Brain Rhythm Fluctuations: Envelope-Phase Modeling and Phase Synchronization
Creators: name: Powanwe, Arthur Sadrack
Publisher: name: Université d'Ottawa / University of Ottawa
Year: 2021
Subjects: Gamma oscillations
Gamma burst
Envelope-phase representation
Phase synchronization

description: Fast neural oscillations known as beta (12-30Hz) and gamma (30-100Hz) rhythms are recorded across several brain areas of various species. They have been linked to diverse functions like perception, attention, cognition, or interareal brain communication. The majority of the tasks performed by the brain involves communication between brain areas. To efficiently perform communication, mathematical models of brain activity require representing neural oscillations as sustained and coherent rhythms. However, some recordings show that fast oscillations are not sustained or coherent. Rather they are noisy and appear as short and random epochs of sustained activity called bursts. Therefore, modeling such noisy oscillations and investigating their ability to show interareal coherence and phase synchronization are important questions that need to be addressed. In this thesis, we propose theoretical models of noisy oscillations in the gamma and beta bands with the same properties as those observed in in \textit{vivo}. Such models should exhibit dynamic and statistical features of the data and support dynamic phase synchronization. We consider networks composed of excitatory and inhibitory populations. Noise is the result of the finite size effect of the system or the synaptic inputs. The associated dynamics of the Local Field Potentials (LFPs) are modeled as linear equations, sustained by additive and/or multiplicative noises. Such oscillatory LFPs are also known as noise-induced or quasi-cycles oscillations. The LFPs are better described using the envelope-phase representation. In this framework, a burst is defined as an epoch during which the envelope magnitude exceeds a given threshold. Fortunately, to the lowest order, the envelope dynamics are uncoupled from the phase dynamics for both additive and multiplicative noises. For additive noise, we derive the mean burst duration via a mean first passage time approach and uncover an optimal range of parameters for healthy rhythms. Multiplicative noise is shown theoretically to further synchronize neural activities and better explain pathologies with an excess of neural synchronization. We used the stochastic averaging method (SAM) as a theoretical tool to derive the envelope-phase equations. The SAM is extended to extract the envelope-phase equations of two coupled brain areas. The goal is to tackle the question of phase synchronization of noise-induced oscillations with application to interareal brain communication. The results show that noise and propagation delay are essential ingredients for dynamic phase synchronization of quasi-cycles. This suggests that the noisy oscillations recorded in \textit{vivo} and modeled here as quasi-cycles are good candidates for such neural communication. We further extend the use of the SAM to describe several coupled networks subject to white and colored noises across the Hopf bifurcation ie in both quasi-cycle and limit cycle regimes. This allows the description of multiple brain areas in the envelope-phase framework. The SAM constitutes an appropriate and flexible theoretical tool to describe a large class of stochastic oscillatory phenomena through the envelope-phase framework.