Unstable dynamics of a periodically driven oscillator in the presence of noise

Multi-scale modelling of the epileptic brain: advantages of computational therapy exploration

Objective: Epilepsy is a complex disease spanning across multiple scales, from ion channels in neurons to neuronal circuits across the entire brain. Over the past decades, computational models have been used to describe the pathophysiological activity of the epileptic brain from different aspects. Traditionally, each computational model can aid in optimizing therapeutic interventions, therefore, providing a particular view to design strategies for treating epilepsy. As a result, most studies are concerned with generating specific models of the epileptic brain that can help us understand the certain machinery of the pathological state. Those specific models vary in complexity and biological accuracy, with system-level models often lacking biological details.Approach: Here, we review various types of computational model of epilepsy and discuss their potential for different therapeutic approaches and scenarios, including drug discovery, surgical strategies, brain stimulation, and seizure prediction. We propose that we need to consider an integrated approach with a unified modelling framework across multiple scales to understand the epileptic brain. Our proposal is based on the recent increase in computational power, which has opened up the possibility of unifying those specific epileptic models into simulations with an unprecedented level of detail.Main results: A multi-scale epilepsy model can bridge the gap between biologically detailed models, used to address molecular and cellular questions, and brain-wide models based on abstract models which can account for complex neurological and behavioural observations.Significance: With these efforts, we move toward the next generation of epileptic brain models capable of connecting cellular features, such as ion channel properties, with standard clinical measures such as seizure severity.

Kölliker-Fuse nuclei regulate respiratory rhythm variability via a gain-control mechanism

Respiration varies from breath to breath. On the millisecond timescale of spiking, neuronal circuits exhibit variability due to the stochastic properties of ion channels and synapses. Does this fast, microscopic source of variability contribute to the slower, macroscopic variability of the respiratory period? To address this question, we modeled a stochastic oscillator with forcing; then, we tested its predictions experimentally for the respiratory rhythm generated by the in situ perfused preparation during vagal nerve stimulation (VNS). Our simulations identified a relationship among the gain of the input, entrainment strength, and rhythm variability. Specifically, at high gain, the periodic input entrained the oscillator and reduced variability, whereas at low gain, the noise interacted with the input, causing events known as "phase slips", which increased variability on a slow timescale. Experimentally, the in situ preparation behaved like the low-gain model: VNS entrained respiration but exhibited phase slips that increased rhythm variability. Next, we used bilateral muscimol microinjections in discrete respiratory compartments to identify areas involved in VNS gain control. Suppression of activity in the nucleus tractus solitarii occluded both entrainment and amplification of rhythm variability by VNS, confirming that these effects were due to the activation of the Hering-Breuer reflex. Suppressing activity of the Kölliker-Fuse nuclei (KFn) enhanced entrainment and reduced rhythm variability during VNS, consistent with the predictions of the high-gain model. Together, the model and experiments suggest that the KFn regulates respiratory rhythm variability via a gain control mechanism.

Nodal recovery, dual pathway physiology, and concealed conduction determine complex AV dynamics in human atrial tachyarrhythmias

The genesis of complex ventricular rhythms during atrial tachyarrhythmias in humans is not fully understood. To clarify the dynamics of atrioventricular (AV) conduction in response to a regular high-rate atrial activation, 29 episodes of spontaneous or pacing-induced atrial flutter (AFL), covering a wide range of atrial rates (cycle lengths from 145 to 270 ms), were analyzed in 10 patients. AV patterns were identified by applying firing sequence and surrogate data analysis to atrial and ventricular activation series, whereas modular simulation with a difference-equation AV node model was used to correlate the patterns with specific nodal properties. AV node response at high atrial rate was characterized by 1) AV patterns of decreasing conduction ratios at the shortening of atrial cycle length (from 236.3 ± 32.4 to 172.6 ± 17.8 ms) according to a Farey sequence ordering (conduction ratio from 0.34 ± 0.12 to 0.23 ± 0.06; P < 0.01); 2) the appearance of high-order alternating Wenckebach rhythms, such as 6:2, 10:2, and 12:2, associated with ventricular interval oscillations of large amplitude (407.7 ± 150.4 ms); and 3) the deterioration of pattern stability at advanced levels of block, with the percentage of stable patterns decreasing from 64.3 ± 35.2% to 28.3 ± 34.5% ( P < 0.01). Simulations suggested these patterns to originate from the combined effect of nodal recovery, dual pathway physiology, and concealed conduction. These results indicate that intrinsic nodal properties may account for the wide spectrum of AV block patterns occurring during regular atrial tachyarrhythmias. The characterization of AV nodal function during different AFL forms constitutes an intermediate step toward the understanding of complex ventricular rhythms during atrial fibrillation.

Cortical pyramidal cells as non-linear oscillators: experiment and spike-generation theory

Cortical neurons are capable of generating trains of action potentials in response to current injections. These discharges can take different forms, e.g., repetitive firing that adapts during the period of current injection or bursting behaviors. We have used a combined experimental and computational approach to characterize the dynamics leading to action potential responses in single neurons. Specifically we investigated the origin of complex firing patterns in response to sinusoidal current injections. Using a reduced model, the theta-neuron, alongside recordings from cortical pyramidal cells we show that both real and simulated neurons show phase-locking to sine wave stimuli up to a critical frequency, above which period skipping and 1-to-x phase-locking occurs. The locking behavior follows a complex "devil's staircase" phenomena, where locked modes are interleaved with irregular firing. We further show that the critical frequency depends on the time scale of spike generation and on the level of spike frequency adaptation. These results suggest that phase-locking of neuronal responses to complex input patterns can be explained by basic properties of the spike-generating machinery.

A minimal model for the respiratory sinus arrhythmia

The cardiac and respiratory rhythms in humans are known to be coupled by several mechanisms. In particular, the first rhythm is deeply modulated by the second. In this report we propose a simple operational model for heart rate variability which, taking such modulation into account, reproduces the main features of some experimental sequences of RR intervals recorded from healthy subjects in the resting condition. Also, peer analysis of the model performance allows us to answer the question whether the observed behaviour should be ascribed to phase synchronisation of the heart beating to the respiratory rhythm. Lastly, the changes of the model activity brought about by changing its relevant parameters are analysed and discussed.

Spiking dynamics of interacting oscillatory neurons

Spiking sequences emerging from dynamical interaction in a pair of oscillatory neurons are investigated theoretically and experimentally. The model comprises two unidirectionally coupled FitzHugh-Nagumo units with modified excitability (MFHN). The first (master) unit exhibits a periodic spike sequence with a certain frequency. The second (slave) unit is in its excitable mode and responds on the input signal with a complex (chaotic) spike trains. We analyze the dynamic mechanisms underlying different response behavior depending on interaction strength. Spiking phase maps describing the response dynamics are obtained. Complex phase locking and chaotic sequences are investigated. We show how the response spike trains can be effectively controlled by the interaction parameter and discuss the problem of neuronal information encoding.

Mechanisms for synchrony and alternation in song interactions of the bushcricket Mecopoda elongata (Tettigoniidae: Orthoptera)

Males of the bushcricket Mecopoda elongata synchronise or alternate their chirps with their neighbours in an aggregation. Since synchrony is imperfect, leader and follower chirps are established in song interactions; females prefer leader chirps in phonotactic trials. Using playback experiments and simulations of song oscillator interactions, we investigate the mechanisms that result in synchrony and alternation, and the probability for the leader role in synchrony. A major predictor for the leader role of a male is its intrinsic chirp period, which varies in a population from 1.6 to 2.3 s. Faster singing males establish the leader role more often than males with longer chirp periods. The phase-response curve (PRC) of the song oscillators differs to other rhythmically calling or flashing insects, in that only the disturbed cycle is influenced in duration by a stimulus. This results in sustained leader or follower chirps of one male, when the intrinsic chirp periods of two males differ by 150 ms or more. By contrast, the individual shape of the male's PRC has only little influence on the outcome of chirp interactions. The consequences of these findings for the evolution of synchrony in this species are discussed.

Amplitude and frequency dependence of spike timing: implications for dynamic regulation

The spike-time reliability of motoneurons in the Aplysia buccal motor ganglion was studied as a function of the frequency content and the relative amplitude of the fluctuations in the neuronal input, calculated as the coefficient of variation (CV). Measurements of spike-time reliability to sinusoidal and aperiodic inputs, as well as simulations of a noisy leaky integrate-and-fire neuron stimulated by spike trains drawn from a periodically modulated process, demonstrate that there are three qualitatively different CV-dependent mechanisms that determine reliability: noise-dominated (CV < 0.05 for Aplysia motoneurons) where spike timing is unreliable regardless of frequency content; resonance-dominated (CV approximately 0.05-0.25) where reliability is reduced by removal of input frequencies equal to motoneuron firing rate; and amplitude-dominated (CV >0.35) where reliability depends on input frequencies greater than motoneuron firing rate. In the resonance-dominated regime, changes in the activity of the presynaptic inhibitory interneuron B4/5 alter motoneuron spike-time reliability. The increases or decreases in reliability occur coincident with small changes in motoneuron spiking rate due to changes in interneuron activity. Injection of a hyperpolarizing current into the motoneuron reproduces the interneuron-induced changes in reliability. The rate-dependent changes in reliability can be understood from the phase-locking properties of regularly spiking motoneurons to periodic inputs. Our observations demonstrate that the ability of a neuron to support a spike-time code can be actively controlled by varying the properties of the neuron and its input.

Firing pattern modulation by oscillatory input in supragranular pyramidal neurons

Neocortical neurons in vivo receive periodic stimuli due to feedforward input from the periphery as well as local cellular and circuit properties. In order to understand how neurons process such information, the responses of neurons to periodic sine wave current stimuli of varying frequencies and amplitudes were investigated. Sine wave stimuli were injected into pyramidal cells of young adult ferret visual cortical slices in vitro using sharp microelectrodes. To simulate higher resting membrane potentials observed in vivo a slight depolarizing current was injected to bring the neuron just to threshold. Initially, neurons discharged at least one action potential per sine wave cycle, but as the frequency was increased, a point was reached where this one-to-one responsiveness was lost. This critical frequency was dependent upon the injected sine wave amplitude and the magnitude of the underlying steady-state depolarization, and was correlated with spike width. Larger steady-state depolarizations and thinner action potentials corresponded to higher critical frequencies. Thus, when a neuron was very active it could respond in a one-to-one fashion over a greater range of frequencies than with the smallest DC offset. The results suggest that the frequency-following characteristics of individual cortical neurons can be modulated by the activity state of the neuron itself.

Complex patterns of abnormal heartbeats

Individuals having frequent abnormal heartbeats interspersed with normal heartbeats may be at an increased risk of sudden cardiac death. However, mechanistic understanding of such cardiac arrhythmias is limited. We present a visual and qualitative method to display statistical properties of abnormal heartbeats. We introduce dynamical "heartprints" which reveal characteristic patterns in long clinical records encompassing approximately 10(5) heartbeats and may provide information about underlying mechanisms. We test if these dynamics can be reproduced by model simulations in which abnormal heartbeats are generated (i) randomly, (ii) at a fixed time interval following a preceding normal heartbeat, or (iii) by an independent oscillator that may or may not interact with the normal heartbeat. We compare the results of these three models and test their limitations to comprehensively simulate the statistical features of selected clinical records. This work introduces methods that can be used to test mathematical models of arrhythmogenesis and to develop a new understanding of underlying electrophysiologic mechanisms of cardiac arrhythmia.

Oscillations and noise: inherent instability of pressure support ventilation?

Pressure support ventilation (PSV) is almost universally employed in the management of actively breathing ventilated patients with acute respiratory failure. In this partial support mode of ventilation, a fixed pressure is applied to the airway opening, and flow delivery is monitored by the ventilator. Inspiration is terminated when measured inspiratory flow falls below a set fraction of the peak flow rate (flow cutoff); the ventilator then cycles to a lower pressure and expiration commences. We used linear and nonlinear mathematical models to investigate the dynamic behavior of pressure support ventilation and confirmed the predicted behavior using a test lung. Our mathematical and laboratory analyses indicate that pressure support ventilation in the setting of airflow obstruction can be accompanied by marked variations in tidal volume and end-expiratory alveolar pressure, even when subject effort is unvarying. Unstable behavior was observed in the simplest plausible linear mathematical model and is an inherent consequence of the underlying dynamics of this mode of ventilation. The mechanism underlying the observed instability is "feed forward" behavior mediated by oscillatory elevation in end-expiratory pressure. In both mathematical and mechanical models, unstable behavior occurred at impedance values and ventilator settings that are clinically realistic.

External noise synchronizes forced oscillators

Periodic pulsatile perturbation of nonlinear oscillators generates phase-locking, quasiperiodic, and chaotic responses. This work shows that the application of external noise to ensembles of such forced systems can synchronize oscillations, even in regimes where neither the noise nor the periodic forcing, when applied alone, would lead to such a phenomenon.

Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors

Weakly electric fish generate a periodic electric field as a carrier signal for active location and communication tasks. Highly sensitive P-type receptors on their surface fire in response to carrier amplitude modulations (AM's) in a noisy phase locked fashion. A simple generic model of receptor activity and signal encoding is presented. Its suprathreshold dynamics, memory and receptor noise reproduce observed firing interval distributions and correlations. The model ultimately explains how smooth responses to AM's are compatible with its nonlinear phase locking properties, and reveals how receptor noise can sometimes enhance the encoding of small yet suprathreshold AM's.

Synchronization in the human cardiorespiratory system

We investigate synchronization between cardiovascular and respiratory systems in healthy humans under free-running conditions. For this aim we analyze nonstationary irregular bivariate data, namely, electrocardiograms and measurements of respiratory flow. We briefly discuss a statistical approach to synchronization in noisy and chaotic systems and illustrate it with numerical examples; effects of phase and frequency locking are considered. Next, we present and discuss methods suitable for the detection of hidden synchronous epochs from such data. The analysis of the experimental records reveals synchronous regimes of different orders n:m and transitions between them; the physiological significance of this finding is discussed.

Resonance effect for neural spike time reliability

The spike timing reliability of Aplysia motoneurons stimulated by repeated presentation of periodic or aperiodic input currents is investigated. Two properties of the input are varied, the frequency content and the relative amplitude of the fluctuations to the mean (expressed as the coefficient of variation: CV). It is shown that, for small relative amplitude fluctuations (CV approximately 0.05-0.15), the reliability of spike timing is enhanced if the input contains a resonant frequency equal to the firing rate of the neuron in response to the DC component of the input. This resonance-related enhancement in reliability decreases as the relative amplitude of the fluctuations increases (CV-->1). Similar results were obtained for a leaky integrate-and-fire neuronal model, suggesting that these effects are a general property of encoders that combine a threshold with a leaky integrator. These observations suggest that, when the magnitude of input fluctuations is small, changes in the power spectrum of the current fluctuations or in the spike discharge rate can have a pronounced effect on the ability of the neuron to encode a time-varying input with reliably timed spikes.

Age-structured and two-delay models for erythropoiesis

An age-structured model is developed for erythropoiesis and is reduced to a system of threshold-type differential delay equations using the method of characteristics. Under certain assumptions, this model can be reduced to a system of delay differential equations with two delays. The parameters in the system are estimated from experimental data, and the model is simulated for a normal human subject following a loss of blood. The characteristic equation of the two-delay equation is analyzed and shown to exhibit Hopf bifurcations when the destruction rate of erythrocytes is increased. A numerical study for a rabbit with autoimmune hemolytic anemia is performed and compared with experimental data.

Dysrhythmias of the respiratory oscillator

Breathing is regulated by a central neural oscillator that produces rhythmic output to the respiratory muscles. Pathological disturbances in rhythm (dysrhythmias) are observed in the breathing pattern of children and adults with neurological and cardiopulmonary diseases. The mechanisms responsible for genesis of respiratory dysrhythmias are poorly understood. The present studies take a novel approach to this problem. The basic postulate is that the rhythm of the respiratory oscillator can be altered by a variety of stimuli. When the oscillator recovers its rhythm after such perturbations, its phase may be reset relative to the original rhythm. The amount of phase resetting is dependent upon stimulus parameters and the level of respiratory drive. The long-range hypothesis is that respiratory dysrhythmias can be induced by stimuli that impinge upon or arise within the respiratory oscillator with certain combinations of strength and timing relative to the respiratory cycle. Animal studies were performed in anesthetized or decerebrate preparations. Neural respiratory rhythmicity is represented by phrenic nerve activity, allowing use of open-loop experimental conditions which avoid negative chemical feedback associated with changes in ventilation.In animal experiments, respiratory dysrhythmias can be induced by stimuli having specific combinations of strength and timing. Newborn animals readily exhibit spontaneous dysrhythmias which become more prominent at lower respiratory drives. In human subjects, swallowing was studied as a physiological perturbation of respiratory rhythm, causing a pattern of phase resetting that is characterized topologically as type 0. Computational studies of the Bonhoeffer-van der Pol (BvP) equations, whose qualitative behavior is representative of many excitable systems, supports a unified interpretation of these experimental findings. Rhythmicity is observed when the BvP model exhibits recurrent periods of excitation alternating with refractory periods. The same system can be perturbed to a state in which amplitude of oscillation is attenuated or abolished. We have characterized critical perturbations which induce transitions between these two states, giving rise to patterns of dysrhythmic activity that are similar to those seen in the experiments. We illustrate the importance of noise in initiation and termination of rhythm, comparable to normal respiratory rhythm intermixed with spontaneous dysrhythmias. In the BvP system the incidence and duration of dysrhythmia is shown to be strongly influenced by the level of noise. These studies should lead to greater understanding of rhythmicity and integrative responses of the respiratory control system, and provide insight into disturbances in control mechanisms that cause apnea and aspiration in clinical disease states. (c) 1995 American Institute of Physics.

Mechanisms of stochastic phase locking

Periodically driven nonlinear oscillators can exhibit a form of phase locking in which a well-defined feature of the motion occurs near a preferred phase of the stimulus, but a random number of stimulus cycles are skipped between its occurrences. This feature may be an action potential, or another crossing by a state variable of some specific value. This behavior can also occur when no apparent external periodic forcing is present. The phase preference is then measured with respect to a time scale internal to the system. Models of these behaviors are briefly reviewed, and new mechanisms are presented that involve the coupling of noise to the equations of motion. Our study investigates such stochastic phase locking near bifurcations commonly present in models of biological oscillators: (1) a supercritical and (2) a subcritical Hopf bifurcation, and, under autonomous conditions, near (3) a saddle-node bifurcation, and (4) chaotic behavior. Our results complement previous studies of aperiodic phase locking in which noise perturbs deterministic phase-locked motion. In our study however, we emphasize how noise can induce a stochastic phase-locked motion that does not have a similar deterministic counterpart. Although our study focuses on models of excitable and bursting neurons, our results are applicable to other oscillators, such as those discussed in the respiratory and cardiac literatures. (c) 1995 American Institute of Physics.

Bistability and the dynamics of periodically forced sensory neurons

Many neurons at the sensory periphery receive periodic input, and their activity exhibits entrainment to this input in the form of a preferred phase for firing. This article describes a modeling study of neurons which skip a random number of cycles of the stimulus between firings over a large range of input intensities. This behavior was investigated using analog and digital simulations of the motion of a particle in a double-well with noise and sinusoidal forcing. Well residence-time distributions were found to exhibit the main features of the interspike interval histograms (ISIH) measured on real sensory neurons. The conditions under which it is useful to view neurons as simple bistable systems subject to noise are examined by identifying the features of the data which are expected to arise for such systems. This approach is complementary to previous studies of such data based, e.g., on non-homogeneous point processes. Apart from looking at models which form the backbone of excitable models, our work allows us to speculate on the role that stochastic resonance, which can arise in this context, may play in the transmission of sensory information.

Use of the frequency-tracking locus in estimating the degree of respiratory entrainment in preterm infants

In order to define the complex interactions between external stimuli and non-linear physiological systems, a technique (the frequency-tracking locus, FTL) was devised that describes the cycle-by-cycle changes in phase angle and amplitude between two signals. Qualitative assessment of the nature of interactions between the signals can be made by examining the FTL. Quantitation of the extent of entrainment of the spontaneous physiological rhythm is possible after deriving a numerical index (the path-length index, PLI) describing the departure of the system from a fully entrained state. The FTL was applied to the study of interactions between spontaneous respiratory effort and mechanical inflation in preterm newborn babies undergoing mechanical ventilation. Stable and unstable states of 1:1 interaction were noted while integer-ratio relationships were seen at low rates of mechanical ventilation. Stable states of entrainment corresponded to a PLI value near unity, and the value of PLI increased rapidly as interactions became unstable. The FTL may be used to describe complex interactions in physiological systems, and may be used as a guide to baby-ventilator matching during mechanical ventilation of the newborn.

Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance

In linear information theory, electrical engineering and neurobiology, random noise has traditionally been viewed as a detriment to information transmission. Stochastic resonance (SR) is a nonlinear, statistical dynamics whereby information flow in a multi-state system is enhanced by the presence of optimized, random noise. A major consequence of SR for signal reception is that it makes possible substantial improvements in the detection of weak periodic signals. Although SR has recently been demonstrated in several artificial physical systems, it may also occur naturally, and an intriguing possibility is that biological systems have evolved the capability to exploit SR by optimizing endogenous sources of noise. Sensory systems are an obvious place to look for SR, as they excel at detecting weak signals in a noisy environment. Here we demonstrate SR using external noise applied to crayfish mechanoreceptor cells. Our results show that individual neurons can provide a physiological substrate for SR in sensory systems.

Nonlinear dynamics of rate-dependent activation in models of single cardiac cells

Recent studies in isolated cardiac tissue preparations have demonstrated the applicability of a one-dimensional difference equation model describing the global behavior of a driven nonpacemaker cell to the understanding of rate-dependent cardiac excitation. As a first approximation to providing an ionic basis to complex excitation patterns in cardiac cells, we have compared the predictions of the one-dimensional model with those of numerical simulations using a modified high-dimensional ionic model of the space-clamped myocyte. Stimulus-response ratios were recorded at various stimulus magnitudes, durations, and frequencies. Iteration of the difference equation model reproduced all important features of the ionic model results, including a wide spectrum of stimulus-response locking patterns, period doubling, and irregular (chaotic) dynamics. In addition, in the parameter plane, both models predict that the bifurcation structure of the cardiac cell must change as a function of stimulus duration, because stimulus duration modifies the type of supernormal excitability present at short diastolic interval. We conclude that, to a large extent, the bifurcation structure of the ionic model under repetitive stimulation can be understood by two functions: excitability and action potential duration. The characteristics of these functions depend on the stimulus duration.

Entrainment in pacemakers characterized by a V-shaped PRC

The behaviour of a class of pacemakers characterized by a V-shaped PRC has been determined, for all possible frequencies and amplitudes of stimulation. The analytical study of the phase transition equation reveals that all rhythmic stimuli, but for a set of measure zero, give rise to entrainment. The ratio between firing and stimulation frequencies is a generalized Cantor function of the ratio between spontaneous and stimulation frequencies. A procedure to compute the detailed input/output pattern that underlies each entrainment ratio is given. Finally, the neurophysiological assumptions and implications of the results obtained are discussed.

Chaos in biological systems

Chaos is a widespread and easily recognizable phenomenon that hardly anybody took notice of until recently. The reason may be that chaos has something profoundly counterintuitive about it. It will not fit easily into any familiar cause–effect frame. The best introduction to chaos is by the way of an example. Consider a leaking faucet (Shaw, 1984). When the weight of the accumulating drop exceeds the surface tension the drop falls and a new drop begins to form. If the leak is small and the pressure in the faucet is constant, the time taken for the drop to reach the critical weight is constant. The dripping is perfectly periodic, the period depending on the leak rate. If the leak is slightly increased, the period of dripping will decrease slightly and vice versa. However, somewhere beyond this point the leaking faucet becomes a nuisance. When the leak is increased beyond a certain point the dripping looses its regularity. The time interval between the drops will first alternate periodically between a short and a long time interval. After a further increase of the leak this double periodic pattern will become unstable and change into a new pattern where four different time intervals between the drops alternate periodically. As the leak is further increased the period will double again and again and finally the dripping becomes completely irregular without any repeating pattern. When this occurs we are observing chaos. At the same time we are posed with the problem of understanding how such a ridiculously simple system can show random behaviour.

Period multupling-evidence for nonlinear behaviour of the canine heart

Although there has recently been considerable interest in applying the theory of nonlinear dynamics to the analysis of complex systems, as yet applications of the theory to biological systems in vivo have been very limited. We report here evidence of nonlinear behaviour in the electrocardiogram and arterial blood pressure traces of the noradrenaline-treated dog. Noradrenaline produces variations in these traces that repeat themselves with regular periods of integral numbers of heart-beats (period multupling), an effect that resembles the 'period-doubling' and other 'bifurcative' behaviour observed when the driving frequency of a nonlinear oscillator is increased above a critical value. The simplest type of periodic variation that we report is the so-called 'electrical alternans', which has long been known as one response of cardiac electrical activity to certain stresses and disease states.

Difference equation model of ventricular parasystole as an interaction between cardiac pacemakers based on the phase response curve

A model of extended ventricular parasystole proposed by Moe et al. (1977) was formulated as a system of nonlinear difference equations by using the phase response curve of myocardial pacemakers. A number of ECG patterns of ventricular arrhythmia such as bigeminy, trigeminy etc. were explained from the property of periodic solutions of the equation. Characteristic properties of special kinds of arrhythmia called "concealed bigeminy" and "concealed trigeminy" were derived mathematically by assuming the model, in relation to the equation of the analog neuron model. The present study was considered to be of clinical significance as a theoretical foundation for the study of genesis of cardiac arrhythmias.

Entrainment of a bursting neuron--II. Atypical cases

Entrainment to external rhythms was analyzed in atypical RPA-1 neurons of Helix pomatia L. stimulated orthodromically. Neurons with long-lasting bursts were tested only with supraharmonic frequencies. All cycle-stimulus relations described for typical oscillators did occur, with particularities due to the frequency of stimulation and the strong fluctuations of cycle parameters. Activity in pairs or triplets was refractory to entrainment. Instead, rhythmical pulses may transform the bimodal activity into a monomodal one. Continuously active neurons and silent ones were also tested. Fragmentation into bursts and long-period bimodal activity could be evoked respectively, suggesting the presence of a latent bimodal pacemaker behind these activities.

Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: a theory for the entrainment of biological oscillators and the generation of cardiac dysrhythmias

A mathematical model for the perturbation of a biological oscillator by single and periodic impulses is analyzed. In response to a single stimulus the phase of the oscillator is changed. If the new phase following a stimulus is plotted against the old phase the resulting curve is called the phase transition curve or PTC (Pavlidis, 1973). There are two qualitatively different types of phase resetting. Using the terminology of Winfree (1977, 1980), large perturbations give a type 0 PTC (average slope of the PTC equals zero), whereas small perturbations give a type 1 PTC. The effects of periodic inputs can be analyzed by using the PTC to construct the Poincaré or phase advance map. Over a limited range of stimulation frequency and amplitude, the Poincaré map can be reduced to an interval map possessing a single maximum. Over this range there are period doubling bifurcations as well as chaotic dynamics. Numerical and analytical studies of the Poincaré map show that both phase locked and non-phase locked dynamics occur. We propose that cardiac dysrhythmias may arise from desynchronization of two or more spontaneously oscillating regions of the heart. This hypothesis serves to account for the various forms of atrioventricular (AV) block clinically observed. In particular 2:2 and 4:2 AV block can arise by period doubling bifurcations, and intermittent or variable AV block may be due to the complex irregular behavior associated with chaotic dynamics.

Phase locking, period-doubling bifurcations, and irregular dynamics in periodically stimulated cardiac cells

The spontaneous rhythmic activity of aggregates of embryonic chick heart cells was perturbed by the injection of single current pulses and periodic trains of current pulses. The regular and irregular dynamics produced by periodic stimulation were predicted theoretically from a mathematical analysis of the response to single pulses. Period-doubling bifurcations, in which the period of a regular oscillation doubles, were predicted theoretically and observed experimentally.