Quantifying harmony between direct and indirect pathways in the basal ganglia: healthy and Parkinsonian states

Abstract

The basal ganglia (BG) show a variety of functions for motor and cognition. There are two competitive pathways in the BG; direct pathway (DP) which facilitates movement and indirect pathway (IP) which suppresses movement. It is well known that diverse functions of the BG may be made through “balance” between DP and IP. But, to the best of our knowledge, so far no quantitative analysis for such balance was done. In this paper, as a first time, we introduce the competition degree ${{\mathcal {C}}}_d$ between DP and IP. Then, by employing ${{\mathcal {C}}}_d$, we quantify their competitive harmony (i.e., competition and cooperative interplay), which could lead to improving our understanding of the traditional “balance” so clearly and quantitatively. We first consider the case of normal dopamine (DA) level of $\phi ^*=0.3$. In the case of phasic cortical input (10 Hz), a healthy state with ${{\mathcal {C}}}_d^* = 2.82$ (i.e., DP is 2.82 times stronger than IP) appears. In this case, normal movement occurs via harmony between DP and IP. Next, we consider the case of decreased DA level, $\phi = \phi ^*(=0.3)~x_{DA}$ ($1 > x_{DA} \ge 0$). With decreasing $x_{DA}$ from 1, the competition degree ${{\mathcal {C}}}_d$ between DP and IP decreases monotonically from ${{\mathcal {C}}}_d^*$, which results in appearance of a pathological Parkinsonian state with reduced ${{\mathcal {C}}}_d$. In this Parkinsonian state, strength of IP is much increased than that in the case of normal healthy state, leading to disharmony between DP and IP. Due to such break-up of harmony between DP and IP, impaired movement occurs. Finally, we also study treatment of the pathological Parkinsonian state via recovery of harmony between DP and IP.

Appendices

Single neuron models and DA effects

Table 3 9 single-cell parameter values in the X (= D1 SPN, D2 SPN, STN, GP, SNr) population

The Izhikevich neuron models are considered as single neuron models in the BG SNN (Izhikevich 2003, 2004, 2007a, b). Evolution of dynamical states of individual cells in the X population [$X=$ D1 (SPN), D2 (SPN), STN, GP, and SNr] is governed by the following equations:

$$\begin{aligned} C_X \frac{dv_i^{(X)}}{dt}= & {} k_X (v_i^{(X)} - v_r^{(X)}) (v_i^{(X)} - v_t^{(X)}) - u_i^{(X)} +I_i^{(X)}, \end{aligned}$$

(8)

$$\begin{aligned} \frac{du_i^{(X)}}{dt}= & {} a_X \left\{ b_X (v_i^{(X)} - v_r^{(X)}) - u_i^{(X)} \right\} ; i = 1,..., N_X, \end{aligned}$$

(9)

with the auxiliary after-spike resetting:

$$\begin{aligned} \text{if}~ v_i^{(X)} \ge v_{peak}^{(X)}, ~\text{then}~ v_i^{(X)} \leftarrow c_X ~\text{and}~ u_i^{(X)} \leftarrow u_i^{(X)}+d_X, \end{aligned}$$

(10)

where $N_X$ and $I_i^{(X)}(t)$ are the total number of cells and the current into the ith cell in the X population, respectively. In Eqs. (8) and (9), the dynamical state of the ith cell in the X population at a time t (msec) is characterized by its membrane potential $v_i^{(X)}(t)$ (mV) and the slow recovery variable $u_i^{(X)}(t)$ (pA). When $v_i^{(X)}(t)$ reaches a threshold $v_{peak}^{(X)}$ (i.e., spike cutoff value), firing a spike occurs, and then $v_i^{(X)}$ and $u_i^{(X)}$ are reset in accord with the rules of Eq. (10).

There are 9 intrinsic parameters in each X population; $C_X$ (pF): membrane capacitance, $v_r^{(X)}$ (mV): resting membrane potential, $v_t^{(X)}$ (mV): instantaneous threshold potential, $k_X$ (nS/mV): parameter associated with the cell’s rheobase, $a_X$ ($\text{msec}^{-1}$): recovery time constant, $b_X$ (nS): parameter associated with the input resistance, $c_X$ (mV): after-spike reset value of $v_i^{(X)}$, $d_X$ (pA): after-spike jump value of $u_i^{(X)}$, and $v_{peak}^{(X)}$ (mV): spike cutoff value. Table 3 shows the 9 intrinsic parameter values of the BG cells. Along with the parameter values of the D1/D2 SPNs provided in Humphries et al. (2009a); Tomkins et al. (2014), we get the parameter values of the other cells (STN, GP, SNr), founded on the work in Fountas and Shanahan (2017). In the case of GP and STN, we consider the major subpopulations of high frequency pauser (85 $\%$) and short rebound bursts (60 $\%$), respectively. Also, we use the standard 2-variable Izhikevich neuron model for the STN, instead of the 3-variable Izhikevich neuron model in Fountas and Shanahan (2017); these two models give nearly the same results for the STN.

We also consider influences of DA modulation on the D1 and D2 SPNs (Humphries et al. 2009a; Tomkins et al. 2014; Fountas and Shanahan 2017). D1 receptors activation has two opposing influences on intrinsic ion channels. It enhances the inward-rectifying potassium current (KIR), leading to hyperpolarization of the D1 SPN. In contrast, it lowers the activation threshold of the L type $\text{Ca}^{2+}$ current, resulting in depolarization of the D1 SPN. These two hyperpolarization and depolarization influences are modelled via variations in intrinsic parameters of the D1 SPN:

$$\begin{aligned} v_r\leftarrow & {} v_r (1+\beta _1^{(\text{D1})} \phi _1), \end{aligned}$$

(11)

$$\begin{aligned} d\leftarrow & {} d(1-\beta _2^{(\text{D1})} \phi _1). \end{aligned}$$

(12)

Here, Eq. (11) models the hyperpolarizing effect of the increasing KIR by upscaling $v_r$, while Eq. (12) models enhanced depolarizing effect of the L type $\text{Ca}^{2+}$ current by downscaling d. The parameters $\beta _1^{\rm (D1)}$ and $\beta _2^{\rm (D1)}$ represent the amplitudes of their respective influences, and $\phi _1$ is the DA level (i.e., fraction of active DA receptors) for the D1 SPNs.

Table 4 Effects of DA modulation on intrinsic parameters of the D1/D2 SPNs

Table 5 In-vivo firing activities of BG cells in awake resting state with tonic cortical input (3 Hz) in the case of the normal DA level of $\phi =0.3$. Spontaneous current $I_{vivo}^{(X)}$, firing rates $f_{vivo}^{(X)}$, and random background input $D_X^*$ ($X=$ D1 SPN, D2 SPN, STN, GP, and SNr)

Next, D2 receptors activation has small inhibitory influence on the slow A-type potassium current, leading to decrease in the cell’s rheobase current. This depolarizing effect is well modelled by downscaling the parameter, k:

$$\begin{aligned} k \leftarrow k (1-\beta ^{(\text{D2})} \phi _2), \end{aligned}$$

(13)

where $\phi _2$ is the DA level for the D2 SPNs, and the parameter $\beta ^{\rm (D2)}$ represents the downscaling degree in k. Table 4 shows DA modulation on the intrinsic parameters of the D1/D2 SPNs where the parameter values of $\beta _1^{\rm (D1)}$, $\beta _2^{\rm (D1)}$, and $\beta ^{\rm (D2)}$ are given (Humphries et al. 2009a; Tomkins et al. 2014; Fountas and Shanahan 2017). In this paper, we consider the case of $\phi _1 = \phi _2 = \phi$.

Time-evolution of $v_i^{(X)}(t)$ and $u_i^{(X)}(t)$ in Eqs. (8) and (9) is governed by the current $I_i^{(X)}(t)$ into the ith cell in the X population:

$$\begin{aligned} I_i^{(X)}(t) = I_{ext,i}^{(X)}(t) - I_{syn,i}^{(X)}(t) + I_{stim}^{(X)}(t). \end{aligned}$$

(14)

Here, $I_{ext,i}^{(X)}$, $I_{syn,i}^{(X)}(t)$, and $I_{stim}^{(X)}(t)$ denote the external current from the external background region (which is not considered in the modeling), the synaptic current, and the injected stimulation current, respectively. In the BG SNN, we consider the case of $I_{stim}=0$ (i.e., no injected stimulation DC current).

The external current $I_{ext,i}^{(X)}(t)$ may be modeled in terms of $I_{spon,i}^{(X)}$ [spontaneous current for spontaneous firing activity, corresponding to time average of $I_{ext,i}^{(X)}(t)$] and $I_{back,i}^{(X)}(t)$ [random background input, corresponding to fluctuation from time average of $I_{ext,i}^{(X)}(t)$]. In the BG population, $I_{spon}^{(X)}$ (independent of i) is just the spontaneous in-vivo current, $I_{vivo}^{(X)}$, to get the spontaneous in-vivo firing rate $f_{vivo}^{(X)}$ in the presence of synaptic inputs in the resting state (in-vivo recording in awake resting state with tonic cortical input). The random background current $I_{back,i}^{(X)}(t)$ is given by:

$$\begin{aligned} I_{back,i}^{(X)}(t) = D_X \cdot \xi _i^{(X)}(t). \end{aligned}$$

(15)

Here, $D_X$ is the parameter controlling the noise intensity and $\xi _i^{(X)}$ is the Gaussian white noise, satisfying the zero mean and the unit variance (Kim and Lim 2018a, b, 2020):

$$\begin{aligned} \langle \xi _i^{(X)}(t) \rangle = 0 ~\text{and}~ \langle \xi _i^{(X)}(t) \xi _j^{(X)}(t') \rangle = \delta _{ij}\delta (t-t'). \end{aligned}$$

(16)

Table 5 shows in-vivo firing activities of BG cells in awake resting state with tonic cortical input for the normal DA level of $\phi =0.3$; spontaneous in-vivo currents $I_{vivo}^{(X)}$, in-vivo firing rates $f_{vivo}^{(X)}$, and random background inputs $D_X^*$ for Humphries et al. (2006); Lindahl et al. (2013); Fountas and Shanahan (2017) are given.

Synaptic currents and DA effects

We explain the synaptic current $I_{syn,i}^{(X)}(t)$ in Eq. (14). There are two kinds of excitatory synaptic currents, $I_{\text{AMPA},i}^{(X,Y)}(t)$ and $I_{\text{NMDA},i}^{(X,Y)}(t)$, which are are the AMPA ($\alpha$-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid) receptor-mediated and NMDA (N-methyl-D-aspartate) receptor-mediated currents from the presynaptic source Y population to the postsynaptic ith cell in the target X population, respectively. In addition to these excitatory synaptic currents, there exists another inhibitory synaptic current, $I_{\text{GABA},i}^{(X,Z)}(t)$, which is the $\mathrm GABA_A$ ($\gamma$-aminobutyric acid type A) receptor-mediated current from the presynaptic source Z population to the postsynaptic ith cell in the target X population.

Here, we follow the “canonical” formalism for the synaptic currents, as in our previous works in the cerebellum (Kim and Lim 2021a, b) and the hippocampus (Kim and Lim 2022a, b, c, 2023). The synaptic current $I_{R,i}^{(T,S)}(t)$ R (= AMPA, NMDA, or GABA) obeys the following equation:

$$\begin{aligned} I_{R,i}^{(T,S)}(t) = g_{R,i}^{(T,S)}(t)~(v_{i}^{(T)}(t) - V_{R}^{(S)}), \end{aligned}$$

(17)

where $g_{(R,i)}^{(T,S)}(t)$ and $V_R^{(S)}$ are synaptic conductance and synaptic reversal potential, respectively.

The synaptic conductance $g_{R,i}^{(T,S)}(t)$ is given by:

$$\begin{aligned} g_{R,i}^{(T,S)}(t) = {{\widetilde{g}}}_{max,R}^{(T,S)} \sum _{j=1}^{N_S} w_{ij}^{(T,S)} ~ s_{j}^{(T,S)}(t), \end{aligned}$$

(18)

where ${{\widetilde{g}}}_{max,R}^{(T,S)}$ and $N_S$ are the maximum synaptic conductance and the number of cells in the source population S, respectively. Here, the connection weight $w_{ij}^{(T,S)}$ is 1 when the jth presynaptic cell is connected to the ith postsynaptic cell; otherwise (i.e., in the absence of such synaptic connection), $w_{ij}^{(T,S)}=0$.

We note that, $s^{(T,S)}(t)$ in Eq. (18) denotes fraction of open postsynaptic ion channels which are opened through binding of neurotransmitters (emitted from the source population S). A sum of exponential-decay functions $E_{R}^{(T,S)} (t - t_{f}^{(j)}-\tau _{R,l}^{(T,S)})$ provides time evolution of $s_j^{(T,S)}(t)$ of the jth cell in the source S population:

$$\begin{aligned} s_{j}^{(T,S)}(t) = \sum _{f=1}^{F_{j}^{(S)}} E_{R}^{(T,S)} (t - t_{f}^{(j)}-\tau _{R,l}^{(T,S)}), \end{aligned}$$

(19)

where $F_j^{(S)},$ $t_f^{(j)},$ and $\tau _{R,l}^{(T,S)}$ are the total number of spikes and the fth spike time of the jth cell, and the synaptic latency time constant, respectively.

Similar to our previous works in the cerebellum (Kim and Lim 2021a, b), we use the exponential-decay function $E_{R}^{(T,S)} (t)$:

$$\begin{aligned} E_{R}^{(T,S)}(t) = e^{-t/\tau _{R,d}^{(T,S)}} \cdot \varTheta (t), \end{aligned}$$

(20)

where $\tau _{R,d}^{(T,S)}$ is the synaptic decay time constant and the Heaviside step function satisfies $\varTheta (t)=1$ for $t \ge 0$ and 0 for $t <0$.

Table 6 Synaptic parameter values. S: source population, T: target population, R: receptor, ${\tilde{g}}_{max,R}^{(T,S)}$: maximum synaptic conductances, $\tau _{R,d}^{(T,S)}$: synaptic decay times, $\tau _{R,l}^{(T,S)}$: synaptic delay times, and $V_R^{(S)}$: synaptic reversal potential

Table 7 Effects of DA modulation on synaptic currents into the target population (T); T: D1 SPN, D2 SPN, STN, and GP

We also note that, in the case of NMDA-receptor, the positive magnesium ions Mg$^{2+}$ block some of the postsynaptic NMDA channels. For this case, fraction of non-blocked NMDA channels is given by a sigmoidal function $f(v^{(T)})$ (Jahr and Stevens 1990; Humphries et al. 2009a; Fountas and Shanahan 2017),

$$\begin{aligned} f(v^{(T)}(t)) = \frac{1}{1+0.28 \cdot [\text{Mg}^{2+}] \cdot e^{-0.062 v^{(T)}(t)}}, \end{aligned}$$

(21)

where $v^{(T)}$ is the membrane potential of a cell in the target population T and $[\text{Mg}^{2+}]$ is the equilibrium concentration of magnesium ions ($[\text{Mg}^{2+}]$ = 1 mM). Then, the synaptic current into the ith cell in the target X population becomes

$$\begin{aligned} I_{syn,i}^{(X)}(t) = I_{\text{AMPA},i}^{(X,Y)}(t) + f(v_i^{(X)}(t)) \cdot I_{\text{NMDA},i}^{(X,Y)}(t) + I_{\text{GABA},i}^{(X,Z)}(t). \end{aligned}$$

(22)

Table 6 shows the synaptic parameters; synaptic parameter values. S: source population, T: target population, R: receptor, ${\tilde{g}}_{max,R}^{(T,S)}$: maximum synaptic conductances, $\tau _{R,d}^{(T,S)}$: synaptic decay times, $\tau _{R,l}^{(T,S)}$: synaptic delay times, and $V_R^{(S)}$: synaptic reversal potential.

We also take into consideration the influence of DA modulation on the synaptic currents into D1 SPN, D2 SPN, STN, and GP cells in Fig. 1 (Humphries et al. 2009a; Tomkins et al. 2014; Fountas and Shanahan 2017). For the synaptic currents into the D1 SPNs, influence of DA modulation ma be modeled by upscaling the NMDA receptor-mediated current $I_\text{NMDA}$ with the factor $\beta ^{(\text{D1})}$:

$$\begin{aligned} I_\text{NMDA} \leftarrow I_\text{NMDA} (1+\beta ^{(\text{D1})} \phi _1). \end{aligned}$$

(23)

Here, $\phi _1$ is the DA level for the D1 SPNs. (There is no DA influence on $I_\text{AMPA}$ for the D1 SPNs.) On the other hand, for the synaptic currents into the D2 SPNs, influence of DA modulation could be modeled by downscaling the AMPA receptor-mediated current $I_\text{AMPA}$ with the factor $\beta ^{(\text{D2})}$:

$$\begin{aligned} I_\text{AMPA} \leftarrow I_\text{AMPA} (1-\beta ^{(\text{D2})} \phi _2). \end{aligned}$$

(24)

Here, $\phi _2$ is the DA level for the D2 SPNs. (There is no DA influence on $I_\text{NMDA}$ for the D2 SPNs.) The scaling factors $\beta ^{(\text{D1})}$ and $\beta ^{(\text{D2})}$ are given in Table 7. Also, effects of DA modulation on synaptic currents into STN cells and GP cells are well given in Table 7. In these cases, all excitatory and inhibitory synaptic currents, $I_\text{AMPA}$, $I_\text{NMDA}$, and $I_\text{GABA}$, are downscaled with their scaling factors, depending on $\phi _2$. Here, $\phi _1 = \phi _2 = \phi$.