d𝑉𝑚
d𝑡 = −(𝐼Na++ 𝐼K++ ⋯ )
𝐶𝑚 (1.1)
as a function of the different ion currents 𝐼𝑋. These are in turn regulated by ion channels in the cell membrane with their specificgatingcharacteristics, which are often dependent on 𝑉𝑚 but may also be activated/deactivated by certain concentrations or even physical strain [10]. Other important components of the ionic system are active membrane enzymes (such as the Na+/K+-ATPase). These act as pumps which create the concentration difference between inside and outside of the cell. Through the Nernst equation this concentration difference explains the resting potential of the cardiac cell.
Models of the cardiac cell vary widely in complexity ranging from only two to dozens of variables [8]. However, as mentioned before, this thesis is focused on studying the heart from a dynamical systems point of view. So in the next sections I will describe excitable systems in more depth.
1.2 Excitable Media
The cardiac tissue constitutes what is called an excitable medium and as such is a part of nonlinear dynamics research. Besides the heart, other examples include Mexican waves, forest fires, or bacterial cultures.
To form a media, neighbours needs to be coupled in some form. Additionally, the basic properties of the local behaviour are the existence of:
• a quiescent or resting state, in which a cell may be excited by neighbours
• an excited stated which may also cause excitation in the surrounding medium
• a refractory state, in which a new excitation is not possible and no excitation of neighbouring cells will occur.
To give some examples, different excitable systems and their corresponding states are given in Table 1.1. At the limit of small cells, these systems can be described by partial differential equations. When describing the system as partial differential equa-tions, at least two variables are necessary. The first variable describes the excitation dynamics while the second variable represents the refractory state and follows the first variable.
One such model is the Bär-Eiswirth model [11] which will be used later in this thesis. The local behaviour of the Bär-Eiswirth model is described by the differential
Table 1.1 Examples of excitable systems and the corresponding states they exhibit.
System Resting Excited Refractory Coupling
cardiac resting state depolarization ion concentrations prevent further
sitting person standing person “tired” person observing neighbours
To understand the implications of these equations it is helpful to plot the nullclines, the lines where d𝑢/d𝑡 = 0 or d𝑣/d𝑡 = 0 as shown in Figure 1.3. Additionally, the arrows indicate the derivatives given in Equation 1.2. Inspecting the figure shows that the system has a stable point when both variables have the value zero. However, when a small stimulus shifts 𝑢 to the green triangle indicated in the figure the self excitation causes a rise in 𝑢 until it plateaus when 𝑢 ≲ 1. Next, the slower dynamics of 𝑣 take over which causes 𝑣 to drop again and the system goes back to stable fixed point – the quiescent state. The local dynamics thus are akin an oscillator only that the oscillation requires an initial perturbation to start.
The parameter 𝑏 directly controls the excitation threshold and negative values would lead to oscillatory behaviour by shifting up the nullcline and making the (0, 0) point unstable. The parameter 𝑎 further modifies the position of the nullcline and thus trajectory. The excitability 𝜀 is important, since it separates the time scales by ensuring that the dynamics in 𝑢 are generally faster than in 𝑣. For the upstroke to occur 𝑢 has to increase in value fast while 𝑣 is still small since otherwise an increase in𝑣would counteract the increase in𝑢. Thus 𝜀 ≪ 1is necessary for the characteristic dynamics.
1.2 Excitable Media 9
Fig. 1.3 a) Phase portrait of the Bär-Eiswirth model with parameters 𝜀 = 0.08, 𝑎 = 0.84, and𝑏 = 0.07. The blue and orange lines show the𝑢- and𝑣-nullclines (zero derivatives). The arrows indicate the derivative of the system. The green line shows a trajectory starting from the triangle and ending at the stable fixed point marked by the star. b+c) The temporal evolution of the 𝑢 and 𝑣variable for the line shown in (a).
1.2.1 Local Dynamics to Spatial Organization
To model spatial coupling in the partial differential equation of an extended medium a diffusive term is introduced into Equation 1.2 (or eq. 1.1 for general cardiac models):
𝜕𝑢
𝜕𝑡 = 𝑓(𝑢, 𝑣) + ∇(D∇𝑢). (1.3)
D is the diffusion tensor and the equation can be simplified with ∇(D∇𝑢) = 𝐷∇2𝑢 for isotropic diffusion with the corresponding diffusion constant 𝐷. In this model the coupling is limited to the membrane potential, although for excitable media diffusion can in general occur for all variables.
A sketch of an excitable medium in the form of a circle is shown in Figure 1.4. In (a) we see the circle fully prepared in the excitable state. A stimulus is initiated at the position indicated by the green triangle. This stimulus travels upward on both sides of the circle and finally collides at the top. On the other hand, in (b), the system is prepared with a refractory region on the left side. This means that the wave initiated from the green triangle cannot travel to the left and a wave only propagates on the right side. However, if the refractory state goes back to the resting state within the time until the wave reaches this area from the other side then the final behaviour is a perpetually counterclockwise rotating wave.
This circular representation is an example of a so called reentry in the cardiac sciences, since such circles may also occur inside the heart. Similar reentry patterns are
a)
xb)
Fig. 1.4 Schematic of a circular excitable medium. Turquoise indicates the resting, green excited, and dark red the refractory state. a) A stimulus is given at the green triangle position into a fully resting system. b) A stimulus is given when the system is prepared with a refractory area on the left side. (Further explanation in the text.)
0 5 10
x 0
5 10
y
0.0 0.2 0.4 0.6 0.8 1.0
u
a)
0 5 10
x 0
5 10
y
−π 0 π
phaseofoscillation
b)
Fig. 1.5 Example of a spiral wave activation in the two-dimenional Bär-Eiswirth model (eq. 1.2) with 𝑎 = 0.84, 𝑏 = 0.07, 𝜀 = 0.07, and 𝐷 = 0.1 (see eq. 1.3). a) The 𝑢 variable, which corresponds roughly to the cardiac membrane potential and b) the phase of the oscillation which is defined as the angle around the point at 𝑢 = 0.65 and 𝑣 = 0.45 in Figure 1.3. In this phase image, the spiral has a well defined core where all phases meet – the phase singularity (PS).
1.3 Cardiac Arrhythmia 11