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#### This Brain Is a Mess: Inference, Random Graphs, and Biophysics to Disentangle Neuronal Networks

At first glance, the neuronal network seems like a tangled web in many areas throughout the nervous system. Often, our best guess is that such “messy” connections are close to random, while obeying certain statistical constraints, e.g. the number of connections per neuron. However, neuronal wiring is coordinated across larger ...

#### Riemann-Hilbert Problems, Their Numerical Solution and the Computation of Nonlinear Special Functions

(2013-11-14)

The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painleve transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are ...

#### Closing the Loop: Optimal Stimulation of Neuronal Networks via Adaptive Control Algorithms

The Caenorhabditis elegans (C. elegans) worm is a well-studied biological organism model. The nervous system of C. elegans is particularly appealing to study, since it is a tractable fully functional neuronal network for which electro-physical connectivity map (connectome) is fully resolved [1,2]. In this work, we use a ...

#### Some Boundary-Value Problems for Water Waves

(2012-09-13)

Euler's equations describe the evolution of waves on the surface of an ideal incompressible fluid. In this dissertation, I discuss some boundary-value problems associated with Euler's equations. My approach is motivated by the ideas generated by Fokas and collaborators, particularly the notion of a global relation for ...

#### Interface Problems using the Fokas Method

Interface problems for partial differential equations are initial boundary value problems for which the solution of an equation in one domain prescribes boundary conditions for the equations in adjacent domains. These types of problems occur widely in applications including heat transfer, quantum mechanics, and mathematical ...

#### Stochastic Dynamics: Markov Chains, Random Transformations and Applications

Stochastic dynamical systems, as a rapidly growing area in applied mathematics, has been a successful modeling framework for biology, chemistry and data science. Depending upon the origin of uncertainties in an application problem, the theory of stochastic dynamics has two different mathematical representations: stochastic ...