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.     

A pacemaker cell pair model based on the phase response curve

A pacemaker cell pair model and the dynamic interaction between the two pacemaker cells is described in this paper. It is an extension of our single pacemaker cell model, in which we studied its response to repetitive external depolarization stimulations. This model is a simple model based on the two most important functional properties of the cardiac pacemaker cells: its intrinsic pacemaker cycle length, which is an `internal' parameter of the cell, and the phase response curve (PRC), which is an `overall collective' function. The PRC contains all the `information' about the possible interactions of the pacemaker cell with the outside world (interaction with surrounding cells, external stimulus, etc.). First, we examined the properties and solutions of 1:1 synchronization between two pacemaker cells. We found that in order to achieve synchronization between two pacemaker cells, there should be limitations on the PRC parameters, which depend on the cells intrinsic cycle lengths. Next, we investigated the 2:1 entrainment state between two interacting pacemaker cells. We found that there is not necessarily a unique solution for this state as there was for the 1:1 state. Finally, we ran our computer model to investigate the properties of more complex patterns of entrainment between two pacemaker cells. As a result of our analytical study, we unveil two new important parameters, which are fully defined as a function of the PRC parameters: (1) the `accelerator factor' which describes the tendency of a pair of interacting pacemaker cells to synchronize at a common cycle length, which is closer to the faster cycle of the pair; (2) the `degree of coupling', which describes the range of the 1:1 synchronization and the `strength' of the interaction between a pair of interacting pacemaker cells. Those two interaction parameters arise as helpful `tools' for the understanding of synchronization and mutual entrainment mechanisms between pacemaker cells. Therefore, this study establishes the PRC as an important determinant and a useful approach for the understanding of the dynamic interaction of pacemaker cells among themselves and with the outside world. Received: 12 May 1997 / Accepted in revised form: 22 April 1998     

Phase resetting and phase singularity of an insect circannual oscillator

In circadian rhythms, the shape of the phase response curves (PRCs) depends on the strength of the resetting stimulus. Weak stimuli produce Type 1 PRCs with small phase shifts and a continuous transition between phase delays and advances, whereas strong stimuli produce Type 0 PRCs with large phase shifts and a distinct break point at the transition between delays and advances. A stimulus of an intermediate strength applied close to the break point in a Type 0 PRC sometimes produces arrhythmicity. A PRC for the circannual rhythm was obtained in pupation of the varied carpet beetle, Anthrenus verbasci, by superimposing a 4-week long-day pulse (a series of long days for 4 weeks) over constant short days. The shape of this PRC closely resembles that of the Type 0 PRC. The present study shows that the PRC to 2-week long-day pulses was Type 1, and that a 4-week long-day pulse administered close to the PRC’s break point induced arrhythmicity in pupation. It is, therefore, suggested that circadian and circannual oscillators share the same mode in phase resetting to the stimuli.     

Synaptic and intrinsic determinants of the phase resetting curve for weak coupling

A phase resetting curve (PRC) keeps track of the extent to which a perturbation at a given phase advances or delays the next spike, and can be used to predict phase locking in networks of oscillators. The PRC can be estimated by convolving the waveform of the perturbation with the infinitesimal PRC (iPRC) under the assumption of weak coupling. The iPRC is often defined with respect to an infinitesimal current as z_i(ϕ), where ϕ is phase, but can also be defined with respect to an infinitesimal conductance change as z_g(ϕ). In this paper, we first show that the two approaches are equivalent. Coupling waveforms corresponding to synapses with different time courses sample z_g(ϕ) in predictably different ways. We show that for oscillators with Type I excitability, an anomalous region in z_g(ϕ) with opposite sign to that seen otherwise is often observed during an action potential. If the duration of the synaptic perturbation is such that it effectively samples this region, PRCs with both advances and delays can be observed despite Type I excitability. We also show that changing the duration of a perturbation so that it preferentially samples regions of stable or unstable slopes in z_g(ϕ) can stabilize or destabilize synchrony in a network with the corresponding dynamics.     

Synchronization of two coupled pacemaker cells based on the phase response curve

In this paper, the synchronization of a pair of pacemaker cells as Sino-Atrial (SA) and Atrio-Ventricullar (AV) nodes have been studied and a new approach for synchronization, based on the concept of Phase Response Curve (PRC), has been proposed. The paper starts with presenting the necessary and sufficient conditions for synchronization in terms of the PRC parameters. Such conditions are time dependent and thus, the paper proceeds with deriving some sufficient conditions, which are not time dependent. The time-delay between the firing time of SA node and when it reaches the AV node is also considered. When the conditions for spontaneous synchronization are not valid, the synchronization is achieved by applying pulses to the AV or the SA nodes or to both of the nodes, depending on the accessibility. The subject has been investigated and sufficient conditions were achieved for all three cases. In each case, the dynamical equations of coupled pacemakers have been determined and the stability analyses of delay dynamical equations between discharges of two pacemakers were performed. The number of excitation pulses and the time intervals for applying them to accessible pacemaker(s) were obtained and eventually some numerical examples were simulated to approve the accuracy of the theoretical results and conditions.     

Circadian phase resetting via single and multiple control targets

Circadian entrainment is necessary for rhythmic physiological functions to be appropriately timed over the 24-hour day. Disruption of circadian rhythms has been associated with sleep and neuro-behavioral impairments as well as cancer. To date, light is widely accepted to be the most powerful circadian synchronizer, motivating its use as a key control input for phase resetting. Through sensitivity analysis, we identify additional control targets whose individual and simultaneous manipulation (via a model predictive control algorithm) out-perform the open-loop light-based phase recovery dynamics by nearly 3-fold. We further demonstrate the robustness of phase resetting by synchronizing short- and long-period mutant phenotypes to the 24-hour environment; the control algorithm is robust in the presence of model mismatch. These studies prove the efficacy and immediate application of model predictive control in experimental studies and medicine. In particular, maintaining proper circadian regulation may significantly decrease the chance of acquiring chronic illness.     

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.     

Circadian phase resetting in response to light-dark and dark-light transitions

Phase shifting of circadian systems by light has been attributed both to parametric effects on angular velocity elicited by a tonic response to the luminance level and to nonparametric instantaneous shifts induced by a phasic response to the dark-light (D>L) and light-dark (L>D) transitions. Claims of nonparametric responses are partly based on "step-PRCs," that is, phase response curves derived from such transitions. Step-PRCs in nocturnal mammals show mostly delays after lights-on and advances after lights-off, and therefore appear incompatible with phase delays generated by light around dusk and advances by light around dawn. We have pursued this paradox with 2 experimental protocols in mice. We first use the classic step-PRC protocol on wheel running activity, using the center of gravity as a phase marker to minimize the masking effects of light. The experiment was done for 3 different light intensities (1, 10, and 100 lux). D>L transitions evoke mostly delays and L>D transitions show no clear tendency to either delay or advance. Overall there is little or no circadian modulation. A 2nd protocol aimed to avoid the problem of masking by assessing phase before and after the light stimuli, both in DD. Light stimuli consisted of either a slow light intensity increase over 48 h followed by abruptly switching off the light, or an abrupt switch on followed by a slow decrease toward total darkness during 48 h. If the abrupt transitions were responsible for phase shifting, we expected large differences between the 2 stimuli. Both light stimuli yielded similar PRCs characterized by delays only with circadian modulation. The results can be adequately explained by a model in which all PRCs evoked by steps result in fact from tonic responses to the light following a step-up or preceding a step-down. In this model only the response reduction of tonic velocity change after the 1st hour is taken into account. The data obtained in both experiments are thus compatible with tonic velocity responses. Contrary to standard interpretation of step-PRCs, nonparametric responses to the transitions are unlikely since they would predict delays in response to lights-off, advances in response to lights-on, while the opposite was found. Although such responses cannot be fully excluded, parsimony does not require invocation of a role for transitions, since all the data can readily be explained by tonic velocity (parametric) effects, which must exist because of the dependence of tau on light intensity.     

Short-term activity cycles in ants: A phase-response curve and phase resetting in worker activity

Single, isolated worker ants are known to become spontaneously active and to respond to interactions with other, active ants. Here I explore the consequences of an interaction between two worker ants on the timing of activity. Isolated worker ants become active after an interval that is characteristic for each individual. The effect of an interaction between two worker ants is strongly dependent on when the interaction takes place. The effect of an interaction is always to decrease the expected interval until the onset of activity. By studying the effect of an interaction on subsequent intervals of activity, it is possible to reject the hypothesis that the change in timing of activity is due to a change in the characteristic period of activity. Rather, the data are consistent with the hypothesis that an interaction causes a phase shift in the normal activity oscillation. The phase-response curve is derived from the observational data. A knowledge of the dynamics of the interactions of individual ants is necessary in order to begin to reconstruct the patterns of colony behavioral activity.     

Phase resetting and phase locking in hybrid circuits of one model and one biological neuron

To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.     

Phase resetting properties of cardiac pacemaker cells

Aggregates of heart cells from chicken embryos beat spontaneously. We used intracellular microelectrodes to record the periodic behavior of the membrane potential that triggers the contractions. Every 5-12 beats, a short current pulse was applied at various points in the cycle to study the phase-dependent resetting of the rhythm. Pulses stronger than 2.5 nA caused the final rhythm to be reset to almost the same point in the cycle regardless of the phase at which the pulse was applied (type zero resetting). Pulses of less than or equal to 1 nA only caused a slight change of the phase. Increasing current intensities to between 1 and 2.5 nA gave rise to an increasing steepness in a small part of the phase-response curve. The observation of type zero resetting implies the existence of a critical stimulation that might annihilate the rhythm. Although we did find a phase at which more or less random responses occurred, the longest pause in the rhythm was 758 ms, 2.4 times the spontaneous interval. This suggests that the resting membrane potential was unstable, at least against the internal noise of the system. The conclusions are discussed in terms of the concepts of classical cardiac electrophysiology.     

Difference equation model of ventricular parasystole as an interaction between cardiac pacemakers based on the phase response curve

A model of extended ventricular parasystole proposed by Moe et al. (1977) was formulated as a system of nonlinear difference equations by using the phase response curve of myocardial pacemakers. A number of ECG patterns of ventricular arrhythmia such as bigeminy, trigeminy etc. were explained from the property of periodic solutions of the equation. Characteristic properties of special kinds of arrhythmia called “concealed bigeminy” and “concealed trigeminy” were derived mathematically by assuming the model, in relation to the equation of the analog neuron model. The present study was considered to be of clinical significance as a theoretical foundation for the study of genesis of cardiac arrhythmias.     

Phase resetting and coupling of noisy neural oscillators

A number of experimental groups have recently computed Phase Response Curves (PRCs) for neurons. There is a great deal of noise in the data. We apply methods from stochastic nonlinear dynamics to coupled noisy phase-resetting maps and obtain the invariant density of phase distributions. By exploiting the special structure of PRCs, we obtain some approximations for the invariant distributions. Comparisons to Monte-Carlo simulations are made. We show how phase-dependence of the noise can move the peak of the invariant density away from the peak expected from the analysis of the deterministic system and thus lead to noise-induced bifurcations. B. Ermentrout supported in part by NIMH and NSF. Action Editor: Wulfram Gerstner     

Existence and stability criteria for phase-locked modes in ring neural networks based on the spike time resetting curve method

We developed a systematic and consistent mathematical approach to predicting 1:1 phase-locked modes in ring neural networks of spiking neurons based on the open loop spike time resetting curve (STRC) and its almost equivalent counterpart—the phase resetting curve (PRC). The open loop STRCs/PRCs were obtained by injecting into an isolated model neuron a triangular shaped time-dependent stimulus current closely resembling an actual synaptic input. Among other advantages, the STRC eliminates the confusion regarding the undefined phase for stimuli driving the neuron outside of the unperturbed limit cycle. We derived both open loop PRC and STRC-based existence and stability criteria for 1:1 phase-locked modes developed in ring networks of spiking neurons. Our predictions were in good agreement with the closed loop numerical simulations. Intuitive graphical methods for predicting phase-locked modes were also developed both for half-centers and for larger ring networks.     

Phase resetting associated with changes of burst shape

Based on our stochastic approach to phase resetting of an ensemble of oscillators, in this article we investigate two stimulation mechanisms which exhibit qualitatively different dynamical behaviour as compared with the stimulation mechanism analysed in a previous study. Both the old as well as one of the new stimulation mechanisms give rise to a characteristic desynchronization behaviour: A stimulus of a given (non-vanishing) intensity administered at a critical initial ensemble phase for a critical duration T _crit annihilates the ensemble's synchronized oscillation. When the stimulation duration exceeds T _crit a transition from type 1 resetting to type 0 resetting occurs. The second new stimulation mechanism does not cause a desynchronization which is connected with a phase singularity. Correspondingly this mechanism only leads to type 1 resetting. In contrast to the stimulation mechanism analysed in a previous study, both new stimulation mechanisms cause burst splitting. According to our results, in this case peak or onset detection algorithms are not able to reveal a correct estimate of the ensemble phase. Thus, whenever stimulation induced burst splitting occurs, phase-resetting curves determined by means of peak or onset detection may be nothing but artifacts. Therefore it is necessary to understand burst splitting in order to develop reliable phase detection algorithms, which are e.g. based on detecting bursts' centers of mass. Our results are important for experimentalists: Burst splitting is, for instance, well-known from tremor resetting experiments. Thus, it often turned out to be at least rather difficult to derive reliable phase-resetting curves from experimental data.     

Interpreting the human phase response curve to multiple bright-light exposures

Czeisler and his colleagues have recently reported that bright light can induce strong (Type O) resetting of the human circadian pacemaker. This surprising result shows that the human clock is more responsive to light than has been previously thought. The interpretation of their results is subtle, however, because of an unconventional aspect of their experimental protocol: They measured the phase shift after three cycles of the bright-light stimulus, rather than after the usual single pulse. A natural question is whether the apparent Type O response could reflect the summation of three weaker Type 1 responses to each of the daily light pulses. In this paper I show mathematically that repeated Type 1 resetting cannot account for the observed Type O response. This finding corroborates the strong resetting reported by Czeisler et al., and supports their claim that bright light induces strong resetting by crushing the amplitude of the circadian pacemaker. Furthermore, the results indicate that back-to-back light pulses can have a cooperative effect different from that obtained by simple iteration of a phase response curve (PRC). In this sense the resetting response of humans is similar to that of Drosophila, Kalanchoe, and Culex, and is more complex than that predicted by conventional PRC theory. To describe the way in which light resets the human circadian pacemaker, one needs a theory that includes amplitude resetting, as pioneered by Winfree and developed for humans by Kronauer.     

Possible functional roles of phase resetting during walking

The walking rhythm is known to show phase shift or "reset" in response to external impulsive perturbations. We tried to elucidate functional roles of the phase reset possibly used for the neural control of locomotion. To this end, a system with a double pendulum as a simplified model of the locomotor control and a model of bipedal locomotion were employed and analyzed in detail. In these models, a movement corresponding to the normal steady-state walking was realized as a stable limit cycle solution of the system. Unexpected external perturbations applied to the system can push the state point of the system away from its limit cycle, either outside or inside the basin of attraction of the limit cycle. Our mathematical analyses of the models suggested functional roles of the phase reset during walking as follows. Function 1: an appropriate amount of the phase reset for a given perturbation can contribute to relocating the system's state point outside the basin of attraction of the limit cycle back to the inside. Function 2: it can also be useful to reduce the convergence time (the time necessary for the state point to return to the limit cycle). In experimental studies during walking of animals and humans, the reset of walking rhythm induced by perturbations was investigated using the phase transition curve (PTC) or the phase resetting curve (PRC) representing phase-dependent responses of the walking. We showed, for the simple double-pendulum model, the existence of the optimal phase control and the corresponding PTC that could optimally realize the aforementioned functions in response to impulsive force perturbations. Moreover, possible forms of PRC that can avoid falling against the force perturbations were predicted by the biped model, and they were compared with the experimentally observed PRC during human walking. Finally, physiological implications of the results were discussed.