With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.
John von Neumann Utilizing numerical algorithms in the field of nonlinear dynamics is in most cases inevitable, since the underlying equations which describe the dynamics of the system can usually not be solved analytically. For example, simple generic systems (e.g. low-dimensional maps) are often used, to study basic dynamics and effects. Here, the equations or maps themselves are the objects of interest. In other cases, differential equations are derived on the basis of experimental observations, in order to model the real system (e.g. the Lorenz system [54]
describing atmospheric convection).
2.3.1 Interplay of Experiments and Numerical Simulations
In the field of cardiac dynamics numerical simulations play a significant role concerning the understanding of basic mechanisms and features of the dynamics of the cardiac muscle. The strong demand for simulation studies has different reasons: Current methods and techniques used in ex-vivo animal heart experiments are able to highlight the electrical wave dynamics on the surface of the heart. The optical mapping technique uses for this purpose fluorescent dyes which are sensitive to voltage or the local calcium concentration [55]. However, the electrical activity in the bulk of the cardiac muscle remains mostly unknown. Also, in the experiment different hearts can exhibit distinct varieties e.g in terms of anatomical sizes or the ability to induce ventricular fibrillation. For ethical reasons the number of experiments involving the sacrifice of an animal should be as low as possible. Thus, experiments using the heart of an animal can also be considered as probing or sampling a very high-dimensional (parameter) space with very limited data points. Still, for the validation of a scientific hypothesis, an adequate number of measurements is needed. Hence, numerical simulations are the ideal environment to test first ideas and hypothesis, which can later be validated in experiments.
Compared with the experimental limitations, in numerical simulations the quantity of results is mainly limited by the computational power. Here, the full information about the state of the system is available at any time, and the exact same simulation can be reproduced if this is desired. In addition, the researcher can interact with the system in a way which is perhaps not (yet) possible in the real world, thus new methods can be tested here.
2.3. The Role of Numerical Simulations The overwhelming13 complexity of a real heart can be broken down to the main governing processes. The governing features, mechanisms and their interaction can be studied sepa-rately (e.g. geometrical effects, the influence of heterogeneities or the role of diseases), and specific regions in the parameter space can be (quasi continuously) analyzed.
However, in order to ensure that the performed simulations reproduce to some extent the dynamics of a real heart, two key features are essential:
• The interaction with the experiments (and with the experimentalists) is crucial: In most cases the motivation for a simulation study originates from experimental results.
In addition, each simulation should provide quantities which can be linked (or gauged) to corresponding observables from the experiment (e.g. length/time scales of action potential durations or equivalent numbers of spiral waves in the simulation domain and inside the heart). Only this connection justifies a possible validity of the numerical results.
• As already stated before, different mechanisms can be investigated separately in simu-lations. The process which simplifies the original complexity to its key features which should be investigated is called modeling. This process is essential for the significance of a simulation study, and will be discussed explicitly in the next section.
2.3.2 The Process of Modeling
Creating a model means, to simplify the full complexity and diversity of functionalities of the real (biological) system, while reproducing essential features and dynamics. That means, the model can describe the real system at different levels of details, as illustrated by the lithographs Der Stier by Pablo Picasso (Fig. 2.19).14
How the dynamics of the heart can be modeled will be discussed in the next sections, includ-ing for example the underlyinclud-ing differential equations describinclud-ing the excitation wave propa-gation, but also the choice of the simulation domain (two-dimensional, three-dimensional, rectangular, realistic geometry, etc.). Today, there exist a variety of models of different levels of complexity, which aim at modeling the diverse mechanisms and dynamics of the heart. In theory, a numerical study could therefore take all the most sophisticated models present into account in order to create a simulation as realistic as possible. The upper limit for the complexity of the simulation is the computer power which is available.
However, we know from experiments that various hearts can be very different in their anatomical size and their dynamic behavior. Hence, modeling a specific real heart in every detail in simulations does not ensure that the obtained results are generally applicable (thus are valid for every heart). Rather, the scientific objective is often to find and investigate robust phenomena, which play a relevant role in most clinical cases. For this reason, sim-ulations need to be performed, which cover a broad range of parameters. Concerning the limitation due to the computational cost it is therefore (in most cases) reasonable to perform
13At least for a physicist.
14Inspired by a talk of Peter Kohl at the 26th of January 2016, in the colloquium of the SFB 937 with the titleSystems Biology of the Heart: Model or Muddle?.
Figure 2.19: Pablo Picasso: Der Stier, Zustand I-XI, Lithographien, 1945-46. c Succes-sion Picasso/VG Bild-Kunst, Bonn 2018, [56]. The differences between each drawing can be viewed as the process of modeling various levels of details of a complex object.
simulations including models of moderate levels of complexity but with varying parameters instead of very few realistic simulations.15
The governing question is, which dynamical models and features the simulation should com-prise. The answer to this question is essential for the general field of numerical simulations of complex (biological) systems:
• The ingredients (the choice of the main features and models) of a numerical simulation should always be determined by the scientific objective of the study.
The above statement involves, that a numerical study with the purpose of investigating a certain scientific hypothesis should comprise primarily those dynamics and features (in particular the corresponding models) which are presumably relevant for the investigation of the objective.
2.3.3 Designing a Simulation
The conclusion from the previous section, that a numerical study should comprise those features which are relevant for the respective scientific objective, leads to the issue how to actually evaluate the relevance of a specific feature (e.g. whether to use a simple cell model, or a more sophisticated ionic model).
15In case of, for example, patient specific studies, this is of course not the case.
2.4. Numerical Simulations