Intrinsic bursting neurons represent a common neuronal type that displays bursting patterns upon depolarization stimulation. These neurons can be described by a system of seven-dimensional equations, which pose a challenge for dynamical analysis. To overcome this limitation, we employed the projection reduction method to reduce the dimensionality of the model. Our approach demonstrated that the reduced model retained the inherent bursting characteristics of the original model. Following reduction, we investigated the bi-parameter bifurcation of the equilibrium point in the reduced model. Specifically, we analyzed the Bogdanov-Takens bifurcation that arises in the reduced system. Notably, the topological structure of the neuronal model near the bifurcation point can be effectively revealed with our proposed method. By leveraging the proposed projection reduction method, we could explore the bursting mechanism in the reduced Pospischil model with greater precision. Our approach offers an effective foundation for generating theories and hypotheses that can be tested experimentally. Furthermore, it enables links to be drawn between neuronal morphology and function, thereby facilitating a deeper understanding of the complex dynamical behaviors that underlie intrinsic bursting neurons.

Parkinson's disease is associated with bursting of the thalamic (TC) neuron, which receives the inhibitory synaptic current of the basal ganglia composed of multiple nuclei; deep brain stimulation (DBS) applied to the basal ganglia can eliminate the bursting to recover to the normal state. In this paper, the complex nonlinear dynamics for the appearance and disappearance of the bursting are obtained in a widely used theoretical model of a neuronal network. First, through a bifurcation analysis, isolated TC neurons exhibit paradoxical bursting induced from the resting state by enhanced inhibitory effect, which is different from the common view that the enhanced inhibitory effect should suppress the electrical behaviors. Second, the mechanism for the appearance of bursting is obtained by analyzing the electrical activities of the basal ganglia. The inhibitory synaptic current from the external segment of the globus pallidus (GPe) induces a reduced firing rate of the subthalamic nucleus (STN); then, an excitatory synaptic current from the STN induces the bursting behaviors of the GPe. The excitatory current of STN neurons and the inhibitory current of the GPe cause bursting behaviors of the internal segment of the globus pallidus (GPi), thus resulting in an enhanced inhibition from the GPi to the TC, which can induce the paradoxical bursting similar to the isolated TC neurons. Third, the cause for the disappearance of paradoxical bursting is acquired.The high frequency pulses of DBS induces enhanced firing activity of the STN and GPe neurons and enhanced inhibitory synaptic current from the GPe to the GPi, resulting in a reduced inhibitory effect from the GPi to the TC, which can eliminate the paradoxical bursting. Finally, the fast-slow dynamics of the paradoxical bursting of isolated TC neurons are acquired, which is related to the saddle-node and saddle-homoclinic orbit bifurcations of the fast subsystem of the TC neuron model. The results provide theoretical support for understanding the mechanism of Parkinson's disease and treatment methods such as DBS.

Bursting is a crucial form of firing in neurons, laden with substantial information. Studying it can aid in understanding the neural coding to identify human behavioral characteristics conducted by these neurons. However, the high-dimensionality of many neuron models imposes a difficult challenge in studying the generative mechanisms of bursting. On account of the high complexity and nonlinearity characteristic of these models, it becomes nearly impossible to theoretically study and analyze them. Thus, this paper proposed to address these issues by focusing on the midbrain dopamine neurons, serving as the central neuron model for the investigation of the bursting mechanisms and bifurcation behaviors exhibited by the neuron. In this study, we considered the dimensionality reduction of a high-dimensional neuronal model and analyzed the dynamical properties of the reduced system. To begin, for the original thirteen-dimensional model, using the correlation between variables, we reduced its dimensionality and obtained a simplified three-dimensional system. Then, we discussed the changing characteristics of the number of spikes within a burst by simultaneously varying two parameters. Finally, we studied the co-dimension-2 bifurcation in the reduced system and presented the bifurcation behavior near the Bogdanov-Takens bifurcation.