Rotating wave solutions of the FitzHugh–Nagumo equations

This paper will treat the bifurcation and numerical simulation of rotating wave (RW) solutions of the FitzHugh-Nagumo (FHN) equations. These equations are often used as a simple mathematical model of excitable media. The dependence of the solutions on a uniformly applied current, and also on the diffusion coefficients or domain size will be studied. Ranges of applied current and/or diffusion coefficients in which RW solutions are observed will be described using bifurcation theory and continuation methods. The bifurcation of time-periodic solutions of these FHN equations without diffusion is described first. Similar methods are then used to find RW solutions on a circular ring and numerical simulations are described. These results are then extended to investigate RW solutions on annular rings of finite cross-section. Scaling arguments are used to show how the existence of solutions depends on either the diffusion coefficient or on the size of the region.     

Rhythms of high-grade block in an ionic model of a strand of regionally ischemic ventricular muscle

Electrical alternans, a beat-to-beat alternation in the electrocardiogram or electrogram, is frequently seen during the first few minutes of acute myocardial ischemia, and is often immediately followed by malignant cardiac arrhythmias such as ventricular tachycardia and ventricular fibrillation. As ischemia progresses, higher-order periodic rhythms (e.g., period-4) can replace the period-2 alternans rhythm. This is also seen in modelling work on a two-dimensional (2-D) sheet of regionally ischemic ventricular muscle. In addition, in the experimental work, ventricular arrhythmias are overwhelmingly seen only after the higher-order rhythms arise. We investigate an ionic model of a strand of ischemic ventricular muscle, constructed as a 3-cm-long 1-D cable with a centrally located 1-cm-long segment exposed to an elevated extracellular potassium concentration ([K(+)](o)). As [K(+)](o) is raised in this "ischemic segment" to represent one major effect of ongoing ischemia, the sequence of rhythms {1:1-->2:2 (alternans)-->2:1} is seen. With further increase in [K(+)](o), one sees higher-order periodic 2N:M rhythms {2:1-->4:2-->4:1-->6:2-->6:1-->8:2-->8:1}. In a 2N:M cycle, only M of the 2N action potentials generated at the proximal end of the cable successfully traverse the ischemic segment, with the remaining ones being blocked within the ischemic segment. Finally, there is a transition to complete block {8:1-->2:0-->1:0} (in an n:0 rhythm, all action potentials die out within the ischemic segment). Changing the length of the ischemic segment results in different rhythms and transitions being seen: e.g., when the ischemic segment is 2 cm long, the period-6 rhythms are not seen; when it is 0.5 cm long, there is a 3:1 rhythm interposed between the 2:1 and 1:0 rhythms. We discuss the relevance of our results to the experimental observations on the higher-order rhythms that presage reentrant ischemic ventricular arrhythmias.     

Transition from polar to duplicate patterns

The existence of symmetric nonuniform solutions in nonlinear reaction-diffusion systems is examined. In the first part of the paper, we establish systematically the bifurcation diagram of small amplitude solutions in the vicinity of the two first bifurcation points. It is shown that:

1. The system can adopt a stable symmetric solution (basic wave number 2) if the value of the bifurcation parameter is changed or if the initial polar structure (basic wave number 1) is sufficiently perturbed.
2. This behavior is independent of the particular reaction-diffusion model proposed and of the number of intermediate components (?2) involved.

In the second part of the paper, analogies are established between the possibilities offered by the bifurcation diagrams, involving only the two first primary branches, and the observation that in the early development of different organisms, appropriate experimental manipulations may switch the normal (polar) developmental pattern to a duplicate structure.     

Ionic channel density of excitable membranes can act as a bifurcation parameter

As the maximal K⁺-conductance (or K⁺-channel density) of the Hodgkin-Huxley equations is reduced, the stable resting membrane potential bifurcates at a subcritical Hopf bifurcation into small amplitude unstable oscillations. These small amplitude solutions jump to large amplitude periodic solutions that correspond to a repetitive discharge of action potentials. Thus the specific channel density can act as a bifurcation parameter, and can control the excitability and autorhythmicity of excitable membranes.     

Biological Experimental Observations of an Unnoticed Chaos as Simulated by the Hindmarsh-Rose Model

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.     

Period-2 cycles and 2:1 phase locking in a biological clock driven by temperature pulses

Crassulacean acid metabolism (CAM) serves as a botanical model system for the investigation of circadian rhythmicity. In a new set of experiments with the obligatory CAM plant Kalancho? daigremontiana the response to periodic stimulations with temperature pulses has been studied. On the basis of an experimental phase-response curve of net CO(2)-gas exchange the effect of periodic stimulation has been simulated using a finite-difference equation. These simulations revealed the locations of two period-2 cycles in the CO(2) uptake of the CAM plant. In subsequent experiments based upon the simulated bifurcation diagram the position and amplitude of one of these cycles were confirmed, while experimental evidence for the second cycle could be found. Possible roles of such dynamics for the functioning of the biological clock are discussed.     

Electrical instability in cardiac muscle: phase singularities and rotors

A dynamical system is "excitable" at some stage in its behavior (e.g. at a rest state or while it is nearly at rest prior to a spontaneous event) if a small, but not too small, stimulus of the right kind elicits an immediate big reaction that eventually leads back to the original state. During this return to excitability a typical system is not excitable. An excitable system need not have an attracting rest state; a spontaneous oscillator can be excitable, too, as is common in biological and in chemical excitable kinetics. In a medium characterized by such excitable dynamics at every point, the excitation can propagate as a travelling pulse. Undamaged cardiac muscle shares with other excitable media certain features of such pulse propagation in two and three dimensions. Among the new electrophysiological phenomena thus anticipated are paired mirror-image vortices ("rotors") organized around phase singularities. These should arise in the myocardium near the intersection of a moving critical contour of phase in the normal cycle of excitation and recovery with a momentary critical contour of local stimulus strength. Such intersections, and the corresponding aftermath of paired rotors, should only occur following certain combinations of stimulus size and stimulus timing. Plotting those combinations on a "vulnerability diagram", one delineates a domain for creation of rotors (corresponding to tachycardia) surrounded on all sides by a halo of combinations at which just a few repetitive responses follow stimulation. The experiments called for to check these implications have now been carried out in the special case of electrically-induced tachycardia in healthy canine ventricle. They support the two-dimensional theory, so a new experiment is suggested to demonstrate wholly intramural three-dimensional vortex filaments.     

Studies on re-entrant arrhythmias and ectopic beats in excitable tissues by bifurcation analyses.

A phase-plane bifurcation analysis is a useful way to theoretically understand how various types of arrhythmias may arise from excitable tissues. In this paper, we have performed phase-plane bifurcation analysis to characterize arrhythmogenic states in excitable tissues. To achieve this, we have first formulated a model which is simple enough to be mathematically tractable, yet captures the non-linear features of cardiac excitation and conduction. In this model, single cells are connected in a circular fashion by gap conductances. Each cell carries the following two types of currents: a passive outward current and an inward "excitable" current which contains an activation and an inactivation gate. The activation gate is responsible for the upstroke of action potential and inactivation gate is responsible for the termination of the plateau potential. With this model, we have constructed bifurcation diagrams as a function of a bifurcation parameter. The parameter chosen as the bifurcation parameter has the property of raising maximum diastolic potential while shorting the refractory period. Our analysis revealed the existence of three distinct multi-stable phases in certain ranges of the bifurcation parameter: (1) bistability between a rotor and a quiescent state, (2) bistability between rotor and ectopic beats, and (3) three stable states co-existing among quiescent state, rotor, and ectopic beats. In these three regions, external impulses exert very distinct effects: In region 1, a brief current pulse can annihilate a re-entrant arrhythmia to quiescence. To initiate re-entry from a quiescent tissue, however, it takes two pulses (a primary pulse followed by a premature pulse at a site different from the "primary" site). In region 2, a brief pulse can convert a re-entrant arrhythmia to ectopic beats. To convert the ectopic beats back to circus movement, these beats have to be suppressed by a few brief current pulses to initiate one-way propagation. Depending on the frequency and strength of impulses in region 3, the tissue can switch back and forth among quiescence, circus movement, and ectopic beats. For comparison, we have also included a more complete Beeler-Reuter cardiac cell model in our analysis and obtained essentially the same results. From the behavioral similarities of these models, we conclude that re-entrant and ectopic arrhythmias must be intrinsic properties of excitable tissues and external stimuli can convert one mode of arrhythmia to another in the multistability regions.(ABSTRACT TRUNCATED AT 400 WORDS)     

The phases of small networks of chemical reactors and neurons

We present an experimental study of the phase relationships observed in small reactor networks consisting of two and three continuous flow stirred tank reactors. In the three-reactor network one chemical oscillator is coupled to two other reactors in parallel in analogy to a small neural net. Each reactor contains an identical reaction mixture of the excitable Belousov-Zhabotinsky reaction which is characterized by its bifurcation diagram, where the electrical current is the bifurcation parameter. Coupling between the reactors is electrical via Pt-working electrodes and it can be either repulsive (inhibitory) or attractive (excitatory). An external electrical stimulus is applied to all three reactors in the form of an asymmetric electrical current pulse which sweeps across the bifurcation diagram. As a consequence, all three reactors oscillate with characteristic oscillation patterns or remain silent in analogy to the firing of neurons. The observed phase behavior depends on the type of coupling in a complex way. This situation is analogous to the in vivo measurements on single neurons (local neurons and projection neurons) performed by G. Laurent and co-workers on the olfactory system of the locust. We propose a simple neural network similar to the reactor network using the Hodgkin-Huxley model to simulate the action potentials of the coupled single neurons. Analogies between the reactor network and the neural network are discussed.     

Subharmonic resonance and chaos in forced excitable systems

 Forced excitable systems arise in a number of biological and physiological applications and have been studied analytically and computationally by numerous authors. Existence and stability of harmonic and subharmonic solutions of a forced piecewise-linear Fitzhugh-Nagumo-like system were studied in Othmer ad Watanabe (1994) and in Xie et al. (1996). The results of those papers were for small and moderate amplitude forcing. In this paper we study the existence of subharmonic solutions of this system under large-amplitude forcing. As in the case of intermediate-amplitude forcing, bistability between 1 : 1 and 2 : 1 solutions is possible for some parameters. In the case of large-amplitude forcing, bistability between 2 : 2 and 2 : 1 solutions, which does not occur in the case of intermediate-amplitude forcing, is also possible for some parameters. We identify several new canonical return maps for a singular system, and we show that chaotic dynamics can occur in some regions of parameter space. We also prove that there is a direct transition from 2 : 2 phase-locking to chaos after the first period-doubling bifurcation, rather than via the infinite sequence of period doublings seen in a smooth quadratic interval map. Coexistence of chaotic dynamics and stable phase-locking can also occur. Received: 6 July 1998 / Revised version: 2 October 1998     

Efficient control of transient wave forms to prevent spreading depolarizations

In various neurological disorders spatio-temporal excitation patterns constitute examples of excitable behavior emerging from pathological pathways. During migraine, seizure, and stroke an initially localized pathological state can temporarily spread indicating a transition from non-excitable to excitable behavior. We investigate these transient wave forms in the generic FitzHugh-Nagumo (FHN) system of excitable media. Our goal is to define an efficient control minimizing the volume of invaded tissue. The general point of such a therapeutic optimization is how to apply control theory in the framework of structures in differential geometry by regarding parameter plane M of the FHN system as a differentiable manifold endowed with a metric. We suggest to equip M with a metric given by pharmacokinetic-pharmacodynamic models of drug receptor interaction.     

Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.     

Bifurcation of periodic solutions of Hodgkin-Huxley model for the squid giant axon

The Hodgkin-Huxley model of the space-clamped squid giant axon is shown to admit unstable periodic solutions for current stimuli less than the stimulus at which the rest state becomes linearly unstable. The periodic solutions are demonstrated both by bifurcation theory and by numerical integration. The presence of subcritical unstable oscillations explains the discontinuous behaviour of the amplitude of the repetitive response as a function of current stimulus     

Investigation of electrical responses recorded from the fenestra cochleae in cats

Electrical responses of the fenestra cochleae to stimulation by clicks of different intensity, polarity, and frequency, were studied in anesthetized cats. The absolute values of amplitude and latent period of the neural component of the response reflect the physiological state of the auditory nerve. Besides ordinary potentials characterized by peaks N1 and N2, specific responses were observed when clicks with an intensity of 85 dB or "rarefaction" clicks were used. Dependence of the amplitude of these responses on the intensity of acoustic stimuli of different polarity was investigated during a change in the rhythm of the stimulation; the effect of different rhythms of stimulation on the gradient of the curve reflecting this relationship was examined. The possible mechanisms of the effect of stimulus frequency are discussed.Scientific-Research Institute of Otolaryngology, Ministry of Health of the Ukrainian SSR, Kiev. Translated from Neirofiziologiya, Vol. 11, No. 2, pp. 151–157, March–April, 1979.     

An analysis of a dendritic neuron model with an active membrane site

We formulate and analyze a mathematical model that couples an idealized dendrite to an active boundary site to investigate the nonlinear interaction between these passive and active membrane patches. The active site is represented mathematically as a nonlinear boundary condition to a passive cable equation in the form of a space-clamped FitzHugh-Nagumo (FHN) equation. We perform a bifurcation analysis for both steady and periodic perturbation at the active site. We first investigate the uncoupled space-clamped FHN equation alone and find that for periodic perturbation a transition from phase locked (periodic) to phase pulling (quasiperiodic) solutions exist. For the model coupling a passive cable with a FHN active site at the boundary, we show for steady perturbation that the interval for repetitive firing is a subset of the interval for the space-clamped case and shrinks to zero for strong coupling. The firing rate at the active site decreases as the coupling strength increases. For periodic perturbation we show that the transition from phase locked to phase pulling solutions is also dependent on the coupling strength.This work was supported in part by NSF Grants MCS 83-00562 and MDS 85-01535     

Analysis of variance delimitation of factors influencing studies of acoustic evoked potentials]

The effect of the variables "stimulus intensity", "residual effect of previous stimulus intensity", "period effect" and "subject effect" on the amplitudes and on the peak latencies of the N1 and P1 components of the acoustical EEG evoked potentials was studied in human subjects with the use of an orthogonal experimental design. The stimulus intensity (30, 50 and 80 dB) accounts for the greatest part of the total variance of the amplitudes and of the peak latencies of N1.-The existence of a totalvariance component due to differences between the subjects is also statistically significant (P equals 5%), As far as the period effect (habituation effect) is concerned, only its effect on the amplitude of N1 - P2 is statistically significant. - The residual effect of the stimulus intensity used in the previous series was not significant.     

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.     

Spontaneous secondary spiking in excitable cells

Kepler & Marder (1993, Biol. Cybern.68, 209-214) proposed a model describing the electrical activity of a crab neuron in which a train of directly induced action potentials is sometimes followed by one or more spontaneous action potentials, referred to as spontaneous secondary spikes. We reduce their five-dimensional model to three dimensions in two different ways in order to gain insight into the mechanism underlying the spontaneous spikes. We then treat a slowly varying current as a parameter in order to give a qualitative explanation of the phenomenon using phase-plane and bifurcation analysis. We demonstrate that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model. The exact dynamics of the excitable system are not important, but the relative time constant and amplitude of the slow inward current are crucial. Using the numerical bifurcation analysis package AUTO (Doedel & Kernevez, 1986, AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology), we compute bifurcation diagrams using the maximum amplitude of the slow inward current as the bifurcation parameter. The full and reduced models have a stable resting potential for all values of the bifurcation parameter. At a critical value of the bifurcation parameter, a stable tonic firing mode arises via a saddle-node of periodics bifurcation. Whether or not the models can exhibit transient or continuous spontaneous spiking depends on their position in parameter space relative to this saddle-node of periodics.     

A Model for the Propagation of Action Potentials in Non-Uniformly Excitable Media

Electrophysiological properties of cardiac tissue change as a function of position. We define the "excitability" as the propagation velocity of an excitation pulse through the tissue, and study a simple FitzHugh-Nagumo (FHN) model of heart tissue whose excitability changes with position. The propagation velocity is shown to be a good continuous measure of the excitability for both limit cycle and excitable tissue. The influence of the spatial dependence of the excitability is examined for several normal and pathological situations. A novel transient effect is observed for a train of pulses propagating across an excitability step. Copyright 1999 Academic Press.     

Event-Related Potentials Dissociate Effects of Salience and Space in Biased Competition for Visual Representation

Background

Selective visual attention is the process by which the visual system enhances behaviorally relevant stimuli and filters out others. Visual attention is thought to operate through a cortical mechanism known as biased competition. Representations of stimuli within cortical visual areas compete such that they mutually suppress each others'' neural response. Competition increases with stimulus proximity and can be biased in favor of one stimulus (over another) as a function of stimulus significance, salience, or expectancy. Though there is considerable evidence of biased competition within the human visual system, the dynamics of the process remain unknown.

Methodology/Principal Findings

Here, we used scalp-recorded electroencephalography (EEG) to examine neural correlates of biased competition in the human visual system. In two experiments, subjects performed a task requiring them to either simultaneously identify two targets (Experiment 1) or discriminate one target while ignoring a decoy (Experiment 2). Competition was manipulated by altering the spatial separation between target(s) and/or decoy. Both experimental tasks should induce competition between stimuli. However, only the task of Experiment 2 should invoke a strong bias in favor of the target (over the decoy). The amplitude of two lateralized components of the event-related potential, the N2pc and Ptc, mirrored these predictions. N2pc amplitude increased with increasing stimulus separation in Experiments 1 and 2. However, Ptc amplitude varied only in Experiment 2, becoming more positive with decreased spatial separation.

Conclusions/Significance

These results suggest that N2pc and Ptc components may index distinct processes of biased competition—N2pc reflecting visual competitive interactions and Ptc reflecting a bias in processing necessary to individuate task-relevant stimuli.