persistenthomology-basedtopologicalanalysisonthegestalt

Research ArticlePersistent Homology-Based Topological Analysis on the GestaltPatterns during Human Brain Cognition Process

Zaisheng Liu 1 Fei Ni 1 Rongpeng Li 1 Honggang Zhang 1 Chang Liu 2

Jiefang Zhang 2 and Songyun Xie 3

1College of Information Science and Electronic Engineering Zhejiang University Zheda Road 38 Hangzhou 310027 China2Communication University of Zhejiang Hangzhou 310018 China3Northwestern Polytechnical University Xirsquoan 710129 China

Correspondence should be addressed to Rongpeng Li lirongpengzjueducn

Received 19 May 2021 Revised 26 August 2021 Accepted 17 September 2021 Published 1 November 2021

Academic Editor Siti Anom Ahmad

Copyright copy 2021 Zaisheng Liu et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)e neuropsychological characteristics inside the brain are still not sufficiently understood in previous Gestalt psychologicalanalyses In particular the extraction and analysis of human brain consciousness information itself have not received enoughattention for the time being In this paper we aim to investigate the features of EEG signals from different conscious thoughtsSpecifically we try to extract the physiologically meaningful features of the brain responding to different contours and shapes inimages in Gestalt cognitive tests by combining persistent homology analysis with electroencephalogram (EEG))e experimentalresults show that more brain regions in the frontal lobe are involved when the subject perceives the random and disorderedcombination of images compared to the ordered Gestalt images Meanwhile the persistence entropy of EEG data evoked byrandom sequence diagram (RSD) is significantly different from that evoked by the ordered Gestalt (GST) images in severalfrequency bands which indicate that the human cognition of the shape and contour of images can be separated to some extentthrough topological analysis )is implies the feasibility to digitize the neural signals while preserving the whole and local featuresof the original signals which are further verified by our extensive experiments In general this paper evaluates and quantifiescognitively related neural correlates by persistent homology features of EEG signals which provides an approach to realizing thedigitization of neural signals Preliminary verification of the analyzability of human consciousness signals provides reliableresearch ideas and directions for the realization of feature extraction and analysis of human brain consciousness cognition

1 Introduction

In recent years with the development of neural networksresearchers are committed to explaining the intrinsic natureof human consciousness generation and artificial intelli-gence (AI) One of the research directions is to explore thelaws of human brain cognition and consciousness genera-tion process to promote the development of machinelearning technology In communication technology therealization of brain-to-brain communication (B2BC) underthe support of future 6G technology also urgently needs amethod to realize the digitization of human brain nervesignals to support the development of its research )e mosttypical analysis method of electroencephalogram (EEG)signals is based on the brain signalsrsquo characteristics by

filtering artifacts removing event-related potentials (ERP)analysis and brain domain heat map with respect to theoriginal time-domain signals )e complex and dynamicmultichannel time-domain signal is not an ideal carrier forinformation transmission Currently various digital analysismethods based on the EEG signals are constantly beingproposed and improved [1ndash4] such as single-trial analysisand other diverse methods which take into account thesignificant differences in EEG signals between differentsubjects Furthermore [3] attempts to extract the digitalfeatures that may bemore relevant and simpler regarding thesignal which coincides with the first step of B2BC thedigitization process of neural signals Combining the rele-vance enlightenment of B2BC and AI the analysis of thehuman brainrsquos cognitive process is of forward-looking value

HindawiJournal of Healthcare EngineeringVolume 2021 Article ID 2334332 11 pageshttpsdoiorg10115520212334332

Gestalt psychology theory is the pioneering foundation ofmodern cognitive learning theory which was established in theearly twentieth century by psychologists WestheimerW Kohlerand K Koffka based on similitude study [5] )ey believe thatthinking is a holistic and meaningful perception rather than asimple collection of connected representations and argue thatlearning lies in forming a gestalt which aims to change onegestalt into another )is cognition process is fundamentallydifferent from the current image recognition model of deeplearning empowered by artificial neural networks (ANNs)

At present with the rapid development of deep learningnewer and stronger algorithm models emerge endlessly andtheir computing power and learning ability for specifiedtasks become increasingly powerful On top of this basissome researchers initiate ambitious new goals and turn tofocus on making AI more ldquointelligentrdquo that is achievingbrain-like intelligence )ey expect machine learning toachieve what the human brain can do and solve problems orrecognize things like the human brain It is well known thatconventional neural networks are designed by the inspira-tion of the fundamental principle of signal transmission in asingle nerve cell )erefore the linkage nature of the wholebiological neural network is the direction we need to exploreas the next frontier It can inspire us to build brain-likeintelligence and transfer from the traditional machinelearning process to a more advanced consciousness level Byexploring the act of brain cognition it is potentially possibleto probe into the generation of consciousness [6] not limitedto what kind of consciousness the brain produces

Researchers have made attempts to explore neural net-works and the human brainrsquos biological patterns [7ndash15] Togain insight into the brainrsquos response to external stimuliscientists have developed functional network analysis meth-odology because they assert that brain functionality is de-termined by the internal interaction between differentneurons as well as different brain regions [11] )ey set out toanalyze the neural signals displayed by the brain as a whole)e spatial and connective relationship of neurons within thebrain structure is a complex connection model which hasbeen used to analyze the human brainrsquos activity with topo-logical tools for a long time )e initial focus of this kind ofresearch is on the somatosensory sensations (e g hot and coldsensations and pain sensations) that are easy to recognize inbrain signals [12] Afterward abnormal EEG responses(epilepsy seizures) [13] steady-state visually evoked potential(SSVEP) movement intention detection [14] and emotionclassification [15] were analyzed We compared the resultsobtained by various EEG analysis methods in various fieldsAmong them the detection accuracy of schizophreniareached a balanced accuracy of 8959 the detection ofmoving images reached a recognition rate of 649ndash795and the classification of emotional EEG signals based ongender reached 904 (SVM) and 92 (KNN) [16]

In particular inspired by the recent research on Gestaltrecognition for which Baker et al [17] and Been Kim et al [18]provided totally conflicting conclusions the discussion on thedifferences between artificial neural networks and the humanbrain cognition process has motivated us to follow researchon the cognitive process at the level of consciousness

Nevertheless it is well realized that the human brainrsquos cog-nition on the overall outline of geometric patterns faces localand global problems in the previous Gestalt experiments

)e mathematic tools of algebraic topology are uniquelyequipped to provide quantitative information about both thelocal and global properties of an arbitrary graph [8] Ac-cordingly topological data analysis (TDA) is capable ofproviding a series of new topological and geometric methodsto analyze the brainrsquos neural networks covering EEG signalsamong which persistent homology is one of the key ap-proaches [19ndash24] Persistent homology analysis providesefficient algorithms for calculating the Betti number of eachcomplex graph in the network families under considerationand encodes the evolution of the nested complex homologygroups at different networking scales Consequently it helpsunderstand the EEG data better and keeps analytical stabilityconcerning perturbations or noise in the EEG signals

In our study 20 participants were considered in tradi-tional visual stimulation experimental methods and col-lecting EEG signals at the same time)e neurophysiologicalevaluation of the contour in the Gestalt experiment wasinvestigated by exploiting Euler characteristics and persis-tent homology features of the EEG signal On that basis howthe regions of the brain are involved in contour recognitionwas interpreted by selecting VietorisndashRips filtration

)e main contributions of this work are as follows

(1) )e topology calculation method adopted in thisexperiment provides effective separability of the EEGsignals of contour cognitive behavior and realizes thedigital feature extraction of EEG signals

(2) We provide reference significance and a referencemethod for B2BC and other work that needs torealize the digital feature extraction of EEG signals

(3) We demonstrated the feasibility of using persistenthomology modeling to analyze EEG signals

2 ExperimentsandMethods forAssessingBrainCognitionrsquos Gestalt Patterns

)e general framework for the neurophysiological assessmentprocess andmethod of Gestalt contour cognition is illustratedin Figure 1 which is based on topological data analysis en-abled by persistent homology During the brain cognitionprocess subjects first watch random sequence diagram (RSD)pictures repeatedly at fixed intervals and then watch Gestalt(GST) images in the same manner Meanwhile the EEG dataare collected by a special cap with sensing electrodes syn-chronously (Step I) )en two methods are used for calcu-lating the correlation coefficient one is to calculate the phasecorrelation coefficient (0ndash1) of the EEG signals between thesensors through the algorithm based on Hilbert transform toconstruct the correlation matrix and the other is to calculatethe standardized Euclidean distance between the sensors toconstruct a distance matrix (Step II) [13] for obtaining thetopological VietorisndashRips simplex (Step III) Finally persis-tent homology methodology is applied to analyze the brainrsquosneurophysiological features stimulated by various picturesacross different qualities (Steps IV and V)

2 Journal of Healthcare Engineering

21 Stimuli As shown in Figure 2 we have selected tworepresentative types of Gestalt pictures with the existenceand nonexistence of a specific outline of a standard triangleOne is the picture of GSTthat people can easily recognize theoutline of the triangle and the others are the pictures of RSD)e size and quality of the two types of pictures are the sameand both are 1440times1080 resolution To explore thecharacteristics of changing consciousness in a subjectrsquoscognition process we repeat the RSD 30 times and thenrepeated the GST 10 times to increase the samplesrsquo amountand eliminate potential experimental errors

22 Procedure After a general introduction to the experi-ment and the EEG cap preparation the subjects start the testwith EEG recording which is described in Figure 3)e EEGrecordings of two cognition periods correspond to twocontinuous stages in the whole process )e first stage is tocollect the EEG signals when the subject does not have aclear cognition of RSD and the second stage is to collect thesignals when the subject recognizes the outline of the tri-angle from the GST When the subjects start to identify thetrianglersquos intrinsic outline from the GST each trial beganwith a fixed time slot that lasts 1 second and then the RSD orGST image appear for 10 seconds After that a rest time slotappears to remind the subjects that they could take a breakfor 1 second One by one the subsequent trials start to run

23 Subjects and Equipment )e EEG data are measuredfrom 20 healthy volunteers (9 males and 11 females in theage group 19ndash27) with normal (or corrected to normal)vision Volunteers are mainly sophomores and juniors )emain age group is 22 years old with an average age of 224and a standard deviation of 171 )e experiment equipmentis a standard Neuracle 64 System which includes a 64-channel adult-sized head cap with the sensor array EEGrecorder with EEG acquisition software and amplifier(NSW364) )e sampling rate is 1000Hz for the EEGsignals and the filtering window is changed with frequencyfrom 03 to 100Hz

24 Topological Data Analysis for EEG Data Topologicaldata analysis for the EEG data has been summarized inFigure 1 and the following provides the correspondingdetails

241 EEG Signals Acquisition and Preprocessing )e EEGdata are collected by the EEG cap and downsampled to250Hz Filtered EEG signals of different wavebands areobtained by a set of filters including δ band (1ndash3Hz) θ band(4ndash7Hz) α band (8ndash13Hz) β band (14ndash30Hz) and thewhole band (1ndash45Hz)

)e specific operations are as follows during the entireacquisition process we mark the EEG signals correspondingto different events to facilitate subsequent trial segmenta-tion Since the entire acquisition process is continuousconsidering the activity frequency of the human brain undernormal conditions we first perform (1ndash45Hz) filtering onthe entire time-domain signal and then try segment toextract the target data we need based on this performsubsequent data adjustments such as baseline calibrationand downsampling

)e filtered signals from each electrode of the EEG capcorrespond to a set of measuring points G As explainedbefore two data analysis methods are used for characterizingthe brain cognition process one is to calculate the corre-lation matrix through the real-time phase relationship andthe other is to define the distance matrix for each pointthrough the signal-level correlation

242 Correlation Matrix Computing )e calculation stepsare as follows

(1) After the key feature extraction and preprocessing ofthe EEG signals we get the signal of each trial periodas follows

FEEG

f11 middot middot middot f1N

⋮ ⋱ ⋮

fM1 middot middot middot fMN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

GST

EEG Acquisition

Step I

Correlation matrixDistance matrix

Step II

Simplex

Step III

Barcode frompersistent homology

Step IV

Persistent entropy

Step V

Dimension 0 Dimension 1RSD

0 0105 105

C11

D11 D1N

DNNDN1

C1N

CN1 CNN

Figure 1 )e neurophysiological process and method of Gestalt contour cognition with topological data analysis

Journal of Healthcare Engineering 3

where N is the total data length which is equal to thesampling rate multiplied by the measurement timeand M is the total number of electrodes that collectthe EEG signals Each row in the matrix FEEG rep-resents the signals collected by one electrode

(2) Hilbert transform [25] is performed on each signal inFEEG that is each row to obtain a new matrix H(FEEG)

(3) H (FEEG) obtained by step 2 is used to calculate theinstantaneous phase of each electrode

ϕ arctanH FEEG( 1113857

FEEG1113888 1113889 (2)

(4) )e value of the corresponding element of the in-cidence matrix is calculated by equation (3) )eabsolute value is taken and then combined to obtainthe incidence matrix equation (4)

Cpq

1N

1113944

N

n1exp j ϕp

(n) minus ϕq(n)1113864 1113865( 1113857

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 pne q

0 p q

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

CMtimesM

C11 middot middot middot C1M

⋮ ⋱ ⋮CM1 middot middot middot CMM

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)

where j is an imaginary unit and ϕp(n) and ϕq(n)

represent the n-th instantaneous phase in the elec-trode p and q respectively

243 Distance Matrix Computing )e filtered signal fromeach electrode in the EEG cap constitutes a set of samplingpoints G and the distance between electrodes with differentchannels is calculated by [13]

d(r t)

1113944N

k1

r|k

sk

minust|k

sk

1113888 1113889

211139741113972

(5)

where r|k and t|k stand for the y-component of differentelectrodes in (xk yk) and sk is the sample standard deviationcalculated from all y-components at position k in channel r

244 Simplicial Complexes Construction Simplicial com-plexes are constructed by VietorisndashRips filtration accordingto either the correlation matrix or the distance matrix ob-tained in Step II which is illustrated in Figure 4

245 Euler Characteristics Before calculating the persistententropy we supplement Euler entropy to do a preliminaryanalysis of the separability of the topological properties ofthe data which also provides a basis for the persistenthomology separability )e topological structures of originalEEG signals are constructed by VietorisndashRips filtration oneuses the phase-locked value (PLV) of the EEG signals data as

GST RSD

Figure 2 Two types of Gestalt pictures used for the brain cognition experiments GST and RSD

rest rest

Repeat 30 times

1 s 10 s 1 s 1 s 10 s 1 s

Repeat 10 times

Trial 1 Trial N

RSD

RSD GST

Figure 3 )e procedure of Gestalt pattern cognition


the normalized correlation coefficient (C-matrix) betweenelectrodes and the other uses the level correlation distance asthe normalized correlation coefficient (D-matrix) Eulerentropy can be calculated according to the Betti numbers Inthe process of VietorisndashRips filtration the Betti numberschange all the time so we can restore an Euler entropy curve)e Euler entropies of brain networks for different values ofe are calculated and it is a remarkable fact that Euler entropyhas a negative peak with the change of ε Since the ε valuecorresponding to the negative peak of Euler entropy isdifferent between the clear and unclear situation we furthercalculate the ε value when the negative peak of Euler entropyappears which is taken as a phase transition point in thiswork In the topological modeling of human brain structurethe phase transition point is shown by Euler entropy oftenrepresents a critical point change in brain activity [8]

246 Persistent Homology Analysis Persistent homology isan algebraic topology methodology that counts the numberof n-dimensional holes in a topological space that is Bettinumber )e Betti number of a generic topological space S iscomposed of β0 β1 and β2 in this paper β0 is the number ofconnected components in S β1 is the number of holes in Sand β2 is the number of voids in S During the filtration thetime when a k-dimensional hole appears in the simplicialcomplex is recorded as Tstart while Tend is the time when thek-dimensional hole disappears Accordingly the k-dimen-sional Betti interval is defined by [Tstart Tend] and thecorresponding persistence barcode is its graphical repre-sentation of it [8 26 27] On the other hand persistententropy (PE) provides a new entropy measure to extract thefeature of topological space by persistence barcode In thispaper B (xi yi)|iεI1113864 1113865 is set to the persistent barcode groupassociated with the filtration of topological space S where i isa set of indexes (Figure 5) Accordingly the persistent en-tropy H of the simplicial complex filtration is calculated bythe following equation

H minus 1113944iεI

pilog pi( 1113857 (6)

where pi yi minus xiL and L 1113936iεI(yi minus xi) MoreoverH canbe rescaled and 1113954H is treated as the persistent homologyfeature of the EEG data in this paper and expressed asfollows

1113954H H

log ℓmax (7)

where ℓmax is the maximum interval in the consideredpersistent barcode group

Topological patterns of the EEG data evoked by the RSDGST pictures are constructed by VietorisndashRips filtration asshown in Figure 6 To examine the relationship betweendifferent brain regions and the perception of image shapeand contour the EEG mapping results are supplementedand drawn in Figure 6 When the subjects perceive irreg-ularly distributed images more brain regions are involvedwith nonprominent features but when they perceive or-dered images there will be clear reaction areas with moreprominent features )erefore we hypothesize that vaguecognitive goals make the task more difficult and lead to moremental activities In addition due to the intense brain ac-tivity observed in the frontal lobe the frontal lobersquos functionneeds further investigation )e frontal lobe is the physio-logical basis of the most complex mental activities It isresponsible for planning regulating and controlling humanmental activities which plays a vital role in humanrsquos ad-vanced and purposeful behaviors Figure 6 indicates thecorrelation between human perception of shapes andhigher-level cognitive processes covering Gestalt patterns)ese results verify our methodrsquos effectiveness in describingthe intrinsic correlation between the EEG signals and shapecognition and our method is closer to the actual biologicalresponse process

In this regard our preliminary Euler entropy analysisdiagram is shown in Figure 7 )e two types of calculatedEuler entropy show the difference in their phase transitionpoints)e red line represents the RSD in the state of unclearrecognition and the blue line represents GST that canrecognize the outline of a triangle It can be seen that thephase transition point of the GST sample will appear earlier

r

Figure 4 VietorisndashRips filtration in persistent homology-based topological data analysis

Dim 1

yixi

I

r

Figure 5 Persistent barcode group


than the RSD )e data result of the overall sample is shownin Figure 8

3 Result and Discussions

It can be observed from Figure 8 that the phase transitionpoint of GST trials of most samples is before the RSD whichintuitively shows the separability of the overall topologicalcharacteristics of the whole brain signal so further con-tinuous coherence analysis can be carried out on this basis

To reduce the computation time of analyzing topologicalfeatures the change of persistent entropy of the subjects atdifferent time latencies after the picture appearance is in-vestigated first )ere is a difference between the RSD andGST trials Without being informed of the experimentrsquospurpose the subjects observed the RSD pictures first whichare disorderly and random Accordingly the overall EEGlevels are shown in each trial )ey were all at a certain level

in a relatively balanced manner while regarding the GSTpictures it was intuitively reflected within 2 s and the EEGlevel was stable afterward )erefore the EEG signals of twoseconds after displaying the image are selected for thepersistent homology analysis in this paper

Table 1 shows the range mean value and maximumvalue of the distinguishing degree of the two types ofcognitive behaviors investigated by adopting persistententropy as the overall experimentrsquos discrimination standard)e average distinguishing rates of each frequency bandclassified by C- and D-matrix were all greater than 70 andthe optimal distinguishing rates reached 90 and 85 for C-and D-matrix respectively

Figure 9 shows the respective performance and com-parison of the persistent entropy obtained by the two dif-ferent matrix calculation methods )e blue line representsthe participantsrsquo response to GST and the red line representstheir response to RSD Based on the statistical classification

EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5

Figure 6 EEG mapping and the corresponding VietorisndashRips simplex Red indicates high levels of neuronal activation in EEG mapping


Phase Transition Point in Euler Entropy

C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100

Figure 7 Phase transition points in Euler entropy curve

075

08

085

09

095

1Phase Transition Point by Matrix C

GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055

Phase Transition Point by Matrix D GST-RSD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005

Phase Transition Point by Matrix D

GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue

Phase Transition Point by Matrix C

GST RSD

Figure 8 Comparison of phase transition points by C- and D-matrix


of the 20 subjects and the frequency comparison it is clearthat both methods clearly describe the separation feature interms of persistent entropy between the two topologicalpatterns in the two types of brain cognition situations Wefurther draw a comparison chart of the GSTand RSD valuescalculated by the correlation matrix as shown in Figure 10According to Figures 9 and 10 almost all the persistententropy values for the topological structure of the EEGsignals induced by GST (blue) are higher than that inducedby RSD (red) the opposite is the case when the persistententropy is calculated by the distance matrix From themeasurement and analysis results it is evident that the bandswith significant differences are the α and θ bands which arein line with the corresponding trend of the overall originalsignal However in the β band the properties of the twotypes of methods are similar and the results of the com-parison of RSD and GSTare similar as depicted in Figure 11

As a summary we have proposed a neurophysiologicalapproach for cognitive assessment of the shape and contourof the Gestalt images via EEG When the subjects perceiveRSD images compared with the GST image more brainregions are involved in the cognition process It can beunderstood that the human brain is in a state of randomnessin this case TDA is used to extract the physiological featuresof EEG signals induced by the shape contours )e resultsverify that the EEG data induced by the GST image are in thebeta band and the persistent entropy values obtained by the

two calculation methods are lower than that of the RSDimage )e persistent entropy values in the α and θ bandsand the overall 1ndash45Hz band consistently show thatPEGST gtPERSD with the correlation matrix calculationmethod and PEGST ltPERSD with the distance matrix cal-culation method

Compared with the conventional neurophysiologicalmethods based on evoked potentials (requiring a specificexperimental paradigm) our approach provides a general-izable method that can extract the overall information fromthe whole brain signal not just the characteristic perfor-mance Our approach focuses not only on the brain responseto external stimuli but also on the algorithms designed tonormalize and extract numeric features that can be reliablyclassified and represent different cognitive perceptions )ealgebraic topology is used to explain the coordination re-lationship between various neural regions in the humanbrain)is work can serve as an inspiration for the analyticalapproach to the collaborative work of complex neuralnetworks )e dimensionality of the complex neural net-work model is reduced to one-dimensional persistent en-tropy to measure its characteristics

Since this paper focuses on a specific case of Gestaltcontour cognition future research may extend to moreanalysis of different Gestalt contour cognitions and evencolor or content cognition and progressively try to leverageTDA to explain the cognitive process Unlike the previous

Table 1 )e table of distinguishing accuracy

Distinguishing rate range () Average () Max ()C-matrix 65ndash90 7333 90D-matrix 55ndash85 7167 85

thetabetaalphaoriginalCorrelation matrix to PE

Distance matrix to PE

655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD

thetabetaalphaoriginal

GST RSD GST RSD GST RSD

GST RSDGST RSDGST RSDRSDGST

Figure 9 Comparison of persistent entropy calculated by C- and D-matrix (1)


original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD


Correlation matrix to PE


0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14

Figure 11 Beta bands


EEG signal analysis experiments this work implies thefeasibility to interpret the human brain consciousness pat-terns in a divisible manner and it is a preliminary explo-ration from feeling to consciousness Furthermore thedeepening grasp of the brain neural networkrsquos linkage be-havior in the brain response process from external stimuli todigital features may inspire us to build a new artificial neuralnetwork structure which requires further research andexperimentation

4 Conclusions and Future Work

In this paper we have proposed an approach physiologicalevaluation of contour cognition from EEG by using per-sistent homology of brain network and extracted its sepa-rable digital feature persistent entropy (PE) Our approachhas acquired cognitively related neural information via in-tegrating the EEG collection with the traditional Gestaltpsychology test procedure and obtained physiologicallymeaningful features of brain responses to different shapeoutlines by topological data analysis (TDA) )e validationexperiment results show that when subjects perceive cha-otically distributed images more brain regions are involvedbut the level values are more average and when they per-ceive ordered images there will be clear reaction areas withmore prominent features )e PE calculated by using twodifferent EEG correlation feature extraction matrices is allseparable In the α θ and (1ndash45Hz) bands the overallperformance is consistent and the two types of calculationsin the β band have reached a unified result of the calculatedvalue and the classification situation )e above results canintuitively show that in some specific B2BC interactionscenarios the transmission of a specific human brain nervesignal into a characteristic signal (PE) can be achieved

)e neurophysiological assessment process of Gestaltcontour cognition is a preliminary study to explore humanconsciousness formation )e experimental results showthat the conventional EEG signal can be digitized andconverted into the matrix relationship between the electrodepoints and then the VietorisndashRips complex is constructed touse the topological calculation to express the characteristicsIt encouragingly shows good separation which provides apossibility for the development of B2BC

At present there still exist some limitations to beaddressed specifically in the following two aspects

(1) One is that noninvasive EEG acquisition equipmentcannot completely restore the spatial location gen-erated by electrical signals which means that theaccuracy of our topological reconstruction con-struction cannot restore the original signals gener-ated by consciousness

(2) Second the use of algebraic topology is still in thepreliminary stage and more experiments are re-quired to verify the robustness of the method

)e outlook for future work can be expanded from twodimensions of breadth and depth )e breadth is that thereare many forms of conscious thinking because the project isan exploratory experiment and we use contour recognition

as the starting point Subsequent work can be developed tothe consciousness analysis of more advanced cognitive be-haviors such as the calculation of simple mathematicalproblems the judgment of the right and wrong of simplelogic and so on )e depth requires us to further enhanceand strengthen this method on the basis of existing researchWe can try to refine the research based on gender differ-ences brain region selection more detailed trial segmen-tation and frequency band selection to verify the robustnessand reliability of the method proposed in this paper )erealization and gradual advancement of these tasks will lay asolid foundation for our future realization of brain-com-puter interconnection and brain-to-brain interconnectiontechnology and this is also an effective means to simulateand realize human intelligence )e analysis of humanconsciousness and thinking activities in this work also ex-pands the breadth and depth of EEG analysis )e researchin this area is still in the preliminary stage for the time beingand we provide enlightening significance for reference

Data Availability

)e data in this study are collected by our own experimentand are available from the first author (liuzszjueducn)upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)is work was supported in part by the National Key RampDProgram of China under Grant 2020YFB1804800 in part bythe National Natural Science Foundation of China underGrants 61731002 and 62071425 in part by the Zhejiang KeyResearch and Development Plan under Grants 2019C01002and 2019C03131 in part by Huawei Cooperation Project inpart by project sponsored by Zhejiang Lab under Grant2019LC0AB01 in part by project sponsored by Ministry ofIndustry and Information Technology under Grant 2019-00891-2-1 and in part by the Zhejiang Provincial NaturalScience Foundation of China under Grant LY20F010016

References

[1] S Siuly S K Khare V Bajaj H Wang and Y Zhang ldquoAcomputerized method for automatic detection of schizo-phrenia using EEG signalsrdquo IEEE Transactions on NeuralSystems and Rehabilitation Engineering vol 28 no 11pp 2390ndash2400 2020

[2] Y Wang H Ombao and M K Chung ldquoTopological dataanalysis of single-trial electroencephalographic signalsrdquo An-nals of Applied Statistics vol 12 no 3 pp 1506ndash1534 2018

[3] S Scholler S Bosse M S Treder et al ldquoToward a directmeasure of video quality perception using EEGrdquo IEEETransactions on Image Processing vol 21 no 5 pp 2619ndash2629 2012

[4] C Liu X Ma J Wang et al ldquoNeurophysiological assessmentof image quality from EEG using persistent homology of brainnetworkrdquo in Proceeding of the 2021 IEEE International


Conference on Multimedia and Expo pp 1ndash6 ICMEShenzhen China July 2021

[5] K Koffka ldquoPerception an introduction to the gestalt-trdquoPsychological Bulletin vol 19 no 10 pp 531ndash585 1922

[6] D George M Lacuteazaro-Gredilla and J S Guntupalli ldquoFromcaptcha to commonsense how brain can teach us about ar-tificial intelligencerdquo Frontiers in Computational Neurosciencevol 14 p 97 2020

[7] A Lavin J S Guntupalli M Lacuteazaro-Gredilla W Lehrachand D George ldquoExplaining visual cortex phenomena usingrecursive cortical networkrdquo 2018 httpswwwbiorxivorgcontentearly20180730380048

[8] F A N Santos E P Raposo M D Coutinho-FilhoM Copelli C J Stam and L Douw ldquoTopological phasetransitions in functional brain networksrdquo Physical Reviewvol 100 no 3 Article ID 032414 2019

[9] A Subasi and E Erccedilelebi ldquoClassification of EEG signals usingneural network and logistic regressionrdquo Computer Methodsand Programs in Biomedicine vol 78 no 2 pp 87ndash99 2005

[10] V Srinivasan C Eswaran and N Sriraam ldquoApproximateentropy-based epileptic EEG detection using artificial neuralnetworksrdquo IEEE Transactions on Information Technology inBiomedicine vol 11 no 3 pp 288ndash295 2007

[11] A Fornito A Zalesky and E Bullmore Fundamentals ofBrain Network Analysis Academic Press Cambridge MAUSA 2016

[12] A C N Chen P Rappelsberger and O Filz ldquoTopology ofEEG coherence changes may reflect differential neural net-work activation in cold and pain perceptionrdquo Brain Topog-raphy vol 11 no 2 pp 125ndash132 1998

[13] M Piangerelli M Rucco L Tesei and EMerelli ldquoTopologicalclassifier for detecting the emergence of epileptic seizuresrdquoBMC Research Notes vol 11 no 1 pp 1ndash7 2018

[14] D Zhang L Yao K Chen S Wang P D Haghighi andC Sullivan ldquoA graph-based hierarchical attention model formovement intention detection from EEG signalsrdquo IEEETransactions on Neural Systems and Rehabilitation Engi-neering vol 27 no 11 pp 2247ndash2253 2019

[15] R Gonzalez-Diaz E Paluzo-Hidalgo and J F QuesadaldquoTowards emotion recognition a persistent entropy appli-cationrdquo in Proceedings of the International Workshop onComputational Topology in Image Context pp 96ndash109Springer Cham 2019

[16] N K Al-Qazzaz M K Sabir and K Grammer ldquoCorrelationindices of electroencephalogram-based relative powers duringhuman emotion processingrdquo in Proceedings of the 2019 9thInternational Conference on Biomedical Engineering andTechnology pp 64ndash70 Tokyo Japan March 2019

[17] N Baker G Erlikhman P J Kellman and H Lu ldquoDeepconvolutional networks do not perceive illusory contoursrdquoCognitive Science 2018

[18] B Kim E Reif M Wattenberg S Bengio and M C MozerldquoNeural networks trained on natural scenes exhibit gestaltclosurerdquo Computational Brain amp Behavior vol 4 no 3pp 251ndash263 2020

[19] A Zomorodian and G Carlsson ldquoComputing persistenthomologyrdquo Discrete amp Computational Geometry vol 33no 2 pp 249ndash274 2005

[20] H Edelsbrunner and J Harer Computational Topology AnIntroduction American Mathematical Society Ann ArborMI USA 2010

[21] R Hartshorne Algebraic Geometry Springer Science ampBusiness Media Berlin Germany 2013

[22] M Petti J Toppi F Babiloni F Cincotti D Mattia andL Astolfi ldquoEeg resting-state brain topological reorganizationas a function of agerdquo Computational Intelligence and Neu-roscience vol 2016 Article ID 6243694 10 pages 2016

[23] M Rucco R Gonzalez-Diaz M-J Jimenez et al ldquoA newtopological entropy-based approach for measuring similari-ties among piecewise linear functionsrdquo Signal Processingvol 134 pp 130ndash138 2017

[24] A Khalid B S Kim M K Chung J C Ye and D JeonldquoTracing the evolution of multi-scale functional networks in amouse model of depression using persistent brain networkhomologyrdquo NeuroImage vol 101 pp 351ndash363 2014

[25] M Johansson ldquo)eHilbert transformrdquoMathematics Masterrsquosthesis Vumlaxjumlo University Vaxjo Sweden 1999

[26] H Adams T Emerson M Kirby et al ldquoPersistence images astable vector representation of persistent homologyrdquo Journalof Machine Learning Research vol 18 pp 1ndash35 2017

[27] Y Wang H Ombao and M K Chung ldquoStatistical persistenthomology of brain signalsrdquo in Proceeding of the ICASSP 2019-2019 IEEE International Conference on Acoustics Speech andSignal Processing (ICASSP) IEEE Brighton UK May 2019


Gestalt psychology theory is the pioneering foundation ofmodern cognitive learning theory which was established in theearly twentieth century by psychologists WestheimerW Kohlerand K Koffka based on similitude study [5] )ey believe thatthinking is a holistic and meaningful perception rather than asimple collection of connected representations and argue thatlearning lies in forming a gestalt which aims to change onegestalt into another )is cognition process is fundamentallydifferent from the current image recognition model of deeplearning empowered by artificial neural networks (ANNs)

At present with the rapid development of deep learningnewer and stronger algorithm models emerge endlessly andtheir computing power and learning ability for specifiedtasks become increasingly powerful On top of this basissome researchers initiate ambitious new goals and turn tofocus on making AI more ldquointelligentrdquo that is achievingbrain-like intelligence )ey expect machine learning toachieve what the human brain can do and solve problems orrecognize things like the human brain It is well known thatconventional neural networks are designed by the inspira-tion of the fundamental principle of signal transmission in asingle nerve cell )erefore the linkage nature of the wholebiological neural network is the direction we need to exploreas the next frontier It can inspire us to build brain-likeintelligence and transfer from the traditional machinelearning process to a more advanced consciousness level Byexploring the act of brain cognition it is potentially possibleto probe into the generation of consciousness [6] not limitedto what kind of consciousness the brain produces

Researchers have made attempts to explore neural net-works and the human brainrsquos biological patterns [7ndash15] Togain insight into the brainrsquos response to external stimuliscientists have developed functional network analysis meth-odology because they assert that brain functionality is de-termined by the internal interaction between differentneurons as well as different brain regions [11] )ey set out toanalyze the neural signals displayed by the brain as a whole)e spatial and connective relationship of neurons within thebrain structure is a complex connection model which hasbeen used to analyze the human brainrsquos activity with topo-logical tools for a long time )e initial focus of this kind ofresearch is on the somatosensory sensations (e g hot and coldsensations and pain sensations) that are easy to recognize inbrain signals [12] Afterward abnormal EEG responses(epilepsy seizures) [13] steady-state visually evoked potential(SSVEP) movement intention detection [14] and emotionclassification [15] were analyzed We compared the resultsobtained by various EEG analysis methods in various fieldsAmong them the detection accuracy of schizophreniareached a balanced accuracy of 8959 the detection ofmoving images reached a recognition rate of 649ndash795and the classification of emotional EEG signals based ongender reached 904 (SVM) and 92 (KNN) [16]

In particular inspired by the recent research on Gestaltrecognition for which Baker et al [17] and Been Kim et al [18]provided totally conflicting conclusions the discussion on thedifferences between artificial neural networks and the humanbrain cognition process has motivated us to follow researchon the cognitive process at the level of consciousness

Nevertheless it is well realized that the human brainrsquos cog-nition on the overall outline of geometric patterns faces localand global problems in the previous Gestalt experiments

)e mathematic tools of algebraic topology are uniquelyequipped to provide quantitative information about both thelocal and global properties of an arbitrary graph [8] Ac-cordingly topological data analysis (TDA) is capable ofproviding a series of new topological and geometric methodsto analyze the brainrsquos neural networks covering EEG signalsamong which persistent homology is one of the key ap-proaches [19ndash24] Persistent homology analysis providesefficient algorithms for calculating the Betti number of eachcomplex graph in the network families under considerationand encodes the evolution of the nested complex homologygroups at different networking scales Consequently it helpsunderstand the EEG data better and keeps analytical stabilityconcerning perturbations or noise in the EEG signals

In our study 20 participants were considered in tradi-tional visual stimulation experimental methods and col-lecting EEG signals at the same time)e neurophysiologicalevaluation of the contour in the Gestalt experiment wasinvestigated by exploiting Euler characteristics and persis-tent homology features of the EEG signal On that basis howthe regions of the brain are involved in contour recognitionwas interpreted by selecting VietorisndashRips filtration

)e main contributions of this work are as follows

(1) )e topology calculation method adopted in thisexperiment provides effective separability of the EEGsignals of contour cognitive behavior and realizes thedigital feature extraction of EEG signals

(2) We provide reference significance and a referencemethod for B2BC and other work that needs torealize the digital feature extraction of EEG signals

(3) We demonstrated the feasibility of using persistenthomology modeling to analyze EEG signals

2 ExperimentsandMethods forAssessingBrainCognitionrsquos Gestalt Patterns

)e general framework for the neurophysiological assessmentprocess andmethod of Gestalt contour cognition is illustratedin Figure 1 which is based on topological data analysis en-abled by persistent homology During the brain cognitionprocess subjects first watch random sequence diagram (RSD)pictures repeatedly at fixed intervals and then watch Gestalt(GST) images in the same manner Meanwhile the EEG dataare collected by a special cap with sensing electrodes syn-chronously (Step I) )en two methods are used for calcu-lating the correlation coefficient one is to calculate the phasecorrelation coefficient (0ndash1) of the EEG signals between thesensors through the algorithm based on Hilbert transform toconstruct the correlation matrix and the other is to calculatethe standardized Euclidean distance between the sensors toconstruct a distance matrix (Step II) [13] for obtaining thetopological VietorisndashRips simplex (Step III) Finally persis-tent homology methodology is applied to analyze the brainrsquosneurophysiological features stimulated by various picturesacross different qualities (Steps IV and V)











FEEG


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

GST

EEG Acquisition

Step I


Step II

Simplex

Step III


Step IV

Persistent entropy

Step V


0 0105 105

C11

D11 D1N

DNNDN1

C1N

CN1 CNN







FEEG1113888 1113889 (2)


Cpq

1N

1113944

N

n1exp j ϕp

(n) minus ϕq(n)1113864 1113865( 1113857

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 pne q

0 p q

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

CMtimesM



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)




d(r t)

1113944N

k1

r|k

sk

minust|k

sk

1113888 1113889

211139741113972

(5)




GST RSD


rest rest

Repeat 30 times

1 s 10 s 1 s 1 s 10 s 1 s

Repeat 10 times

Trial 1 Trial N

RSD

RSD GST





H minus 1113944iεI

pilog pi( 1113857 (6)


1113954H H

log ℓmax (7)




r


Dim 1

yixi

I

r










EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5




C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References








































FEEG


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

GST

EEG Acquisition

Step I


Step II

Simplex

Step III


Step IV

Persistent entropy

Step V


0 0105 105

C11

D11 D1N

DNNDN1

C1N

CN1 CNN







FEEG1113888 1113889 (2)


Cpq

1N

1113944

N

n1exp j ϕp

(n) minus ϕq(n)1113864 1113865( 1113857

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 pne q

0 p q

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

CMtimesM



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)




d(r t)

1113944N

k1

r|k

sk

minust|k

sk

1113888 1113889

211139741113972

(5)




GST RSD


rest rest

Repeat 30 times

1 s 10 s 1 s 1 s 10 s 1 s

Repeat 10 times

Trial 1 Trial N

RSD

RSD GST





H minus 1113944iεI

pilog pi( 1113857 (6)


1113954H H

log ℓmax (7)




r


Dim 1

yixi

I

r










EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5




C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References



































FEEG1113888 1113889 (2)


Cpq

1N

1113944

N

n1exp j ϕp

(n) minus ϕq(n)1113864 1113865( 1113857

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 pne q

0 p q

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(3)

CMtimesM



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (4)




d(r t)

1113944N

k1

r|k

sk

minust|k

sk

1113888 1113889

211139741113972

(5)




GST RSD


rest rest

Repeat 30 times

1 s 10 s 1 s 1 s 10 s 1 s

Repeat 10 times

Trial 1 Trial N

RSD

RSD GST





H minus 1113944iεI

pilog pi( 1113857 (6)


1113954H H

log ℓmax (7)




r


Dim 1

yixi

I

r










EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5




C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References

































H minus 1113944iεI

pilog pi( 1113857 (6)


1113954H H

log ℓmax (7)




r


Dim 1

yixi

I

r










EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5




C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References






































EEG

Map

ping

Viet

oris-

Rips

Sim

plex

GST RSD

215

107

0

-107

-215

20

15

10

5

0

-5

-10

-15

-20

22

0

-22

-43

5432

01

-1-2-3-4-5




C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References
































C-M

atrix

D-M

atrix

8

6

4

4

2

2

0

0

20 40 60 80 100 40 60 80

10

8

6

4

2

0

10

8

6

4

2

00 50 100 0 50 100


075

08

085

09

095


GST RSD

GST RSD

GST-RSD

C-M

atrix

D-M

atrix

06062064066068

07072074076078

0808

075

07

065

06

055


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-01 -005 0 005

-01 -005 0 005


GST RSD

075

08

085

09

095

1

The fi

lter v

alue

The fi

lter v

alue


GST RSD












655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References








































655

545

435

325

2

5 59

585

585

58

575

57

565

56

555

55

59585

58575

57565

56555

55

58

575

57

565

56

555

8

5

7

5

6

5

6595857565554

6

59

58

57

56

55

54

6

61

59

58

57

56

55

54

535251

0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 200 2 4 6 8 10 12 14 16 18 20

GST RSD






original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References































original

original

alpha

alpha

theta

original

alpha

theta

theta

originalalphatheta

0 20 40 60 80 100

0 20 40 60 80 100

GST

13 13 187 7 2

911

515

317

RSD

GSTRSD




0 10 20 30 40 50 60 70 80 90 100

C-beta

D-beta

C-beta D-beta

GSTgtRSD

7 6

RSDgtGST

13 14












Data Availability




Acknowledgments


References








































Data Availability




Acknowledgments


References































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