Continuation and bifurcation analysis of a periodically forced excitable system

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh-Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different "M:N" periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main "tree" of period-2(P) branches, where the 1:1, 1:0, 2:2, 2:1,... rhythms are located, and of several closed loops made up of period-{N x 2(P)} branches (N>2), isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms (N>2) on one hand, and N:N-1 or Wenckebach rhythms (N>2) on the other hand, are located. Stable M:N and M:N-1 rhythms (M>or=N) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

Bifurcation phenomena appearing in the Lotka-Volterra competition equations: a numerical study

Bifurcation phenomena appearing in the Lotka-Volterra competition equations with periodically varying coefficients are studied numerically. We assume sinusoidal oscillations of the coefficients and use phase differences between them as free parameters. We are mainly concerned with the case where a pair of stable and unstable positive periodic solutions exists, although one of the trivial periodic solutions is stable and the other is unstable. We obtain a very curious bifurcation diagram in which two branches of stable and unstable positive periodic solutions are connected at both ends, but are connected with no other branches. We show how this unusual diagram can be viewed as a cross-section of a multidimensional bifurcation diagram. The region in a 3-dimensional parameter space where a pair of stable and unstable positive periodic solutions exists is shown in an example, and the ecological meaning of the phase differences necessary for stable coexistence of two species is considered. Finally, a bifurcation problem with the average intrinsic growth rate as a parameter is also dealt with numerically, in relation with Cushing's result.     

Transient oscillations induced by delayed growth response in the chemostat

In this paper, in order to try to account for the transient oscillations observed in chemostat experiments, we consider a model of single species growth in a chemostat that involves delayed growth response. The time delay models the lag involved in the nutrient conversion process. Both monotone response functions and nonmonotone response functions are considered. The nonmonotone response function models the inhibitory effects of growth response of certain nutrients when concentrations are too high. By applying local and global Hopf bifurcation theorems, we prove that the model has unstable periodic solutions that bifurcate from unstable nonnegative equilibria as the parameter measuring the delay passes through certain critical values and that these local periodic solutions can persist, even if the delay parameter moves far from the critical (local) bifurcation values.When there are two positive equilibria, then positive periodic solutions can exist. When there is a unique positive equilibrium, the model does not have positive periodic oscillations and the unique positive equilibrium is globally asymptotically stable. However, the model can have periodic solutions that change sign. Although these solutions are not biologically meaningful, provided the initial data starts close enough to the unstable manifold of one of these periodic solutions they may still help to account for the transient oscillations that have been frequently observed in chemostat experiments. Numerical simulations are provided to illustrate that the model has varying degrees of transient oscillatory behaviour that can be controlled by the choice of the initial data.Mathematics Subject Classification: 34D20, 34K20, 92D25Research was partially supported by NSERC of Canada.This work was partly done while this author was a postdoc at McMaster.     

Effects of quarantine in six endemic models for infectious diseases

Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence. For most of these models, the endemic equilibrium is asymptotically stable, but for the SIQR model with the quarantine-adjusted incidence, the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation. The Hopf bifurcation surface and stable periodic solutions are found numerically.     

Application of the describing function to the Danziger-Elmegreen equations

The describing function method is used as a guide to the behaviour of the solutions of the equations of Danziger and Elmergreen, proposed as a model of periodic catatonia. The method suggests that whenever the equilibrium point is unstable it is surrounded by a stable closed periodic orbit. This is confirmed in specific cases by computation.     

Phase resetting and bifurcation in the ventricular myocardium.

With the dynamic differential equations of Beeler, G. W., and H. Reuter (1977, J. Physiol. [Lond.]. 268:177-210), we have studied the oscillatory behavior of the ventricular muscle fiber stimulated by a depolarizing applied current I app. The dynamic solutions of BR equations revealed that as I app increases, a periodic repetitive spiking mode appears above the subthreshold I app, which transforms to a periodic spiking-bursting mode of oscillations, and finally to chaos near the suprathreshold I app (i.e., near the termination of the periodic state). Phase resetting and annihilation of repetitive firing in the ventricular myocardium were demonstrated by a brief current pulse of the proper magnitude applied at the proper phase. These phenomena were further examined by a bifurcation analysis. A bifurcation diagram constructed as a function of I app revealed the existence of a stable periodic solution for a certain range of current values. Two Hopf bifurcation points exist in the solution, one just above the lower periodic limit point and the other substantially below the upper periodic limit point. Between each periodic limit point and the Hopf bifurcation, the cell exhibited the coexistence of two different stable modes of operation; the oscillatory repetitive firing state and the time-independent steady state. As in the Hodgkin-Huxley case, there was a low amplitude unstable periodic state, which separates the domain of the stable periodic state from the stable steady state. Thus, in support of the dynamic perturbation methods, the bifurcation diagram of the BR equation predicts the region where instantaneous perturbations, such as brief current pulses, can send the stable repetitive rhythmic state into the stable steady state.     

Traveling Wave Solutions of a Nerve Conduction Equation

We consider a pair of differential equations whose solutions exhibit the qualitative properties of nerve conduction, yet which are simple enough to be solved exactly and explicitly. The equations are of the FitzHugh-Nagumo type, with a piecewise linear nonlinearity, and they contain two parameters. All the pulse and periodic solutions, and their propagation speeds, are found for these equations, and the stability of the solutions is analyzed. For certain parameter values, there are two different pulse-shaped waves with different propagation speeds. The slower pulse is shown to be unstable and the faster one to be stable, confirming conjectures which have been made before for other nerve conduction equations. Two periodic waves, representing trains of propagated impulses, are also found for each period greater than some minimum which depends on the parameters. The slower train is unstable and the faster one is usually stable, although in some cases both are unstable.     

Bifurcation of orbits and synchrony in inferior olive neurons

Inferior olive neurons (IONs) have rich dynamics and can exhibit stable, unstable, periodic, and even chaotic trajectories. This paper presents an analysis of bifurcation of periodic orbits of an ION when its two key parameters (a, μ) are varied in a two-dimensional plane. The parameter a describes the shape of the parabolic nonlinearity in the model and μ is the extracellular stimulus. The four-dimensional ION model considered here is a cascade connection of two subsystems (S(a) and S(b)). The parameter plane (a - μ) is delineated into several subregions. The ION has distinct orbit structure and stability property in each subregion. It is shown that the subsystem S(a) or S(b) undergoes supercritical Poincare-Andronov-Hopf (PAH) bifurcation at a critical value μ(c)(a) of the extracellular stimulus and periodic orbits of the neuron are born. Based on the center manifold theory, the existence of periodic orbits in the asymptotically stable S(a), when the subsystem S(b) undergoes PAH bifurcation, is established. In such a case, both subsystems exhibit periodic orbits. Interestingly when S(b) is under PAH bifurcation and S(a) is unstable, the trajectory of S(a) exhibits periodic bursting, interrupted by periods of quiescence. The bifurcation analysis is followed by the design of (i) a linear first-order filter and (ii) a nonlinear control system for the synchronization of IONs. The first controller uses a single output of each ION, but the nonlinear control system uses two state variables for feedback. The open-loop and closed-loop responses are presented which show bifurcation of orbits and synchronization of oscillating neurons.     

Digital Computer Solutions for Excitation and Propagation of the Nerve Impulse

The Hodgkin-Huxley model of the nerve axon describes excitation and propagation of the nerve impulse by means of a nonlinear partial differential equation. This equation relates the conservation of the electric current along the cablelike structure of the axon to the active processes represented by a system of three rate equations for the transport of ions through the nerve membrane. These equations have been integrated numerically with respect to both distance and time for boundary conditions corresponding to a finite length of squid axon stimulated intracellularly at its midpoint. Computations were made for the threshold strength-duration curve and for the repetitive firing of propagated impulses in response to a maintained stimulus. These results are compared with previous solutions for the space-clamped axon. The effect of temperature on the threshold intensity for a short stimulus and for rheobase was determined for a series of values of temperature. Other computations show that a highly unstable subthreshold propagating wave is initiated in principle by a just threshold stimulus; that the stability of the subthreshold wave can be enhanced by reducing the excitability of the axon as with an anesthetic agent, perhaps to the point where it might be observed experimentally; but that with a somewhat greater degree of narcotization, the axon gives only decrementally propagated impulses.     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Desynchronization of brain rhythms with soft phase-resetting techniques

 Composite stimulation techniques are presented here which are based on a soft (i.e., slow and mild) reset. They effectively desynchronize a cluster of globally coupled phase oscillators in the presence of noise. A composite stimulus contains two qualitatively different stimuli. The first stimulus is either a periodic pulse train or a smooth, sinusoidal periodic stimulus with an entraining frequency close to the cluster's natural frequency. In the course of several periods of the entrainment, the cluster's dynamics is reset (restarted), independently of its initial dynamic state. The second stimulus, a single pulse, is administered with a fixed delay after the first stimulus in order to desynchronize the cluster by hitting it in a vulnerable state. The incoherent state is unstable, and thus the desynchronized cluster starts to resynchronize. Nevertheless, resynchronization can effectively be blocked by repeatedly delivering the same composite stimulus. Previously designed stimulation techniques essentially rely on a hard (i.e., abrupt) reset. With the composite stimulation techniques based on a soft reset, an effective desynchronization can be achieved even if strong, quickly resetting stimuli are not available or not tolerated. Accordingly, the soft methods are very promising for applications in biology and medicine requiring mild stimulation. In particular, it can be applied to effectively maintain incoherency in a population of oscillatory neurons which try to synchronize their firing. Accordingly, it is explained how to use the soft techniques for (i) an improved, milder, and demand-controlled deep brain stimulation for patients with Parkinson's disease or essential tremor, and for (ii) selectively blocking gamma activity in order to manipulate visual binding. Received: 3 July 2001 / Accepted in revised form: 7 February 2002     

Basilar membrane nonlinearity]

The nonlinear correction of the cochlear partitions movement equation becomes the main part in the resonant section. In the neighbourhood of this section the periodic solution, having the drawing force frequency omega, gets the region of the non-stability in the interval (see abstract), where a is the vibration amplitude and h is the basilar membrane thickness. The amplitude jump in the unstable region may be a stimulus exciting the hair cells and also the cause of cochlear eddies phenomenon.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Construction of an associative memory using unstable periodic orbits of a chaotic attractor

Unstable periodic orbits are the skeleton of a chaotic attractor. We constructed an associative memory based on the chaotic attractor of an artificial neural network, which associates input patterns to unstable periodic orbits. By processing an input, the system is driven out of the ground state to one of the pre-defined disjunctive areas of the attractor. Each of these areas is associated with a different unstable periodic orbit. We call an input pattern learned if the control mechanism keeps the system on the unstable periodic orbit during the response. Otherwise, the system relaxes back to the ground state on a chaotic trajectory. The major benefits of this memory device are its high capacity and low-energy consumption. In addition, new information can be simply added by linking a new input to a new unstable periodic orbit.     

On the stability of the stationary state of a population growth equation with time-lag

Summary If in the Verhulst equation for population growth the reproduction factor depends on the history then the equilibrium may become unstable and oscillations and even non-constant periodic solutions may occur. It is shown that the equilibrium is unstable if the reproduction factor at time t is, up to a sufficiently large factor, an arbitrary average of the population densities in the interval (t–2, t–1].     

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.     

Dynamics of the noisy neural network

The dynamic features of interspike interval sequences and structures of spatiotemporal patterns of firing in a coupled noisy neural network are investigated. The system displays complex dynamics under periodic external stimuli. The dynamics is modulated by a periodic impulse-like synaptic current which relates to a global coupling interaction between neurons. The firings of the stimulated neurons are phase-locked to this current. In addition, the interspike interval histograms are studied for the case of frozen noise which does not change its value within a time interval once it has been distributed onto the network. It is found that the peaks in these histograms are located at integer multiples of the period of the external stimulus, and the heights of these peaks decay exponentially, which corresponds to the experimental results. Received: 12 April 1997 / Accepted in revised form: 7 May 1997     

交流外电场下映射神经元放电节律的分析

神经元不同的放电节律承载着不同的刺激信息。文章基于神经元映射模型,研究低频交流电场对神经元放电节律的影响。在外部刺激下映射模型表现出丰富的放电模式,包括周期簇放电、周期峰放电、交替放电和混沌放电。神经元对刺激频率和振幅的变化极为敏感,随着频率的增大,放电节律表现出从簇放电到峰放电和混沌放电的反向加周期分岔序列;在周期节律转迁过程中存在一种新的交替节律,其放电序列为两种周期放电模式的交替,峰峰间期序列具有整数倍特征。外电场的频率影响细胞内、外离子振荡周期,导致神经元放电与刺激信号同步,对放电节律的影响更为明显。研究结果揭示了交流外电场对神经元放电节律的作用规律,有助于探寻外电场对生物神经系统兴奋性的影响和神经系统疾病的致病机理。