Reliable signal transmission among cells is important for long-range coordination. While higher organisms have designated structures for signal transmission, such as axons, it remains unclear how simpler communities of cells are organized to relay signals. Furthermore, many biological systems exhibit spatial heterogeneity, which can interrupt signal propagation. In this thesis, we investigate this problem by modeling the spatial organization and dynamics of electrochemical signaling, and we compare our results to experiments from our collaborators on Bacillus subtilis bacterial biofilms. The experiments show that only a fraction of cells participates in signal propagation and that these cells are spatially clustered with a size distribution that follows a power-law decay. These observations suggest that the fraction of participating cells is just at the tipping point between a disconnected and a fully connected conduit for signal transmission. We utilize percolation theory and a minimal FitzHugh-Nagumo-type excitable dynamics model to test this hypothesis, and genetically modified biofilms with altered structure and dynamics to validate our modeling. Our results suggest that the biofilm is organized near the critical percolation point in order to negotiate the benefit and cost of long-range signal transmission. Then, more detailed experiments show that the participation probability is correlated from cell to cell and varies in space. We use these observations to develop an enhanced percolation model, and show using simulations and a renormalization argument that the main conclusions are unaffected by these features. Finally, we use our dynamic model to investigate the effects of heterogeneity beyond the radial wave regime and into the spiral wave regime. We find that spatial correlations in the heterogeneity promote or suppress spiraling depending on the parameters, a surprising feature that we explain by demonstrating that these spirals form by distinct mechanisms. We characterize the dependence of the spiral period on the heterogeneity using techniques from percolation theory. Taken together, our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities.<br>

The aim of this master thesis is to create a simple 3D electro-anatomical model of cardiac tissue that will be able to simulate the electrical activation in both a healthy heart and a heart with arrhytmogenic substrate. The model of electrical activation is realized in the COMSOL Multiphysics, simulation software for modelling using the finite element method. The Fitzhugh-Nagumo equation was used to model the excitatory feature of the myocardium and 2D models of myocardial tissue describing the propagation of action potential in healthy tissue, ischemic tissue, spontaneous action potential formation in the SA node, and spiral wave formation were first developed based on appropriate parameters. Subsequently, simplified 3D models of the heart describing the spread of excitement in a healthy heart, in the presence of accessory pathway and in third-degree atrioventricular block were created. The simplified 3D heart model offers a compromise between computational load and model complexity and can be used as a diagnostic tool for tissue and whole heart adjustment with appropriate equation parameter settings.

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The aim of this master thesis is to create a 3D electroanatomical model of a heart atria, which would be able to perform atrial fibrillation. To control the model, the differential equations of the FitzHugh-Nagumo model were chosen. These equations describe the change of voltage on the cell membrane. The equations have established parameters. The modification of them leads to changes in the behavior of the model. The simulations were performed in the COMSOL Multiphysics environment. In the first step, the simulations were performed on 2D models. Simulations of healthy heart, atrial flutter and atrial fibrillation were created. The acquired knowledge served as a basis for the creation of a 3D model on which atrial fibrillation was simulated on the basis of ectopic activity and reentry mechanism. Convincing results were obtained in accordance with the used literature. The advantages of computational modeling are its availability, zero ethical burden and the ability to simulate even rarer arrhythmias. The disadvantage of the procedure is the need to compromise between accuracy and computational complexity of simulations.