Generation of Very Slow Neuronal Rhythms and Chaos Near the Hopf Bifurcation in Single Neuron Models

Slow passage through a Hopf-like bifurcation in piecewise linear systems: Application to elliptic bursting

The phenomenon of slow passage through a Hopf bifurcation is ubiquitous in multiple-timescale dynamical systems, where a slowly varying quantity replacing a static parameter induces the solutions of the resulting slow-fast system to feel the effect of the Hopf bifurcation with a delay. This phenomenon is well understood in the context of smooth slow-fast dynamical systems; in the present work, we study it for the first time in piecewise linear (PWL) slow-fast systems. This special class of systems is indeed known to reproduce all features of their smooth counterpart while being more amenable to quantitative analysis and offering some level of simplification, in particular, through the existence of canonical (linear) slow manifolds. We provide conditions for a PWL slow-fast system to exhibit a slow passage through a Hopf-like bifurcation, in link with possible connections between canonical attracting and repelling slow manifolds. In doing so, we fully describe the so-called way-in/way-out function. Finally, we investigate this slow passage effect in the Doi-Kumagai model, a neuronal PWL model exhibiting elliptic bursting oscillations.

Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model

In this article, we describe and analyze the chaotic behavior of a conductance-based neuronal bursting model. This is a model with a reduced number of variables, yet it retains biophysical plausibility. Inspired by the activity of cold thermoreceptors, the model contains a persistent Sodium current, a Calcium-activated Potassium current and a hyperpolarization-activated current (I_h) that drive a slow subthreshold oscillation. Driven by this oscillation, a fast subsystem (fast Sodium and Potassium currents) fires action potentials in a periodic fashion. Depending on the parameters, this model can generate a variety of firing patterns that includes bursting, regular tonic and polymodal firing. Here we show that the transitions between different firing patterns are often accompanied by a range of chaotic firing, as suggested by an irregular, non-periodic firing pattern. To confirm this, we measure the maximum Lyapunov exponent of the voltage trajectories, and the Lyapunov exponent and Lempel-Ziv's complexity of the ISI time series. The four-variable slow system (without spiking) also generates chaotic behavior, and bifurcation analysis shows that this is often originated by period doubling cascades. Either with or without spikes, chaos is no longer generated when the I_h is removed from the system. As the model is biologically plausible with biophysically meaningful parameters, we propose it as a useful tool to understand chaotic dynamics in neurons.

Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators

To elucidate how a biological rhythm is regulated, the extended (three-dimensional) Bonhoeffer-van der Pol or FitzHugh-Nagumo equations are employed to investigate the dynamics of a population of neuronal oscillators globally coupled through a common buffer (mean field). Interesting phenomena, such as extraordinarily slow phase-locked oscillations (compared to the natural period of each neuronal oscillator) and the death of all oscillations, are observed. We demonstrate that the slow synchronization is due mainly to the existence of "fast" oscillators. Additionally, we examine the effect of noise on the synchronization and variability of the interspike intervals. Peculiar phenomena, such as noise-induced acceleration and deceleration, are observed. The results herein suggest that very small noise may significantly influence a biological rhythm.

Experimental study of firing death in a network of chaotic FitzHugh-Nagumo neurons

The FitzHugh-Nagumo neurons driven by a periodic forcing undergo a period-doubling route to chaos and a transition to mixed-mode oscillations. When coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems.

Firing patterns in a conductance-based neuron model: bifurcation, phase diagram, and chaos

Responding to various stimuli, some neurons either remain resting or can fire several distinct patterns of action potentials, such as spiking, bursting, subthreshold oscillations, and chaotic firing. In particular, Wilson's conductance-based neocortical neuron model, derived from the Hodgkin-Huxley model, is explored to understand underlying mechanisms of the firing patterns. Phase diagrams describing boundaries between the domains of different firing patterns are obtained via extensive numerical computations. The boundaries are further studied by standard instability analyses, which demonstrates that the chaotic neural firing could develop via period-doubling and/or period- adding cascades. Sequences of the firing patterns often observed in many neural experiments are also discussed in the phase diagram framework developed. Our results lay the groundwork for wider use of the model, especially for incorporating it into neural field modeling of the brain.

Chaos at the border of criticality

The present paper points out a novel scenario for the formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincare map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincare map and those observed in the numerical experiments of the continuous system show a very good agreement.

Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model

It is a challenging problem to establish safe and simple therapeutic methods for various complicated diseases of the nervous system, particularly dynamical diseases such as epilepsy, Alzheimer's disease, and Parkinson's disease. From the viewpoint of nonlinear dynamical systems, a dynamical disease can be considered to be caused by a bifurcation induced by a change in the values of one or more regulating parameter. Therefore, the theory of bifurcation control may have potential applications in the diagnosis and therapy of dynamical diseases. In this study, we employ a washout filter-aided dynamic feedback controller to control the onset of Hopf bifurcation in the Hodgkin-Huxley (HH) model. Specifically, by the control scheme, we can move the Hopf bifurcation to a desired point irrespective of whether the corresponding steady state is stable or unstable. In other words, we are able to advance or delay the Hopf bifurcation, so as to prevent it from occurring in a certain range of the externally applied current. Moreover, we can control the criticality of the bifurcation and regulate the oscillation amplitude of the bifurcated limit cycle. In the controller, there are only two terms: the linear term and the nonlinear cubic term. We show that while the former determines the location of the Hopf bifurcation, the latter regulates the criticality of the Hopf bifurcation. According to the conditions of the occurrence of Hopf bifurcation and the bifurcation stability coefficient, we can analytically deduce the linear term and the nonlinear cubic term, respectively. In addition, we also show that mixed-mode oscillations (MMOs), featuring slow action potential generation, which are frequently observed in both experiments and models of chemical and biological systems, appear in the controlled HH model. It is well known that slow firing rates in single neuron models could be achieved only by type-I neurons. However, the controlled HH model is still classified as a type-II neuron, as is the original HH model. We explain that the occurrence of MMOs can be related to the presence of the canard explosion where a small oscillation grows through a sequence of canard cycles to a relaxation oscillation as the control parameter moves through an interval of exponentially small width.

The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales

In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in the classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode oscillations (MMOs). This theory is only rigorously valid, however, for epsilon sufficiently small, where epsilon is a parameter that arises from nondimensionalization of the HH system. Here, we illustrate how the parameter regime over which MMOs exist, and the features of the MMO patterns within this regime, vary with respect to several key parameters in the nondimensionalized HH equations, including epsilon. Moreover, we explain our findings in terms of the effects that these parameters are expected to have on certain organizing structures within the corresponding flow, generalized from analysis done previously in the singular limit.

Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation theory to analyze the model's dynamics under systematic variation of the parameters tau( h ), tau( n ), and applied current I. As expected, we find that for fixed (tau( h ), tau( n )), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if tau( h ) or tau( n ) is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit.