation reveals that all rhythmic stimuli, but for a set of measure zero, give rise to entrainment. The ratio between firing and stimulation frequencies is a generalized Cantor function of the ratio between spontaneous and stimulation frequencies. A procedure to compute the detailed input/output pattern that underlies each entrainment ratio is given. Finally, the neurophysiological assumptions and implications of the results obtained are discussed.     

一端受外力的交联振荡器链中的内部传输

本文研究了二类一端受外力的交联振荡器链：最邻近多相位交联振荡器链,以及多重交联振荡器链,讨论了它们产生内部传输,即各振荡器与外力具有相同频率的现象。文中近似相位差方程、指数二分性理论和中心流形理论被应用于系统的渐近近似。研究。本文得到了更符合于实际情况的神经网络CPG链动态特性分析结论。     

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

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

Entrainment of two coupled van der pol oscillators by an external oscillation

A system composed of two coupled internal oscillators and an external oscillation is studied as a model of biological control systems. The type of interaction between the internal oscillators is a mutual and dissipative one. Three macroscopic states of the internal oscillators are demonstrated in the absence of the external oscillation. Strict and loose entrainment regions of the internal oscillators by the external oscillation are shown in respect to the intensity of the mutual interaction, the intrinsic frequency difference of the internal oscillations, and the magnitude and frequency of the external oscillation. On the other hand, an idea of holonic system is introduced and the fundamental properties of the model as a holonic system are elucidated.     

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.     

Differential contributions of synaptic and intrinsic inhibitory currents to speech segmentation via flexible phase-locking in neural oscillators

Current hypotheses suggest that speech segmentation—the initial division and grouping of the speech stream into candidate phrases, syllables, and phonemes for further linguistic processing—is executed by a hierarchy of oscillators in auditory cortex. Theta (∼3-12 Hz) rhythms play a key role by phase-locking to recurring acoustic features marking syllable boundaries. Reliable synchronization to quasi-rhythmic inputs, whose variable frequency can dip below cortical theta frequencies (down to ∼1 Hz), requires “flexible” theta oscillators whose underlying neuronal mechanisms remain unknown. Using biophysical computational models, we found that the flexibility of phase-locking in neural oscillators depended on the types of hyperpolarizing currents that paced them. Simulated cortical theta oscillators flexibly phase-locked to slow inputs when these inputs caused both (i) spiking and (ii) the subsequent buildup of outward current sufficient to delay further spiking until the next input. The greatest flexibility in phase-locking arose from a synergistic interaction between intrinsic currents that was not replicated by synaptic currents at similar timescales. Flexibility in phase-locking enabled improved entrainment to speech input, optimal at mid-vocalic channels, which in turn supported syllabic-timescale segmentation through identification of vocalic nuclei. Our results suggest that synaptic and intrinsic inhibition contribute to frequency-restricted and -flexible phase-locking in neural oscillators, respectively. Their differential deployment may enable neural oscillators to play diverse roles, from reliable internal clocking to adaptive segmentation of quasi-regular sensory inputs like speech.     

Evidence for the entrainment of breathing by locomotor pattern in human

In human, it has been shown that interactions between locomotor and respiratory patterns may lead to locomotor-respiratory couplings termed entrainment. In order to prove that this coupling is really an entrainment, we tried to show that it obeys one of the expected rules, i.e. that it evolves and is not present for all imposed locomotor frequencies. For that purpose, seventeen healthy volunteers were asked to run on a treadmill at 14 different locomotor rates (instead of 2 or 3 in previous works) for 40 s. All the subjects did not exhibit the same coupling and different relationships could be obtained: the most commonly observed was 2:1 (2 locomotor activities for a respiratory one) but other forms could appear (4:1 and even 5:2 or 3:2). When the coupling evolution was followed in the same subject, it did not appear for all locomotor frequencies but only for locomotor periods close to harmonics of respiratory ones (absolute coordination). On both sides of these values, it progressively evolved to relative coordination and to the lack of coordination. When two forms of absolute coordination were observed in a same subject, the phase relationships followed the rules of the entrainment. Compared to data obtained in quadrupeds, these results suggest that the entrainment of breathing frequency by the locomotor activity is due to central interactions between the respiratory and locomotor pattern generators and does not depend on a chemical regulation avoided here by short locomotor sequences.     

Model of bidirectional interaction between myocardial pacemakers based on the phase response curve

As a basis for the study of sinus rhythm determination, a model is proposed of bidirectionally-coupled oscillators as a system of difference equations based on the phase response curve of sinoatrial pacemaker cells. Solutions corresponding to the one-to-one synchronization of the two pacemakers are obtained, and the relation among those solutions is examined: It is revealed that two different solutions with different cycle length coexist, and the synchronized frequency can be higher or lower than the original intrinsic frequencies of the two pacemaker cells. The experimental results of the cultured cells of cardiac pacemakers are interpreted by the analytical result of the model.     

Entrainment boundaries of relaxation oscillators coupled by low pass filters

Abstract

A simple electronic network employing unijunction transistors has been used to obtain qualitative information regarding the entrainment behavior of relaxation oscillators coupled by low pass filters. Using arbitrary criteria, entrainment boundaries have been determined over the frequency range from 0.14 to 0.5 Hz for filters having time constants of 0.55,1.1 and 1.7 s. It is shown that the efficacy of entrainment is related to filter time constant and the Fourier structure of the bilaterally coupled signals     

A comparison of circatidal rhythmicity and entrainment by hydrostatic pressure cycles in the rock goby, Gobius paganellus L. and the shanny, Lipophrys pholis (L.)

The intertidal teleosts Gobius paganellus and Lipophrys pholis show endogenous circatidal activity rhythms when recorded in constant conditions. Under these conditions, the rhythm of L. pholis is the more precise which may indicate stronger coupling between underlying circalunadian oscillators in this species. In G. pagunellus the inter-oscillator coupling may be weaker and this could enable a more subtle interpretation of tidal fluctuations than in L. pholis . The oscillators may, however, be fundamentally different in the two species; circalunadian in G. paganellus and circatidal in L. pholis .
When exposed to hydrostatic pressure cycles of tidal frequency both species responded pre- dominantly to increasing pressure, which suggests that in the wild they are likely to be most active on the rising tide. Hydrostatic pressure cycles are confirmed as a zeitgeber for both species by the successful entrainment of some individuals. The lack of entrainment of others impIies that additional zeitgebers are required for complete entrainment.     

"Rapid" rhythmic discharges of sympathetic nerves: sources, mechanisms of generation, and physiological relevance

Like virtually all other physiological control systems, the sympathetic nervous system controlling cardiovascular function is characterized by the presence of rhythmic activity. These include slow rhythms with frequencies at or below that of the respiration and rapid rhythms with frequencies at or above that of the heart beat. The rapid rhythms are the subject of this review. The specific questions entertained are as follows: (1) Are the rapid cardiac-related and 10-Hz rhythms inherent to central sympathetic networks, or are they imposed on sympathetic nerve discharge (SND) by extrinsic periodic inputs? (2) Does basal SND arise from an anatomically circumscribed "vasomotor center" composed of pacemaker neurons in the rostral ventrolateral medulla or from an anatomically distributed network oscillator composed of different types of brainstem neurons, none of which necessarily have intrinsic pacemaker properties? (3) Are the rapid rhythms generated by single circuits or by systems of coupled oscillators, each with a separate target? (4) Are the rapid rhythms in SND simply by-products of the sympathetic generating mechanisms, or do they subserve selective and special functions, such as the formulation of differential patterns of spinal sympathetic outflow that support particular behaviors? The controversial aspects of these issues and the state-of-the-art analytical methods used to study them are stressed in this review.     

Sensitivity of basic oscillatory mechanisms for pattern generation and detection

 Intrinsic oscillators are the basic building blocks of central pattern generators, which model the neural circuits underlying pattern generation. Coupled intrinsic oscillators have been shown to synchronize their oscillatory frequencies and to maintain a characteristic pattern of phase relationships. Recently, oscillatory neurons have also been identified in sensory systems that are involved in decoding phase information. It has been hypothesized that the neural oscillators are part of neural circuits that implement phase-locked loops (PLLs), which are well-known electrical circuits for temporal decoding. Thus, there is evidence that intrinsic neural oscillators participate in both temporal pattern generation and temporal pattern decoding. The present paper investigates the dynamics underlying forced oscillators and forced PLLs, using a single framework, and compares both their stability and sensitivity characteristics. In particular, a method for assessing whether an oscillatory neuron is forced directly or indirectly, as part of a PLL, is developed and applied to published data. Received: 17 July 2000 / Accepted in revised form: 14 March 2001     

Phase angle difference alters coupling relations of functionally distinct circadian oscillators revealed by rhythm splitting

The interactions (i.e., coupling) between multiple oscillators of a circadian system determine basic properties of the integrated pacemaker. Unfortunately, there are few experimental models to investigate the putative interactions of functionally defined oscillators comprising the mammalian circadian pacemaker. Here the authors induce in hamsters a novel circadian entrainment pattern that is characterized by the daily expression of robust wheel-running activity in each scotophase of a 24-h light:dark:light:dark cycle. The daily activity bouts are mediated by 2 circadian oscillators, here designated "daytime" and "nighttime," that have been temporally dissociated under this light regime. To assess the phase dependence of interactions between oscillatory components, the phase relationship of the 2 daily scotophases was manipulated over a 4-h range, and the timing of activity of the daytime and nighttime components under entrained and probe conditions was examined. The average phase angle of entrainment and the day-to-day variability of activity onset of each activity component depended on the phase relationship of the respective scotophases and not on whether the component occurred in the daytime or the nighttime. Short-term denial of wheel access subsequently influenced amount and duration of wheel running but not timing of its onset, suggesting that only the former measures depend on a homeostatic mechanism sensitive to the time elapsed since prior intense running. Replacement of individual photophases with darkness revealed phase attraction between oscillators that was not dependent on the phase relationship of component oscillators but differed for daytime versus nighttime activity components. Entrainment patterns shown here cannot be accounted for by only nonparametric actions of light. Instead, the phase-dependent interactions of oscillators strongly influence entrainment properties, whereas intrinsic functional differences in dissociated oscillators apparently influence their attraction in darkness. This model system may be ideal for identifying genomic and physiological factors that mediate these interactions and thus contribute importantly to system properties of the mammalian circadian clock.     

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.     

Simulation of circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators

In mammals, circadian rhythms are driven by a pacemaker located in the suprachiasmatic nuclei (SCN) of the anterior hypothalamus. The firing rate of neurons within the SCN exhibits a circadian rhythm. There is evidence that individual neurons within the SCN act as circadian oscillators. Rhythm generation in the SCN was therefore modeled by a system of self-sustained oscillators. The model is composed of up to 10000 oscillatory elements arranged in a square array. Each oscillator has its own (randomly determined) intrinsic period reflecting the widely dispersed periods observed in the SCN. The model behavior was investigated mainly in the absence of synchronizing zeitgebers. Due to local coupling the oscillators synchronized and an overall rhythm emerged. This indicates that a locally coupled system is capable of integrating the output of individual clock cells with widely dispersed periods. The period of the global output (average of all oscillators) corresponded to the average of the intrinsic periods and was stable even for small amplitudes and during transients. Noise, reflecting biological fluctuations at the cellular level, distorted the global rhythm in small arrays. The period of the rhythm could be stabilized by increasing the array size, which thus increased the robustness against noise. Since different regions of the SCN have separate output pathways, the array of oscillators was subdivided into four quadrants. Sudden deviations of periodicity sometimes appeared in one quadrant, while the periods of the other quadrants were largely unaffected. This result could represent a model for splitting, which has been observed in animal experiments. In summary, the multi-oscillator model of the SCN showed a broad repertoire of dynamic patterns, revealed a stable period (even during transients) with robustness against noise, and was able to account for such a complex physiological behavior as splitting.     

The mathematical modeling of entrained biological oscillators

In this paper perturbation methods are used for the mathematical analysis of coupled relaxation oscillators. This study covers entrainment by an external periodic stimulus as well as mutual entrainment of coupled oscillators with different limit cycles. The oscillators are of a type one meets in the modeling of biological oscillators by chemical reactions and electronic circuits. Special attention is given to entrainment different from 1∶1. The results relate to phenomena occurring in physiological experiments, such as the periodic stimulation of neural and cardiac cells, and in the non-regular functioning of organs and organisms, such as the AV-block in the heart.     

Mutual entrainment of two pacemaker cells. A study with an electronic parallel conductance model

Repetitive firing of pacemaker cells was simulated with the use of an electronic Hodgkin-Huxley-like membrane model having a capacitance, a sodium-, a potassium- and a leakage conductance connected in parallel. The frequency of firing of the model cells was controlled by the leakage conductance. Two such model cells of sometimes widely different intrinsic frequencies were coupled through a coupling resistance R_c. Mutual synchronization was allowed to occur by decreasing R_c in small steps from high to low values.The lower R_c, the more the ratio of the mean frequencies of both cells approaches unity. Around certain R_c values stable entrainment occurs during which the mean frequencies are related as simple integral values m: n (e.g. 2:1, 3:2, 4:3, 1:1 for decreasing R_c). Within the range of a fixed m: n entrainment the phase difference between the oscillations of the two cells declines with R_c. When R_c is close to the range ofm: n entrainment, “almost entrainment” occurs in which m: n entrainment is periodic. But also further away from the range of m: n entrainment periodic variations in action potential interval occur.The model cells were adjusted in such a way that a depolarizing current pulse can only advance the first occurring and all subsequent action potentials. This property is used to explain effects of R_c on the mean frequency of the two cells both within and without the R_c ranges of entrainment. Furthermore, it is illustrated how in certain cases the synchronized frequency at R_c = 0 ω follows from simple electrical considerations of the properties of the uncoupled cells.Several of the entrainment phenomena observed are well known in the fields of electronic relaxation oscillators, clinical electrocardiography and circadian rhythms, though with different terminologies. The present model results exemplify the general nature of these phenomena and clarify entrainment phenomena, recently observed in coupling experiments with pacemaker cells from embryonic heart tissue.     

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.     

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.