Spatial and temporal patterns in neural networks: Cortical map development and traveling waves in neural tissue

Spatial and temporal patterns of activity in the visual cortex are the subject of mathematical analysis and computer simulation in this thesis.; We used numerical simulations to show that two biologically-inspired reduced models produce realistic ocular dominance and orientation selectivity maps. Both models use local normalization for weight growth and positive correlations between inputs from different eyes. For one of these models, it is possible to obtain ocular dominance maps even with monotone cortical-cortical interactions only. Parameter regimes leading to segregated ocular dominance patterns were derived analytically. Furthermore, we showed how the periodicity of these ocular dominance maps is determined by measurable entities such as retinal and cortical correlation functions. Disinhibited cortical slices exhibit a vast array of temporal patterns of activity such as regular traveling waves, spiral waves and lurching waves. Modeling neural tissue as networks of theta neurons, we used numerical simulations to explore the existence, initiation and stability of these waves.; We analytically proved the existence of constant speed synaptically generated traveling waves in a one-dimensional network of theta neurons. We established that some form of synaptic depression or other adaptive mechanism must exist in order to attain single-spike traveling waves. We present results on how the behavior of solutions to this network depends on the maximal synaptic coupling strength and the speed of the wave.; We investigated the onset and evolution of single-spike waves in an integrate-and-fire network of synaptically coupled neurons in one- and two-dimensional domains. We determined the critical size of initial excitation domain of the network necessary for the onset of propagation and we derived an integro-differential equation for the evolution of the firing time as a function of spatial position.