The effects of DBS patterns on basal ganglia activity and thalamic relay

Abstract

Thalamic neurons receive inputs from cortex and their responses are modulated by the basal ganglia (BG). This modulation is necessary to properly relay cortical inputs back to cortex and downstream to the brain stem when movements are planned. In Parkinson’s disease (PD), the BG input to thalamus becomes pathological and relay of motor-related cortical inputs is compromised, thereby impairing movements. However, high frequency (HF) deep brain stimulation (DBS) may be used to restore relay reliability, thereby restoring movements in PD patients. Although therapeutic, HF stimulation consumes significant power forcing surgical battery replacements, and may cause adverse side effects. Here, we used a biophysical-based model of the BG-Thalamus motor loop in both healthy and PD conditions to assess whether low frequency stimulation can suppress pathological activity in PD and enable the thalamus to reliably relay movement-related cortical inputs. We administered periodic pulse train DBS waveforms to the sub-thalamic nucleus (STN) with frequencies ranging from 0–140 Hz, and computed statistics that quantified pathological bursting, oscillations, and synchronization in the BG as well as thalamic relay of cortical inputs. We found that none of the frequencies suppressed all pathological activity in BG, though the HF waveforms recovered thalamic reliability. Our rigorous study, however, led us to a novel DBS strategy involving low frequency multi-input phase-shifted DBS, which successfully suppressed pathological symptoms in all BG nuclei and enabled reliable thalamic relay. The neural restoration remained robust to changes in the model parameters characterizing early to late PD stages.

Appendix

Below we present the details of the RT model that we used in our study. The network topology used in this model is shown in Fig. 11.

Fig. 11

The network topology of our model. The synaptic connections between different neurons in the basal ganglia thalamus network are shown. The network is divided into groups of four neurons which project to group of four neurons in a different structure. Solid arrows depict a strong synaptic connection whereas dashed arrows depict a weaker synaptic connection. Note that internal synaptic connections between GPe neurons are not shown due to their random nature

1.1 A. STN

$$ \begin{array}{rll} v'&=&-(I_{l}(v) + I_{na}(v,h)+I_{k}(v,n)+I_{ahp}(v,ca)+I_{ca}(v)\\ &&\quad +\,I_{t}(v,r))-I_{gpe\rightarrow stn}+ hets + dbs(t) \end{array} $$

$$ h'\!=\!\frac{\phi(h_{\infty}(v)\!-\!h)}{\tau_{h}(v)}; n'\!=\!\frac{\phi(n_{\infty}(v)\!-\!h)}{\tau_{n}(v)}; r'\!=\!\frac{\phi_{r}(r_{\infty}(v)\!-\!r)}{\tau_{r}(v)} $$

$$ ca'= \phi \times eps (-I_{ca}(v) - I_{t}(v,r) - k_{ca} \times ca) $$

$$ s'=\alpha(1-s) \times s_{\infty}(v+ab) - \beta \times s $$

$$ si'=\alpha i \ast (1-si)\times s_{\infty}(v+ab)-\beta i \times si $$

Here, $I_{l}(v)\!=\!g_{l}(v\!-\!v_{l});\! I_{na}(v,h)\!=\!g_{na}(m_{\infty}(v))^{3}h(v\!-\!v_{na})$; $I_{k}(v,n)\!=\!g_{k}n^{4}(v\!-\!v_{k})$; $I_{ahp}(v,ca)=\frac{g_{ahp}(v-v_{k})ca}{ca+k1}$; $i_{ca}(v)=g_{ca}((s_{\infty}(v))^{2}) \!\times\! (v\!-\!v_{ca})$; $i_{t}(v,r)\!=\!g_{t}(t_{\infty}(v)^{3}) r_{new}(r)^{2}(v\!-\!vca)$; I _gpe→stn  = g _gpe→stn (Σsg) × (v − v _syngg )

$$ \begin{array}{rll} s_{\infty}(v)&=&\frac{1}{1+\exp\left(\frac{v+\theta_{s}}{ss}\right)}; m_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+\theta_{m}}{sm}\right)};\\ h_{\infty}(v)&=&\frac{1}{1+\exp\left(\frac{v-\theta_{h}}{sh}\right)}; n_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v-\theta_{n}}{sn}\right)}; \end{array} $$

$$ \begin{array}{rll} r_{\infty}(v)&=&\frac{1}{1+\exp\left(\frac{v-\theta_r}{kr}\right)}; t_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v-\theta_t}{kt}\right)};\\ r_{new}(r)&=&\frac{1}{1+\exp\left(\frac{r-r_{th}}{r_{sig}}\right)}- \frac{1}{1+\exp\left(-\frac{r_{th}}{r_{sig}}\right)} \end{array} $$

$$ \begin{array}{rll} \tau_{n}(v)&=&\tau_{n0}+\frac{\tau_{n1}}{1+\exp\left(\frac{v+th_{n}}{\sigma_{n}}\right)};\\ \tau_{h}(v)&=&\tau_{h0}+\frac{\tau_{h1}}{1+\exp\left(\frac{v+th_{h}}{\sigma_{h}}\right)};\\ \tau_{r}(v)&=&\tau_{r0}+\frac{\tau_{r1}}{1+\exp\left(\frac{v+th_{r}}{\sigma_{r}}\right)}; \end{array} $$

Parameter values used are:

$$ \begin{array}{rll} &&{\kern-6pt} v_{l}=-60, v_{na}=55, v_{k}=-80, \theta_{m}=30, sm=15, \\ &&{\kern-6pt} g_{l}=2.25, g_{na}=37.5, g_{k}=45, tn=1, th=0.05, \\ &&{\kern-6pt} g_{ahp}=9, g_{ca}=0.5, v_{ca}=140, k1=15, eps=5e-05,\\ &&{\kern-6pt} kca=22.5, \theta_s=39, ss=8, xp=1, i=0, \theta_h=-39,\\ &&{\kern-6pt} sh\!=\!3.1, \theta_n\!=\!-32, sn=-8, \tau_{n0}\!=\!1, \tau_{n1}\!=\!100, th_{n}\!=\!80, \\ &&{\kern-6pt}\sigma_{n}=26, \tau_{h0}=1, \tau_{h1}=500, th_{h}=57, \sigma_{h}=3, \phi=0.75 \end{array} $$

$$ \begin{array}{rll} &&{\kern-6pt}\theta_{t}\!=\!-63, kt=-7.8, g_{t}=0.5, \phi_{r}=0.5; \theta_{r}=-67, kr\!=\!2, \\ &&{\kern-6pt}\tau_{r0}=7.1, \tau_{r1}=17.5, th_{r}=-68, \sigma_{r}=2.2, \alpha=5, \beta=1,\\ &&{\kern-6pt}ab=-30, g_{syn}=0.9, v_{syn}=-100, r_{th}=0.25,\\ &&{\kern-6pt}r_{sig}=-0.07, \rho1=0.097, a1=0.6, i1=200, hets=2, \\ &&{\kern-6pt}\alpha i =1, \beta i = .2, g_{gpe\rightarrow stn} = 0.1, v_{syngg}=-80 \end{array} $$

1.2 B. GPe and GPi

$$ \begin{array}{rll} v'&=&-(I_{l}(v)+I_{na}(v,h)+I_{k}(v,n)+I_{ahp}(v,ca)+I_{ca}(v)\\ &&+\,I_{t}(v,r)) + I_{\rm app} - I_{gpe\rightarrow gp} - I_{stn\rightarrow gp} + hetg \end{array} $$

$$ \begin{array}{rll} h'\!=\!\frac{\delta_{h}(h_{\infty}(v)\!-\!h)}{\tau_{h}(v)}; n'\!=\!\frac{\delta_{n}(n_{\infty}(v)\!-\!n)}{\tau_{n}(v)}; r'\!=\!\frac{\phi_{r}(r_{\infty}(v)\!-\!r)}{\tau_{r}(v)} \end{array} $$

$$ ca'=\phi \times eps (-I_{ca}(v)- I_{t}(v,r)-k_{ca} \times ca) $$

$$ sg'=\alpha(1-sg) \times s_{\infty}(v+ab) - \beta \times s $$

$$ \begin{array}{rll} sggi' &=& \alpha i \times (1-sggi) \times s_{\infty} (v+ab) - \beta i \times sggi \\ &&{\mbox{(only for GPe)}} \end{array} $$

Here,

$$ \begin{array}{rll} &&{\kern-6pt} I_{l}(v)=g_{l}(v-v_{lg}); I_{nag}(v,h)=g_{na}(m_{\infty}(v))^{3} h(v-v_{na}); \\ &&{\kern-6pt} I_{k}(v,n)=g_{k}n^{4}(v-v_{k}); I_{ahp}(v,ca)=\frac{g_{ahp}(v-v_k)ca}{ca+k1};\\ &&{\kern-6pt} i_{ca}(v)=g_{ca}((s_{\infty}(v))^{2}) \times (v-v_{ca});\\ &&{\kern-6pt} i_{t}(v,r)=g_{t}(s_{\infty}(v)^{3})r(v-vca);\\ &&{\kern-6pt} I_{gpe\rightarrow gpe} = g_{syngg}(\Sigma sg) \times (v-v_{syngg});\\ &&{\kern-6pt}I_{stn\rightarrow gpe} = g_{stn\rightarrow gpe} (\Sigma s) \times (v-v_{syng}) \end{array} $$

For GPi, the synaptic currents are:

$$ \begin{array}{rll} &&{\kern-6pt}I_{stn\rightarrow gpi} = g_{stn\rightarrow gpi} si (v-v_{syng});\\ &&{\kern-6pt}I_{gpe\rightarrow gpi}=g_{synggi} sggi(v-vsyn) \end{array} $$

Different functions above are:

$$ \begin{array}{rll} &&{\kern-6pt} s_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+\theta_s}{ks}\right)}; s1_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+\theta_{s1}}{ks1}\right)};\\ &&{\kern-6pt} m_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+\theta_m}{sm}\right)}; h_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v-\theta_h}{\sigma_h}\right)};\\ &&{\kern-6pt} n_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v-\theta_n}{sn}\right)}; r_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v-\theta_r}{kr}\right)};\\ &&{\kern-6pt} \tau_{n}(v)=\tau_{n0}+\frac{\tau_{n1}}{1+\exp\left(\frac{v+th_n}{sn}\right)};\\ &&{\kern-6pt} \tau_{h}(v)=\tau_{h0}+\frac{\tau_{h1}}{1+\exp\left(\frac{v+th_h}{sh}\right)}; hv(v)=\frac{1}{1+\exp\left(\frac{v}{.001}\right)}; \end{array} $$

Parameter values used are:

$$ \begin{array}{rll} &&{\kern-6pt}g_{na}=120, g_{k}=30, g_{ahp}=30, g_{t}=.5, g_{ca}=.1, g_{l}=.1,\\ &&{\kern-6pt}v_{na}=55, v_{k}=-80, v_{ca}=120, v_{l}=-55, \theta_s=-57,\\ &&{\kern-6pt}ks=2, \theta_{s1}=-35, ks1=2, \theta_r=-70, k_r=-2, \tau_r=30,\\ &&{\kern-6pt}\theta_{m}=-37, \sigma_{m}=10, \theta_n=-50, \sigma_n=14, \tau_{n0}=.05, \\ &&{\kern-6pt}\tau_{n1}=.27, thn=-40, sn=-12, \theta_h=-58, \sigma_h=-12, \\ &&{\kern-6pt}\tau_{h0}=.05, \tau_{h1}=.27, thh=-40, sh=-12, k1=30, \\ &&{\kern-6pt}kca=20, eps=0.0001, \phi=1, \delta_{n}=.05, \delta_h=.05,\\ &&{\kern-6pt}I_{\rm app}=-0.5, g_{gpe\rightarrow gpe}=2, v_{synng}=-80, g_{stn\rightarrow gpe}=0.3,\\ &&{\kern-6pt}v_{syn}=0,\alpha=2, \beta=.04, \alpha i =0.1, \beta i = .08, ab=-20,\\ &&{\kern-6pt}thresg=0.0, hetg =0.3 \end{array} $$

For PD we use:

$$ I_{\rm app}=-3,g_{gpe\rightarrow gpe}=.1 $$

Parameters that differ for GPi:

$$ \begin{array}{rll} &&{\kern-6pt}I_{\rm app}=-2.8, \beta=.08, kca=20, \\&&{\kern-6pt}g_{stn\rightarrow gpi}=0.7,g_{gpe\rightarrow gpi}=1 \end{array} $$

1.3 C. Thalamus

$$ \begin{array}{rll} v'&=&-(I_{l}(v)+ I_{na}(v,h)+I_{k}(v,h)+ I_{t}(v,r))\\&&+ I_{sm}- g_{gpi \rightarrow tc}\Sigma sg \times (v-v_{syntc})\\ h'&=&\frac{tadj (h_\infty (v)-h)}{\tau_{h}(v)}; r'\frac{qht(r_\infty (v)-r)}{\tau_{r}(v)} \end{array} $$

Different Currents in above equation are:

$$ \begin{array}{lll}I_{na}(v,h)&=&g_{na}m_{\infty}(v)^{3}h(v-e_{na});\\[3pt] I_k(v,h)&=&g_k(0.75(1-h))^4(v-e_k);\\[3pt] I_l(v)&=&g_l(v-e_l);I_t(v,r)=g_{t}mt_{\infty}(v)^{2}r\times{v} \end{array} $$

Different function in above equations are:

$$ \begin{array}{rll} &&{\kern-6pt}m_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+37}{7}\right)};mt_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+60}{6.2}\right)};\\ &&{\kern-6pt}ah(v)=0.128*\exp\left(-\frac{46+v}{18}\right);\\ &&{\kern-6pt}bh(v)=\frac{apr}{1+\exp\left(\frac{23+v}{5}\right)};\\ &&{\kern-6pt}\tau_{h}(v)=\frac{1}{ah(v)+bh(v)};h_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+41}{4}\right)};\\ &&{\kern-6pt}r_{\infty}(v)=\frac{1}{1+\exp\left(\frac{v+84}{4}\right)};\\ &&{\kern-6pt}\tau_{r}(v)=0.4\left(28+apt * \exp\left(\frac{v+25}{-10.5}\right)\right) \end{array} $$

Parameter values are:

$$ \begin{array}{rll} &&{\kern-6pt}v_{syn}=-85, g_{gpi\rightarrow{tc}}=.15, asg=200, bsg=.4,itc=6.5,\\ &&{\kern-6pt}shi=-80,dur=5,g_{na}=3, g_{k}=5,g_{l}=.05,e_{na}=50,\\ &&{\kern-6pt}e_{k}=-90,e_{l}=-70,g_{t}=5,qht=2.5,tadj=1,\\&&{\kern-6pt}apr=4,apt=1 \end{array} $$