# Break-up and Recovery of Harmony between Direct and Indirect Pathways in The
Basal Ganglia; Huntington’s Disease and Treatment

Sang-Yoon Kim<EMAIL_ADDRESS>Woochang Lim<EMAIL_ADDRESS>Institute for
Computational Neuroscience and Department of Science Education, Daegu National
University of Education, Daegu 42411, Korea

###### Abstract

The basal ganglia (BG) in the brain exhibit diverse functions for motor,
cognition, and emotion. Such BG functions could be made via competitive
harmony between the two competing pathways, direct pathway (DP) (facilitating
movement) and indirect pathway (IP) (suppressing movement). As a result of
break-up of harmony between DP and IP, there appear pathological states with
disorder for movement, cognition, and psychiatry. In this paper, we are
concerned about the Huntington’s disease (HD), which is a genetic
neurodegenerative disorder causing involuntary movement and severe cognitive
and psychiatric symptoms. For the HD, the number of D2 SPNs ($N_{\rm D2}$) is
decreased due to degenerative loss, and hence, by decreasing $x_{\rm D2}$
(fraction of $N_{\rm D2}$), we investigate break-up of harmony between DP and
IP in terms of their competition degree ${\cal C}_{d}$, given by the ratio of
strength of DP (${\cal S}_{DP}$) to strength of IP (${\cal S}_{IP}$) (i.e.,
${\cal C}_{d}={\cal S}_{DP}/{\cal S}_{IP}$). In the case of HD, the IP is
under-active, in contrast to the case of Parkinson’s disease with over-active
IP, which results in increase in ${\cal C}_{d}$ (from the normal value). Thus,
hyperkinetic dyskinesia such as chorea (involuntary jerky movement) occurs.
Treatment of HD, based on optogenetics, is also investigated through recovery
of harmony between DP and IP. Finally, we study effect of loss of healthy
synapses of all the BG cells on HD.

Basal ganglia, Huntington’s disease, Direct pathway (DP), Indirect
pathways(IP), Harmony between DP and IP, Competition degree

###### pacs:

87.19.lj, 87.19.lu, 87.19.rs

##  I Introduction

The basal ganglia (BG) (called the dark basement of the brain) are a group of
subcortical deep-lying nuclei, receiving excitatory cortical input from most
areas of cortex, and they provide inhibitory output to the thalamus and
brainstem Luo ; Kandel ; Squire ; Bear . The BG exhibit a variety of functions
for motor control and regulation of cognitive and emotional processes Luo ;
Kandel ; Squire ; Bear ; GPR1 ; GPR2 ; Hump1 ; Hump2 ; Hump3 ; Man .
Dysfunction in the BG is related to movement disorder [e.g., Parkinson’s
disease (PD) and Huntington’s disease (HD)] and cognitive and psychiatric
disorders Luo ; Kandel ; Squire ; Bear .

In this paper, we are concerned about the HD. It is a rare hereditary
neurodegenerative disease with severe symptoms for motor, cognition, and
emotion HD1 ; HD2 ; HD3 ; HD4 ; HD5 ; HD6 ; HD7 . As is well known, patients
with HD show hyperkinetic dyskinesia such as chorea (involuntary jerky dance-
like movement) as well as cognitive (e.g., dementia) and psychiatric (e.g,,
depression and anxiety) disorders. In contrast, patients with PD show
hypokinetic disorder such as slowed movement (bradykinesia) PD1 ; PD2 ; PD3 ;
PD4 ; PD5 ; PD6 . Thus, if PD lies at one end of the spectrum of movement
disorders in the BG, HD lies at the other end. We note that HD is caused by a
mutated huntingtin (HTT) gene on chromosome 4 HTT1 ; HTT2 . As a result of
mutation in HTT gene, the defective HTT gene has abnormal excessive repeats of
a three-base (CAG) DNA sequence; in the mutant gene, the repeat occurs over
and over again, from 40 times to more than 80. The greater the number of CAG
repeats, the earlier the onset and severity of HD. This kind of trinucleotide
repeat expansion results in production of abnormal HTT protein that
accumulates, resulting in creation of toxic HTT protein aggregates damaging
neurons (e.g., death of striatal cells in the BG). Thus, the primary
pathological feature of HD is appearance of toxic HTT protein aggregates,
causing the characteristic neurodegeneration seen in HD, in contrast to the
case of PD where DA deficiency is a major cause.

In our previous work for the PD in the BG KimPD , we developed a spiking
neural network (SNN) for the BG, based on anatomical and physiological data
derived from rat-based studies Ana4 ; Ana5 ; Ana6 ; Ana7 ; Phys1 ; Phys2 ;
Phys3 ; Phys4 ; Phys5 ; Phys6 ; Phys7 ; Phys8 ; Phys9 ; Phys10 ; Phys11 ;
Phys12 ; Phys13 ; Phys14 ; Phys15 ; Phys16 ; Phys17 ; Phys18 ; Phys19 ; Phys20
. Here, we use rat-brain terminology throughout. The BG receive excitatory
cortical input from most regions of cortex via the input nuclei [striatum and
subthalamic nucleus (STN)] and project inhibitory output via the output
nucleus [substantia nigra pars reticulata (SNr)], through the thalamus to the
motor area of the cortex Hump1 ; Man . We also note that, the principal input
nucleus, striatum, is the primary recipient of dopamine (DA), arising from the
substantia nigra pars compacta (SNc). Within the striatum, spine projection
neurons (SPNs), comprising up to 95 $\%$ of the whole striatal population, are
the only primary output neurons Str1 ; Str2 . There are two types of SPNs with
D1 and D2 receptors for the DA. The DA modulates firing activity of the D1 and
D2 SPNs in a different way SPN1 ; SPN2 ; CN6 . In the early stage of HD,
degenerative loss of D2 SPNs occurs due to mutation in the HTT gene, while DA
level in the striatum is nearly normal Degen1 ; Degen2 ; Degen3 ; Degen4 .

There are two competing pathways, direct pathway (DP) and indirect pathway
(IP), in the BG DIP1 ; DIP2 ; DIP3 ; DIP4 . D1 SPNs in the striatum make
direct inhibitory projection to the output nucleus, SNr, through DP, and then
the thalamus becomes disinhibited. Consequently, movement facilitation occurs.
In contrast, D2 SPNs are connected to the SNr through IP, crossing the
intermediate control nucleus, GP (globus pallidus), and the STN. In the case
of IP, the firing activity of the SNr becomes enhanced mainly because of
excitatory input from the STN. As a result, firing activity of the thalamus
becomes decreased, resulting in movement suppression.

Diverse functions of the BG could be made via “balance” of DP and IP. So far,
a variety of subjects for the BG have been investigated in many computational
works GPR1 ; GPR2 ; Hump1 ; Hump2 ; Hump3 ; Man ; Str2 ; SPN1 ; SPN2 ; Frank1
; Frank2 ; CN1 ; CN2 ; CN3 ; CN4 ; CN5 ; CN6 ; CN7 ; CN8 ; CN9 ; CN10 ; CN11 ;
CN12 ; CN13 ; CN14 ; CN15 ; CN16 ; CN17 ; CN18 ; CN19 ; CN20 ; CN21 ; CN22 ;
CNHD1 ; CNHD2 ; CNHD3 . But, no quantitative analysis for balance between DP
and IP was made. As a first time, in our recent work KimPD , we made
quantitative analysis for competitive harmony (i.e., competition and
cooperative interplay) between DP and IP by introducing their competition
degree ${\cal C}_{d}$, given by the ratio of strength of DP (${\cal S}_{DP}$)
to strength of IP (${\cal S}_{IP}$) (i.e., ${\cal C}_{d}={\cal S}_{DP}/{\cal
S}_{IP}$); ${\cal S}_{DP}$ $({\cal S}_{IP})$ is given by the magnitude of the
total time-averaged synaptic current into the output nucleus, SNr, through DP
(IP).

In this paper, we take into consideration of degenerative loss of D2 SPNs for
the HD; $N_{\rm D2}$ (number of D2 SPNs) = $N_{\rm D2}^{*}$ (normal value)
$x_{\rm D2}$ [$1>x_{\rm D2}$ (fraction of number of D2 SPNs) $\geq 0$] Degen1
; Degen2 ; Degen3 ; Degen4 ; Degen5 ; Degen6 . By decreasing $x_{\rm D2}$ from
1, we investigate break-up of harmony between DP and IP for the HD by
employing the competition degree ${\cal C}_{d}$ in the case of normal DA level
($\phi=0.3$). Due to degenerative loss of D2 SPNs, IP becomes under-active
(i.e., weakened), leading to increase in ${\cal C}_{d}$ from normal value.
Thus, hyperkinetic dyskinesia such as chorea occurs, which is in contrast to
the case of PD with reduced ${\cal C}_{d}$, causing hypokinetic disorder.
Next, based on optogenetics OG1 ; OG2 , treatment of HD is also studied via
recovery of harmony between DP and IP. Through activation of D2 SPN (STN) and
deactivation of GP, IP becomes strengthened and thus harmony between DP and IP
may be recovered. Finally, we investigate effect of loss of healthy synapses
of all the BG cells on HD Loss1 ; Loss2 ; Loss3 ; Loss4 ; Loss5 ; Loss6 .

This paper is organized as follows. In Sec. II, we describe our SNN for the
BG. Then, in the main Sec. III, we make quantitative analysis of break-up and
recovery of harmony between DP and IP for the HD. Finally, we give summary and
discussion in Sec. IV.

##  II BG Spiking Neural Network

In this section, we describe our SNN for the BG. In our recent work KimPD , we
developed our BG SNN, composed of D1/D2 SPNs, STN neurons, GP neurons, and SNr
neurons, based on the anatomical and the physiological properties of the BG
Ana4 ; Ana5 ; Ana6 ; Ana7 ; Phys1 ; Phys2 ; Phys3 ; Phys4 ; Phys5 ; Phys6 ;
Phys7 ; Phys8 ; Phys9 ; Phys10 ; Phys11 ; Phys12 ; Phys13 ; Phys14 ; Phys15 ;
Phys16 ; Phys17 ; Phys18 ; Phys19 ; Phys20 . In our BG SNN, we also consider
the modulation effect of DA on D1/D2 SPN and afferent synapses into the D1/D2
SPNs, the STN, and the GP SPN1 ; SPN2 ; CN6 . Here, for the sake of
completeness and understanding of readers, we provide description of our BG
SNN; for more details, refer to Sec. II in Ref. KimPD .

Figure 1: Box diagram of our spiking neural network for the basal ganglia (BG). Excitatory and inhibitory connections are denoted by lines with triangles and circles, respectively, and dopamine-modulated cells and connections are represented in blue color. Striatum and STN (subthalamic nucleus), receiving the excitatory cortical input, are two input nuclei to the BG. In the striatum, there are two kinds of inhibitory spine projection neurons (SPNs); SPNs with the D1 receptors (D1 SPNs) and SPNs with D2 receptors (D2 SPNs). The D1 SPNs make direct inhibitory projection to the output nuclei SNr (substantia nigra pars reticulate) through the direct pathway (DP; green color). In contrast, the D2 SPNs are connected to the SNr through the indirect pathway (IP; red color) crossing the GP (globus pallidus) and the STN. The inhibitory output from the SNr to the thalamus/brainstem is controlled through competition between the DP and IP.  Table 1: Intrinsic parameter values for each BG cell in the $X$ [= D1 (SPN), D2 (SPN), STN, GP, SNr] population. Parmeters |  D1/D2 SPN |  STN |  GP |  SNr
---|---|---|---|---
$C_{X}$ | 16.1 | 23.0 | 68.0 | 172.1
$v_{r}^{(X)}$ | -80.0 | -56.2 | -53.0 | -64.58
$v_{t}^{(X)}$ | -29.3 | -41.4 | -44.0 | -51.8
$k_{X}$ | 1 | 0.439 | 0.943 | 0.7836
$a_{X}$ | 0.01 | 0.021 | 0.0045 | 0.113
$b_{X}$ | -20 | 4 | 3.895 | 11.057
$c_{X}$ | -55 | -47.7 | -58.36 | -62.7
$d_{X}$ | 84.2 | 17.1 | 0.353 | 138.4
$v_{peak}^{(X)}$ | 40 | 15.4 | 25 | 9.8

###  II.1 Framework of Our SNN of The BG

Figure 1 shows a box diagram of major neurons and synaptic connections in our
BG SNN. Our BG SNN consists of striatum and STN (input nuclei), SNr (output
nucleus), and GP (intermediate controller). Here, all the BG neurons except
the excitatory STN neuron are inhibitory ones.

Both striatum and STN receive cortical inputs from most regions of the cortex.
Here, we model cortical inputs in terms of 1,000 independent Poisson spike
trains with firing rate $f_{i}$ $(i=1,\cdots,1000)$. Tonic cortical input in
the resting state has $f=3$ Hz. On the other hand, the phasic cortical input
in the phasically-active state has $f=10$ Hz, independently of $i$ Hump1 ; CI1
; CI2 ; CI3 ; CI4 ; CI5 ; Str2 ; CN6 ; CN14 . The striatum is also the primary
recipient of the DA (arising from the SNc). Two kinds of SPNs with D1 and D2
receptors for the DA exist within the striatum. They comprise up to 95 $\%$ of
the whole striatal population Str1 ; Str2 . We note that these D1 and D2 SPNs
show different firing behaviors because of DA modulation SPN1 ; SPN2 ; CN6 .

There are two competing pathways, DP and IP, in the BG DIP1 ; DIP2 ; DIP3 ;
DIP4 . The D1 SPNs make direct inhibitory projection to the output nucleus,
SNr, via DP (green color in Fig. 1). Then, the thalamus is disinhibited,
resulting in movement facilitation. On the other hand, the D2 SPNs are
connected to the SNr via IP (red color in Fig. 1), crossing the GP and the
STN. Here, the GP plays a role of intermediate controller to control the
firing behavior of the STN. In the case of IP, the firing activity of the SNr
becomes increased mainly due to excitatory input from the STN. Consequently,
firing activity of the thalamus becomes decreased, leading to movement
suppression. Thus, the firing activity of the output nucleus, SNr, is
controlled through competition between DP (green) and IP (red).

We choose the numbers of the striatal neurons, the STN neurons, the GP
neurons, and the SNr neurons in the BG, based on the anatomical information
Ana6 . A scaled-down SNN where the total number of striatal neurons is
$2,791$, corresponding to $\frac{1}{1000}$ of the $2,791\cdot 10^{3}$ striatal
cells found in the rat BG, is developed. In this way, we scale down with ratio
$10^{-3}$ for all the BG neurons CN11 ; CN17 ; $N_{\rm D1}=N_{\rm D2}=1325,$
$N_{\rm STN}=14,$ $N_{\rm GP}=46,$ and $N_{\rm SNr}=26$ ($N_{X}$ is the number
of neurons in the $X$ population). 90-97 $\%$ of the whole striatal population
corresponds to the major subpopulation of D1/D2 SPNs CN11 ; in our BG SNN, we
choose 95 $\%$. The remaining 5 $\%$ subpopulation of fast spiking
interneurons are not considered in our SNN.

The cortex (Ctx) provides the external excitatory inputs randomly to the D1/D2
SPNs and the STN neurons with the connection probabilities, $p_{c}^{({\rm
SPN,Ctx})}$ = 0.084 (8.4 $\%$) and $p_{c}^{({\rm STN,Ctx})}$ = 0.03 (3 $\%$),
respectively CN6 . Here, we consider random synaptic connections between BG
cells in Fig. 1. The synaptic connection probabilities $p_{c}^{(T,S)}$ from a
presynaptic neuron in the source population ($S$) to a postsynaptic neuron in
the target population ($T$) in the BG are as follows CN14 : $p_{c}^{({\rm
SNr,D1})}=0.033$ (3.3 $\%$), $p_{c}^{({\rm GP,D2})}=0.033$ (3.3 $\%$),
$p_{c}^{({\rm GP,STN})}=0.3$ (30 $\%$), $p_{c}^{({\rm GP,GP})}=0.1$ (10 $\%$),
$p_{c}^{({\rm STN,GP})}=0.1$ (10 $\%$), $p_{c}^{({\rm SNr,STN})}=0.3$ (30
$\%$), and $p_{c}^{({\rm SNr,GP})}=0.1066$ (10.66 $\%$).

###  II.2 Izhikevich Spiking Neuron Models and DA Effects in Our BG SNN

The Izhikevich spiking neuron model (which is not only biologically plausible,
but also computationally efficient) is used as elements of our BG SNN Izhi1 ;
Izhi2 ; Izhi3 ; Izhi4 . Unlike the Hodgkin-Huxley-type conductance-based
models, the Izhikevich model matches neuronal dynamics by tuning its
parameters.

Our BG SNN consists of 5 populations of D1 SPNs, D2 SPNs, STN neurons, GP
neurons, and SNr neurons. The following equations govern evolution of
dynamical states of individual neurons in the $X$ population [$X=$ D1 (SPN),
D2 (SPN), STN, GP, and SNr]:

$\displaystyle C_{X}\frac{dv_{i}^{(X)}}{dt}$ $\displaystyle=$ $\displaystyle
k_{X}(v_{i}^{(X)}-v_{r}^{(X)})(v_{i}^{(X)}-v_{t}^{(X)})$ (1) $\displaystyle-
u_{i}^{(X)}+I_{i}^{(X)},$ $\displaystyle\frac{du_{i}^{(X)}}{dt}$
$\displaystyle=$ $\displaystyle
a_{X}\left\\{b_{X}(v_{i}^{(X)}-v_{r}^{(X)})-u_{i}^{(X)}\right\\};$ (2)
$\displaystyle~{}~{}~{}~{}~{}~{}i=1,...,N_{X},$

with the auxiliary after-spike resetting:

${\rm if}~{}v_{i}^{(X)}\geq v_{peak}^{(X)},~{}{\rm
then}~{}v_{i}^{(X)}\leftarrow c_{X}~{}{\rm and}~{}u_{i}^{(X)}\leftarrow
u_{i}^{(X)}+d_{X}.$ (3)

Here, $N_{X}$ and $I_{i}^{(X)}(t)$ are the total number of neurons and the
current into the $i$th neuron in the $X$ population, respectively. The
dynamical state of the $i$th neuron in the $X$ population at a time $t$ (msec)
is given by its membrane potential $v_{i}^{(X)}(t)$ (mV) and the slow recovery
variable $u_{i}^{(X)}(t)$ (pA). As the membrane potential $v_{i}^{(X)}(t)$
arrives at its apex $v_{peak}^{(X)}$, the neuron fires, and then both the
membrane potential $v_{i}^{(X)}$ and the recovery variable $u_{i}^{(X)}$ are
reset according to the rules of Eq. (3).

The Izhikevich neuron model has 9 intrinsic parameters in each $X$ population;
$C_{X}$ (membrane capacitance, pF), $v_{r}^{(X)}$ (resting membrane potential,
mV), $v_{t}^{(X)}$ (instantaneous threshold potential, mV), $k_{X}$ (parameter
associated with the neuron’s rheobase, nS/mV), $a_{X}$ (recovery time
constant, ${\rm msec}^{-1}$), $b_{X}$ (parameter associated with the input
resistance, nS), $c_{X}$ (after-spike reset value of $v_{i}^{(X)}$, mV),
$d_{X}$ (after-spike jump value of $u_{i}^{(X)}$, pA), and $v_{peak}^{(X)}$
(spike cutoff value, mV). The 9 intrinsic parameter values of D1 SPN, D2 SPN,
STN, GP, and SNr are shown in Table 1. We obtain the parameter values of the
STN neuron, GP neuron, and SNr neuron, based on the work in CN6 , in addition
to the parameter values of the D1/D2 SPNs given in SPN1 ; SPN2 . These
parameter values are based on physiological properties of the D1/D2 SPNs, the
STN neurons, the GP neurons, and the SNr neurons Phys1 ; Phys2 ; Phys3 ; Phys4
; Phys5 ; Phys6 ; Phys7 ; Phys8 ; Phys9 ; Phys10 ; Phys11 .

Time-evolution of the dynamical state [$v_{i}^{(X)}(t),$ $u_{i}^{(X)}(t)$] is
governed by the current $I_{i}^{(X)}(t)$ into the $i$th neuron in the $X$
population in Eq. (1), given by:

$I_{i}^{(X)}(t)=I_{ext,i}^{(X)}(t)-I_{syn,i}^{(X)}(t)+I_{stim}^{(X)}(t),$ (4)

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

We model the external current $I_{ext,i}^{(X)}(t)$ in terms of its time
average, $I_{spon,i}^{(X)}$ (spontaneous current for spontaneous firing
activity), and fluctuation from the time average, $I_{back,i}^{(X)}(t)$
(random background input). Here, $I_{spon}^{(X)}$ (independent of $i$) is just
the spontaneous in-vivo current, $I_{vivo}^{(X)}$, to obtain the spontaneous
in-vivo firing rate $f_{vivo}^{(X)}$ in the presence of synaptic inputs in the
tonic cortical resting state. We also model the random background current
$I_{back,i}^{(X)}(t)$ as follows:

$I_{back,i}^{(X)}(t)=D_{X}\cdot\xi_{i}^{(X)}(t).$ (5)

Here, $D_{X}$ is the noise intensity parameter and $\xi_{i}^{(X)}$ is the
Gaussian white noise, which satisfies the zero mean and the unit variance Kim1
; Kim2 ; Kim3 :

$\langle\xi_{i}^{(X)}(t)\rangle=0~{}{\rm
and}~{}\langle\xi_{i}^{(X)}(t)\xi_{j}^{(X)}(t^{\prime})\rangle=\delta_{ij}\delta(t-t^{\prime}).$
(6)

Finally, we consider the effects of DA modulation on the D1 and D2 SPNs SPN1 ;
SPN2 ; CN6 . As a result of D1 receptors activation, there occur two opposing
effects on intrinsic ion channels. Enhancement of the inward-rectifying
potassium current (KIR) results in hyperpolarization of the D1 SPN. On the
other hand, decrease in the activation threshold of the L type ${\rm Ca}^{2+}$
current leads to depolarization of the D1 SPN. These two competing
hyperpolarization and depolarization effects may be modeled through changes in
intrinsic parameters of the D1 SPN:

$v_{r}(1+\beta_{1}^{({\rm D1})}\phi_{1})~{}~{}~{}{\rm
and}~{}~{}~{}d(1-\beta_{2}^{({\rm D1})}\phi_{1}),$ (7)

where $\beta_{1}^{({\rm D1})}=0.0289,$ $\beta_{2}^{({\rm D1})}=0.331,$ and
$\phi_{1}$ is the DA level (i.e., fraction of active DA receptors) for the D1
SPNs. Here, the hyperpolarizing effect of the increasing KIR is modeled by
upscaling $v_{r}$ in Eq. (7). On the other hand, enhanced depolarizing effect
of the L type ${\rm Ca}^{2+}$ current is modelled by downscaling $d$ in Eq.
(7). The parameters $\beta_{1}^{({\rm D1})}$ and $\beta_{2}^{({\rm D1})}$
denote the amplitudes of their respective effects.

Table 2: Spontaneous in-vivo current $I_{vivo}^{(X)}$, in-vivo firing rates $f_{vivo}^{(X)}$, and random background input $D_{X}^{*}$ for in-vivo firing activities of BG cells in awake resting state with tonic cortical input (3 Hz) for the normal DA level of $\phi=0.3$; $X=$ D1 (SPN), D2 (SPN), STN, GP, and SNr Parmeters | D1/D2 SPN |  STN |  GP |  SNr
---|---|---|---|---
$I_{vivo}^{(X)}$ | 0 | 56.5 | 84.0 | 292.0
$f_{vivo}^{(X)}$ | 1 | 9.9 | 29.9 | 25.5
$D_{X}^{*}$ | 246 | 11.9 | 274 | 942

In contrast, small inhibitory effect on the slow A-type potassium current
through D2 receptors activation results in decrease in the neuron’s rheobase
current. We model the depolarizing effect by downscaling the parameter, $k$:

$k(1-\beta^{({\rm D2})}\phi_{2}),$ (8)

where $\phi_{2}$ is the DA level for the D2 SPNs, and the parameter
$\beta^{({\rm D2})}~{}(=~{}0.032)$ denotes the downscaling degree in $k$.

In this paper, we consider the case of normal DA level throughout;
$\phi_{1}=\phi_{2}=\phi=0.3$. In-vivo firing activities of BG neurons in awake
resting state with tonic cortical input (3 Hz) are shown in Table 2;
spontaneous in-vivo currents $I_{vivo}^{(X)}$, in-vivo firing rates
$f_{vivo}^{(X)}$, and random background inputs $D_{X}^{*}$ in each $X$
population [$X=$ D1 (SPN), D2 (SPN), STN, GP, and SNr] are given.

###  II.3 Synaptic Currents and DA Effects in Our BG SNN

We consider the synaptic current $I_{syn,i}^{(X)}(t)$ into the $i$th neuron in
the $X$ population in Eq. (4). As in our previous works Cere1 ; Cere2 ; Hippo1
; Hippo2 ; Hippo3 ; Hippo4 , we follow the “canonical” formalism for the
synaptic currents. There are 3 kinds of synaptic currents, $I_{{\rm
AMPA},i}^{(X,Y)}(t)$, $I_{{\rm NMDA},i}^{(X,Y)}(t)$, and $I_{{\rm
GABA},i}^{(X,Z)}(t)$. Here, $I_{{\rm AMPA},i}^{(X,Y)}(t)$ and $I_{{\rm
NMDA},i}^{(X,Y)}(t)$ are the excitatory 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 $i$th neuron in the
target $X$ population. In contrast, $I_{{\rm GABA},i}^{(X,Z)}(t)$ is the
inhibitory $\rm GABA_{A}$ ($\gamma$-aminobutyric acid type A) receptor-
mediated current from the presynaptic source $Z$ population to the
postsynaptic $i$th neuron in the target $X$ population.

The $R$ (= AMPA, NMDA, or GABA) receptor-mediated synaptic current
$I_{R,i}^{(T,S)}(t)$ from the presynaptic source $S$ population to the $i$th
postsynaptic neuron in the target $T$ population is given by:

$I_{R,i}^{(T,S)}(t)=g_{R,i}^{(T,S)}(t)~{}(v_{i}^{(T)}(t)-V_{R}^{(S)}).$ (9)

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

We get the synaptic conductance $g_{R,i}^{(T,S)}(t)$ from:

$g_{R,i}^{(T,S)}(t)={\widetilde{g}}_{max,R}^{(T,S)}\sum_{j=1}^{N_{S}}w_{ij}^{(T,S)}~{}s_{j}^{(T,S)}(t).$
(10)

Here, ${\widetilde{g}}_{max,R}^{(T,S)}$ is the maximum synaptic conductance
and $N_{S}$ is the number of neurons in the source population. The
interpopulation synaptic connection from the source $S$ population to the
target $T$ population is given by the connection weight matrix $W^{(T,S)}$
($=\\{w_{ij}^{(T,S)}\\}$) where $w_{ij}^{(T,S)}=1$ in the presence of synaptic
connection from the $j$th neuron in the source $S$ population to the $i$th
neuron in the target $T$ population; otherwise, in the absence of synaptic
connection, $w_{ij}^{(T,S)}=0$.

Fraction of open postsynaptic ion channels (opened via binding of
neurotransmitters emitted from the source $S$ population) is represented by
$s^{(T,S)}(t)$ in Eq. (10). Time evolution of $s^{(T,S)}(t)$ is given by a sum
of exponential-decay functions
$E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)})$:

$s_{j}^{(T,S)}(t)=\sum_{f=1}^{F_{j}^{(S)}}E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)}).$
(11)

Here, $t_{f}^{(j)}$ and $F_{j}^{(S)}$ are the $f$th spike time and the total
number of spikes of the $j$th cell in the source $S$ population, respectively,
and $\tau_{R,l}^{(T,S)}$ is the synaptic latency time constant for
$R$-mediated synaptic current.

As in our previous works in the cerebellum Cere1 ; Cere2 , the exponential-
decay function $E_{R}^{(T,S)}(t)$ (corresponding to contribution of a
presynaptic spike occurring at $t=0$ in the absence of synaptic latency) is
given by:

$E_{R}^{(T,S)}(t)=e^{-t/\tau_{R,d}^{(T,S)}}\cdot\Theta(t).$ (12)

Here, $\tau_{R,d}^{(T,S)}$ is the synaptic decay time constant, and
$\Theta(t)$ is the Heaviside step function: $\Theta(t)=1$ for $t\geq 0$ and 0
for $t<0$.

Table 3: Parameters for the synaptic currents from the source population ($S$) to the target population ($T$): Maximum synaptic conductances $\tilde{g}_{max,R}^{(T,S)}$, synaptic decay times $\tau_{R,d}^{(T,S)}$, synaptic delay times $\tau_{R,l}^{(T,S)}$, and synaptic reversal potential $V_{R}^{(S)}$. $S\rightarrow T$ | $R$ | $\tilde{g}_{max,R}^{(T,S)}$ | $\tau_{R,d}^{(T,S)}$ | $\tau_{R,l}^{(T,S)}$ | $V_{R}^{(S)}$
---|---|---|---|---|---
Ctx $\rightarrow$ D1/D2 SPN | AMPA | 0.6 | 6 | 10 | 0
NMDA | 0.3 | 160 | 10 | 0
Ctx $\rightarrow$ STN | AMPA | 0.388 | 2 | 2.5 | 0
NMDA | 0.233 | 100 | 2.5 | 0
D1 SPN $\rightarrow$ SNr | GABA | 4.5 | 5.2 | 4 | -80
D2 SPN $\rightarrow$ GP | GABA | 3.0 | 6 | 5 | -65
STN $\rightarrow$ GP | AMPA | 1.29 | 2 | 2 | 0
NMDA | 0.4644 | 100 | 2 | 0
GP $\leftrightarrow$ GP | GABA | 0.765 | 5 | 1 | -65
GP $\rightarrow$ STN | GABA | 0.518 | 8 | 4 | -84
STN $\rightarrow$ SNr | AMPA | 12 | 2 | 1.5 | 0
NMDA | 5.04 | 100 | 1.5 | 0
GP $\rightarrow$ SNr | GABA | 73 | 2.1 | 3 | -80
Figure 2: Involuntary jerky movement due to degenerative loss of D2 SPNs in
the tonic pathological state for the tonic cortical input (3 Hz) in the
resting state. Colors: parts, related to DP (green), while parts, associated
with IP (red). (a) Raster plot of spikes and IPSR (instantaneous population
spike rate) $R_{\rm D1}(t)$ of D1 SPNs. Raster plots of spikes and IPSRs
$R_{\rm D2}(t)$ of D2 SPNs for (b1) $x_{\rm D2}$ = 1.0, (b2) 0.8, (b3) 0.5,
and (b4) 0.2. (c) Plot of population-averaged MFR (mean firing rate) $\langle
f_{i}^{({\rm D2})}\rangle$ of D2 SPNs versus $x_{\rm D2}$. Raster plots of
spikes and IPSRs $R_{\rm STN}(t)$ of STN neurons for (d1) $x_{\rm D2}$ = 1.0,
(d2) 0.8, (d3) 0.5, and (d4) 0.2. Raster plots of spikes and IPSRs $R_{\rm
GP}(t)$ of GP neurons for (e1) $x_{\rm D2}$ = 1.0, (e2) 0.8, (e3) 0.5, and
(e4) 0.2. Plots of population-averaged MFRs of (f) STN neurons $\langle
f_{i}^{({\rm STN})}\rangle$ and (g) GP neurons $\langle f_{i}^{({\rm
GP})}\rangle$ versus $x_{\rm D2}$. (h) Plots of strengths of DP
${\cal{S}}_{DP}$ and IP ${\cal{S}}_{IP}$ versus $x_{\rm D2}$. (i) Plot of the
competition degree ${\cal{C}}_{d}$ versus $x_{\rm D2}$. Raster plot of spikes
and IPSR $R_{\rm SNr}(t)$ of SNr neurons for (j1) $x_{\rm D2}$ =1, (j2) 0.8,
(j3) 0.5, and (j4) 0.2. (k) Plot of population-averaged MFR $\langle
f_{i}^{({\rm SNr})}\rangle$ of SNr neurons versus $x_{\rm D2}$.

In the NMDA-receptor case, some of the postsynaptic NMDA channels are known to
be blocked by the positive magnesium ion Mg2+. In this case, fraction of NMDA
channels that are not blocked by the Mg2+ ion is given by a sigmoidal function
$f(v^{(T)})$ SPN1 ; CN6 ; NMDA ,

$f(v^{(T)}(t))=\frac{1}{1+0.28\cdot[{\rm Mg}^{2+}]\cdot
e^{-0.062v^{(T)}(t)}}.$ (13)

Here, $v^{(T)}$ is the membrane potential of a neuron in the target population
$T$ and $[{\rm Mg}^{2+}]$ is the equilibrium concentration of magnesium ions
($[{\rm Mg}^{2+}]$ = 1 mM).

Thus, the synaptic current into the $i$th neuron in the target $X$ population
becomes

$I_{syn,i}^{(X)}(t)=I_{{\rm AMPA},i}^{(X,Y)}(t)+f(v_{i}^{(X)}(t))\cdot I_{{\rm
NMDA},i}^{(X,Y)}(t)+I_{{\rm GABA},i}^{(X,Z)}(t).$ (14)

Synaptic parameters of the synaptic currents from the source population $S$ to
the target population $T$: maximum synaptic conductance
$\tilde{g}_{max,R}^{(T,S)}$, synaptic decay time $\tau_{R,d}^{(T,S)}$,
synaptic delay time $\tau_{R,l}^{(T,S)}$, and synaptic reversal potential
$V_{R}^{(S)}$ are shown in Table 3. These parameter values are also based on
the physiological properties of the relevant neurons SPN2 ; CN6 ; CN14 ; Phys2
; Phys12 ; Phys13 ; Phys14 ; Phys15 ; Phys16 ; Phys17 ; Phys18 ; Phys19 ;
Phys20 .

Finally, we consider the effect of DA modulation on the synaptic currents into
D1 SPN, D2 SPN, STN, and GP neurons in Fig. 1 SPN1 ; SPN2 ; CN6 . In the case
of synaptic currents into the D1 SPNs, DA modulation effect is modeled by up-
scaling the NMDA receptor-mediated current $I_{\rm NMDA}$ with the factor
$\beta^{({\rm D1})}$:

$I_{\rm NMDA}(1+\beta^{({\rm D1})}\phi_{1}),$ (15)

where $\beta^{({\rm D1})}=0.5$ and $\phi_{1}$ is the DA level for the D1 SPNs.
(In the case of D1 SPNs, there is no effect of DA modulation on the AMPA
receptor-mediated current $I_{\rm AMPA}.$) On the other hand, in the case of
synaptic currents into the D2 SPNs, DA modulation effect is modeled by down-
scaling the AMPA receptor-mediated current $I_{\rm AMPA}$ with the factor
$\beta^{({\rm D2})}$:

$I_{\rm AMPA}(1-\beta^{({\rm D2})}\phi_{2}),$ (16)

where $\beta^{({\rm D2})}=0.3$ and $\phi_{2}$ is the DA level for the D2 SPNs.
(There is no DA effect on $I_{\rm NMDA}$ for the D2 SPNs.)

Unlike the above cases of D1 and D2 SPNs, there are DA effects on all the
synaptic currents, $I_{\rm AMPA},$ $I_{\rm NMDA},$ and $I_{\rm GABA}$ into the
STN neurons and GP neurons. In these cases, all excitatory and inhibitory
synaptic currents, $I_{\rm AMPA}$, $I_{\rm NMDA}$, and $I_{\rm GABA}$, are
down-scaled with their scaling factors, depending on $\phi_{2}$ (DA level for
the D2 SPNs):

$\displaystyle(I_{AMPA}+f(v)\cdot I_{NMDA})(1-\beta_{1}^{({\rm
STN})}\phi_{2})$ $\displaystyle+I_{GABA}(1-\beta_{2}^{({\rm STN})}\phi_{2}),$
(17) $\displaystyle(I_{AMPA}+f(v)\cdot I_{NMDA})(1-\beta_{1}^{({\rm
GP})}\phi_{2})$ $\displaystyle+I_{GABA}(1-\beta_{2}^{({\rm GP})}\phi_{2}).$
(18)

Here, the scaling factors $\beta_{1}^{(\rm{STN})}=\beta_{2}^{(\rm{STN})}=0.5$
and $\beta_{1}^{(\rm{GP})}=\beta_{2}^{(\rm{GP})}=0.5$.

##  III Quantitative Analysis of Break-up and Recovery of Harmony between DP
and IP for the HD

In this section, we quantitatively analyze competitive harmony (i.e.,
competition and cooperative interplay) between DP and IP for the HD in terms
of the competition degree ${\cal C}_{d}$ between them, introduced in our
previous work KimPD . ${\cal C}_{d}$ is given by the ratio of strength of DP
(${\cal S}_{DP}$) to strength of IP (${\cal S}_{IP}$) (i.e., ${\cal
C}_{d}={\cal S}_{DP}/{\cal S}_{IP}$).

We consider the early stage of HD where neurodegerative loss of D2 SPNs
occurs; $N_{\rm D2}$ (number of D2 SPNs) = $N_{\rm D2}^{*}$ (=1325; normal
value) $x_{\rm D2}$ [$1>x_{\rm D2}$ (fraction of number of D2 SPNs) $\geq 0$]
Degen1 ; Degen2 ; Degen3 ; Degen4 ; Degen5 ; Degen6 . By decreasing $x_{\rm
D2}$ from 1, we investigate break-up of harmony between DP and IP in both
cases of tonic cortical input (3 Hz) in the resting state and phasic cortical
input (10 Hz) in the phasically active state. In these cases, the IP becomes
weakened, and thus ${\cal C}_{d}$ becomes larger than normal ones.
Consequently, involuntary jerky movement and abnormal hyperkinetic movement
occur in the tonic and phasic cases, respectively. Next, we study treatment of
HD through recovery of harmony between DP and IP. We strengthen the IP via
activation of D2 SPNs and STN neurons and deactivation of GP neurons, based on
optogenetics OG1 ; OG2 . Consequently, harmony between DP and IP becomes
recovered, leading to normal movement. Finally, we also investigate the effect
of loss of healthy synapses in the BG neurons on the HD.

###  III.1 Break-up of Harmony between DP and IP for the HD

In the early stage of HD, we consider the case of normal DA level of
$\phi_{1}=\phi_{2}=\phi=0.3$ for the D1 and D2 SPNs. As explained in Sec.
II.1, cortical inputs are modeled in terms of 1,000 independent Poisson spike
trains with firing rate $f$. We first consider the case of tonic cortical
input with $f=3$ Hz in the resting state Hump1 ; CI1 ; CI2 ; CI3 ; CI4 ; CI5 ;
Str2 ; CN6 ; CN14 .

Population firing behavior of BG neurons could be well visualized in the
raster plot of spikes, corresponding to a collection of spike trains of
individual BG neurons. Figure 2(a) shows the raster plot of spikes for D1
SPNs, associated with DP (green color). In contrast to the case of D1 SPNs,
degenerative loss of D2 SPNs occurs. With decreasing $x_{\rm D2}$ (i.e.,
fraction of number of D2 SPNs) from 1, we also get the raster plots of spikes
of D2 SPNs [Figs. 2(b1)-2(b4)], the STN neurons [Figs. 2(d1)-2(d4)], and the
GP neurons [Figs. 2(e1)-2(e4)], related to the IP (red color) for $x_{\rm
D2}=$ 1.0, 0.8, 0.5, and 0.2.

As a collective quantity showing population behaviors, we employ an IPSR
(instantaneous population spike rate) which may be obtained from the raster
plot of spikes IPSR1 ; IPSR2 ; IPSR3 ; IPSR4 ; IPSR5 . Each spike in the
raster plot is convoluted with a kernel function $K_{h}(t)$ to get a smooth
estimate of IPSR $R_{X}(t)$ Kernel :

$R_{X}(t)=\frac{1}{N_{X}}\sum_{i=1}^{N_{X}}\sum_{s=1}^{n_{i}^{(X)}}K_{h}(t-t_{s,i}^{(X)}).$
(19)

Here, $N_{X}$ is the number of the neurons in the $X$ population, and
$t_{s,i}^{(X)}$ and $n_{i}^{(X)}$ are the $s$th spiking time and the total
number of spikes for the $i$th neuron, respectively. We use a Gaussian kernel
function of band width $h$:

$K_{h}(t)=\frac{1}{\sqrt{2\pi}h}e^{-t^{2}/2h^{2}},~{}~{}~{}~{}-\infty<t<\infty,$
(20)

where the band width $h$ of $K_{h}(t)$ is 20 msec. The IPSRs $R_{X}(t)$ for
$X=$ D1 (SPN), D2 (SPN), STN, GP, and SNr are also shown below their
respective raster plots of spikes. Here, the case of $x_{\rm D2}=1$
corresponds to the normal one without degenerative loss of D2 SPNs. With
decreasing $x_{\rm D2}$ from 1, the population firing activities of the D2
SPNs, the STN neurons, and the GP neurons, associated with IP (red), are
changed, while that of the D1 SPN, related to DP (green), is unchanged.

We also study the population-averaged mean firing rate (MFR) of the neurons
$\langle f_{i}^{(X)}\rangle$ in the $X$ population [$X=$ D1 (SPN), D2 (SPN),
STN, and GP]; $f_{i}^{(X)}$ is the MFR of the $i$th neuron in the $X$
population, and $\langle\cdots\rangle$ represents a population average over
all the neurons. For the D1 and D2 SPNs, $\langle f_{i}^{({\rm
D1})}\rangle=1.03$ Hz and $\langle f_{i}^{({\rm D2})}\rangle=0.97$ Hz,
independently of $x_{\rm D2}$, because there is no change in cortical inputs
to the D1/D2 SPNs; see Fig. 2(c) for the D2 SPNs. As $x_{\rm D2}$ is decreased
from 1, $\langle f_{i}^{({\rm GP})}\rangle$ of the GP neurons is increased
from 29.9 to 38.8 Hz, due to decrease in inhibitory projection from the D2
SPNs, as shown in Fig. 2(g). In contrast, because of increased inhibitory
projection from the GP, $\langle f_{i}^{({\rm STN})}\rangle$ of the STN
neurons is decreased from 9.9 to 6.5 Hz [see Fig. 2(f)].

We note that, there are two types of synaptic currents into the (output) SNr
neurons, $I_{DP}$ and $I_{IP}$, via DP (green) and IP (red) in Fig. 1,
respectively. The DP current, $I_{DP}(t),$ is just the (inhibitory) synaptic
current from the D1 SPNs to the SNr neurons:

$I_{DP}(t)=-I_{syn}^{({\rm SNr,D1})}(t).$ (21)

There is no change in $I_{DP}(t),$ independently of $x_{\rm D2}$.

The IP current, $I_{IP}(t),$ is composed of the excitatory component,
$I_{IP}^{(E)}(t),$ and the inhibitory component, $I_{IP}^{(I)}(t):$

$I_{IP}(t)=I_{IP}^{(E)}(t)+I_{IP}^{(I)}(t).$ (22)

Here, $I_{IP}^{(E)}(t)$ [$I_{IP}^{(I)}(t)$] is just the synaptic current from
the STN (GP) to the SNr:

$I_{IP}^{(E)}(t)=-I_{syn}^{({\rm SNr,STN})}(t)~{}~{}~{}{\rm
and}~{}~{}~{}I_{IP}^{(I)}(t)=-I_{syn}^{({\rm SNr,GP})}(t).$ (23)

Unlike the case of $I_{DP}(t),$ with decreasing $x_{\rm D2}$ from 1,
$I_{IP}(t)$ becomes decreased due to decrease in $I_{IP}^{(E)}(t)$ and
increase in $|I_{IP}^{(I)}(t)|$ ($|\cdots|$: absolute magnitude).

Firing activity of the (output) SNr neurons is determined through competition
between $I_{DP}(t)$ (DP current) and $I_{IP}(t)$ (IP current) into the SNr.
The strengths of DP and IP, ${\cal S}_{DP}$ and ${\cal S}_{IP}$, are given by
the magnitudes of their respective time-averaged synaptic currents:

${\cal S}_{DP}=|\overline{I_{DP}(t)}|~{}~{}~{}{\rm and}~{}~{}~{}{\cal
S}_{IP}=|\overline{I_{IP}(t)}|,$ (24)

where the overline denotes the time averaging and $|\cdots|$ represents the
absolute magnitude. Then, the competition degree ${\cal C}_{d}$ between DP and
IP (given by the ratio of ${\cal S}_{DP}$ to ${\cal S}_{IP}$) was introduced
in KimPD :

${\cal C}_{d}=\frac{{\cal S}_{DP}}{{\cal S}_{IP}}.$ (25)

Figure 3: Abnormal hyperkinetic movement due to degenerative loss of D2 SPNs
in the phasic pathological state for the phasic cortical input (10 Hz) in the
phasically-active state. Colors: parts, related to DP (green), while parts,
associated with IP (red). (a) Raster plot of spikes and IPSR (instantaneous
population spike rate) $R_{\rm D1}(t)$ of D1 SPNs. Raster plots of spikes and
IPSRs $R_{\rm D2}(t)$ of D2 SPNs for (b1) $x_{\rm D2}$ = 1.0, (b2) 0.8, (b3)
0.5, and (b4) 0.2. (c) Plot of population-averaged MFR (mean firing rate)
$\langle f_{i}^{({\rm D2})}\rangle$ of D2 SPNs versus $x_{\rm D2}$. Raster
plots of spikes and IPSRs $R_{\rm STN}(t)$ of STN neurons for (d1) $x_{\rm
D2}$ = 1.0, (d2) 0.8, (d3) 0.5, and (d4) 0.2. Raster plots of spikes and IPSRs
$R_{\rm GP}(t)$ of GP neurons for (e1) $x_{\rm D2}$ = 1.0, (e2) 0.8, (e3) 0.5,
and (e4) 0.2. Plots of population-averaged MFRs of (f) STN neurons $\langle
f_{i}^{({\rm STN})}\rangle$ and (g) GP neurons $\langle f_{i}^{({\rm
GP})}\rangle$ versus $x_{\rm D2}$. (h) Plots of strengths of DP
${\cal{S}}_{DP}$ and IP ${\cal{S}}_{IP}$ versus $x_{\rm D2}$. (i) Plot of the
competition degree ${\cal{C}}_{d}$ versus $x_{\rm D2}$. Raster plot of spikes
and IPSR $R_{\rm SNr}(t)$ of SNr neurons for (j1) $x_{\rm D2}$ =1, (j2) 0.8,
(j3) 0.5, and (j4) 0.2. (k) Plot of population-averaged MFR $\langle
f_{i}^{({\rm SNr})}\rangle$ of SNr neurons versus $x_{\rm D2}$.

For $x_{\rm D2}=1$ (without degenerative loss of D2 SPNs), ${\cal S}_{DP}$ =
23.1 and ${\cal S}_{IP}$ = 23.4, and hence DP and IP become nearly balanced
(i.e., ${\cal C}_{d}=0.99$). In this non-degenerative case, the SNr neurons
fire very actively with $\langle f_{i}^{({\rm SNr})}\rangle=25.5$ Hz. Due to
strong inhibitory projection from the SNr, the thalamic cells become silent,
resulting in no movement (i.e., the BG door to the thalamus is locked in the
normal tonic default state).

Figure 4: Treatment of tonic pathological state by strengthening IP for the
tonic cortical input (3 Hz). Colors: parts, related to DP (green), while
parts, associated with IP (red). (1) Activation of D2 SPN: Plots of (a1)
${\cal{S}}_{DP}$ (green circles) and ${\cal{S}}_{IP}$ (red crosses), (a2)
${\cal{C}}_{d}$, and (a3) $\langle f_{i}^{({\rm SNr})}\rangle$ versus $\Delta
I_{ion}^{({\rm D2})}$ for $x_{\rm D2}=0.5$. (b) Plot of threshold $\Delta
I_{ion}^{({\rm D2})*}$ versus $x_{\rm D2}$. (2) Activation of STN: Plots of
(c1) ${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (c2) ${\cal{C}}_{d}$, and (c3)
$\langle f_{i}^{({\rm SNr})}\rangle$ versus $\Delta I_{ion}^{({\rm STN})}$ for
$x_{\rm D2}=0.5$. (d) Plot of threshold $\Delta I_{ion}^{({\rm D2})*}$ versus
$x_{\rm D2}$. (3) Deactivation of GP: Plots of (e1) ${\cal{S}}_{DP}$ and
${\cal{S}}_{IP}$, (e2) ${\cal{C}}_{d}$, and (e3) $\langle f_{i}^{({\rm
SNr})}\rangle$ versus $\Delta I_{ion}^{({\rm GP})}$ for $x_{\rm D2}=0.5$. (f)
Plot of threshold $\Delta I_{ion}^{({\rm GP})*}$ versus $x_{\rm D2}$. (4)
Ablation of GP: Plots of (g1) ${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (g2)
${\cal{C}}_{d}$, and (g3) $\langle f_{i}^{({\rm SNr})}\rangle$ versus $x_{\rm
GP}$ for $x_{\rm D2}=0.5$. (h) Plot of threshold $x_{\rm GP}^{*}$ versus
$x_{\rm D2}$. Horizontal dashed lines in (a2), (c2), (e2), and (g2) represent
${\cal{C}}_{d}^{*}$ (= 1.0) for the default healthy tonic state when $x_{\rm
D2}=1$. Horizontal dashed lines in (a3), (c3), (e3), and (g3) represent
$\langle f_{i}^{({\rm SNr})}\rangle$ (=25.5 Hz) for the default healthy tonic
state when $x_{\rm D2}=1$.

But, with decreasing $x_{\rm D2}$ from 1 (degenerative case), as shown in Fig.
2(h), ${\cal S}_{IP}$ is rapidly decreased from 23.4 to 1.4 , while there is
no change in ${\cal S}_{DP}$ (= 23.1). In this way, IP for the HD becomes
weakened. Thus, as $x_{\rm D2}$ is decreased from 1, the competition degree
${\cal C}_{d}$ between DP and IP is found to increase from 0.99 to 16.5 [see
Fig. 2(i)]. Thus, balance between DP and IP becomes broken up in the
degenerative tonic case.

Figures 2(j1)-2(j4) show raster plots of spikes and IPSRs $R_{\rm SNr}(t)$ of
the (output) SNr neurons for $x_{\rm D2}$ = 1.0, 0.8, 0.5, and 0.2,
respectively. We note that, firing activity of the SNr neurons becomes reduced
with decreasing $x_{\rm D2}$ because of weakened IP. As a result of decrease
in ${\cal S}_{IP}$ (strength of IP), the population-averaged MFR $\langle
f_{i}^{\rm(SNr)}\rangle$ is found to decrease from 25.5 to 6.9 Hz with
decreasing $x_{\rm D2}$ from 1, as shown in Fig. 2(k). Thus, the BG gate to
the thalamus becomes opened even in the case of tonic cortical input (3 Hz) in
the resting state via break-up of balance between DP and IP. Consequently, a
tonic pathological state with involuntary jerky movement occurs, in contrast
to the tonic default state without movement.

Next, we consider the case of phasic cortical input (10 Hz) in the phasically
active state Hump1 ; CI1 ; CI2 ; CI3 ; CI4 ; CI5 ; Str2 ; CN6 ; CN14 , which
is shown in Fig. 3. Population firing behavior of D1 SPNs, associated with DP
(green color), is shown in their raster plot of spikes and the IPSR $R_{\rm
D1}$(t) in Fig. 3(a). In comparison to the tonic case with the population-
averaged MFR $\langle f_{i}^{\rm(D1)}\rangle$ = 1.03 Hz in Fig. 2(a), firing
activity of the D1 SPNs become very active with $\langle
f_{i}^{\rm(D1)}\rangle$ = 30.7 Hz, independently of $x_{\rm D2}$.

But, due to degenerative loss of D2 SPNs, population firing activities of the
D2 SPNs, the STN neurons, and the GP neurons [related to the IP (red color)]
are changed with decreasing $x_{\rm D2}$, as shown in their raster plots of
spikes and IPSRs in Fig. 3. The population-averaged MFRs of the D2 SPNs, the
STN neurons, and the GP neurons are also shown in Figs. 3(c), 3(f), and 3(g),
respectively. For the D2 SPNs, $\langle f_{i}^{\rm(D2)}\rangle$ = 24.1 Hz
[much larger than that (0.97 Hz) in the tonic case], independently of $x_{\rm
D2}$, because there is no change in cortical input to the D2 SPNs. As a result
of decreased inhibitory projection from the D2 SPNs, $\langle f_{i}^{({\rm
GP})}\rangle$ of the GP neurons is rapidly increased from 7.3 to 66.1 Hz with
decreasing $x_{\rm D2}$ from 1; the increasing rate is higher than the tonic
case. On the other hand, due to increase in inhibitory projection from the GP,
$\langle f_{i}^{({\rm STN})}\rangle$ of the STN neurons decreases from 39.8 to
17.6 Hz; the decreasing rate is also larger than that in the tonic case.

We consider the case of $x_{\rm D2}=1$ without degeneration. In this non-
degenerative case, ${\cal S}_{DP}$ = 2309.7 and ${\cal S}_{IP}$ = 815.6. Thus,
the competition degree becomes ${\cal C}_{d}=2.82$ [i.e., ${\cal S}_{DP}$
(strength of DP) is 2.82 times larger than ${\cal S}_{IP}$ (strength of IP)].
In this case, $\langle f_{i}^{({\rm SNr})}\rangle$ of the (output) SNr neurons
are decreased to 5.5 Hz (cf., in the tonic case, 25.5 Hz). Consequently, the
BG door to the thalamus becomes opened, leading to normal movement. This
phasic healthy state with ${\cal C}_{d}=2.82$ is in contrast to the tonic
healthy state with ${\cal C}_{d}\simeq 1.0$ resulting in no movement.

However, as $x_{\rm D2}$ is decreased from 1 (degenerative case), ${\cal
S}_{IP}$ is rapidly decreased from 815.6 to 92.3 , while there is no change in
${\cal S}_{DP}$ (= 2309.7) [see Fig. 3(h)]. Thus, IP becomes rapidly weakened.
Due to such under-activity of IP, the competition degree ${\cal C}_{d}$
increases from 2.82 (healthy state) to 25.0, as shown in Fig. 3(i).
Consequently, harmony between DP and IP becomes broken up in the degenerative
case with $x_{\rm D2}<1$, and then a phasic pathological state with abnormal
hyperkinetic movement appears, in contrast to the phasic healthy state with
normal movement.

Raster plots of spikes and IPSRs $R_{\rm SNr}(t)$ of the (output) SNr neurons
for $x_{\rm D2}$ = 1.0, 0.8, 0.5, and 0.2 are shown in Figs. 3(j1)-3(j4),
respectively. Due to under-activity of IP, firing activity of the SNr neurons
becomes decreased with decreasing $x_{\rm D2}$ from 1. Due to decreased ${\cal
S}_{IP}$ (strength of IP), the population-averaged MFR $\langle
f_{i}^{\rm(SNr)}\rangle$ decreases from 5.5 (healthy state) to 0.7 Hz with
decreasing $x_{\rm D2}$ from 1 [see Fig. 3(k)]. In this phasic pathological
state with ${\cal C}_{d}>2.82$ (where harmony between DP and IP is broken up),
abnormal hyperkinetic movement disorder occurs, in contrast to the normal
movement for the phasic healthy state with ${\cal C}_{d}=2.82$ (where there is
harmony between DP and IP).

To sum up the above results briefly, it is shown that, for the HD,
pathological states (where harmony between DP and IP is broken up) appear due
to degenerative loss of D2 SPNs in the cases of both tonic and phasic cortical
inputs. On the other hand, for the PD, pathological states appear because of
DA deficiency PD1 ; PD2 ; PD3 ; PD4 ; PD5 ; PD6 ; KimPD . In the case of HD,
IP is under-active, in contrast to the case of PD with over-active IP. Thus,
patients with HD exhibit abnormal hyperkinetic movement disorder, while
patients with PD show abnormal hypokinetic movement disorder. Consequently, HD
lies at one end of the spectrum of movement disorders in the BG, while PD lies
at the other end.

###  III.2 Treatment of HD via Recovery of Harmony between DP and IP

For the pathological state in the HD, IP is under-active due to degenerative
loss of D2 SPNs, in comparison to the healthy state. Thus, harmony between DP
and IP is broken up (i.e. occurrence of disharmony between DP and IP), leading
to abnormal hyperkinetic movement disorder. Here, we investigate treatment of
the pathological state with enhanced competition degree ${\cal C}_{d}$ (than
the normal one for the healthy state) in both cases of tonic and phasic
cortical inputs via recovery of harmony between DP and IP.

Activation and deactivation of the target neurons via optogenetics are studied
in KimPD . When light-sensitive proteins (called the opsins) are activated
through light stimulation, variation in the intrinsic ionic currents of the
neurons in the target population $X$, $\Delta I_{ion}^{(X)}$, takes place. If
$\Delta I_{ion}^{(X)}$ is positive (negative), firing activity of the target
neurons is increased (decreased), leading to their activation (deactivation)
OG1 ; OG2 . As discussed in KimPD , we simulate the effects of optogenetics by
adding $\Delta I_{ion}^{(X)}$ in Eq. (1), in addition to the current,
$I_{i}^{(X)}$, into the target $X$ population. With increasing the intensity
of light stimulation, the magnitude of $\Delta I_{ion}^{(X)}$ also increases.

We first consider tonic pathological states with enhanced competition degree
${\cal C}_{d}$ [larger than that (1) for the tonic healthy state (with
balanced DP and IP)], occurring due to degenerative loss of D2 SPNs, in the
case of tonic cortical input (3 Hz) (see Fig. 2). As an example, we consider
the tonic pathological case of $x_{\rm D2}=0.5$ with ${\cal C}_{d}=1.53$. In
this pathological case, IP is under-active in comparison to the tonic healthy
case (with balanced DP and IP); firing activity of D2 SPNs is under-active,
leading to over-activity of GP neurons, which then results in under-activity
of the STN neurons. Hence, for recovery of balance between DP and IP, we try
to strengthen the IP via activation of D2 SPNs and STN neurons and
deactivation of GP neurons.

We first strengthen the IP through activation of the target (under-active) D2
SPNs. Figure 4(a1) shows plots of ${\cal S}_{IP}$ (strength of IP) and ${\cal
S}_{DP}$ (strength of DP) versus $\Delta I_{ion}^{({\rm D2})}$. As $\Delta
I_{ion}^{({\rm D2})}$ is increased from 0, ${\cal S}_{IP}$ (red) increases
from 15.1, while ${\cal S}_{DP}$ (green) remains unchanged (i.e., 23.1). As a
result of increase in ${\cal S}_{IP}$, the competition degree ${\cal C}_{d}$
between DP and IP is found to decrease from 1.53 [Fig. 4(a2)]. Also, the
population-averaged MFR of the output SNr neurons, $\langle f_{i}^{({\rm
SNr})}\rangle$, is found to increase from 16.1 Hz [Fig. 4(a3)].

We note that, as $\Delta I_{ion}^{({\rm D2})}$ passes a threshold $\Delta
I_{ion}^{({\rm D2})*}$ (= 262 pA), ${\cal C}_{d}~{}=~{}{\cal
C}_{d}^{*}~{}(=~{}1.0)$ and $\langle f_{i}^{({\rm SNr})}\rangle~{}=~{}\langle
f_{i}^{({\rm SNr})*}\rangle$ (= 25.5 Hz); ${\cal C}_{d}^{*}$ and $\langle
f_{i}^{({\rm SNr})*}\rangle$ are those for the tonic healthy state, and they
are represented by the horizontal dashed lines in Figs. 4(a2) and 4(a3). Thus,
for $x_{\rm D2}=0.5$, the pathological state with ${\cal C}_{d}=1.53$ may have
${\cal C}_{d}$ (= 1.0) via activation of D2 SPNs for the threshold, $\Delta
I_{ion}^{({\rm D2})*}$ (= 262 pA); DP and IP becomes balanced, as in the case
of tonic healthy state. In this way, balance between DP and IP is recovered
for $\Delta I_{ion}^{({\rm D1})*}$ = 262 pA. Figure 4(b) shows the plot of
$\Delta I_{ion}^{({\rm D2})*}$ versus $x_{\rm D2}$. As $x_{\rm D2}$ is
decreased from 1, the threshold $\Delta I_{ion}^{({\rm D2})*}$ is increased;
with decreasing $x_{\rm D2}$, more $\Delta I_{ion}^{({\rm D2})*}$ is necessary
for recovery of balance between DP and IP.

Figure 5: Treatment of phasic pathological state by strengthening IP for the
phasic cortical input (10 Hz). Colors: parts, related to DP (green), while
parts, associated with IP (red). (1) Activation of D2 SPN: Plots of (a1)
${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (a2) ${\cal{C}}_{d}$, and (a3) $\langle
f_{i}^{({\rm SNr})}\rangle$ versus $\Delta I_{ion}^{({\rm D2})}$ for $x_{\rm
D2}=0.5$. (b) Plot of threshold $\Delta I_{ion}^{({\rm D2})*}$ versus $x_{\rm
D2}$. (2) Activation of STN: Plots of (c1) ${\cal{S}}_{DP}$ and
${\cal{S}}_{IP}$, (c2) ${\cal{C}}_{d}$, and (c3) $\langle f_{i}^{({\rm
SNr})}\rangle$ versus $\Delta I_{ion}^{({\rm STN})}$ for $x_{\rm D2}=0.5$. (d)
Plot of threshold $\Delta I_{ion}^{({\rm STN})*}$ versus $x_{\rm D2}$. (3)
Deactivation of GP: Plots of (e1) ${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (e2)
${\cal{C}}_{d}$, and (e3) $\langle f_{i}^{({\rm SNr})}\rangle$ versus $\Delta
I_{ion}^{({\rm GP})}$ for $x_{\rm D2}=0.5$. (f) Plot of threshold $\Delta
I_{ion}^{({\rm GP})*}$ versus $x_{\rm D2}$. (4) Ablation of GP: Plots of (g1)
${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (g2) ${\cal{C}}_{d}$, and (g3) $\langle
f_{i}^{({\rm SNr})}\rangle$ versus $x_{\rm GP}$ for $x_{\rm D2}=0.5$. (h) Plot
of threshold $x_{\rm GP}^{*}$ versus $x_{\rm D2}$. Horizontal dashed lines in
(a2), (c2), (e2), and (g2) represent ${\cal{C}}_{d}^{*}$ (= 2.82) for the
healthy phasic state when $x_{\rm D2}=1$. Horizontal dashed lines in (a3),
(c3), (e3), and (g3) represent $\langle f_{i}^{({\rm SNr})}\rangle$ (= 5.5 Hz)
for the healthy phasic state when $x_{\rm D2}=1$.

We also strengthen the IP via activation of the target (under-active) STN
neurons, which is shown in Figs. 4(c1)-4(c3) for $x_{\rm D2}=0.5$. All the
behaviors are qualitatively the same as those in the case of activation of D2
SPNs. With increasing $\Delta I_{ion}^{({\rm STN})}$ from 0, ${\cal S}_{IP}$
(strength of IP) increases, leading to decrease in the competition degree
${\cal C}_{d}$, and the population-averaged MFR of the output SNr neurons,
$\langle f_{i}^{({\rm SNr})}\rangle$, also increases. But, the threshold
$\Delta I_{ion}^{({\rm STN})*}$ (= 14 pA), where balance between DP and IP is
recovered (i.e., ${\cal C}_{d}$ = 1 and $\langle f_{i}^{({\rm SNr})}\rangle$ =
25.5 Hz, as in the case of tonic healthy state), is smaller than that (262 pA)
in the case of activation of D2 SPNs. The mono-synaptic effect of STN neurons
on the output SNr neurons is more direct than the bi- or tri-synaptic effect
of D2 SPNs, which could result in the smaller threshold $\Delta I_{ion}^{({\rm
STN})*}$ in the case of STN neurons. Figure 4(d) shows the plot of $\Delta
I_{ion}^{({\rm STN})*}$ versus $x_{\rm D2}$. With decreasing $x_{\rm D2}$ from
1, the threshold $\Delta I_{ion}^{({\rm STN})*}$ increases, as shown in Fig.
4(d); As $x_{\rm D2}$ is decreased, more $\Delta I_{ion}^{({\rm STN})*}$ is
necessary for recovery of balance between DP and IP.

Unlike the cases of activation of (under-active) D2 SPNs and STN neurons, IP
may be strengthened via deactivation of (over-active) GP neurons; in the case
of deactivation, $\Delta I_{ion}^{(GP)}$ is negative, in contrast to the case
of activation with $\Delta I_{ion}^{(X)}>0$ [$X$ = D2 (SPN) and STN]. Figures
4(e1)- 4(e3) and 4(f) show the case of deactivation of GP neurons. As the
magnitude of $\Delta I_{ion}^{({\rm GP})}$ is increased (i.e., more negative),
strength of IP, ${\cal S}_{IP}$ (red), is found to increase from 15.1, while
${\cal S}_{DP}$ (green) remains constant (= 23.1). Thus, when passing a
threshold $\Delta I_{ion}^{({\rm GP})*}=-28$ pA, balance between DP and IP
becomes recovered (i.e., the competition degree ${\cal C}_{d}$ becomes 1 and
the population-averaged MFR of output SNr neurons $\langle f_{i}^{({\rm
SNr})}\rangle$ becomes 25.5 Hz) [see Figs. 4(e2)- 4(e3)]. As shown in Fig.
4(f), with decreasing $x_{\rm D2}$ from 1, the threshold $\Delta
I_{ion}^{({\rm GP})*}$ is decreased (i.e., its magnitude increases); as
$x_{\rm D2}$ is decreased from 1, more negative $\Delta I_{ion}^{({\rm GP})*}$
is required for recovery of balance between DP and IP.

Instead of the above deactivation of GP neurons via optogenetics, we also
consider ablation of (over-active) GP neurons in the pathological state for
$x_{\rm D2}=0.5$ to reduce the over-activity of GP neurons. In the case of
ablation, the number of GP neurons, $N_{{\rm GP}}$, is reduced to $N_{\rm
GP}^{(n)}$ $x_{\rm GP}$ ($1>x_{\rm GP}>0)$, where $N_{\rm GP}^{(n)}$ (= 46) is
the normal number of GP neurons and $x_{\rm GP}$ is the fraction of number of
GP neurons. As shown in Figs. 4(g1)- 4(g3) and 4(h), the effect of decreasing
$x_{\rm GP}$ via ablation is similar to that of deactivation of GP neurons via
optogenetics. As $x_{\rm GP}$ is decreased from 1, strength of IP, ${\cal
S}_{IP}$ (red), is found to increase from 15.1 (i.e., IP becomes strengthened)
[see Fig. 4(g1)]. When passing a threshold, $x_{\rm GP}^{*}~{}(\simeq 0.78)$,
balance between DP and IP becomes recovered (i.e., ${\cal C}_{d}$ = 1.0 and
$\langle f_{i}^{({\rm SNr})}\rangle$ = 25.5 Hz), as shown in Figs.
4(g2)-4(g3). Figure 4(h) shows the plot of $x_{\rm GP}^{*}$ versus $x_{\rm
D2}$. With decreasing $x_{\rm D2}$ from 1, $x_{\rm GP}^{*}$ decreases; more
ablation (i.e., smaller $x_{\rm GP}$) is necessary for balance between DP and
IP.

Next, we consider phasic pathological states with enhanced competition degree
${\cal C}_{d}$ [larger than that (2.82) for the phasic healthy state (with
harmony between DP and IP)], occurring due to degenerative loss of D2 SPNs, in
the case of phasic cortical input (10 Hz) (see Fig. 3). As an example, we
consider the pathological case of $x_{\rm D2}=0.5$ with ${\cal C}_{d}=7.19$.
In this phasic pathological case, IP is under-active in comparison to the case
of phasic healthy state. For the phasic healthy state with ${\cal C}_{d}^{*}$
= 2.82 (i.e., harmony between DP and IP), the population-averaged MFR of
output STr neurons, $\langle f_{i}^{({\rm SNr})*}\rangle,$ is much reduced to
5.5 Hz, leading to normal movement, in contrast to the case of tonic healthy
state with ${\cal C}_{d}\simeq 1.0$ and $\langle f_{i}^{({\rm SNr})}\rangle$ =
25.5 Hz without movement. As in the above tonic pathological state, firing
activity of D2 SPNs is under-active, resulting in over-activity of GP neurons,
which then leads to under-activity of the STN neurons. Hence, for recovery of
harmony between DP and IP, we strengthen the IP through activation of D2 SPNs
and STN neurons and deactivation of GP neurons by employing optogenetic
technique and via ablation of GP neurons.

Figure 5 shows treatment of phasic pathological state for $x_{\rm D2}=0.5$
with ${\cal C}_{d}=7.19$; (1) activation of D2 SPNs, (2) activation of STN
neurons, (3) deactivation of GP neurons, and (4) ablation of GP neurons. The
overall results of these treatments are qualitatively the same as those in the
above case of tonic pathological state in Fig. 4. Only the corresponding
thresholds are quantitatively different; (1) $\Delta I_{ion}^{({\rm D2})*}$ =
1,636 pA, (2) $\Delta I_{ion}^{({\rm STN})*}$ = 405 pA, (3) $\Delta
I_{ion}^{({\rm GP})*}=-540$ pA, and (4) $x_{\rm GP}^{*}~{}(\simeq 0.52)$. When
passing a threshold for each treatment, harmony between DP and IP becomes
recovered (i.e., ${\cal C}_{d}$ = 2.82 and $\langle f_{i}^{({\rm
SNr})}\rangle$ = 5.5 Hz), resulting in normal movement. Finally, we note that,
with decreasing $x_{\rm D2}$, the thresholds, $\Delta I_{ion}^{({\rm D2})*}$
and $\Delta I_{ion}^{({\rm STN})*}$, for activations of D2 SPNs and STN
neurons are increased (i.e., more positive), and the threshold, $\Delta
I_{ion}^{({\rm GP})*}$ for deactivation of GP neurons becomes more negative,
as shown in Figs. 5(b), 5(d), and 5(f). Thus, as $x_{\rm D2}$ is decreased,
more light stimulation for activation and deactivation is necessary for
recovery of harmony between DP and IP. Also, in the case of ablation of GP
neurons, with decreasing $x_{\rm D2}$, more ablation is required to get
harmony between DP and IP.

Figure 6: Effect of loss of healthy synapses in all the BG neurons on the HD
for $x_{\rm D2}=0.5$. Colors: parts, related to DP (green), while parts,
associated with IP (red). (1) Tonic pathological state: Plots of (a)
${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (b) ${\cal{C}}_{d}$, and (c) $\langle
f_{i}^{({\rm SNr})}\rangle$ versus $x_{c}$. (2) Phasic pathological state:
Plots of (d) ${\cal{S}}_{DP}$ and ${\cal{S}}_{IP}$, (e) ${\cal{C}}_{d}$, and
(f) $\langle f_{i}^{({\rm SNr})}\rangle$ versus $x_{c}$.

###  III.3 Effect of Loss of Healthy Synapses on The HD

In the HD, loss of healthy synapses occurs not only in the striatum, but also
in other regions of the BG, including STN, GP, and SNr Loss1 ; Loss2 ; Loss3 ;
Loss4 ; Loss5 ; Loss6 . Such loss of synapses in the BG neurons is an
important feature of HD, and it is thought to contribute to the motor and
cognitive symptoms of the disease. Here, we study effect of loss of healthy
synapses of all the BG neurons on HD.

As examples, we consider pathological states for $x_{\rm D2}$ = 0.5 in both
cases of tonic (3 Hz) and phasic (10 Hz) cortical inputs. Loss of synapses in
the BG neurons is modeled in terms of decreased synaptic connection
probability, $p_{c}=p_{c}^{(n)}~{}x_{c}$; $p_{c}^{(n)}$ is the normal synaptic
connection probability (depending on the type of BG neurons and given in the
subsection II.1) and $x_{c}$ represents the fraction in $p_{c}$ ($1>x_{c}>0$).

We first consider a tonic pathological state in Figs. 6(a)-6(c). As a result
of loss of synapses, decreased cortical inputs into D1 SPNs leads to reduction
in their firing activity $\langle f_{i}^{({\rm D1})}\rangle$. Then, strength
of DP, ${\cal S}_{DP}$, becomes decreased. As shown in Fig. 6(a), ${\cal
S}_{DP}$ (green color) is found to monotonically decrease from 23.1 with
decreasing $x_{c}$ from 1. Also, due to reduced cortical synaptic inputs into
D2 SPNs, firing activity of D2 SPNs, $\langle f_{i}^{({\rm D2})}\rangle$,
becomes decreased, leading to increased firing activity of GP neurons
($\langle f_{i}^{({\rm GP})}\rangle$), which then results in decrease in the
firing activity of STN neurons ($\langle f_{i}^{({\rm STN})}\rangle$).
Consequently, strength of IP, ${\cal S}_{IP}$, becomes decreased. In this case
of IP, with decreasing $x_{c}$ from 1, ${\cal S}_{IP}$ (red color) is found to
more rapidly decrease from 15.1 than the case of DP. Then, the competition
${\cal C}_{d}$ between DP and IP increases rapidly from 1.53 with decreasing
$x_{c}$ from 1 [see Fig. 6(b)]. Thus, as $x_{c}$ is decreased from 1,
population-averaged MFR of the output SNr neurons ($\langle f_{i}^{({\rm
SNr})}\rangle$) decreases from 16.1 Hz. In this way, with decreasing $x_{c}$,
the degree of disharmony between DP and IP becomes increased, resulting in
more severe involuntary jerky movement in the tonic pathological case.

Next, we consider a phasic pathological state in Figs. 6(d)-6(f). With
decreasing $x_{c}$, tendency in the phasic pathological case is qualitatively
the same as that in the above tonic pathological case. Based on the same
reasoning given in the tonic pathological case, as $x_{c}$ is decreased from
1, strength of IP (${\cal S}_{IP}$; red color) is found to decreases much more
rapidly than strength of DP (${\cal S}_{DP}$; green color), as shown in Fig.
5(d). Then, the competition degree ${\cal C}_{d}$ increases from 7.19 with
decreasing $x_{c}$ from 1 [see Fig. 5(e)]. Consequently, firing activity of
the output SNr neurons ($\langle f_{i}^{({\rm SNr})}\rangle$) decreases from
2.7 Hz with $x_{c}$, as shown in Fig. 5(f). In this way, as $x_{c}$ is
decreased from 1, the broken-up degree of harmony between DP and IP becomes
increased, leading to more severe abnormal hyperkinetic movement disorder in
the phasic pathological case.

Overall, in both tonic and phasic pathological cases, as a result of loss of
healthy synapses in the BG neurons, symptoms of the HD become more severe with
decreasing $x_{c}$, because the disharmony degree between DP and IP becomes
increased.

##  IV Summary and Discussion

The BG exhibit a variety of functions for motor, cognition, and emotion.
Dysfunction in the BG is associated with movement disorder (e.g., HD and PD)
and cognitive and psychiatric disorders. There are two competing pathways in
the BG, DP (facilitating movement) and IP (suppressing movement) DIP1 ; DIP2 ;
DIP3 ; DIP4 . In our recent work KimPD , as a first time, we made quantitative
analysis of competitive harmony between DP and IP in the default tonic state
and the phasic healthy and pathological states by introducing their
competition degree, ${\cal C}_{d},$ between DP and IP, given by the ratio of
strength of DP (${\cal S}_{DP})$ to strength of IP (${\cal S}_{IP})$ (i.e.,
${\cal C}_{d}={\cal S}_{DP}/{\cal S}_{IP}$).

In the case of normal DA level. a healthy state with normal movement was found
to appear, while in the case of lower DA level, a pathological state (PD) with
reduced competition degree was found to occur. In PD, DP is under-active,
while IP is over-active, resulting in abnormal hypokinetic movement.

In this paper, we are concerned in the HD which is a genetic neurodegenerative
disease. As a result of mutant HTT gene, toxic HTT protein aggregates appear,
causing the characteristic neurodegeneration seen in HD. We considered
degenerative loss of D2 SPNs in the case of normal DA level. By decreasing
$x_{\rm D2}$ (i.e. fraction of number of D2 SPNs) from 1, we quantitatively
analyzed break-up of harmony between DP and IP. IP was found to be under-
active (i.e., weakened), in contrast to the case of PD with over-active IP.
Thus, the competition degree ${\cal C}_{d}$ becomes increased than normal one.
Consequently, abnormal hyperkinetic movement such as chorea occurs, in
contrast to the case of PD with hypokinetic disorder.

Unfortunately, at present there is no cure for HD. The available treatments
for HD primarily aim to control and alleviate its symptoms, resulting from
weakened IP: medication treatment Treat1 ; Treat2 ; Treat3 ; Treat4 , reducing
symptoms, deep brain stimulation in research and clinical trials Treat5 ;
Treat6 ; Treat7 , and experimental surgery Treat8 . We studied treatment of HD
via recovery of harmony between DP and IP by activating D2 SPNs and STN
neurons and deactivating GP neurons, based on optogenetics OG1 ; OG2 . Through
the treatment process, the IP becomes strengthened, and thus harmony between
DP and IP may be regained. We also studied effects of loss of healthy synapses
of the BG cells on the HD. As a result of such synaptic loss, the HD status
was found to become more severe.

Finally, we discuss limitations of our present work and future works. In the
present work, we considered early stage of HD where degenerative loss of D2
SPNs occurs in the nearly normal DA level. But, in the late stage of HD,
degenerative loss of D1 SPN also occurs along with decrease in DA level,
leading to hypokinetic disorder (e.g., rigidity and bradykinesia) due to
weakened DP, as in the case of PD Dege-D1 . Moreover, in addition to deaths of
D1/D2 SPNs, degeneration of cortical pyramidal cells occurs Dege-C1 ; Dege-C2
. Hence, as a future work, it would be interesting to investigate consequences
of degeneration of D1 SPNs and cortical pyramidal cells, in addition to
degenerative loss of D2 SPNs.

Next, we would like to consider more realistic striatal circuit in the BG. In
our present striatal circuit, we considered only the D1/D2 SPNs (95 $\%$ major
population). But, the minor population of fast interneurons (FSIs) in the
striatum are known to exert strong effects on firing activities of the D1/D2
SPNs Str2 ; FSI . Hence, in future, it would be worth while to contain the
FSIs in our BG SNN. In our present BG SNN, cortical inputs were modelled by
Poisson spike trains. Such SNN could be extended to the cortico-BG-thalamo-
cortical (CBGTC) loop by including the cortical and the thalamic neurons for
more complete computational work CN1 ; CBGTC .

We also make discussion on application of the optogenetics to human patients
for treatment of a pathological state OG3 ; OG4 . In the case of HD, harmony
between DP and IP is broken up due to under-active IP. As shown in Sec. III.2,
harmony between DP and IP could be recovered by strengthening IP. To this end,
optogenetic techniques may be employed. Activation of D2 SPNs and STN neurons
via optogenetics results in strengthening IP. We hope that, in near future,
safe clinical applications of optogenetics to human patients with HD could be
successfully available via collaboration of researchers and clinicians. Then,
it would take a substantial step forward for treatment of HD.

## Acknowledgments

This research was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (Grant No. 20162007688).

## References

  * (1) L. Luo, Principles of Neurobiology (Garland Science, New York, 2016).
  * (2) E. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science (McGraw-Hill, New York, 1991).
  * (3) L. R. Squire, F. E. Bloom, S. K. McConnell, J. L. Roberts, N. C. Spitzer, and M. J. Zigmond, Fundamental Neuroscience (Academic Press, New York, 2003).
  * (4) M. F. Bear, B. M. Connors, and M. A. Paradiso, Neuroscience: Exploring the Brain (Lippincott Williams $\&$ Wikins, Philadelphia, 2007).
  * (5) K. Gurney, T. J. Prescott, and P. Redgrave, Biol. Cybern. 84, 401 (2001).
  * (6) K. Gurney, T. J. Prescott, and P. Redgrave, Biol. Cybern. 84, 411 (2001).
  * (7) M. D. Humphries, R. D. Stewart, and K. N. Gurney, J. Neurosci. 26, 12921 (2006).
  * (8) M. D. Humphries, “Basal ganglia: Mechanisms for action selection,” in Encyclopedia of Computation Neuroscience (Springer, New York, 2014).
  * (9) M. D. Humphries and K. Gurney, Biol. Cybern. 115, 323 (2021).
  * (10) A. Mandali, M. Rengaswamy, V. S. Chakravarthy, and A. A. Moustafa, Front. Neurosci. 9, 191 (2015).
  * (11) G. P. Bates, R. Dorsey, J. F. Gusella, M. R. Hayden, C. Kay, B. R. Leavitt, M. Nance, C. A. Ross, R. I. Scahill, R. Wetzel, E. J. Wild, and S. J. Tabrizi, Nat. Rev. Dis. Primers 1, 15005 (2015).
  * (12) C. A Ross and S. J. Tabrizi, Lancet Neurol. 10, 83 (2011).
  * (13) J. S. Paulsen, J. D. Long, H. J. Johnson, E. H. Aylward, C. A. Ross, J. K. Williams, M. A. Nance, C. J. Erwin, H. J. Westervelt, D. L. Harrington, H. J. Bockholt, Y. Zhang, E. A. McCusker, E. M. Chiu, and P. K. Panegyres; PREDICT-HD Investigators and Coordinators of the Huntington Study Group, Front. Aging Neurosci. 6, 78 (2014).
  * (14) F. O. Walker, Lancet 369, 218 (2007).
  * (15) M. E. MacDonald, C. M. Ambrose, M. P. Duyao, R. H. Myers, C. Lin, L. Srinidhi, G. Barnes, S. A. Taylor, M. James, N. Groot, H. MacFarlane, B. Jenkins, M. A. Anderson, N. S. Wexler, J. F. Gusella, G. P. Bates, S. Baxendale, H. Hummerich, S. Kirby, M. North, S. Youngman, R. Mott, G. Zehetner, Z. Sedlacek, A. Poustka, A. M. Frischauf, H. Lehrach, A. J. Buckler, D. Church, L. Doucettestamm, M. C. Odonovan, W. Riba-rodriguez, M. Shah, V. P. Stanton, S. A. Strobel, K. M. Draths, J. L. Wales, P. Dervan, D. E. Housman, M. Altherr, R. Shiang, L. Thompson, T. Fielder, and J. J. Wasmuth, Cell 72, 971 (1993).
  * (16) N. G. Rojas, M. E. Cesarini, G. Peker, G. A. Da Prat, J. L. Etcheverry, and E. Gatto, J. Neurol. Res. 12, 93 (2022).
  * (17) G. P. Bates, Nat. Rev. Genet. 6, 766 (2005).
  * (18) M. J. Armstrong and M. S. Okun, JAMA 323. 548 (2020).
  * (19) B. L. B. Marino, L. R. de Souza, K. P. A. Sousa, J. V. Ferreira, E. C. Padilha, C. H. T. P. da Silva, C. A. Taft and L. I. S. Hage-Melim, Mini. Rev. Med. Chem. 20, 754 (2020).
  * (20) J. Guridi, R. González-Redondo, and J. A. Obeso, Parkinsons Dis., 943159 (2012).
  * (21) M. D. Humphries, J. A. Obeso, and J. A. Dreyer, J. Neurol. Neurosurg. Psychiatry 89, 1181 (2018).
  * (22) J. A. Obeso, M. Rodriguez-Oroz, C. Marın, P. Alonso, I. Zamarbide, J. L. Lanciego, and M. Rodriguez-Diaz, Neurology 62, S17 (2004).
  * (23) J. A. Obeso, C. Marin, C. Rodriguez-Oroz, J. Blesa, B. Benitez-Temiño, J. Mena-Segovia, M. Rodríguez, and C. W. Olanow, Ann. Neurol. 64, S30 (008).
  * (24) J. F. Gusella, N. S. Wexler, P. M. Conneally, S. L. Naylor, M. A. Anderson, R. E. Tanzi, P. C. Watkins, K. Ottina, M. R. Wallace, A. Y. Sakaguchi, A. B. Young, I. Shoulson, E. Bonilla, and J. B. Martin, Nature 306, 234 (1983).
  * (25) M. E. MacDonald, C. M. Ambrose, M. P. Duyao, R. H. Myers, C. Lin, L. Srinidhi, G. Barnes, S. A. Taylor, M. James, N. Groat, H. MacFarlane, B. Jenkins, M. A. Anderson, N. S. Wexler, J. F. Gusella. G. P. Bates, S. Baxendale, H. Hummerich, S. Kirby, M. North, S. Youngman, R. Mott, G. Zehetner, Z. Sedlacek, A. Poustka, A.-M. Frischauf, H. Lehrach, A. J. Buckler, D. Church, L. Doucette-Stamm, M. C. O’Donovan, L. Riba-Ramirer, M. Shah, V. P. Stanton, S. A. Strobe, K. M. Draths, J. L. Wales, P. Dervan, D. E. Housman, M. Altherr, R. Shiang, L. Thompson, T. Fielder, J. J. Wasmuth, D. Tagle, J. Valdes, L. Elmer, M. Allard, L. Castilla, M. Swaroop, K. Blanchard, and F. S. Collins, R. Snell, T. Holloway, K. Gillespie, N. Datson, D. Shaw, and P. S. Harper; The Huntington’s Disease Collaborative Research Group, Cell 72, 971 (1993).
  * (26) S.-Y. Kim and W. Lim, “Ying-yang competitive harmony between direct and indirect pathways in a spiking neural network of the basal ganglia,” (2023). bioRxiv doi: https://doi.org/10.1101/2023.09.19.558549.
  * (27) A.R. Sadek, P. J. Magill, and J. P. Bolam, J. Neurosci. 27, 6352 (2007).
  * (28) P. Mailly, S. Charpier, A. Menetrey, and J. M. Deniau, J. Neurosci. 23, 5247 (2003).
  * (29) D. E. Oorschot, J Comp. Neurol. 366, 580 (1996).
  * (30) I. Bar-Gad, G. Morris, and H. Bergman, Prog. Neurobiol. 71, 439 (2003).
  * (31) T. S. Gertler, C. S. Chan, and D. J. Surmeier, J. Neurosci, 28, 10814 (2008).
  * (32) J. Moyer, J. A. Wolf, and L. H. Finkel, J Neurophysiol. 98, 3731 (2007).
  * (33) J. A. Wolf, J. T. Moyer, M. T. Lazarewicz, D. Contreras, M. Benoit-Marand, P. O’Donnell, and L. H. Finkel, J. Neurosci. 25, 9080 (2005).
  * (34) N. E. Hallworth, C. J. Wilson, and M. D. Bevan, J. Neurosci. 23, (2003).
  * (35) J. Bugaysen, M. Bronfeld, H. Tischler, I. Bar-Gad, and A. Korngreen, PLoS One 5, e12001 (2010).
  * (36) C. Richards, T. Shiroyama, and S. Kitai, Neurosci. 80, 545 (1997).
  * (37) M. D. Bevan and C. J. Wilson, J. Neurosci. 19, 7617 (1999).
  * (38) M. D. Bevan, C. J. Wilson, J. P. Bolam, and P. J. Magill, J. Neurophysiol. 83, 3169 (2000).
  * (39) M. D. Bevan, P. J. Magill, N. E. Hallworth, J. P. Bolam, and C. J. Wilson, J. Nuerophysiol. 87, 1348 (2002).
  * (40) J. L. Góngora-Alfaro, S. Hernández-López, J. Flores-Hernández, and E. Galarraga, Neurosci. Res. 29, 225 (1997).
  * (41) X. Liu, Q. Zhang, Y. Wang, and F. Chen, J. Shanghai Jiaotong Univ. (Sci.) 27, 505 (2022).
  * (42) J. Baufreton, J. F. Atherton, D.J. Surmeier, and M. D. Bevan, J. Neurosci. 25, 8505 (2005).
  * (43) W. M. Connelly, J. M. Schulz, G. Lees, and J. N. Reynolds, J. Neurosci. 30, 14854 (2010).
  * (44) K. Fujimoto and H. Kita, Brain Res. 609, 185 (1993).
  * (45) K.-Z. Shen and S. W. Johnson, J. Physiol. 573, 697 (2006).
  * (46) R. Ammari, B. Bioulac, L. Garcia, and C. Hammond, Front. Syst. Neurosci. 5, 43 (2011).
  * (47) H. Nakanishi, H. Kita, and S. T. Kitai, Brain Res. 527, 81 (1990).
  * (48) M. R. Park, W. M. Falls, and S. T. Kitai, J. Comp. Neurol. 211, 284 (1982).
  * (49) P. Dayan, L. F. Abbott, Theoretical Neuroscience (MIT, Cambridge, 2001).
  * (50) T. Götz, U. Kraushaar, J. Geiger, J. Lübke, T. Berger, P. Jonas, J. Neurosci. 17, 204 (1997).
  * (51) J. P. Bolam, H. Bergman, A. M. Graybiel, M. Kimura, D. Plenz, H. S. Seung, D. J. Surmeier, and J. R. Wickens, “Microcircuits in the striatum,” in S. Grillner and A. M. Graybiel (eds.), Microcircuits: The Interface Between Neurons and Global Brain Function (MIT Press, Cambridge, 2006) pp. 165–190.
  * (52) M. D. Humphries. R. Wood, and K. Gurney, Neural Netw. 22, 1174 (2009).
  * (53) M. D. Humphries. N. Lepora, R. Wood, and K. Gurney, Front. Comput. Neurosci. 3, 26 (2009).
  * (54) A. Tomkins, E. Vasilaki, C. Beste, K. Gurney, and M. D. Humphries, Front. Comput. Neurosci. 7, 192 (2014).
  * (55) Z. Fountas and M. Shanahan, PLoS ONE 12, e0189109 (2017).
  * (56) I. Han, Y. M. You, J. H. Kordower, S. T. Brady, and G. A. Morfini, J. Neurochem. 113, 1073 (2010).
  * (57) A. Reiner, R. L. Albin, K. D. Anderson, C. J. D’Amato, J. B. Penney, and A. B. Young, Proc. Natl. Acad. Sci. U S A. 85, 5733 (1988).
  * (58) R. L. Albin, A. Reiner, K. D. Anderson, L. S. Dure, B. Handelin, R. Balfour, W. O. Whetsell Jr, J. B. Penney, and A. B. Young, Ann. Neurol. 31, 425 (1992).
  * (59) E. K. Richfield, K. A. Maguire-Zeiss, H. E. Vonkeman, and P. Voorn, Ann. Neurol. 38, 852 (1995).
  * (60) R. L. Albin, A. B. Young, and J. B. Penne, Trends Neurosci. 12, 366 (1989).
  * (61) G. E. Alexander and M. D. Crutcher, Trends Neurosci. 13, 266 (1990).
  * (62) M. R. DeLong, Trends Neurosci. 13, 281 (1990).
  * (63) A. V. Kravitz, B. S. Freeze, P. R. L. Parker, K. Kay, M. T. Thwin, K. Deisseroth, and A. C. Kreitzer, Nature 466, 622 (2010).
  * (64) M. J. Frank, L. C. Seeberger, and R. C. O’Reilly, Science 306, 1940 (2004).
  * (65) M. J. Frank, J. Cogn. Neurosci. 17, 51 (2005).
  * (66) E. M. Navarro-López, U. Celikok, and N. S. Sengör, Cogn. Neurodyn. 15, 693 (2021).
  * (67) E. M. Navarro-López, U. Celikok, and N. S. Sengör, “Chapter 9 - Hybrid systems neuroscience,” in A. E. Hady (ed.) Closed Loop Neuroscience (Elsevier, London, 2016) pp. 113-129.
  * (68) U. Celikok, E. M. Navarro-López, and N. S. Sengör (2016) arXiv:1601.07740.
  * (69) U. Celikok and N. S. Sengör, Conference paper in the International Conference on Artificial Neural Networks (2016).
  * (70) Y. Cakir, J. Cogn. Syst. 4, 38 (2019).
  * (71) Z. Fountas and M. Shanahan, International Joint Conference on Neural Networks (IJCNN) 1 (2014).
  * (72) Z. Fountas and M. Shanahan, (2017) BioRxiv DOI:10.1101/197129.
  * (73) Z. Fountas and M. Shanahan, International Joint Conference on Neural Networks (IJCNN), 2407 (2014).
  * (74) Z. Fountas, Action Selection in The Rhythmic Brain: The Role of The Basal Ganglia and Tremor (2016) PhD thesis.
  * (75) M. D. Mark, R. Wood, and K. Gurney, PLoS Comput. Biol. 6, e1001011 (2010).
  * (76) M. D. Humphries, J. A. Obeso, and J. K. Dreyer, J. Neurol. Neurosurg. Psychiatry 89, 1181 (2018).
  * (77) M. J. Frank, B. Loughry, and R. C. O’Reilly, Cogn. Affect Behav. Neurosci. 1, 137 (2001).
  * (78) M. Lindahl, I. K. Sarvestani, O. Ekeberg, and J. H. Kotaleski, Front. Comput. Neurosci. 7, 76 (2013).
  * (79) M. Lindahl and J. H. Kotaleski, eNeuro 3, e0156-16 (2016).
  * (80) K. P. Michmizos, K. S. Nikita, “Local field potential driven Izhikevich model predicts a subthalamic nucleus neuron activity,” in: Engineering in Medicine and Biology Society, EMBC, 2011 Annual International Conference of the IEEE (IEEE, 2011). p. 5900-5903.
  * (81) B. Sen-Bhattacharya, S. James, O. Rhodes, I. Sugiarto, A. Rowley, A, B. Stokes, K. Gurney, and S. B. Furber, IEEE Trans. Cogn. Develop. Syst. 10, 823 (2018).
  * (82) K. Kumaravelu, D. T. Brocker, and W. M. Grill, J. Comput. Neurosci. 40, 207 (2016).
  * (83) C. M. Thibeault and N. Srinivasa, Front. Comput. Neurosci. 7, 88 (2013).
  * (84) J. E. Rubin, Curr. Opin. Neurobiol. 46, 127 (2017).
  * (85) D. S. Andres and O. Darbin, J. Neuropsychiatry Clin. Neurosci. 30, 101 (2018).
  * (86) O. Maith, F. V. Escudero, H. Ü. Dinkelbach, J. Baladron, A. Horn, F. Irmen, A. A. Kühn, and F. H. Hamker, Eur. J. Neurosci. 53, 2278 (2021).
  * (87) Á. González-Redondo, F. Naveros, E. Ros, and J. A. Garrido, Int. J. Neural Syst. 30, 2050057 (2020).
  * (88) J. W. Goodliffe, H. Song, A. Rubakovic, W. Chang, M. Medalla, C. M. Weaver, and J. I. Lübke, PLOS ONE 13, e0200626 (2018).
  * (89) G. Mulcahy, B. Atwood, and A. Kuznetsov, PLOS ONE 15, e0228081 (2020).
  * (90) A. R. Crossman, Neurosci. 21, 1 (1987).
  * (91) A. R. Crossman, I. J. Mitchell, M. A. Sambrook, and A. Jackson. Brain 111, 1211 (1988).
  * (92) A. V. Kravitz, B. S. Freeze, P. R. L. Parker, K. Kay, M. T. Thwin, K. Deisseroth, and A. C. Kreitzer, Nature 466, 622 (2010).
  * (93) F. Tecuapetla, S. Matias, G. P. Dugue, Z. F. Mainen, and R. M. Costa, Nat. Commun. 5, 4315 (2014).
  * (94) G. J. Klapstein, R. S. Fisher, H. Zanjani, C. Cepeda, E. S. Jokel, M. F. Chesselet, and M. S. Levine, J. Neurophysiol. 86, 2667 (2001).
  * (95) A. Reiner, R. L. Albin, K. D. Anderson, C. J. D’Amato, J. B. Penney, and A. B. Young, Proc. Natl. Acad. Sci. U S A. 85, 5733 (1988).
  * (96) A. J. Milnerwood and L. A. Raymond, Trends Neurosci. 33, 513 (2010).
  * (97) J. L. Plotkin, and D. J. Surmeier, Curr. Opin. Neurobiol. 33, 53 (2015).
  * (98) C. Cepeda, N. Wu, V. M. André, D. M. Cummings, and M. S. Levine, Prog. Neurobiol. 81, 253 (2007).
  * (99) A. J. Milnerwood, C. M. Gladding, M. A. Pouladi, A. M. Kaufman, R. M. Hines, J. D. Boyd, R. W. Ko, O. C. Vasuta, R. K. Graham, M. R. Hayden, T. H. Murphy, and L. A. Raymond, Neuron 65, 178 (2010).
  * (100) R. S. Turner and M. R. DeLong, J. Neurosci. 20, 7096 (2000).
  * (101) E. Bauswein, C. Fromm, and A. Preuss, Brain Res. 493, 198 (1989).
  * (102) J. E. Belforte, V. Zsiros, E. R. Sklar, Z. Jiang, G. Yu, Y. Li, E. M. Quinlan, and K. Nakazawa, Nat. Neurosci. 13, 76 (2010).
  * (103) J. L. Reed, H. Z. Qi, Z. Zhou, M. R. Bernard, M. J. Burish, A. Bonds, and J. H. Kaas, J. Neurophysiol. 103, 2139 (2010).
  * (104) T. Zheng and C. J. Wilson, J. Neurophysiol. 87, 1007 (2002).
  * (105) E. M. Izhikevich, IEEE Trans. Neural. Netw. 14, 1569 (2003).
  * (106) E. M. Izhikevich, IEEE Trans. Neural Netw. 15, 1063 (2004).
  * (107) E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting (MIT Press, Cambridge, 2007).
  * (108) E. M. Izhikevich, Cereb. Cortex 17, 2443 (2007).
  * (109) S.-Y. Kim and W. Lim, Cogn. Neurodyn. 14, 535 (2020).
  * (110) S.-Y. Kim and W. Lim, Neural Netw. 106, 50 (2018).
  * (111) S.-Y. Kim and W. Lim, Neural Netw. 97, 92 (2018).
  * (112) S.-Y. Kim and W. Lim, Neural Netw. 134, 173 (2021).
  * (113) S.-Y. Kim and W. Lim, Cogn. Neurodyn. 15, 1067 (2021).
  * (114) S.-Y. Kim and W. Lim, Phys. Rev. E 105, 014418 (2022)
  * (115) S.-Y. Kim and W. Lim, Cogn. Neurodyn. 16, 643 (2022).
  * (116) S.-Y. Kim and W. Lim, Cogn. Heurodyn. 16, 1427 (2022).
  * (117) S.-Y. Kim and W. Lim, Cogn. Neurodyn. DOI: 10.1007/s11571-023-09985-5 (2023).
  * (118) C. E. Jahr and C. F. Stevens, J. Neurosci. 10, 3178 (1990).
  * (119) X. J. Wang, Physiol. Rev. 90, 1195 (2010).
  * (120) N. Brunel and X. J. Wang, J Neurophysiol. 90, 415 (2003).
  * (121) C. Geisler, N. Brunel, and X. J. Wang, J. Neurophysiol. 94, 4344 (2005).
  * (122) N. Brunel and V. Hakim, Chaos 18, 015113 (2008).
  * (123) S.-Y. Kim and W. Lim W, J. Neurosci. Meth. 226, 161 (2014).
  * (124) H. Shimazaki and S. Shinomoto, J. Comput. Neurosci. 29, 171 (2010).
  * (125) A. C. Bachoud-Lévi, J. Ferreira, R. Massart, K. Youssov, A. Rosser, M. Busse, D. Craufurd, R. Reilmann, G. D. Michele, D. Rae, F. Squitieri, K. Seppi, C. Perrine, C. Scherer-Gagou, O. Audrey, C. Verny, and J.-M. Burgunder, Front. Neurol. 10, 710 (2019).
  * (126) D. Luis-Ravelo, H. Estévez-Silva, P. Barroso-Chinea, D. Afonso-Oramas, J. Salas-Hern’andez, J. Rodríguez-Núñez, A. Acevedo-Arozena, D. Marcellino, T. González-Hernández, Exp. Neurol. 299, 137 (2018).
  * (127) M. R. Hayden, B. R. Leavitt, U. Yasothan, and P. Kirkpatrick, Nat. Rev. Drug Discov. 8, 17 (2009).
  * (128) H. Bashir and J. Jakovic, Expert Rev. Neurother. 18, 51 (2018).
  * (129) S. J. Nagel, A. G. Machado, John T. Gale, D. A. Lobel, and M. Pandya, Front. Syst. Neurosci. 9, 32 (2015).
  * (130) M. Sharma and M. Deogaonkar, J. Clin. Neurosci. 22, 812 (2015).
  * (131) L. Wojtecki, S. J. Groiss, C. J. Hartmann, S. Elben, S. Omlor, A. Schnitzler, and J. Vesper, Brain Sci. 6, 38 (2016).
  * (132) J. Demeestere and W. Vandenberghe, CNS Nuerosci. Ther. 17, 705 (2011).
  * (133) A. Berardelli, J. Noth, P. D. Thompson, E. L. Bollen, A. Curra, G. Deuschl, J. G. van Dijk, R. Topper, M. Schwarz, and R. A. Roos, Mov. Disord. 14, 398 (1999).
  * (134) M. Cudkowicz and N. W. Kowall, Ann. Neurol. 27, 200 (1990).
  * (135) J. C. Hedreen, C. E. Peyser, S. E. Folstein, and C. A. Ross, Neurosci. Lett. 133, 257 (1991).
  * (136) T. Koos and J. M. Tepper, Nature Neurosci. 2, 467 (1999).
  * (137) S. Narayanan, Neurocomput. 52–54, 605 (2003).
  * (138) A. H. Gittis and E. A. Yttri, Curr. Opin. Biomed. Eng. 8, 14 (2018).
  * (139) Y. Shen, R. E. Campbell, D. C. Côté and M.-E. Paquet, Front. Neural Circuits 14, article 41 (2020).