Anti-phase Solutions Research Articles

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.

ANTI-PHASE SYNCHRONIZATION OF INHIBITORILY COUPLED NEURONS

Some new insights into the anti-phase synchronization of a network of inhibitorily coupled neurons are presented by means of qualitative analysis and numerical simulation. The network, which satisfies three significant conditions, gives rise to typical anti-phase synchronous solutions. By using the geometric method of dynamical systems, each corresponding reduced network model is analyzed, and a special and important parameter region, called P*-jumping up region, is identified. In terms of some basic properties of the P*-jumping up region, two conditions for the neuron states to enter this region are given. These conditions can be satisfied by adjusting a parameter K that controls the decay speed of the inhibitory coupling. Furthermore, it is concluded that if the parameter K is chosen so as to satisfy the two conditions, then two neurons will achieve anti-phase synchronous oscillation, independent of the initial conditions.

BIFURCATIONS, STABILITY AND SYNCHRONIZATION IN DELAYED OSCILLATORY NETWORKS

Current studies in neurophysiology award a key role to collective behaviors in both neural information and image processing. This fact suggests to exploit phase locking and frequency entrainment in oscillatory neural networks for computational purposes. In the practical implementation of artificial neural networks delays are always present due to the non-null processing time and the finite signal propagation speed. This manuscript deals with networks composed by delayed oscillators, we show that either long delays or constant external inputs can elicit oscillatory behavior in the single neural oscillator. Using center manifold reduction and normal form theory, the equations governing the whole network dynamics are reduced to an amplitude-phase model (i.e. equations describing the evolution of both the amplitudes and the phases of the oscillators). The analysis of a network with a simple architecture reveals that different kind of phase locked oscillations are admissible, and the possible coexistence of in-phase and anti-phase locked solutions.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

Forced oscillation modes in a birhythmic system of two coupled relaxation oscillators near Andronoy - Hopf bifurcation

A system of two identical relaxation oscillators of the FitzZHugh-Nagumo type with the parameters chosen in the vicinity of a bifurcation of limit cycle emergence was examined for the dynamic modes arising in the presence of a weak harmonic signal applied to both elements. Slow variable exchange between them gives rise to three stable limit cycles called in-phase, anti-phase, and extremely asymmetrical, in which only one of the oscillators generates spikes. In this study, we show that slow variable exchange also causes this system to respond to weak harmonic forcing in a manner quite different from what is known in the classical dynamics of forced oscillations. In addition to the expected synchronization tongues generated by interaction of the signal with the in-phase attractor, we observed at least three other consequences of the coexistence of different solutions: (i) there appeared broad bands of synchronization of the signal with the anti-phase solution at high frequencies multiple to the frequency of the anti-phase oscillations, whereas the base synchronization frequency band became much narrower; (ii) signal period ranges were found in which the limit cycles were such that several spikes were produced over the complete period and each oscillator was characterized with the same set of discrete

Limit Cycles Arising in a Chain of Inhibitorily Coupled Identical Relaxation Oscillators Near the Self-Oscillation Threshold

We consider the dynamic regimes arising in a linear chain of four identical stiff FitzHugh- Nagumo oscillators existing in the vicinity of the bifurcation of limit cycle emergence. It is shown that in a broad range of coupling forces, slow-variable exchange between the oscillators gives rise to multiple limit cycles with different periods and different phase relations. In addition to the expected antiphase solutions, three families of stable limit cycles that differ in the number of bursts of the fast variable in the neighboring elements and in the number of bursts per period are detected. The boundaries of attractor stability are calculated and the parameter regions of their coexistence are found.

Role of synaptic delay in organizing the behavior of networks of self-inhibiting neurons.

We consider a pair of mutually coupled inhibitory neurons in which each neuron is also self-inhibitory. We show that the size of the synaptic delay determines the existence and stability of solutions. For small delays, there is no synchronous solution, but a stable antiphase and a stable on-state solution. For long delays, only the synchronous solution is stable. For intermediate delays, either the antiphase or synchronous solutions are stable. In contrast to prior work, for stability of synchrony, we only require the existence of a single slow process.

Anti-phase solutions in relaxation oscillators coupled through excitatory interactions.

Relaxation oscillators interacting via models of excitatory chemical synapses with sharp thresholds can have stable anti-phase as well as in-phase solutions. The mechanism for anti-phase demonstrated in this paper relies on the fact that, in a large class of neural models, excitatory input slows down the receiving oscillator over a portion of its trajectory. We analyze the effect of this "virtual delay" in an abstract model, and then show that the hypotheses of that model hold for widely used descriptions of bursting neurons.

Minimal equations for antiphase dynamics in multimode lasers

We consider two models of multimode lasers: the Tang-Statz-deMars equations modeling a solid-state Fabry-P\'erot laser [C.L. Tang, H. Statz, and G. deMars, J. Appl. Phys. 34, 2289 (1963)] and the rate equations modeling intracavity second-harmonic generation in a Nd:YAG ring laser (where YAG denotes yttrium aluminum garnet). In both models, the dynamics is dominated by a global coupling of the modes of the electromagnetic field in the cavity. Although the equations for these two problems are fairly different, we prove that a dominant asymptotic approximation can be determined in each case that leads to the same conservative problem. It depends on one parameter, which measures the strength of the global coupling, and admits a class of antiphase periodic solutions.

Weak coupling of strongly nonlinear, weakly dissipative identical oscillators

This paper considers a network of coupled oscillators with (a) weak coupling and (b) weak dissipation (but strongly nonlinear response). By changing to energy- angle coordinates, equations for the slow evolution of the amplitudes and phases of the individual oscillators are obtained. Restricting to problems with identical oscillators and coupling weaker than dissipation, an approximation to an attracting invariant torus for two oscillators is constructed and averaging is performed on this to reduce to a single equation for the phase difference. For two van der Pol-Duffing type oscillators coupled by a cubic function this method is used to predict bifurcation from in-phase stability (through a region where neither in-phase nor antiphase solutions are stable) to antiphase stability as a coupling parameter is changed. Furthermore, the method is used to investigate analytically the destabilization of the in-phase solution when two oscillators are dissipatively coupled through one state variable. For a large class of systems the stability of the in-phase solution depends on the sign of the shear or dependence of frequency on amplitude of the individual oscillators.

Amplitude response of coupled oscillators

We investigate the interaction of a pair of weakly nonlinear oscillators (eg. each near a Hopf bifurcation) when the coupling strength is comparable to the attraction of the limit cycles. Changes in amplitude cannot then be ignored, and there are new phenomena. We show that even for simple forms of these equations, there are parameter regimes in which the interaction causes the system to stop oscillating, and the rest state at zero, stabilized by the interaction, is the only stable solution. When the uncoupled oscillators have a local frequency that is independent of amplitude (zero shear), and the coupling is scalar, we give a complete description of the behavior. We then analyze more complicated equations to show the effects of amplitude-dependent frequency in the uncoupled equations (nonzero shear) and nonscalar coupling. For example, when there is shear, there can be bistability between a phase-locked solution and an unlocked “drift” solution, in which the oscillators go at different average frequencies. There can also be bistability between a phase locked solution and an equilibrium, nonoscillatory, solution. The equations for the zero shear case have a pair of degenerate bifurcation points, and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities. When the coupling is nonscalar, there are a variety of new behaviors, including bistability of an in-phase and an anti-phase solution for two identical oscillators, parameters for which only the anti-phase solution is stable, and “phase-trapping,” i.e. nonlocked solutions in which the oscillators still have the same average frequency. It is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system.

Two coupled neural oscillators as a model of the circadian pacemaker

Pittendrigh first found that the circadian rhythm of locomotor activity in nocturnal rodents split into two components. Hoffman then reported that the splitting phenomenon was even more reproducible in the small diurnal primate Tupaia. These “splitting” experiments and many other experiments suggest that two coupled oscillators may constitute the circadian pacemaker system. Pittendrigh proposed a phenomenological two-oscillator model. Daan and Berde developed a quantitative model assuming that the interaction between the two constituent oscillators is by instantaneous resets. Their model system can simulate several qualitative features in the experimental data. As the assumption of instantaneous resets seems to be unnatural, we study two limit cycle oscillators, which are coupled continuously to each other, as a model of the circadian pacemaker. We assume the following points, (i) One oscillator in a resting state does not affect another oscillator, (ii) Two oscillators are identical, (iii) The coupling is symmetrical. By the theory of Hopf bifurcation it is found that the general two-oscillator system has two stable periodic solutions. One is the in-phase solution where the two constituent oscillators oscillate in phase synchrony. Another is the anti-phase solution where the two oscillators oscillate 180 ° out of phase. The former corresponds to a single pattern of locomotor activity and the latter corresponds to a splitting pattern. Furthermore, we study specific two-neural oscillators, which are linearly coupled to each other. By the method of secondary bifurcation we find that the model shows simultaneous stability of the two alternative phase relationships and the hysteresis phenomena found in Tupaia. A natural period of the uncoupled constituent oscillator is longer than that of the in-phase solution but it is shorter than that of the anti-phase solution. This is in agreement with the data of Tupaia.

Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junction, a model system consisting of two electrically coupled neurons was studied. It was revealed that the model system generates both the in-phase pattern and the anti-phase pattern stably. Physiological condierations revealed the following. The in-phase discharge pattern (synergism) is a unique output of a electrically coupled pacemaker neurons. However, both the in-phase (synergism) and the anti-phase (antagonism) discharge patterns are possible for electrically coupled bursting neurons. We may expect other discharge patterns, such as a random discharge pattern (chaos), than the inphase and the anti-phase pattern in electrically coupled neurons. Random firing patterns in the inferior olive may be attributed to electrical synapses. Until now, we have assumed that excitation periods of two neurons were almost the same. When two periods are greatly different, various phenomena are expected. Determination of the stability of the anti-phase solution by the analytical methods, when diffusion constants are very small, is one of our future problems.