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Changes in Neuronal Entropy in a Network Model of the Cortico-Basal

Ganglia during Deep Brain Stimulation

John E. Fleming1,Student Member, IEEE and Madeleine M. Lowery1,Member, IEEE

Abstract— Neuronal entropy changes are observed in the

basal ganglia circuit in Parkinson’s disease (PD). These changes

are observed in both single unit recordings from globus pallidus

(GP) neurons and in local ﬁeld potential (LFP) recordings from

the subthalamic nucleus (STN). These changes are hypothesized

as representing changes in the information coding capacity of

the network, with PD resulting in a reduction in the coding

capacity of the basal ganglia network. Entropy changes in the

LFP and in single unit recordings are investigated in a detailed

physiological model of the cortico-basal ganglia network during

STN deep brain stimulation (DBS). The model incorporates

extracellular stimulation of STN afferent ﬁbers, with both

orthodromic and antidromic activation, and simulation of the

LFP detected at a differential recording electrode. LFP sample

entropy and beta-band oscillation power were found to be

altered following the application of DBS. The ﬁring pattern

entropy of GP neurons in the network were observed to

decrease during high frequency stimulation and increase during

low frequency stimulation. Simulation results were consistent

with experimentally reported changes in neuronal entropy

during DBS.

I. INTRODUCTION

Parkinson’s disease (PD) is a neurodegenerative disease

characterized by a triad of motor symptoms; bradykinesia,

akinesia, and tremor. Recent research has focused on identi-

fying signals from the central and peripheral nervous system

which can be used to quantify the disease state and symptom

severity. These signals are commonly referred to as disease

‘biomarkers’. Clinical and experimental studies have identi-

ﬁed several potential biomarkers for PD, such as increased

oscillatory activity in the beta frequency band (13-30 Hz)

recorded from the cortico-basal ganglia circuit [1], beta-

gamma band (60-200 Hz) phase-amplitude coupling in the

primary motor cortex [2], and changes in neuronal entropy

in the basal ganglia network [3]. Deep brain stimulation is

an effective treatment for PD which has been shown to have

measurable effects on these biomarkers. These effects include

reducing beta-band oscillatory activity [1], reducing beta-

gamma band phase-amplitude coupling [2], and regularizing

neural ﬁring rates in the basal ganglia network [4], [5].

Firing pattern entropy, calculated using single unit record-

ings from the basal ganglia network, and sample entropy,

calculated from the STN local ﬁeld potential (LFP), are two

entropy measures with observable changes during PD. Firing

pattern entropy quantiﬁes an upper bound on the information

*Research supported by the European Research Council (ERC) under

the European Union’s Horizon 2020 research and innovation programme

(ERC-2014-CoG-646923-DBSModel).

1John E. Fleming and Madeleine M. Lowery are with the Neuromus-

cular Systems Laboratory, School of Electrical & Electronic Engineering,

University College Dublin, Ireland (e-mail: john.ﬂeming@ucdconnect.ie).

embedded in a spike train. Spike trains from globus pallidus

(GP) neurons are shown to have increased ﬁring pattern

entropy during PD, with this being reduced to near healthy

levels during effective DBS [3]–[6]. Sample entropy is a

measure for assessing the complexity of physiological time

series signals. In [7], it was shown that there was an inverse

relationship between beta-band oscillation power and beta-

band sample entropy in the STN LFP. Furthermore, in [7]

LFP sample entropy was utilized to distinguish between PD

patients who experienced freezing of gait episodes and those

who did not.

Computational modelling allows the investigation of net-

work dynamics which may be difﬁcult to access during

clinical practice. Here a computational model of the cortico-

basal ganglia network is utilized to investigate changes in

basal ganglia neuronal entropy during DBS. Single unit

recordings from GP neurons are utilized to assess changes

in ﬁring pattern entropy, while the STN LFP is simulated to

assess changes in sample entropy and beta-band oscillatory

activity during DBS. The results from the computational

model are compared with results from clinical and experi-

mental studies. Quantifying changes in basal ganglia entropy

may lead to an improved understanding of the relationship

between oscillatory activity and entropy in the network, and

how this relationship is modiﬁed during PD and DBS.

II. METHODS

A physiologically based model of the cortico-basal ganglia

network incorporating extracellular DBS and simulation of

the STN LFP was utilized [8]. The structure of the network

model is presented in Fig. 1 and includes the closed loop

formed between the cortex, basal ganglia and thalamus.

The major model components include single compartment,

conductance-based biophysical models of the STN, globus

pallidus externa (GPe), globus pallidus interna (GPi) and

thalamus, each of which have been validated and employed

in previous modelling studies [8]–[10]. The cortex is repre-

sented by a network of interneurons and multi-compartment

cortical neurons. Each component is described in greater

detail below.

Six hundred cells consisting of one hundred STN, GPe,

GPi, thalamic, interneuron and cortical neurons were con-

nected through excitatory and inhibitory synapses, AMPA

and GABAa, respectively. The STN neurons received direct

excitatory input from the cortex via the hyperdirect path-

way and inhibitory input from the GPe. Each STN neuron

received excitatory input from ﬁve cortical neurons and

inhibitory input from two GPe neurons. Each GPe neuron

received inhibitory input from one other GPe neurons and

excitatory input from two STN neurons. Each GPi neuron

received excitatory input from a single STN neuron and

inhibitory input from a single GPe neuron. Each thalamic

neuron received inhibitory input from a single GPi neuron.

Cortical neurons received excitatory input from a single tha-

lamic neuron and inhibitory input from a single interneuron.

Interneurons received excitatory input from a single cortical

axon. All connections within the network were randomly

assigned.

Cortex

Interneurons Soma

AIS

Axon

Collateral

CorticalNeuron

STN

GPe

GPi

Thalamus

+

+

+

+

+

Figure 1. Schematic diagram of the cortico-basal ganglia

model. Excitatory and inhibitory connections are indicated

with a + or –, respectively.

The presence of pathologically exaggerated beta oscilla-

tions in the cortico-basal ganglia network, typically observed

in PD, were simulated by varying synaptic gains within the

network in accordance with [11]. An increased cortical drive

to the STN, due to strengthening of the hyperdirect pathway,

led to the emergence of beta oscillations within the network

and the STN LFP.

A. Cortex

The model used to simulate the cortex consisted of cor-

tical neurons and interneurons. The cortical neuron model

included a soma, axon initial segment (AIS), main axon, and

axon collateral. The cortical neuron soma and interneuron

models are based on the regular spiking neuron model

developed by Pospischil et al. [12]. The model used to

simulate the AIS, main axon, and axon collateral is based

on results from the experimental and modeling study in [13].

The membrane potentials of the cortical compartments and

interneurons are described by

Cm

dvm

dt

=—Il—INa —IK—IK d

—IM—X

k

Ik

syn (1)

Where Cmis the membrane capacitance, Ilis the leak

current, INa is the sodium current, IKis the potassium

current, IKd is D potassium current, IMis a slow, voltage-

dependent potassium current, and Isyn are synaptic currents.

The cortical soma model excluded the IKd current. The

cortical AIS, main axon and axon collateral segments did

not include the IMcurrent. Finally, cortical interneurons did

not include either the IKd or IMcurrents. Further details

regarding the parameters used can be found in [12], [13].

B. Subthalamic Nucleus

The STN model incorporates a physiological representa-

tion of STN neurons developed by Otsuka et al. [14]. The

model captures the generation of plateau potentials, which

are believed to play an important role in generating STN

bursting activity in PD. The membrane potential of an STN

neuron is given by

Cm

dvm

dt

=—Il—INa —IK—IA

—IL—IT—ICa—K—X

k

Ik

syn (2)

Where Cmis the membrane capacitance, Ilis the leak

current, INa is a sodium current, IKis a Kv3-type potassium

current, IAis a voltage dependent A-type potassium current,

ILis an L-type long lasting calcium current, ICa−Kis a

calcium activated potassium current, and Isyn are synaptic

currents. Further details can be found in [14].

C. Globus Pallidus and Thalamus

The models used to simulate GPe, GPi, and thalamic

neurons are based on those presented by Rubin and Terman

in [15]. The membrane potential of a GP neuron is described

by

Cm

dvm

dt

=—Il—INa —IK—IT

—ICa —IAH P —X

k

Ik

syn (3)

Where Cmis the membrane capacitance, Ilis the leak

current, INa is the sodium current, IKis a potassium current,

INa is a sodium current, ITis a low-threshold T-type calcium

current, ICa is a voltage-dependent afterhyperpolarizaation

potassium current, and Isyn are synaptic currents. Thalamic

neurons were modelled similarly, with the exception of

excluding ICa and IAH P in the thalamic model. Further

details regarding the GPe, GPi, and thalamus models can

be found in [15].

D. Synapses

Individual synaptic currents, Ik

syn, were described by

Ik

syn =Rk(Vm—Erev ) (4)

Where Ik

syn is the kth synaptic current, Rkrepresents

the kinetics of the onset and decay of current following a

presynaptic spike for synapse k, and Erev is the reversal

potential for the appropriate type of synapse. Further details

regarding the synaptic models can be found in [16].

E. Application of DBS and LFP Simulation

The extracellular potential due to a current source, Ix, at

time twas calculated as

Vx(t) = Ix(t)

4πσrx

(5)

Where σis the conductivity of a homogenous, isotropic

medium representing brain tissue. The distance from a point

in extracellular space to the current source Ix, or vice versa,

is given as rx.

For simulating the voltage applied to cortical collaterals

due to a monopolar stimulation electrode, rxwas the dis-

tance between each collateral segment and the stimulation

electrode, while Ixwas a square wave current source with

130 Hz frequency, 60 s pulse width and varying amplitude.

Cortical collaterals were assigned a random position in a

2 mm radius of extracellular space around the stimulation

electrode.

To simulate the recording of the LFP using a differ-

ential recording electrode, STN neurons, like the cortical

collaterals, were assigned positions in a 2 mm radius of

extracellular space around the stimulation electrode. Each

recording electrode was positioned 1.365 mm away from the

stimulation electrode, with each recording electrode being

placed either side of the stimulation electrode. The LFP

recorded at each recording electrode was then calculated

as the summation of the total extracellular voltages due to

each STN neuron’s synaptic currents in the extracellular

space, where Ixcorresponds to the synaptic currents of an

STN neuron of distance rxaway from one of the recording

electrodes.

F. LFP Sample Entropy

Sample Entropy was calculated as the negative natural

logarithm of the estimated conditional probability that two

sequences similar for mpoints remain similar at the next

point, where self-matches are not included in calculating the

probability [17]. It is deﬁned as

SampEn(m, r, N) = —ln[Am+1(r)/Am(r)] (6)

Where Am+1(r)represents the number of vector pairs

(within the time series) of length m+ 1 whose mutual

distance is less than a tolerance r, and Am(r)equals the

number of vector pairs (within the time series) of length

mwhose mutual distance is less than r. Here the length

of the vector pairs, m, denotes the embedding dimension.

The mutual distance between the vector pairs was calculated

using the Chebyshev distance between the pairs, with mand

rset to 4 and 20% of the standard deviation of the data

respectively.

G. Firing Pattern Entropy

The ﬁring pattern entropy of a spike train was calculated

by binning the inter spike intervals of the train in logarithmic

time, as in [18]. The leftmost and rightmost bin edges were

set just below, or just above, the smallest and largest inter

spike intervals observed, respectively, in each population.

The entropy of the spike train was then calculated using

Shannon Entropy

H(X) = —

N—1

X

i=0

PIS Iilog2(PISIi) (7)

Where His the entropy of spike train X,PISIiis the

probability of inter spike interval ioccurring in the spike

train, and Nis the number of inter spike interval bins.

H. Simulation Details

The model was implemented in Python using the API

package PyNN [19] with NEURON v7.6.5 as the model

simulator. A timestep of 0.01 ms was used for simulations.

Post-processing was done using custom scripts in MATLAB

(The MathWorks, Inc., Natick, MA). To examine LFP sample

entropy the LFP was ﬁrst down-sampled and low-pass ﬁl-

tered at 100Hz to remove stimulation artifact. To examine the

magnitude of beta-band oscillations in the LFP the LFP was

band-pass ﬁltered between 10 and 35 Hz, full-wave rectiﬁed

and averaged by low-pass ﬁltering at 2 Hz.

III. RESULTS

A. LFP Sample Entropy

The effect of varying stimulation amplitude on the STN

LFP sample entropy was investigated using a ﬁxed frequency

and pulse width of 130 Hz and 60 µs, respectively, Fig. 2 (a).

A progressive increase in the sample entropy was observed

as the stimulation amplitude increased. For comparison, the

corresponding magnitude of beta-band oscillations in the

LFP is given in Fig. 2 (b).

(b)

(a)

DBS Amplitude (mA)

Figure 2. Normalized STN LFP (a) sample entropy and (b)

beta-band oscillation power as a function of DBS amplitude.

B. Firing Pattern Entropy

The effect of varying stimulation frequency on the ﬁring

pattern entropy of GPe and GPi neurons was investigated

using a ﬁxed amplitude and pulse width of 3 mA and 60

µs, respectively. Fig. 3 shows the cumulative distribution

of the ﬁring pattern entropy for each population. Firing

pattern entropy was reduced following the application of

high frequency stimulation (HFS), with a frequency of 130

Hz, and increased following the application of low frequency

stimulation (LFS), with a frequency of 20 Hz.

(b)

(a)

Entropy (bits/spike)

Cumulative Distribution Function

No DBS

HFS

LFS

Figure 3. Cumulative distributions of the ﬁring pattern en-

tropy for the (a) GPe and (b) GPi neuron populations due to

high frequency and low frequency stimulation.

IV. DISCUSSION AND CONCLUSION

The aim of this study was to investigate changes in

neuronal entropy due to extracellular DBS in a computational

model of the cortico-basal ganglia network during PD. The

model includes extracellular stimulation of cortical afferent

ﬁbers projecting to the STN and simulation of the resulting

LFP. This allows for comparison with clinical and exper-

imental results which have previously investigated entropy

changes in the cortico-basal ganglia network during PD.

Sample entropy was observed to have an inverse rela-

tionship with beta-band oscillation power, Fig. 2. In [7],

an inverse relationship was observed between beta-band

sample entropy and beta-band oscillation power taken from

STN LFP recordings in freely moving patients during three

movement tasks. Here, a distinction was not made between

frequency bands when calculating sample entropy. However,

effective DBS did result in similar behaviour, with beta-band

power decreasing, and sample entropy increasing in the LFP

for increasing DBS amplitude.

Firing pattern entropy in GP neurons decreased during

HFS, and increased during LFS of the STN, Fig. 3. These

results agree with those presented in [4]–[6] and support the

hypothesis that effective DBS regularizes ﬁring patterns in

GP neurons.

The computational model presented displays changes in

neuronal entropy consistent with those presented in clinical

and experimental literature. These results suggest that inves-

tigation into basal ganglia entropy changes during PD and

DBS may elucidate the relationship between network entropy

and oscillation power during disease progression. Moreover,

these results support further investigation of the utilization

of entropy-based measures in closed-loop DBS strategies.

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