Brain rhythms in small and large networks of neurons
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I studied two neuronal networks, one small to investigate the interaction of brain rhythms and one large, to investigate the effects of multiple connectivity types on resonance in a target network. Theta (4 − 8 Hz) and gamma (30 − 80 Hz) rhythms are commonly associated with memory and learning. The precision of co-firing between neurons and incoming inputs is critical in these cognitive functions. To understand the interaction of the two rhythms, I considered a single model neuron with an inhibitory autapse and M- current, under forcing from gamma pulses and a sinusoidal current of theta frequency. The M-current has a long time constant (~90 ms) and generates resonance at theta frequencies. I found that this slow M-current contributes to the precise co-firing between the network and fast gamma pulses in the presence of a slow sinusoidal forcing. This current expands the range of phase-locking frequency to the gamma input, counteracts the slow theta forcing, and admits bistability in some parameter range. The effects of the M-current balancing the theta forcing are reduced if the sinusoidal current is faster than the theta frequency band. For these results I used averaging methods, geometric singular perturbation theory, and bifurcation analysis. Beta rhythms (10 − 30 Hz) are associated with motor functions; patients with Parkinson’s Disease display prominent pathological beta rhythms in the basal ganglia. Research has suggested that a sub-circuit of the basal ganglia, subthalamic nucleus- globus pallidus externus (STN-GPe), is a potential generator of beta rhythms. The anatomical structure of STN-GPe also suggests that it may act as an amplifier of incoming rhythms. I considered a model of this sub-circuit based on the work of Kumar et al. (2011) and studied the mechanism of its intrinsic oscillation and how it might amplify inputs from the striatum. Through parameter sweeps, I found that the network manifests a robust intrinsic beta oscillation, not changeable by moderate parameter variation. Surprisingly, this STN-GPe network only amplifies rhythms of or close to the intrinsic oscillatory frequency, regardless of three different connection structures simulated. However, introducing heterogeneity into the network can make the network amplify rhythms of a wide range of frequencies.