The phase-space dynamics of systems of spiking neurons

This thesis investigates the dynamics of systems of neurons in the brain. It considers two questions: (1) Are there coherent spatiotemporal structures in the dynamics of neuronal systems that can denote discrete computational states, and (2) If such structures exist, what restrictions do the dynamics of the system at the physical level impose on the dynamics of the system at the corresponding These problems are addressed by way of an investigation of the phase-space dynamics of a general model of local systems of biological neurons.; realistic assumptions about the biological neuron. The system, in consequence, accommodates a wide range of neuronal models.; Appropriate instantiations of the system are used to simulate the dynamics of a typical column in the neocortex. The results demonstrate that the dynamical behavior of the system is akin to that observed in neurophysiological experiments.; Formal analysis of local properties of flows reveals contraction, expansion, and folding in different sections of the phase-space. A stochastic process is formulated in order to determine the salient properties of the dynamics of a generic column in the neocortex. The process is analyzed and the criterion for the dynamics of the system to be sensitive to initial conditions is identified. Based on physiological parameters, it is then deduced that periodic orbits in the region of the phase-space corresponding to “normal operational conditions” in the neocortex are almost surely (with probability 1) unstable, those in the region corresponding to “seizure-like conditions” in the neocortex are almost surely stable, and trajectories in the region of the phase-space corresponding to “normal operational conditions” in the neocortex are almost surely sensitive to initial conditions.; Next, a procedure is introduced that isolates from the phase-space all basic sets, complex sets, and attractors incrementally.; Based on the two sets of results, it is concluded that chaotic attractors that are potentially anisotropic play a central role in the dynamics of such systems. Finally, the ramifications of this result with regard to the computational nature of neocortical neuronal systems are discussed.