Computational study of associations between the synaptic conductance of STN and GPe and the development of Parkinson’s disease

Abstract

There is evidence that the subthalamic nucleus (STN) and globus pallidus pars externa (GPe) involve in the development of Parkinson’s disease, a neurodegenerative disorder characterized by motor and non-motor symptoms and loss of dopaminergic neurons in which the error index (EI) in firing patterns is widely used to address the related issues. Whether and how this interaction mechanism of STN and GPe affects EI in Parkinson’s disease is uncertain. To account for this, we propose a kind of basal ganglia-thalamic network model associated with Parkinson’s disease coupled with neurons, and investigate the effect of synaptic conductance of STN and GPe on EI in this network, as well as their internal relationship under EI as an index. The results show a relationship like a piecewise function between the error index and the slope of the state transition function of synaptic conductance from STN to GPe ($g_{\mathit{snge}}$) and from GPe to STN ($g_{\mathit{gesn}}$). And there is an approximate negative correlation between EI and $g_{\mathit{gesn}}$. Increasing $g_{\mathit{snge}}$ and decreasing $g_{\mathit{gesn}}$ can improve the fidelity of thalamus information transmission and alleviate Parkinson’s disease effectively. These obtained results can give some theoretical evidence that the abnormal synaptic releases of STN and GPe may be the symptoms of the development of Parkinson’s disease, and further enrich the understanding of the pathogenesis and treatment mechanism of Parkinson’s disease.

Introduction

Parkinson’s disease (PD) is the second-most neurodegenerative disorder which has tremor, rigidity, slowness of movement, loss of balance, difficulty in speaking and writing, psychological disturbance, and other motor and non-motor symptoms (Balestrino and Schapira 2020; Armstrong and Okun 2020). Apart from pharmacological treatments (Connolly and Lang 2014), various surgical interventions have shown efficacy in terms of therapeutic effects (Armstrong and Okun 2020; Perlmutter and Mink 2006). However, the understanding of these pathogenesis and treatment mechanisms is still relatively lacking. Researches in these areas are even more important than the therapy itself.

Pathologically, Parkinson’s disease is characterized by the loss of dopaminergic neurons in the substantia nigra pars compacta and the accumulation of misfolded α-synuclein (Poewe et al. 2017). In electrophysiological characteristics, it mainly shows that the thalamus (TH) responds incorrectly to the signal transmission of the sensorimotor cortex. Thus, selecting an index that can effectively reflect the correctness of neural activity in the basal ganglia-thalamic circuit is crucial for subsequent research. Rubin and Terman (2004) proposed to quantify the fidelity of the thalamus response to sensorimotor cortical information input by calculating the error rate of two types of errors (false positives and misses), and to show whether the state is Parkinson’s disease and its degree. So et al. (2012) divided the error into three types: miss, burst, and spurious. With the aid of the new error index, they investigated the therapeutic effect of deep brain stimulation (DBS) with different proportions of local cells and passing fibers in Parkinson’s disease. Subsequent researchers also recognized and used this indicator many times as a basis for measuring the condition of Parkinson’s disease to conduct various researches (Fan et al. 2016).

Current research generally believes that the subthalamic nucleus and the globus pallidus pars externa are two important nuclei in the basal ganglia-thalamic circuit, which have a major impact on the formation and treatment of Parkinson’s disease. Based on anatomical and electrophysiological studies, Holgado et al. (2010) built an STN-GPe network to verify that the interaction of STN and GPe caused the abnormal oscillation of basal ganglia with frequencies in the beta band (13–30 Hz) and pointed out several necessary conditions that the range of some parameters like synaptic connections between STN and GPe should satisfy to form this phenomenon. Shouno et al. (2017) thought that the recurrent subthalamo-pallidal circuit generates and maintains the Parkinsonian abnormal oscillations. They developed a spiking neuron model of the STN-GPe circuit by incorporating electrophysiological properties of neurons and synapses, and identified regions in the space of the intrinsic excitability of GPe neurons and synaptic strength from the GPe to the STN that reproduce normal and Parkinsonian states. Wang et al. (2023) provided a subthalamopallidal network containing two classes of GPe neurons (PV, Lhx6 GPe). They confirmed that the STN neurons can stabilize the spiking sequence of GPe neurons and inhibit abnormal synchronous oscillations both in control and pathological conditions. Besides, the synaptic coupling in heterogenous pallidal and STN-GPe affect the propagation of abnormal rhythms in pallidal and subthalamopallidal networks, respectively. Fan et al. (2012) found that the loss of dopamine increases the number of synaptic connections at each STN-GPe axon terminal, leads to a marked enhancement of the GPe-STN pathway, then, produces Parkinson’s abnormal firing patterns. Moran et al. (2011) discovered that chronic dopamine depletion reorganized the cortico-basal ganglia-thalamocortical circuit. It increased the number of effective connections from cortex to STN and decreased connectivity from STN to GPe. The above studies have determined that the changes in material generation and transfer, firing patterns and modes, and connection relationship especially synaptic form between STN and GPe lead to the generation of Parkinson’s disease. These effects will manifest in the form of changes in synaptic connections such as synaptic conductance. However, in the modeling process of some of the previous classic studies (Rubin and Terman 2004; So et al. 2012), some parameters such as synaptic conductance were set as fixed constants in the healthy state and Parkinson’s state, which did not reflect the relationship and influence between them. This is inconsistent with the reality and current research understanding, and may have a greater impact on the research results.

This paper will improve the BG (basal ganglia-thalamic) network model and introduce the error index to study the bidirectional relationship between the synaptic conductance of STN and GPe and its influences on the formation and treatment of Parkinson’s disease. These findings will enrich the understanding of the pathogenesis and treatment mechanism, and provide a theoretical reference for subsequence research and treatment.

Model and methods

Network model

Parkinson’s disease is a neurodegenerative disorder caused by the degeneration of dopaminergic neurons in the substantia nigra pars compacta (SNc) (Beuter and Vasilakos 1995). The SNc is a key part of the basal ganglia-thalamocortical motor circuit which is closely related to pathophysiological movement disorders (Obeso et al. 2000). In the basal ganglia-thalamocortical motor circuit (see Fig. 1), the basal ganglia is mainly composed of the subthalamic nucleus (STN), globus pallidus pars externa (GPe), globus pallidus pars interna (GPi), the striatum, the substantia nigra pars compacta and the reticular (SNr). The striatum and STN are the main input nuclei for the basal ganglia to receive information from the cortex, and the main output nuclei GPi and SNr are considered as the same structure because of their similar connections and cytological functions (Bar-Gad et al. 2003). The damage of the SNc first results in a decrease in dopamine, which can regulate the information flow received by the subcortical structure from the cortex through direct and indirect pathways. The depletion of dopamine finally influences the function of synapses from GPi to Thalamus, and furtherly affects the ability of the thalamus to relay information from the cerebral cortex.

Fig. 1.

Basal ganglia-thalamocortical model, modified from Obeso et al. (2000). It consists of the cerebral cortex, basal ganglia, and thalamus. Excitatory connections are indicated by red arrow lines, and inhibitory connections by blue round-headed lines. The striatum can be connected with GPi/SNr through the direct pathway on the right side, or the indirect pathway composed of GPe and STN on the left side

This paper establishes a computational model based on the improved basal ganglia-thalamic network proposed by So et al. (2012). By improving the RT model proposed by Rubin and Terman (2004), the model takes the sparse regular connection of each nuclei in the basal ganglia-thalamic network as the main body. These projections are chosen in according with the topographical organization and the convergence and divergence of synaptic connectivity within the basal ganglia (Smith et al. 1998). It can reflect the realistic electrophysiological characteristics of each nuclei under Parkinsonian states, healthy states, and some treatment methods. As Fig. 2 shows, in four neuronal population (STN/GPe/GPi/TH) with the same number scale, each STN cell takes excitatory projects to the neighboring 2 GPe cells and 2 GPi cells; each GPe cell takes inhibitory connections to neighboring 2 GPe cells (one on the left and one after next to the right), 2 GPi cells (skip 2 nearest cells) and 2 nearest STN cells; each GPi cell sends inhibitory project to immediate neighboring 1 TH cell. The connectivity between neuronal populations in this paper has a periodic structure that same as the reference model (Terman et al. 2002), that is, the front cell on the left and the rear cell on the right are neighbors.

Fig. 2.

The sparse connection model of each nuclei in the basal ganglia-thalamic network, adapted from the RT model and BG network model. Thalamus, subthalamic nucleus, globus pallidus pars interna and externa with the same number scale form the network through sparse connections

In this model, the projected input currents from other brain regions to these three nuclei (STN, GPe, and GPi) are summed into a positive constant bias current ${\mathit{Iapp}}_{\text{STN}/\text{GPe}/\text{GPi}}$ (So et al. 2012) to maintain three nuclei to fire with the frequency of 10/70/80 Hz under the normal condition (Steigerwald et al. 2008; Boraud et al. 1998).In addition, the firing pattern of Parkinson’s disease is caused by changes in the input of the basal ganglia. Changing the size of $I_{\mathit{app}}$ can simulate the transition of the BG network between the Parkinson’s state and the healthy state. Decreasing the positive constant bias current $I_{\mathit{app}}$ applied to three neuronal population lead the condition from health to Parkinson’s disease. STN, GPe, and GPi show the phenomenon of high-frequency synchronous oscillation. This result is consistent with what is observed in humans with PD (Levy et al. 2002), as well as in dopamine-depleted rodents (Wilson et al. 2006) and monkeys (Bergman et al. 1998). On the contrary, the state changes from Parkinson’s condition to health by increasing the applied current $I_{\mathit{app}}$. To keep the consistency of behavior and theoretical guidance, this paper takes Parkinson's disease as the initial state for research.

Each neuron type

In the BG network model, dynamical behaviors such as the change of membrane potential of four types of neurons can be simulated by the conductance-based Hodgkin-Huxley neurons model. The canonical form can be written as follows,

$$C_m{V}^{\prime }=-\sum I_{\mathit{ion}}-I_{\mathit{syn}}+I_{\mathit{app}}/I_{\mathit{SMC}},$$ 1

where $C_m=1$ μF/cm² is the membrane capacitance for all cell models, $\sum I_{\mathit{ion}}$ represents the total ionic currents, $I_{\mathit{syn}}$ is the synaptic currents, $I_{\mathit{app}}$ is the applied current used in STN/GPe/GPi models, $I_{\mathit{SMC}}$ is the sensorimotor cortex input current used in the TH model.

In the TH model,

$$\sum I_{\mathit{ion}}=I_L+I_{\mathit{Na}}+I_K+I_T.$$ 2

In STN/GPe/GPi models,

$$\sum I_{\mathit{ion}}=I_L+I_{\mathit{Na}}+I_K+I_T+I_{\mathit{Ca}}+I_{\mathit{ahp}},$$ 3

where $I_L$ is the leak current, $I_{\mathit{Na}}$ is the sodium current, $I_K$ is the potassium current, $I_T$ is the low-threshold T-type calcium current, $I_{\mathit{Ca}}$ is the high-threshold calcium current, $I_{\mathit{ahp}}$ is the calcium-activated after hyperpolarization potassium current. The detailed specific equations and parameters for the above model are provided in the following tables (see Table 1, 2, 3 and 4).

Table 1.

TH cell model equations and parameters

Current	Equation	Gating variables		Parameters
$I_L$	$g_L\left(v-E_L\right)$			$g_L=0.05$ $E_L=-70$
$I_{\mathit{Na}}$	$g_{\mathit{Na}}{m}_{\infty }{\left(v\right)}^{3}h\left(v-E_{\mathit{Na}}\right)$	$m_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+37}{7}\right)\right]$	${\text{h}}_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+41}{4}\right)\right]$	$g_{\mathit{Na}}=3$ $E_{\mathit{Na}}=50$
		${\tau }_h\left(v\right)=1/\left[0.128\text{exp}\left(-\frac{v+46}{18}\right)+4/\left(1+\text{exp}\left(-\frac{v+23}{5}\right)\right)\right]$
$I_K$	$g_{K}0.75{\left(1-h\right)}^4\left(v-E_K\right)$	*same $\text{h}$ as in $I_{\mathit{Na}}$		$g_K=5$ $E_K=-75$
$I_T$	$g_T{p}_{\infty }{\left(v\right)}^{2}r\left(v-E_T\right)$	$p_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+60}{6.20}\right)\right]$	$r_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+84}{4}\right)\right]$	$g_T=5$ $E_T=0$
			${\tau }_r\left(v\right)=0.15\left[28+\text{exp}\left(-\frac{v+25}{10.50}\right)\right]$

Table 2.

STN cell model equations and parameters

Current	Equation	Gating variables		Parameters
$I_L$	$g_L\left(v-E_L\right)$			$g_L=2.25$ $E_L=-65$
$I_{\mathit{Na}}$	$g_{\mathit{Na}}{m}_{\infty }{\left(v\right)}^{3}h\left(v-E_{\mathit{Na}}\right)$	$m_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+30}{15}\right)\right]$	$h_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+39}{3.10}\right)\right]$	$g_{\mathit{Na}}=37$ $E_{\mathit{Na}}=55$
		${\tau }_h\left(v\right)=1+500/\left[1+\text{exp}\left(-\frac{v+57}{-3}\right)\right]$
$I_K$	$g_K{n}^4\left(v-E_K\right)$	$n_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+32}{8}\right)\right]$		$g_K=45$ $E_K=-80$
		${\tau }_n\left(v\right)=1+100/\left[1+\text{exp}\left(-\frac{v+80}{-26}\right)\right]$
$I_T$	$g_T{a}_{\infty }{\left(v\right)}^3{b}_{\infty }\left(r\right)\left(v-E_T\right)$	$a_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+63}{7.80}\right)\right]$	$r_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+67}{2}\right)\right]$	$g_T=0.50$ $E_T=0$
		$b_{\infty }\left(r\right)=1/\left[1+exp\left(-\frac{r-0.40}{0.10}\right)\right]$$-\left[1/\left[1+\text{exp}\left(4\right)\right]\right]$	${\tau }_r\left(v\right)=7.1+17.5/\left[1+\text{exp}\left(-\frac{v+68}{-2.20}\right)\right]$
$I_{\mathit{Ca}}$	$g_{\mathit{Ca}}{c}^2\left(v-E_{\mathit{Ca}}\right)$	$c_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+20}{8}\right)\right]$		$g_{\mathit{Ca}}=2$ $E_{\mathit{Ca}}=140$
		${\tau }_c\left(v\right)=1+10/\left[1+\text{exp}\left(\frac{v+80}{26}\right)\right]$
$I_{\mathit{ahp}}$	$g_{\mathit{ahp}}\left(v-E_{\mathit{ahp}}\right)\left(\frac{\mathit{CA}}{CA+15}\right)$			$g_{\mathit{ahp}}=20$ $E_{\mathit{ahp}}=-80$

Table 3.

GP cell model equations and parameters

Current	Equation	Gating variables		Parameters
$I_L$	$g_L\left(v-E_L\right)$			$g_L=0.10$ $E_L=-65$
$I_{\mathit{Na}}$	$g_{\mathit{Na}}{m}_{\infty }{\left(v\right)}^{3}h\left(v-E_{\mathit{Na}}\right)$	$m_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+37}{10}\right)\right]$	$h_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+58}{12}\right)\right]$	$g_{\mathit{Na}}=120$ $E_{\mathit{Na}}=55$
		${\tau }_h\left(v\right)=0.05+0.27/\left[1+\text{exp}\left(-\frac{v+40}{-12}\right)\right]$
$I_K$	$g_K{n}^4\left(v-E_K\right)$	$n_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+50}{14}\right)\right]$		$g_K=30$ $E_K=-80$
		${\tau }_n\left(v\right)=0.05+0.27/\left[1+\text{exp}\left(-\frac{v+40}{-12}\right)\right]$
$I_T$	$g_T{a}_{\infty }{\left(v\right)}^{3}r\left(v-E_T\right)$	$a_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+57}{2}\right)\right]$	$r_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(\frac{v+70}{2}\right)\right]$	$g_T=0.50$ $E_T=0$
$I_{\mathit{Ca}}$	$g_{\mathit{Ca}}{s}_{\infty }{\left(v\right)}^3\left(v-E_{\mathit{Ca}}\right)$	$s_{\infty }\left(v\right)=1/\left[1+\text{exp}\left(-\frac{v+35}{2}\right)\right]$		$g_{\mathit{Ca}}=0.15$ $E_{\mathit{Ca}}=120$
$I_{\mathit{ahp}}$	$g_{\mathit{ahp}}\left(v-E_{\mathit{ahp}}\right)\left(\frac{\mathit{CA}}{CA+10}\right)$			$g_{\mathit{ahp}}=10$ $E_{\mathit{ahp}}=-80$

Table 4.

Applied currents to basal ganglia under healthy and Parkinsonian conditions

Conditions	$I_{\mathit{app}}$ for STN	$I_{\mathit{app}}$ for GPe	$I_{\mathit{app}}$ for GPi
Healthy	33 μA/cm2	20 μA/cm2	21 μA/cm2
Parkinsonian	23 μA/cm2	7 μA/cm2	15 μA/cm2

Membrane potentials (v) of the TH cells are governed by the equations:

$$C_{m}V\prime =-I_L-I_{\mathit{Na}}-I_K-I_T-I_{GPi\to TH}+I_{\mathit{SMC}}$$

$$h\prime =\left[h_{\infty },\left(v\right),-,h\right]/{\tau }_h\left(v\right)$$

$$r\prime =\left[r_{\infty },\left(v\right),-,r\right]/{\tau }_r\left(v\right)$$

Membrane potentials (v) of the STN cells are governed by the equations:

$$C_{m}V\prime =-I_L-I_{\mathit{Na}}-I_K-I_T-I_{\mathit{Ca}}-I_{\mathit{ahp}}-I_{GPe\to STN}+I_{\mathit{app}}$$

$$h\prime =0.75\left[h_{\infty },\left(v\right),-,h\right]/{\tau }_h\left(v\right)$$

$$n\prime =0.75\left[n_{\infty },\left(v\right),-,n\right]/{\tau }_n\left(v\right)$$

$$r\prime =0.20\left[r_{\infty },\left(v\right),-,r\right]/{\tau }_r\left(v\right)$$

$$c\prime =0.08\left[c_{\infty },\left(v\right),-,c\right]/{\tau }_c\left(v\right)$$

$$\mathit{CA}\prime =3.75\times {10}^{-5}\left(-,I_T,-,I_{\mathit{Ca}},-,22.50,\times ,C,A\right)$$

GPe and GPi cells are modeled similarly. Membrane potentials (v) of the GP cells are governed by the equations:

$$C_{m}V\prime =-I_L-I_{\mathit{Na}}-I_K-I_T-I_{\mathit{Ca}}-I_{\mathit{ahp}}-I_{STN\to GP}-I_{GPe\to GPe/GPi}+I_{\mathit{app}}$$

$$h\prime =0.05\left[h_{\infty },\left(v\right),-,h\right]/{\tau }_h\left(v\right)$$

$$n\prime =0.10\left[n_{\infty },\left(v\right),-,n\right]/{\tau }_n\left(v\right)$$

$$r\prime =\left[r_{\infty },\left(v\right),-,r\right]/30$$

$$\mathit{CA}\prime =1\times {10}^{-4}\left(-,I_T,-,I_{\mathit{Ca}},-,15,\times ,C,A\right)$$

The sensorimotor cortex input current $I_{\mathit{SMC}}$ is modeled as a series of monophasic current pulses with an amplitude of 3.5 μA/cm² and a duration of 5 ms. It can be represented by:

$$I_{\mathit{SMC}}=i_{\mathit{SMC}}H\left(sin\left(2\pi t/{\rho }_{\mathit{SMC}}\right)\right)\times \left[1-H\left(sin\left(2\pi \left(t+{\delta }_{\mathit{SMC}}\right)/{\rho }_{\mathit{SMC}}\right)\right)\right],$$ 4

where $i_{\mathit{SMC}}=3.50$ μA/cm² is the amplitude of current input, H is a Heaviside bi-value step function (i.e. $H\left(x\right)=1$ if $x>0$ and $H\left(x\right)=0$ if $x\le 0$), ${\rho }_{\mathit{SMC}}$ is the period of $I_{\mathit{SMC}}$, ${\delta }_{\mathit{SMC}}=5$ ms is the duration of every positive current input. To mimic the natural irregular signal input from the cortex to the thalamus, the instantaneous frequencies of the current pulse satisfy a gamma distribution with an average rate (AR) of 14 Hz and a coefficient of variation (CV) of 0.2. That is,

$${\rho }_{\mathit{SMC}}=1000/\mathrm{\Gamma }\left(A,B\right),$$ 5

where $A=1/{\mathit{CV}}^2$, $B=AR/A$, $\mathrm{\Gamma }$ represents the gamma distribution function, and ${\rho }_{\mathit{SMC}}$ has the unit of ms.

Synaptic currents

Synaptic currents $I_{\mathit{syn}}$ is the transmission current generated by the synaptic connection between nuclei. According to the connection form in the BG network model, the synaptic current $I_{\mathit{syn}}$ can be divided into $I_{STN\to GPe}$, $I_{STN\to GPi}$, $I_{GPe\to STN}$, $I_{GPe\to GPe}$, $I_{GPe\to GPi}$ and $I_{GPi\to TH}$. Referring to the RT model and its improved model, the synaptic current can be modeled using the equation

$$I_{\mathit{syn}}=g_{\mathit{syn}}S\left(v-E_{\mathit{syn}}\right),$$ 6

where S follows a first-order differential equation if the pre-synaptic neuron is GPe

$$\frac{\mathit{dS}}{\mathit{dt}}=2\left(1-S\right)H_{\infty }\left(v-20\right)-0.04S,$$ 7

where $H_{\infty }=1/\left[1,+,e,x,p,\left(-,\frac{V+57}{2}\right)\right]$ is a smooth approximation of the bi-value step function H. If the pre-synaptic neuron is STN or GPi, S follows a second-order alpha synapse model

$$\frac{\mathit{dS}}{\mathit{dt}}=z,$$ 8

$$\frac{\mathit{dz}}{\mathit{dt}}=0.234u\left(t\right)-0.40z-0.04S,$$ 9

where $u\left(t\right)=1$ if the pre-synaptic neuron cell crosses the threshold of −10 mV, that is, it generates an action potential. Otherwise, $u\left(t\right)=0$. The model of synaptic conductance and resting potential are listed in Table 5.

Table 5.

Parameters for synapses

Synapses	Parameters
$I_{STN\to GPe}$	$g_{\mathit{syn}}=0.15$	$E_{\mathit{syn}}=0$
$I_{STN\to GPi}$	$g_{\mathit{syn}}=0.15$	$E_{\mathit{syn}}=0$
$I_{GPe\to STN}$	$g_{\mathit{syn}}=0.50$	$E_{\mathit{syn}}=-85$
$I_{GPe\to GPe}$	$g_{\mathit{syn}}=0.50$	$E_{\mathit{syn}}=-85$
$I_{GPe\to GPi}$	$g_{\mathit{syn}}=0.50$	$E_{\mathit{syn}}=-85$
$I_{GPi\to TH}$	$g_{\mathit{syn}}=0.08$	$E_{\mathit{syn}}=-85$

Error Index (EI)

One of the most important indexes to measure Parkinson’s disease status is the fidelity of the thalamus relaying information input from the sensorimotor cortex. Referring to the RT model and the improved BG network model, we introduce an error index to quantify the performance of the BG network and represent the state and degree of Parkinson. The error index can be expressed as

$$EI=\frac{\left(N_{\mathit{miss}}+N_{\mathit{burst}}+N_{\mathit{spur}}\right)}{N_{\mathit{SMC}}},$$ 10

where $N_{\chi }$ is the total number of times the event $\chi$ occurred, $\chi$ can be miss, burst, spur, and SMC (sensorimotor cortex). As shown in Fig. 3, Miss represents that the thalamus fails to generate a spike. Burst represents the thalamus spikes more than once within 25 ms of a stimulation input. Spurious occurs when the thalamus spikes in the absence of a stimulation input. Error index refers to the proportion of the number of the above three error events in the total number of stimulation inputs from SMC. The network takes a perfect performance when every stimulation input from SMC results in a single action potential in each thalamic neuron, detected as transmembrane voltage crossing a threshold of −40 mV. According to medical tests and clinical records, the error index of Parkinson’s patients is generally greater than 0.30, around 0.34. When EI is less than 0.05, the result is generally defined as a healthy state.

Fig. 3.

Example of a thalamic cell responding to the input from SMC. The yellow line is the input from SMC. The blue lines represent the firing pattern of the thalamic cell. Three types of error are pointed out on the graph, where * represents a miss, + is the burst, and ^ occurs a spurious

Next, we tune the conductance of synaptic current from STN to GPe and from GPe to STN under the Parkinsonian state, stimulate and calculate the EI corresponding to different synaptic conductance to investigate the relationship between EI and conductivity.

Simulations are implemented in Matlab 2016a for its advantages in matrix operations. All equations are solved using the forward Euler method with a time step of 0.01 ms. The program runs on intel(R) Core(TM) i5-7300HQ CPU and NVIDIA GeForce GTX 1050 GPU.

Results

Electrophysiological Characteristics

The above models can simulate the electrophysiological characteristics and firing patterns of each neuronal population under the Parkinsonian state and healthy state. As shown in Fig. 4, in the healthy state, STN, GPe, and GPi spike in a relatively irregular way with low synchronization. The TH discharge condition is basically consistent with the information input from SMC. And there are no three types of errors. It correctly responds to the stimulation from the SMC with high fidelity. Under the Parkinsonian state, high-frequency oscillations burst in GPe and GPi, and STN fires in a periodic tremor-like manner. TH occurs a lot of firing error events, and cannot correctly relay the information input from SMC. The EI is higher than 0.30. It can be seen from the above results that the BG network model can perform the electrophysiological characteristics and firing patterns related to Parkinson’s disease.

Fig. 4.

Firing patterns of TH, STN, GPe, and GPi. a In the healthy state, TH responds to the SMC inputs correctly, each nuclei fire irregularly. b Under the Parkinsonian state, TH occurs errors, and each nuclei fire with high synchronization

The dynamics of EI under different $g_{\mathit{snge}}$ and $g_{\mathit{gesn}}$

As the main information input nuclei in the BG network, STN has a far-reaching impact on subsequent neuronal population directly or indirectly. The information transmission between STN and GPe is achieved by the excitatory synaptic current from STN to GPe and the inhibitory synaptic current from GPe to STN. We introduce a state transition function of synaptic conductance to explore its relationship:

$$g_{\mathit{gesn}}=k·g_{\mathit{snge}}+b,$$ 11

where $g_{\mathit{gesn}}$ is the synaptic conductance from GPe to STN, and $g_{\mathit{snge}}$ is the synaptic conductance from STN to GPe. Under the Parkinsonian state, we adjust the synaptic conductance $g_{\mathit{snge}}$ and $g_{\mathit{gesn}}$ from 0 to 0.50 with the step of 0.005, to get the error index corresponding to different synaptic conductivity. As shown in Fig. 5, the graph has some obvious characteristics and distinctions.

Fig. 5.

The error index of the BG network corresponding to different $g_{\mathit{snge}}$ and $g_{\mathit{gesn}}$. The blue point shows a low error index, and the yellow one shows a high error index. There is a clear and relatively regular boundary between blue and yellow areas

In terms of color representation, the dark blue area is mainly concentrated in a small area in the top left corner. At this time, the EI is less than 0.05, showing a healthy state. The yellow area has an EI greater than 0.30, showing a Parkinsonian state. There is a distinct boundary between yellow and blue areas with a regular form, the parameters are concise and explicit. When $g_{\mathit{snge}}$ approaches to 0.50, $g_{\mathit{gesn}}$ approaches to 0, the EI gets smaller and gradually approaches to 0 (dark blue area in the upper left corner). On the contrary, the EI becomes larger, reaching 0.30 and even exceeding 0.34 when $g_{\mathit{snge}}$ approaches to 0 and $g_{\mathit{gesn}}$ approaches to 0.50. It is relatively simple and rough to conclude that increasing $g_{\mathit{snge}}$ and reducing $g_{\mathit{gesn}}$ can reduce the error index. This assumption is also in line with the understanding of the pathogenic mechanism of Parkinson’s disease in the basal ganglia-thalamic circuit that the reduced inhibitory effect from GPe to STN and the enhanced excitatory effect from STN to GPe result in the Parkinsonian state.

During the research, we find that when the EI is at a relatively large level ($EI>0.20$), there is an orderly change in the corresponding situation. Some parts of the process figures are listed in Fig. 6. With the increase of EI, corresponding synaptic conductance gradually shifts to the left, $g_{\mathit{gesn}}$ gets smaller. EI and $g_{\mathit{gesn}}$ appear a similar negative correlation phenomenon.

Fig. 6.

Synaptic conductance for several specific EI ranges. Horizontal axis is $g_{\mathit{gesn}}$, and the vertical axis is $g_{\mathit{snge}}$. a Synaptic conductivity in $EI\in [0.20:0.21]$; b Synaptic conductivity in $EI\in [0.25:0.26]$; c Synaptic conductivity in $EI\in [0.30:0.31]$; d Synaptic conductivity in $EI\in [0.35:0.36]$. There is a negative correlation between EI and $g_{\mathit{gesn}}$

Numerical simulation of Parkinson’s states and development

Adjusting the synaptic conductance can simulate the Parkinson’s state and its error index. We extract the synaptic conductance corresponding to every small range of EI. By taking linear fitting and some other methods on them, we can get the slope of the state transition function k at different EI levels. And after analyzing the slope of the state transition function k at different EI levels, we find some interesting results (see Fig. 7).

Fig. 7.

The relationship between the slope of state transition function k and error index (EI). The fitting result shows a three-stage piecewise function. The network keeps stable in the first stage. In the second stage, the slope k gradually increases and accumulates with the increase of EI. With the increase of k to a certain extent, when EI > 0.20, the level of k rises in a burst manner, and then there is a quadratic correlation with EI

At the first stage, when EI is smaller than 0.05, k is less than 0.10 and varies at a low level, the network keeps in a stable condition. This low error index and stable state are ordinarily considered as a healthy state. There is almost no error occurs. And the thalamus can respond to the information correctly and effectively. At this time, $k=0.6728EI+0.03112$, there is an approximate linearity correlation between the slope of state transition function and error index. When EI is larger than 0.05, the property of the network is getting to change. At the second stage, the level of k has a sharp increase and manifests as a first-order linearity correlation at $EI\in [0.05,0.20]$. At this moment, $k=3.892EI+0.06748$, and it gets larger with the increase of EI. It seems like a progress of the accumulation of the error and the break of the stable state. There are some complex and flexible changes in the basal-ganglia thalamic network which will finally lead to a state change. As this process accumulates, a threshold is reached. At the third stage, when the error index is greater than 0.20, k gets one more rise significantly in a burst manner and reaches its peak nearly at EI = 0.30. This high error index is ordinarily considered as a classic Parkinsonian state in physiology and pathology. In this condition, $k={-114.10\text{EI}}^2+65.89EI-7.713$, $EI\in [0.20,0.40]$.

In summary, k is a piecewise function of the error index:

$$\text{k}=\left\{\begin{array}{c}0.6728EI+0.03112,EI\in \left[0,0.05\right].\\ 3.892EI+0.06748,EI\in \left[0.05,0.20\right].\\ -114.10E{I}^2+65.89EI-7.713,EI\in \left[0.20,0.40\right].\\ \end{array}\right)$$ 12

When EI < 0.05, $g_{\mathit{gesn}}$ and $g_{\mathit{snge}}$ have a relationship as Eq. (11), where k = 0.1267, b = 0.0548. Under this relationship, when we take $g_{\mathit{snge}}\in [0.25,0.35]$ and $g_{\mathit{gesn}}\in (0,0.10]$, the EI is as small as possible, and the alleviation effect of Parkinson’s disease is better.

Discussion and conclusion

According to the RT model and improved BG network model, the simulation reflects the electrophysiological characteristics of each neuronal population in the basal ganglia-thalamic network under Parkinsonian state and healthy state. The analysis of the fidelity of thalamic information transmission under different synaptic current conductance conditions revealed a clear relationship between synaptic conductance and EI. Different pathogenic states have significant differences. EI converges to 0 (healthy state) when $g_{\mathit{snge}}$ is close to 0.50 and $g_{\mathit{gesn}}$ is close to 0. When $g_{\mathit{snge}}$ tends to 0 and $g_{\mathit{gesn}}$ becomes larger, the EI is larger, showing a Parkinsonian state. In addition, we discover the relationship between the state transition function of $g_{\mathit{snge}}$ and $g_{\mathit{gesn}}$ and EI. The slope of function k leads to different manners in different ranges of error index. When EI is less than 0.05 with a healthy state, k varies slowly at a low level. It rises substantially and has a first-order positive correlation with EI when EI > 0.05. When the EI increases to 0.20, the slope of the state transition function k has a quadratic correlation with the EI and reaches the maximum when the EI is about 0.30, which is pathological.

This paper discovers a clear change relationship between synaptic conductance and EI and a significant difference in parameters between the healthy state and Parkinsonian state by simulating the firing patterns in the BG network. Increasing $g_{\mathit{snge}}$ and decreasing $g_{\mathit{gesn}}$ can improve the fidelity of thalamus information transmission and alleviate Parkinson’s disease effectively. This finding is also consistent with our understanding of the mechanisms underlying the transition between healthy and Parkinsonian states in the basal ganglia-thalamocortical motor circuit. From the perspective of pathogenic mechanism, increasing $g_{\mathit{snge}}$ and decreasing $g_{\mathit{gesn}}$ can restore the enhancement in excitatory projections from STN to GPe and the decrease in inhibitory projections from GPe to STN which lead to Parkinson’s disease (Fan et al. 2016). It can recover the increase in the number of synaptic connections per GPe-STN axon terminal and the substantial strengthening of the GPe-STN pathway which are caused by the depletion of dopamine (Fan et al. 2012).

The results of this paper can be used as a supplement to the pathogenesis and treatment mechanism of Parkinson’s disease. It shows a relatively clear numerical expression that there is an interaction mechanism of STN and GPe affects EI in Parkinson’s disease. It can guide medical staff to choose appropriate treatment methods based on the above ideas, and to change the synaptic conductance and other parameters of some nuclei with specific goals to alleviate and treat Parkinson’s disease.

In addition, this paper mainly uses simulation as a tool, thus, there may be some errors caused by accuracy. The initial state parameters of Parkinson’s disease selected in the study cannot cover the diversity of patients’ conditions, and there is a certain gap between them and the actual level. Furthermore, this paper just selects only a few synaptic conductance as the research subjects according to the importance of the influence in the circuit mechanism, which cannot fully reflect the overall picture of the Parkinson’s disease network. Expanding the selection of research subjects is also the direction that subsequent research needs to supplement.

Declarations

Conflict of interest

The authors declare that they have no conflict of interest.