Stochastic Oscillations Motivated by the Noisy and Rhythmic Firing Activity of Neurons

A human brain contains billions of neurons. These are extremely complex dynamical systems that are affected by intrinsic channel noise and extrinsic synaptic noise. Oscillatory behaviour is a phenomena that arises in single neurons as well as in neuronal networks. Based on the noisy and rhythmic firing activity of nerve cells, the aim of this thesis is to provide an overview on the existing theory of stochastic oscillations. The main question is how oscillations can be defined mathematically in a stochastic setting. This work suggests two very different definitions of a stochastic oscillator according to certain well-known deterministic tools. The first one states that the solution of a specific two-dimensional stochastic differential equation is a stochastic oscillator if it has infinitely many simple zeros almost surely. The second definition of a stochastic oscillator corresponds to the concept of a random periodic solution that is based on the cocycle property. Two particular stochastic equations are introduced each satisfying one of these definitions. For this purpose, two detailed proofs on the validity of the first and the second definition, respectively, are presented. A second topic addressed by this work is the stability of stochastic equations. The system energy carries information on how the trajectory propagates in the phase plane. Besides, Lyapunov exponents describe the asymptotic exponential growth of random dynamical systems. Finally, this work provides an application of the developed theory to specific standard oscillatory models as well as to the Van der Pol oscillator on which the FitzHugh- Nagumo neuron model is based. Sample path simulations of the stochastic equations are provided by the implementation of the Euler-Maruyama method in MATLAB. Furthermore, an exact simulation method applied to the stochastic harmonic oscillator equation is introduced.