Stability Of Phase-locked Solutions Research Articles

Dynamics of coupled Kuramoto oscillators with distributed delays.

This paper studies the effects of two different types of distributed-delay coupling in the system of two mutually coupled Kuramoto oscillators: one where the delay distribution is considered inside the coupling function and the other where the distribution enters outside the coupling function. In both cases, the existence and stability of phase-locked solutions is analyzed for uniform and gamma distribution kernels. The results show that while having the distribution inside the coupling function only changes parameter regions where phase-locked solutions exist, when the distribution is taken outside the coupling function, it affects both the existence, as well as stability properties of in- and anti-phase states. For both distribution types, various branches of phase-locked solutions are computed, and regions of their stability are identified for uniform, weak, and strong gamma distributions.

Entrainment Maps.

Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

Effects of synaptic plasticity on phase and period locking in a network of two oscillatory neurons.

We study the effects of synaptic plasticity on the determination of firing period and relative phases in a network of two oscillatory neurons coupled with reciprocal inhibition. We combine the phase response curves of the neurons with the short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network. Fixed points of these maps correspond to the phase-locked modes of the network. These maps allow us to analyze the dependence of the resulting network activity on the properties of network components. Using a combination of analysis and simulations, we show how various parameters of the model affect the existence and stability of phase-locked solutions. We find conditions on the synaptic plasticity profiles and the phase response curves of the neurons for the network to be able to maintain a constant firing period, while varying the phase of locking between the neurons or vice versa. A generalization to cobwebbing for two-dimensional maps is also discussed.

Using feed-forward networks to infer the activity of feed-back neuronal networks

Phase-locked activity of neurons is important for producing the proper activation sequence and coordination of neurons within Central Pattern Generating (CPG) networks. As it is often difficult to analyze the phase-locked activity of a large feedback network directly, it is critical to develop analytic methods that are lower-dimensional which involve different pieces of the full network. Here, we provide a technique to study smaller feed-forward networks and then combine the obtained information to understand the activity of feedback networks. As a case study, we study the phase of activity of two reciprocally inhibitory bursting neurons in the crustacean stomatogastric ganglion (STG) in which the firing time of one neuron has an effect on the period of the other, and vice versa. Each of these two processes defines a feed-forward map and we combine them to determine the period and activity phase of the feedback network. We examine the conditions on the existence and stability of phase-locked solutions when the synapses display short-term depression, a common form of synaptic plasticity. We find that the stability is lost in some circumstances in the feedback network with the existence of the synaptic depression. Consider two neurons coupled via inhibitory synapses. Neuron A is modeled as oscillatory while neuron B as tonically active. The synapse from A to B is depressing but B to A is not. The cycle period of neuron A depends on the firing time of B relative to A, which affects the recovery and strength of the synapse from A to B. This synapse, in turn, determines the firing time of B relative to A. We derive a one-dimensional Poincare map Π , defined at the onset of A activity, that measures the strength sn of the A to B synapse (Fig. ​(Fig.1).1). The model is built as in [1]. Here d measures the extent of depression of the A to B synapse, tn the firing time of B relative to A and Pn the period of A in each cycle. A fixed point of this map determines the unique period and phase at which A and B lock. Its existence corresponds to the intersection point of the two feed-forward maps (Fig. ​(Fig.2),2), each of which is obtained from a different feed-forward sub-network. One map, H, determines how tn affects sn+1, while the other map, G, determines how sn affects tn. The map H is defined separately in three different regions. In Region I, neuron B fires when A is still active but its activity is not terminated by B; Region II is similar to I but the activity of A is terminated by B; in Region III, B fires when A is inactive. In biological networks Region I and II are small and the graphs of H and G typically intersect in Region III, as observed in the two mutually inhibitory neurons PD and LP in the STG. We find that the phase-locked solution is stable when it occurs in Regions I and III but the stability is lost in some circumstances in Region II. Figure 1 Reciprocally inhibitory network. Figure 2 Intersection of feed-forward curves H and G correspond to a fixed point of the one-dimensional feedback map.

Phase locking in Josephson ladders and the discrete sine-Gordon equation: the effects of boundary conditions and current-induced magnetic fields.

We report on the stability of phase-locked solutions to ladder arrays of underdamped Josephson junctions under both periodic and open boundary conditions and in the presence of current-induced magnetic fields. We calculate the Floquet exponents based on the resistively and capacitively shunted junction (RCSJ) model, as well as on a simplified model of the ladder that leads to a discrete sine-Gordon (DSG) equation for the horizontal, current-biased junctions. In the case of zero induced magnetic fields, we find the DSG equation (commonly applied to parallel arrays) appreciably overestimates the exponents of the full ladder in the overdamped regime (corresponding to the limit of small junction capacitance, beta(c)), and that difference physically results from differing spectra for small-amplitude phase oscillations of the DSG and RCSJ equations. mutual inductance between plaquettes is included we find there are ranges of values for the mutual inductance for which the ladder is in fact unstable. To understand the cause of the observed instabilities, it is crucial to consider the behavior of the vertical junctions.

Nonlinear dynamics of microwave synthesizers-stability and noise

The nonlinear dynamics of microwave synthesizers based on type-II third-order loops is analyzed in this paper. Instead of using standard simplified models, realistic models are considered for the loop filter, phase detector (PD), and voltage-controlled oscillator based on experimental characterization. The new models enable the simulation of incidental frequency modulation and the accurate prediction of the synthesizer operation ranges, including possible hysteresis phenomena. The stability of phase-locked solutions is analyzed, enabling the prediction of possible chaotic behavior. For an accurate determination of the output spectrum, a phase-noise simulation is also carried out, considering the noise contributions from the loop elements. The sidebands inherent to the synthesizer solution are taken into account for this analysis. The analysis strategy has been applied to a microwave synthesizer, operating in the 2-3 GHz band, with very good results. Two types of PDs are considered: the JK flip-flop PD and frequency mixer, comparing the resulting loop performance in terms of stability and phase noise.

Symmetry and phase-locking in a ring of pulse-coupled oscillators with distributed delays

Phase-locking in a ring of pulse-coupled integrate-and-fire oscillators with distributed delays is analysed using group theory. The period of oscillation of a solution and those related by symmetry is determined self-consistently. Numerical continuation of maximally symmetric solutions in characteristic system length and timescales yields bifurcation diagrams with spontaneous symmetry breaking. The stability of phase-locked solutions is determined via a linearisation of the oscillator firing map. In the weak-coupling regime, averaging leads to an effective phase-coupled model with distributed phase-shifts and the analysis of the system is considerably simplified. In particular, the collective period of a solution is now slaved to the relative phases. For odd numbered rings, spontaneous symmetry breaking can lead to a change of stability of a travelling wave state via a simple Hopf bifurcation. The resulting non-phase-locked solutions are constructed via numerical continuation at these bifurcation points. The corresponding behaviour in the integrate-and-fire system is explored with simulations showing bifurcations to quasiperiodic firing patterns.