Abstract
An important class of results generated by biophysical research in the twentieth century is that of mathematical models, both of complete biological systems and of their constituting parts.
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Notes
- 1.
The molecular basis of this variable permeability of the membrane (ion channels, cf. Sect. 1.2) was not known at the time.
- 2.
Incidentally, this case brings the model of cardiac tissue very close to the neuronal cable equation from which it branched off. There, the assumption of an infinite conductivity in the extracellular space is justified by the large amount of fluid that surrounds the axon.
- 3.
Other solutions are possible, viz. all those for which a vector potential \(\mathbf {A}\) exists with \({\underline{\sigma }}_e^*\nabla v_e=\nabla \times \mathbf {A}\). For \(v_e= -\mathbf {E}\cdot \mathbf {x}\), choose \(\mathbf {A}=-(({\underline{\sigma }}_i\mathbf {E})_3x_2,({\underline{\sigma }}_i\mathbf {E})_1x_3,({\underline{\sigma }}_i\mathbf {E})_2x_1)^T\).
- 4.
\({\underline{D}}\) constant has to be required only in the interface region, because it is reasonable to assume that the dynamical system, uninfluenced by the phase field, only produces finite spatial gradients in the physical domain \(\mathcal D\). Anything else would render numerical simulations (even without the phase field method) impossible in any case. Therefore, here, it is sufficient to show that these gradients are not amplified by the choice of the steepness parameter \(\xi \) of \(\phi \). Furthermore, the assumption of a nearly straight boundary is probably not as stringent as it might first seem, as on the spatial scale of \(\xi \), every boundary with finite curvature becomes straight for \(\xi \rightarrow 0\). However, this generalization will not be shown here.
- 5.
- 6.
This distance is indeed finite and depends on the parameter \(\epsilon \), which determines, for a fixed starting point of the trajectory, whether the \(u\) dynamics will be successful in driving the system even more to the right and thus to performing an action potential, or whether the \(v\) dynamics is dominant and brings the system back over the \(u\)-nullcline and thus directly to the fixed point.
- 7.
In allusion to some of the excitable media models used in this thesis, the letter \(\mathbf {u}\) refers here to a state in phase space (instead of the more widely-used \(\mathbf {x}\)) to avoid confusion with a position \(\mathbf {x}\in \mathbb {R}^d\) in space (with \(d=1,2,3\)).
- 8.
Exceptions are, e.g., unstable fixed points or unstable periodic orbits embedded into an attractor.
- 9.
The fourth-order accuracy of the nine-point stencil sometimes referred to is only present under special circumstances (for example, for the solution of the Poisson equation), because the fourth-order error has a special form that can be corrected for in those cases.
- 10.
See Sect. 1.2.
- 11.
- 12.
- 13.
Optical properties of di-4-ANEPPS were mainly extracted from a recent publication on ratiometric optical mapping [84], which contains collected and combined figures from a number of earlier studies (see also references therein). The properties of the LEDs can be found on the data sheet available at the manufacturer’s website http://www.luxeonstar.com/v/vspfiles/downloadables/DS30.pdf (note, however, that the product line has been discontinued).
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Bittihn, P. (2015). Methods. In: Complex Structure and Dynamics of the Heart. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-12232-8_2
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