Single Cell Dynamics – Cardiac Tissue as an Excitable Medium

the voltage between the inside and the outside of a cell) is composed and what happens during an action potential on a cellular level. The electrical potential inside and outside of

Figure 2.1: The ion channels which are mainly responsible for the resting potentialV_m^rest. In (a) the leak channels are depicted, which allow a steady transmembrane current of sodium and potassium channels. In addition, the Na⁺−K⁺pump provides for an exchange of three sodium ions from the inside to the outside, and at the same time a propagation of two potassium ions from the outside to the inside of the cell. This figure was published in [24], Copyright Elsevier (2018).

a cardiac cell is determined by ion concentrations of specific ions (e.g. sodium (Na⁺), potas-sium (K⁺), chloride (Cl⁻) or calcium (Ca²⁺)). The membrane, which separates the inside from the outside of the cell, is permeable for these ions under the consideration of different mechanisms. Goldman derived already in 1943 an expression for the membrane potentialV_m depending on ion concentrations [23] (denoted by square brackets [x]_i/o for concentrations inside and outside the cell, respectively) and specific ion permeabilities (denoted by Px), called the Goldman equation:

V_m= RT

F ·P_Na·[Na⁺]o+P_K·[K⁺]o+P_Cl·[Cl⁻]i

P_Na·[Na⁺]_i+P_K·[K⁺]_i+P_Cl·[Cl⁻]_o , (2.1) whereR is the ideal gas constant,T is the temperature andF is Faraday’s constant.

Resting Potential

The resting membrane potential is mainly determined by the sodium and potassium con-centrations, other ions and proteins do only play a minor role here. The actual resting potential is primarily composed by two effects: leak channels of the cell allow the diffusion of sodium and potassium due to a non-zero concentration gradient of the ions (Fig. 2.1(a)).

In addition, thesodium–potassium pump, also called Na⁺/K⁺-ATPase is an enzyme which pumps three sodium ions from the inside to the outside and at the same time two potassium ions from the outside to the inside of the cell (Fig. 2.1(b)). These two effects combined

2.1. Complexity of the Heart

provide a resting potential of aboutV_m^rest≈ −90 mV.¹

Action Potential

When an excitation wave travels through the tissue (which subsequently triggers the me-chanical contraction (see section 2.1.3 on page 20)), the local membrane potentialV_mdiffers from the resting potentialV_m^restand one can observe an action potential (Fig. 2.2(a)), which is characterized by a rapid depolarization of the membrane potential (ii) from the resting potential V_m^rest ≈ −90 mV (i) to approximately 20 mV (upstroke), followed by a relatively stable plateau, which determines the length of the action potential (iii). Subsequently, the membrane potential repolarizes (iv) back to the initial resting potential (v). The whole dynamics during this process is determined by the sophisticated behavior of voltage gated ion channels. By going step by step through the stages (i)-(v) in Fig. 2.2, we examine the respective underlying dynamics in each part.

Initially, the action potential is in the resting state ((i) in Fig. 2.2). The action potential is then triggered by a slight increase of the membrane potential V_m over the excitation threshold of aboutV_m^th≈ −70 mV (vertical orange line in Fig. 2.2(a)) by nearby cells (details about the propagation of the electrical signal are given in section 2.1.2 on page 17). This causes the opening of specific voltage gated sodium channels.

Figure 2.3 sketches the dynamics of these ion channels for sodium (a) and potassium (b).

The voltage gated sodium channels have two gates, an activation gate and an inactivation gate. During the resting state, the activation gate is closed, whereas the inactivation gate is open. When the membrane potential exceeds the excitation threshold, the activation gate opens and sodium ions pour (due to a concentration gradient) from the outside to the inside (sketched in Fig. 2.2(b), by definition a current from the outside to the inside is positive, the other way negative). This abrupt change in the ion concentrations leads to the upstroke of the membrane potential (phase (ii) in Fig. 2.2(a)). The inactivation gate is also triggered by the initial increase of the membrane potential, but it closes a few 10000ths of a second later than the opening of the activation gate. This mechanism leads to a temporal limitation of the sodium current. In contrast to the fast reaction of the sodium channels to a voltage change (therefore also denoted as “fast” channels), the voltage gated potassium channels (Fig. 2.3(b)) open a bit later, approximately at the same time as the sodium channels close.

Potassium ions can then propagate from the inside to the outside of the cell (sketched in Fig. 2.2(b), the negative sign is due to the definition of the direction of the flow). This effect alone would cause a continuous decrease of the membrane potential, which can also be observed in many other types of cells, e.g. in nerve fibers. However, in cardiac muscle cells, the membrane potential remains relatively constant after the upstroke ((iii) in Fig. 2.2(a)), due to additional voltage gated calcium channels. These “slow” channels, allow calcium ions to propagate from the outside to the inside of the cell (Fig. 2.2(b)), and therefore counteract the decrease of the membrane potential caused by the propagation of potassium

1By definition, the membrane potential is counted from the inside (more negative) to the outside of the cell. This causes the negative sign of the resting potential.

−100 0 100 200 300 400 Time [ms]

−100

−50 0 50

Vm[mV]

(i) (ii) (iii) (iv) (v)

Threshold

(a)

(b)

IonicCurrent[a.u.]

Na⁺ Ca²⁺

K⁺

Figure 2.2: The action potential and its underlying mechanism. Subfigure (a) shows a generic action potential of cardiac muscle with a typical action potential duration of about 250 ms and a voltage threshold of aboutV_m^th≈ −70 mV. The evolution of the membrane po-tentialV_mcan be subdivided into the initial resting state (i), the upstroke (ii), the following approximately constant plateau (iii), the repolarization (iv) and the return to the resting state (v). In (b), the transmembrane ionic currents are schematically depicted, which mainly determine the underlying dynamics of the action potential. The initial increase ofV_m over the excitation threshold causes the voltage gated sodium channels to open for a short time (Fig. 2.3), resulting in the upstroke ofV_m (ii). The “slow” potassium and calcium channels open later, and allow the propagation of calcium ions from the outside to the inside (blue curve) and potassium ions from the inside to the outside of the cell (green dashed curve).

The direction of the flow is clarified by the negative sign of the potassium current. The two currents approximately annihilate each other (in terms of an effective change of V_m) and determine the plateau of the action potential (iii). Eventually, the calcium channels close earlier than the potassium channels (iv), thus the potassium current predominates and causes the repolarization of the membrane potential back to the resting state (v).

2.1. Complexity of the Heart

Figure 2.3: A basic description of the mechanism of voltage gated ion channels. In (a) three stages of voltage gated sodium channels are shown, respective to different states during an action potential. During the resting state, the activation gate is closed and the inactivation gate is open (left). When the membrane potential of the cell exceeds the excitation threshold, the activation gate opens and allows sodium ions to propagate (due to a concentration gradient) from the outside to the inside of the cell (middle). After less than a millisecond, the inactivation gate closes (right). The channel returns to the initial state (left) only, when the membrane voltage reaches the resting state again. The voltage gated potassium channels (b) behave similar, despite the fact that they possess only one gate. This gate opens and closes also triggered by the membrane potential, but this process happens much slower than the voltage gated sodium channels. This figure was published in [24], Copyright Elsevier (2018).

ions. Finally, the calcium channels slowly close and the potassium channels predominate, leading to the repolarization of the membrane potential (iv) back to the resting state (v).

The resulting action potential can be characterized by a certain action potential duration (APD) (Fig. 2.2(a)) of about APD≈200 ms to 300 ms. Due to the heterogeneous substrate of the heart, the exact lengths and durations and related details of the respective ion channel dynamics can vary from region to region (e.g. endocardium, epicardium, pacemaker cells).

Another distinctive feature of the dynamics is the existence of arefractory period, an amount of time (subsequently to the upstroke of the action potential, and of comparable length as the action potential duration), where the cell cannot be excited again. The inactivation gates of the voltage gated sodium channels do not open again, if the membrane potential returns to (or almost to) the resting potential of approximately −90 mV, even if a second signal arrives from another cell. This behavior stabilizes the proper dynamics of the heart and impedes disorganized contraction patterns. With the existence of a resting state, an excited state, and a refractory state, cardiac tissue can be recognized as anexcitable medium. As a remark, the ion channel dynamics discussed here is mainly responsible for the action potential, but also other kinds of ion channel dynamics contributes to the final shape of the action potential (e.g. chloride ions or “fast” potassium channels [25]). The desired level of detail describing the actual dynamics is, however, in particular essential for the process of modeling (for the role of models in numerical simulations see section 2.3 on page 40).

2.1. Complexity of the Heart

Figure 2.4: The anatomy of cardiac muscle fibers. The lower part of the figure shows a schematic drawing of cardiac muscle fibers and its constituents. Magnified sketches of the gap junctions (left plot of the upper part) and the desmosomes (right plot of the upper part) are depicted. Reprinted by permission of Pearson Education, Inc., New York, New York [1].