High-order synchronization, transitions, and competition among Arnold tongues in a rotator under harmonic forcing

High-order synchronization in identical neurons with asymmetric pulse coupling

The phenomenon of high-order (p/q) synchronization, induced by two different frequencies in the system, is well known and studied extensively in forced oscillators including neurons and to a lesser extent in coupled oscillators. Their frequencies are locked such that for every p cycles of one oscillator there are q cycles of the other. We demonstrate this phenomenon in a pair of coupled neurons having identical frequencies but asymmetric coupling. Specifically, we focus on an excitatory-inhibitory (E-I) neuron pair where such an asymmetry is naturally present even with equal reciprocal synaptic strengths (g) and inverse time constant (α). We thoroughly investigate the asymmetric coupling-induced p/q frequency-locking structure in (g,α) parameter space through simulations and analysis. Simulations display quasiperiodicity, devil staircase, a Farey arrangement of spike sequences, and presence of reducible and irreducible p/q regions. We introduce an analytical method, based on event-driven maps, to determine the existence and stability of any spike sequence of the two neurons in a p/q frequency-locked state. Specifically, this method successfully deals with nonsmooth bifurcations and we could utilize it to obtain solutions for the case of identical E-I neuron pair under arbitrary coupling strength. In contrast to the so-called Arnold tongues, the p/q regions obtained here are not structureless. Instead they have their own internal bifurcation structure with varying levels of complexity. Intrasequence and intersequence multistability, involving spike sequences of same p/q state, are found. Additionally, multistability also arises by overlap of p/q with p^{'}/q^{'}. The boundaries of both reducible and irreducible p/q regions are defined by saddle node and nonsmooth grazing bifurcations of various types.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

Arnold tongues in oscillator systems with nonuniform spatial driving

Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing qualitative implications of our results for controlling coupled oscillator systems such as the social amoebae Dictyostelium and populations of Belousov Zhabotinsky (BZ) catalytic particles using spatially structured external drives.

Stabilization of dynamics of oscillatory systems by nonautonomous perturbation

Synchronization and stability under periodic oscillatory driving are well understood, but little is known about the effects of aperiodic driving, despite its abundance in nature. Here, we consider oscillators subject to driving with slowly varying frequency, and investigate both short-term and long-term stability properties. For a phase oscillator, we find that, counterintuitively, such variation is guaranteed to enlarge the Arnold tongue in parameter space. Using analytical and numerical methods that provide information on time-variable dynamical properties, we find that the growth of the Arnold tongue is specifically due to the growth of a region of intermittent synchronization where trajectories alternate between short-term stability and short-term neutral stability, giving rise to stability on average. We also present examples of higher-dimensional nonlinear oscillators where a similar stabilization phenomenon is numerically observed. Our findings help support the case that in general, deterministic nonautonomous perturbation is a very good candidate for stabilizing complex dynamics.

On cross-frequency phase-phase coupling between theta and gamma oscillations in the hippocampus

Phase-amplitude coupling between theta and multiple gamma sub-bands is a hallmark of hippocampal activity and believed to take part in information routing. More recently, theta and gamma oscillations were also reported to exhibit phase-phase coupling, or n:m phase-locking, suggesting an important mechanism of neuronal coding that has long received theoretical support. However, by analyzing simulated and actual LFPs, here we question the existence of theta-gamma phase-phase coupling in the rat hippocampus. We show that the quasi-linear phase shifts introduced by filtering lead to spurious coupling levels in both white noise and hippocampal LFPs, which highly depend on epoch length, and that significant coupling may be falsely detected when employing improper surrogate methods. We also show that waveform asymmetry and frequency harmonics may generate artifactual n:m phase-locking. Studies investigating phase-phase coupling should rely on appropriate statistical controls and be aware of confounding factors; otherwise, they could easily fall into analysis pitfalls.

DOI:http://dx.doi.org/10.7554/eLife.20515.001

Neuroscientists have long sought to understand how the brain works by analyzing its electrical activity. Placing electrodes on the scalp or lowering them into the brain itself reveals rhythmic waves of activity known as oscillations. These arise when large numbers of neurons fire in synchrony. Recordings reveal that the frequency of these oscillations – the number of cycles of a wave per second, measured in Hertz – can vary between brain regions, and within a single region over time. Moreover, oscillations with different frequencies can co-exist and interact with one another.

Within the hippocampus, an area of the brain involved in memory, two types of oscillations dominate: theta waves and gamma waves. Theta waves are relatively slow waves, with a frequency between 5 and 10 Hertz. Gamma waves are faster, with a frequency of up to 100 Hertz. Recent work has suggested that gamma waves and theta waves show a phenomenon called phase-phase coupling. Since gamma waves are faster than theta waves, multiple cycles of gamma can occur during a single cycle of theta. Phase-phase coupling is the idea that gamma and theta waves align themselves, such that gamma waves always begin at the same relative position within a theta wave. This was thought to help the hippocampus to encode memories.

Using computer simulations and recordings from the rat hippocampus, Scheffer-Teixeira and Tort have now reexamined the evidence for theta-gamma phase-phase coupling. The new results suggest that previous reports describing the phenomenon may have relied on inadequate statistical techniques. Using stringent control analyses, Scheffer-Teixeira and Tort find no evidence for prominent theta-gamma phase-phase coupling in the hippocampus. Instead, the simulations suggest that what appeared to be statistically significant coupling may in reality be an artifact of the previous analysis.

Phase-phase coupling of theta and gamma waves has also been reported in the human hippocampus. The next step therefore is to apply these more robust analysis techniques to data from the human brain. While revisiting previously accepted findings may not always be popular, it will likely be essential if neuroscientists want to accurately understand how new memories are formed.

DOI:http://dx.doi.org/10.7554/eLife.20515.002

Brain rhythms and neural syntax: implications for efficient coding of cognitive content and neuropsychiatric disease

The perpetual activity of the cerebral cortex is largely supported by the variety of oscillations the brain generates, spanning a number of frequencies and anatomical locations, as well as behavioral correlates. First, we review findings from animal studies showing that most forms of brain rhythms are inhibition-based, producing rhythmic volleys of inhibitory inputs to principal cell populations, thereby providing alternating temporal windows of relatively reduced and enhanced excitability in neuronal networks. These inhibition-based mechanisms offer natural temporal frames to group or "chunk" neuronal activity into cell assemblies and sequences of assemblies, with more complex multi-oscillation interactions creating syntactical rules for the effective exchange of information among cortical networks. We then review recent studies in human psychiatric patients demonstrating a variety alterations in neural oscillations across all major psychiatric diseases, and suggest possible future research directions and treatment approaches based on the fundamental properties of brain rhythms.

Cross-frequency phase-phase coupling between θ and γ oscillations in the hippocampus

Neuronal oscillations allow for temporal segmentation of neuronal spikes. Interdependent oscillators can integrate multiple layers of information. We examined phase-phase coupling of theta and gamma oscillators in the CA1 region of rat hippocampus during maze exploration and rapid eye movement sleep. Hippocampal theta waves were asymmetric, and estimation of the spatial position of the animal was improved by identifying the waveform-based phase of spiking, compared to traditional methods used for phase estimation. Using the waveform-based theta phase, three distinct gamma bands were identified: slow gamma(S) (gamma(S); 30-50 Hz), midfrequency gamma(M) (gamma(M); 50-90 Hz), and fast gamma(F) (gamma(F); 90-150 Hz or epsilon band). The amplitude of each sub-band was modulated by the theta phase. In addition, we found reliable phase-phase coupling between theta and both gamma(S) and gamma(M) but not gamma(F) oscillators. We suggest that cross-frequency phase coupling can support multiple time-scale control of neuronal spikes within and across structures.

Cross-frequency phase-phase coupling between θ and γ oscillations in the hippocampus

Neuronal oscillations allow for temporal segmentation of neuronal spikes. Interdependent oscillators can integrate multiple layers of information. We examined phase-phase coupling of theta and gamma oscillators in the CA1 region of rat hippocampus during maze exploration and rapid eye movement sleep. Hippocampal theta waves were asymmetric, and estimation of the spatial position of the animal was improved by identifying the waveform-based phase of spiking, compared to traditional methods used for phase estimation. Using the waveform-based theta phase, three distinct gamma bands were identified: slow gamma(S) (gamma(S); 30-50 Hz), midfrequency gamma(M) (gamma(M); 50-90 Hz), and fast gamma(F) (gamma(F); 90-150 Hz or epsilon band). The amplitude of each sub-band was modulated by the theta phase. In addition, we found reliable phase-phase coupling between theta and both gamma(S) and gamma(M) but not gamma(F) oscillators. We suggest that cross-frequency phase coupling can support multiple time-scale control of neuronal spikes within and across structures.