A biophysical model of the cortex-basal ganglia-thalamus network in the 6-OHDA lesioned rat model of Parkinson’s disease

Abstract

Electrical stimulation of sub-cortical brain regions (the basal ganglia), known as deep brain stimulation (DBS), is an effective treatment for Parkinson’s disease (PD). Chronic high frequency (HF) DBS in the subthalamic nucleus (STN) or globus pallidus interna (GPi) reduces motor symptoms including bradykinesia and tremor in patients with PD, but the therapeutic mechanisms of DBS are not fully understood. We developed a biophysical network model comprising of the closed loop cortical-basal ganglia-thalamus circuit representing the healthy and parkinsonian rat brain. The network properties of the model were validated by comparing responses evoked in basal ganglia (BG) nuclei by cortical (CTX) stimulation to published experimental results. A key emergent property of the model was generation of low-frequency network oscillations. Consistent with their putative pathological role, low-frequency oscillations in model BG neurons were exaggerated in the parkinsonian state compared to the healthy condition. We used the model to quantify the effectiveness of STN DBS at different frequencies in suppressing low-frequency oscillatory activity in GPi. Frequencies less than 40 Hz were ineffective, low-frequency oscillatory power decreased gradually for frequencies between 50 Hz and 130 Hz, and saturated at frequencies higher than 150 Hz. HF STN DBS suppressed pathological oscillations in GPe/GPi both by exciting and inhibiting the firing in GPe/GPi neurons, and the number of GPe/GPi neurons influenced was greater for HF stimulation than low-frequency stimulation. Similar to the frequency dependent suppression of pathological oscillations, STN DBS also normalized the abnormal GPi spiking activity evoked by CTX stimulation in a frequency dependent fashion with HF being the most effective. Therefore, therapeutic HF STN DBS effectively suppresses pathological activity by influencing the activity of a greater proportion of neurons in the output nucleus of the BG.

Appendix

All transmembrane potentials $(v)$ are expressed in $mV$, intrinsic and synaptic conductances in $mS/{cm}^2$, currents in $\mu A/{cm}^2$, and time constants in $msec$. For all cell models the membrane capacitance is $1\;\mu A/{cm}^2$.

1.1 Thalamic neuron model

$$ {C}_m\frac{d{v}_{Th}}{dt} = -{I}_l-{I}_K-{I}_{Na}-{I}_t-{I}_{gith}+{I}_{appth} $$

$$ \frac{dh}{dt}=\frac{h_{\infty}\left({v}_{Th}\right)-h}{\tau_h\left({v}_{Th}\right)} $$

$$ \frac{dr}{dt}=\frac{r_{\infty}\left({v}_{Th}\right)-r}{\tau_r\left({v}_{Th}\right)} $$

Table 2 TH neuron model equations

1.2 External globus pallidus neuron model

$$ {C}_m\frac{d{v}_{GPe}}{dt} = -{I}_l-{I}_K-{I}_{Na}-{I}_t-{I}_{Ca}-{I}_{ahp}-{I}_{snge, ampa}-{I}_{snge, nmda}-{I}_{gege}-{I}_{strgpe}+{I}_{appgpe} $$

$$ \frac{dn}{dt}=\frac{0.1*\left({n}_{\infty}\left({v}_{GPe}\right)-n\right)}{\tau_n\left({v}_{GPe}\right)} $$

$$ \frac{dh}{dt}=\frac{0.05*\left({h}_{\infty}\left({v}_{GPe}\right)-h\right)}{\tau_h\left({v}_{GPe}\right)} $$

$$ \frac{dr}{dt}=\frac{r_{\infty}\left({v}_{GPe}\right)-r}{\tau_r\left({v}_{GPe}\right)} $$

$$ \frac{dCA}{dt}={10}^{-4}*\left(-{I}_{ca}-{I}_t-15*CA\right) $$

Table 3 GPe neuron model equations

1.3 Internal globus pallidus neuron model

$$ {C}_m\frac{d{v}_{GPi}}{dt} = -{I}_l-{I}_K-{I}_{Na}-{I}_t-{I}_{Ca}-{I}_{ahp}-{I}_{sngi}-{I}_{gegi}-{I}_{strgpi}+{I}_{appgpi} $$

$$ \frac{dn}{dt}=\frac{0.1*\left({n}_{\infty}\left({v}_{GPi}\right)-n\right)}{\tau_n\left({v}_{GPi}\right)} $$

$$ \frac{dh}{dt}=\frac{0.05*\left({h}_{\infty}\left({v}_{GPi}\right)-h\right)}{\tau_h\left({v}_{GPi}\right)} $$

$$ \frac{dr}{dt}=\frac{r_{\infty}\left({v}_{GPi}\right)-r}{\tau_r\left({v}_{GPi}\right)} $$

$$ \frac{dCA}{dt}={10}^{-4}*\left(-{I}_{ca}-{I}_t-15*CA\right) $$

Table 4 GPi neuron model equations

1.4 Subthalamic nucleus neuron model

$$ {C}_m\frac{d{v}_{STN}}{dt} = -{I}_{Na}-{I}_K-{I}_a-{I}_L-{I}_t-{I}_{Cak}-{I}_l-{I}_{gesn}-{I}_{cosn, ampa}-{I}_{cosn, nmda}+{I}_{dbs} $$

$$ \frac{dn}{dt}=\frac{\left({n}_{\infty}\left({v}_{STN}\right)-n\right)}{\tau_n\left({v}_{STN}\right)} $$

$$ \frac{dh}{dt}=\frac{h_{\infty}\left({v}_{STN}\right)-h\Big)}{\tau_h\left({v}_{STN}\right)} $$

$$ \frac{dm}{dt}=\frac{\left({m}_{\infty}\left({v}_{STN}\right)-m\right)}{\tau_m\left({v}_{STN}\right)} $$

$$ \frac{da}{dt}=\frac{\left({a}_{\infty}\left({v}_{STN}\right)-a\right)}{\tau_a\left({v}_{STN}\right)} $$

$$ \frac{db}{dt}=\frac{\left({b}_{\infty}\left({v}_{STN}\right)-b\right)}{\tau_b\left({v}_{STN}\right)} $$

$$ \frac{dc}{dt}=\frac{\left({c}_{\infty}\left({v}_{STN}\right)-c\right)}{\tau_c\left({v}_{STN}\right)} $$

$$ \frac{dd1}{dt}=\frac{\left(d{1}_{\infty}\left({v}_{STN}\right)-d1\right)}{\tau_{d1}\left({v}_{STN}\right)} $$

$$ \frac{dd2}{dt}=\frac{\left(d{2}_{\infty}\left({v}_{STN}\right)-d2\right)}{\tau_{d2}\left({v}_{STN}\right)} $$

$$ \frac{dp}{dt}=\frac{\left({p}_{\infty}\left({v}_{STN}\right)-p\right)}{\tau_p\left({v}_{STN}\right)} $$

$$ \frac{dq}{dt}=\frac{\left({q}_{\infty}\left({v}_{STN}\right)-q\right)}{\tau_q\left({v}_{STN}\right)} $$

$$ \frac{dr}{dt}=\frac{\left({r}_{\infty}\left({v}_{STN}\right)-r\right)}{\tau_r\left({v}_{STN}\right)} $$

$$ \frac{dC{a}_i}{dt}=-5.18*{10}^{-6}*\left({I}_L+{I}_t\right)-\left(2*{10}^{-3}*C{a}_i\right) $$

Table 5 STN neuron model equations

1.5 Striatum medium spiny neuron model

$$ {C}_m\frac{d{v}_{Str}}{dt} = -{I}_l-{I}_K-{I}_{Na}-{I}_m-{I}_{gaba}-{I}_{costr} $$

$$ \frac{dm}{dt}={\alpha}_m\left({v}_{Str}\right)*\left(1-m\right)-{\beta}_m\left({v}_{Str}\right)*m $$

$$ \frac{dh}{dt}={\alpha}_h\left({v}_{Str}\right)*\left(1-h\right)-{\beta}_h\left({v}_{Str}\right)*h $$

$$ \frac{dn}{dt}={\alpha}_n\left({v}_{Str}\right)*\left(1-n\right)-{\beta}_n\left({v}_{Str}\right)*n $$

$$ \frac{dp}{dt}={\alpha}_p\left({v}_{Str}\right)*\left(1-p\right)-{\beta}_p\left({v}_{Str}\right)*p $$

$$ \frac{dS}{dt}={G}_{gaba}\left({v}_{Str}\right)*\left(1-S\right)-\left(\frac{S}{ta{u}_i}\right) $$

Table 6 MSN neuron model equations

1.6 Cortical regular spiking projection neuron model

$$ \frac{d{v}_{rs}}{dt} = 0.04*{v}_{rs}^2+5*{v}_{rs}+140-{u}_{rs}-{I}_{ie}-{I}_{thco} $$

$$ \frac{d{u}_{rs}}{dt} = {a}_{rs}*\left(\left({b}_{rs}*{v}_{rs}\right)-{u}_{rs}\right) $$

$$ if\ {v}_{rs}\ge 30\ mV,\ then $$

$$ {v}_{rs}={c}_{rs} $$

$$ {u}_{rs}={u}_{rs}+{d}_{rs} $$

Table 7 CTX regular spiking neuron model parameters

Table 8 CTX Regular Spiking neuron model equations

1.7 Cortical fast spiking interneuron model

$$ \frac{d{v}_{fsi}}{dt} = 0.04*{v}_{fsi}^2+5*{v}_{fsi}+140-{u}_{fsi}-{I}_{ei} $$

$$ \frac{d{u}_{fsi}}{dt} = {a}_{fsi}*\left(\left({b}_{fsi}*{v}_{fsi}\right)-{u}_{fsi}\right) $$

$$ if\ {v}_{fsi}\ge 30\ mV,\ then $$

$$ {v}_{fsi}={c}_{fsi} $$

$$ {u}_{fsi}={u}_{fsi}+{d}_{fsi} $$

Table 9 CTX fast spiking interneuron model parameters

Table 10 CTX fast spiking interneuron model equations

Table 11 Healthy and PD state parameters