Computational Models of Closed–Loop Deep Brain Stimulation

Abstract

Deep Brain Stimulation (DBS) is a neurosurgical intervention that sends electrical signals to the brain to effectively alleviate the symptoms of neurological disorders such as Parkinson’s disease. Although the conventional high frequency DBS shows remarkable therapeutic success, it is desirable to overcome the downsides of such a form of open–loop DBS. Using computational models, we explore a closed–loop DBS paradigm, multi–site delayed feedback stimulation (MDFS), that may potentially overcome the drawbacks of constant high frequency DBS. We first develop a biological-faithful computational network model of basal ganglia and thalamus in parkinsonian conditions. The model mimics the pathological neuronal activity observed in the basal ganglia in parkinsonian conditions, such as increased firing rate, bursting patterns, and synchronization. We then evaluate the outcome of closed–loop MDFS being applied to the parkinsonian network by examining both quantitative measures of neurons in the basal ganglia and the relay error of thalamocortical (TC) neurons. Our computational results show that closed–loop MDFS significantly diminish TC relay error by breaking the bursting pattern and desynchronizing the synchronized clusters in the basal ganglia. The design of MDFS suggests that it is superior to open–loop stimulation in that not only the stimulation signal is guided by changes in neuronal activities specific to disorders being treated, but also MDFS shows much lower energy consumption compared with the conventional high frequency DBS. To support the computational results and feasibility of closed–loop DBS, we further review some previous work that validates the evaluation measure of TC relay error and some recent experimental studies that validate the on–demand type of DBS.

Appendix

In the following text, we use ${{g}_{i}}$ to denote conductance in $mS/c{{m}^{2}}$ and ${{v}_{i}}$ or ${{V}_{i}}$ to denote reversal potentials in $mV$, where the subscript $i$ are from the set $\left\{L,Na,K,Ca,AHP,T,E,Gi,Ge\to Ge,Ge\to Sn,Ge\to Gi,Sn\to Ge,Sn\to Gi \right\}$. $\tau$ with a subscript or both a superscript and a subscript is a time constant in units of msec. All $\alpha$ and $\beta$ with subscripts are rate constants in units of ms⁻¹ $^{-1}$. Other parameters either are constants without units or with units given in the following text.

Functions for TC neurons in system (4.1):

$$ {{m}_{\infty }}(v)=1/(1+{{e}^{-(v+37)/7}}),\ {{p}_{\infty }}(v)=1/(1+{{e}^{-(v+60)/6.2}}), $$

$$ {{h}_{\infty }}(v)=1/(1+{{e}^{(v+41)/4}}),\ {{r}_{\infty }}(v)=1/(1+{{e}^{(v+84)/4}}), $$

$$ {{\tau }_{h}}(v)=1/({{a}_{h}}(v)+{{b}_{h}}(v)),\ {{\tau }_{r}}(v)=0.4(28+{{e}^{-(v+25)/10.5}}), $$

$$ {{a}_{h}}(v)=0.128{{e}^{-(46+v)/18}},\ {{b}_{h}}(v)=4/(1+{{e}^{-(23+v)/5}}), $$

Parameters for TC neurons:

$$ {{g}_{L}}=0.14,\ {{g}_{Na}}=3,\ {{g}_{K}}=5,\ {{g}_{T}}=5,\ {{g}_{E}}=0.018,\ {{g}_{Gi}}=.009, $$

$$ {{v}_{L}}=-72,\ {{v}_{Na}}=50,\ {{v}_{K}}=-90,\ {{v}_{T}}=90,\ {{v}_{E}}=0,\ {{V}_{Gi}}=-85, $$

$$ p=50\,\text{ms,}\ d=5\,\text{ms,}\ wi{{n}_{off}}=12\,\text{ms}. $$

GPi currents:

$$ {{I}_{L}}(v)={{g}_{L}}(v-{{v}_{L}}),\ {{I}_{Na}}={{g}_{Na}}(m_{\infty }^{3}(v))h(v-{{v}_{Na}}),\ {{I}_{K}}={{g}_{K}}{{n}^{4}}(v-{{v}_{K}}), $$

$$ {{I}_{T}}={{g}_{T}}a_{\infty }^{3}(v)r(v-{{v}_{Ca}}),\ {{I}_{Ca}}={{g}_{Ca}}s_{\infty }^{2}(v)(v-{{v}_{Ca}}) $$

$$ {{I}_{AHP}}={{g}_{AHP}}(v-{{v}_{K}})([Ca]/([Ca]+k)), $$

$$ {{I}_{Sn\to Gi}}={{g}_{Sn\to Gi}}{{s}_{Sn\to Gi}}(v-{{v}_{Sn\to Gi}}), $$

where ${{s}_{Sn\to Gi}}$ is listed under STN equations,

and

$$ {{I}_{Ge\to Gi}}={{g}_{Ge\to Gi}}{{s}_{Ge\to Gi}}(v-{{v}_{Ge\to Gi}}), $$

where ${{s}_{Ge\to Gi}}$ is listed under GPe equations.

${{I}_{appi}}=-1\,{}\mu\text{A}$ is a constant applied current.

GPi equations and functions:

$$ {n}'={{\varphi }_{n}}({{n}_{\infty }}(v)-n)/{{\tau }_{n}}(v),\ {h}'={{\varphi }_{h}}({{h}_{\infty }}(v)-h)/{{\tau }_{h}}(v),\ {r}'=\varphi ({{r}_{\infty }}(v)-r)/{{\tau }_{r}}, $$

$$ [Ca{]}'=\varepsilon (-{{I}_{Ca}}-{{I}_{T}}-{{k}_{Ca}}[Ca]),\ {{s}_{Gi}}'=\alpha (1-{{s}_{Gi}}){{S}_{\infty }}(v)-{{\beta }_{Gi}}{{s}_{Gi}}, $$

where ${{S}_{\infty }}(v)$ is given in section “Model Equations for STN, GPe and GPi Neurons”.

$$ {{X}_{\infty }}(v)=1/(1+{{e}^{-(v-{{\theta }_{X}})/{{\sigma }_{X}}}}), $$

where $X=m,n,h,r,a,s$,

and

$$ {{\tau }_{X}}(v)=\tau_{X}^{0}+\tau_{X}^{1}/(1+{{e}^{-(v-\theta_{X}^{\tau })/\sigma_{X}^{\tau }}}), $$

where $X=n,h$.

GPi parameters:

$$ {{g}_{L}}=0.1,\ {{g}_{Na}}=120,\ {{g}_{K}}=30,\ {{g}_{T}}=0.5,\ {{g}_{Ca}}=0.1,\ {{g}_{AHP}}=30,\ {{g}_{Sn\to Gi}}=0.5,\ {{g}_{Ge\to Gi}}=1, $$

$$ {{v}_{L}}=-55,\ {{v}_{Na}}=55,\ {{v}_{K}}=-80,\ {{v}_{Ca}}=120,\ {{v}_{Ge\to Gi}}=-100,\ {{v}_{Sn\to Gi}}=0, $$

$$ \tau_{n}^{0}=0.05,\ \tau_{n}^{1}=0.27,\ \tau_{h}^{0}=0.05,\ \tau_{h}^{1}=0.27,\ {{\tau }_{r}}=30, $$

$$ {{\phi }_{r}}=1,\ {{\phi }_{n}}=0.1,\ {{\phi }_{h}}=0.05 $$

$$ {{k}_{1}}=30,\ {{k}_{Ca}}=15,\ \varepsilon =0.0001\,\text{m}{{\text{s}}^{-1}},$$

$$ {{\theta }_{r}}=-70,\ {{\theta }_{m}}=-37,\ {{\theta }_{n}}=-50,\ {{\theta }_{h}}=-58,\ {{\theta }_{a}}=-57,\ {{\theta }_{s}}=-35,\ \alpha =2,\ \theta_{n}^{\tau }=-40,\ \theta_{h}^{\tau }=-40, $$

$$ {{\sigma }_{m}}=10,\ {{\sigma }_{n}}=14,\ {{\sigma }_{h}}=-12,\ {{\sigma }_{r}}=-2,\ {{\sigma }_{a}}=2,\ {{\sigma }_{s}}=2,\ \sigma_{n}^{\tau }=-12,\ \sigma_{h}^{\tau }=-12, $$

$$ {{\beta }_{Gi}}=.08,\ {{k}_{Ca}}=15. $$

STN currents:

${{I}_{L}}$, ${{I}_{Na}}$, ${{I}_{K}}$, ${{I}_{Ca}}$, ${{I}_{AHP}}$ are as given above for the GPi neuron, and ${{I}_{T}}={{g}_{T}}a_{\infty }^{3}(v){{b}_{\infty }}(r)(v-{{v}_{Ca}})$. The synaptic currents from GPe to STN are the following:

${{I}_{Ge\to Sn}}={{g}_{Ge\to Sn}}\sum\nolimits_{j\in \Lambda }{}s_{Ge\to Sn}^{j}(v-{{v}_{Sn\to Ge}})$, where $\Lambda$ is a subgroup of GPe cells and ${{s}_{Ge\to Sn}}$ is given in GPe equations.

For the stimulation current ${{I}^{stim}}$, see details in sections “Generation of Stimulation Current from Feedback” and “Comparison Between Open–Loop and Closed–Loop Stimulations”.

STN equations and functions:

$n$, $h$, $r$, $[Ca]$ equations and functions ${{X}_{\infty }}(v),{{\tau }_{X}}(v)$ are the same as given above for GPi neuron, except there is no ${{r}_{\infty }}(v)$ used and we introduce ${{b}_{\infty }}(r)=1/(1+{{e}^{(r-{{\theta }_{b}})/{{\sigma }_{b}}}})-1/(1+{{e}^{-{{\theta }_{b}}/{{\sigma }_{b}}}})$.

The synaptic input from STN to GPe and GPi is described as:

$$ {{s}_{Sn\to Ge}}'={{\alpha }_{Sn\to Ge}}(1-{{s}_{Sn\to Ge}}){{s}_{\infty }}(v-30)-{{\beta }_{Sn\to Ge}}{{s}_{Sn\to Ge}}, $$

$$ {{s}_{Sn\to Gi}}'={{\alpha }_{Sn\to Gi}}(1-{{s}_{Sn\to Gi}}){{s}_{\infty }}(v-30)-{{\beta }_{Sn\to Gi}}{{s}_{Sn\to Gi}}. $$

STN Parameters:

$$ {{g}_{L}}=2.25,\ {{g}_{Na}}=37.5,\ {{g}_{K}}=45,\ {{g}_{T}}=0.5,\ {{g}_{Ca}}=0.5,\ {{g}_{AHP}}=9,\ {{g}_{Ge\to Sn}}=0.9, $$

$$ {{v}_{L}}=-60,\ {{v}_{Na}}=55,\ {{v}_{K}}=-80,\ {{v}_{Ca}}=140,\ {{v}_{Ge\to Sn}}=-100, $$

$$ \tau_{n}^{0}=1,\ \tau_{n}^{1}=100,\ \tau_{h}^{0}=1,\ \tau_{h}^{1}=500,\ \tau_{r}^{0}=7.1,\ \tau_{r}^{1}=17.5, $$

$$ {{\phi }_{r}}=0.5,\ {{\phi }_{n}}=0.75,\ {{\phi }_{h}}=0.75, $$

$$ {{k}_{1}}=15,\ {{k}_{Ca}}=22.5,\ \varepsilon =5\times {{10}^{-5}}, $$

$$ {{\theta }_{r}}=-67,\ {{\theta }_{m}}=-30,\ {{\theta }_{n}}=-32,\ {{\theta }_{h}}=-39,\ {{\theta }_{a}}=-63,\ {{\theta }_{s}}=-39,\ {{\theta }_{b}}=0.25, $$

$$ \theta_{n}^{\tau }=-80,\ \theta_{h}^{\tau }=-57,\ \theta_{r}^{\tau }=68 $$

$$ {{\sigma }_{m}}=15,\ {{\sigma }_{n}}=8,\ {{\sigma }_{h}}=-3.1,\ {{\sigma }_{r}}=-2,\ {{\sigma }_{a}}=7.8,\ {{\sigma }_{s}}=8,\ {{\sigma }_{b}}=0.07 $$

$$ \sigma_{n}^{\tau }=-26,\ \sigma_{h}^{\tau }=-3,\ \sigma_{r}^{\tau }=-2.2, $$

$$ {{\alpha }_{Sn\to Ge}}=5,\ {{\alpha }_{Sn\to Gi}}=1,\ {{\beta }_{Sn\to Ge}}=1,\ {{\beta }_{Sn\to Gi}}=0.05,\ wk=0.45. $$

GPe currents:

${{I}_{L}}$, ${{I}_{Na}}$, ${{I}_{K}}$, ${{I}_{Ca}}$, ${{I}_{AHP}}$ are modeled as given above for the GPi neuron. The synaptic currents to GPe are:

${{I}_{Sn\to Ge}}={{g}_{Sn\to Ge}}\sum\nolimits_{j\in \Lambda }{}{{s}_{Sn\to Ge}}(v-{{v}_{Sn\to Ge}})$, where $\Lambda$ is a subgroup of STN neurons and ${{s}_{Sn\to Ge}}$ is given under STN equations.

${{I}_{Ge\to Ge}}={{g}_{Ge\to Ge}}\sum\nolimits_{j\in \Lambda }{}{{s}_{Ge\to Ge}}(v-{{v}_{Ge\to Ge}})$ where $\Lambda$ is a subgroup of STN neurons and ${{s}_{Ge\to Ge}}$ is the same as ${{s}_{Ge\to Sn}}$ given under GPe equations.

${{I}_{app}}=-1.2$ is a constant applied current.

GPe equations and functions:

$n$, $h$, $r$, $[Ca]$ equations and function ${{X}_{\infty }}(v),{{\tau }_{X}}(v)$ are as given above for the GPi neuron.

The synaptic input from STN to GPe and GPi is described as:

$$ {{s}_{Ge\to Sn}}'={{\alpha }_{Ge\to Sn}}(1-{{s}_{Ge\to Sn}}){{s}_{\infty }}(v-20)-{{\beta }_{Ge\to Sn}}{{s}_{Ge\to Sn}}, $$

$$ {{s}_{Ge\to Gi}}'={{\alpha }_{Ge\to Gi}}(1-{{s}_{Ge\to Gi}}){{s}_{\infty }}(v-20)-{{\beta }_{Ge\to Gi}}{{s}_{Ge\to Gi}}. $$

GPe parameters:

Most parameters for GPe are the same as those for GPi. We only list those that have different values and the additional ones not present in the GPi model.

$$ {{g}_{Sn\to Ge}}=0.18,\ {{g}_{Ge\to Ge}}=0.01,\ {{v}_{Sn\to Ge}}=0,\ {{v}_{Ge\to Ge}}=-80, $$

$$ {{\alpha }_{Ge\to Sn}}=2,\ {{\alpha }_{Ge\to Gi}}=1 $$

$$ {{\beta }_{Ge\to Sn}}=0.04,\ {{\beta }_{Ge\to Gi}}=0.1. $$

CRS parameters: $\rho =0.7$, ${{a}_{1}}=0.9$, ${{a}_{0}}=58$, ${{\tau }_{0}}=30$, which is the phase shift in ${{f}_{k}},$ $k=1,2,3,4$.

MDFS parameters: $a=0.0025,$,$b=0.00136,$,$\mu =0.0002$, $\tau =25.$