A computational modelling approach to investigate different targets in deep brain stimulation for Parkinson’s disease

Abstract

We investigated by a computational model of the basal ganglia the different network effects of deep brain stimulation (DBS) for Parkinson’s disease (PD) in different target sites in the subthalamic nucleus (STN), the globus pallidus pars interna (GPi), and the globus pallidus pars externa (GPe). A cellular-based model of the basal ganglia system (BGS), based on the model proposed by Rubin and Terman (J Comput Neurosci 16:211–235, 2004), was developed. The original Rubin and Terman model was able to reproduce both the physiological and pathological activities of STN, GPi, GPe and thalamo-cortical (TC) relay cells. In the present study, we introduced a representation of the direct pathway of the BGS, allowing a more complete framework to simulate DBS and to interpret its network effects in the BGS. Our results suggest that DBS in the STN could functionally restore the TC relay activity, while DBS in the GPe and in the GPi could functionally over-activate and inhibit it, respectively. Our results are consistent with the experimental and the clinical evidences on the network effects of DBS.

Appendix

Basic sets of equations of the eRTM are presented here. See Rubin and Terman (2004) and Terman et al. (2002) for more details on parameters and equations used.

The membrane potential of each STN neuron obeys the current balance equation:

$$\begin{array}{*{20}l} {{C_{{\text{m}}} V^{\prime }_{{{\text{STN}}}} = - I_{{\text{L}}} - I_{{{\text{Na}}}} - I_{{\text{K}}} - I_{{\text{T}}} - I_{{{\text{Ca}}}} - I_{{{\text{AHP}}}} + } \hfill} \\ {{ + I_{{{\text{GPe}} \to {\text{STN}}}} + I_{{{\text{MoreSTN}}}} + I_{{{\text{DBS}}}} } \hfill} \\ \end{array} $$

The model features: potassium and sodium spike-producing currents I _K e I _Na; a low-threshold T-type Ca²⁺ current (I _T); a high-threshold Ca²⁺ current (I _Ca); a Ca²⁺ activated, voltage-independent after-hyperpolarization K⁺ current (I _AHP); and a leak current (I _L). These currents (for STN, GPe, GPi and thalamic cells) are described by Hodgkin–Huxley formalism. Constant, depolarizing current I _MoreSTN were introduced in Rubin and Terman (2004) to simulate diffuse excitatory input from cortex to STN cells.

Models for single GPe and GPi cells are very similar:

$$\begin{array}{*{20}l} {{C_{{\text{m}}} V^{\prime }_{{{\text{GPe}}}} = - I_{{\text{L}}} - I_{{{\text{Na}}}} - I_{{\text{K}}} - I_{{\text{T}}} - I_{{{\text{Ca}}}} - I_{{{\text{AHP}}}} + } \hfill} \\ {{ + I_{{{\text{STN}} \to {\text{GPe}}}} + I_{{{\text{GPe}} \to {\text{GPe}}}} + I_{{{\text{striatum}} \to {\text{GPe}}}} + I_{{{\text{MoreGPe}}}} } \hfill} \\ \end{array} $$

$$\begin{array}{*{20}l} {{C_{{\text{m}}} V^{\prime }_{{{\text{GPi}}}} = - I_{{\text{L}}} - I_{{{\text{Na}}}} - I_{{\text{K}}} - I_{{\text{T}}} - I_{{{\text{Ca}}}} - I_{{{\text{AHP}}}} + } \hfill} \\ {{ + I_{{{\text{STN}} \to {\text{GPi}}}} + I_{{{\text{GPe}} \to {\text{GPi}}}} + I_{{{\text{striatum}} \to {\text{GPi}}}} + I_{{{\text{MoreGPi}}}} + I_{{{\text{DBS}}}} } \hfill} \\ \end{array} $$

Constant, depolarizing currents I _MoreGPe and I _moreGPi were introduced to simulate more diffuse excitation from STN to pallidal cells than that reproducible by synaptic connections. I _{striatum→GPe} and I _{striatum→GPi} represent the constant inhibitory current input from striatum to pallidal cells.

Thalamo-cortical relay cells obey the following equation:

$$C_{{\text{m}}} V^{\prime }_{{{\text{TH}}}} = - I_{{\text{L}}} - I_{{{\text{Na}}}} - I_{{\text{K}}} - I_{{\text{T}}} - I_{{{\text{GPi}}}} - I_{{{\text{SM}}}} $$

I _SM represents a train of rectangular depolarizing current pulses from the cortex, identified by pulse amplitude (6 pA/μm²), pulse length (6 ms), and pulse repetition frequency.

Synaptic currents I _α→β from α to β cells were modelled as follows:

$$I_{{\alpha \to \beta }} = g_{{\alpha \to \beta }} \cdot {\left[ {V_{\beta } - E_{{\alpha \to \beta }} } \right]}{\sum\limits_j {s^{j}_{\alpha } } }$$

The summation is taken over the presynaptic α cells.

The input currents to the cells are negative if inhibitory, positive if excitatory under the sign convention used here.