The human brain contains on the order of $10^9$ neurons with each neuron having on the order of $10^4$ synaptic connections with other neurons. Within each neuron, there are protein channels that dictate when ions can flow through them. It is the flow of these ions that is the basis for action potential generation, and these action potentials are the source of neural communication and information. These channels exist in various configurations some of which are conducting (``open'') and some of which are non-conducting (``closed''). Moreover, these channels can stochastically switch between the open and closed states. It is nothing short of remarkable that the brain functions as it does despite the randomness present within each neuron.

What role these microscopic fluctuations, herein known as channel noise, have on macroscopic neural network properties is an open area of neuroscience that has generated a great deal of interest in recent years due to the advancement of computational methods. In this thesis, we first introduce the Hodgkin-Huxley model and mathematical equations which incorporate this channel noise in the Hodgkin-Huxley model. We then study the role of channel noise on properties of small neural networks which begins in Chapter 3. The first property we will look at is how channel noise affects the timing of the first action potential after stimulus onset. This property, known as first spike latency, is believed to be a coding mechanism used by neurons to communicate information between stimuli and brain processing. We will then look at the role of channel noise on neural synchronization. Abnormal synchronization has been strongly correlated with a number of neural disorders such as Alzheimer's disease and Parkinson's disease.

One area of research in neuroscience that is of fundamental interest is the relationship between neural spiking and cognitive processing. For this thesis, in addition to the small neural network models for first spike latency and synchronization, we will consider a recently developed model for cognition and study the model's behavior when subjected to noise. We will conclude with a brief summary of the results obtained as well as discuss ways to extend the research to larger neural network systems.

We present a numerical study of classical particles obeying a Langevin equation and moving on a solid crystalline surface under an external force that may either be constant or modulated by periodic oscillations. We focus on the particle drift velocity and diffusion. The roles of friction and equilibrium thermal fluctuations are studied for two nonlinear dynamical regimes corresponding to low and to high but finite friction. We identify a number of resonances and antiresonances, and provide phenomenological interpretations of the observed behaviour.