The beta oscillation conditions in a simplified basal ganglia network

A model description of beta oscillations in the external globus pallidus

The external globus pallidus (GPe), a subcortical nucleus located in the indirect pathway of the basal ganglia, is widely considered to have tight associations with abnormal beta oscillations (13-30 Hz) observed in Parkinson's disease (PD). Despite that many mechanisms have been put forward to explain the emergence of these beta oscillations, however, it is still unclear the functional contributions of the GPe, especially, whether the GPe itself can generate beta oscillations. To investigate the role played by the GPe in producing beta oscillations, we employ a well described firing rate model of the GPe neural population. Through extensive simulations, we find that the transmission delay within the GPe-GPe pathway contributes significantly to inducing beta oscillations, and the impacts of the time constant and connection strength of the GPe-GPe pathway on generating beta oscillations are non-negligible. Moreover, the GPe firing patterns can be significantly modulated by the time constant and connection strength of the GPe-GPe pathway, as well as the transmission delay within the GPe-GPe pathway. Interestingly, both increasing and decreasing the transmission delay can push the GPe firing pattern from beta oscillations to other firing patterns, including oscillation and non-oscillation firing patterns. These findings suggest that if the transmission delays within the GPe are at least 9.8 ms, beta oscillations can be produced originally in the GPe neural population, which also may be the origin of PD-related beta oscillations and should be regarded as a promising target for treatments for PD.

The mechanism of Parkinson oscillation in the cortex: Possible evidence in a feedback model projecting from the globus pallidus to the cortex

The origin, location and cause of Parkinson's oscillation are not clear at present. In this paper, we establish a new cortex-basal ganglia model to study the origin mechanism of Parkinson beta oscillation. Unlike many previous models, this model includes two direct inhibitory projections from the globus pallidus external (GPe) segment to the cortex. We first obtain the critical calculation formula of Parkinson's oscillation by using the method of Quasilinear analysis. Different from previous studies, the formula obtained in this paper can include the self-feedback connection of GPe. Then, we use the bifurcation analysis method to systematically explain the influence of some key parameters on the oscillation. We find that the bifurcation principle of different cortical nuclei is different. In general, the increase of the discharge capacity of the nuclei will cause oscillation. In some special cases, the sharp reduction of the discharge rate of the nuclei will also cause oscillation. The direction of bifurcation simulation is consistent with the critical condition curve. Finally, we discuss the characteristics of oscillation amplitude. At the beginning of the oscillation, the amplitude is relatively small; with the evolution of oscillation, the amplitude will gradually strengthen. This is consistent with the experimental phenomenon. In most cases, the amplitude of cortical inhibitory nuclei (CIN) is greater than that of cortical excitatory nuclei (CEX), and the two direct inhibitory projections feedback from GPe can significantly reduce the amplitude gap between them. We calculate the main frequency of the oscillation generated in this model, which basically falls between 13 and 30 Hz, belonging to the typical beta frequency band oscillation. Some new results obtained in this paper can help to better understand the origin mechanism of Parkinson's disease and have guiding significance for the development of experiments.

Interaction of Indirect and Hyperdirect Pathways on Synchrony and Tremor-Related Oscillation in the Basal Ganglia

Low-frequency oscillatory activity (3-9 Hz) and increased synchrony in the basal ganglia (BG) are recognized to be crucial for Parkinsonian tremor. However, the dynamical mechanism underlying the tremor-related oscillations still remains unknown. In this paper, the roles of the indirect and hyperdirect pathways on synchronization and tremor-related oscillations are considered based on a modified Hodgkin-Huxley model. Firstly, the effects of indirect and hyperdirect pathways are analysed individually, which show that increased striatal activity to the globus pallidus external (GPe) or strong cortical gamma input to the subthalamic nucleus (STN) is sufficient to promote synchrony and tremor-related oscillations in the BG network. Then, the mutual effects of both pathways are analysed by adjusting the related currents simultaneously. Our results suggest that synchrony and tremor-related oscillations would be strengthened if the current of these two paths are in relative imbalance. And the network tends to be less synchronized and less tremulous when the frequency of cortical input is in the theta band. These findings may provide novel treatments in the cortex and striatum to alleviate symptoms of tremor in Parkinson's disease.

Emergence of Beta Oscillations of a Resonance Model for Parkinson's Disease

In Parkinson's disease, the excess of beta oscillations in cortical-basal ganglia (BG) circuits has been correlated with normal movement suppression. In this paper, a physiologically based resonance model, generalizing an earlier model of the STN-GPe circuit, is employed to analyze critical dynamics of the occurrence of beta oscillations, which correspond to Hopf bifurcation. With the experimentally measured parameters, conditions for the occurrence of Hopf bifurcation with time delay are deduced by means of linear stability analysis, center manifold theorem, and normal form analysis. It is found that beta oscillations can be induced by increasing synaptic transmission delay. Furthermore, it is revealed that the oscillations originate from interaction among different synaptic connections. Our analytical results are consistent with the previous experimental and simulating findings, thus may provide a more systematic insight into the mechanisms underlying the transient beta bursts.

Bipolar oscillations between positive and negative mood states in a computational model of Basal Ganglia

Bipolar disorder is characterized by mood swings-oscillations between manic and depressive states. The swings (oscillations) mark the length of an episode in a patient's mood cycle (period), and can vary from hours to years. The proposed modeling study uses decision making framework to investigate the role of basal ganglia network in generating bipolar oscillations. In this model, the basal ganglia system performs a two-arm bandit task in which one of the arms (action responses) leads to a positive outcome, while the other leads to a negative outcome. We explore the dynamics of key reward and risk related parameters in the system while the model agent receives various outcomes. Particularly, we study the system using a model that represents the fast dynamics of decision making, and a module to capture the slow dynamics that describe the variation of some meta-parameters of fast dynamics over long time scales. The model is cast at three levels of abstraction: (1) a two-dimensional dynamical system model, that is a simple two variable model capable of showing bistability for rewarding and punitive outcomes; (2) a phenomenological basal ganglia model, to extend the implications from the reduced model to a cortico-basal ganglia setup; (3) a detailed network model of basal ganglia, that incorporates detailed cellular level models for a more realistic understanding. In healthy conditions, the model chooses positive action and avoids negative one, whereas under bipolar conditions, the model exhibits slow oscillations in its choice of positive or negative outcomes, reminiscent of bipolar oscillations. Phase-plane analyses on the simple reduced dynamical system with two variables reveal the essential parameters that generate pathological 'bipolar-like' oscillations. Phenomenological and network models of the basal ganglia extend that logic, and interpret bipolar oscillations in terms of the activity of dopaminergic and serotonergic projections on the cortico-basal ganglia network dynamics. The network's dysfunction, specifically in terms of reward and risk sensitivity, is shown to be responsible for the pathological bipolar oscillations. The study proposes a computational model that explores the effects of impaired serotonergic neuromodulation on the dynamics of the cortico basal ganglia network, and relates this impairment to abstract mood states (manic and depressive episodes) and oscillations of bipolar disorder.