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A biophysically inspired microelectrode recording-based model for the subthalamic nucleus activity in Parkinson's disease

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A biophysically inspired microelectrode recording-based model for thesubthalamic nucleus activity in Parkinson’s disease
Sabato Santaniello
a,
*, Giovanni Fiengo
a
, Luigi Glielmo
a
, Giuseppe Catapano
b
a
GRACE – Group for Research on Automatic Control Engineering, Department of Engineering, Universita´ degli Studi del Sannio, 82100 Benevento, Italy
b
Department of Neurosciences, Division of Neurosurgery, Azienda Ospedaliera ‘‘G. Rummo’’, 82100 Benevento, Italy
1. Introduction
The subthalamic nucleus (STN) and all the other nuclei of thebasalgangliahavereceivedgreatattentioninthelast30yearsbothunder the neuro-anatomic (e.g., [1,2]), neuro-physiologic (e.g., [3–
6])andneuro-functional[7]pointofview.Inthelastdecadeagreat
interest by the Neurocomputating community has also grownaround such areas in thebrain and many efforts havebeen done inordertonumericallyreplicatethecorrespondingbehavior(e.g.,[8–10]). Such interest is due to the fact that the underneathnetworking mechanisms are still partly unknown and that theextraordinary rich and various range of ﬁring patterns that thebasal ganglia cells are able to produce [11] have not yet beenfaithfully modeled.But the main reason for the attention given to the STN is surelyduetoitsroleinthemanifestationofthemotorsymptomsofsomespread neuro-degenerative pathologies, like Parkinson’s disease(PD), dystonia and essential tremor. It is quite well demonstrated,in fact, that they produce signiﬁcant alterations of the STNphysiological electrical activity and of the basal ganglia ﬁringpatterns (e.g., [12–15]). Such interest is even more increased sinceit has been observed that the typical Parkinsonian movementdisorders (i.e. tremor, rigidity, akynesia and postural instability)considerably reduce when the usually adopted L-dopa basedpharmacological therapies are associated with the modulation of the subthalamic activity through a suitable locally appliedelectrical stimulation. To this aim, the deep brain stimulation(DBS) is the technique currently used in Neurosurgery both for PDand other pathologies [16–18]. Brieﬂy, the DBS consists in squarepulse trains provided to selected targets inside the brain andgenerated by an artiﬁcial stimulator surgically implanted in thepatient. Frequency and duty cycle of the stimulation are usuallyﬁxed in order to guarantee safety for the patients while theamplitude is set by the surgeon during the implantation accordingto a clinical procedure consisting of the injectionof sample stimuliand the evaluation of the correspondingly induced effects on thepatient’s motor symptoms. Associated with the appropriatepharmacological therapy, the DBS greatly reduces most of themotor symptoms, limits drug-induced dyskinesia and frequentlyimproves patients’ ability to perform activities of daily living withlessencumbrancefrommotorﬂuctuations[19,20].Atasubcorticallevel, the clinical improvements in Parkinson’s disease correlate
Biomedical Signal Processing and Control 3 (2008) 203–211
A R T I C L E I N F O
Article history:
Received 28 October 2007
Received in revised form 20 March 2008
Accepted 20 March 2008
Available online 16 May 2008
Keywords:
STN modelingParkinson’s diseaseNonlinear systemsSpike detectionContinuous wavelet transform
A B S T R A C T
The subthalamic nucleus (STN) plays a central role in movement actuation and manifestation of movement disorders (i.e., tremor, rigidity, akynesia and postural instability) in Parkinson’s disease (PD)patients. Moreover, it has been recently revealed that an opportune electrical stimulation of the STN,called deep brain stimulation (DBS), can strongly contribute to the annihilation of the PD-related motordisorders. Currently, a great effort is made both in Medicine, Neurosciences and Engineering forunderstandingand modelingin details how theSTN works, howPDdetermines itspathological behaviorand DBS restores the correct motor function.Thepaperisorganizedintwoparts.FirstlysomestochasticpropertiesoftheSTNelectricalactivityareobtained by analyzing a preliminary set of experimental data coming from microelectrode recordings(MERs)intwoPDpatientswhounderwentthesurgicalimplantationofDBSelectrodes.Then,anonlinear,stochastic, continuous-state model describing the global electrical behavior of the STN in PD patients isproposed. It is inspired by the fundamental physiologic features of the subthalamic cells and a ﬁctitiousvector state is introduced to represent the main dynamics. Its numerical parameters and stochasticproperties are chosen by ﬁtting the available data.
2008 Elsevier Ltd. All rights reserved.
* Correspondingauthor at: Dipartimento di Ingegneria, Universita´ degliStudi delSannio, Via C. Rampone, 9, 82100 Benevento, Italy. Tel.: +39 0824 305585;fax: +39 0824 325246.
E-mail address:
sabato.santaniello@unisannio.it (S. Santaniello).
Contents lists available at ScienceDirect
Biomedical Signal Processing and Control
journal homepage: www.elsevier.com/locate/bspc
1746-8094/$ – see front matter
2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.bspc.2008.03.001
with the modulation of the oscillatory activity induced by the DBSalongthebasalganglia[21].Itisproved,infact,thattheDBSintheSTN increases the subthalamic low-frequency oscillations (i.e., 1–1.5 Hz, [22]), modulates the beta ones [21], forces the pallidal
neuronal elements to synchronize at frequencies greater than70 Hz [23], and spreads its effects until the motor thalamus due tothe central role of the STN in the coordination of the basal gangliafunction [24]. Such results suggest that the DBS works at thecellular level by modifying the ﬁring patterns running from theextrapyramidalsystemtowardthecorticalareas,i.e.,itreducestheoscillations of the dopamine-depleted pathways in the tremorband(i.e., 2–7 Hz), reshapesthose inthebetaones (i.e.,13–35 Hz),and enhances high frequency oscillation (
>
70 Hz). Thus, it issupposedtointroducearegularizationintheﬁringpatternsandaninformational lesion of the stimulated nucleus [25].However, a detailed and exhaustive description of themodiﬁcations induced by PD in the ﬁring patterns of the basalganglia and of the dynamics according to which DBS affects bothSTN behavior and its pallidal and nigral projections is unfortu-nately still far to come, even if important results and interestingmodels have been reported in Neurosciences (e.g., [26–28]) andNeuroengineering [29]. The great variety of synaptic connectionshapes,projectionﬁbers,dendriticarborizationsandioniccurrentsinvolved in the functional interactions of the basal ganglia is achallenging hurdle toward a full comprehension of such mechan-isms since it introduces a level of complexity hard to manage orrepresent through simpliﬁed models. Moreover, while it is quiteclear how the single subthalamic cell electro-chemically behaves(see [3,4,30]) and different dynamical models have been proposedin correspondence (e.g., [31–33]), a lot of work is still required tounderstand how cells generate the macroscopic subthalamicactivity, as it results from in vivo microelectrode recordings(MERs) during DBS devices implantation (see Fig. 1, Trace 1). Tothis aim a signal processing approach has been recently proposedinliterature to characterizetheintraoperative localpotentials[34]and to analyze the inter-spike intervals (ISIs) distribution [35] of the MERs, but few information have been extracted, while greatefforts have dealt with ﬁnite-elements models reproducing theeffect of DBS at the cellular level [29]. Such results may be helpfulinanalysisbutcannotbeusedforDBScontroltasks,whicharenowemerging as fundamental topics in Biomedical Engineering. Itshouldbeconsidered,infact,thatthesubthalamicresponsetoDBSstimuli changes in time either physiologically or as a consequenceof the electrodes corrosion or other external events, making thestimulation no more suited or effective and, consequently, thecomfort for the patient and the device energy management notoptimal.Inthatcase,itwouldbehopefultoautomaticallytunetheparameters of the stimulation in order to recover the best possibleeffect. To this aim a sufﬁciently fast model of STN activity can be
Nomenclature
B
(
d
,
b
) beta function with parameters
d
and
b
C
r
r
-th contiguous region in
D
*
D
{0,1,
. . .
,
N
s
–1}
D
*
S
g
c
2
G
D
c
D
c
subset of
D
for which (4) is satisﬁedeCDF
i
empirical CDF for
T
si
f
(
p
f
,
t
) Generic CDF with parameters
p
f
2
D
f
f
A
optimal ﬁtting
f
ð
p
MEAN
f
;
A
;
t
Þ
for
A
in (15)
f
i
optimal ﬁtting
f
ð
p
MLE
f
;
i
;
t
Þ
for
T
si
in (13)
F
B
(
t
) beta CDF
g
c
c
-th temporal scale
G
{
g
0
,
g
1
,
. . .
,
g
J
}
G
r
f
g
c
2
G
:
S
ð
c
;
d
Þ
>
Q
c
;
d
2
C
r
g
L
i
number of spikes for the
i
-th trace
N
c
number of contiguous regions in
D
*
N
s
number of temporal translations
p
MEAN
f
;
A
optimal parameters for
f
ﬁtting times in
A p
MLE
f
;
i
MLE parameters for
f
ﬁtting times in
T
si
PDF
f
ð
p
f
;
t
ik
Þ
PDF of
f
evaluated with parameters
p
f
atsample
t
ik
.
s
(
t
) generic signal to be analyzed
s
1
,
. . .
,
s
M
traces available for analysis
S
(
c
,
d
) wavelettransformcoefﬁcientofthesignal
s
(
t
)withrespect to
t
d
and
g
c
t
d
d
-th temporal translation
t
ik
k
-th spike arrival time for the
i
-th traceThr threshold for the Quiroga’s algorithm
T
(
t
) time series of the inter-times between consecutivespikes
T
c r
arg max
t
d
;
d
2
C
r
f
S
ð
c
;
d
Þ
:
S
ð
c
;
d
Þ
>
Q
c
g
.
T
si
set of the spike arrival times for
i
-th trace
T
r
r
-th spike arrival time
u
DBS related input
v
activation level of inhibitory synapses
w
(
t
) white noise with 0 mean, standard deviation
s
x
activation level of excitatory synapses
y
noise-affected signal generated by recorded cells
z
noise-free signal generated by recorded cells
a
rate for
x
and
v
g
,
a
i
,
b
i
,
u
i
Model parameters (with
i
=
R
,
D
,
H
).
u
arctan(
x
,
v
)
Q
c
acceptance threshold for the hypothesis that aspike is detectable at scale
g
c
s
n
median
ðj
s
j
=
0
:
6745
Þ
v
(
t
) angular velocity stochastically deﬁned
c
(
t
) generic wavelet mother function
c
c
,
d
(
t
) waveletmotherfunctiontranslatedof
t
d
andscaledby
g
c
according to (3)
Fig. 1.
Typical shape of intraoperative microelectrode traces both far and close totheSTN.Trace1hasbeenrecordedattheplannedtheoreticaltarget.Trace24.0 mmbefore it. Data are from Patient 1, hemisphere ‘‘right’’, electrode A.
S. Santaniello et al./Biomedical Signal Processing and Control 3 (2008) 203–211
204
useful to design control strategies. More generally, if any real timeand feasible control action would be introduced in the DBS, asimpliﬁed and computationally reasonable model is required todescribe the behavior of the whole basal ganglia in PD patientswith and without DBS. In that case, in order to faithfully matchexperimental data and reliably predict the effects of the stimula-tion, modeling efforts should focus on the STN behavior asmacroscopically resulted from intraoperative MERs before andafter DBS.In this scenario, we present a nonlinear, stochastic, continuousstate model of the STN electrical behavior at the MERs level whoseparameters have been customized to PD patients and whosestructurehasbeeninspiredbythemainphysiologicfeaturesofthesubthalamiccells. In particular,it has been observed thatthe spikearrival times tend to distribute according to a beta cumulativedistribution function (CDF) when normalized to the interval [0, 1],independently of the procedure used for the spike detection.Preliminary results of the work illustrated in this paper havebeen presented in [36].
2. Methods
ThetypicalshapeofSTNintraoperativerecordings(Fig.1,Trace1)isduetothesuperpositionofmanyspikes(oractionpotentials–APs in the followings –) generated by several neurons placed atdifferent distances from the microelectrodes. While it has beenmodeled by [37] how such superposition physically takes place,information about the actual network topology of the recordedneurons cannot still be derived from the available signals. In thispaper we want to globally describe the STN behavior near themicroelectrodes and mathematically interpret the shape of theintraoperative recordings. It means to detect each spike from theavailableMERsoftheSTNandinvestigatethestochasticpropertiesof the corresponding temporal occurrence.Since it is unknown where these spikes exactly are located inthe available recordings, the application of a spike detectionalgorithm rather than another can be neither signiﬁcant norreliable.Forthisreason,wehaveprocessedtheavailabledatatwiceby adopting two different spike detection algorithms: we haveapplied a well known continuous wavelet transform-basedalgorithm [38] and a diffused threshold-based algorithm [39] on
the same dataset and conducted similar stochastic analysis on thecorresponding results separately. Quite identical conclusions havebeen drawn (see Section 3) in the two cases, thus proving that theactivity from the neurons recorded by the microelectrodes is welldetectable independently from the adopted algorithm. This mightbe due to a relatively high signal-to-noise ratio (SNR) for themembrane voltage signals, as plausible considered that themeasurement noise has low amplitude (compare Trace 1 and 2in Fig. 1) and that the recorded neural currents decrease when themicroelectrode-neuron distance increases [37]. The assumedshape of the spikes (see Fig. 2), instead, has been inspired bothby the neuron electrophysiology [40] and a visual inspection of experimental data.
2.1. Wavelet-based spike detection
A wavelet-based detection algorithm has been chosen due tothe similarity between the selected spike shape and thebiorthogonal wavelets and the excellent performance of thesealgorithmsbothwithhighandlowSNRsignals(e.g.,[38,41,42]).Inparticular, it has been used the Nenadic and Burdick’s algorithm[38] since it further provides an automatic estimation of temporalarrival of detected spikes showing good performance for a widerange of SNR values.It can be brieﬂy resumed as follows. Let
s
(
t
) denote the genericsignal to be analyzed and
c
(
t
) the chosen wavelet (mother)function[43],i.e.afunctionofﬁniteenergy(i.e.,
c
2
L
2
(
R
))andzeroaverage
Z
R
c
ð
t
Þ
d
t
¼
0
:
(1)Let us choose
N
s
time instants
t
d
,
d
2
D
= {0,1,
. . .
,
N
s
1}, uniformlydistributed in the domain of
s
(
t
) and called ‘‘translations’’ in thefollowing,deﬁnetheset
G
= {
g
0
,
g
1
,
. . .
,
g
J
}of
J
+ 1ﬁxedpositiverealnumbers, called ‘‘temporal scales’’,
1
and calculate the correspond-ing
s
(
t
) wavelet transform coefﬁcients
S
ð
c
;
d
Þ ¼
Z
R
s
ð
t
Þ
c
c
;
d
ð
t
Þ
d
t
8
d
2
D
;
c
¼
0
;
. . .
;
J
(2)where
c
c
;
d
ð
t
Þ ¼
1
ﬃﬃﬃﬃﬃ
g
c
p
c
t
t
d
g
c
:
(3)Then, for each temporal scale
g
c
2
G
, the algorithm tests thehypothesis
j
S
ð
c
;
d
Þj
>
Q
c
d
2
D
(4)where
Q
c
isaconstantcalculatedforthe
c
-thscale.
Q
c
canbeviewedas an acceptance threshold for the hypothesis that a spike isdetectable at the scale
g
c
and its numerical value follows from thesolution of a binary hypothesis testing problem, as detailed in [38].If for some
d
(4) is satisﬁed, a new AP is detected at the
c
-thscale. Since, for a ﬁxed scale
g
c
and translation
t
d
,
S
(
c
,
d
) representsthe resemble index of
s
(
t
) to the wavelet
c
c
,
d
, and considering thatAPs are highly localized in time (they typically last
1–1.5 ms), itcan be concluded that the wavelet coefﬁcients satisfying (4) andrepresenting the same spike should be neighbors both intranslation and scale, i.e. they should be obtained for smallchanges in the values of
c
and
d
. For that reason, denoted with
D
c
the subset of
D
for which (4) is satisﬁed at scale
g
c
, the subset
D
¼
[
g
c
2
G
D
c
(5)will be structured in a number of contiguous regions (i.e.,subsets of
D
over which (4) is satisﬁed in succession at any of
Fig. 2.
Typical shape of a subthalamic neuron spike recorded by a microelectrode.The refractory, depolarization and hyperpolarization phases correspond to the
R
,
D
and
H
local extremes (see Section 2.3).
1
For details about how to choose the values of
N
s
,
J
,
t
d
for all
d
in
D
and
g
c
with
c
= 0,
. . .
,
J
see [38].
S. Santaniello et al./Biomedical Signal Processing and Control 3 (2008) 203–211
205
the analyzing scales), each of them corresponding to a speciﬁcspike.To estimate the spike arrival times, instead,
D
* is partitionedinto its contiguous regions
C
r
D
¼
[
N
c
r
¼
1
C
r
;
(6)where
N
c
is the number of contiguous regions in
D
*. The arrivaltime
T
r
of the
r
-th spike is calculated as the mean value of itsestimation at each scale where the spike is detected
T
r
¼
1
jj
G
r
jj
X
g
c
2
G
r
T
c r
r
¼
1
;
. . .
;
N
c
;
(7)where
G
r
:
¼f
g
c
2
G
:
S
ð
c
;
d
Þ
>
Q
c
;
d
2
C
r
g
;
jjjj
is the cardinality of aset and
T
c r
:
¼
arg max
t
d
;
d
2
C
r
f
S
ð
c
;
d
Þ
:
S
ð
c
;
d
Þ
>
Q
c
g
:
(8)
2.2. Threshold-based spike detection
A threshold-based algorithm is a natural choice for spikedetection when the spikes are well distinct from the backgroundnoise.Nevertheless,inordertoavoidanyartifact,itisimportanttoselect carefully the threshold value. For that reason, we preferredto set the threshold automatically on the basis of a statisticalanalysisofthedata,asdonebyQuirogaetal.[39].Inparticular,wepreliminary band-pass ﬁltered (8thorder Butterworthﬁlter,band:[1, 1000] Hz) the data in order to remove evident artifacts and,then, calculated the threshold Thr as:Thr
¼
4
s
n
(9)where
s
n
¼
median
j
s
j
0
:
6745
;
(10)
s
is the processed vector of data and the constant 0.6745 hasbeen taken from [39], due to the efﬁciency that such valueguaranteed even on more noisy training data sets in [39]. A newspike is detected every time that the given data overcome thethreshold with positive gradient. Such instants are used asestimation for the detected spike arrival times. In the following,the adopted threshold-based spike detection procedure will besynthetically indicated as the ‘‘Quiroga’s algorithm’’.As in Quiroga et al. [39], phenomena of double detection areavoided by deﬁning a customizable (set to 1.5 ms in our analysis)refractory period during which no spike can be detected.
2.3. Statistical characterization of the detected spikes
For each of the two algorithms, theobtained spike arrival timeshave been successively characterized through stochastic analyses.For this aim, it should be pointed out that all the available datawere organized in 10 s-long voltage traces, with a samplingfrequency of 24 kHz, each one corresponding to the electricalactivity recorded at a known position along the penetrationtrajectory from the skull to the STN. Such traces can be classiﬁedaccording to the patient, the brain hemisphere and the distancefrom the targeted STN, as described in details in Section 3. Thestochastic characterization, then, has been obtained as follows:
1.
denoted with
s
1
,
. . .
,
s
M
all available traces and compressed thearrivaltimesofthedetectedspikestotheinterval[0,1],foreach
s
i
,
i
= 1,
. . .
,
M
, let us deﬁne
T
si
¼ f
t
i
1
;
t
i
2
;
. . .
;
t
iL
i
g
;
(11)where
L
i
is the number of spikes detected in
s
i
and
t
ik
,
k
= 1,
. . .
,
L
i
,their compressed (or, as said in the following, ‘‘normalized’’)arrival times. For each
T
si
the empirical cumulative distributionfunction (eCDF
i
) is calculated;
2.
chosen a set
F
of well known continuous parametric CDFs,
2
f
(
p
f
,
t
),with
p
f
2
D
f
theparametersvector,theoptimal
p
f
isdeterminedfor each
T
si
and each
f
by means of the maximum likelihoodestimation (MLE) algorithm ([44]; implementation as inMATLAB
1
[45])
p
MLE
f
;
i
¼
arg max
p
f
2
D
f
X
L
i
k
¼
1
ln
ð
PDF
f
ð
p
f
;
t
ik
ÞÞ
8
f
2
F
and
8
i
2f
1
;
. . .
;
M
g
;
(12)wherePDF
f
ð
p
f
;
t
ik
Þ
is theProbabilityDensityFunction(PDF)of
f
evaluated with parameters
p
f
at samples
t
ik
. Then, the optimalﬁtting CDF
f
i
on each
T
si
is chosen as
f
i
¼
arg min
f
2
F
1
L
i
X
L
i
k
¼
1
ð
f
ð
p
MLE
f
;
i
;
t
ik
Þ
eCDF
i
ð
t
ik
ÞÞ
2
8
i
2f
1
;
. . .
;
M
g
:
(13)The MLE algorithm was chosen because it is a minimumvarianceunbiasedestimatorifthenumberofprocessedsamplesis large, as in our case, while the use of logarithms in thelikelihood function to maximize in (12) is introduced just fornumerical efﬁciency.
3.
The statistical analysis is then extended to groups of traces inorder to extract the mean features of data coming fromhomogeneous areas in the STNs. In particular, for eachmonitored STN and each recording microelectrode, the STN-related traces are considered together and the corresponding
T
si
are grouped in a set, denoted as
A
. Then the parameter vector ischosen as
p
MEAN
f
;
A
¼
arg max
p
f
2
D
f
X
T
si
2
A
L
i
L
tot
X
L
i
k
¼
1
ln
ð
PDF
f
ð
p
f
;
t
ik
ÞÞ
8
f
2
F
and
8
A
;
(14)where
L
tot
¼
P
T
si
2
A
L
i
is the total number of detected spikes.Again, the optimal approximating CDF is calculated as:
f
A
¼
arg min
f
2
F
X
T
si
2
A
L
i
L
tot
X
Lik
¼
1
ð
f
ð
p
MEAN
f
;
A
;
t
ik
Þ
eCDF
ð
t
ik
ÞÞ
2
:
(15)The proposed algorithm for the stochastic characterizationutilizes both CDFs and PDFs, with the latter considered forparameter estimation of well shaped distributions and the formerfor the selection of the best approximant in a least square sense.SuchdistinctionisoperatedsinceCDFsareusuallyeasierandmoreimmediate to empirically evaluate than PDFs and, then, areindicatedwhen,aswedidforeachtrace,theapproximantfunctionhas to be chosen into a set of predeﬁned parametric already ﬁttedalternatives. In particular, the set of eligible CDFs has been chosenlarge enough to cover all possible cumulative function shapes,since a goal of thepresent work is to investigate if the spike arrival
2
F
={beta, Birnbaum-Saunders, exponential, extreme value, gamma, inverseGaussian, log-logistic, logistic, lognormal, Nakagami, normal, Rayleigh, rician, t-location, uniform, Weibull} [45].
S. Santaniello et al./Biomedical Signal Processing and Control 3 (2008) 203–211
206
times tend to distribute according to repetitive and knownstructures. Moreover, spikes arrival times have been normalizedtotheinterval[0,1]inordertomaximallyenlargetheset
F
withoutaltering their actual distribution. Finally, we note that the thirdstep ofthealgorithmisaimedat extractingcommonfeaturesfromtraces related to different sites in the same (subthalamic) area. Forthis reason, considering that the higher the ﬁring rate (i.e. numberof spikes per trace) the nearer the recording site would be to theSTN, the weights
L
i
/L
tot
in (14) and (15) are useful to distinguish
between the contributions from different sites and focus thestatistics on the actual STN recordings.
2.4. Subthalamic nucleus model
Theﬁringpatternofasingleneuronisgenerallyinﬂuencedbyitsinner chemical state and by the sequence of synaptic inputs(excitatory and/or inhibitory) received from other cells [5,40]. Thisaspecthasinspiredourmodel:sincetheoutputmustmatchasignalwhich is the superposition of the activity of different neurons, andconsidering the mechanisms of synchronization existing in actualneural network topologies (see [6,10]), we introduce a state vectordescribing the global electrical behavior of the subthalamic area incontact with the microelectrodes and the net effects of glutama-tergic
3
excitatory and GABAergic
4
inhibitory inputs. The adoptedmodel is a third order dynamical system, according to
˙ x
¼
a
x
v
v˙ v
¼
a
v
v
x˙ z
¼
u
g
z
þ
X
i
2f
R
;
D
;
H
g
v
a
i
b
i
sinh
ðð
u
u
i
Þ
=
b
i
Þ
cosh
2
ðð
u
u
i
Þ
=
b
i
Þ
(16)whose output is given by
y
¼
z
þ
w
ð
t
Þ
(17)where
a
¼
1
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
2
þ
v
2
p
is the rate for
x
and
v
;
u
= arctan(
v
,
x
)(the four quadrant arctangent of the
x
and
v
values);
g
,
a
i
,
b
i
and
u
i
,
i
=
R
,
D
,
H
, are parameters to be set;
w
(
t
)
WN
(0,
s
), white noisewith0meanandstandarddeviation
s
tobechosen,and
v
=
v
(
t
)isastochasticvariabletobecharacterized.Insuchmodelthestates
x
and
v
canberespectivelyinterpretedasanindexofthegloballevelof activation of glutamatergic excitatory and GABAergic inhibitorysynapses on the modeled subthalamic area, while
z
is the noise-free signal generated by the neurons in contact with themicroelectrodes. In agreement with such interpretation,
a
canbe viewed as the rate of activation of the excitatory and inhibitoryinputsduetoincreasingnumberofcorticalandpallidalprojectionsﬁring toward the STN. It is worth to note that such rate is nevernegative because no kind of synaptic plasticity is modeled, whilesettlement in the activation state of both the inhibitory andexcitatory inputs is achieved by means of the mutual terms (
v
v
intheﬁrstequationof (16)and
v
x
inthesecondone).Suchtermscan be viewed as a consequence of the projections (not explicitlymodeled here) of the STN toward the external globus pallidus(GPe). Finally, for the sake of completeness, we introduced
u
(
t
) asan optional inputto model the modulatingeffect of the DBS. In thefollowinganalysis,however,it willbeassumed
u
= 0, since nodatafrom the patients during the stimulation is currently available.The mathematical structure of the model has been chosen toguarantee that the repetitive occurrence of APs could be depictedby the movement of the trajectory in the state-space around anattractinglimitcyclewithunitradiusinthe
ð
x
;
v
Þ
plane,asdonebyMcSharry et al. [46] for electrocardiogram signals. Distinct pointson the generated signal represent events characterizing the APdynamics:
R
,
D
and
H
(Figs. 2 and 3) are the extreme points of therefraction, depolarization and hyperpolarization phases respec-tively [40] and correspond to negative and positive attractors/repellors in the
z
-direction. These events are placed at ﬁxed anglesalong the unit circle given by
u
R
,
u
D
,
u
H
(Table 1). When thetrajectory approaches one of these events, it is pushed upwards ordownwardsaway from thelimit cycle according to the events. Thetime required for a whole cycle revolution depends on the APsfrequency and is related to
v
, which is the angular velocity of thelimit cycle trajectory.In particular, the statistics of
v
(
t
) are set in order to match thenumber of effective cycle revolutions and their temporaloccurrence distribution with the processed experimental data.In fact, denoted with
T
(
t
) the period representing the inter-timesbetween consecutive spike arrival instants, the angular velocity
v
(
t
) is given by
v
ð
t
Þ ¼
2
p
T
ð
t
Þ
:
(18)The proposed state equations (16) have been integratednumerically using a fourth-order Runge–Kutta method with aﬁxed time step
D
t
= 1/
f
s
where
f
s
is the sampling frequency(
f
s
= 24 kHz).DataanalysisofthetypicalsubthalamictracesinaPDpatient has been used to suggest suitable times (and, therefore,angles
u
i
with
i
=
R
,
D
,
H
) and values of
a
i
and
b
i
for
RDH
points.
3. Results
The previously described algorithms of spike detection andstochastic characterization have been applied to a set of intrao-perative, invivo, subcorticalrecordings comingfrom two Parkinso-nianpatients.ThecorrespondingresultsindicatethattheSTNspikearrivaltimeshavestochasticdistributionswithregularityproperties
Fig.3.
Typicalstate-spacetrajectorygeneratedbytheproposedmodel.The
R
,
D
and
H
correspond to the introduced trajectory attractors/repellors. The trajectory hasbeen generated for
a
R
= 20,
a
D
=
120,
a
H
= 80,
g
= 150 and the other parameters asin Table 1.
Table 1
Parameters of the proposed modelIndex (
i
)
R D H
u
i
(radians)
p
/12 0
p
/12
a
i
40
250 150
b
i
0.1 0.06 0.12
3
GlutamateisaneurotransmitterinvolvedinthesynapsesfromthecortextotheSTN.
4
GABA is a neurotransmitter involved in the synapses from the GPe to the STN.
S. Santaniello et al./Biomedical Signal Processing and Control 3 (2008) 203–211
207

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