Dynamical Analyses on Beta Oscillations in a STN-GPE-GPI Model of Parkinson’s Disease

School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China Inner Mongolia Key Laboratory for Economic Data Analysis and Mining, Hohhot 010070, China School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China State Key Laboratory of Reproductive Regulation and Breeding of Grassland Livestock, College of Life Sciences, Inner Mongolia University, Hohhot 010021, China


Introduction
Parkinson's disease is a chronic neurodegenerative disease with the symptoms of involuntary tremor of the hand and head, muscle rigidity, slow movement, and imbalance of posture [1]. e main pathological causes of Parkinson's disease ascribe to the loss of dopamine neurons in the basal ganglia [2], which consists of the striatum, globus pallidus (GP), subthalamic nucleus (STN), compacta (SNc), and reticular (SNr) structure of the substantia nigra [3][4][5]. e loss of dopamine neurons causes beta oscillations with frequencies ranging from 13 Hz to 30 Hz in the basal ganglia [6,7]. erefore, it is necessary for understanding the mechanism of Parkinson's disease to analyze the conditions of beta oscillations in the basal ganglia [8].
Many researches explored the origin of beta oscillations in the basal ganglia [9][10][11][12]. Van Albada et al. believed that oscillations originated from the cortical-thalamic loop and then spread to the basal ganglia with the development of the disease [13]. Holgado et al. found that the STN-GP loop in the basal ganglia plays an important role in generating oscillations, which are related to connection weight and synaptic transmission time between STN and GP [14]. Furthermore, a model with two STN and one GP populations are considered to get the stability boundary of oscillations [15,16]. However, GP population has not been divided into the external and internal segment (GPe and GPi) in the above models, where GPi is the main output structure of the basal ganglia and also used to treat dystonia by deep brain electrical stimulation in medicine [5]. Actually, GPi and GPe are affected by different intensities from striatum to result in Parkinson's disease. erefore, it is necessary to add both GPe and GPi into the basal ganglia network.
Based on the above considerations, we introduce both GPe and GPi in the above model proposed by Holgado et al. as a new STN-GPe-GPi model. e mechanism of generating beta oscillations for the new model is explored through theoretical analyses and numerical simulation. e model of STN-GPe-GPi loop is given in Section 2. Section 3 shows the results. Stability analyses and bifurcation for this model are given through theory analyses. Also, the effect of inputs from the cortex and striatum and some parameters related to GPi on beta oscillations is discussed in Section 4. Finally, the conclusion is given in Section 5.

Model
e model of STN-GPe-GPi loop，which is an extended STN-GP model, is considered here in order to understand the mechanism of Parkinson's disease. Actually, GP can be divided into two parts, GPe and GPi. ey can receive different excitatory inputs from the cortex in Parkinson's disease, and GPi as the output part of the basal ganglia mainly affects the activity of neurons in thalamic and cortical areas. GPe and GPi receive inhibitory and excitatory stimulations from striatum and STN, respectively, but GPe sends inhibitory signal to GPi and STN. Besides, STN also receives excitatory stimulation from the cortex (see Figure 1). e following firing rate equations are used to describe the dynamical behavior of the model, as shown in Figure 1 [ where G 1 (t), G 2 (t), and S(t) represent the firing rate of GPi，GPe, and STN; τ G and τ S are time constants of GP and STN; W and T are synaptic connection weight and the delay of signal transmission between neural populations, respectively, W ij and T ij are connection weight and time delay between the neural populations i and j, W SGE , W SGI , T SGE , and T SGI denote connection weights and the time delays from STN to GPe and GPi, respectively, Str and Ctx are input constants from the striatum and cortex. F S and F G are activation functions of STN and GP, which are given by the following formulas [14]: where M i is the maximum firing rate of neuron population i and B i is the firing rate of neuron population i with no input. e activation functions F S (x) and F G (x) with their derivatives are shown in Figures 2(a) and 2(b), respectively. e parameters and their source are given in Table 1, and the connection weights between neuron populations in healthy and disease states are given in Table 2. Figure 3 shows time series of the firing rate of healthy and disease states, which oscillates for Parkinson's disease and reaches the steady state in the healthy state.

Stability Analyses for the STN-GPE-GPI Model.
In this section, we obtain the following equations (5)-(7) by simplifying equations (1)-(3) with F(x) � x without inputs from the striatum and cortex and the identical τ and T: Equations (5)-(7) are given in matrix form as follows: Figure 1: Schematic diagram of STN-GPe-GPi. Each rectangular represents neural population, arrows indicate excitatory inputs, lines ending with black dots mean inhibitory inputs, and the striatum and cortex represent their inputs, respectively.

Complexity
Let Using Laplace transform [30],   Complexity 3 Equation (8) can be given as follows: Without loss of generality, let en, where S is the eigenvalue of the characteristic equation and I is the unit matrix. Taking the determinant on both sides of matrix (13), then Let s � iλ and e −2sT be expanded by Euler transformation, and equation (16) is changed into For simplification, let τ � 1 and T � T/τ，equation (17) can be written as follows： Let the real part and the imaginary part be zero, respectively:  Complexity Add the squares of (19) and (20) to get the following equation: Bring equation (21) into equation (19), and we get stability boundary of the linear system： So, the linear model equations (5)-(7) oscillate at the following condition:

Stability Analysis for the Nonlinear Model.
For the nonlinear model with activation functions, we linearize the activation functions at the steady state (G * 1 , G * 2 , S * ) and get the characteristic equation (24) in a matrix form based on the steps in Section 3.1.
where erefore, the stability boundary of the nonlinear model is as follows: e oscillation condition is We analyze the stability of the model shown in Figure 1 and get the stability conditions equations (22) and (26) for the linear and nonlinear model, respectively. Next, we draw the stability boundary curves of equations (22) and (26) and explore the effect of the inputs from the striatum and cortex on oscillation through numerical simulation. Figures 4(a) and 4(b) describe the stability boundary curves of linear and nonlinear models based on equations (22) and (26), respectively. e decreasing boundary curve is infinitely close to x-axis with the increase of W GS W SGE ⟶ ∞. e system oscillates for parameter values above the curve while it is stable for ones below the curve. However, the nonlinear system oscillate for a larger weight W GS W SGE than the one of linear system at the same T/τ. Figures 4(c) and 4(d), respectively, show the influence of the inputs from the cortex and striatum on the stability of the nonlinear system. e boundary curve of the cortex is in the shape of "U," where the system will oscillate in the area of above "U" and it will reach a stable steady state in the area below "U." While the boundary curve of the striatum decreases to x-axis with the increase of the input of the striatum.

The Effect of Parameters Related to GPI on Frequency and Amplitude of Nonlinear Model Oscillation
In this section, the effect of three groups of parameters related to GPi, W GG − T GG , W SGI − T SGI , and W SGE − W SGI , on oscillation frequency ( Figures 5 (a 1 )-(a 3 )) and amplitude ( Figures 5 (b 1 )-(b 3 )) are also considered. Besides, time series of the GPI firing rate are given in Figure 6 for typical parameter values in each group of parameters in order to clearly see the  Table 2. 6 Complexity influence of these parameters on the frequency and amplitude of GPi oscillation. We set the frequency of the stable steady state to be 0. As can be seen from Figures 5 (a 1 ) and (b 1 ), the plane is divided into two parts, where the firing rate of GPi reaches a steady state for parameters taken in the blue area of the upper half while it will oscillate for parameters taken in the red area of the lower half. Frequency of oscillation with 14 Hz almost is not affected by W GG and T GG , while smaller W GG and larger T GG increase the amplitude of oscillation to 180. Besides, Figure 5 (a 3 ) shows that the firing rate of GPi reaches the steady state for smaller W SGE (W SGE < 3.0) while it oscillates for large W SGE (W SGE > 3.0), where the frequency and amplitude increase for smaller W SGE and larger W SGI . Furthermore, time series of the GPi firing rate are given in Figure 6 for three pairs of parameters from each group of parameter plane in Figure 5 to clearly see the influence of these parameters on the frequency and amplitude of the GPi firing rate. It can be seen from Figures 6 (a 1 )-(a 3 ) that the firing rate of GPi oscillates for smaller W GG and larger T GG .
According to Figures 6 (b 1 )-(b 3 ), the firing rate of GPi always oscillates with smaller amplitude and larger frequency for smaller W SGI and T SGI while the case is opposite for larger W SGI and T SGI . Figures 6 (c 1 )-(c 3 ) show that larger W SGE makes the firing rate of GPi oscillate and larger W SGI increases the amplitude of oscillation.
To sum up, the connection weights between neurons have a great influence on oscillation of the GPi firing rate, where smaller W GG and larger W SGE , W SGI easily induce the oscillation of the GPi firing rate. However, transmission delays between neurons affect the amplitude of the GPi firing rate, where larger T GG and smaller T SGI will increase the amplitude.

Conclusions
Analyzing the conditions of beta oscillation in the basal ganglia is helpful to understand the mechanism of Parkinson's disease. In this paper, we analyze the conditions of beta oscillation in the STN-GPe-GPi model for different cases. First, stability analyses give stability boundary condition equations (22) and (26)   Figures 4(a) and 4(b). e nonlinear system oscillates for larger connection weight than the one of the linear system due to activation function. In addition, we consider the influence of the cortex and striatum as external input of the STN-GPe-GPi loop on the stability boundary of the nonlinear system. As can be seen from Figures 4(c) and 4(d), the nonlinear system is in the state of oscillation for 100 < ctx < 400 or 5 < str < 25. Comparing with the results in [15], the stability boundary system in this model moves to the right and other results almost are consistent with the ones in [15]. Furthermore, it can be seen from numerical simulation that the influence of the connection weights and delays is related to GPi on its oscillation. Smaller connection weight from GPe to GPi (W GG ) and larger one from STN to GPe (W SGE ) make the system oscillate easily, regardless of connection weight (W SGI ) and transmission delay (T SGI ) from STN to GPi. We hope that the results may provide guidance for the therapy of reducing pathological oscillations of PD, especially for the operation of the region related to GPi. However, it is necessary to consider a more complete neural network related to Parkinson's disease and explore the conditions of beta oscillation in response to different time delay, noise, and temperature [31]. Furthermore, we will investigate the pathogenesis of Parkinson's disease from the perspective of systems' biology in the future [32].

Data Availability
e data used to support the findings of the study can be obtained from the link https://pan.baidu.com/s/ 1HCTdFTukmkp29_kushR6Bg (password: hypg) and from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.