Hilbert phase entropy imaging of fMRI time series - Taylor & Francis

Computer Methods in Biomechanics and Biomedical Engineering Vol. 13, No. 1, February 2010, 121–133

Hilbert phase entropy imaging of fMRI time series Wei Liao, Huafu Chen* and Zhengyong Pan Key Laboratory for Neuroinformation of Ministry of Education, School of Life Science and Technology, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China (Received 18 March 2009; final version received 23 May 2009) Functional magnetic resonance imaging (fMRI) data-processing methods in the time domain include correlation analysis and the general linear model, among others. Virtually, many fMRI processing strategies utilise temporal information and ignore or pay little attention to phase information, resulting in an unnecessary loss of efficiency. We proposed a novel method named Hilbert phase entropy imaging (HPEI) that used the discrete Hilbert transform of the magnitude time series to detect brain functional activation. The data from two simulation studies and two in vivo fMRI studies that both contained block-design and event-related experiments revealed that the HPEI method enabled the effective detection of brain functional activation and the distinction of different response patterns. Our results demonstrate that this method is useful as a complementary analysis, but hypothesis-constrained, in revealing additional information regarding the complex nature of fMRI time series. Keywords: functional MRI; Hilbert transform; Hilbert phase entropy

1. Introduction Functional magnetic resonance imaging (fMRI) is a noninvasive imaging technique that produces a series of brain images that reflect the time course of neuronal activity in areas engaged during perception, action and cognitive tasks (Faisan et al. 2005). The temporal and magnitude behaviours of each voxel that are measured using blood oxygenation level dependent (BOLD) contrast reflect variations in the concentrations of oxyhemoglobin and deoxyhemoglobin, as well as neuronal activation. A large number of statistical test methods have been proposed to analyse fMRI data in detecting brain activation. Statistical parametric mapping (SPM; Friston et al. 1994) based on the general linear model is a powerful tool for analysing fMRI data; however, these methods require determinate statistical inference and hypothesis tests. Similarly, statistical test methods, such as the t tests and the F-test, require accurate time information that corresponds to the experiment pattern in order to compute the temporal correlation between the reference pattern and the observed BOLD signals. The existing methods used in fMRI brain mapping require strong prior assumptions regarding the shape and timing of the fMRI signal in the activated voxels. Generally, the linear modelling approaches utilise an estimation of the hemodynamic response (Friston et al. 1994; Rosen et al. 1998). However, the information from the original complex time course is ignored in such analyses (Laird et al. 2002; Rowe and Logan 2005). In practice, the shape of the hemodynamic response function (HRF) may vary across brain areas, with age, and

*Corresponding author. Email: [email protected] ISSN 1025-5842 print/ISSN 1476-8259 online q 2010 Taylor & Francis DOI: 10.1080/10255840903062552 http://www.informaworld.com

within different subjects (Handwerker et al. 2004). Given that the linearity assumption underlying the convolution model does not hold when considering short duration stimulation (Faisan et al. 2007), we proposed an alternative method for fMRI data processing that uses the theory of Hilbert phase synchronisation to obtain a more complete understanding of the dynamics of interacting systems (Laird et al. 2002). Synchronisation is a nonlinear phenomenon that is highly relevant when studying the intricate dynamics between oscillators that are either weakly coupled or driven by external force (Laird et al. 2002). The definition of synchronisation is not widely agreed upon; however, it occurs when a functional relationship exists between subsystems. The occurrence of synchronisation has been observed in many studies of man-made systems and in the investigation of brain activity during pathological tremor using magnetoencephalography, electromyogram recording, electrocardiogram and nearinfrared spectroscopy of motor cortex hemodynamics (Laird et al. 2002). Laird et al. (2002) proposed the creation of activation maps between the instantaneous phase of the reference function and the voxel’s time series in whole brain fMRI datasets based on the phase of the reference signal as one of the phase-locking conditions. In this work, however, we have taken into account the inherent Hilbert phase entropy difference between the two simple states of each voxel for two reasons: the optimum method for modelling phase information is unclear, and to avoid bias caused by variation of the form and timing of the HRF in the convolution model.

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In the present study, the Hilbert phase entropy imaging (HPEI) method to detect brain functional activation for fMRI data was used. Although the philosophies of HPEI and the previous phase synchronisation study (Laird et al. 2002) are similar, in that both approaches delineate the instantaneous phase relationships in whole brain fMRI, there are different assumptions about phase synchronisation between the two methods (Laird’s method is based on the phase of the reference signal as one of the phaselocking conditions, whereas the HPEI method is based on the interrelations of the Hilbert phases between different states). We calculated a Hilbert phase time series that provides information concerning the interrelations of the Hilbert phases between the task and control states using the Hilbert transform, which provides additional information regarding the dynamics of functional activation. Then, we created functional activation maps that classified the activated and nonactivated voxels using the entropy difference in the Hilbert phase time series between the task and control states. First, a block-design and an eventrelated simulation experiment were performed and analysed. Then, the data from in vivo fMRI with visual stimulation and synchronous hand movement were analysed, as well as a visual event-related experiment.

2.

Materials and methods

2.1

Principle of HPEI

2.1.1 The Hilbert transform Determination of a relationship between Hilbert phases using the theory of phase synchronisation requires calculation of the Hilbert phase signal for each voxel’s time series. This can also be accomplished using the Hilbert transform, which has previously been used in fMRI as a method of estimating hemodynamic delay response (Laird et al. 2002; Saad et al. 2003). For a real-valued time-domain signal s(t), the Hilbert transform Hs(t) is defined as the convolution internal of s(t) and ð1=ptÞ: sðtÞ 1 HsðtÞ ¼ dt ¼ sðtÞ^ : p ðt 2 t Þ p t 21 ð þ1

ð1Þ

The Hilbert phase time series is defined as

uðtÞ ¼ arctan

HsðtÞ ; sðtÞ

ð2Þ

where u is from 2ðp=2Þ to p=2.

modelled each Hilbert phase time series uðtÞ as a Gaussian probability density function (pdf) under the task- and control-states, respectively. The pdf for the values in uðtÞ was calculated as 2 2 1 pðuðtÞjSj Þ ¼ pffiffiffiffiffiffi e2ðuðtÞ2mj Þ =2sj ; 2 p sj

j [ {0; 1}:

ð3Þ

Assuming that the time samples were normalised, ptask denotes the paradigm TASK periods, and pcontrol denotes the paradigm CONTROL periods; the control-state S0 Gaussian parameters were

m0 ¼

1

X

Lpcontrol t[pcontrol

uðtÞ;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X 1 s0 ¼ ðuðtÞ 2 m0 Þ2 ; Lpcontrol t[pcontrol and the task-state S1 Gaussian parameters were 1 X m1 ¼ uðtÞ; Lptask t[ptask

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X s1 ¼ ðuðtÞ 2 m1 Þ2 ; Lptask t[ptask

where Lpcontrol was the total number of time samples in the CONTROL periods and Lptask was the total number of time samples in the TASK periods. A critical issue in fMRI analysis is how to estimate the correct task-state and control-state distributions. A simplifying assumption in this regard is that all time samples in the paradigm TASK periods belong to the task state, and all samples in the paradigm CONTROL periods belong to the control state (Clare 1997).

2.1.3 Synchronisation difference index In synchronisation theory, the degree of phase synchronisation is determined by an analysis of the distribution of the Hilbert phase difference signal, also termed the relative phase distribution. The relative phase distribution for two tie series that are phase-locked is a peaked distribution (Laird et al. 2002). Laird et al. (2002) took advantage of the reference function that was created by convolving the stimulated time series. Our chosen statistic test was based on entropy (Laird et al. 2002), S¼2

N X

pt ln pt ;

ð4Þ

t¼1

2.1.2

Probability density function

Because our simple block-design and event-related paradigm had only two levels (on and off), we set two states: task-state S1 and control-state S0. We further

where pt derived from Equations (3) was the probability mass of the Hilbert phase fMRI time series. N is equal to Lpcontrol in the CONTROL periods and equal to Lptask in the TASK periods.

Computer Methods in Biomechanics and Biomedical Engineering Task- and control-state entropy were calculated as Stask and Scontrol under the ptask and pcontrol , respectively. The synchronisation difference index between the two states was calculated as Sdif ¼ Stask 2 Scontrol ;

ð5Þ

In our calculation, the value of Sdif ranges from zero to infinite, where the zero corresponds to nonactivation due to the Sdif in the same level, and the higher Sdif corresponding to activation due to Stask will be clearly different from Scontrol . Since the distribution of the indices was unknown, a nonparametric estimation based on bootstrapping techniques (Efron and Tibshirani 1993) was used to assess signification of the obtained Sdif . One of the benefits of the bootstrap procedure is that no distribution assumptions are necessary to use it. The null hypothesis was that Sdif ¼ Stask 2 Scontrol ¼ 0 in our work. 2.1.4

HPEI algorithm process

According to the above theory the HPEI algorithm steps can be summarised as follows: (1) compute the Hilbert phase time series from the measured fMRI signal according to Equations (1) and (2); (2) compute the Hilbert phase time series signal TASK periods pdf ptask and CONTROL periods pdf pcontrol according to Equation (3); (3) compute the task-state entropy Stask and control-state entropy Scontrol according to Equation (4); (4) compute the synchronisation difference index Sdif between the two states according to Equation (5); (5) acquire a functional activation map according to the synchronisation difference index Sdif , that can calculate the p values for all voxels on the brain based on bootstrapping techniques (Efron and Tibshirani 1993); and (6) assess significance thresholds that could be obtained with methods controlling for the false discovery rate (FDR, the expected proportion of false positives within all tests with a positive result; Genovese et al. 2002) for voxelwise tests. 2.2 Statistical inference The previous study (Laird et al. 2002) in which phase synchronisation analysis characterised the instantaneous phase relationships in whole brain fMRI data has suggested that values of the synchronisation index range from zero to one, where zero corresponds to no synchronisation, or a uniform distribution, and one corresponds to a perfect synchronisation, or a Dirac delta distribution from which the statistical inference could easily be gained. In the current study, since the distribution

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of the indices was also unknown, a nonparametric estimation based on bootstrapping techniques (Efron and Tibshirani 1993) was used to assess the significance of the obtained Sdif as follows: (1) Assuming the Sdif of each voxel (in total N voxels) was gained by the HPEI method mentioned above, we sampled these Sdif with replacement, which denotes that some elements (Sdif ) could be picked more than once from the original Sdif data, and computed the mean of them S dif . (2) We repeated step 1 a moderate to large number, B (B ¼ 1000 in our study), of times to come up with ‘bootstrap’ estimates S dif1 , S dif2 , . . . , S difB . In general, the value of B ¼ 200 , 5000 is sufficient (Efron and Tibshirani 1993). (3) Used the standard deviation (mean value) of the B estimates in step 2 to estimate the standard error (mean) to gain the p values. When performing simultaneous statistical tests of all voxels, we could assess significance thresholds employing methods controlled for the FDR (Genovese et al. 2002), the expected proportion of false positives among all tests for which the null hypothesis Sdif ¼ 0 was rejected. This is favourable for dealing with multiple comparison problems while retaining considerable power in the detection of effects and adapting to the noise level in the data. The FDR corrected thresholds, corresponding synchronisation difference index for the term being non-zero Sdif at an accepted FDR level q are obtained from the set of p values obtained by bootstrapping techniques over all voxels. The p values obtained by bootstrapping techniques for a given difference term were then regarded as the proportion of larger values in the null distribution. Subsequently, the FDR corrected base threshold was obtained from the p values as follows: Given the ordered collection of p values, let r be the largest i for which p½i # ði=NÞ £ ðq=ðcðNÞÞÞ, then the threshold was set at the value corresponding to the p-value p½r. The value of the constant cðNÞ was determined by assumptions on the joint distribution P of p values over all voxels. Here, it was set to cðNÞ ¼ Ni¼1 1=i, which applied for any distribution of p values over voxels (Genovese et al. 2002). 2.3

Block-design simulation data generation

The three block-design synthetic datasets, BS1 – BS3, were composed of five active subregions using activation patterns embedded at known locations, which was used in our previous study (Liao et al. 2008). For a more detailed account, we refer the reader to Liao et al. (2008). Dataset BS1 was designed to illustrate variability in the shape of the hemodynamic response to a stimulation block (unsustained activation during task; Faisan et al. 2007; Liao et al. 2008). Five models of BOLD activation were computed by convolving the HRF model proposed by the SPM software (sum of two Gamma functions) with the five deterministic timing functions, as plotted in the Figure 1(c). The time series of all voxels of a subregion

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Figure 1. Results of BS1 under SNR ¼ 0.5: (a) functional activation maps were calculated by the HPEI (left), Laird’s (middle) and SPM method (right) (all p , 0.001, FDR corrected), (b) ROC curves obtained by the three methods for p-value ranging from 0.00001 to 0.01 (FDR corrected) and (c) experiment paradigm used for evaluation (shaded in grey) and BOLD activation models used to build the datasets (thick line).

consists of the given signal mixed with Gaussian noise under signal-to-noise ratio (SNR) ¼ 0.5 (SNR is defined as the standard deviation of the normalised ideal waveform divided by the standard deviation of the noise). These five latter functions were all derived from the same ‘expected’ boxcar-like timing function (120 scan length, 6 blocks TASK interleaved with 6 blocks CONTROL, 10 scans each, beginning from CONTROL). BS2 was designed to illustrate timing variability between the stimulation paradigm and hemodynamic response (activation delay; Liao et al. 2008). Five activation patterns were obtained by convolving the above HRF model with five delayed versions (delay of 2, 4, 6, 8 or 10 s) of the aforementioned ‘expected’ boxcarlike timing function, as depicted in Figure 2(c). The spatial pattern and SNR were identical to those of BS1. Such delays have been observed for real blockdesign fMRI datasets, and could represent shifts in local brain activity: in short, a cognitive process could induce other cognitive processes across the brain (Faisan et al. 2007). BS3 was designed to illustrate signal-to-noise variability (Faisan et al. 2007; Liao et al. 2008). For each dataset, five activation patterns were obtained by convolving the HRF model of SPM with five exact

replications of the aforementioned ‘expected’ boxcar-like timing function, as depicted in Figure 3(c). The differences in SNR (0.25, 0.5, 0.75, 1.0 and 1.25) were calculated repeatedly. 2.4

Event-related simulation data generation

The two event-related synthetic datasets, BS4 and BS5, were also composed of five active subregions using activation patterns embedded at known locations, which was also used in our previous study (Liao et al. 2008). BS4 was designed to illustrate timing variability between the stimulation paradigm and hemodynamic response (activation delay). Five activation patterns were obtained by convolving the above HRF model with five delayed versions (delay of 2, 4, 6, 8 or 10 s), as depicted in Figure 4(c). A practical event stimulus pattern such as this was used in the actual event-related fMRI data collection (totally, 90 volumes). The spatial pattern and SNR were identical to those of BS1. The patterns could represent a shift in local brain activity: in short, a cognitive process could induce other cognitive processes across the brain (Faisan et al. 2007). BS5 was designed to illustrate signal-to-noise variability. For each dataset, five activation patterns were obtained

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Figure 3. Results of BS3: (a) functional activation maps were calculated by the HPEI (left), Laird’s (middle) and SPM method (right) (all p , 0.001, FDR corrected) under the SNR ¼ 0.5, (b) ROC curves obtained by the three methods for SNR value ranging from 0.25 to 1.25 (all fixed p , 0.001, FDR corrected) and (c) experiment paradigm used for evaluation (shaded in grey) and BOLD activation models used to build the datasets (thick line).

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by convolving the HRF model of SPM, as depicted in the Figure 5(c). The differences in SNR (0.25, 0.5, 0.75, 1.0 and 1.25) were calculated repeatedly.

angle, 908. A total of six cycles (12 epochs) were performed; thus, 120 scans were acquired. 2.6

2.5

Block-design fMRI experimental paradigm

An in vivo fMRI multi-task block-design experiment was performed using visual stimulus and synchronous hand movement. That fMRI dataset has been described previously (Liao et al. 2008). The condition for successive blocks alternated between rest and simultaneous visual stimulus and hand movement, staring with rest. Visual stimulation was presented at the centre of the visual field with a frequency of 8 Hz, light intensity of 200 cd/cm2 and visual angle of 28. Visual stimulation signals acted as a trigger to start hand movement. Three healthy subjects participated in the fMRI experiment (two males, one female; mean age, 30 years). The experiment was conducted at the University of Texas Health Center at San Antonio, using a 3T Magnetom TRIO (Siemens, Glasgow, UK). Gradient echo echo-planar imaging (EPI) sequences were acquired using the following parameters: 23 slices; TR, 2000 ms; TE, 30 ms; FOV, 24 cm; matrix, 64 £ 64; in-plane resolution, 3.75 £ 3.75 mm; slice thickness 5 mm without gap; flip

Event-related fMRI experimental paradigm

Each subject performed a visual event-related experiment that presented stimulus three times, at 20, 80 and 140 s; each stimulus was presented for a period of 2000 ms. A blank trial was also performed in which the stimulus was presented at the centre of the visual field, with a frequency of 8 Hz, light intensity of 200 cd/cm2 and visual angle of 28. Gradient echo EPI sequences parameters were identical with block-design experiment. In total 90 scans were acquired. The paradigm reference is shown as the area shaded grey in Figures 4(c) and 5(c). 2.7

Data-processing procedure

First, the experimental data were pre-processed using SPM2 software (www.fil.ion.ucl.uk/spm; Friston et al. 1994) in line with our previous study (Liao et al. 2008). Spatial transformation that included realignment was performed using three-dimensional rigid-body registration to correct for head motion.


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Figure 5. Results of BS5: (a) functional activation maps were calculated by the HPEI (left), Laird’s (middle) and SPM method (right) (all p , 0.001, FDR corrected) under the SNR ¼ 0.5, (b) ROC curves obtained by the three methods for SNR value ranging from 0.25 to 1.25 (all fixed p , 0.001, FDR corrected) and (c) experiment paradigm used for evaluation (shaded in grey) and BOLD activation models used to build the datasets (thick line).

Second, to increase analysis efficiency, only signals in the brain were processed. Voxels with values lower than a certain threshold were regarded as background. Empirically, we set the threshold values to be 1/10 of the highest intensity in the input image (Liao et al. 2008). Third, we systematically removed noise from the considered data after eliminating background noise from the fMRI data. Linear trends were removed from the data to eliminate the effect of gross signal drifts, which could be caused by scanner instabilities and/or gross physiological change in the subject. A high-pass filter with a cutoff frequency of 1/128 Hz was used to remove lowfrequency noise. Each time course was normalised in succession by subtracting its mean and dividing by its standard deviation to give a time course with zero mean and unit variance. Finally, the data were processed using the HPEI method. 2.8 Performance-receiver operating characteristic analysis To validate the performance of our HPEI method and compare it with two prior methods (Laird’s method (Laird et al. 2002) and the SPM method (Friston et al. 1994)) we performed receiver operating characteristic (ROC) analysis

(Sorenson and Wang 1996). We used the Jaccard coefficient (JC ¼ a/(a þ b þ c); Anderberg 1973), where a is the number of correct classifications of activated voxels (true positive ratio, TPR), b is the number of false negative ratio (FNR) found in an activation cluster and c is the number misclassification of inactivated voxels (false positive ratio, FPR), to represent a quantitative measure of the quality of the cluster. The JC should thus provide a quantitative measure of the quality of the activation regions because it emphasises the scores of the TPR. The ROC curves versus p values (FDR corrected) for three methods were depicted for comparison. The p values ranged from 0.00001 to 0.01 in the figure. Note that ROC analysis is only used for the simulation data and not for real fMRI. 3.

Results

3.1 Block-design simulation result The activation detection performance of the HPEI, Laird’s and SPM methods could be compared using the activation maps and ROC analysis (Sorenson and Wang 1996) within 100 simulations, because the ‘ground truth’ was known for all synthetic datasets (BS1 – BS3). The activation detection results obtained from block-design datasets BS1 (Figure 1(a)) according to HPEI, Laird’s and SPM

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methods (in the left, middle and right columns, respectively), BS2 (Figure 2(a)) and BS3 (Figure 3(a)). The activation detection results acquired from BS1 revealed that the HPEI method was not relatively affected by the various forms of the hemodynamic response. For p values ranging from 0.00001 to 0.01 (FDR corrected), the HPEI method achieved JC about 77% in activation detection performance while Laird’s method achieved about 63% and the SPM method about 72% (Figure 1(b)). In particular, when comparing activation maps of the HPEI ( p , 0.001, FDR corrected), Laird’s method ( p , 0.001, FDR corrected) and the SPM method ( p , 0.001, FDR corrected; Figure 1(a)), it is clear that the FPRs for all three methods are low, while the FNRs for Laird’s and the SPM methods are relatively high (Table 1). Activation detection results acquired from BS2 revealed that the HPEI method was not relatively affected by the various hemodynamic response timings coupled with the input stimulation blocked paradigm. For p values ranging from 0.00001 to 0.01 (FDR corrected), the HPEI method achieved JC about 62% in activation detection performance while Laird’s method achieved about 57% and the SPM method about 38% (Figure 2(b)). The detection ability of the HPEI method remained effective when activation delays ranged from 2 to 6 s. In contrast, three activation areas were not found by the SPM method (Figure 2(a), right). Activation detection results acquired from BS3 revealed that the HPEI method was not clearly affected by the various SNR conditions. While the p-value was fixed at 0.001 (FDR corrected) under various SNR conditions, JC of the three methods are showed in the Figure 3(b). Activation maps of three methods are shown in Figure 3(a) (the SNR ¼ 0.5 shown only). 3.2

Event-related simulation result

The activation detection results obtained from eventdesign datasets BS4 (Figure 4(a)) and BS5 (Figure 5(a)) according to the HPEI, Laird’s and SPM methods (in the left, middle and right columns, respectively). Activation detection results acquired from BS4 revealed that the HPEI method was not relatively affected by the various

Table 1.

3.3

Block-design fMRI experiment result

The activation detection results obtained by the HPEI and SPM methods on real block-design data of the visual stimulation and synchronous hand movement task were compared based on the functional activation maps. For each of the subjects, the HPEI activation maps show the presence of neural activity in all cortical visual areas and cortical motor areas that were expected to be active. Figure 6 illustrates the HPEI and SPM methods (all p , 0.001, FDR corrected) activation mapping results obtained by the visual stimulation and synchronous hand movement task, and reveals activation of the subject’s bilateral primary motor cortex, supplementary motor cortex and primary visual cortex. Table 2 shows the significant functional brain activations obtained by HPEI, and the Brodmann’s areas and Talairach coordinates

The TPR, FNR and FPR detected by three methods. HPEI

BS1 BS2 BS3 BS4 BS5

timings of the hemodynamic response coupled with the input stimulation event paradigm. For p values ranging from 0.00001 to 0.01 (FDR corrected), the HPEI method achieved JC about 58% in activation detection performance while Laird’s method achieved about 32% and the SPM method about 10% (Figure 4(b)). The detection ability of the HPEI method remained effective when activation delays ranged from 2 to 8 s. Interestingly, the detection results computed using the SPM method are so dependent on the delay of fMRI signal that four activation areas were not found (Figure 4(a), right). As in the blockdesign simulation results, the HPEI method captured the intrinsic Hilbert phase entropy of the fMRI time series in the event-related simulated test, regardless of activation delay. The activation detection results acquired from BS5 revealed that the HPEI method was not relatively affected by the various SNR conditions. While the p-value was fixed at 0.001 (FDR corrected) under various SNR conditions, the JC of the three methods are shown in Figure 5(b). Activation maps of the three methods are shown in Figure 5(a) (the SNR ¼ 0.5 shown only). These methods validate phase entropy statistical information, in contrast to conventional fMRI processing methods that are based on time-domain behaviour.

Laird’s

SPM

TPR

FPR

FNR

TPR

FPR

FNR

TPR

FPR

FNR

0.77(0.04) 0.63(0.03) 0.91(0.04) 0.71(0.02) 1(0)

0(0) 0.01(0.01) 0(0) 0.25(0.06) 0(0)

0.23(0.04) 0.37(0.03) 0.09(0.04) 0.29(0.02) 0(0)

0.63(0.01) 0.54(0.10) 0.75(0.05) 0.30(0.02) 1(0)

0.01(0.01) 0(0) 0(0) 0.03(0.02) 0.32(0.02)

0.37(0.01) 0.46(0.10) 0.25(0.05) 0.69(0.02) 0(0)

0.72(0.01) 0.31(0.02) 0.85(0.02) 0.01(0.01) 0.21(0.06)

0(0) 0(0) 0(0) 0(0) 0.03(0.02)

0.28(0.01) 0.69(0.02) 0.15(0.02) 0.99(0.01) 0.79(0.06)

Note: The result of BS3 and BS5 under SNR ¼ 0.5 listed only. Standard errors are listed in the brackets.


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Figure 6. Results for real visual stimulation and synchronous hand movement task: (a) functional activation maps gained by SPM ( p , 0.001, FDR corrected) and (b) the activation by HPEI ( p , 0.001, FDR corrected), in which the activation areas are located in the subject’s bilateral primary motor cortex, supplementary motor cortex and primary visual cortex.

Table 2.

Significant functional brain activations obtained by HPEI.

Cluster

Area for local maxima

Brodman’s area

Synchronisation index

Talairach coordinate (mm) of max voxel (x, y, z)

Visual cortex LC OL cuneus RC OL cuneus LC OL lingual gyrus RC OL lingual gyrus LC OL cuneus Sensorimotor areas LC PL postcentral gyrus RC PL postcentral gyrus LC FL precentral gyrus RC FL precentral gyrus LC FL medial frontal gyrus

BA BA BA BA BA

17 17 18 18 19

3.33 4.25 2.93 3.04 2.79

212, 296, 2 10, 2 96, 2 212, 288, 2 12 14, 2 76, 28 220, 280, 2 12

BA BA BA BA BA

3 3 4 4 6

2.68 2.69 3.39 2.43 2.4

242, 222, 54 48, 218, 54 238, 220, 54 34, 2 24, 52 26, 212, 54

Abbreviation used: LC, left cerebrum; RC, right cerebrum; FL, frontal lobe; OL, occipital lobe; PL, parietal lobe.

corresponding statistics of local maxima. From these results, we can see that the bilateral primary motor cortex, supplementary motor cortex and primary visual cortex are effectively highlighted by the SPM and the HPEI methods. We also made the following observations: (1) the HPEI method produces more compact and clearer highlighted regions (left primary motor cortex), which indicate that the Hilbert phase entropy method can successfully distinguish activation and nonactivation voxels by the different synchronisation index between the TASK and CONTROL periods, and is more accurate than conventional methods and (2) the HPEI method can effectively avoid the FPR most commonly located at the periphery of the brain because HPEI uses phase synchronisation in mapping the activation. 3.4

Event-related fMRI experiment result

Activation detection results obtained by the HPEI and SPM methods for real event-related data from the visual stimulation experiments were compared based on the functional activation maps. For each of the three

subjects studied, the HPEI maps show the presence of neural activity in all cortical visual areas that were expected to be activated. These results are identical to those obtained by the SPM method. Figure 7 illustrates the HPEI and SPM (all p , 0.001, FDR corrected) activation mapping results obtained by the visual stimulation task and shows activation of the subject’s primary visual cortexes.

4. Discussion Analysis of the Hilbert phase signals of the time series in fMRI via the HPEI method was used as a new complementary analysis, but hypothesis-constrained, in determining the phase relationship between different states. Experimental results obtained on both synthetic and real data illustrate the relevance of HPEI in fMRI brain mapping. The results clearly demonstrate the robustness of the HPEI mapping method to noise, activation delay and the shape-variability of the hemodynamic response.

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Figure 7. Results for real visual event-related stimulation task: (a) functional activation maps gained by SPM ( p , 0.001, FDR corrected) and (b) the activation by HPEI ( p , 0.001, FDR corrected), in which the activation areas are located in the primary visual cortexes of the subject.

The HPEI mapping method outperformed Laird’s and the SPM method in most of the conducted experiments.

4.1

HPEI theory

In the previous study (Laird et al. 2002), the relative phases of detecting a relationship between the reference function and voxels that were active during a motor task have been introduced. The study of Laird et al. (2002) concluded that the instantaneous phase signals of the time course in fMRI calculated by the Hilbert transform were effectively used in determining the occurrence of an interrelation between phases. Although the philosophy of our study is similar to that work, which detected the active voxels via the Hilbert transform, there are fundamental differences between the two studies, mainly in the synchronisation difference index (in Laird et al. (2002), the reference function was considered; meanwhile, in our work, the task- and control-state entropy were associatively used to distinguish the active and inactive voxels). The reference function was constructed by the HRF and experimental paradigms. But the shape of the HRF may vary across different brain regions, with age, and even in different subjects (Handwerker et al. 2004; Faisan et al. 2007). To the contrary the phase relationship between different states, which does not depend on the HRF model, intrinsically characterises the phase synchronisation fMRI time course of each voxel. On the other hand, the intrinsically neurophysiological meanings between different states may be characterised by the phase synchronisation. Moreover, the underlying neurophysiological points of different states may enable more experiments to be generated for testing in future studies. 4.2

Simulation experiments

We investigated the relationship between the Hilbert phases of TASK and CONTROL states of the time series of each voxel as a method of detecting activation by

determining if the Hilbert phase locking condition was fulfilled (Laird et al. 2002). Five synthetic datasets (BS1– BS5) were used to test the activation detection ability of the HPEI method. The activation detection results obtained from BS1 demonstrated that the HPEI method was relatively insensitive to variations in hemodynamic response in a stimulation block compared to Laird’s and SPM methods (Figure 1 and Table 1). In this context, the HPEI method is used to detect functional activation when the shape of the HRF is unknown. The activation detection results obtained from BS2 demonstrated that the HPEI method was clearly relatively insensitive to timing variation in the hemodynamic response compared to Laird’s and SPM method (Figure 2 and Table 1). Conventional approaches to fMRI analysis that include correlation analysis and the t-test consider only time-domain behaviour; as a result, the temporal fMRI series is highly sensitive to delay. The SPM results (Figure 2(a), right) indicated that three activation areas that were delayed by 6, 8 and 10 s in the assumed time series were not found. The Laird’s results (Figure 2(a), middle) indicate that two activation areas that were delayed by 8 and 10 s in the assumed time series were found, but with higher FNR compared to the HPEI (Figure 2 and Table 2). It is the possible explanation that the reference function considered in Laird’s phase synchronisation analysis was constructed by the HRF and experimental paradigms, but the shape of the HRF may vary across different brain regions, with age, and even within different subjects (Handwerker et al. 2004; Faisan et al. 2007). The event-related simulation experiment showed that the HPEI approach was clearly superior to the other methods with regard to different delay responses. In different delay response synthetic dataset BS4, the averaged correct classification for JC performance is 58% for the HPEI method, compared to 32% for Laird’s method and 10% for the SPM method (Figure 4(b)). Our HPEI results are consistent with

Computer Methods in Biomechanics and Biomedical Engineering findings by Saad et al. (2003) where Hilbert phase estimation is less sensitive to noise. The results of our simulation experiments described above show that the HPEI method was relatively independent of the various timings of the coupled hemodynamic response. We were not truly and completely characterising phase synchronisation in fMRI activation data, but were instead detecting the presence of a relationship between the Hilbert phase entropy of the TASK state and the CONTROL state of each voxels’ time series. Unlike conventional fMRI data-processing methods such as linear correlation, which did not correctly characterise dependencies in multivariate time series data with strong nonlinear sources (Friston et al. 2000), HPEI accounts for nonlinear interactions among regions of the brain and uses the inherent statistical information between the different states of each voxels’ time series. 4.3

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labelled ‘a’, ‘b’, ‘c’ and ‘d’. The SPM approach failed in this regard because the fMRI intensity of the time series was too weak. We also made the following observations: (1) the HPEI method can effectively produce activation areas that are coherent with the SPM method and (2) the HPEI method produces more strongly highlighted regions (extended a, b, c and d), which implies that the Hilbert phase entropy method that successfully distinguishes activated and nonactivated voxels by the different synchronisation index between the TASK and CONTROL periods was more accurate than conventional methods for the case of event-related experiments. The HPEI approach relies on the Hilbert phase information between the two states: it does not rely on the different HRF responses with temporal domain information. These results of in vivo fMRI block-design and event-related experiments consist of our simulation test for different response patterns, response delays and SNR.

fMRI experiments

The HPEI method was then used in characterising Hilbert phase synchronisation in real fMRI data. The HPEI maps show the presence of the expected neural activity in all cortical visual areas and cortical motor areas active for simultaneous hand movement and visual stimulus blockdesign task experiments. These results are in accordance with those obtained by the SPM method. The accuracy of the detection of brain activation areas using conventional methods declined when the fMRI voxel’s time series intensity was relatively low in the event-related experiment paradigm. In contrast, HPEI maps the fMRI voxel temporal series into the Hilbert phase domain, which is insensitive to SNR. As shown in Figure 7, we found that areas a, b, c and d were not detected by the SPM method, but that their activated maps were thresholded at a certain level using HPEI ( p , 0.001, FDR corrected). The extended areas found by HPEI are listed in Table 3. The corresponding averaged voxel’s time series labelled ‘a’, ‘b’, ‘c’ and ‘d’ are shown in Figure 8(a) – (d), respectively, and were consistent with the event-related paradigm. This result demonstrates the importance of finding the areas

Table 3.

4.4 HPEI potential applications In the present study, we presented the HPEI approach that utilised the available Hilbert phase information from the fMRI time series and compared it with Laird’s algorithm and the standard SPM algorithm. The HPEI approach performed better than the temporal domain approaches. The HPEI method found a larger number of activated voxels than the SPM approach (Figures 6 and 7; Tables 2 and 3) and succeeded in the analysis of all voxels for block-design and event-related experiments. One possible reason for the success of the HPEI approach is that it utilised the Hilbert phase information, which is insensitive to different response patterns, response delays and SNR (Saad et al. 2003). A potential application for this method is in revealing additional information on the complex nature of the fMRI time series. Further investigation is necessary to develop the HPEI method and to improve our understanding of its nondeterministic behaviour.

Significant extended functional brain activations from HPEI.

Extended area

Area for local maxima

a b c d

RC OL fusiform gyrus RC TL fusiform gyrus LC OL middle occipital gyrus RC OL lingual gyrus RC OL fusiform gyrus

Brodman’s area

Synchronisation index

Talairach coordinate (mm) of max voxel (x, y, z)

BA 19 BA 37 BA 18 BA 18 BA 19

12.31 11.48 15.23 13.96 13.78

26, 2 82, 212 34, 2 46, 212 2 28, 2 82, 28 32, 2 70, 28 32, 2 68, 28

Abbreviation used: LC, left cerebrum; RC, right cerebrum; OL, occipital lobe; TL, temporal lobe.

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Figure 8. Averaged time series of the four extended activeed areas shown in Figure 7(a) – (d) are four different averaged time series detected by the HPEI method.

5.

Conclusion

In the present study, we proposed a new HPEI method to detect activation for fMRI data. The Hilbert transform was used to create functional maps that provide information regarding interrelations between the Hilbert phases of the task and control states. Block-design and eventrelated simulation experiments and in vivo fMRI data revealed that this method enables brain functional activation to be effectively detected. Our results demonstrate that this method is useful as a complementary analysis, but hypothesis-constrained, in revealing additional information regarding the complex nature of fMRI time series. Acknowledgements The author thanks Xia Zhao, Jia-Hong Gao, et al. at the Research Imaging Center, University of Texas Health Science Center, San Antonio, Texas, USA, for fMRI data acquisition. This work is supported by grants from the Natural Science Foundation of China (90820006 and 30770590) and key research project of science and technology of MOE (107097).

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