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Fractional Diffusion : Numerical Methods and Applications in Neuroscience


In biological contexts, experimental evidence suggests that classical diffusion is not the best description in instances of complex biophysical transport. Instead, anomalous diffusion has been shown to occur in various circumstances, potentially caused by such underlying mechanisms as active transport, macromolecular crowding in a complex and tortuous extracellular or intracellular environment, or complex media geometry. Elegant ways of simulating these complicated transport processes are to connect the spatial characteristics of a medium (porosity or tortuosity of a complex extracellular environment), to fractional order operators. Some approaches include special random walk models representing crowded or disordered media; at the continuum limit, these random walk models approach fractional differential equations (FDEs), including variations of the fractional diffusion equation. Fractional differential equations are an extension of classical integer-order differential equations, and in recent decades have been increasingly used to model the dynamics of complex systems in a wide variety of fields including science, engineering, and finance. However, finding tractable and closed form analytical solutions to FDEs, including the fractional diffusion equation and its variants, is generally extremely difficult and often not feasible, and especially so when integrating these equations into more complex physical models with multiple other components; therefore, the development of stable and accurate numerical methods is vital. In this thesis we explore the topic of anomalous diffusion and the fractional diffusion equation from multiple perspectives. We begin by connecting the micro-molecular behavior of diffusing particles undergoing anomalous diffusion, to the general derivation of the fractional diffusion equation. We then develop numerical approaches to efficiently solve the time-fractional diffusion equation, and characterize these methods in terms of accuracy, stability, and algorithmic complexity. We then make use of these numerical methods by applying fractional diffusion to a model of the signaling events leading up the induction of long-term depression (LTD). We leverage the fact that the fractional diffusion equation can capture the complex geometry in which diffusing particles travel, and use this to simplify an existing model of LTD induction; furthermore, we show that our modified model is capable of retaining the most important functionality of the original model

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