Simulation and Analysis of Gradient Frequency Neural Networks

The methodologies introduced and applied in this work have fundamental roles in connecting the component level descriptions of brain dynamics (single neurons) to population level (neural networks). The synchronization regions of a single neuron with respect to a periodic external stimulus either deterministic or noisy is evaluated. Many neurons in the auditory system of the brain must encode periodic signals. These neurons under periodic stimulation display rich dynamical states including mode-locking and chaotic responses. Periodic stimuli such as sinusoidal waves and amplitude modulated (AM) sounds can lead to various forms of n:m mode-locked states, in which a neuron fires n action potentials per m cycles of the stimulus. The regions of existence of various mode-locked states on the frequency-amplitude plane, called Arnold tongues, are obtained numerically for Izhikevich neurons. Arnold tongues analysis provides useful insight into the organization of mode-locking behavior of neurons under periodic forcing. We found these tongues for both class-1 (integrator) and class-2 (resonator) neurons, with and without noise. These results are applied in real data and Arnold tongues of a real neuron are obtained using methods of circular statistics such as vector strength. Rayleigh statistical test and Monte Carlo method are necessary to detect and confirm phase-locking (in general mode-locking) in noisy data. These are done for a well-isolated inferior colliculus rabbit cell using acoustic stimuli in the rabbit\u27s ear canal and 2:1 mode-locked behavior of the cell is confirmed. Next a canonical model for Wilson-Cowan oscillators is derived by doing step by step mathematical analysis. Among these steps we applied some mathematical theorems in stability analysis and dynamical systems. We took advantage of normal form theory after Henri Poincare, to obtain a canonical model of a neural oscillator model for single and coupled populations. This canonical model exhibits all kinds of bifurcations. As an important case in theoretical neuroscience, we showed how we can simplify the model to obtain Hopf bifurcation. We obtained the governing equations of each oscillator under this bifurcation, which gave us the dynamical behavior and time evolution of each oscillator\u27s amplitude and phase. A novel and straightforward way is presented to solve the average relative phase of two coupled oscillators based on wave solutions applied in wave mechanics via mathematical physics. This approach gives us a complete analytic method to solve for the average relative phase equations of two coupled neurons or neural populations without applying averaging theory, which is lengthy and complicated. These solutions are important in physical sciences, including complex dynamical systems such as theoretical and computational neuroscience