Pattern Formation and Synchronization in Excitale Systems under the Influence of Spatiotemporal Colored Noise

The omnipresence of noise in natural systems is a fact that has led to controversial discussions over the last decades, the question being whether natural system actively make use of ambient fluctuations in synchronization or pattern forming processes. One possibility for a system to make use of noise in a constructive way is the notion of Spatiotemporal Stochastic Resonance, a phenomenon wherein noise induces and/or sustains spatiotemporal patterns, which has been confirmed to exist in chemical and biological systems. Noise in natural systems usually has finite spatiotemporal correlations, it is then said to be `colored'. With the eye on biological systems, the present work investigates the influence of colored noise on the effect of Spatiotemporal Stochastic Resonance and synchronization in excitable and oscillatory media, respectively. In particular, two different types of noise are considered, which are commonly encountered in natural systems. The first type shows short-range, exponentially decaying correlations both in time and space, whereas the second type is power-law noise, having long-range correlations due to the power-law shape of its spatiotemporal power spectral density. First, numerical algorithms for the efficient modeling of spatiotemporal correlated noise are developed and different analysis tools for the detection and extraction of coherent structures from a noisy, spatially extended system are introduced and evaluated. These tools are used to investigate the effect of additive, spatiotemporal correlated noise on a two-dimensional network of biologically motivated elements in its respective noise-sustained pattern forming, its excitable and its oscillatory regime.The results of this thesis lead to the following conclusions: Noise-induced pattern formation is a robust phenomenon with respect to a wide range of spatiotemporal noise correlations. It is seen that a moderate temporal, and in the case of power-law noise also spatial, noise correlation is beneficial to the effect of Spatiotemporal Stochastic Resonance. The effect of spatial amplitude desynchronization in an oscillatory system is enhanced by both intermediate short- and long-range correlated noise for moderate temporal and spatial noise color. The above findings are reproduced analytically by a linear response analysis, which views the respective media in terms of their spatiotemporal frequency filter characteristics. The results are in good agreement with the simulations for any temporally correlated noise, but generally disagree for large spatial noise color due to finite size effects of the system. The analysis shows that the influence of colored noise on the pattern forming and synchronization processes is a consequence of the impact of the noise color on the energy transfer from the noise to the medium. The analytical results deviate from the numerical simulations in the case of spatial power-law noise not only due to finite-size effects, but also due to the particular self-affine properties of the noise, which the theory does not account for. The results put the phenomenon of Spatiotemporal Stochastic Resonance on a broader theoretical and numerical footing. Its robustness to spatiotemporal noise color makes this effect a candidate for noise-induced phenomena in nature. It is to be expected, that noise in neuro-physiological systems can have indeed favorable effects