Modeling shifts in the rate and pattern of subthalamopallidal network activity during deep brain stimulation

Abstract

Deep brain stimulation (DBS) of the subthlamic nucleus (STN) represents an effective treatment for medically refractory Parkinson’s disease; however, understanding of its effects on basal ganglia network activity remains limited. We constructed a computational model of the subthalamopallidal network, trained it to fit in vivo recordings from parkinsonian monkeys, and evaluated its response to STN DBS. The network model was created with synaptically connected single compartment biophysical models of STN and pallidal neurons, and stochastically defined inputs driven by cortical beta rhythms. A least mean square error training algorithm was developed to parameterize network connections and minimize error when compared to experimental spike and burst rates in the parkinsonian condition. The output of the trained network was then compared to experimental data not used in the training process. We found that reducing the influence of the cortical beta input on the model generated activity that agreed well with recordings from normal monkeys. Further, during STN DBS in the parkinsonian condition the simulations reproduced the reduction in GPi bursting found in existing experimental data. The model also provided the opportunity to greatly expand analysis of GPi bursting activity, generating three major predictions. First, its reduction was proportional to the volume of STN activated by DBS. Second, GPi bursting decreased in a stimulation frequency dependent manner, saturating at values consistent with clinically therapeutic DBS. And third, ablating STN neurons, reported to generate similar therapeutic outcomes as STN DBS, also reduced GPi bursting. Our theoretical analysis of stimulation induced network activity suggests that regularization of GPi firing is dependent on the volume of STN tissue activated and a threshold level of burst reduction may be necessary for therapeutic effect.

Appendix

The network model was composed of single compartment, conductance based neuron models. The membrane voltage of each single cell was evaluated in the NEURON simulation environment (v5.8). Action potentials were detected and postsynaptic cells were notified following a set delay that a presynaptic event occurred. Source code for the published network model is available on the NeuronDB database.

1.1 Membrane currents

The active currents included in the single cell models followed a basic Hodgkin-Huxley paradigm. The individual channel kinetics were based on the formulation described in Otsuka et al. (2004), and parameters were defined to reproduce the in vitro firing properties of isolated rodent neurons (Otsuka et al. 2004; Nambu and Llinas 1994; Cooper and Stanford 2000). Corresponding intracellular current-clamp and voltage-clamp recordings from primates do not exist, so we used the next most appropriate animal model. However, we believe this is a relatively minor issue, as isolated neurons of the basal ganglia at are typically tonic firers with highly consistent pacemaker activity. We propose that the vast majority of the modulation seen in vivo is the result of network interactions, and as such we elected to concentrate on the network influence of neural activity rather than the possible nuances of interspecies channel kinetic differences. The maximum conductances for the STN and pallidal models are given in Table 5.

Table 5 Maximal conductances

$$ \begin{array}{*{20}{c}} {{I_{Na}} = {g_{Na}}{m^3}h\left( {{V_m} - {E_{Na}}} \right)} \hfill \\{{I_{Kdr}} = {g_{Kdr}}{n^4}\left( {{V_m} - {E_K}} \right)} \hfill \\{{I_{AHP}} = {g_{AHP}}{r^2}\left( {{V_m} - {E_K}} \right)} \hfill \\{{I_A} = {g_A}{a^2}b\left( {{V_m} - {E_K}} \right)\;\left( {\hbox{STN only}} \right)} \hfill \\{{I_T} = {g_T}{p^2}q\left( {{V_m} - {E_{Ca}}} \right)} \hfill \\{{I_L} = {g_L}{c^2}{d_1}{d_2}\left( {{V_m} - {E_{Ca}}} \right)\;\left( {\hbox{STN only}} \right)} \hfill \\{{I_{leak}} = {g_{leak}}\left( {{V_m} - {E_{leak}}} \right)} \hfill \\\end{array} $$

The gating variables follow first order dynamics of the form

$$ \frac{{dx}}{{dt}} = \frac{{{x_\infty }\left( {{V_m}} \right) - x}}{{{\tau_x}\left( {{V_m}} \right)}} $$

where ${x_\infty }\left( {{V_m}} \right) = {\left[ {1 + \exp \left( {\frac{{{V_m} - {\theta_{\infty, x}}}}{{{\sigma_{\infty, x}}}}} \right)} \right]^{ - 1}}$and

$$ {\tau_\infty }\left( {{V_m}} \right) = {\tau_{0,x}} + {\tau_{1,x}}{\left[ {1 + \exp \left( { - \frac{{{V_m} - {\theta_{1,x}}}}{{{\sigma_{1,x}}}}} \right)} \right]^{ - 1}} + {\tau_{2,x}}\left[ {\exp \left( { - \frac{{{V_m} - {\theta_{2,x}}}}{{{\sigma_{2,x}}}}} \right)} \right] $$

Tables 6 and 7 list the values of the parameters for the STN and pallidal model kinetics. The steady state activation value for the calcium dependent potassium current (r _∞) and the inactivation value for L-type calcium current (d _2,∞) were functions of calcium level (Cai), rather than membrane voltage.

Table 6 STN kinetic parameters

Table 7 GPx kinetic parameters

1.2 Calcium dynamics

Calcium level (Cai) follows the equation

$$ \frac{{dCai}}{{dt}} = {\varepsilon_{Ca}}\left( { - \frac{{{I_T} + {I_L}}}{{2F}} - {k_{Ca}}Cai} \right) $$

The variable Cai was not meant to be a uniform intracellular calcium concentration. Rather, it was the calcium available at or near membrane bound potassium channels. Cai was increased by calcium currents (negative inward by convention) and decreased by calcium pumps. The constant ε _Ca represents the combined effects of intracellular calcium buffering mechanisms and cellular geometry and F is Faraday’s constant. The constant k_Ca is the calcium pump rate. For all simulations ε _Ca  = 337.1 and k _Ca  = .2/ ε _Ca .

1.3 Synaptic currents

Synaptic currents were modeled by the equation ${I_{syn}} = R\left( {{V_m} - {E_{rev}}} \right)$ where R is an activation level and E _rev is the reversal potential (0 for AMPA and −80 for GABA synapses) (Destexhe et al. 1994a,b). Each synapse received events from its presynaptic cells. The conductance of the synapse increased exponentially toward the maximum conductance (gSrcTar set for each pathway) following a presynaptic event, and decreased exponentially following a set time (t _rise ) during which no presynaptic events occurred. If multiple events occurred within a short period of time, R did not decay back to 0 and therefore reached levels closer to gSrcTar. The synapse could be thought of as having two states, “ON”, where $\frac{{dR}}{{dt}} = \alpha \left( {{R_{\max }} - R} \right)$ and “OFF”, where $\frac{{dR}}{{dt}} = - \beta R$ and the initial condition was taken as R at the time of the last spike. This scheme was implemented as a continuous function dependent on the time of the last spike and the count of spikes occurring within a recent interval corresponding to the rise time of the synaptic current (t _rise  = .3).

1.4 Affects of DA level

Changing DA level from the MPTP state to the Normal state effected three parameters for each pathway in the network model, namely p_syn and mean background interspike interval for input pathways, as well as scaling of maximal conductance for most BG pathways. The value of p_syn was 0.25 in the MPTP state and 0.05 in the Normal state, as described in Methods. The mean interspike interval increased from 75 in the MPTP state to 100 in the Normal state in the CtxSTN and StrGPe pathways. The mean interspike interval decreased from 150 in the MPTP state to 100 in the Normal state in the StrGPi pathway. DA level has also been modeled as a scaling factor multiplying maximal synaptic conductances in BG (Humphries et al. 2006). Accordingly, a multiplicative factor for maximal synaptic conductances decreased from 1.25 in the MPTP state to 1.0 in the Normal state for the CtxSTN, StrGPe, GPeSTN and GPeGPi pathways and increased from 1.25 to 2.0 in the STNGPe pathway. These changes enabled the model to more closely follow the cortical beta rhythm in the low DA state.