Phase control of neural pacemakers

Dynamic pattern generation in behavioral and neural systems

In the search for principles of pattern generation in complex biological systems, an operational approach is presented that embraces both theory and experiment. The central mathematical concepts of self-organization in nonequilibrium systems (including order parameter dynamics, stability, fluctuations, and time scales) are used to show how a large number of empirically observed features of temporal patterns can be mapped onto simple low-dimensional (stochastic, nonlinear) dynamical laws that are derivable from lower levels of description. The theoretical framework provides a language and a strategy, accompanied by new observables, that may afford an understanding of dynamic patterns at several scales of analysis (including behavioral patterns, neural networks, and individual neurons) and the linkage among them.

Global bifurcations of a periodically forced nonlinear oscillator

The effects of periodic pulsatile stimulation on a simple mathematical model of biological oscillations, called the radial isochron clock (RIC), are investigated as a function of stimulus frequency and amplitude. This system can be reduced to a two parameter, one-dimensional circle map. Numerical and topological methods are used to give a very detailed picture of the observed bifurcations over the complete range of parameters. The bifurcations are generic for a class of models which generalize the RIC.

Effects of current pulses on the sustained discharge of visual cells of Limulus

Current pulses were used in the eccentric and retinular cells of the Limulus lateral eye to produce changes in the interspike interval of the discharge sustained by a constant light level. The effects on the interspike interval of hyperpolarizing and depolarizing perturbations, applied at various delays from the previous spike, were measured for different intensities and durations of the current pulse. The results show that when the perturbations were applied in the first part of the interval, effects contrary to what is normal were produced (i.e, hyperpolarizing pulses decreased the interspike interval instead of increasing it and vice versa for depolarizing pulses). Here we discuss briefly the implications on neural encoding models.

Phase resetting properties of cardiac pacemaker cells

Aggregates of heart cells from chicken embryos beat spontaneously. We used intracellular microelectrodes to record the periodic behavior of the membrane potential that triggers the contractions. Every 5-12 beats, a short current pulse was applied at various points in the cycle to study the phase-dependent resetting of the rhythm. Pulses stronger than 2.5 nA caused the final rhythm to be reset to almost the same point in the cycle regardless of the phase at which the pulse was applied (type zero resetting). Pulses of less than or equal to 1 nA only caused a slight change of the phase. Increasing current intensities to between 1 and 2.5 nA gave rise to an increasing steepness in a small part of the phase-response curve. The observation of type zero resetting implies the existence of a critical stimulation that might annihilate the rhythm. Although we did find a phase at which more or less random responses occurred, the longest pause in the rhythm was 758 ms, 2.4 times the spontaneous interval. This suggests that the resting membrane potential was unstable, at least against the internal noise of the system. The conclusions are discussed in terms of the concepts of classical cardiac electrophysiology.

Impulse responses of automaticity in the Purkinje fiber

We examined the effects of brief current pulses on the pacemaker oscillations of the Purkinje fiber using the model of McAllister , Noble, and Tsien (1975. J. Physiol. [Lond.]. 251:1-57). This model was used to construct phase-response curves for brief electric stimuli to find "black holes," where rhythmic activity of the Purkinje fiber ceases. In our computer simulation, a brief current stimulus of the right magnitude and timing annihilated oscillations in membrane potential. The model also revealed a sequence of alternating periodic and chaotic regimes as the strength of a steady bias current is varied. We compared the results of our computer simulations with experimental work on Purkinje fibers and pointed out the importance of modeling results of this kind for understanding cardiac arrhythmias.

Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory

Injection of a current pulse of brief duration into an aggregate of spontaneously beating chick embryonic heart cells resets the phase of the activity by either advancing or delaying the time of occurrence of the spontaneous beat subsequent to current injection. This effect depends upon the polarity, amplitude, and duration of the current pulse, as well as on the time of injection of the pulse. The transition from prolongation to shortening of the interbeat interval appears experimentally to be discontinuous for some stimulus conditions. These observations are analyzed by numerical investigation of a model of the ionic currents that underlie spontaneous activity in these preparations. The model consists of: Ix, which underlies the repolarization phase of the action potential, IK2, a time-dependent potassium ion pacemaker current, Ibg, a background or time-independent current, and INa, an inward sodium ion current that underlies the upstroke of the action potential. The steady state amplitude of the sum of these currents is an N-shaped function of potential. Slight shifts in the position of this current-voltage relation along the current axis can produce either one, two, or three intersections with the voltage axis. The number of these equilibrium points and the voltage dependence of INa contribute to apparent discontinuities of phase resetting. A current-voltage relation with three equilibrium points has a saddle point in the pacemaker voltage range. Certain combinations of current-pulse parameters and timing of injection can shift the state point near this saddle point and lead to an interbeat interval that is unbounded . Activation of INa is steeply voltage dependent. This results in apparently discontinuous phase resetting behavior for sufficiently large pulse amplitudes regardless of the number of equilibrium points. However, phase resetting is fundamentally a continuous function of the time of pulse injection for these conditions. These results demonstrate the ionic basis of phase resetting and provide a framework for topological analysis of this phenomenon in chick embryonic heart cell aggregates.

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.

Melanin: the organizing molecule

The hypothesis is advanced that (neuro)melanin (in conjunction with other pigment molecules such as the isopentenoids) functions as the major organizational molecule in living systems. Melanin is depicted as an organizational "trigger" capable of using established properties such as photon-(electron)-phonon conversions, free radical-redox mechanisms, ion exchange mechanisms, and semiconductive switching capabilities to direct energy to strategic molecular systems and sensitive hierarchies of protein enzyme cascades. Melanin is held capable of regulating a wide range of molecular interactions and metabolic processes primarily through its effective control of diverse covalent modifications. To support the hypothesis, established and proposed properties of melanin are reviewed (including the possibility that (neuro)melanin is capable of self-synthesis). Two "melanocentric systems"--key molecular systems in which melanin plays a central if not controlling role--are examined: 1) the melanin-purine-pteridine (covalent modification) system and 2) the APUD (or diffuse neuroendocrine) system. Melanin's role in embryological organization and tissue repair/regeneration via sustained or direct current is considered in addition to its possible control of the major homeostatic regulatory systems--autonomic, neuroendocrine, and immunological.

Entrainment of a bursting neuron--I. typical activity

Highly regular RPA-1 oscillator neurons of the land snail Helix pomatia L. are tested for entrainment by rhythmical stimulation. Both with orthodromic and direct hyperpolarizing pulses the bursting activity could be entrained to frequencies higher or lower than, and equal to, the spontaneous one, in this order of difficulty. Depolarizing pulses give mainly entrainment to higher frequencies, but synchronization to frequencies lower than the spontaneous one was also demonstrable. Each of these different driving relations implies the consecutive stimuli to become locked to a peculiar range of phases and not to a single phase of resonance inside the cycle.

The case for modulated parasystole

Several recently published tracings of parasystolic rhythms with unusual features were studied to determine whether the biphasic response curve characteristic of modulated parasystole could be extracted from the patterns of arrhythmia. Even in cases in which the possibility of modulation had been rejected, the phase response curves could be derived by inverse analysis of the ectopic intervals. When the derived curves were inserted into the program of the computer model, almost exact matches for the clinical patterns were recorded. In one case, in which the right atrium was driven at increasing rates, the patterns of manifest ectopic ventricular responses appeared to be re-entrant rather than parasystolic. In this case, similar patterns as a function of heart rate were recorded from a mathematical model of reflection.

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.

Suppression of pacemaker activity by rapid repetitive phase delay

Spontaneous activity of pacemaker cells of structures may be suppressed by rapid repetitive stimulation. Conditions are that the oscillator's phase reset curve, characterizing the phase resetting effect of single stimuli, has a phase delay part and that the interval between the stimuli falls within a range of values, determined by the form of the phase reset curve. Under these conditions, which appeared the same as those for stable underdrive pacing, the pacemaker becomes stably entrained to the stimuli without firing, i.e. it is kept within a certain part of its limit cycle because the pulses repeatedly delay the next coming action potential. This rapid stimulation suppression of pacemaker activity is demonstrated experimentally on a simple electronic pacemaker cell model for two types of phase reset curves, a biphasic one for depolarizing and a monophasic one for hyperpolarizing pulses. Computer simulations of coupled pacemaker cells, interacting by phase reset curves, illustrate how this type of pacemaker suppression may protect a population of pacemaker cells like the sinus node in the heart against arrhythmias.

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.

Phase plane description of endogenous neuronal oscillators in Aplysia

Phase plane techniques are used to describe graphically the limit cycle behavior of identified endogenous neuronal oscillators in the isolated abdominal ganglion of Aplysia. Intracellularly recorded membrane potential from a bursting neuron and its first derivative with respect to time are used as coordinates (state variables) in phase space. The derivative is either measured electronically or calculated digitally. Each trajectory in phase space represents the entire output of the bursting neuron, i.e., both the rapid action potentials and slow pacemaker potentials. Phase plane portraits are presented for the free run limit cycle before and after a change in a system parameter (applied transmembrane current) and also for phase resetting produced by direct synaptic inhibition from an identified interneuron. The complex topology of the trajectory suggests that the bursting oscillator is a higher order system. Therefore, the second time derivative is used as another state variable. This type of phase plot can help to relate biophysical and mathematical analyses.

Analysis of entrainment of circadian oscillators by skeleton photoperiods using phase transition curves

A skeleton photoperiod consists of two short pulses which are applied on the circadian oscillator at times corresponding to the beginning and to the end of a continuous light stimulus. To study several problems in entrainment of circadian rhythms by skeleton photoperiods, we develop a simple diagrammatic solution of the steady state entrainment making use of phase transition curves which are directly gotten from phase response curves. The graphical method is simple and systematic to study entrainment by light cycles with various day lengths. As the method is also intuitive, we can easily examine three problems. (1) In Drosophila the phase relation (psi) between rhythm and light cycle is a continuous function of day length of skeleton photoperiods up to about 12 h, but a marked discontinuity (psi-jump) sets in between 13 and 14 h. By the diagrammatic method we find that psi-jump is mathematically a bifurcation phenomenon. (2) The action of photoperiods up to about 12 h is fully simulated by two 15-min skeleton pulses. Do 3-min skeleton pulses imitate the complete photoperiods? We find that pulse width is arbitrary to some extent. (3) Why skeleton photoperiods up to about 12 h are good models of complete photoperiods? The reason is the small amplitude and the nearly symmetrical form of phase response curves in the subjective day.

Control of repetitive firing in squid axon membrane as a model for a neuroneoscillator

1. Repetitive firing in space-clamped squid axons bathed in low Ca and stimulated by a just suprathreshold step of current can be annihilated by a brief depolarizing or hyperpolarizing pulse of the proper magnitude applied at the proper phase. 2. In response to such perturbations, membrane potential and ionic currents show damped oscillations toward a steady state. 3. For other, non-annihilating, perturbations repetitive firing resumes with unaltered frequency but with phase resetting. 4. Experimental findings are compared with calculations for the space-and current-clamped Hodgkin-Huxley equations. Annihilation of repetitive firing to a steady state corresponds to a solution trajectory perturbed off a stable limit cycle and into the domain of attraction of a coexistent stable singular point. 5. Experimentally and theoretically the nerve exhibits hysteresis with two different stable modes of operation for a just suprathreshold range of bias current: the oscillatory repetitive firing state and the time-independent steady state. 6. Analogy is made to a brief synaptic input (excitatory or inhibitory) which may start or stop a biological pace-maker.

Two coupled oscillators as a model for the coordinated finger tapping by both hands

Recently, it was found that rhythmic movements (e.g. locomotion, swimmeret beating) are controlled by mutually coupled endogeneous neural oscillators (Kennedy and Davis, 1977; Pearson and Iles, 1973; Stein, 1974; Shik and Orlovsky, 1976; Grillner and Zangger, 1979). Meanwhile, it has been found out that the phase resetting experiment is useful to investigate the interaction of neural oscillators (Perkel et al., 1963; Stein, 1974). In the preceding paper (Yamanishi et al., 1979), we studied the functional interaction between the neural oscillatory which is assumed to control finger tapping and the neural networks which control some tasks. The tasks were imposed on the subject as the perturbation of the phase resetting experiment. In this paper, we investigate the control mechanism of the coordinated finger tapping by both hands. First, the subjects were instructed to coordinate the finger tapping by both hands so as to keep the phase difference between two hands constant. The performance was evaluated by a systematic error and a standard deviation of phase differences. Second, we propose two coupled neural oscillators as a model for the coordinated finger tapping. Dynamical behavior of the model system is analyzed by using phase transition curves which were measured on one hand finger tapping in the preivous experiment (Yamanishi et al., 1979). Prediction by the model is in good agreement with the results of the experiments. Therefore, it is suggested that the neural mechanism which controls the coordinated finger tapping may be composed of a coupled system of two neural oscillators each of which controls the right and the left finger tapping respectively.

Phase resetting and annihilation of pacemaker activity in cardiac tissue

Spontaneous rhythmic activity in isolated cardiac pacemaker cells can be terminated by a brief, subthreshold, depolarizing or hyperpolarizing perturbation of the proper magnitude applied at a specific point in the pacemaker cycle. Evidence is provided in support of a topological theory of the existence of a "singular" point in cardiac oscillators.

A simple model for phase locking of biological oscillators

A mathematical model is presented for phase locking of a biological oscillator to a sinusoidal stimulus. Analytical, numerical and topological considerations are used to discuss the patterns of phase locking as a function of amplitude of the sinusoidal stimulus and the relative frequencies of the osillator and the sinusoidal stimulus. The sorts of experimental data which are needed to make comparisons between theory and experiment are discussed.

Interactions between centrally and peripherally generated neuromuscular oscillations

An experimentally based model of the mammalian neuromuscular system has been extended to include the interaction of sinusoidal inputs generated within the central nervous system and those produced peripherally by reflex pathways, together with muscle properties and external loads. Multiple reflex pathways and pathways having acceleration as well as velocity and length sensitivity and considered. The responses are analyzed for brief inputs (Dirac delta-functions), sinusoidal driving functions and mixtures of the two over ranges in which the model behaves either linearly or non-linearly. Approximate solutions are derived for the non-linear range, and exact numerical solutions are computed for a few examples within the linear range. The extent to which brief inputs can reset ongoing oscillations and the extent to which sinusoidal inputs can entrain these oscillations are of particular interest.