Theoretical neuroscience: from long wavelength cortical patterning to spatial navigation

The science of brain function has a long and vibrant history. Recent technological developments have dramatically improved and facilitated data acquisition from a variety of methodologies to monitor brain activity, ranging from electroencephalography to opto-genetics. This highlights a need for concomitant theories of brain function. Such theories can act as a bridge between descriptions of the brain pertaining to data at different levels, from molecular to behavioural, using methods of mathematics, physics, and computer science.\ud \ud The models presented in this thesis do not incorporate all the biophysical, anatomical and physiological data collected to date. Rather, the focus is on simplified models that contain sufficient detail to explain the essence of the phenomena considered. Moreover, they are constructed to allow the application of analytical mathematical tools to explore their behaviour.\ud \ud In particular, this thesis proposes parsimonious neural models that aim to explain the mechanism by which humans and animals can navigate using spatial memory. The material presented ranges over a number of levels of description, and utilises a variety of mathematical techniques. A common theme throughout is the use of ideas from nonlinear dynamical systems to gain insight into neural mechanisms, ranging from activity patterns of cells underlying navigation, to the derivation of temporal difference reinforcement learning algorithms to solve reward based problems.\ud \ud This work presents three main contributions. Firstly, it analytically determines which model parameters contribute to the observed difference in wavelength scale of the formed activity patterns in computational models for grid cells. Moreover, this thesis explores extensions to these models in order to find a neural mechanism that could account for the difference in wavelength scale. It is shown, after analysing the linear stability of spatially homogeneous steady states to spatio-temporal perturbations, that the addition of axo-dendritic connections provides a mechanism for the difference in wavelength scale.\ud \ud Secondly, based on recent research, this work proposes a different type of model, a network of spiking neurons, to uncover the mechanisms, related to rebound spiking, for variation in scale of grid cell firing fields. Travelling waves are observed on computer simulations of this model. The analytical construction of such waves is accomplished using techniques from the field of non-smooth dynamical systems. Moreover, the dispersion curve, that determines how wave speed varies as a function of the period, is constructed. Such dispersion curve exhibits a wide range of long wavelength solutions. In order to exhibit how the variation of parameters affects the maximum allowed period, a wave stability analysis is developed. This work entails and broadens the use of non-standard analysis techniques.\ud \ud The final part of the thesis makes a direct link to experiments, combining reinforcement learning theory and computer simulations to shed light on the neurocomputational mechanisms underlying behaviour of rats in a variation of the Morris watermaze experiment. Particularly, the simulation employs a continuous time actor-critic framework, in which the actor and critic are represented as firing rate neural networks. The ability of the artificial rats to learn and reach the different goal locations is measured under different variations of the model