Complex activity patterns in arterial wall: Results from a model of calcium dynamics

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Abstract

Using a dynamical model of smooth muscle cells in an arterial wall, defined as a system of coupled five-dimensional nonlinear oscillators, on a grid with cylindrical symmetry, we compare the admissible activity patterns with those known from the heart tissue. We postulate on numerical basis the possibility to induce a stable spiral wave in the arterial wall. Such a spiral wave can inhibit the propagation of the axial calcium wave and effectively stop the vasomotion. We also discuss the dynamics of the circumferential calcium wave in comparison to rotors in venous ostia that are a common source of supraventricular ectopy. We show that the velocity and in consequence the frequency range of the circumferential calcium wave is by orders of magnitude too small compared to that of the rotors. The mechanism of the rotor is not likely to involve the calcium-related dynamics of the smooth muscle cells. The calcium-related dynamics which is voltage-independent and hard to be reset seems to actually protect the blood vessels against the electric activity of the atria. We also discuss the microreentry phenomenon, which was found in numerical experiments in the studied model.

Introduction

It is interesting to compare modeling in two types of active media: the arterial wall tissue consisting of smooth muscle cells (SMCs) and the heart tissue. The key factor in the dynamics of SMC is the dynamics of the calcium Ca2+ ions. Calcium dynamics on both cellular and macroscopic level has been a subject of vivid research, both theoretical and experimental showing a number of important results [1]. The nature of calcium sparks [2] was studied theoretically [3] soon after the first experimental evidence of their presence in quiescent heart cells [4]. The distribution of calcium ions and their dynamics was observed to be nonuniform within a single cell. Hence a cell is sometimes modeled as a spatially extended system [5]. However, in order to model the propagation of activity in the tissue it is convenient to reduce the single cell dynamics to a lumped system: a nonlinear oscillator (which was found to exhibit deterministic chaos [6]) and to consider complex coupling between many oscillators [7].

Section snippets

Modeling the heart tissue

When the heart tissue is treated as an active medium, on a reasonable level of approximation each cell may be modeled as a nonlinear FitzHugh–Nagumo oscillator [8]. The coupling between the cells is only electric and propagation of activity is equivalent to propagation of the action potential [9], [10], [11]. The ionic currents are rather considered as local and “pinned” to each cell. Such an approach inevitably neglects many details of the dynamics of a cell—the aim of such an approach is to

Modeling the smooth muscle tissue

In contrast to the heart, the activity of the arterial walls in particular, and smooth muscle tissue in general, as modeled by [6], [7] is not restricted to sole propagation of electric potential. In the model [7] which we consider, coupling is introduced for calcium concentration and concentration of inositol 1,4,5-trisphosphase (IP3) as well. Both particles diffuse between neighboring cells through the poorly selective gap junctions. From the point of view of nonlinear dynamics it is

Rotors in electrophysiology

There is one specific pattern of activity which is important from the electrophysiological point of view: a rotor [17]. Rotors are commonly found in venous ostia in the atrium. They have a form of reentrant waves propagating around venous ostia of pulmonary veins. The electric activity of the rotor spreads to the atria, being a frequent source of atrial fibrillation. So far the only method used to eliminate a rotor is through isolation of venous ostia using ablation procedure [18]. Up to our

The aim of the paper

The primary aim of current paper is to prove in a numerical experiment the existence of spiral waves in the arterial wall: i.e. an active medium of a rich, high-dimensional dynamics, spatial inhomogeneity and cylindrical topology. The next aim is to study the properties of spiral waves and their possible role in vasomotion and also to study different patterns of activity and their properties. Then, we would like to verify if the rotors may be the result of a circumferential calcium wave

The model

The original model of a smooth muscle cell consisting of membrane oscillator and an intracellular oscillator was introduced by Parthimos [6], who also discussed many of its properties, including the effects of nonlinearity and the presence of deterministic chaos. The variables in the model are: calcium concentration in cytosol ci, and in sarcoplasmatic reticulum (SR) si, cell transmembrane potential vi, and the open state probability of calcium-activated potassium channels wi. The model does

Results

Below we present main findings concerning the admissible and preferred wave topology, velocity of waves with application to rotor substrate analysis and finally the electrically induced microreentry phenomenon. The main difference between the model we study and the activity patterns observed in cardiac muscle is that the whole tissue is in the oscillatory state: i.e. above the Hopf bifurcation, whether in cardiac muscle it is in the passive state (waiting for activation). Therefore, once we

Discussion

As pointed out in Section 7.1, we found it merely impossible to introduce even the simplest activity patterns by short peak of transmembrane potential, being a passive variable [7]. Due to the presence of the intracellular Ca2+-based oscillator, any patterns of activity that are successfully induced, are protected from the electric activity of the surrounding media and thus merely impossible to be electrically reset. Such feature seems valuable in the natural pacemakers of the heart. Such a

Conclusions

In conclusion, we may state that the arterial tissue is an interesting example of active (oscillatory) medium. In this medium all types of waves, known from cardiac tissue, may be observed. Due to the topology of the medium, pairwise patterns of activity—like pair of circumferential waves are preferred (i.e. more stable and easier to be induced). The running axial wave, responsible for vasomotion can annihilate after collision with such a spiral wave. Such a phenomenon, related with the

Conflict of interest statement

None declared.

Acknowledgments

Professor Jan J. Żebrowski is thanked for continuous support. Professor Jean-Jacques Meister and the team of the Laboratory of Cell Biophysics, EPFL is thanked for hospitality (TB). Jens Christian Brings Jacobsen MD, PhD and professor Andrzej Bere¸sewicz are thanked for discussion (TB). The paper was supported by the Polish Ministry of Science, Grant no. 496/N-COST/2009/0.

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