Phase response characteristics of sinoatrial node cells

The virtual sinoatrial node: What did computational models tell us about cardiac pacemaking?

Since its discovery, the sinoatrial node (SAN) has represented a fascinating and complex matter of research. Despite over a century of discoveries, a full comprehension of pacemaking has still to be achieved. Experiments often produced conflicting evidence that was used either in support or against alternative theories, originating intense debates. In this context, mathematical descriptions of the phenomena underlying the heartbeat have grown in importance in the last decades since they helped in gaining insights where experimental evaluation could not reach. This review presents the most updated SAN computational models and discusses their contribution to our understanding of cardiac pacemaking. Electrophysiological, structural and pathological aspects - as well as the autonomic control over the SAN - are taken into consideration to reach a holistic view of SAN activity.

What keeps us ticking? Sinoatrial node mechano-sensitivity: the grandfather clock of cardiac rhythm

The rhythmic and spontaneously generated electrical excitation that triggers the heartbeat originates in the sinoatrial node (SAN). SAN automaticity has been thoroughly investigated, which has uncovered fundamental mechanisms involved in cardiac pacemaking that are generally categorised into two interacting and overlapping systems: the 'membrane' and 'Ca²⁺ clock'. The principal focus of research has been on these two systems of oscillators, which have been studied primarily in single cells and isolated tissue, experimental preparations that do not consider mechanical factors present in the whole heart. SAN mechano-sensitivity has long been known to be a contributor to SAN pacemaking-both as a driver and regulator of automaticity-but its essential nature has been underappreciated. In this review, following a description of the traditional 'clocks' of SAN automaticity, we describe mechanisms of SAN mechano-sensitivity and its vital role for SAN function, making the argument that the 'mechanics oscillator' is, in fact, the 'grandfather clock' of cardiac rhythm.

Use of Phase Transition Curves for Testing Models of the pH-Oscillatory Hydrogen Peroxide-Thiosulfate-Sulfite Reaction

The reaction system hydrogen peroxide-thiosulfate-sulfite in diluted sulfuric acid (HPTS) displays strongly nonlinear dynamics when operated in a continuous-flow stirred tank reactor. Due to a crucial role of hydrogen ion during the reaction, this system is a prime example of an inorganic pH-oscillator. Under specific external conditions the system exhibits multiple steady states, periodic oscillations and chaotic behavior. We focus on evaluating alternative kinetic models by exploring phase resetting of the periodic oscillatory regime caused by a single-pulse perturbation with various reacting species. Phase transition curve (PTC), the plot of phase after the resetting against the phase of perturbation, is a convenient characteristic of the oscillatory dynamics adopted as a major tool in this work. Experimental results for hydrogen ions, hydroxide ions, thiosulfate ions, sulfite ions, and hydrogen sulfite ions used as perturbants are systematically compared with calculations under corresponding conditions using two available reaction mechanisms. In addition, we use the stoichiometric network analysis to identify possible core oscillatory subnetworks in the models and choose the one that corresponds best to the measured PTCs.

Phase-locking behaviors in an ionic model of sinoatrial node cell and tissue

Phase-locking behaviors in sinoatrial node (SAN) are closely related to cardiac arrhythmias. An ionic model considering structural heterogeneity of SAN is numerically investigated. The bifurcations between phase-locking zones are interpreted by the map derived from the phase resetting curve. Furthermore, the validity of the circle map in describing phase locking of the actual SAN system is evaluated and explained. We reveal also how the phase-locking behaviors in heterogeneous tissue depend on the location of stimulating site and the coupling strength of the tissue. All these results may be of suggestive uses for understanding and controlling practical SAN dynamics.

Bifurcation analysis of a periodically forced relaxation oscillator: differential model versus phase-resetting map

We compare the dynamics of the periodically forced FitzHugh-Nagumo oscillator in its relaxation regime to that of a one-dimensional discrete map of the circle derived from the phase-resetting response of this oscillator (the "phase-resetting map"). The forcing is a periodic train of Gaussian-shaped pulses, with the width of the pulses much shorter than the intrinsic period of the oscillator. Using numerical continuation techniques, we compute bifurcation diagrams for the periodic solutions of the full differential equations, with the stimulation period being the bifurcation parameter. The period-1 solutions, which belong either to isolated loops or to an everywhere-unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes, merge together to form a single branch at larger stimulation amplitudes. As a consequence of the fast-slow nature of the oscillator, this merging occurs at virtually the same stimulation amplitude for all the period-1 loops. Again using continuation, we show that this stimulation amplitude corresponds, in the circle map, to a change of topological degree from one to zero. We explain the origin of this coincidence, and also discuss the translational symmetry properties of the bifurcation diagram.

Phase resetting, phase locking, and bistability in the periodically driven saline oscillator: experiment and model

The saline oscillator consists of an inner vessel containing salt water partially immersed in an outer vessel of fresh water, with a small orifice in the center of the bottom of the inner vessel. There is a cyclic alternation between salt water flowing downwards out of the inner vessel into the outer vessel through the orifice and fresh water flowing upwards into the inner vessel from the outer vessel through that same orifice. We develop a very stable (i.e., stationary) version of this saline oscillator. We first investigate the response of the oscillator to periodic forcing with a train of stimuli (period=Tp) of large amplitude. Each stimulus is the quick injection of a fixed volume of fresh water into the outer vessel followed immediately by withdrawal of that very same volume. For Tp sufficiently close to the intrinsic period of the oscillator (T0) , there is 1:1 synchronization or phase locking between the stimulus train and the oscillator. As Tp is decreased below T0 , one finds the succession of phase-locking rhythms: 1:1, 2:2, 2:1, 2:2, and 1:1. As Tp is increased beyond T0 , one encounters successively 1:1, 1:2, 2:4, 2:3, 2:4, and 1:2 phase-locking rhythms. We next investigate the phase-resetting response, in which injection of a single stimulus transiently changes the period of the oscillation. By systematically changing the phase of the cycle at which the stimulus is delivered (the old phase), we construct the new-phase--old-phase curve (the phase transition curve), from which we then develop a one-dimensional finite-difference equation ("map") that predicts the response to periodic stimulation. These predicted phase-locking rhythms are close to the experimental findings. In addition, iteration of the map predicts the existence of bistability between two different 1:1 rhythms, which was then searched for and found experimentally. Bistability between 1:1 and 2:2 rhythms is also encountered. Finally, with one exception, numerical modeling with a phenomenologically derived Rayleigh oscillator reproduces all of the experimental behavior.

Phase Resetting in One-Dimensional Model of the Sinoatrial Node

In this paper, we use a one-dimensional model of the rabbit sinoatrial node (SAN), and we investigate the response of the model to hyperpolarizing and depolarizing stimulus. Depending on the stimulus timing, either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior of the model is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. The main focus of this paper is to compare the dynamic properties of the spatially extended system and the single cell model. The detailed analysis of the results provides new insights in the understanding of the transition from the theoretical single cell models to the spatially extended systems.