Phase response characteristics of sinoatrial node cells

Abstract

In this work, the dynamic response of the sinoatrial node (SAN), the natural pacemaker of the heart, to short external stimuli is investigated using the Zhang et al. model. The model equations are solved twice for the central cell and for the peripheral cell. A short current pulse is applied to reset the spontaneous rhythmic activity of the single sinoatrial node cell. Depending on the stimulus timing either a delay or an advance in the occurrence of next action potential is produced. This resetting behavior is quantified in terms of phase transition curves (PTCs) for short electrical current pulses of varying amplitude which span the whole period. For low stimulus amplitudes the transition from advance to delay is smooth, while at higher amplitudes abrupt changes and discontinuities are observed in PTCs. Such discontinuities reveal critical stimuli, the application of which can result in annihilation of activity in central SAN cells. The detailed analysis of the ionic mechanisms involved in its resetting behavior of sinoatrial node cell models provides new insight into the dynamics and physiology of excitation of the sinoatrial node of the heart.

Introduction

Auto-rhythmic excitation in the cardiac muscle is driven by the sinoatrial node (SAN), the natural pacemaker of the heart. Under physiological or pathological conditions during clinical interventions, such us defibrillation of cardiac muscle or accidental situations like electrocution, the pacemaker of the heart is subjected to external stimulation [1].

Experimental [2], [3] and numerical [4], [5], [6] studies indicate that the application of short current pulses to sinoatrial pacemaker cells result in changes in the cell's cycle length which depend on both the phase and the amplitude of the stimulus [3], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. In most cases, the applied perturbation does not affect the amplitude or the configuration of the pacemaker cell action potential. The magnitude and direction of the phase shift resulting from the alteration of the beat-to-beat interval depends on the timing as well as on the strength of the stimulus. In terms of the resetting effect, the total charge injected into the cell seems to be the crucial factor. Thus, any combinations of various pulse amplitudes and durations, which result in the injection of the same amount of current, are equivalent [5].

The response of biological oscillators to external perturbations has been extensively studied using both theoretical and experimental approaches (reviewed in [32]). Using topological arguments, Winfree [33] greatly contributed to the qualitative understanding of phase resetting responses in biological oscillators and pointed out that, in a spatially extended system like cardiac tissue, such responses play a crucial role in the initiation of arrhythmias. Stimuli of critical amplitude and phase might lead to annihilation of normal rhythmic activity and the initiation of complex spiral wave arrhythmias.

Mathematical models which describe the electrical activity of SAN have been presented by various research groups [1]. The first models produced by Yanagihara et al. [34] and Noble and Noble [35], where the basis for models presented later [36], [37].

The analysis of phase response characteristics provides new insight into the dynamics of the mathematical models, of the electrical activities of SAN, and an experimentally verifiable test for the accurate reconstruction of SAN tissue dynamics. Simulation of the spatial variation in the electrical properties of SAN cells and their phase response profiles is necessary for reconstruction of the complex dynamics of the SAN tissue. Zhang et al. [38] recently presented one-dimensional model which describes all of the above.

In this work, we investigate and characterize the effects of external stimulation on central and peripheral SAN cells using the Zhang et al. model [38]. The equation of the model along with the appropriate initial conditions are solved using the Runge–Kutta method. Short (0.5 ms) depolarizing and hyperpolarizing electrical current pulses of varying amplitude and of timing spanning the whole period are applied and the phase transition curves (PTC) are obtained [22]. Three-dimensional PTC [4], relating old phase and stimulus amplitude to the new phase after stimulation, are generated in order to locate critical stimuli and singularities in the models. Numerical simulations for the electrical activity of the transitional cells lying between the center and the periphery of the SAN [38] are carried out. Simulations are conducted when specific ionic currents vary.

The application of a critical depolarizing stimulus (about 0.4 nA) during the late repolarization phase of action potential resulted in annihilation of the activity in central SAN cells, revealing the existence of a stable singularity in the corresponding model configuration [22]. Detailed analysis of the phase response characteristics of the peripheral cells failed to show a similar singularity and annihilation of normal activity. This difference in phase response behavior of central and peripheral cells is portrayed in the structure of the corresponding PTC.