Analysis and dynamics of multiple-spike waves in neural networks

Spatially structured bursts of propagating neural activity revealed in cortical slice experiments and in vivo tantalise many scientists on their possible functional mechanisms. Theoretical studies suggest waves with complex firing patterns afford a great capacity for the transmission of information across the brain. This thesis develops a framework for analysing the dynamics of such waves within spiking neuronal networks. We seek to investigate important questions concerning how the wave’s spatiotemporal voltage properties, propagation speed and spike time interval distributions depend on the underlying network structure and the intrinsic features of the neurons that make up the network. These are often difficult to extract with biophysically detailed network models. We therefore analyse simplified spiking networks of synaptically connected neurons, capable of supporting a rich repertoire of propagating activity, yet, amenable to mathematical analysis. Useful information is then obtained on the dynamics of waves found in this network in relation to the model’s parameters. These results can be compared to the findings obtained from more detailed computational studies and experimental observations. Numerical simulations in discrete networks of integrate-and-fire neurons reveal localised bumps that can wander diffusively across the network. These wandering bumps are seen to evolve into persistent synchronous coherent propagating structures, where neurons fire multiple times as the wave envelope passes over. We call these structures multiple-spike waves. An intrinsic feature of the neuron, describing how quickly neurons process synaptic current, is shown to be an important determinant in the emergent network activity. Waves with different number of spiking events co-exist across most parameter regimes, and with lateral-inhibition synaptic connectivity structure, can exhibit large variability in wave speed that has not been reported in studies of networks with purely excitatory connectivity. As a result, we investigate the interaction dynamics of multiple-spike waves on a large spatial domain. Here we find that multiple-spikes waves can merge to form a composite system, with greater complexity in the firing patterns, increasing the wave’s information content. Mathematical progress is made by studying a partial integro-differential equation that is equivalent to the discrete network as the number of neurons tends to infinity. We develop a method of solving the wave speed of the multiple spike waves and its set of spike-times, which then allows us to construct the network’s exact voltage and synaptic profiles and formulate a non-local eigenvalue problem to compute asymptotic stability. This is achieved by considering general perturbations around the wave’s firing times. An in-depth numerical study on the multiple-spike wave’s bifurcation structure is performed, uncovering various mechanisms behind propagation failure and how the wave’s dynamics depend on the network’s system parameters. The analysis of waves with a large number of spikes poses interesting questions regarding the existence of stationary bump solutions in the continuum limit. Uncertainty quantification is performed on waves, revealing how different types of uncertainty in system parameters influence the wave solutions statistical properties. This allows for predictions of the spatial regions of the waves profile most vulnerable to destabilisation. We finally analyse synaptically generated waves in a similar spiking network of Morris-Lecar neurons, where we find interesting transitions from single to double spike waves. Also, similar to what was seen in the integrate-and-fire network, the wave’s dynamics at the network level is strongly influenced by the neuron’s intrinsic features