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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Statistical Modeling and Analysis of Laser-Evoked Potentials of Electrocorticogram Recordings from Awake Humans</title><author><name sortKey="Chen, Zhe" sort="Chen, Zhe" uniqKey="Chen Z" first="Zhe" last="Chen">Zhe Chen</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Ohara, Shinji" sort="Ohara, Shinji" uniqKey="Ohara S" first="Shinji" last="Ohara">Shinji Ohara</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Cao, Jianting" sort="Cao, Jianting" uniqKey="Cao J" first="Jianting" last="Cao">Jianting Cao</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Vialatte, Francois" sort="Vialatte, Francois" uniqKey="Vialatte F" first="François" last="Vialatte">François Vialatte</name><affiliation><nlm:aff id="I1">Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi 351-0198, Saitama, Japan</nlm:aff></affiliation></author><author><name sortKey="Lenz, Fred A" sort="Lenz, Fred A" uniqKey="Lenz F" first="Fred A." last="Lenz">Fred A. Lenz</name><affiliation><nlm:aff id="I3">Department of Neurosurgery, Johns Hopkins Hospital, Baltimore, MD 21287-7274, USA</nlm:aff></affiliation></author><author><name sortKey="Cichocki, Andrzej" sort="Cichocki, Andrzej" uniqKey="Cichocki A" first="Andrzej" last="Cichocki">Andrzej Cichocki</name><affiliation><nlm:aff id="I1">Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi 351-0198, Saitama, Japan</nlm:aff></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">18369410</idno><idno type="pmc">2271124</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2271124</idno><idno type="RBID">PMC:2271124</idno><idno type="doi">10.1155/2007/10479</idno><date when="2007">2007</date><idno type="wicri:Area/Pmc/Corpus">000362</idno><idno type="wicri:explorRef" wicri:stream="Pmc" wicri:step="Corpus" wicri:corpus="PMC">000362</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Statistical Modeling and Analysis of Laser-Evoked Potentials of Electrocorticogram Recordings from Awake Humans</title><author><name sortKey="Chen, Zhe" sort="Chen, Zhe" uniqKey="Chen Z" first="Zhe" last="Chen">Zhe Chen</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Ohara, Shinji" sort="Ohara, Shinji" uniqKey="Ohara S" first="Shinji" last="Ohara">Shinji Ohara</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Cao, Jianting" sort="Cao, Jianting" uniqKey="Cao J" first="Jianting" last="Cao">Jianting Cao</name><affiliation><nlm:aff>NONE</nlm:aff></affiliation><affiliation><nlm:aff>NONE</nlm:aff></affiliation></author><author><name sortKey="Vialatte, Francois" sort="Vialatte, Francois" uniqKey="Vialatte F" first="François" last="Vialatte">François Vialatte</name><affiliation><nlm:aff id="I1">Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi 351-0198, Saitama, Japan</nlm:aff></affiliation></author><author><name sortKey="Lenz, Fred A" sort="Lenz, Fred A" uniqKey="Lenz F" first="Fred A." last="Lenz">Fred A. Lenz</name><affiliation><nlm:aff id="I3">Department of Neurosurgery, Johns Hopkins Hospital, Baltimore, MD 21287-7274, USA</nlm:aff></affiliation></author><author><name sortKey="Cichocki, Andrzej" sort="Cichocki, Andrzej" uniqKey="Cichocki A" first="Andrzej" last="Cichocki">Andrzej Cichocki</name><affiliation><nlm:aff id="I1">Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi 351-0198, Saitama, Japan</nlm:aff></affiliation></author></analytic><series><title level="j">Computational Intelligence and Neuroscience</title><idno type="ISSN">1687-5265</idno><idno type="eISSN">1687-5273</idno><imprint><date when="2007">2007</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>This article is devoted to statistical modeling and analysis of electrocorticogram (ECoG) signals induced by painful cutaneous laser stimuli, which were recorded from implanted electrodes in awake humans. Specifically, with statistical tools of factor analysis and independent component analysis, the pain-induced laser-evoked potentials (LEPs) were extracted and investigated under different controlled conditions. With the help of wavelet analysis, quantitative and qualitative analyses were conducted regarding the LEPs' attributes of power, amplitude, and latency, in both averaging and single-trial experiments. Statistical hypothesis tests were also applied in various experimental setups. Experimental results reported herein also confirm previous findings in the neurophysiology literature. In addition, single-trial analysis has also revealed many new observations that might be interesting to the neuroscientists or clinical neurophysiologists. These promising results show convincing validation that advanced signal processing and statistical analysis may open new avenues for future studies of such ECoG or other relevant biomedical recordings.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct><biblStruct></biblStruct></listBibl></div1></back></TEI><pmc article-type="research-article"><pmc-dir>properties open_access</pmc-dir>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Comput Intell Neurosci</journal-id><journal-id journal-id-type="publisher-id">CIN</journal-id><journal-title>Computational Intelligence and Neuroscience</journal-title><issn pub-type="ppub">1687-5265</issn><issn pub-type="epub">1687-5273</issn><publisher><publisher-name>Hindawi Publishing Corporation</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">18369410</article-id><article-id pub-id-type="pmc">2271124</article-id><article-id pub-id-type="doi">10.1155/2007/10479</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title>Statistical Modeling and Analysis of Laser-Evoked Potentials of Electrocorticogram Recordings from Awake Humans</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Chen</surname><given-names>Zhe</given-names></name><xref ref-type="aff" rid="I1"><sup>1, 2</sup></xref><xref ref-type="aff" rid="I2"></xref><xref ref-type="corresp" rid="cor1">*</xref></contrib><contrib contrib-type="author"><name><surname>Ohara</surname><given-names>Shinji</given-names></name><xref ref-type="aff" rid="I3"><sup>3, 4</sup></xref><xref ref-type="aff" rid="I4"></xref></contrib><contrib contrib-type="author"><name><surname>Cao</surname><given-names>Jianting</given-names></name><xref ref-type="aff" rid="I1"><sup>1, 5</sup></xref><xref ref-type="aff" rid="I5"></xref></contrib><contrib contrib-type="author"><name><surname>Vialatte</surname><given-names>François</given-names></name><xref ref-type="aff" rid="I1"><sup>1</sup></xref></contrib><contrib contrib-type="author"><name><surname>Lenz</surname><given-names>Fred A.</given-names></name><xref ref-type="aff" rid="I3"><sup>3</sup></xref></contrib><contrib contrib-type="author"><name><surname>Cichocki</surname><given-names>Andrzej</given-names></name><xref ref-type="aff" rid="I1"><sup>1</sup></xref></contrib></contrib-group><aff id="I1"><sup>1</sup>Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Wako-shi 351-0198, Saitama, Japan</aff><aff id="I2"><sup>2</sup>Neuroscience Statistics Research Lab, Massachusetts General Hospital, Harvard Medical School, Boston, MA 02114, USA</aff><aff id="I3"><sup>3</sup>Department of Neurosurgery, Johns Hopkins Hospital, Baltimore, MD 21287-7274, USA</aff><aff id="I4"><sup>4</sup>Department of Neurosurgery, Kyoto Kizugawa Hospital 26-1, Nishi-Rokutan, Hirakawa Joyo 610-0101, Kyoto, Japan</aff><aff id="I5"><sup>5</sup>Department of Electronics and Information Engineering, Saitama Institute of Technology, Fukaya-shi 369-0293, Saitama, Japan</aff><author-notes><corresp id="cor1">*Zhe Chen: <email>zhechen@neurostat.mgh.harvard.edu</email></corresp><fn fn-type="other"><p>Recommended by Saied Sanei</p></fn></author-notes><pub-date pub-type="ppub"><year>2007</year></pub-date><pub-date pub-type="epub"><day>6</day><month>8</month><year>2007</year></pub-date><volume>2007</volume><elocation-id>10479</elocation-id><history><date date-type="received"><day>25</day><month>12</month><year>2006</year></date><date date-type="accepted"><day>18</day><month>5</month><year>2007</year></date></history><permissions><copyright-statement>Copyright © 2007 Zhe Chen et al.</copyright-statement><copyright-year>2007</copyright-year><license license-type="open-access"><p>This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</p></license></permissions><abstract><p>This article is devoted to statistical modeling and analysis of electrocorticogram (ECoG) signals induced by painful cutaneous laser stimuli, which were recorded from implanted electrodes in awake humans. Specifically, with statistical tools of factor analysis and independent component analysis, the pain-induced laser-evoked potentials (LEPs) were extracted and investigated under different controlled conditions. With the help of wavelet analysis, quantitative and qualitative analyses were conducted regarding the LEPs' attributes of power, amplitude, and latency, in both averaging and single-trial experiments. Statistical hypothesis tests were also applied in various experimental setups. Experimental results reported herein also confirm previous findings in the neurophysiology literature. In addition, single-trial analysis has also revealed many new observations that might be interesting to the neuroscientists or clinical neurophysiologists. These promising results show convincing validation that advanced signal processing and statistical analysis may open new avenues for future studies of such ECoG or other relevant biomedical recordings.</p></abstract></article-meta></front><body><sec sec-type="section" id="sec1"><title>1. INTRODUCTION</title><p>Pain is an essential function for the organism to enable immediate awareness of actual or threatening injury for further
adopting a self-protective behavior. Roughly speaking, pain is a complex and
subjective experience in the brain; it involves sensory, affective, cognitive,
and motivational components and is associated with autonomous activity,
nocifensive reflexes and reactions. In clinical practice, neurophysiological evaluation of pain in humans
has been an important subject of research in the last decade (Bromm and Lorenz
[<xref ref-type="bibr" rid="B2">1</xref>]).</p><p>In the literature, there are many approaches for
monitoring and measuring the pain-related brain activities, including
electroencephalogram (EEG), magnetoencephalogram (MEG), and fMRI. In
particular, the electrocorticogram (ECoG) records directly the cortical
(electrical) activities from subdural electrode grids that are implanted in the
human subjects for collecting information for surgical treatments of medically
intractable epilepsy (i.e., patients in the hospital upon approval). As an
invasive recording tool, ECoG offers some superior features that are
unavailable for EEG or MEG recordings. Specifically, unlike EEG that measures
the electrical potentials recorded from the scalp, ECoG directly records the
potentials from the cortical surface, thereby having a higher signal-to-noise
ratio (SNR) and higher spatial resolution (because of closer electrode
spacing). Consequently, activities in beta or gamma bands are better recorded
in ECoG due to less spatial summation and phase cancelation (or high-cut filter
effect) than in scalp EEG recordings.</p><p>Since the energy of the infrared laser can be used to
produce a brief thermal stimulus applied to the skin such as to selectively
activate the skin nociceptor, the recordings of brain responses to short laser
pulses (the so-called laser-evoked potentials, or LEPs) have increasingly
become a useful method for evaluating the function of central nociceptive
pathways. The roles of LEPs for detecting abnormalities in patients have been
noted (García-Larrea et al. [<xref ref-type="bibr" rid="B15">2</xref>]). Generally, there are two
or three major peaks in the pain-evoked LEPs, which may be generated in
multiple regions. In the literature, most research efforts focused on two peaks
of the LEPs, the so-called N2 and P2, which correspond to the vertex
negative-positive complex<xref ref-type="fn" rid="FN1"><sup>1</sup></xref> . The timing when the peak of the LEP appears
is referred to the latency of LEPs. Typically, N2 was found around 150–400
milliseconds, and P2 was found around 230–500 milliseconds, depending on the
laser pulse duration and intensity, as well as the stimulus site or area (Bromm
and Lorenz [<xref ref-type="bibr" rid="B2">1</xref>]). The
difference in latency is essentially related to the response differences in peripheral
conduction distance. Specifically, LEP reflects an integrative cortical
response to the painful laser stimuli rather than a simple reaction of the
sensory cortex to it; thus, in the healthy subject the amplitude of cortical
LEPs correlates with the subjective sensation of pain, rather than with the
physical stimulus intensity (García-Larrea et al. [<xref ref-type="bibr" rid="B15">2</xref>]). For
instance, paying attention to the laser stimulus simultaneously increases the subjective
pain sensation and the LEP amplitude, both of which decrease in turn when the
subject is distracted from the stimulus (García-Larrea et al. [<xref ref-type="bibr" rid="B16">3</xref>]). In addition to the amplitude, the latencies of the LEPs are often important for the neurophysiological evaluation of pain (Bromm
and Lorenz [<xref ref-type="bibr" rid="B2">1</xref>]). In a
later section, we will analyze the amplitudes and latencies of LEP components
N2 and P2 in detail. As suggested in the literature, the negative component
(N2) seems to be induced mainly by the activation in the bilateral
operculoinsular cortices and contralateral primary somatosensory cortex (SI)
(e.g., Tarkka and Treede [<xref ref-type="bibr" rid="B42">4</xref>], Iannetti
[<xref ref-type="bibr" rid="B18">5</xref>]), and the
positive component (P2) is mainly generated by the cingulate gyrus (e.g.,
Tarkka and Treede [<xref ref-type="bibr" rid="B42">4</xref>],
Lenz et al. [<xref ref-type="bibr" rid="B26">6</xref>], Iannetti et al. [<xref ref-type="bibr" rid="B18">5</xref>]). However, it should also be noted that both N2 and P2 could be recorded and observed at multiple cortical
regions simultaneously (e.g., Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>–<xref ref-type="bibr" rid="B36">9</xref>]); therefore, although there may be some evidence that one LEP is more related to a particular region than
the other, a full understanding of their underlying mechanisms remains unclear.</p><p>In the previous studies (Ohara et al. [<xref ref-type="bibr" rid="B35">8</xref>]) of the ECoG recordings from the awake humans, it was found that attention to painful cutaneous laser stimuli enhances pain-related
LEPs in cortical regions receiving nociceptive input, typically at multiple
cortical sites (Ohara et al. [<xref ref-type="bibr" rid="B36">9</xref>]). Specifically, it was
observed that at primary somatosensory (SI), parasylvian (PS), and medial
frontal (MF: anterior cingulate and supplementary motor area) cortex areas, the
amplitudes of the negative (N2<sup>∗</sup>) and positive (P2<sup>∗</sup><sup>∗</sup>) LEP components<xref ref-type="fn" rid="FN2"><sup>2</sup></xref> were enhanced by attention to
(counting stimuli), in comparison with distraction from the stimuli (reading
for comprehension). It was suggested therein that attention controls both early
(N2<sup>∗</sup>) and late (P2<sup>∗</sup><sup>∗</sup>) pain-related input to SI (and other) cortical regions, while the late positive deflections (that follow the P2<sup>∗</sup><sup>∗</sup> peak) are specifically related to attention. It was also reported in other independent
EEG studies (e.g., Legrain et al. [<xref ref-type="bibr" rid="B22">10</xref>, <xref ref-type="bibr" rid="B23">11</xref>]) that LEPs can be modulated by selective spatial attention. In [<xref ref-type="bibr" rid="B34">7</xref>], Ohara et al. observed
that attention to painful stimuli leads to enhanced event-related desynchronization (ERD) in cortical regions receiving input from nociceptors, and the alpha ERD is more widespread and more intense during attention to the
laser than distraction from the stimuli. This was also consistent with the
observations from other studies using EEG or MEG recordings (Mouraux et al. [<xref ref-type="bibr" rid="B32">12</xref>], Ploner et al. [<xref ref-type="bibr" rid="B39">13</xref>]).</p><p>In recent years, many statistical tools, such as principal component analysis (PCA), independence component analysis (ICA),
parallel factor analysis (PARAFAC), common spatial subspace decomposition (CSSD), statistical wavelet thresholding (SWT), and Kalman filtering, have been used for analyzing biological or biomedical data, including EEG, MEG, and fMRI
(e.g., Lee et al. [<xref ref-type="bibr" rid="B25">14</xref>], Cao et al. [<xref ref-type="bibr" rid="B6">15</xref>, <xref ref-type="bibr" rid="B5">16</xref>], Makeig et al. [<xref ref-type="bibr" rid="B27">17</xref>], Anemüller et al. [<xref ref-type="bibr" rid="B1">18</xref>], Miwakeichi et al. [<xref ref-type="bibr" rid="B30">19</xref>], Browne
and Cutmore [<xref ref-type="bibr" rid="B4">20</xref>], Wang [<xref ref-type="bibr" rid="B44">21</xref>], Galka [<xref ref-type="bibr" rid="B14">22</xref>], Cichocki [<xref ref-type="bibr" rid="B9">23</xref>, <xref ref-type="bibr" rid="B8">24</xref>]). The common goal of these mathematical tools is to discover the hidden components underlying the data and
extract the markers for characterizing specific events (e.g., event-related
potentials). In addition, combing ICA or other statistical tools with advanced
time-frequency analysis methods has also been advocated in cognitive
neuroscience and neuroimaging (e.g., Makeig et al. [<xref ref-type="bibr" rid="B28">25</xref>],
Mørup et al. [<xref ref-type="bibr" rid="B31">26</xref>]).</p><p>In this paper, we conduct both quantitative and
qualitative analyses of ECoG data induced by pain stimuli controlled by a laser
pulse. The investigation is focused on two selected human subjects under
several different controlled stimuli conditions: attention, distraction, as
well as under different laser intensity levels. Statistical analysis was
conducted for both averaging trials and single trials. The averaging-trial
study attempts to find out the dominant and common components (especially LEPs)
by averaging all trials (of one subject) under the same conditions. In
contrast, the single-trial study aims to search for instantaneous brain waves
and to analyze the corresponding LEP properties (such as the amplitude and
latency). The signal-trial analysis is important because the spontaneous brain
activities that are regarded as “noise” are often diminished by averaging. We
believe that the results obtained from the single trials, if analyzed appropriately,
often offer extra information that is unavailable in the averaging-trial study
(e.g., Makeig et al. [<xref ref-type="bibr" rid="B28">25</xref>]).</p><p>To achieve our goal, we select proper processing
procedures and mathematical tools, including factor analysis (FA) and ICA, to
the experimental recordings. This builds on the assumption that within a short
timescale the ECoG recordings are approximated by an instantaneous linear
generative model that is corrupted by additive noise. The LEPs of interest and
other instantaneous brain activities are assumed to be mutually independent. To
blindly separate the sources of interest (i.e., LEPs), we first resort on a
dimensionality reduction procedure followed by an efficient and robust ICA
estimation method. In addition, with eigenvalue decomposition, an energy ratio
threshold is defined to reject nonsignificant components, which are regarded as
the interfering noise from the raw ECoG recordings. The values of these two
statistical analysis methods have been demonstrated in both averaging and
single trials (e.g., Cao et al. [<xref ref-type="bibr" rid="B6">15</xref>, <xref ref-type="bibr" rid="B5">16</xref>]). In terms of single-trial analysis, wavelet-based time-frequency analysis is also used to assist the
quantitative analysis of <italic>Z</italic>-score
transformed power across different frequency bands. Whilst these statistical
methods are not new the contribution of this paper is to integrate these
methods with careful computational procedures and present a systematic study of
the ECoG recordings for their LEP characterizations, which might offer some
insights for the neurophysiological or clinical practice. To our best
knowledge, we are in the first position or for the first time, to employ the
statistical ICA tools to pain-related ECoG recordings. We describe the
computational modeling and analysis in detail and present some interpretations
and discussions from our experimental results. On the one hand, we strive to
relate the results to the reported neurophysiological observations in the
literature; on the other hand, we also pinpoint several interesting findings
and observations in our single-trial data analysis.</p></sec><sec sec-type="section" id="sec2"><title>2. DATA</title><sec sec-type="subsection" id="subsec2.1"><title>2.1. Recordings</title><p>To obtain the ECoG recordings, special grid electrodes
were implanted on the cortical surface of the subjects (i.e., patients for
surgical treatment of epilepsy). The grid consisted of platinum-iridium
circular electrodes (2.3mm diameter) with a center-to-center distance between
electrodes of 1cm (Ad-Tech, Racine, Wis, USA). The LEPs were recorded with the
implanted grid electrodes over the SI, PS, and MF regions; see <xref ref-type="fig" rid="fig1">Figure 1</xref> for an
illustration. During recordings, the subjects wore goggles and reclined on a
bed, quietly wakeful with eyes open. Painful heat stimulation was delivered to
the contralateral hand dorsum (contralateral to the grid) by a Thulium YAG
laser (Neurotest, Wavelight Inc., Starnberg, Germany). The duration of each
pulse was 1 millisecond and the beam diameter was 6mm. Laser energy level was determined
to produce a painful sensation of 3-4/10 on a decimal scale (with 0 denoting no
pain, and 10 denoting the most intense pain). The ECoG signals were recorded
with sampling frequency 1000Hz. The recordings were carried out at the Johns
Hopkins Hospital between 1999 and 2003 (Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>–<xref ref-type="bibr" rid="B36">9</xref>]). The protocol was reviewed and approved annually by the Institutional Review Board of the Johns Hopkins
Hospital and all subjects signed an informed consent for the studies.</p></sec><sec sec-type="subsection" id="subsec2.2"><title>2.2. Subjects</title><p>For the purpose of presentation clarification and due
to space limit, we have chosen two human subjects in the current study. The
statistics of the recording setup regarding the selected two subjects are
listed in <xref ref-type="table" rid="tab1">Table 1</xref>. Specifically, the first subject was a 21-year old woman with
medically intractable seizures since age 10; her neurological examinations and
brain magnetic resonance images (MRIs) were normal. Subdural electrode grids
were planted over the frontal-central-parasylvian cortex (no. 1–64 channels)
and the medial wall of the left hemisphere (no. 65–80 channels). The second
subject was a 21-year old man with complex partial seizures since age 4, whose
MRI showed a small cavernoma in the right parietal lobe (contralateral to the
side of the implantation). The ECoG signals were recorded from the left
fronto-parietal lobe (64 channels) and medial frontal lobe (16 channels). All
the signals were recorded with reference to one intracranial electrode.</p></sec><sec sec-type="subsection" id="subsec2.3"><title>2.3. Experimental paradigm</title><p>There are two types of experimental protocols designed
for subjects: <italic>attention/distraction</italic> , and <italic>intensity</italic> . In the
attention condition, the subject was asked to count the number of painful
stimuli and to report both that number and the average pain intensity after
each run of laser pulses; in the distraction condition, the subject read a
magazine article and answered questions about it after the run. In these two
conditions, constant level of laser intensity was used for the subject, and 38
laser pulses were delivered with an interstimulus interval that was randomly
varied between 50 and 10 seconds within each run. Additionally, in the intensity
experiment, varying levels of laser stimuli were delivered to the subject, and
the subject was asked to rate the subjective pain sensation according to the
decimal scale.</p></sec><sec sec-type="subsection" id="subsec2.4"><title>2.4. Filtering</title><p>Upon loading the raw ECoG recordings to the computer,
the data were amplified and band-pass filtered at 0.1–300Hz (Astro-Med, Inc.,
West Warwick, RI, USA). Subsequently, we conducted a simple notch filtering
procedure to filter out the AC components of power supply (60Hz).</p></sec></sec><sec sec-type="section" id="sec3"><title>3. MATHEMATICAL MODELING AND ANALYSIS</title><sec sec-type="subsection" id="subsec3.1"><title>3.1. Generative model</title><p>The experimental data are assumed to be generated by a
probabilistic generative model that is described by two equations as
follows:<disp-formula id="eq1"><label>(1)</label><mml:math id="M1"><mml:mrow><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">x</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle mathvariant="bold" stretchy="true"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">B</mml:mi></mml:mrow></mml:mstyle><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">z</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold" stretchy="true"><mml:mrow><mml:mi>ε</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="eq2"><label>(2)</label><mml:math id="M2"><mml:mrow><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">z</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">A</mml:mi></mml:mrow></mml:mstyle><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">s</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>t</italic> denotes the time index. Equation (<xref ref-type="disp-formula" rid="eq1">1</xref>) is essentially a factor analysis (FA) model, where <bold><italic>z</italic></bold><italic><sub>t</sub></italic> ∈ <italic>ℝ</italic><sup><italic>n</italic></sup> is the hidden
variable called “factor,” the <italic>m</italic> × <italic>n</italic> matrix <bold>B</bold> is called the
“loading matrix, ” <bold>x</bold><italic><sub>t</sub></italic> ∈ <italic>ℝ</italic><sup><italic>m</italic></sup> denotes the
observed multi-channel signals measured in the electrodes, <italic><bold>μ</bold></italic> ∈ <italic>ℝ</italic><italic><sup>m</sup></italic> denotes the
constant mean vector that is often assumed to be zero, and <bold>ε</bold><italic><sub>t</sub></italic> ∈ <italic>ℝ</italic><italic><sup>m</sup></italic> denotes the
additive uncorrelated noise that corrupts the measurements. Equation (<xref ref-type="disp-formula" rid="eq2">2</xref>) describes a linear
mixture model that is related to the blind source separation (BSS) problem of
our interest, where <italic><bold>s</bold></italic><italic><sub>t</sub></italic> ∈ <italic>ℝ</italic><italic><sup>N</sup></italic> denotes the
independent source signals originated from the brain, <bold>A</bold> denotes a
linear mixing matrix that roughly models the mixing process and the stationary
propagation or scattering effect within a short timescale (say, 200 to
600 mesc); and the mixed signals consist of the hidden factor <bold><italic>z</italic></bold><italic><sub>t</sub></italic> obtained in (<xref ref-type="disp-formula" rid="eq1">1</xref>). In the
current setting of this paper, we assume <italic>m</italic> > <italic>n</italic> = <italic>N</italic>.</p><p>No doubt that the generative model described by
(<xref ref-type="disp-formula" rid="eq1">1</xref>)
and (<xref ref-type="disp-formula" rid="eq2">2</xref>) is somewhat oversimplified for the ECoG data.
However, we believe that the instantaneous linear mixing model is rather
reasonable at a short timescale and therefore can be used in the first step. In
addition, we assume that matrices <bold><italic>A</italic></bold> and <bold><italic>B</italic></bold> are constant
within the a short duration of measurements. Now, the statistical estimation
problem is to infer the independent sources <bold><italic>s</italic></bold><italic><sub>t</sub></italic> given the
observed <bold><italic>x</italic></bold><italic><sub>t</sub></italic>. We will tackle this problem via these two
statistical tools as described below. Notably, similar methodology has been
applied to MEG or EEG recordings with successes in some other real-life recordings
(e.g., Cao et al. [<xref ref-type="bibr" rid="B6">15</xref>, <xref ref-type="bibr" rid="B5">16</xref>]).</p></sec><sec sec-type="subsection" id="subsec3.2"><title>3.2. Factor analysis</title><p>Without loss of generality, we assume that <italic>μ</italic> = <bold>0</bold>, and the factor variables satisfy 𝔼[<bold><italic>z</italic></bold><italic><sub>t</sub></italic>] = <bold>0</bold> and 𝔼[<bold><italic>z</italic></bold><italic><sub>t</sub></italic><bold><italic>z</italic></bold><italic><sub>t</sub></italic><italic><sup>T</sup></italic>] = <bold>C</bold><sub><bold>z</bold></sub>, where <bold>C</bold><sub><bold>z</bold></sub> is the covariance matrix; and the noise is Gaussian distributed with zero mean and
covariance matrix <bold>Σ</bold>, which we denote by <bold>ε</bold> ∼ <italic>𝒩</italic>(<bold>0</bold>,<bold>Σ</bold>). In light of (<xref ref-type="disp-formula" rid="eq1">1</xref>), we
have
<disp-formula id="eq3"><label>(3)</label><mml:math id="M3"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>𝔼</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mn mathvariant="bold">0</mml:mn></mml:mstyle><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mi>𝔼</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:msubsup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo stretchy="false">]</mml:mo><mml:mo>≡</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">C</mml:mi></mml:mstyle><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">C</mml:mi></mml:mstyle><mml:mi>z</mml:mi></mml:msub><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:mi>T</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mstyle mathvariant="bold" stretchy="true"><mml:mi>Σ</mml:mi></mml:mstyle><mml:mn>.</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula>If <bold>z</bold><italic><sub>t</sub></italic> is Gaussian
distributed, then <bold>x</bold><italic><sub>t</sub></italic> is also
Gaussian distributed. If we further restrict that the factor <bold>z</bold><italic><sub>t</sub></italic> is whitened,
then <bold>C<sub>z</sub></bold> = <bold>I</bold> (where <bold>I</bold> denotes the
identity matrix); this assumption is reasonable since we can always scale the
loading factors <bold>B</bold> to satisfy the
original model equation. Typically, dim(<bold>x</bold>) > dim(<bold>z</bold>), therefore FA is also a dimensionality-reduction
method. A close examination of our experimental multielectrode recordings
indicates that there are strong correlations between adjacent electrodes, which
therefore justifies the necessity of dimensionality reduction.</p><p>From a probabilistic point of view, we can write <italic>p</italic>(<bold>z</bold><italic><sub>t</sub></italic>) = <italic>𝒩</italic>(<bold>0</bold>,<bold>I</bold>), then <italic>p</italic>(<bold>x</bold><italic><sub>t</sub></italic>) = <italic>𝒩</italic>(<bold>0</bold>, <bold>BB</bold><italic><sup>T</sup></italic> + <bold>Σ</bold>). Under the Gaussian assumption of the factor analyzer, the posterior probability <italic>p</italic>(<bold>z</bold><italic><sub>t</sub></italic> | <bold>x</bold><italic><sub>t</sub></italic>) is also Gaussian, with mean and covariance, respectively, defined by
<disp-formula id="eq4"><label>(4)</label><mml:math id="M4"><mml:mrow><mml:mi mathvariant="double-struck">E</mml:mi><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="bold">B</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="bold">Σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="bold">B</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="bold">I</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi mathvariant="bold">B</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="bold">Σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula><disp-formula id="eq5"><label>(5)</label><mml:math id="M5"><mml:mrow><mml:mtext>Cov</mml:mtext><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi mathvariant="bold">B</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi mathvariant="bold">Σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant="bold">B</mml:mi><mml:mo>+</mml:mo><mml:mi mathvariant="bold">I</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:mrow></mml:math></disp-formula></p><p>Now the goal of FA is to estimate the unknown matrices <bold>B</bold> and Σ, given the observed data {<bold>x</bold><italic><sub>t</sub></italic>}. In the literature, two types of estimation
procedures can be employed.</p><statement id="head1"><title>Maximum likelihood estimation</title><p>By deriving the log likelihood function (see the
appendix) with respect to the unknown variables, we can use iterative
optimization procedures, such as the gradient ascent or
expectation-maximization (EM) algorithm, to obtain the optimal solution. Upon
obtaining the maximum likelihood estimates of <bold>B</bold> and Σ, we can further calculate the hidden factor <bold>z</bold><italic><sub>t</sub></italic> by (<xref ref-type="disp-formula" rid="eq4">4</xref>).</p></statement><statement id="head2"><title>Least-squared estimation</title><p>Given observed samples {<bold>x</bold><italic><sub>t</sub></italic>}<sub><italic>t</italic> = 1</sub><sup><italic>T</italic></sup>, we can calculate the sample covariance matrix
(assuming zero mean) and conduct its eigenvalue decomposition (EVD) as
follows:<disp-formula id="eq6"><label>(6)</label><mml:math id="M6"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">C</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>x</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>T</mml:mi></mml:mfrac><mml:mstyle><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover></mml:mstyle><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:msubsup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">U</mml:mi><mml:mi>Λ</mml:mi></mml:mstyle><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">U</mml:mi></mml:mstyle><mml:mi>T</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <bold>U</bold> is the <italic>m</italic> × <italic>m</italic> orthogonal
matrix that consists of eigenvectors as its column vectors, <bold>Λ</bold> is a diagonal
matrix that consists of the diagonal entries as eigenvalues. Note that when the
noise is zero or the noise is negligible and has a diagonal covariance matrix,
then FA reduces to PCA as a special case. Upon PCA, we can empirically estimate
the noise covariance. Let <bold>U</bold><sub><italic>n</italic></sub> denote an <italic>m</italic> × <italic>n</italic> matrix that consists of the first <italic>n</italic> dominant eigenvectors, then we can estimate the noise covariance by
<disp-formula id="eq7"><label>(7)</label><mml:math id="M7"><mml:mrow><mml:mover accent="true"><mml:mstyle mathvariant="bold" stretchy="true"><mml:mi>Σ</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">C</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>x</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">U</mml:mi></mml:mstyle><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mstyle mathvariant="bold" stretchy="true"><mml:mi>Λ</mml:mi></mml:mstyle><mml:mi>n</mml:mi></mml:msub><mml:msubsup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">U</mml:mi></mml:mstyle><mml:mi>n</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>and estimate the loading matrix
by<disp-formula id="eq8"><label>(8)</label><mml:math id="M8"><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">U</mml:mi></mml:mstyle><mml:mi>n</mml:mi></mml:msub><mml:msubsup><mml:mstyle mathvariant="bold" stretchy="true"><mml:mi>Λ</mml:mi></mml:mstyle><mml:mi>n</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup><mml:mn>.</mml:mn></mml:mrow></mml:math></disp-formula>Finally, the factor variable <bold>z</bold><italic><sub>t</sub></italic> is produced by
a linear transformation:
<disp-formula id="eq9"><label>(9)</label><mml:math id="M9"><mml:mrow><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">Q</mml:mi></mml:mstyle><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where
<inline-formula><mml:math id="M10"><mml:mrow><mml:mi mathvariant="bold">Q</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mover accent="true"><mml:mi mathvariant="bold">B</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mover accent="true"><mml:mi>Σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mover accent="true"><mml:mi mathvariant="bold">B</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mover accent="true"><mml:mi mathvariant="bold">B</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mover accent="true"><mml:mi>Σ</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>. Note that in this case, the dimensionality of <bold>z</bold><italic><sub>t</sub></italic> can be
determined by PCA with dimensionality reduction, whereas the remaining
components are considered to be “significant” in terms of variance or energy
contribution.</p></statement></sec><sec sec-type="subsection" id="subsec3.3"><title>3.3. Independent component analysis</title><p>Upon performing the model reduction using FA, we
further aim to apply the blind source separation (BSS) approach, using the tool
of ICA (e.g., Cichocki and Amari [<xref ref-type="bibr" rid="B17">27</xref>]), to recover the hidden sources in (<xref ref-type="disp-formula" rid="eq2">2</xref>). Roughly speaking,
ICA is built upon the assumption that the hidden sources in <bold>s</bold><italic><sub>t</sub></italic> are mutually
independent and subject to an instantaneous linear mixing.</p><p>There are many ICA/BSS algorithms available in the
literature. To our interest, two kinds of batch (i.e., noniterative) ICA/BSS
algorithms are considered.</p><statement id="head3"><title>Time-domain method</title><p>Specifically, we focus on the BSS algorithms based on generalized EVD of the time-delayed cross-correlation
matrices or cumulant statistics, such as the SOBI (second-order blind
identification) and JADE (joint approximate diagonalization of eigen-matrices)
algorithms. These methods are fast and noniterative (thereby independent of the
initial conditions). In our experiments, we have tried and compared the SOBI
and JADE algorithms, and found that their results were qualitatively similar.
However, JADE is more desirable and preferred since it incorporates
higher-order statistics.</p></statement><statement id="head4"><title>Time-frequency method</title><p>Specifically, the source separation criterion of this method is conducted in time-frequency domain based
on joint diagonalization of the spatial time-frequency distribution (TFD). A
representative example is the algorithm described by Févotte and Doncarli
[<xref ref-type="bibr" rid="B13">28</xref>]. This method is
more intuitively appealing (by taking into account of the information in both
time and frequency) and has been demonstrated to be robust to noise.<xref ref-type="fn" rid="FN3"><sup>3</sup></xref></p></statement><p>Notably, although the hidden factor <bold>z</bold> is whitened
(with zero mean and unit variance), it is still likely that the mixing matrix
is ill-conditioned, which thereby makes the estimation of its inverse (or
Moore-Penrose pseudoinverse), the demixing matrix <bold>W</bold> = <bold>A</bold><sup>−1</sup> (or <bold>W</bold> = <bold>A</bold><sup>†</sup> ≡ (<bold>A</bold><sup><italic>T</italic></sup><bold>A</bold>)<sup>−1</sup><bold>A</bold><sup><italic>T</italic></sup>), rather
difficult, especially in single-trial experiments. One way to overcome this
problem is to conduct a two-stage ICA procedure. The essence of the two-stage
ICA is as follows: the role of the first-stage ICA is “rough tuning,” which
produces a guess (or poor estimate) of the ill-conditioned mixing matrix; and
the final “fine tuning” job is accomplished by the second-stage ICA routine.
The trick of such a two-stage ICA often helps to recover the hidden components
in many ill-conditioned scenarios if it is not the case, the
second-stage ICA simply produce improved or similar results as in the
first-stage ICA.</p><p>The significance of the (uncorrelated or independent)
components is determined by their relative energy (or variance).
Physiologically, we believe those sources that have relative great energy are
more meaningful in terms of repeatability. In practice, selecting the number of
principal components is done by EVD followed by a threshold selection. In our
experiments, five to eight principal components were typically selected, which
account for about 97–99% of the total energy. Specifically, let <bold>9</bold> = diag{<italic>λ</italic><sub>1</sub>,<italic>λ</italic><sub>2</sub>,…,<italic>λ</italic><sub>n</sub>} denote the
diagonal matrix that contains the nondecreasing eigenvalues <italic>λ</italic><sub>1</sub> ≥ <italic>λ</italic><sub>2</sub> ≥⋯≥<italic>λ</italic><sub>n</sub>≥ 0 , the number of significant components, <italic>k</italic>, is chosen according to the following criterion:
<disp-formula id="eq10"><label>(10)</label><mml:math id="M11"><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mi>arg</mml:mi><mml:mo></mml:mo><mml:mtext> </mml:mtext><mml:munder><mml:mstyle><mml:mrow><mml:mi>min</mml:mi><mml:mo></mml:mo></mml:mrow></mml:mstyle><mml:mi>i</mml:mi></mml:munder><mml:msub><mml:mi>L</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mtext> </mml:mtext><mml:mtext> </mml:mtext><mml:mtext>s</mml:mtext><mml:mtext>.t</mml:mtext><mml:mtext>.</mml:mtext><mml:msub><mml:mi>L</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mstyle><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>i</mml:mi></mml:msubsup></mml:mstyle><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow><mml:mrow><mml:mstyle><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup></mml:mstyle><mml:msub><mml:mi>λ</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:mfrac><mml:mo>></mml:mo><mml:mi>T</mml:mi><mml:mi>h</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula> </p><p>in which the threshold Th was empirically
set as 0.97; the nonnegative eigenvalue indicates the relative significance of
specific component in terms of its energy contribution.</p></sec><sec sec-type="subsection" id="subsec3.4"><title>3.4. Identification of interested source by deflation</title><p>Let <bold>y</bold><sup>(1)</sup> = <bold>W</bold><sup>(1)</sup><bold>z</bold> and <bold>y</bold> = <bold>W</bold><sup>(2)</sup><bold>y</bold><sup>(1)</sup> denote,
respectively, the first- and second-stage ICA unmixing equations, where <bold>W</bold><sup>(1)</sup> and <bold>W</bold><sup>(2)</sup> denote the
associated unmixing matrices; then the final unmixed signals, <bold>y</bold><italic><sub>t</sub></italic>, can be estimated as<disp-formula id="eq11"><label>(11)</label><mml:math id="M12"><mml:msub><mml:mstyle><mml:mrow><mml:mi mathvariant="bold">y</mml:mi></mml:mrow></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mstyle><mml:mrow><mml:mi mathvariant="bold">W</mml:mi><mml:mi>z</mml:mi></mml:mrow></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mstyle><mml:mrow><mml:mi mathvariant="bold">W</mml:mi></mml:mrow></mml:mstyle><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mstyle><mml:mrow><mml:mi mathvariant="bold">W</mml:mi></mml:mrow></mml:mstyle><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:msub><mml:mstyle><mml:mrow><mml:mi mathvariant="bold">z</mml:mi></mml:mrow></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:math></disp-formula>where <bold>W</bold> = <bold>W</bold><sup>(2)</sup><bold>W</bold><sup>(1)</sup> denote the
global (combined) unmixing matrix.<xref ref-type="fn" rid="FN4"><sup>4</sup></xref> Notably, each column of <bold>W</bold><sup>−1</sup> contains the relative strengths of a source component at the individual scalp electrodes,
which can be used to identify the interested source component.</p><p>Given the estimated <bold>y</bold><italic><sub>t</sub></italic> = [<italic>y</italic><sub>1</sub>(<italic>t</italic>),<italic>y</italic><sub>2</sub>(<italic>t</italic>),…,<italic>y</italic><sub>n</sub>(<italic>t</italic>)]<sup><italic>T</italic></sup>, we can also reconstruct the <italic>partial</italic> hidden
factor by projecting the <italic>i</italic>th component of <bold>y</bold><italic><sub>t</sub></italic>, denoted by <italic>y</italic><italic><sub>i</sub></italic>(<italic>t</italic>), backward onto the subspace<xref ref-type="fn" rid="FN5"><sup>5</sup></xref><disp-formula id="eq12"><label>(12)</label><mml:math id="M13"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">W</mml:mi></mml:mstyle><mml:mo>†</mml:mo></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mn>,0,</mml:mn><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>, 0, </mml:mtext><mml:mo>…</mml:mo><mml:mn>,0</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:msup><mml:mo>≡</mml:mo><mml:msub><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">W</mml:mi></mml:mstyle><mml:mo>†</mml:mo></mml:msup><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>i</mml:mi></mml:msub><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mn>,</mml:mn></mml:mrow></mml:math></disp-formula>where [<bold>W</bold><sup>†</sup>]<italic><sub>i</sub></italic> denotes the <italic>i</italic>th column
vector of the matrix <bold>W</bold><sup>†</sup>. Furthermore, we can reconstruct the specific source
of interest in the observed data space (i.e., the scalp signals contributed
merely to the <italic>i</italic>th
source)
<disp-formula id="eq13"><label>(13)</label><mml:math id="M14"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">Q</mml:mi></mml:mstyle><mml:mo>†</mml:mo></mml:msup><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">Q</mml:mi></mml:mstyle><mml:mo>†</mml:mo></mml:msup><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">W</mml:mi></mml:mstyle><mml:mo>†</mml:mo></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mn>,0,</mml:mn><mml:msub><mml:mi>y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mtext>, 0, </mml:mtext><mml:mo>…</mml:mo><mml:mn>,0</mml:mn><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:msup><mml:mn>.</mml:mn></mml:mrow></mml:math></disp-formula> By projecting <mml:math id="M15"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math> to the original
channels' positions (i.e., the 8 × 8 electrode
layout), we essentially <italic>identify</italic> the source(s) of interest. It should be
noted that the “source identification” here is only limited to the
two-dimensional scalp surface, and does not refer to localization of the
three-dimensional spatial position of the “voxel.”</p><p>In addition, in order to evaluate the <italic>relative</italic> contribution of every electrode to the extracted independent component (especially for the LEP), we need to consider the <italic>joint</italic> effect of <mml:math id="M16"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math> and <bold>W</bold>. For this purpose, we may also calculate the <italic>weighted</italic> estimate of the sensor space <mml:math id="M17"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math> as follows:
<disp-formula id="eq14"><label>(14)</label><mml:math id="M18"><mml:mrow><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mo stretchy="true">˜</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msubsup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">w</mml:mi></mml:mstyle><mml:mi>i</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:mo>⊙</mml:mo><mml:msub><mml:mrow><mml:mover accent="true"><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mn>,</mml:mn><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mn>,</mml:mn><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo stretchy="true">^</mml:mo></mml:mover></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>T</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where ⊙ denotes the Hadamard (elementwise) product, <italic>w</italic><italic><sub>i</sub></italic> = [<italic>w</italic><italic><sub>i</sub></italic><sub>1</sub>,<italic>w</italic><italic><sub>i</sub></italic><sub>2</sub>,…,<italic>w</italic><italic><sub>i</sub></italic><sub>n</sub>] denotes the <italic>i</italic>th row vector
of the matrix <bold>W</bold>, and <inline-formula><mml:math id="M19"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mn>2</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mi>T</mml:mi></mml:msup></mml:mrow></mml:math></inline-formula> is the back-projected sensor space from the <italic>i</italic>th independent
source via (<xref ref-type="disp-formula" rid="eq13">13</xref>). As a distinction, we call the reconstructed <inline-formula><mml:math id="M20"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in the sensory
space as “unweighted map” and the reconstructed <inline-formula><mml:math id="M21"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="bold">x</mml:mi><mml:mo>˜</mml:mo></mml:mover><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> in the sensory space as “weighted map.” Notably, because of the degeneracy of <bold>W</bold>, the “weighted map” is subject to the scaling and algebraic sign uncertainties.</p></sec><sec sec-type="subsection" id="subsec3.5"><title>3.5. Time-frequency analysis</title><p>In addition to analyzing temporal signals, we also
resort on time-frequency analysis tools (such as the short-time Fourier
transform, or Wigner-Ville distribution, and wavelet transform) to extract more
information for quantitative comparisons. Specifically, wavelet analysis is
appealing and considered superior to the short-time Fourier transform for
nonstationary signals, including EEG (e.g., Mallat et al. [<xref ref-type="bibr" rid="B29">29</xref>]; Tallon-Baudry et al. [<xref ref-type="bibr" rid="B41">30</xref>], Düzel et al. [<xref ref-type="bibr" rid="B12">31</xref>],
Mouraux et al. [<xref ref-type="bibr" rid="B32">12</xref>], Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>]). Here, we choose the continuous wavelet transform for our purpose because of its adaptive time-frequency analysis via multiscale decomposition.
However, because of the <italic>uncertainty principle</italic> , in order to obtain a good
frequency resolution, sufficient time samples are required. In the experiments,
we will use the Wigner-Ville distribution for an illustration purpose, while in
the quantitative analysis we will use the continuous wavelet transform.</p><p>For a temporal signal <italic>x</italic>(<italic>t</italic>) (i.e., the raw
recordings from one electrode channel), the power of its continuous-time
wavelet transform is described by<disp-formula id="eq15"><label>(15)</label><mml:math id="M22"><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">|</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∗</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where ∗ denotes
convolution product between the signal and the mother wavelet function, and <italic>ψ</italic>(<italic>t</italic>,<italic>ω</italic><sub>0</sub>) is a
complex-valued Morlet mother function:<disp-formula id="eq16"><label>(16)</label><mml:math id="M23"><mml:mrow><mml:mi>ψ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>π</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup><mml:mi>exp</mml:mi><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mrow><mml:mo>−</mml:mo><mml:msup><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mi>exp</mml:mi><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mn>,</mml:mn></mml:mrow></mml:math></disp-formula>where <inline-formula><mml:math id="M24"><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msqrt></mml:mrow></mml:math></inline-formula>, and <italic>σ</italic> is the
bandwidth parameter. The width of the Morlet wavelet, defined by 2<italic>π</italic><italic>σ</italic><italic>ω</italic><sub>0</sub>, is set to 7 in our study.<xref ref-type="fn" rid="FN6"><sup>6</sup></xref> The central frequency <italic>ω</italic><sub>0</sub> ranges from 1
to 60 Hz in steps of 1 Hz. To analyze the specific temporal window of interest,
we select a 100-millisecond prestimulus period and a 500-millisecond
poststimulus period, with a total window length 600 milliseconds.</p><p>To compare the power change between the prestimulus
and poststimulus periods, we need to introduce some “relative” measures to obtain
a baseline for the poststimulus power. This is important because we are not
interested in the “absolute” power statistic <italic>per se</italic> , but interested in
the stimulus-induced relative power change. In the literature, the measure of
event-related band power change (ERBP) was defined as (e.g., Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>])
<disp-formula id="eq17"><label>(17)</label><mml:math id="M25"><mml:mrow><mml:mi>E</mml:mi><mml:mi>R</mml:mi><mml:mi>B</mml:mi><mml:mi>P</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>10</mml:mn><mml:mi>log</mml:mi><mml:mo></mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mfrac><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>m</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>d</mml:mi><mml:mi>B</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mn>,</mml:mn></mml:mrow></mml:math></disp-formula>where <italic>m</italic>(<italic>ω</italic><sub>0</sub>) denotes the <italic>median</italic> power envelope during the prestimulus period. Alternatively, we can use another
measure, which we refer to as “ <italic>Z</italic>-score
transformed poststimulus power,” by using the <italic>Z</italic>-score
transformation (e.g., Browne and Cutmore [<xref ref-type="bibr" rid="B4">20</xref>]):<disp-formula id="eq18"><label>(18)</label><mml:math id="M26"><mml:mrow><mml:mover accent="true"><mml:mi>X</mml:mi><mml:mo stretchy="true">˜</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>X</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>ω</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>where <italic>μ</italic>(<italic>ω</italic><sub>0</sub>) and <italic>σ</italic>(<italic>ω</italic><sub>0</sub>) are,
respectively, the <italic>mean</italic> and <italic>standard deviation</italic> of the power in a
specific channel band (with center frequency <italic>ω</italic><sub>0</sub>) during the
prestimulus period. The motivation of (<xref ref-type="disp-formula" rid="eq18">18</xref>) is to introduce
baseline power values across different frequency bands for the poststimulus
power statistics, which are used for standardized comparisons. In doing so, the
low-amplitude component at high frequency will be highlighted, which also makes
the time-frequency atom in the <italic>gamma</italic> (>32 Hz) band more visible. See <xref ref-type="fig" rid="fig2">Figure 2</xref> for an
illustrative example. Note that the <italic>Z</italic>-score power
value can be negative; the positive values indicate the <italic>event-related
synchronization</italic> (ERS), and the negative values indicate the <italic>event-related
desynchronization</italic> (ERD), both between the prestimulus and
poststimulus periods. Hence, the <italic>Z</italic>-score
transformation provides a clearer understanding of the time-frequency map (in
terms of relative power change).</p><p>In some cases, the resulted <italic>Z</italic>-score
transformed poststimulus power will be converted to a two-dimensional
time-frequency distribution map, denoted by <italic>E</italic>(<italic>t</italic>, <italic>ω</italic>), and further normalized to unity such that <italic>∬</italic><italic>E</italic>(<italic>t</italic>,<italic>ω</italic>)<italic>d</italic> <italic>ω</italic> <italic>d</italic> <italic>t</italic> = 1, which we refer to as the normalized power. In doing
so, each time-frequency atom can be interpreted by a nonnegative probability in
the time-frequency plane.</p></sec></sec><sec sec-type="section" id="sec4"><title>4. COMPARATIVE EXPERIMENTS FOR AVERAGING TRIALS</title><p>We first apply the above described computational
procedure and statistical tools for averaging trials, the signal-trial
experiments will be treated later in more detail. The experimental results
reported in this section will be illustrated for subject 1; two kinds of
conditions, counting and reading, are considered.</p><sec sec-type="subsection" id="subsec4.1"><title>4.1. Extraction of laser-evoked potentials</title><p>First, we aim at extracting LEPs for the
averaging-trial experiments. Specifically, according to the laser onset tag,
the ECoG recordings (of all channels) were averaged upon the total number of
trials at each run. By doing so, the effect of the visual or muscle artifacts
may be greatly reduced. However, it is difficult to identify the LEPs from the
averaging ECoG waveforms of all channels (see <xref ref-type="fig" rid="fig3">Figure 3</xref>). Not only the peaks of
the LEPs are less evident, the averaging waveforms still suffer from noise and
artifact corruption.</p><p>To overcome these issues, we then apply the
statistical tools (FA and ICA) to further process these trial-averaging
signals. In the averaging-trial experiments for subject 1, we selected five
independent components for the purpose of extracting LEPs. These five
independent components are considered to be “significant” because they
contribute mostly to the averaged ECoG data in terms of variance or
energy.<xref ref-type="fn" rid="FN7"><sup>7</sup></xref> Due to the
averaging/smoothing effect, one-stage ICA procedure (with the JADE algorithm)
was found typically sufficient in the experiments<xref ref-type="fn" rid="FN8"><sup>8</sup></xref>The experimental results for the subject 1, in the time domain as well
as in the time-frequency domain, are illustrated in Figures <xref ref-type="fig" rid="fig4">4</xref> and <xref ref-type="fig" rid="fig5">5</xref>. As observed
in the figures, we can extract typical peaks around 150 milliseconds and 200
milliseconds, which might correspond to the hypothetic N2 and P2 peaks (or N2<sup>∗</sup> and P2<sup>∗</sup><sup>∗</sup>) of LEPs,
which we also refer to as N150 and P200, (a), 5(a), 13, 15(a), and 21. Please
check. respectively; the other components can be viewed as other significant
independent spontaneous brain activities. These findings were confirmed in both
attention (counting) and distraction (reading) conditions.</p><p>Next, we conduct the task of LEP source
identification. This is done by back-projecting the <italic>i</italic>th independent
component (i.e., the estimated LEP) back to the observed sensor space.
Specifically, the power contour maps of N2 (N150) and P2 (P200) under the
attention and distraction conditions are illustrated in Figures <xref ref-type="fig" rid="fig6">6</xref> and <xref ref-type="fig" rid="fig8">8</xref>, respectively. The results are qualitatively close (but not identical) to the
previous study (Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>, <xref ref-type="bibr" rid="B35">8</xref>]), in which the LEP peak was found over the interhemispheric (medial) surface. We also plot the <italic>combined</italic> contributions of the power
contour map for N150 and P200, averaged from 120 milliseconds to 240 milliseconds, as shown in <xref ref-type="fig" rid="fig7">Figure 7</xref>. As seen in the figure, in the LEP-N2 (i.e., N150), the greatest brain activities happen around the vertex (Cz)—the upper right corner of the 8 × 8 electrode layout (see Figures <xref ref-type="fig" rid="fig1">1</xref>, <xref ref-type="fig" rid="fig6">6</xref>, and <xref ref-type="fig" rid="fig7">7</xref>), these observations were consistent with our
early result (Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>, <xref ref-type="bibr" rid="B35">8</xref>]), as well as other
independent findings using EEG and fMRI with a similar setup (e.g., see <xref ref-type="fig" rid="fig1">Figure
1</xref> of Iannetti et al. [<xref ref-type="bibr" rid="B18">5</xref>]). Similarly, we also
obtained the LEPs' mappings for subject 2 (see <xref ref-type="fig" rid="fig9">Figure 9</xref>).</p></sec><sec sec-type="subsection" id="subsec4.2"><title>4.2. Relative power</title><p>We compare the time-frequency distribution (TFD) power
between the prestimulus and poststimulus periods. The averaged total power (per
channel) and the averaged power (per channel) of specific frequency bands,
including <italic>theta</italic> (4–7.5 Hz), <italic>alpha</italic> (8–12Hz), <italic>beta</italic> (12.5–32Hz), and <italic>gamma</italic> (32–60Hz), are all calculated. In <xref ref-type="table" rid="tab2">Table 2</xref>, we
summarize the statistics of two subjects under the attention (counting) and
distraction (reading) conditions. The corresponding scatter plots of
prestimulus and poststimulus power (of selected frequency bands) of all
channels are shown in Figures <xref ref-type="fig" rid="fig10">10</xref> and <xref ref-type="fig" rid="fig11">11</xref>.</p><p>From <xref ref-type="table" rid="tab2">Table 2</xref>, several observations are
noteworthy.
<list list-type="roman-lower"><list-item><p>The power in
the poststimulus period is generally greater than that in the prestimulus
period, which is obviously evidenced in terms of total power, <italic>θ</italic> and α power.</p></list-item><list-item><p>The <italic>θ</italic> power increase
(or ERS) is relatively more pronounced in the attention condition than in the
distraction condition.</p></list-item><list-item><p>The <italic>β</italic> power remains
roughly the same level after the laser stimulus, regardless of the undertaken
tasks.</p></list-item><list-item><p>The <italic>γ</italic> power is typically
small in all conditions, with slightly greater value in the attention condition
than in the distraction condition.</p></list-item></list>
</p><p>It is noteworthy that the above observations are consistent with the findings
reported in neuroscience and neurophysiology (to name a few, Bromm and Lorenz
[<xref ref-type="bibr" rid="B2">1</xref>], García-Larrea et al. [<xref ref-type="bibr" rid="B15">2</xref>], Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>]). Although
the statistics summarized in <xref ref-type="table" rid="tab2">Table 2</xref> are calculated based on the averaging
trials, statistical test (see the next subsection) on single trials also
reveals statistical significance.</p></sec><sec sec-type="subsection" id="subsec4.3"><title>4.3. Statistical hypothesis testing</title><p>In order to evaluate the results of the averaging
trials, we conduct some statistical hypothesis tests in order to confirm the
“statistical meaning” of the extracted LEPs. This procedure is necessary
because the result of the extracted LEPs in the averaging trials does not tell
anything <italic>in statistical sense</italic> about each single trial; namely, we need
to be sure if the results we are tempted to interpret are due to random effects
from averaging, or due to the consistent causality in all or most of individual
single trials.</p><p>Two popular hypothesis testing methods we consider
here are the ANOVA (analysis of variance, or <italic>F</italic>-test) and
Mann-Whitney test (or <italic>U</italic>-test). In our
experiments, we first use the Mann-Whiteny test to calculate the so-called <italic>P</italic>-value. Second,
we also apply a logarithm transformation of the raw samples in attempt to
obtain the Gaussianity (i.e., the raw samples are lognormal distributed, as
confirmed by the Shapiro-Wilk test), and then apply the ANOVA to calculate the <italic>P</italic>-values.<xref ref-type="fn" rid="FN9"><sup>9</sup></xref></p><p>To conduct the statistical tests, we apply the
estimated unmixing matrix <bold>W</bold> from the
averaged trial to each single trial; then we obtain the <italic>surrogate</italic> “single-trial LEP”<xref ref-type="fn" rid="FN10"><sup>10</sup></xref> for individual single
trials, for either LEP-N150 or LEP-P200. For a specific LEP component, we
expect that there is a consistent and significant difference between the
prestimulus and poststimulus periods in terms of their absolute values. In our
case, the statistical test was conducted in the time-frequency domain. For
instance, for the LEP-N150 (or LEP-P200), according to its time-frequency map,
we empirically choose a window around the maximum power value (i.e., the
magnitude) in the time-frequency map,<xref ref-type="fn" rid="FN11"><sup>11</sup></xref> and
further conducted the Mann-Whitney test for each extracted LEP component in all
single trials, in which the comparison was done in the time-frequency domain.
Specifically, we compared the <italic>average mean</italic> of the power value inside the
time-frequency window centered around the maximum point (which in the time
domain corresponds to the extracted LEP peak) with that of the prestimulus
period (with the same time-frequency window size), both averaged across all
frequency bins. Consequently, we may expect that the signal amplitude in the
region of interest is significantly greater than that of the baseline. The
statistical hypothesis testing results are summarized in <xref ref-type="table" rid="tab3">Table 3</xref>, and the
corresponding boxplots are shown in <xref ref-type="fig" rid="fig12">Figure 12</xref>. As seen in the table, the <italic>P</italic>-values of <italic>U</italic>-test are all
smaller than .05, and three of them are much smaller than .01, consequentially,
they are statistically significant. For the sake of completeness and sanity
check, we also calculated the <italic>P</italic>-values that
are not associated with the LEPs, (i.e., the other independent components
extracted from the averaged-trials), we have consistently observed that their <italic>P</italic>-values are
greater than .2 (around .2<italic>∼</italic>.6); hence, we
can conclude that these non-LEP components obtained in the averaged trials are
ascribed by the random effect that is not consistent in each single trial.</p></sec></sec><sec sec-type="section" id="sec5"><title>5. QUALITATIVE AND QUANTITATIVE ANALYSES OF SINGLE-TRIAL RECORDINGS</title><p>The averaging-trial experiments and statistical tests
described above present an informative baseline and guideline for further
single-trial experiments. As we mentioned earlier, it is well known that by
averaging the ECoG recordings, we might lose some valuable information due to
cancelation. For this reason, single-trial experimental findings would be also
interesting. Nevertheless, single-trial analysis is more challenging because of
the random background activities and artifacts; hence, obtaining consistent yet
interpretable results is quite difficult. To succeed, we may require additional
care or more sophisticated processing. <xref ref-type="table" rid="tab4">Table 4</xref> lists the operation comparisons between the averaging and single-trial analyses at each stage of procedure.</p><p>Notably, in contrast to the averaging-trial experiments in which the artifact effects are greatly reduced, strong artifacts may exist in the single-trial experiments. In practice, artifacts (often with
low-frequency components) are sometimes observed by visual inspection. In this
case, we will be cautioned about using these “bad” channels. A simple solution
is to discard them or average with their neighboring channels. Selection of bad
channels is often assisted with the reference of averaging trials. For
instance, channels with extremely high amplitude and low frequency are
generally regarded as eye movement artifacts. Since the FA/ICA statistical
methods described above can somehow reduce these effects, hence only those
channels with obvious artifacts were removed in the experimental procedure.</p><p>In the sequel, we will conduct qualitative and
quantitative comparisons of single-trial recordings for different measurements
listed in <xref ref-type="table" rid="tab1">Table 1</xref>.</p><sec sec-type="subsection" id="subsec5.1"><title>5.1. Setup</title><p>In single-trial experiments, the number of independent
components usually varies from trial to trial (for the purpose of extracting
LEPs), and we typically choose the number between 5 and 8. This is because in
individual single trials, some small-amplitude but potentially important
components at high frequency may play a crucial role, which is also interesting
to observe. For the same purpose, we will use the two-stage ICA procedure (JADE
algorithm followed by TFD joint diagonalization) described earlier in Section
3.</p><p>Upon extracting the LEP of interest, we further
identify the LEP localization in the sensor space and focus on the analysis on <italic>one</italic> specific channel (in contrast to the analysis of <italic>all</italic> channels in the
averaging trials). Specifically, we will examine the single-trial recordings
under attention and distraction conditions, as well as the statistics of the
LEP attributes (latency and amplitude) with varying pain levels (i.e., given
different laser intensities).</p></sec><sec sec-type="subsection" id="subsec5.2"><title>5.2. Single trials versus averaging trials</title><p>In the single-trial experiments, we apply the
above-described procedure with the goal of extracting the LEPs under different
conditions, and the results obtained in the averaging trials are considered to
be the baselines for qualitative comparison.</p><p>Typically, not all single trials have good quality
recordings compared with the averaging trials. Here we show a few successful
examples that are capable of identifying the markers of the LEPs. Notably, in
our experiments, it was observed that the LEP-N2 can be easily identified,
while the LEP-P2 is more difficult to separate. See <xref ref-type="fig" rid="fig13">Figure 13</xref> for two
illustrations under different setting conditions.</p><p>In order to evaluate the variability between different
single trials, we apply the estimated demixing matrix <bold>W</bold> obtained from
averaging trials to all individual single trials, by which we obtain a set of
LEP components for N150 and P200 (one pair for each single trial). Furthermore,
we may use the available tools of the EEGLAB toolbox
(<ext-link ext-link-type="uri" xlink:href="http://www.sccn.ucsd.edu/eeglab">http://www.sccn.ucsd.edu/eeglab</ext-link>; Delorme and Makeig [<xref ref-type="bibr" rid="B10">32</xref>], Delorme et al. [<xref ref-type="bibr" rid="B11">33</xref>], Makeig et al.
[<xref ref-type="bibr" rid="B28">25</xref>]) to visualize the event-related spectral perturbation (ERSP) and the intertrial
coherence (ITL) for the specific LEP components, as well as the cross-coherence
between the independent LEP components. Specifically, the ERSP shows the
spectral power change from prestimulus baseline (in dB) relative to the
stimulus onset; and the ITL measures the consistency or reproducibility of the
phase of stimulus-locked trial activity in the selected independent components.
For instance, see <xref ref-type="fig" rid="fig14">Figure 14</xref> for an illustrative example of two LEP components
obtained from the attention task (recalling <xref ref-type="fig" rid="fig4">Figure 4</xref>). As seen in the figure,
the cross-coherence magnitude (from 0 and 1) indicates the degree of
synchronization between two independent LEP components, and the cross-coherence
phase (from −180 to 180 degree)
indicates that the LEP-N150 component is leading ahead of the LEP-P200
component.</p></sec><sec sec-type="subsection" id="subsec5.3"><title>5.3. Attention versus distraction</title><p>For subjects 1 and 2, consistent alpha waves were
found among many (but not all) single trials in the reading task (i.e.,
distraction condition); whereas in the counting task (i.e., attention
condition), the significant alpha component was not observed in most of single
trials. In some reading tasks, no obvious LEP was identified, while the
dominant alpha waves can be observed. See <xref ref-type="fig" rid="fig15">Figure 15</xref> for an illustration. In
such cases, since there are no clear LEP peaks being observed, it remains an
open question that whether this phenomenon is ascribed to “ <italic>habituation to
the pain</italic> ” or “ <italic>loss of attention</italic> .” The reason that alpha rhythms
appear frequently in the reading task might be due to the fact that the subject
was in a relatively relaxed mood (especially compared with the counting task).</p><p>In addition, we also measure the coherence of
signal-trial ECoG data under different conditions. In <xref ref-type="fig" rid="fig16">Figure 16</xref>, the coherency
of alpha (8–12Hz) and beta (12.5–32Hz) bands between pairwise channels during
the poststimulus period is illustrated. In order to visualize the coherency, putting
all connections in one plot will be informative. Specifically, the complete
8-by-8 layout illustrates the first 64 electrodes' positions; at each
electrode's position, we also plot aN 8-by-8 contour plot that represents the
pairwise coherence between a specific electrode and the other electrodes, in
which the specific electrode is marked by a relatively big filled circle. As
seen, typically there is strong coherence in the range of neighboring
electrodes. Comparing <xref ref-type="fig" rid="fig16">Figure 16(a)</xref> with <xref ref-type="fig" rid="fig16">Figure 16</xref>(b), and <xref ref-type="fig" rid="fig16">Figure 16(c) </xref>with
<xref ref-type="fig" rid="fig16">Figure 16(d)</xref>, we can observe that there is stronger coherence in the alpha and
beta bands in the distraction condition than in the attention condition.</p></sec><sec sec-type="subsection" id="subsec5.4"><title>5.4. LEP-component power versus laser intensity</title><p>For the same human subject in a series of single
trials, it is expected that varying the level of stimuli (by changing the laser
intensity), the <italic>amplitude</italic> and <italic>latency</italic> of the LEPs will
consequently vary, so does the <italic>power</italic> of the LEP components in the
time-frequency map. For this purpose of analyzing the power of LEP components
at difference frequency bands, we have conducted quantitative and comparative
analysis for subject 2 under varying controlled conditions.</p><p>The power statistics are summarized in <xref ref-type="table" rid="tab5">Table 5</xref>. It
should be noted that the power values in <xref ref-type="table" rid="tab5">Table 5</xref> refer to the “ <italic>Z</italic>-score
transformed” poststimulus power according to (<xref ref-type="disp-formula" rid="eq18">18</xref>), and all the
values are averaged over the total number of trials in each run. The statistics
are calculated for the first 64 channels including the one that has the highest
power contribution (no. 14 channel) at each trial. The scatter plots of the <italic>Z</italic>-score
transformed power for <italic>theta, alpha, beta</italic>, and <italic>gamma</italic> bands are
shown in <xref ref-type="fig" rid="fig17">Figure 17</xref>. In the off-diagonal subplots, the scatter plots of
cross-band power are shown; whereas the diagonal subplots show the histograms
of the power distribution in the relative frequency bands. In each subplot, the
correlation coefficient between the power across different frequency bands is
also calculated (stored in matrix <bold>C</bold>), as well as
the associated <italic>P</italic>-values for the
student's <italic>t</italic>-test (stored
in matrix <bold>P</bold>). As seen,
with different levels of laser intensities, the <italic>Z</italic>-score
transformed power across different bands is correlated to certain degree: as
the laser intensity increases, the degree of correlation at certain frequency
bands (e.g., between <italic>theta</italic> and <italic>alpha</italic>) tends to decrease. A
cut-off correlation coefficient of 0.7 was considered as a sign of
significance. Each <italic>P</italic>-value
indicates the probability of testing the hypothesis of no correlation, or the probability of getting a correlation as large as the observed
value by random chance, when the true correlation is zero. If <italic>P</italic>(<italic>i</italic>, <italic>j</italic>) is small (say,
less than 0.05), then the correlation <bold>C</bold>(<italic>i</italic>, <italic>j</italic>) is
statistically significant.</p><p>From our data analysis, several observations are
noteworthy.
<list list-type="roman-lower"><list-item><p>Compared to the
prestimulus period, the power across different frequency bands in the
poststimulus period mostly (or in majority) increases, as evidenced by the <italic>positive</italic> mean values of the <italic>Z</italic>-score
transformed (relative) power, although their standard deviations are relative
large.</p></list-item><list-item><p>In one specific
run, the general trend is that the <italic>Z</italic>-score
transformed <italic>θ</italic> power increases
as the laser intensity increases; it seems that no general rule can be found
for α, <italic>β</italic>, and <italic>γ</italic> power among our
experiments.</p></list-item><list-item><p>In different
runs (i.e., 1a, 1b, 2a, 2b), the mean power statistics with the same laser intensity
often vary. This is not unreasonable because in each run the conditions of the
subject may be different; in addition, the (random) order for presenting the
laser stimuli is also different in each run (see <xref ref-type="fig" rid="fig18">Figure 18</xref>), their overall
effects (say, e.g., between 480 → 640 → 800 and 640 → 800 → 480) would be
certainly distinct. Such a “hysteresis” phenomenon is well known in
psychology and psychophysics. In an effort to investigate this phenomenon, we
take the 800mJ intensity level as an example. According to <xref ref-type="fig" rid="fig18">Figure 18</xref>, the total
numbers of 480mJ, 640mJ, and 800mJ preceding 800mJ are 10, 16, and 14,
respectively. In order to compare their effects on the <italic>Z</italic>-score power,
we calculate the means and standard deviations of different frequency bands
under these three different conditions (namely, 480 → 800 , 640 → 800 , 800 → 800 ), and the
results are shown in <xref ref-type="fig" rid="fig19">Figure 19</xref>. It is interesting to observe from the figure
that their <italic>Z</italic>-score power
statistics are quite different especially at the low-frequency (theta and
alpha) bands. Generally, the <italic>Z</italic>-score power
are highest for 480 → 800 , followed by 640 → 800 , and then lowest for 800 → 800 this is
not surprising considering the sensation habituation effect. Statistical tests
show at the theta and alpha bands, the pairwise comparison of <italic>Z</italic>-scored power
among three conditions is statistically significant (ANOVA, <italic>P</italic> < .01).</p></list-item></list></p></sec><sec sec-type="subsection" id="subsec5.5"><title>5.5. LEP amplitude and latency versus laser intensity</title><p>Consistent with the previous studies (Ohara et al. [<xref ref-type="bibr" rid="B34">7</xref>, <xref ref-type="bibr" rid="B35">8</xref>]), the peak amplitudes were measured from the baseline
value, which was defined as the averaged value during the prestimulus period.
Latencies were measured as the time of the peak amplitude (except for the artifact)
for each component; and peak was regarded as significant when the peak
amplitude was above the mean ± SD prestimulus
level. However, in the previous studies, peak amplitudes and latencies were
both measured from reproducible, averaged waveforms; here, we attempt to
measure the latencies from single trials, while the amplitude will still be
measured from averaging (over the trials at each run) because of its strong
randomness; and the standard deviation of the amplitude estimate is calculated
based on 4 independent runs among the recordings. In the meantime, we will
focus the measurements on the first 64 electrodes (channels) for the primary
somatosensory (SI) region, while the analyses for the parasylvian and medial
frontal (MF) regions are ignored here. As observed in our experiments (<xref ref-type="table" rid="tab6">Table 6</xref>), the averaged amplitudes of the LEPs (for both N2 and P2) increase as
increasing laser intensity, except for one case of P2 under the 800mJ
condition; however, the mean statistic is also accompanied with a relatively
large standard deviation, which reflects the random variations of measurement
and/or subject conditions.</p><p>In our single-trial experiments, it was found that the
latencies of the LEPs vary from trials to trials, evidenced by a large standard
deviations (see <xref ref-type="fig" rid="fig20">Figure 20</xref>). In addition, by varying the laser intensity, the
LEP-N2 and LEP-P2 also exhibit different attributes in terms of latency and
amplitude. The corresponding statistics are summarized in <xref ref-type="table" rid="tab6">Table 6</xref> and <xref ref-type="fig" rid="fig20">Figure
20</xref>. Specifically, several observations are noteworthy.</p><p><list list-type="roman-lower"><list-item><p>As seen in
<xref ref-type="table" rid="tab6">Table 6</xref>, the stronger is the laser intensity, the sooner the LEP appears;
namely, the value of the LEP latency is smaller. See <xref ref-type="fig" rid="fig21">Figure 21</xref> for two
illustrative results.</p></list-item><list-item><p>When the laser
intensity is small (e.g., 480mJ), it is quite difficult to extract the LEP
(either one or two) with the available ICA technique. This is partly because
the LEP is so weak that it is overwhelmed in the background “noise” (brain
activities). Indeed, it is even difficult to identify the peaks via visual
inspection from the averaging recordings. </p></list-item></list></p><p>Generally, the amplitude of the LEP is a reflection of the sensation of the pain. Although it
seems difficult to discover quantitative relationship between the intensity of
the laser beam and the amplitude/latency of the LEPs, it is qualitatively clear
that there exists correlation between them, especially when the intensity
difference is large. This phenomenon might serve as a useful evaluation tool in
the clinical practice.</p><p>To evaluate the statistical significance of the LEP
peak amplitude and latency, we also conduct statistical tests based on their
single-trial measurements. We first conduct a robust linear regression fit
(using the MATLAB function “robustfit”) between the laser intensity value
(regression variable) and the measure of interest (amplitude or latency of the
LEP), and then obtain Pearson's correlation statistic <italic>r</italic>. Next, we calculate the <italic>t</italic> statistic as follows:</p><p><disp-formula><mml:math id="M27"><mml:mrow><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mi>r</mml:mi><mml:msqrt><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msqrt></mml:mrow><mml:mrow><mml:msqrt><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mrow><mml:mi>r</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula>where ℓ denotes the number of regression sample pairs. From
the <italic>t</italic>-statistic, we can further evaluate the statistical significance (i.e., <italic>P</italic>-value) from the <italic>t</italic>-table. In our case, we found the linear fit for LEP's latency is significant (<italic>r</italic> = 0.87, <italic>P</italic> < .05); however, the linear fit for LEP's
amplitude is not significant. </p></sec><sec sec-type="subsection" id="subsec5.6"><title>5.6. Subjective sensation versus laser intensity</title><p>Finally, we follow the procedure of Ohara et al. [<xref ref-type="bibr" rid="B37">34</xref>] to analyze
the relationship between the subjective sensation (in
terms of pain rating) and the laser intensity. Specifically, the
subject was asked to rate the pain level in decimal scale (0
no pain, 10 the most intense pain sensation). The mean and
standard deviation statistics are calculated based on all single
trials given three different laser intensities, as shown in
<xref ref-type="fig" rid="fig22">Figure 22</xref>. Generally, it is seen that the average subjective
pain sensation increases as the level of the laser intensity
increases. Statistical tests show significant sensation differences
between different levels of laser intensities (ANOVA,
P < .001 between 480 mJ and 800 mJ; P < .05 between
640 mJ and 800 mJ). Moreover, we also evaluate the correlation
between subjective sensation and LEP amplitude; however,
no significant correlation was observed between the
pain sensation rating and LEP amplitude for subject 2. We
suspect this is partially due to the large variations among
the subjective pain rating, even with the same laser intensity
(specifically, the mean ± SD of the pain rating value for
laser intensities 480 mJ, 640 mJ, and 800 mJ are 0.15 ± 0.70,
1.05 ± 1.49, and 2.10 ± 2.47, resp.). Although our data here
seem to suggest that the subjective pain sensation and the
objective LEP attribute observation might not be necessarily
correlated, we should also be cautioned that the pain is a very
complex sensation and is susceptible to many human factors
and experimental conditions. Verification of any claim in this
matter require more data and careful analysis.</p></sec></sec><sec sec-type="section" id="sec6"><title>6. DISCUSSION AND CONCLUSION</title><p>In this paper, we have used the statistical tools of FA/ICA for
extracting and analyzing the LEPs. To our best knowledge,
the statistical analysis and quantitative results reported here
are among the new (if not the first) reports that apply sophisticated
and systematic statistical analyses to the laser-induced
pain data in the literature. In both averaging and single trials,
we have demonstrated that the pain-evoked event potentials
can be extracted and further analyzed with careful design of
statistical procedure, and that the ICA/BSS approaches show
a promising role in analyzing the multichannel ECoG data
recorded from the awake human subjects. Our results here
have also validated our previous findings in the early investigations
and the reported neurophysiological observations in
the literature. This is encouraging in that it justifies the merits
of blind signal processing for neurobiological or physiological
data analysis. The next challenge of this line of research
is to extract consistently less-dominant (in terms of
power) and potentially important pain-related components
that are beyond the LEPs from single trials, which will be the
subject of future study.</p><p>We have focused on one particular type of blind signal
processing tool (namely, ICA) in this paper. However,
we make no claim that the choice is unique or optimal. Indeed,
we have been aware of the strengthes and weaknesses of
the ICA during the experimental investigations (e.g., Makeig
et al. [<xref ref-type="bibr" rid="B28">25</xref>]), although other improved ICA models, such as the
spatially constrained ICA (Ille et al. [<xref ref-type="bibr" rid="B19">35</xref>], Hesse and James
[<xref ref-type="bibr" rid="B17">36</xref>]) or the temporally constrained ICA (James and Gibson
[<xref ref-type="bibr" rid="B20">37</xref>]), can be considered. It is also noteworthy to point out several other powerful blind signal processing tools and statistical
algorithms, whichmight be valuable for the future investigation:
<list list-type="roman-lower"><list-item><p>nonnegative matrix factorization (NMF) (e.g., Lee
and Seung [<xref ref-type="bibr" rid="B24">38</xref>]), which is an approximate matrix
factorization method for nonnegative data (e.g.,
spectra, or time-frequency map). Unlike ICA, the
independence assumption is relaxed or unnecessary
in NMF, on the other hand, extra constraints (such as the smoothness or sparsity) can be imposed for this
statistical model.<xref ref-type="fn" rid="FN12"><sup>12</sup></xref></p></list-item><list-item><p>parallel factor analysis (PARAFAC) (e.g., Bro [<xref ref-type="bibr" rid="B3">39</xref>]),
which is a well-suited method for analyzing highdimensional
tensorial data; PARAFAC can be viewed
as a generalization of higher-order FA or highdimensional
NMF (if additional nonnegativity
constraint is imposed).</p></list-item><list-item><p>common spatial subspace decomposition (CSSD)
(Wang et al. [<xref ref-type="bibr" rid="B44">21</xref>]), which is a spatial filtering method
for extracting signal components specific to one
condition from multichannel electrode recordings
given multiple task conditions. This kind of common
spatial pattern algorithm may be used for evaluating
the ECoG recordings under different task conditions;
however, unlike the ICA method, it is a supervised
algorithm that uses labeled data for classification.</p></list-item></list>
</p><p>In addition to the above-mentioned statistical tools, it would
be also interesting to investigate the instantaneous brain
activities and dynamics (Makeig et al. [<xref ref-type="bibr" rid="B28">25</xref>]), which may provide
useful information of interactions inside the brain for
specific patients with ECoG recordings. Finally, we believe
what we reported here is only the first step towards a complete
“statistical” understanding of the pain-evoked ECoG
data, the substantiation of our observations, claims, and conclusionmade
in this article would require more experimental
verification of ECoG recordings in the future.</p></sec></body><back><app-group><app><title>APPENDIX</title><sec sec-type="section" id="sec7"><title>MAXIMUM LIKELIHOOD ESTIMATION OF FACTOR ANALYSIS</title><p>Let us consider a general factor analysis(FA) model as follows:
<disp-formula id="eqA.1"><label>(A.1)</label><mml:math id="M28"><mml:mrow><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">x</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">B</mml:mi></mml:mrow></mml:mstyle><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi mathvariant="bold">z</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mstyle stretchy="true"><mml:mrow><mml:mi>ε</mml:mi></mml:mrow></mml:mstyle></mml:mrow><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math></disp-formula>
where <bold>x</bold><italic><sub>t</sub></italic> ∈ ℝ<sup><italic>m</italic></sup> denotes the
observed variable, <italic>μ</italic> denotes the
mean vector, <bold>z</bold><italic><sub>t</sub></italic> ∈ ℝ<sup><italic>n</italic></sup> denotes the
hidden variable called “factor,” and <bold>B</bold> is an <italic>m</italic> × <italic>n</italic> “loading matrix.” With the probabilistic assumptions that <bold>z</bold><italic><sub>t</sub></italic> ∼ <italic>𝒩</italic> (<bold>0</bold>,<bold>I</bold>), ε<italic><sub>t</sub></italic> ∼ <italic>𝒩</italic> (<bold>0</bold>, Σ), and 𝔼[<bold>z</bold><italic><sub>t</sub>ε<sub>t</sub></italic>] = <bold>0</bold>, then we may derive that
<disp-formula id="eqA.2"><label>(A.2)</label><mml:math id="M29"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:mi>𝔼</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi>μ</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mi>𝔼</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi>μ</mml:mi></mml:mstyle><mml:mo>+</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:mtext>Var</mml:mtext><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:msup><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:mi>T</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mstyle stretchy="true"><mml:mi>Σ</mml:mi></mml:mstyle><mml:mo>,</mml:mo><mml:mtext>Cov</mml:mtext><mml:mo stretchy="false">[</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">x</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">z</mml:mi></mml:mstyle><mml:mi>t</mml:mi></mml:msub><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mstyle stretchy="true"><mml:mi mathvariant="bold">B</mml:mi></mml:mstyle><mml:mn>.</mml:mn></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula></p><p>Let <italic>θ</italic> = (<italic>μ</italic>, <bold>B</bold>, Σ) denote the unknown parameters, then the log likelihood of the FA model is written as
<disp-formula id="eqA.3"><label>(A.3)</label><mml:math id="M30"><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>ℒ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>θ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mrow><mml:mtext>In</mml:mtext><mml:mtext> </mml:mtext><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>|</mml:mo><mml:mi>θ</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mtext>In</mml:mtext><mml:mrow><mml:mo>|</mml:mo><mml:mi>Σ</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mrow><mml:msubsup><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:mstyle></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant="bold">B</mml:mi><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant="bold">B</mml:mi><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mfrac><mml:mi>T</mml:mi><mml:mn>2</mml:mn></mml:mfrac><mml:mtext>In</mml:mtext><mml:mrow><mml:mo>|</mml:mo><mml:mi>Σ</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mrow><mml:mtext>tr</mml:mtext><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:msubsup><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi><mml:mi>T</mml:mi></mml:msubsup></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"></mml:mtd><mml:mtd columnalign="left"><mml:mrow><mml:mo>−</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>T</mml:mi></mml:munderover><mml:mrow><mml:mtext>tr</mml:mtext></mml:mrow></mml:mstyle><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant="bold">B</mml:mi><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi mathvariant="bold">x</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi mathvariant="bold">B</mml:mi><mml:msub><mml:mi mathvariant="bold">z</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>T</mml:mi></mml:msup><mml:msup><mml:mi>Σ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math></disp-formula>where tr[·] denotes the trace operator, and |Σ| denotes the determinant of <bold>Σ</bold>. Maximizing <italic>ℒ</italic>(<italic>θ</italic>) with respect to the unknown parameters yields the maximum likelihood estimate. An elegant solution can be obtained by using the iterative EM algorithm.</p></sec></app></app-group><ack><title>ACKNOWLEDGMENTS</title><p>The authors thank two anonymous reviewers for their
critical and valuable comments. This project was done in collaboration between
the Laboratory for Advanced Brain Signal Processing at RIKEN Brain Science
Institute and the Department of Neurosurgery and Neurology at Johns Hopkins
University. The experimental data reported in this article were recorded in the
Johns Hopkins Hospital. We thank Liangyu Zhao for assistance during the early
investigation. The ICA software “ICALAB” based on MATLAB platform is
available for download in our website <ext-link ext-link-type="uri" xlink:href="http://www.bsp.brain.riken.jp/ICALAB">http://www.bsp.brain.riken.jp/ICALAB</ext-link>.</p></ack><fn-group><fn id="FN1"><p><sup>1</sup>This is in contrast to the earliest component N1, which is a lateralized,
relatively small negative peak.</p></fn><fn id="FN2"><p><sup>2</sup>In the previous studies, the authors were not sure if the ECoG-LEPs correspond to the scalp-recorded LEPs (N2 and P2), therefore they used the nomenclatures N2* and P2**. Here, for simplicity, we use N2 and P2 for referring to these two LEPs. The LEPs can also be labeled by polarity and latency; when the latency is known, we also use N150 or P200 for the same reference purpose.</p></fn><fn id="FN3"><p><sup>3</sup>The Matlab code is available at <ext-link ext-link-type="uri" xlink:href="http://www-sigproc.eng.cam.ac.uk/~cf269">http://www-sigproc.eng.cam.ac.uk/~cf269</ext-link>.</p></fn><fn id="FN4"><p><sup>4</sup>If only one-stage ICA is used, then <bold>W</bold><sup>(1)</sup> = <bold>I</bold>, <bold>y</bold><sup>(1)</sup> = <bold>z</bold>, and <bold>W</bold> = <bold>W</bold><sup>(2)</sup>.</p></fn><fn id="FN5"><p><sup>5</sup>If the demixing matrix is square, then the matrix pseudoinverse <bold>W</bold><sup>†</sup> will
reduce to the matrix inverse <bold>W</bold><sup>−1</sup>.</p></fn><fn id="FN6"><p><sup>6</sup>Generally, the greater the width parameter is, the better frequency resolution
we can obtain; nevertheless, this is at the cost of sacrificing temporal
resolution. The temporal resolution has a reciprocal relationship with respect
to the frequency resolution.</p></fn><fn id="FN7"><p><sup>7</sup>Specifically, the two LEP components are more significant and relatively
robust in that they remain approximately unchanged when we vary (increase
or decrease) the selected number of components by 1 or 2.</p></fn><fn id="FN8"><p><sup>8</sup>This is in contrast to the two-stage ICA procedure in single-trials; this is
often witnessed by the observation that the outcomes of the second-stage
ICA are not much different from the results of the first-stage ICA</p></fn><fn id="FN9"><p><sup>9</sup>It should be noteworthy that it is also possible to apply the logarithm
transformation to the samples before theMann-Whitney test; in this case,
the <italic>P</italic>-values will remain unchanged, except that the standard deviation
will become smaller after the logarithm transformation.</p></fn><fn id="FN10"><p><sup>10</sup>We call it surrogate single-trial LEP because the LEP is not estimated directly
from individual single trial alone; instead, its recovery arises from
the unmixing matrix that is estimated based on averaging all single trials.</p></fn><fn id="FN11"><p><sup>11</sup>Typically, the window of temporal axis is centered at 150 milliseconds
(or 200 milliseconds) with width 30 milliseconds, and the window of frequency
axis covers from 4.5 to 6.5 Hz (with the resolution of 0.25 Hz for
each frequency bin).</p></fn><fn id="FN12"><p><sup>12</sup>For various implementation, see online resources and MATLAB toolboxes
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earliest component N1,
which is a lateralized, relatively small negative peak.</p></fn><fn id="fn2"><label>2</label><p>In the previous studies, the authors
were not sure if the ECoG-LEPs correspond to the scalp-recorded LEPs (N2<sup>∗</sup> and
P2), therefore they used the nomenclatures N2<sup>∗</sup> and P2<sup>∗</sup><sup>∗</sup>. Here, for simplicity, we use N2 and P2 for referring
to these two LEPs. The LEPs can also be labeled by polarity and latency; when
the latency is known, we also use N150 or P200 for the same reference
purpose.</p></fn><fn id="fn3"><label>3</label><p>The Matlab
code is available at <ext-link ext-link-type="uri" xlink:href="http://www-sigproc.eng.cam.ac.uk/∼">http://www-sigproc.eng.cam.ac.uk/∼</ext-link> f269.</p></fn><fn id="fn4"><label>4</label><p>If only one-stage ICA is used, then <bold>W</bold><sup>(1)</sup> = <bold>I</bold>, <bold>y</bold><sup>(1)</sup> = <bold>z</bold>, and <bold>W</bold> = <bold>W</bold><sup>(2)</sup>.</p></fn><fn id="fn5"><label>5</label><p>If the demixing matrix is
square, then the matrix pseudoinverse <bold>W</bold><sup>†</sup> will reduce to
the matrix inverse <bold>W</bold><sup>−1</sup>.</p></fn><fn id="fn6"><label>6</label><p>.Generally, the greater the
width parameter is, the better frequency resolution we can obtain;
nevertheless, this is at the cost of sacrificing temporal resolution. The
temporal resolution has a reciprocal relationship with respect to the frequency
resolution.</p></fn><fn id="fn7"><label>7</label><p>Specifically, the two LEP components are more significant and relatively
robust in that they remain approximately unchanged when we vary (increase or
decrease) the selected number of components by 1 or 2.</p></fn><fn id="fn8"><label>8</label><p>.This is in contrast to the
two-stage ICA procedure in single-trials; this is often witnessed by the
observation that the outcomes of the second-stage ICA are not much different
from the results of the first-stage ICA.</p></fn><fn id="fn9"><label>9</label><p>It
should be noteworthy that it is also possible to apply the logarithm
transformation to the samples before the Mann-Whitney test; in this case, the <italic>P</italic>-values will
remain unchanged, except that the standard deviation will become smaller after
the logarithm transformation.</p></fn><fn id="fn10"><label>10</label><p>We call it surrogate single-trial LEP because the LEP is
not estimated directly from individual single trial alone; instead, its
recovery arises from the unmixing matrix that is estimated based on averaging
all single trials.</p></fn><fn id="fn11"><label>11</label><p>Typically, the window of temporal axis is
centered at 150 milliseconds (or 200 milliseconds) with width 30 milliseconds,
and the window of frequency axis covers from 4.5 to 6.5Hz (with the resolution
of 0.25Hz for each frequency bin).</p></fn><fn id="fn12"><label>12</label><p>For various implementation, see online resources and MATLAB toolboxes developed in our lab: http://www.bsp.brain.riken.jp/ICALAB/nmflab.html.</p></fn></fn-group></back><floats-wrap><fig id="fig1" position="float"><label>Figure 1</label><caption><p>The
implanted electrodes' layout; the somatosensory cortex that is associated with
the sensation of the pain is located in the parietal lobe of the brain. (a)
subject 1; (b) subject 2 (where CS and SF correspond to no. 8 and no. 64
channels, resp.); (c), (d) implanted grids imposed on the reconstructed 3D
magnetic resonance images of two subjects. Note that the number of implanted
grids shown on the 3D images is more than the number of the available channels
shown in <xref ref-type="table" rid="tab1">Table 1</xref>; because of the limitation in data acquisition, only a subset
of the grids were selected (CS: central sulcus; SF: sylvian fissure; CiS:
cingulate sulcus; MCiS: marginal CiS).</p></caption><graphic xlink:href="CIN2007-10479.001"></graphic></fig><fig id="fig2" position="float"><label>Figure 2</label><caption><p>The original (upper panel, unit <italic>μ</italic><bold>V</bold><sup>2</sup>) versus <italic>Z</italic>-score
transformed (bottom panel, unitless) wavelet scalogram of one selected channel
in a single trial (subject 2, attention task, laser intensity 720mJ). The white
dash lines indicate the laser stimulus onset. As seen, the ERS and ERD are
highlighted more clearly by the <italic>Z</italic>-score
transformation given by (<xref ref-type="disp-formula" rid="eq18">18</xref>).</p></caption><graphic xlink:href="CIN2007-10479.002"></graphic></fig><fig id="fig3" position="float"><label>Figure 3</label><caption><p>The averaging waveforms (arbitrary scaling) from
averaging trials for both subjects in two tasks. As seen, the averaging evoked
potentials are not clearly evident in these plots.</p></caption><graphic xlink:href="CIN2007-10479.003"></graphic></fig><fig id="fig4" position="float"><label>Figure 4</label><caption><p><italic>Left panels:</italic> five estimated significant independent components (ICs) extracted from
averaging-trial experiment of the counting task (attention situation) for
subject 1. <italic>Right panels:</italic> the associated time-frequency distribution
(TFD) maps.</p></caption><graphic xlink:href="CIN2007-10479.004"></graphic></fig><fig id="fig5" position="float"><label>Figure 5</label><caption><p><italic>Left panels:</italic> five estimated significant independent components (ICs) extracted from
averaging-trial experiment of the reading task (distraction situation) for
subject 1. <italic>Right panels:</italic> the associated time-frequency distribution
(TFD) maps.</p></caption><graphic xlink:href="CIN2007-10479.005"></graphic></fig><fig id="fig6" position="float"><label>Figure 6</label><caption><p>Source identification in the averaging trial of
subject 1: the back-projected 8 × 8 (first 64
channels) scaled amplitude contour map of the LEP peak at N150 (the 5th
independent source at 150 milliseconds, left panel) and P200 (the 4th
independent source at 200 milliseconds, right panel) in the counting task
(attention condition).</p></caption><graphic xlink:href="CIN2007-10479.006"></graphic></fig><fig id="fig7" position="float"><label>Figure 7</label><caption><p><italic>Left panel:</italic> the “weighted” map of LEP-N150 (compared to the “unweighted” map the left panel of <xref ref-type="fig" rid="fig5">Figure 5</xref>) from (<xref ref-type="disp-formula" rid="eq14">14</xref>). <italic>Right panel</italic>: the back-projected 8-by-8
(first 64 channels) power (i.e., the absolute value of the amplitude) contour
map of the two LEPs, N150 and P200, averaged between 120 milliseconds to 240
milliseconds (subject 1, attention condition).</p></caption><graphic xlink:href="CIN2007-10479.007"></graphic></fig><fig id="fig8" position="float"><label>Figure 8</label><caption><p>Source identification in the averaging trial of
subject 1: the back-projected 8-by-8 (first 64 channels) scaled amplitude
contour map of the LEP peak at N150 (the 3rd independent source at 150
milliseconds, left panel) and P200 (the 4th independent source at 200
milliseconds, right panel) in the reading task (distraction condition).</p></caption><graphic xlink:href="CIN2007-10479.008"></graphic></fig><fig id="fig9" position="float"><label>Figure 9</label><caption><p>Source identification in the averaging trial of
subject 2: the back-projected 8-by-8 (first 64 channels) scaled amplitude
contour map of the LEP peak at N2 and P2 in the counting (top 2 panels) and
reading (bottom 2 panels) tasks.</p></caption><graphic xlink:href="CIN2007-10479.009"></graphic></fig><fig id="fig10" position="float"><label>Figure 10</label><caption><p>The scatter plots of prestimulus and poststimulus
power comparisons in averaging trials for 89 channels (subject 1, left:
counting task, right: reading task).</p></caption><graphic xlink:href="CIN2007-10479.010"></graphic></fig><fig id="fig11" position="float"><label>Figure 11</label><caption><p>The scatter plots of prestimulus and poststimulus
power comparisons in averaging trials for 80 channels (subject 2, left:
counting task, right: reading task).</p></caption><graphic xlink:href="CIN2007-10479.011"></graphic></fig><fig id="fig12" position="float"><label>Figure 12</label><caption><p>Boxplots of the absolute value of raw samples,
together with their Mann-Whitney test <italic>P</italic>-values on the
counting (left panel) and reading (right panel) tasks (subject 1, laser
intensity 720 mJ).</p></caption><graphic xlink:href="CIN2007-10479.012"></graphic></fig><fig id="fig13" position="float"><label>Figure 13</label><caption><p>The extracted independent components (including LEPs)
in signal-trials experiments for counting (left) and reading (right) tasks
(subject 1, laser intensity 720 mJ).</p></caption><graphic xlink:href="CIN2007-10479.013"></graphic></fig><fig id="fig14" position="float"><label>Figure 14</label><caption><p><italic>Left and middle panels:</italic> event-related (log) power spectral perturbation
(ERSP, in dB, top row) and inter-trial coherence (ITC, from 0 to 1, bottom row)
changes time locked to the LEP components in single trials (subject 1,
attention task). <italic>Right panel:</italic> cross-coherence between LEP N150 and P200,
with magnitude plot (from 0 to 1; top row) and phase plot (from −180 to 180 degree;
bottom row).</p></caption><graphic xlink:href="CIN2007-10479.014"></graphic></fig><fig id="fig15" position="float"><label>Figure 15</label><caption><p><italic>Left panels:</italic> the 5 estimated sources extracted from a single-trial experiment of the reading
task (subject 2). The 5th independent source contains typical alpha waves. <italic>Right
panels:</italic> the corresponding time-frequency representation.</p></caption><graphic xlink:href="CIN2007-10479.015"></graphic></fig><fig id="fig16" position="float"><label>Figure 16</label><caption><p>Pairwise coherence
maps between the first 64 channels (subject 1, laser intensity 720 mJ) averaged
over all single trials within a duration of 800 milliseconds in poststimulus
period. (a) <italic>alpha</italic>-range coherence in the counting task; (b) <italic>alpha</italic>-range
coherence in the reading task. (c) <italic>beta</italic>-range coherence in the counting
task; (d) <italic>beta</italic>-range coherence in the reading task.</p></caption><graphic xlink:href="CIN2007-10479.016"></graphic></fig><fig id="fig17" position="float"><label>Figure 17</label><caption><p>The scatter plots (using the MATLAB function
“plotmatrix”) of the <italic>Z</italic>-score
transformed power for the <italic>theta, alpha, beta</italic> , and <italic>gamma</italic> bands:
(a) 480 mJ, (b) 640 mJ, (c) 800 mJ, each based upon 40, 44, and 40 single trials,
respectively. At each panel, the diagonal plots show the histograms of <italic>Z</italic>-score power of
the associated frequency bands (from left to right, theta, alpha, beta, and
gamma); the off-diagonal plots show the scatter plots of <italic>Z</italic>-score power
across different frequency bands. Matrix <bold>C</bold> contains the
correlation coefficients, and matrix <bold>P</bold> contains the
associated <italic>P</italic>-values from
the student's <italic>t</italic>-test.</p></caption><graphic xlink:href="CIN2007-10479.017"></graphic></fig><fig id="fig18" position="float"><label>Figure 18</label><caption><p>The graphical
illustration of the laser intensity presentation orders at different runs (1a,
1b, 2a, and 2b). Note that the combined 62 trial sequences of “1a + 1b” and
“2a + 2b” are of identical order.</p></caption><graphic xlink:href="CIN2007-10479.018"></graphic></fig><fig id="fig19" position="float"><label>Figure 19</label><caption><p>The <italic>Z</italic>-score power
comparison of three different hysteresis effects. (a) <italic>theta</italic> band, (b) <italic>alpha</italic> band, (c) <italic>beta</italic> band, (d) <italic>gamma</italic> band.</p></caption><graphic xlink:href="CIN2007-10479.019"></graphic></fig><fig id="fig20" position="float"><label>Figure 20</label><caption><p>The error bars (mean ± SD) of the
estimated latencies (from single trials) for LEP-N2 (left panel) and LEP-P2
(right panel) with varying laser intensities. The dotted lines indicate the
estimated latencies from the averaging trials.</p></caption><graphic xlink:href="CIN2007-10479.020"></graphic></fig><fig id="fig21" position="float"><label>Figure 21</label><caption><p>The 6 estimated sources extracted from averaged trials
(subject 2) with different laser intensities (left panel: 640mJ, right panel:
800mJ).</p></caption><graphic xlink:href="CIN2007-10479.021"></graphic></fig><fig id="fig22" position="float"><label>Figure 22</label><caption><p>Pain sensation rating at different levels of laser
intensity (left panel, averaged over all 124 single trials for subject 2). This
is compared with another subject (right panel, averaged over 124 single trials,
data from Ohara et al. [<xref ref-type="bibr" rid="B37">34</xref>]).</p></caption><graphic xlink:href="CIN2007-10479.022"></graphic></fig><table-wrap id="tab1" position="float"><label>Table 1</label><caption><p>Summary of the experimental recordings of two human subjects.</p></caption><table frame="hsides" rules="groups"><thead><tr><th align="center" rowspan="1" colspan="1">Subject</th><th align="center" rowspan="1" colspan="1">Condition</th><th align="center" rowspan="1" colspan="1">Laser intensity</th><th align="center" rowspan="1" colspan="1">No. of electrodes</th><th align="center" rowspan="1" colspan="1">No. of runs</th><th align="center" rowspan="1" colspan="1">No. of trials at each run</th></tr></thead><tbody><tr><td align="center" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">Attention</td><td align="center" rowspan="1" colspan="1">720 mJ</td><td align="center" rowspan="1" colspan="1">89</td><td align="center" rowspan="1" colspan="1">3</td><td align="center" rowspan="1" colspan="1">38, 38, 38</td></tr><tr><td align="center" rowspan="1" colspan="1">1</td><td align="center" rowspan="1" colspan="1">Distraction</td><td align="center" rowspan="1" colspan="1">720 mJ</td><td align="center" rowspan="1" colspan="1">89</td><td align="center" rowspan="1" colspan="1">3</td><td align="center" rowspan="1" colspan="1">38, 38, 38</td></tr><tr><td colspan="6" rowspan="1"><hr></hr></td></tr><tr><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">Attention</td><td align="center" rowspan="1" colspan="1">720 mJ</td><td align="center" rowspan="1" colspan="1">80</td><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">38, 38</td></tr><tr><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">Distraction</td><td align="center" rowspan="1" colspan="1">720 mJ</td><td align="center" rowspan="1" colspan="1">80</td><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">38, 38</td></tr><tr><td colspan="6" rowspan="1"><hr></hr></td></tr><tr><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">Intensity</td><td align="center" rowspan="1" colspan="1">480 mJ</td><td align="center" rowspan="1" colspan="1">80</td><td align="center" rowspan="1" colspan="1">4</td><td align="center" rowspan="1" colspan="1">8, 12, 10, 10</td></tr><tr><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">Intensity</td><td align="center" rowspan="1" colspan="1">640 mJ</td><td align="center" rowspan="1" colspan="1">80</td><td align="center" rowspan="1" colspan="1">4</td><td align="center" rowspan="1" colspan="1">11, 11, 12, 10</td></tr><tr><td align="center" rowspan="1" colspan="1">2</td><td align="center" rowspan="1" colspan="1">Intensity</td><td align="center" rowspan="1" colspan="1">800 mJ</td><td align="center" rowspan="1" colspan="1">80</td><td align="center" rowspan="1" colspan="1">4</td><td align="center" rowspan="1" colspan="1">10, 10, 10, 10</td></tr></tbody></table></table-wrap><table-wrap id="tab2" position="float"><label>Table 2</label><caption><p>The relative power comparisons between prestimulus period (100 milliseconds) and
poststimulus period (500 milliseconds) in the averaging-trials. The statistics
are averaged over the total number of channels (namely, divided by 89 and 80
for subjects 1 and 2, resp.) and the relative time period. The values are
unitless, reflecting the ratio among the normalized energy of the
time-frequency map.</p></caption><table frame="hsides" rules="groups"><thead><tr><th rowspan="2" colspan="1"></th><th align="center" rowspan="1" colspan="1">Subject 1</th><th align="center" rowspan="1" colspan="1">Subject 1</th><th align="center" rowspan="1" colspan="1">Subject 2</th><th align="center" rowspan="1" colspan="1">Subject 2</th></tr><tr><th align="center" rowspan="1" colspan="1">(counting)</th><th align="center" rowspan="1" colspan="1">(reading)</th><th align="center" rowspan="1" colspan="1">(counting)</th><th align="center" rowspan="1" colspan="1">(reading)</th></tr></thead><tbody><tr><td rowspan="1" colspan="1">Ave. prestimulus total power</td><td align="center" rowspan="1" colspan="1">0.0667</td><td align="center" rowspan="1" colspan="1">0.0796</td><td align="center" rowspan="1" colspan="1">0.0748</td><td align="center" rowspan="1" colspan="1">0.0969</td></tr><tr><td rowspan="1" colspan="1">Ave. prestimulus <italic>θ</italic> power</td><td align="center" rowspan="1" colspan="1">0.0160</td><td align="center" rowspan="1" colspan="1">0.0265</td><td align="center" rowspan="1" colspan="1">0.0175</td><td align="center" rowspan="1" colspan="1">0.0333</td></tr><tr><td rowspan="1" colspan="1">Ave. prestimulus α power</td><td align="center" rowspan="1" colspan="1">0.0048</td><td align="center" rowspan="1" colspan="1">0.0105</td><td align="center" rowspan="1" colspan="1">0.0112</td><td align="center" rowspan="1" colspan="1">0.0102</td></tr><tr><td rowspan="1" colspan="1">Ave. prestimulus <italic>β</italic> power</td><td align="center" rowspan="1" colspan="1">0.0047</td><td align="center" rowspan="1" colspan="1">0.0078</td><td align="center" rowspan="1" colspan="1">0.0136</td><td align="center" rowspan="1" colspan="1">0.0087</td></tr><tr><td rowspan="1" colspan="1">Ave. prestimulus <italic>γ</italic> power</td><td align="center" rowspan="1" colspan="1">0.0021</td><td align="center" rowspan="1" colspan="1">0.0019</td><td align="center" rowspan="1" colspan="1">0.0026</td><td align="center" rowspan="1" colspan="1">0.0025</td></tr><tr><td colspan="5" rowspan="1"><hr></hr></td></tr><tr><td rowspan="1" colspan="1">Ave. poststimulus total power</td><td align="center" rowspan="1" colspan="1">0.1867</td><td align="center" rowspan="1" colspan="1">0.1841</td><td align="center" rowspan="1" colspan="1">0.1850</td><td align="center" rowspan="1" colspan="1">0.1806</td></tr><tr><td rowspan="1" colspan="1">Ave. poststimulus <italic>θ</italic> power</td><td align="center" rowspan="1" colspan="1">0.0486</td><td align="center" rowspan="1" colspan="1">0.0617</td><td align="center" rowspan="1" colspan="1">0.0539</td><td align="center" rowspan="1" colspan="1">0.0502</td></tr><tr><td rowspan="1" colspan="1">Ave. poststimulus α power</td><td align="center" rowspan="1" colspan="1">0.0074</td><td align="center" rowspan="1" colspan="1">0.0124</td><td align="center" rowspan="1" colspan="1">0.0154</td><td align="center" rowspan="1" colspan="1">0.0160</td></tr><tr><td rowspan="1" colspan="1">Ave. poststimulus <italic>β</italic> power</td><td align="center" rowspan="1" colspan="1">0.0051</td><td align="center" rowspan="1" colspan="1">0.0073</td><td align="center" rowspan="1" colspan="1">0.0134</td><td align="center" rowspan="1" colspan="1">0.0104</td></tr><tr><td rowspan="1" colspan="1">Ave. poststimulus <italic>γ</italic> power</td><td align="center" rowspan="1" colspan="1">0.0024</td><td align="center" rowspan="1" colspan="1">0.0024</td><td align="center" rowspan="1" colspan="1">0.0033</td><td align="center" rowspan="1" colspan="1">0.0023</td></tr></tbody></table></table-wrap><table-wrap id="tab3" position="float"><label>Table 3</label><caption><p>Statistical
hypothesis testing statistics of various extracted LEPs in averaging trials for
subject 1. The Mann-Whitney <italic>U</italic>-test was
applied to the “absolute value” of the raw samples, and the ANOVA <italic>F</italic>-test was
applied to the logarithm transformation of the absolute value of the raw
samples. The N/A implies that the samples are neither normally nor log-normally
distributed and therefore cannot be used for ANOVA.</p></caption><table frame="hsides" rules="groups"><thead><tr><th rowspan="2" colspan="1"><italic>P</italic>-value</th><th align="center" rowspan="1" colspan="1">Counting</th><th align="center" rowspan="1" colspan="1">Counting</th><th align="center" rowspan="1" colspan="1">Reading</th><th align="center" rowspan="1" colspan="1">Reading</th></tr><tr><th align="center" rowspan="1" colspan="1">(N150)</th><th align="center" rowspan="1" colspan="1">(P200)</th><th align="center" rowspan="1" colspan="1">(N150)</th><th align="center" rowspan="1" colspan="1">(P200)</th></tr></thead><tbody><tr><td rowspan="1" colspan="1"><italic>U</italic>-test</td><td align="center" rowspan="1" colspan="1">.0029</td><td align="center" rowspan="1" colspan="1">.0013</td><td align="center" rowspan="1" colspan="1">7 × 10<sup>−5</sup></td><td align="center" rowspan="1" colspan="1">.0269</td></tr><tr><td rowspan="1" colspan="1"><italic>F</italic>-test</td><td align="center" rowspan="1" colspan="1">.0003</td><td align="center" rowspan="1" colspan="1">N/A</td><td align="center" rowspan="1" colspan="1"> 6 × 10<sup>−5</sup></td><td align="center" rowspan="1" colspan="1">.0183</td></tr></tbody></table></table-wrap><table-wrap id="tab4" position="float"><label>Table 4</label><caption><p>A comparison of main operations between the averaging and single-trial analyses.</p></caption><table frame="hsides" rules="groups"><thead><tr><th rowspan="1" colspan="1">Routine</th><th align="center" rowspan="1" colspan="1">Averaging trials</th><th align="center" rowspan="1" colspan="1">Single trials</th><th align="center" rowspan="1" colspan="1">Purpose</th></tr></thead><tbody><tr><td rowspan="1" colspan="1">Averaging</td><td align="center" rowspan="1" colspan="1">Yes</td><td align="center" rowspan="1" colspan="1">No</td><td rowspan="1" colspan="1">Smoothing</td></tr><tr><td rowspan="1" colspan="1">FA+PCA</td><td align="center" rowspan="1" colspan="1">Yes</td><td align="center" rowspan="1" colspan="1">Yes</td><td rowspan="1" colspan="1">Noise and dimensionality reduction</td></tr><tr><td rowspan="1" colspan="1">first-stage ICA</td><td align="center" rowspan="1" colspan="1">Yes</td><td align="center" rowspan="1" colspan="1">Yes</td><td rowspan="1" colspan="1">Extracting independent sources</td></tr><tr><td rowspan="1" colspan="1">second-stage ICA</td><td align="center" rowspan="1" colspan="1">No</td><td align="center" rowspan="1" colspan="1">Optional</td><td rowspan="1" colspan="1">Fine tuning of the sources</td></tr><tr><td rowspan="1" colspan="1">Source identification</td><td align="center" rowspan="1" colspan="1">Yes</td><td align="center" rowspan="1" colspan="1">Yes</td><td rowspan="1" colspan="1">Locating the LEPs of interest</td></tr><tr><td rowspan="1" colspan="1">WVD</td><td align="center" rowspan="1" colspan="1">Optional</td><td align="center" rowspan="1" colspan="1">Optional</td><td rowspan="1" colspan="1">Visualization</td></tr><tr><td rowspan="1" colspan="1">Wavelet transform</td><td align="center" rowspan="1" colspan="1">Yes</td><td align="center" rowspan="1" colspan="1">Optional</td><td rowspan="1" colspan="1"><italic>Z</italic>-score transform</td></tr><tr><td rowspan="1" colspan="1">Statistical test</td><td align="center" rowspan="1" colspan="1">Optional</td><td align="center" rowspan="1" colspan="1">Optional</td><td rowspan="1" colspan="1">Testing hypothesis</td></tr></tbody></table></table-wrap><table-wrap id="tab5" position="float"><label>Table 5</label><caption><p>Comparative statistics of the relative power of the normalized wavelet scalogram followed
by <italic>Z</italic>-score transformation (for subject 2, no. 14 electrode) in single-trial analysis. The
mean and standard deviation (mean ±SD) statistics are calculated by averaging the number
of trials in each run.</p></caption><table frame="hsides" rules="groups"><thead><tr><th rowspan="1" colspan="1">Run</th><th align="center" rowspan="1" colspan="1">No. of trials</th><th align="center" rowspan="1" colspan="1">Intensity</th><th align="center" rowspan="1" colspan="1"><italic>θ</italic> power</th><th align="center" rowspan="1" colspan="1"><italic>α</italic> power</th><th align="center" rowspan="1" colspan="1"><italic>β</italic> power</th><th align="center" rowspan="1" colspan="1"><italic>γ</italic> power</th></tr></thead><tbody><tr><td rowspan="1" colspan="1">1a</td><td align="center" rowspan="1" colspan="1">8</td><td align="center" rowspan="1" colspan="1">480 mJ</td><td align="center" rowspan="1" colspan="1"> 0.10 ± 0.76 </td><td align="center" rowspan="1" colspan="1"> 0.24 ± 0.76 </td><td align="center" rowspan="1" colspan="1"> 0.33 ± 0.70 </td><td align="center" rowspan="1" colspan="1"> 0.27 ± 0.62 </td></tr><tr><td rowspan="1" colspan="1">1a</td><td align="center" rowspan="1" colspan="1">11</td><td align="center" rowspan="1" colspan="1">640 mJ</td><td align="center" rowspan="1" colspan="1"> 0.22 ± 0.83 </td><td align="center" rowspan="1" colspan="1"> 0.09 ± 0.48 </td><td align="center" rowspan="1" colspan="1"> 0.03 ± 0.30 </td><td align="center" rowspan="1" colspan="1"> 0.10 ± 0.49</td></tr><tr><td rowspan="1" colspan="1">1a</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">800 mJ</td><td align="center" rowspan="1" colspan="1"> 0.57 ± 2.92 </td><td align="center" rowspan="1" colspan="1"> 0.09 ± 0.72 </td><td align="center" rowspan="1" colspan="1"> −0.03 ± 0.39 </td><td align="center" rowspan="1" colspan="1"> 0.14 ± 0.67 </td></tr><tr><td colspan="7" rowspan="1"><hr></hr></td></tr><tr><td rowspan="1" colspan="1">1b</td><td align="center" rowspan="1" colspan="1">12</td><td align="center" rowspan="1" colspan="1">480 mJ</td><td align="center" rowspan="1" colspan="1"> 0.38 ± 1.63 </td><td align="center" rowspan="1" colspan="1"> 0.19 ± 0.73 </td><td align="center" rowspan="1" colspan="1"> 0.25 ± 0.44 </td><td align="center" rowspan="1" colspan="1"> 0.22 ± 0.40 </td></tr><tr><td rowspan="1" colspan="1">1b</td><td align="center" rowspan="1" colspan="1">11</td><td align="center" rowspan="1" colspan="1">640 mJ</td><td align="center" rowspan="1" colspan="1"> 0.58 ± 0.91 </td><td align="center" rowspan="1" colspan="1"> 0.22 ± 0.57 </td><td align="center" rowspan="1" colspan="1"> 0.08 ± 0.58 </td><td align="center" rowspan="1" colspan="1"> 0.05 ± 0.48 </td></tr><tr><td rowspan="1" colspan="1">1b</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">800 mJ</td><td align="center" rowspan="1" colspan="1"> 0.66 ± 1.23 </td><td align="center" rowspan="1" colspan="1"> 0.59 ± 0.79 </td><td align="center" rowspan="1" colspan="1"> 0.47 ± 1.39 </td><td align="center" rowspan="1" colspan="1"> 0.41 ± 0.92 </td></tr><tr><td colspan="7" rowspan="1"><hr></hr></td></tr><tr><td rowspan="1" colspan="1">2a</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">480 mJ</td><td align="center" rowspan="1" colspan="1"> 0.29 ± 0.61 </td><td align="center" rowspan="1" colspan="1"> 0.40 ± 0.57 </td><td align="center" rowspan="1" colspan="1"> 0.53 ± 1.26 </td><td align="center" rowspan="1" colspan="1"> 0.43 ± 0.66 </td></tr><tr><td rowspan="1" colspan="1">2a</td><td align="center" rowspan="1" colspan="1">12</td><td align="center" rowspan="1" colspan="1">640 mJ</td><td align="center" rowspan="1" colspan="1"> 0.76 ± 1.62 </td><td align="center" rowspan="1" colspan="1"> 0.35 ± 0.93 </td><td align="center" rowspan="1" colspan="1"> 0.21 ± 0.66 </td><td align="center" rowspan="1" colspan="1"> 0.11 ± 0.61 </td></tr><tr><td rowspan="1" colspan="1">2a</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">800 mJ</td><td align="center" rowspan="1" colspan="1"> 0.77 ± 2.11 </td><td align="center" rowspan="1" colspan="1"> 0.24 ± 0.69 </td><td align="center" rowspan="1" colspan="1"> 0.12 ± 0.33 </td><td align="center" rowspan="1" colspan="1"> 0.36 ± 0.51 </td></tr><tr><td colspan="7" rowspan="1"><hr></hr></td></tr><tr><td rowspan="1" colspan="1">2b</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">480 mJ</td><td align="center" rowspan="1" colspan="1"> −0.16 ± 0.30 </td><td align="center" rowspan="1" colspan="1"> −0.04 ± 0.31 </td><td align="center" rowspan="1" colspan="1"> 0.17 ± 0.28 </td><td align="center" rowspan="1" colspan="1"> 0.31 ± 0.38 </td></tr><tr><td rowspan="1" colspan="1">2b</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">640 mJ</td><td align="center" rowspan="1" colspan="1"> 0.30 ± 0.93 </td><td align="center" rowspan="1" colspan="1"> 0.34 ± 0.85 </td><td align="center" rowspan="1" colspan="1"> 0.28 ± 0.84 </td><td align="center" rowspan="1" colspan="1"> 0.05 ± 0.44 </td></tr><tr><td rowspan="1" colspan="1">2b</td><td align="center" rowspan="1" colspan="1">10</td><td align="center" rowspan="1" colspan="1">800 mJ</td><td align="center" rowspan="1" colspan="1"> 0.64 ± 1.56 </td><td align="center" rowspan="1" colspan="1"> 0.21 ± 0.59 </td><td align="center" rowspan="1" colspan="1"> 0.10 ± 0.27 </td><td align="center" rowspan="1" colspan="1"> 0.20 ± 0.33 </td></tr></tbody></table></table-wrap><table-wrap id="tab6" position="float"><label>Table 6</label><caption><p>Comparative
results of the estimated amplitudes and latencies of the LEPs (subject 2, under
rating condition) from single and averaging trials. The last row indicate the
selected number of single trials (by excluding some bad trials) used to
evaluate the latencies.</p></caption><table frame="hsides" rules="groups"><thead><tr><th rowspan="1" colspan="1"></th><th colspan="3" align="center" rowspan="1">N2 (SI region)</th><th colspan="3" align="center" rowspan="1">P2 (SI region)</th></tr></thead><tbody><tr><td rowspan="1" colspan="1">Intensity (mJ)</td><td align="center" rowspan="1" colspan="1">480</td><td align="center" rowspan="1" colspan="1">640</td><td align="center" rowspan="1" colspan="1">800</td><td align="center" rowspan="1" colspan="1">480</td><td align="center" rowspan="1" colspan="1">640</td><td align="center" rowspan="1" colspan="1">800</td></tr><tr><td colspan="7" rowspan="1"><hr></hr></td></tr><tr><td rowspan="1" colspan="1">Latency (milliseconds)</td><td align="center" rowspan="1" colspan="1"> 260 </td><td align="center" rowspan="1" colspan="1">178</td><td align="center" rowspan="1" colspan="1"> 122 </td><td align="center" rowspan="1" colspan="1"> 300 </td><td align="center" rowspan="1" colspan="1"> 248 </td><td align="center" rowspan="1" colspan="1"> 171 </td></tr><tr><td rowspan="1" colspan="1">Amplitude (<italic>μ</italic>V)</td><td align="center" rowspan="1" colspan="1"> −121 ± 18 </td><td align="center" rowspan="1" colspan="1"> −125 ± 23 </td><td align="center" rowspan="1" colspan="1"> −150 ± 41 </td><td align="center" rowspan="1" colspan="1"> 112 ± 31 </td><td align="center" rowspan="1" colspan="1"> 126 ± 55 </td><td align="center" rowspan="1" colspan="1"> 98 ± 20 </td></tr><tr><td rowspan="1" colspan="1">No. of trials</td><td align="center" rowspan="1" colspan="1">28</td><td align="center" rowspan="1" colspan="1">36</td><td align="center" rowspan="1" colspan="1">34</td><td align="center" rowspan="1" colspan="1">22</td><td align="center" rowspan="1" colspan="1">26</td><td align="center" rowspan="1" colspan="1">26</td></tr></tbody></table></table-wrap></floats-wrap></pmc></record>Pour manipuler ce document sous Unix (Dilib)
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