Abstract and Figures

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 ring 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.
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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
CorticalNeuron
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
=IlINa IKIK d
IMX
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
=IlINa IKIA
ILITICaKX
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, ICaKis 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
=IlINa IKIT
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(VmErev ) (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) =
N1
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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... Axon collaterals were positioned within the STN, parallel to one another, and orientated to project in the direction. The axon and collateral model was based on that used previously by Lowery and Kang 27,38 with membrane parameters based on an experimental study in mice developed by Foust et al. 39 . This model captures the threshold of activation of the axon and collateral by considering the properties of the model neuron along with the electrode geometry, the electrode-tissue-interface, and the heterogeneity and electrical properties of brain tissue incorporated in the rat DBS FEM model. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 A c c e p t e d M a n u s c r i p t 16 Multi-compartment axon collaterals were simulated to be 500 µm long with a diameter of 0.5 µm and were comprised of 11 segments with 10 nodes. ...
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