Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

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.     

Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.     

Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.     

Multiple pulse interactions and averaging in systems of coupled neural oscillators

Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University     

Phase model-based neuron stabilization into arbitrary clusters

Deep brain stimulation (DBS) is a common method of combating pathological conditions associated with Parkinson’s disease, Tourette syndrome, essential tremor, and other disorders, but whose mechanisms are not fully understood. One hypothesis, supported experimentally, is that some symptoms of these disorders are associated with pathological synchronization of neurons in the basal ganglia and thalamus. For this reason, there has been interest in recent years in finding efficient ways to desynchronize neurons that are both fast-acting and low-power. Recent results on coordinated reset and periodically forced oscillators suggest that forming distinct clusters of neurons may prove to be more effective than achieving complete desynchronization, in particular by promoting plasticity effects that might persist after stimulation is turned off. Current proposed methods for achieving clustering frequently require either multiple input sources or precomputing the control signal. We propose here a control strategy for clustering, based on an analysis of the reduced phase model for a set of identical neurons, that allows for real-time, single-input control of a population of neurons with low-amplitude, low total energy signals. After demonstrating its effectiveness on phase models, we apply it to full state models to demonstrate its validity. We also discuss the effects of coupling on the efficacy of the strategy proposed and demonstrate that the clustering can still be accomplished in the presence of weak to moderate electrotonic coupling.     

Phase locking of biological clocks

Radial isochron clocks (RICs) and their response to external signals and coupling with other RICs are studied. RICs are derived as phase approximations to self-sustained oscillators. Their response to single impulses (phase resetting) and to repetitive impulses is determined. This response may be harmonic or chaotic. Finally, the effect of coupling between clocks is studied. Simple coupling is shown to exhibit rhythm splitting like that observed in fish and small mammals. New phase locking results for general weakly coupled RIC systems are also derived.Supported in part by the National Science Foundation under grants MCS-80-15359 (FCH) and MCS-79-02505 (JPK)     

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.     

Effect of Sharp Jumps at the Edges of Phase Response Curves on Synchronization of Electrically Coupled Neuronal Oscillators

We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.     

Neural mechanisms potentially contributing to the intersegmental phase lag in lamprey

Factors contributing to the production of a phase lag along chains of oscillatory networks consisting of Hodgkin-Huxley type neurons are analyzed by means of simulations. Simplified network configurations are explored consisting of the basic building blocks of the spinal central pattern generator (CPG) generating swimming in the lamprey. It consists of reciprocally coupled crossed inhibitory C interneurons and ipsilateral excitatory E interneurons that activate C neurons and other E neurons. Oscillatory activity in the model network can, in the simplest case, be produced by a pair of reciprocally coupled C interneurons oscillating through an escape mechanism. Different levels of tonic excitation drive the network over a wide burst frequency range. In this type of network, powerful frequency-regulating factors are the effective inhibition produced by the active side, in combination with the tendency of the inactive side to escape from the inhibition. These two mechanisms can be affected by several factors, e.g. spike frequency adaptation (calcium-dependent K(+) channels), N-methyl-D-aspartate membrane properties as well as presence of low-voltage activated calcium channels. A rostrocaudal phase lag can be produced either by extending the contralateral inhibitory projections or the ipsilateral excitatory projections relatively more in the caudal than the rostral direction, since both an increased inhibition and a phasic excitation slow down the receiving network. The phase lag becomes decreased if the length of the intersegmental projections is increased or if the projections are extended symmetrically in both the rostral and the caudal directions. The simulations indicate that the conditions in the ends of an oscillator chain may significantly affect sign, magnitude and constancy of the phase lag. Also, with short and relatively weak intersegmental connections, the network remains robust against perturbations as well as intrinsic frequency differences along the chain. The phase lag (percentage of cycle duration) increases, however, with burst frequency also when the coupling strength is comparatively weak. The results are discussed and compared with previous "phase pulling" models as well as relaxation oscillators.     

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。     

Phase response curves of a molecular model oscillator: implications for mutual coupling of paired oscillators

Increasing evidence indicates that the accessory medulla is the circadian pacemaker controlling locomotor activity rhythms in insects. A prominent group of neurons of this neuropil shows immunoreactivity to the peptide pigment-dispersing hormone (PDH). In Drosophila melanogaster, the PDH-immunoreactive (PDH-ir) lateral neurons, which also express the clock genes period and timeless, are assumed to be circadian pacemaker cells themselves. In other insects, such as Leucophaea maderae, a subset of apparently homologue PDH-ir cells is a candidate for the circadian coupling pathway of the bilaterally symmetric clocks. Although knowledge about molecular mechanisms of the circadian clockwork is increasing rapidly, very little is known about mechanisms of circadian coupling. The authors used a computer model, based on the molecular feedback loop of the clock genes in D. melanogaster, to test the hypothesis that release of PDH is involved in the coupling between bilaterally paired oscillators. They can show that a combination of all-delay- and all-advance-type interactions between two model oscillators matches best the experimental findings on mutual pacemaker coupling in L. maderae. The model predicts that PDH affects the phosphorylation rate of clock genes and that in addition to PDH, another neuroactive substance is involved in the coupling pathway, via an all-advance type of interaction. The model suggests that PDH and light pulses, represented by two distinct classes of phase response curves, have different targets in the oscillatory feedback loop and are, therefore, likely to act in separate input pathways to the clock.     

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.     

Automatic classification of interference patterns in driven event series: application to single sympathetic neuron discharge forced by mechanical ventilation

This study proposes a method for the automatic classification of nonlinear interactions between a strictly periodical event series modelling the activity of an exogenous oscillator working at a fixed and well-known rate and an event series modelling the activity of a self-sustained oscillator forced by the exogenous one. The method is based on a combination of several well-known tools (probability density function of the cyclic relative phase, probability density function of the count of forced events per forcing cycle, conditional entropy of the cyclic relative phase sequence and a surrogate data approach). Classification is reached via a sequence of easily applicable decision rules, thus rendering classification virtually user-independent and fully reproducible. The method classifies four types of dynamics: full uncoupling, quasiperiodicity, phase locking and aperiodicity. In the case of phase locking, the coupling ratio (i.e. n:m) and the strength of the coupling are calculated. The method, validated on simulations of simple and complex phase-locking dynamics corrupted by different levels of noise, is applied to data derived from one anesthetized and artificially ventilated rat to classify the nonlinear interactions between mechanical ventilation and: (1) the discharges of two (contemporaneously recorded) single postganglionic sympathetic neurons innervating the caudal ventral artery in the tail and (2) arterial blood pressure. Under central apnea, the activity of the underlying sympathetic oscillators is perturbed by means of five different lung inflation rates (0.58, 0.64, 0.76, 0.95, 1.99 Hz). While ventilation and arterial pressure are fully uncoupled, ventilation is capable of phase locking sympathetic discharges, thus producing 40% of phase-locked patterns (one case of 2:5, 1:1, 3:2 and 2:2) and 40% of aperiodic dynamics. In the case of phase-locked patterns, the coupling strength is low, thus demonstrating that this pattern is sliding. Non-stationary interactions are observed in 20% of cases. The two discharges behave differently, suggesting the presence of a population of sympathetic oscillators working at different frequencies.     

Slow Noise in the Period of a Biological Oscillator Underlies Gradual Trends and Abrupt Transitions in Phasic Relationships in Hybrid Neural Networks

In order to study the ability of coupled neural oscillators to synchronize in the presence of intrinsic as opposed to synaptic noise, we constructed hybrid circuits consisting of one biological and one computational model neuron with reciprocal synaptic inhibition using the dynamic clamp. Uncoupled, both neurons fired periodic trains of action potentials. Most coupled circuits exhibited qualitative changes between one-to-one phase-locking with fairly constant phasic relationships and phase slipping with a constant progression in the phasic relationships across cycles. The phase resetting curve (PRC) and intrinsic periods were measured for both neurons, and used to construct a map of the firing intervals for both the coupled and externally forced (PRC measurement) conditions. For the coupled network, a stable fixed point of the map predicted phase locking, and its absence produced phase slipping. Repetitive application of the map was used to calibrate different noise models to simultaneously fit the noise level in the measurement of the PRC and the dynamics of the hybrid circuit experiments. Only a noise model that added history-dependent variability to the intrinsic period could fit both data sets with the same parameter values, as well as capture bifurcations in the fixed points of the map that cause switching between slipping and locking. We conclude that the biological neurons in our study have slowly-fluctuating stochastic dynamics that confer history dependence on the period. Theoretical results to date on the behavior of ensembles of noisy biological oscillators may require re-evaluation to account for transitions induced by slow noise dynamics.     

Modeling the dual pacemaker system of the tau mutant hamster

Circadian pacemakers in many animals are compound. In rodents, a two-oscillator model of the pacemaker composed of an evening (E) and a morning (M) oscillator has been proposed based on the phenomenon of "splitting" and bimodal activity peaks. The authors describe computer simulations of the pacemaker in tau mutant hamsters viewed as a system of mutually coupled E and M oscillators. These mutant animals exhibit normal type 1 PRCs when released into DD but make a transition to a type 0 PRC when held for many weeks in DD. The two-oscillator model describes particularly well some recent behavioral experiments on these hamsters. The authors sought to determine the relationships between oscillator amplitude, period, PRC, and activity duration through computer simulations. Two complementary approaches proved useful for analyzing weakly coupled oscillator systems. The authors adopted a "distinct oscillators" view when considering the component E and M oscillators and a "system" view when considering the system as a whole. For strongly coupled systems, only the system view is appropriate. The simulations lead the authors to two primary conjectures: (1) the total amplitude of the pacemaker system in tau mutant hamsters is less than in the wild-type animals, and (2) the coupling between the unit E and M oscillators is weakened during continuous exposure of hamsters to DD. As coupling strength decreases, activity duration (alpha) increases due to a greater phase difference between E and M. At the same time, the total amplitude of the system decreases, causing an increase in observable PRC amplitudes. Reduced coupling also increases the relative autonomy of the unit oscillators. The relatively autonomous phase shifts of E and M oscillators can account for both immediate compression and expansion of activity bands in tau mutant and wild-type hamsters subjected to light pulses.     

Modeling circadian rhythm generation in the suprachiasmatic nucleus with locally coupled self-sustained oscillators: phase shifts and phase response curves

Circadian rhythm generation in the suprachiasmatic nucleus was modeled by locally coupled self-sustained oscillators. The model is composed of 10,000 oscillators, arranged in a square array. Coupling between oscillators and standard deviation of (randomly determined) intrinsic oscillator periods were varied. A stable overall rhythm emerged. The model behavior was investigated for phase shifts of a 24-h zeitgeber cycle. Prolongation of either the dark or the light phase resulted in a lengthening of the period, whereas shortening of the dark or the light phase shortened the period. The model's response to shifts in the light-dark cycle was dependent only on the extent of the shift and was insensitive to changes in parameters. Phase response curves (PRC) and amplitude response curves were determined for single and triple 5-h light pulses (1000 lux). Single pulses lead to type 1 PRCs with larger phase shifts for weak coupling. Triple pulses generally evoked type 1 PRCs with the exception of weak coupling, where a type 0 PRC was observed.     

Simple models for excitable and oscillatory neural networks

 Chains of coupled oscillators of simple “rotator” type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin–Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of “body shapes” generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Received: 25 November 1996 / Revised version: 9 December 1997     

Synchronization dynamics of two coupled neural oscillators receiving shared and unshared noisy stimuli

The response of neurons to external stimuli greatly depends on the intrinsic dynamics of the network. Here, the intrinsic dynamics are modeled as coupling and the external input is modeled as shared and unshared noise. We assume the neurons are repetitively firing action potentials (i.e., neural oscillators), are weakly and identically coupled, and the external noise is weak. Shared noise can induce bistability between the synchronous and anti-phase states even though the anti-phase state is the only stable state in the absence of noise. We study the Fokker-Planck equation of the system and perform an asymptotic reduction ρ ₀. The ρ ₀ solution is more computationally efficient than both the Monte Carlo simulations and the 2D Fokker-Planck solver, and agrees remarkably well with the full system with weak noise and weak coupling. With moderate noise and coupling, ρ ₀ is still qualitatively correct despite the small noise and coupling assumption in the asymptotic reduction. Our phase model accurately predicts the behavior of a realistic synaptically coupled Morris-Lecar system.