Excitable media are an important class of systems, examples of which include epidemics, predator-prey interactions, nervous systems, and heart muscle. Aggregating cellular slime moulds are an example of an excitable medium. The species of cellular slime mould Dictyostelium discoideum is an important model organism that many science laboratories use. Studying the aggregation of slime moulds increases knowledge about excitable media generally. One method of studying the aggregation of slime mould is to simulate theft behaviour on a computer model. This thesis presents the author's computer model of cellular slime mould Dictyostelium discoideum and the results of experiments carried out using the computer model. The experiments investigate the relation between the aggregation patterns and the various parameters of the model. These parameters are the density of artificial slime moulds, the acrasin threshold, the acrasin degradation rate, and the rate of acrasin secretion. Randomness has an effect on the aggregation patterns produced. Results of experiments are presented that examine the effect of randomness. Two forms of randomness are investigated: random secretion of acrasin by the artificial slime moulds; random initial reactivity of the artificial slime moulds. The computer model describes an artificial environment in which artificial slime mould amoebae interact with each other and their environment. Out of these individual interactions the global patterns that characterize slime mould aggregations emerge. The model facilitates the study of these individual interactions and hence the global patterns that emerge. The model and the experimental results described in this thesis contribute to the study of the aggregation phase of the life cycle of Dictyosteliuni discoideum. The author proposes mechanism that could underlie certain classes of aggregation patterns. These patterns include net-like aggregations and loop aggregations. The computer model presented in this thesis is successful in emulating the behaviour of the cellular slime mould Dictyostelium discoideum. In its present form the model is a useful tool to biologists. The results of experiments conducted with the model suggest mechanisms that may underlie certain pattern produced by living slime moulds. A result of particular interest is the initiation of the spiral wave pattern from a loop wave, which produces a loop aggregation.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

Self-oscillating reactions have been widely observed and studied since the last century because they exhibit unique behaviors different from the traditional chemical reactions. Self-oscillating systems, such as the Belousov-Zhabotinsky (BZ) reaction, oxidation reaction of CO on single crystal Pt, and calcium waves in the heart tissue, are of great interest in a variety of scientific areas. This thesis contributes to the understanding of wave transition in BZ reaction, and to possible applications of non-equilibrium behaviors of polymer systems. In BZ reaction, two types of wave patterns, target and spiral, are frequently observed. The transition from one to another is not fully understood. Hence, a systematic investigation has been performed here to investigate the mechanism by which heterogeneity affects the formation of wave patterns. A BZ reaction catalyst was immobilized in ion exchange polystyrene beads to form active beads. Then active and inactive beads with no catalyst loading were mixed together with various ratios to achieve various levels of heterogeneity. In the same reaction environment, different wave patterns were displayed for the bead mixtures. We observed a transition from target patterns to spiral patterns as the percentage of the active beads in the beads mixture decreased. The increase of the heterogeneity led to wave pattern transition. Heterogeneity hindered the propagation of target waves and broke them into wavelets that generated spiral waves. In an effort to develop practical applications based on non-equilibrium phenomena, we have established a novel drug delivery system. A proton generator Zirconium Phosphate (ZrP) was imbedded inside a pH sensitive polymer matrix, poly acrylic acid (PAA). Through the ion exchange with sodium cation (Na+), ZrP generates protons to control the swelling/shrinking behaviors of PAA. The drug encapsulated in the matrix can be released in a controlled manner by adjusting the supply of Na+. This system might be developed into vehicles to deliver drugs to specific targets and release at a proper time. This new delivery technique will be convenient and significantly increase the efficiency of medicines.

This thesis investigates the origin of spontaneous activity in the uterus. This organ does not show any activity until shortly before delivery, where fast and efficient contractions are generated. The aim of this work is to provide insight into the origin of spontaneous oscillations and into the transition from asynchronous to synchronized activity in the pregnant uterus. One intriguing aspect in the uterus is the absence of any pacemaker cell. The organ is composed of muscular cells, which are excitable, and connective cells, whose behavior is purely passive; None of these cells, taken in isolation, spontaneously oscillates. We develop an hypothesis based on the observed strong increase in the electrical coupling between cells in the last days of pregnancy. The study is based on a mathematical model of excitable cells, coupled to each other on a regular lattice, and to a fluctuating number of passive cells, consistent with the known structure of the uterus. The two parameters of the model, the coupling between excitable cells, and between excitable and passive cells, grow during pregnancy.Using both a model based on measured electrophysiological properties, and a generic model of excitable cell, we demonstrate that spontaneous oscillations can appear when increasing the coupling coefficients, ultimately leading to coherent oscillations over the entire tissue. We study the transition towards a coherent regime, both numerically and semi-analytically, using the simple model of excitable cells. Last, we demonstrate that, the realistic model reproduces irregular action potential propagation patterns as well as the bursting behavior, observed in the in-vitro experiments.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

In this thesis we investigate the mechanisms underlying the generation of abnormal EEG rhythms in epilepsy, which is a crucial step towards better treatment of this disorder in the future. To this end, macroscopic scale mathematical models of the interactions between neuronal populations are examined. In particular, the role of interactions between neural masses that are spatially distributed in cortical networks are explored. In addition, two other important aspects of the modelling process are addressed, namely the conversion of macroscopic model variables into EEG output and the comparison of multivariate, spatio-temporal data. For the latter, we adopt a vectorisation of the correlation matrix of windowed data and subsequent comparison of data by vector distance measures. Our modelling studies indicate that excitatory connectivity between neural masses facilitates self-organised dynamics. In particular, we report for the first time the production of complex rhythmic transients and the generation of intermittent periods of 'abnormal' rhythmic activity in two different models of epileptogenic tissue. These models therefore provide novel accounts of the spontaneous, intermittent transition between normal and pathological rhythms in primarily generalised epilepsies and the evocation of complex, self-terminating, spatio-temporal dynamics by brief stimulation in focal epilepsies. Two key properties of these models are excitability at the macroscopic level and the presence of spatial heterogeneities. The identification of neural mass excitability as an important processes in spatially extended brain networks is a step towards uncovering the multi-scale nature of the pathological mechanisms of epilepsy. A direct consequence of this work is therefore that novel experimental investigations are proposed, which in itself is a validation of our modelling approach. In addition, new considerations regarding the nature of dynamical systems as applied to problems of transitions between rhythmic states are proposed and will prompt future investigations of complex transients in spatio-temporal excitable systems.

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>

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