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.     

Difference equation model of the entrainment of myocardial pacemaker cells based on the phase response curve

As a method for investigating entrainment phenomena found in biological oscillations, a model is presented which is formulated as a set of difference equations on the basis of phase response curves of biological oscillators. Properties of the model are discussed and synchronization of cardiac pacemaker cells to the external periodic stimuli is analyzed by use of the model. It is shown that the model is reducible to an equivalent system to the mathematical neuron model of Nagumo et al. under a specific parametric condition of the phase response curve. As an application to the cardiac arrhythmias, the model of ventricular parasystoles studied by Moe et al. is described mathematically in terms of a system of difference equations. In addition, a more comprehensive model of ventricular parasystoles is proposed.     

A pacemaker cell pair model based on the phase response curve

A pacemaker cell pair model and the dynamic interaction between the two pacemaker cells is described in this paper. It is an extension of our single pacemaker cell model, in which we studied its response to repetitive external depolarization stimulations. This model is a simple model based on the two most important functional properties of the cardiac pacemaker cells: its intrinsic pacemaker cycle length, which is an `internal' parameter of the cell, and the phase response curve (PRC), which is an `overall collective' function. The PRC contains all the `information' about the possible interactions of the pacemaker cell with the outside world (interaction with surrounding cells, external stimulus, etc.). First, we examined the properties and solutions of 1:1 synchronization between two pacemaker cells. We found that in order to achieve synchronization between two pacemaker cells, there should be limitations on the PRC parameters, which depend on the cells intrinsic cycle lengths. Next, we investigated the 2:1 entrainment state between two interacting pacemaker cells. We found that there is not necessarily a unique solution for this state as there was for the 1:1 state. Finally, we ran our computer model to investigate the properties of more complex patterns of entrainment between two pacemaker cells. As a result of our analytical study, we unveil two new important parameters, which are fully defined as a function of the PRC parameters: (1) the `accelerator factor' which describes the tendency of a pair of interacting pacemaker cells to synchronize at a common cycle length, which is closer to the faster cycle of the pair; (2) the `degree of coupling', which describes the range of the 1:1 synchronization and the `strength' of the interaction between a pair of interacting pacemaker cells. Those two interaction parameters arise as helpful `tools' for the understanding of synchronization and mutual entrainment mechanisms between pacemaker cells. Therefore, this study establishes the PRC as an important determinant and a useful approach for the understanding of the dynamic interaction of pacemaker cells among themselves and with the outside world. Received: 12 May 1997 / Accepted in revised form: 22 April 1998     

Synchronization of two coupled pacemaker cells based on the phase response curve

In this paper, the synchronization of a pair of pacemaker cells as Sino-Atrial (SA) and Atrio-Ventricullar (AV) nodes have been studied and a new approach for synchronization, based on the concept of Phase Response Curve (PRC), has been proposed. The paper starts with presenting the necessary and sufficient conditions for synchronization in terms of the PRC parameters. Such conditions are time dependent and thus, the paper proceeds with deriving some sufficient conditions, which are not time dependent. The time-delay between the firing time of SA node and when it reaches the AV node is also considered. When the conditions for spontaneous synchronization are not valid, the synchronization is achieved by applying pulses to the AV or the SA nodes or to both of the nodes, depending on the accessibility. The subject has been investigated and sufficient conditions were achieved for all three cases. In each case, the dynamical equations of coupled pacemakers have been determined and the stability analyses of delay dynamical equations between discharges of two pacemakers were performed. The number of excitation pulses and the time intervals for applying them to accessible pacemaker(s) were obtained and eventually some numerical examples were simulated to approve the accuracy of the theoretical results and conditions.     

Characterizing dynamic interactions between ultradian glucocorticoid rhythmicity and acute stress using the phase response curve

The hypothalamic-pituitary-adrenal (HPA) axis is a dynamic oscillatory hormone signalling system that regulates the pulsatile secretion of glucocorticoids from the adrenal glands. In addition to regulation of basal levels of glucocorticoids, the HPA axis provides a rapid hormonal response to stress that is vitally important for homeostasis. Recently it has become clear that glucocorticoid pulses encode an important biological signal that regulates receptor signalling both in the central nervous system and in peripheral tissues. It is therefore important to understand how stressful stimuli disrupt the pulsatile dynamics of this system. Using a computational model that incorporates the crucial feed-forward and feedback components of the axis, we provide novel insight into experimental observations that the size of the stress-induced hormonal response is critically dependent on the timing of the stress. Further, we employ the theory of Phase Response Curves to show that an acute stressor acts as a phase-resetting mechanism for the ultradian rhythm of glucocorticoid secretion. Using our model, we demonstrate that the magnitude of an acute stress is a critical factor in determining whether the system resets via a Type 1 or Type 0 mechanism. By fitting our model to our in vivo stress-response data, we show that the glucocorticoid response to an acute noise stress in rats is governed by a Type 0 phase-resetting curve. Our results provide additional evidence for the concept of a deterministic sub-hypothalamic oscillator regulating the ultradian glucocorticoid rhythm, which constitutes a highly responsive peripheral hormone system that interacts dynamically with hypothalamic inputs to regulate the overall hormonal response to stress.     

Interpreting the human phase response curve to multiple bright-light exposures

Czeisler and his colleagues have recently reported that bright light can induce strong (Type O) resetting of the human circadian pacemaker. This surprising result shows that the human clock is more responsive to light than has been previously thought. The interpretation of their results is subtle, however, because of an unconventional aspect of their experimental protocol: They measured the phase shift after three cycles of the bright-light stimulus, rather than after the usual single pulse. A natural question is whether the apparent Type O response could reflect the summation of three weaker Type 1 responses to each of the daily light pulses. In this paper I show mathematically that repeated Type 1 resetting cannot account for the observed Type O response. This finding corroborates the strong resetting reported by Czeisler et al., and supports their claim that bright light induces strong resetting by crushing the amplitude of the circadian pacemaker. Furthermore, the results indicate that back-to-back light pulses can have a cooperative effect different from that obtained by simple iteration of a phase response curve (PRC). In this sense the resetting response of humans is similar to that of Drosophila, Kalanchoe, and Culex, and is more complex than that predicted by conventional PRC theory. To describe the way in which light resets the human circadian pacemaker, one needs a theory that includes amplitude resetting, as pioneered by Winfree and developed for humans by Kronauer.     

On the impact of the distance between two genes on their interaction curve

We analyze a basic building block of gene regulatory networks using a stochastic/geometric model in search of a mathematical backing for the discrete modeling frameworks. We consider a network consisting only of two interacting genes: a source gene and a target gene. The target gene is activated by the proteins encoded by the source gene. The interaction is therefore mediated by activator proteins that travel, like a signal, from the source to the target. We calculate the production curve of the target proteins in response to a constant-rate production of activator proteins. The latter has a sigmoidal shape (like a simple delay line) that is sharper and taller when the two genes are closer to each other. This provides further support for the use of discrete models in the analysis gene regulatory networks. Moreover, it suggests an evolutionary pressure towards making the interacting genes closer to each other to make their interactions more efficient and more reliable.     

Modulation of Hydra attenuata rhythmic activity: phase response curve.

We investigated the effect of photic stimulation on the frequency of Hydra attenuata column contractions. We used positive or negative abrupt light transitions, single or repetitive light or darkness pulses, and alternation of light and darkness periods. The main results are: (a) The frequency of the contraction pulse trains (CPTs) varies transiently in response to an abrupt variation of the light intensity. (b) CPTs in progress can be inhibited by different types of photic stimuli. (c) The response time to a single photic stimulus varies during the inter-CPT interval and depends also on the polarity of the stimulus. (d) The CPTs are entrainable with repetitive light stimulation of various frequencies. (e) Long-lasting variations of the frequency of CPTs occur after the end of a repetitive light stimulation. We suggest that the mechanism responsible for the rhythym of column contractions is quite similar to that on which other biological rhythmic phenomena are based.     

A honey bee (Apis mellifera) light phase response curve

The authors report a phase response curve (PRC) for individual honey bees (Apis mellifera) to single 1-h light pulses (1000 lux) using an Aschoff Type 1 protocol (n = 134). The bee PRC is a weak (Type 1) PRC with a maximum advance of 1.5 h between circadian time (CT) 18 and 3 and a maximum delay of 1.5 h between CT 12 and 18. This is the first published honey bee light PRC and provides an important resource for chronobiologists and honey bee researchers. It may also have practical applications for what is an economically important species frequently transported across different time zones.     

Reciprocal interaction between IK1 and If in biological pacemakers: A simulation study

Pacemaking dysfunction (PD) may result in heart rhythm disorders, syncope or even death. Current treatment of PD using implanted electronic pacemakers has some limitations, such as finite battery life and the risk of repeated surgery. As such, the biological pacemaker has been proposed as a potential alternative to the electronic pacemaker for PD treatment. Experimentally and computationally, it has been shown that bio-engineered pacemaker cells can be generated from non-rhythmic ventricular myocytes (VMs) by knocking out genes related to the inward rectifier potassium channel current (I_K1) or by overexpressing hyperpolarization-activated cyclic nucleotide gated channel genes responsible for the “funny” current (I_f). However, it is unclear if a bio-engineered pacemaker based on the modification of I_K1- and I_f-related channels simultaneously would enhance the ability and stability of bio-engineered pacemaking action potentials. In this study, the possible mechanism(s) responsible for VMs to generate spontaneous pacemaking activity by regulating I_K1 and I_f density were investigated by a computational approach. Our results showed that there was a reciprocal interaction between I_K1 and I_f in ventricular pacemaker model. The effect of I_K1 depression on generating ventricular pacemaker was mono-phasic while that of I_f augmentation was bi-phasic. A moderate increase of I_f promoted pacemaking activity but excessive increase of I_f resulted in a slowdown in the pacemaking rate and even an unstable pacemaking state. The dedicated interplay between I_K1 and I_f in generating stable pacemaking and dysrhythmias was evaluated. Finally, a theoretical analysis in the I_K1/I_f parameter space for generating pacemaking action potentials in different states was provided. In conclusion, to the best of our knowledge, this study provides a wide theoretical insight into understandings for generating stable and robust pacemaker cells from non-pacemaking VMs by the interplay of I_K1 and I_f, which may be helpful in designing engineered biological pacemakers for application purposes.     

Changes in the phase response curve of the circadian clock to a phase-shifting stimulus.

Experiments were conducted in hamsters to determine whether the phase response curve (PRC) to injections of the short-acting benzodiazepine triazolam is a fixed or a labile property of the circadian clock. The results indicated that (1) both the shape and the amplitude of the PRC to triazolam generated on the first day of transfer from a light-dark cycle (LD 14:10) to constant darkness (DD) (i.e., PRCLD) were different from those of the PRC generated after many days in DD (PRCDD); and (2) the phase-shifting effects of triazolam on the activity rhythms of hamsters transferred from LD 14:10 or 12:12 to DD changed dramatically within the first 8-9 days spent in DD. In an attempt to accelerate the resynchronization of the circadian clock of hamsters subjected to an 8-hr advance in the LD cycle, triazolam was given to the animals at a time selected on the basis of the characteristics of PRCLD. The activity rhythms of five of eight triazolam-treated animals were resynchronized to the new LD cycle within 2-4 days after the shift, whereas those of most of the control animals were resynchronized 21-29 days after the shift. These findings suggest that attempts to use pharmacological or nonpharmacological tools to phase-shift circadian clocks under entrained conditions should take into account information derived from PRCs generated at the time of transition from entrained to free-running conditions.     

Photic phase response curve in Octodon degus: assessment as a function of activity phase preference

Light exposure during the early and late subjective night generally phase delays and advances circadian rhythms, respectively. However, this generality was recently questioned in a photic entrainment study in Octodon degus. Because degus can invert their activity phase preference from diurnal to nocturnal as a function of activity level, assessment of phase preference is critical for computations of phase reference [circadian time (CT) 0] toward the development of a photic phase response curve. After determining activity phase preference in a 24-h light-dark cycle (LD 12:12), degus were released in constant darkness. In this study, diurnal (n = 5) and nocturnal (n = 7) degus were randomly subjected to 1-h light pulses (30-35 lx) at many circadian phases (CT 1-6: n = 7; CT 7-12: n = 8; CT 13-18: n = 8; and CT 19-24: n = 7). The circadian phase of body temperature (Tb) onset was defined as CT 12 in nocturnal animals. In diurnal animals, CT 0 was determined as Tb onset + 1 h. Light phase delayed and advanced circadian rhythms when delivered during the early (CT 13-16) and late (CT 20-23) subjective night, respectively. No significant phase shifts were observed during the middle of the subjective day (CT 3-10). Thus, regardless of activity phase preference, photic entrainment of the circadian pacemaker in Octodon degus is similar to most other diurnal and nocturnal species, suggesting that entrainment mechanisms do not determine overt diurnal and nocturnal behavior.     

Apparent discontinuities in the phase-resetting response of cardiac pacemakers

Injection of a brief stimulus pulse resets the spontaneous periodic activity of a sinoatrial node cell: a stimulus delivered early in the cycle generally delays the time of occurrence of the next action potential, while the same stimulus delivered later causes an advance. We investigate resetting in two models, one with a slow upstroke velocity and the other with a fast upstroke velocity, representing central and peripheral nodal cells, respectively. We first formulate each of these models as a classic Hodgkin-Huxley type of model and then as a model representing a population of single channels. In the Hodgkin-Huxley-type model of the slow-upstroke cell the transition from delay to advance is steep but continuous. In the corresponding single-channel model, due to the channel noise then present, repeated resetting runs at a fixed stimulus timing within the transitional range of coupling intervals lead to responses that span a range of advances and delays. In contrast, in the fast-upstroke model the transition from advance to delay is very abrupt in both classes of model, as it is in experiments on some cardiac preparations ("all-or-none" depolarization). We reduce the fast-upstroke model from the original seven-dimensional system to a three-dimensional system. The abrupt transition occurs in this reduced model when a stimulus transports the state point to one side or the other of the stable manifold of the trajectory corresponding to the eigendirection associated with the smaller of two positive eigenvalues. This stable manifold is close to the slow manifold, and so canard trajectories are seen. Our results demonstrate that the resetting response is fundamentally continuous, but extremely delicate, and thus suggest one way in which one can account for experimental discontinuities in the resetting response of a nonlinear oscillator.     

Hormone response to bidirectional selection on social behavior

Behavior is a quantitative trait determined by multiple genes. Some of these genes may have effects from early development and onward by influencing hormonal systems that are active during different life-stages leading to complex associations, or suites, of traits. Honey bees (Apis mellifera) have been used extensively in experiments on the genetic and hormonal control of complex social behavior, but the relationships between their early developmental processes and adult behavioral variation are not well understood. Bidirectional selective breeding on social food-storage behavior produced two honey bee strains, each with several sublines, that differ in an associated suite of anatomical, physiological, and behavioral traits found in unselected wild type bees. Using these genotypes, we document strain-specific changes during larval, pupal, and early adult life-stages for the central insect hormones juvenile hormone (JH) and ecdysteroids. Strain differences correlate with variation in female reproductive anatomy (ovary size), which can be influenced by JH during development, and with secretion rates of ecdysteroid from the ovaries of adults. Ovary size was previously assigned to the suite of traits of honey bee food-storage behavior. Our findings support that bidirectional selection on honey bee social behavior acted on pleiotropic gene networks. These networks may bias a bee's adult phenotype by endocrine effects on early developmental processes that regulate variation in reproductive traits.     

Biological oscillators can be stopped—Topological study of a phase response curve

Many biological oscillators are stable against noise and perturbation (e.g. circadian rhythms, biochemical oscillators, pacemaker neurons, bursting neurons and neural networks with periodic outputs). The experiment of phase shifts resulting from discrete perturbation of stable biological rhythms was developed by Perkel and coworkers (Perkel et al., 1964). By these methods, they could get important insights into the entrainment behaviors of biological rhythms. Phase response curves, which are measured in these experiments, can be classified into two types. The one is the curve with one mapping degree (Type 1), and the other is that with zero mapping degree (Type 0) (Winfree, 1970). We define the phase response curve mathematically, and explain the difference between these two types by the homotopy theory. Moreover, we prove that, if a Type 0 curve is obtained at a certain magnitude of perturbation, there exists at least one lower magnitude for which the phase response curve cannot be measured. Some applications of these theoretical results are presented.