www.elsevier.com/locate/ynimg
NeuroImage 28 (2005) 227 – 237
Measuring temporal dynamics of functional networks using phase
spectrum of fMRI data
Felice T. Sun,a,* Lee M. Miller,b and Mark D’Espositoa
a
Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, USA
Section of Neurobiology, Physiology, and Behavior, and the Center for Mind and Brain, University of California, Davis, CA 95616, USA
b
Received 11 February 2005; revised 19 May 2005; accepted 20 May 2005
Available online 12 July 2005
We present a novel method to measure relative latencies between
functionally connected regions using phase-delay of functional magnetic
resonance imaging data. Derived from the phase component of coherency,
this quantity estimates the linear delay between two time-series. In
conjunction with coherence, derived from the magnitude component of
coherency, phase-delay can be used to examine the temporal properties of
functional networks. In this paper, we apply coherence and phase-delay
methods to fMRI data in order to investigate dynamics of the motor
network during task and rest periods. Using the supplementary motor
area (SMA) as a reference region, we calculated relative latencies between
the SMA and other regions within the motor network including the dorsal
premotor cortex (PMd), primary motor cortex (M1), and posterior parietal
cortex (PPC). During both the task and rest periods, we measured
significant delays that were consistent across subjects. Specifically, we
found significant delays between the SMA and the bilateral PMd, bilateral
M1, and bilateral PPC during the task condition. During the rest
condition, we found that the temporal dynamics of the network changed
relative to the task period. No significant delays were measured between
the SMA and the left PM and left M1; however, the right PM, right M1,
and bilateral PPC were significantly delayed with respect to the SMA.
Additionally, we observed significant map-wise differences in the dynamics
of the network at task compared to the network at rest. These differences
were observed in the interaction between the SMA and the left M1, left
superior frontal gyrus, and left middle frontal gyrus. These temporal
measurements are important in determining how regions within a network
interact and provide valuable information about the sequence of cognitive
processes within a network.
D 2005 Elsevier Inc. All rights reserved.
Keywords: Temporal dynamics; Phase-delay; Coherence; fMRI
Introduction
Functional magnetic resonance imaging (fMRI) is a promising tool
for studying neural networks because it provides researchers with the
ability to image whole-brain activity with a fine spatial resolution
* Corresponding author.
E-mail address: [email protected] (F.T. Sun).
Available online on ScienceDirect (www.sciencedirect.com).
1053-8119/$ - see front matter D 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.neuroimage.2005.05.043
(¨3 mm, Kim et al., 1999). While the predominant method of analysis
for fMRI data has been to identify which brain regions are involved in
the performance of a specific task, researchers have recently begun to
take advantage of the multivariate structure of fMRI data to monitor
how brain regions interact to perform a task. One way to characterize
these interactions is through the temporal correlation or coherence
between regions. These types of interactions are referred to as functional
connectivity. In a recent paper (Sun et al., 2004), we presented a spectral
method using coherence of fMRI data to investigate functional
connectivity. Here, we extend the method to detect relative latency
differences between interacting regions.
The standard method of analysis of fMRI data is to estimate the
relationship between the signal in each spatial subunit (voxel) and a
reference function, which is based on the stimulus paradigm. This
parameter estimate indicates the amount of task-related activity in
each voxel. Statistical parametric maps for the entire brain volume
are generated in this way by estimating the activity independently for
each voxel. While there is often an implicit assumption that brain
regions interact during the performance of a task, this method
provides no means of measuring their relationship and therefore
cannot provide a direct statistical measure of the inter-regional
interactions. Moreover, this method is highly dependent on the
reference function, which assumes a specific shape and timing of the
hemodynamic response.
In an earlier paper, we presented a method to address these
shortcomings using coherence, the magnitude component of coherency,
which is a spectral measure of the linear time-invariant relationship
between regions. We demonstrated that in this application, coherence is
superior to correlation because correlation is sensitive to shape and
temporal shift differences in the hemodynamic response, while
coherence is invariant in these differences. Additionally, because
coherence does not require a reference function or model of the
response, the method can be extended to many types of experimental
paradigms, including those with no specific stimuli, such as rest periods
and delay periods. Here, we build upon the method and demonstrate
how the phase-delay, derived from the phase component of coherency,
can be used to investigate shifts in activation timing between regions.
Measuring these temporal delays between regions is an important
component in determining how regions within a network interact and
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F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
for providing valuable information about the sequence of cognitive
processes within a network (Formisano and Goebel, 2003). Combined,
coherence and phase-delay can be used to estimate the degree of linear
relation and the relative difference in latency between brain regions.
In this paper, we apply these methods to fMRI data acquired
during the performance of a bimanual task and during fixation to
examine how the inter-regional dynamics of the motor network
change between task and rest. During the task period, subjects are
visually cued to execute a sequence of key presses that alternate
between the left and right hands. The distributed network of motor
cortical regions considered to be highly involved in such bimanual
tasks includes the primary sensorimotor cortex (S1/M1), the dorsal
premotor cortex (PMd), the supplementary motor area (SMA), and
the posterior parietal cortex (PPC) (for a review, see Swinnen, 2002).
During the rest phase, subjects are asked to maintain fixation and
refrain from practicing or rehearsing motor sequences. While there
are no ‘‘activations’’ during the rest period, fMRI studies have
examined a distributed network of motor regions coupled to the
primary motor cortex at rest. This network, similar to the network
during task, includes M1 (contralateral to the reference region),
SMA, and PMd (Biswal et al., 1995; Xiong et al., 1999).
While previous imaging and lesion studies have provided
substantial information about each region’s functionality in the
motor network, the inter-regional interactions, in particular the
temporal dynamics of the network, are not well understood. The
most extensively studied temporal relationship within the motor
network has been between the SMA and M1. Previous imaging
and electrophysiology studies have consistently shown that the
activity in the SMA increases prior to the activity in M1 during
the execution of a motor task (Ikeda et al., 1992; Kansaku et al.,
1998; Menon et al., 1998; Weilke et al., 2001). However, little is
known about the interactions between any other nodes in the
network during task or how the temporal dynamics compare to
that of the network during rest. Using the methods described
above, we measure the functional coupling and temporal latencies
of all regions in the motor network with the SMA as a reference
region.
A number of studies have suggested that the SMA may play
a central role in coordinating the motor network during the
execution of this task. Electrophysiology and neuroimaging
studies have shown that the SMA is highly active in tasks that
require bimanual coordination, movement preparation, and
motor sequencing (Roland et al., 1980; Kazennikov et al.,
1999; Stephan et al., 1999a; Lee and Quessy, 2003). In
addition, lesion studies have shown that bimanual coordination,
particularly the temporal coordination between the left and right
hands, is impaired in subjects with lesions that affect the SMA
(Laplane et al., 1977; Halsband et al., 1993; Stephan et al.,
1999b). Thus, by selecting the SMA as a reference region, we
can compare our findings to previous measures of the SMA –
M1 latency and gain insight in the dynamics of the bimanual
network.
Methods
Coherency
Magnitude and phase
Coherency, a measure of the linear time-invariant relationship
between two time-series, is a complex-valued function of
frequency. The coherency, R, of time-series, x and y, is defined
as:
fxy ðkÞ
Rxy ðkÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
fxx ðkÞI fyy ðkÞ
ð1Þ
where f xy (k) is the cross-spectrum of x and y at frequency k, and
f xx (k) and f yy (k) are the respective power-spectra of x and y
(Brillinger, 2001). In order to investigate specific features of the
linear time-invariant relationship, we decompose the coherency
function into its magnitude and phase components.
From the magnitude component of coherency, we derive
coherence, which is a measure of the linear association
between two time-series. The coherence, Coh, of x and y is
defined as:
Cohxy ðkÞ ¼ jRxy ðkÞj2 ¼
jfxy ðkÞj2
:
fxx ðkÞI fyy ðkÞ
ð2Þ
This real-valued function of frequency is bounded by 0 and 1,
where 0 indicates that the two time-series have no linear relationship, and 1 indicates that one time-series can perfectly predict the
other in a linear fashion. Note that the use of prediction here is not
limited to causal prediction. In this case, prediction can be forward
and/or backward.
From the phase component of coherency, we derive the phasespectrum, which can be used to determine the relative timing
between two coherent time-series. The phase-spectrum, u, is
defined as:
uðkÞ ¼ arg Rxy ðkÞ ¼ arg fxy ðkÞ ;
ð3Þ
where arg{ } represents the argument. The group delay, s(k),
which is a measure of the linear delay between two time-series is
defined to be proportional to the derivative of the phase-spectrum:
2p4sðkÞ ¼ d
uðkÞ:
dk
ð4Þ
Thus, for a frequency band of interest, we can estimate a linear
phase-delay as the average slope of the phase-spectrum within the
band.
In summary, we extract two real-valued functions, coherence
and phase-delay, from the magnitude and phase components of
coherency. Coherence measures how well one time-series predicts
another and phase-delay estimates the temporal lead or lag between
them.
Estimation
To apply these spectral measures to fMRI data, we limit the
frequencies of interest to those contained in the power-spectrum
of the hemodynamic response, typically (0 – 0.15 Hz) (Aguirre
et al., 1997; Sun et al., 2004). This band of frequencies enables
us to measure the linear coupling and delay for a wide range of
hemodynamic shapes, not just for the canonical HRF. The
estimation of the band-averaged coherence was presented in Sun
et al. (2004) and is summarized in Appendix A of this paper.
Here, we present the estimation of the phase-delay from the
phase-spectrum. From Eq. (3), we see that the phase-spectrum is
simply the argument of the cross-spectrum. In order to minimize
variance of the estimate, we find the cross-spectrum using
Welch’s periodogram averaging method (see Appendix A). The
phase-spectrum estimate is then simply the phase of the crossspectrum estimate. The phase-delay is estimated by calculating
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
the average slope of the unwrapped phase-spectrum. This can
be described as:
Rk
ŝs ¼ 2
d
k1 dk
uðkÞdk
2pðk2 k1 Þ
;
ð5Þ
which reduces to
ŝs ¼ 1 uðk2 Þ uðk1 Þ
:
2p
k2 k1
ð6Þ
In our example, we choose k 1 = 0, and due to the mean
centering of the time-series, u (k 1) is also restricted to zero.
Thus,
ŝs ¼ 1 uðk2 Þ
;
2p k2
ð7Þ
which can be estimated by
~ uðki Þ
ŝs ¼ i
2p ~ ki
;
ð8Þ
i
(choosing i such that k i is within the frequency band of interest)
such that the estimate error is minimized.
In order to determine the goodness of the phase-delay estimate,
we calculate the root mean squared error (RMSE) or residual, the
most common evaluation for an estimated linear fit. The error in
RMSE is defined to be the difference between the phase-spectrum
values and the line whose slope was derived by Eq. (8) (Fig. 1D,
shaded region). For a given false positive rate (alpha), NFFT, and
frequency band, the confidence interval of the phase-delay estimate
is proportional to the RMSE (Draper and Smith, 1981).
229
Notably, because the segments used in Welch’s periodogramaveraging method are mean centered, the value of the phasespectrum at k = 0 is defined to be either zero or p. Theoretically,
the value is p if one time-series is the inverse of the other –
however, in practice, the value may flip due to noise in the signal.
Therefore, in order to determine the most likely value at 0 Hz, we
estimate the slope using both 0 and p as values of the phasespectrum at 0 Hz and select the value that results in the smallest
RMSE for the estimate of the slope.
In Fig. 1, we present an example of these measures using
simulated fMRI data. A model fMRI signal was generated by
convolving an impulse train with the canonical hemodynamic
response function (Josephs et al., 1997). This time-series was
then replicated, and a delay of 1 s was introduced to one of the
time-series (Fig. 1A). Independent Gaussian noise was added to
both time-series, resulting in the two time-series displayed in Fig.
1B. Fig. 1C presents the coherence function of these two timeseries. Because the hemodynamic response function only has
power in the low-frequency band, we focus our analysis on the
0 – 0.15 Hz band (the shaded region in Figs. 1C and D). Within
this band of frequencies, the coherence value is high, indicating
that there is a linear relationship between the signals. The low
coherence values in higher frequencies are due to the added
noise, which spans all frequencies and is independent across the
two time-series. Fig. 1D presents the unwrapped phase-spectrum.
The phase within the low-frequency range exhibits a roughly
linear relationship with frequency. The slope of the fitted line is
the estimate of the linear phase-delay for the low-frequency
range. Here, the slope is calculated to be .9954 for a true delay of
1 s, and the RMSE of the fit is 0.0805, which results in a 95%
confidence interval of T 0.11 s.
Fig. 1. Illustration of coherence and phase-spectra as a measure of the linear time-invariant relationship and delay between two time-series. In this simulation, a
time-series representative of the hemodynamic response, x 0(t), is passed through a linear delay of 1 s (A). Both x(t) and y(t) (the original signal and the delayed
signal, respectively) are corrupted with independent sources of noise to result in the time-series shown in panel (B). (C) The coherence of x and y is near 1
within the 0 – 0.15 Hz band and near zero elsewhere. (D) The phase-spectrum reveals a linear relationship between the frequency and phase within the same
band. The phase-delay estimated from the slope of the phase-spectrum within the band (0 – .15 Hz) is .9954 s with a RMSE of 0.0805.
230
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
Experimental methods
Subjects
Fourteen right-handed subjects (4 female; ages 19 – 29 years;
mean age = 23.5) participated after giving informed consent
according to procedures approved by the University of California.
The subjects reported no history of neurological or psychiatric
disorders and were taking no medications at the time of the study.
Experimental paradigm
In the MRI scanner, five 8-min functional runs were acquired
for each subject using a mixed block/event-related paradigm
(Visscher et al., 2003). The runs were composed of 4 types of
condition blocks, each presented twice in a pseudorandom order.
The condition types were a rest condition and three types of
sequence conditions: ‘‘novel’’, ‘‘learned’’, and ‘‘random’’. The
‘‘random’’ condition is presented in this report as the task
condition. For each of the sequence blocks, five novel 8-item
sequences were presented to the subject; sequences were separated
with a pseudorandomized inter-trial interval (ITI) of 2.2, 4.4, or 6.6
s for a total block length of 58 s.
For each sequence, a series of eight consecutive visual cues
were presented for a duration 725 ms each, resulting in a total
sequence time of 5800 ms. Subjects were instructed to respond to
each visual cue with the corresponding key within the time of the
cue presentation (see Fig. 2). During the rest block, also of length
58 s, subjects were presented with a centered fixation cross. For
this condition, subjects were instructed not to perform or rehearse
any of the sequences. In total, there were 10 task blocks and 10 rest
blocks, which result in 580 s per condition (Fig. 2).
Stimulus presentation
The stimuli were designed and presented using Eprime
presentation software (http://www.pstnet.com). They were then
back-projected onto a custom-designed non-magnetic projection
screen that the subject viewed via a mirror. Responses were
collected using a pair of five-fingered MR-compatible keyboards.
MRI data acquisition
All images were acquired with a 4 T Varian INOVA MR
scanner (http://www.varianinc.com) and a TEM send-and-receive
RF head coil (http://www.mrinstruments.com). Functional images
were acquired using a 2-shot gradient-echo echo-planar image
(GE-EPI) sequence with a repetition time (TR) of 543 ms per half
k-space, an echo time of 28 ms, and flip angle of 20-, resulting in
432 total volumes acquired per run (864 after time-interpolation).
Each volume, covering the top of the brain, consisted of ten 5-mmthick axial slices with a 0.5 mm inter-slice gap. Each slice was
acquired with a 22.4 cm2 field of view with a 64 64 matrix size,
resulting in an in-plane resolution of 3.5 3.5 mm. High
resolution (.875 .875 mm) in-plane T1-weighted anatomical
images were also acquired using a gradient-echo multi-slice
(GEMS) sequence for anatomical localization. Finally, MPFlash
3D T1-weighted scans were acquired so that functional data could
be normalized to the Montreal Neurological Institute (MNI) atlas
space.
Fig. 2. Task diagram. Sequences were presented to subjects using visual cues (see inset labeled Sequence Presentation). The stimuli were presented using four
possible cue positions on either side of the fixation cross for a total of eight possible positions. Here, the horizontal dashes (F – _) are placeholders representing
finger position. During the sequence presentation, an Fx_ appears in place of a dash, indicating to the subject the key to press. The Fx_ changes position every
725 ms, for a total of eight different keys per sequence. During scanning, subjects are presented with five sequences per block, separated by an ITI of 2200,
4400, or 6600 ms. An example of the task condition is diagrammed. During the rest condition, a fixation cross was presented for the duration of the block. Two
blocks of each sequence condition and two blocks of rest were presented in a pseudorandom order during each run.
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
MRI data analyses
Preprocessing. Functional images acquired from the scanner
were reconstructed from k-space using a linear time-interpolation
algorithm across shots of equal ordinal rank to double the effective
sampling rate and corrected for slice-timing skew using temporal
sinc-interpolation. Images were then corrected for movement using
rigid-body transformation parameters and smoothed with an 8 mm
full width at half maximum (FWHM) Gaussian kernel using SPM2
(http://www.fil.ion.ucl.ac.uk/spm).
Univariate analyses. Standard univariate analyses were performed to identify regions with high task-related activity. To
model task-related activity, we used the canonical hemodynamic
response function (HRF) (Josephs et al., 1997) convolved with
independent variables for the onset and duration of each sequence.
Here, the duration of each sequence was modeled as an epoch of
5800 ms. These covariates were entered into the modified general
linear model for analysis. Parameter estimates, reflecting the
percent signal change relative to baseline, were estimated for each
covariate. T statistics of the parameter estimates were used to
functionally identify the reference region and the regions of interest
for the coherency analysis. All voxels selected for the reference
region and regions of interest surpassed a threshold of P < .05,
corrected for multiple comparisons using family-wise error
(Nichols and Hayasaka, 2003).
Coherency magnitude and phase analysis. To identify networks
of functional connectivity, we generated coherence, phase-delay,
and residual (RMSE) maps by calculating the task-specific
coherency between a reference seed located within the SMA
and all other regions in the brain. Anatomical masks of the SMA
were drawn for each subject to include cortex on the medial wall,
dorsal to the cingulate gyrus, and between the central sulcus and
the vertical plane through the anterior commissure (VAC) (Picard
and Strick, 2001). Within the anatomical mask, the reference seed
was chosen as the most significant voxel (based on univariate
statistics) and all surrounding significant voxels within a 6 mm
radius. For all subjects, these seeds were located on the left
(dominant) hemisphere.
The time-series for every voxel were segmented and concatenated with segments of the same condition to generate
condition-specific time-series. Each segment was mean-centered
and windowed using a 4-point split-cosine bell (Bloomfield, 1976).
Windowing reduces spectral leakage from any discontinuities
introduced by segmenting and concatenating the time-series. Both
the task and rest conditions had a total of 1040 data points.
Coherency magnitude and phase estimates were computed
using a fast Fourier transform algorithm implemented in Matlab
6.5 (http://www.mathworks.com). We used Welch’s periodogramaveraging method to estimate the condition-specific coherence of
the seed ROI with all other voxels in the brain (using a 64-point
discrete Fourier transform, Hanning window, and overlap of 32
points). We then generated condition-specific coherence magnitude, delay, and residual maps for the seed ROI using the
estimate of the band-averaged coherence within the 0 – 0.15 Hz
band (Sun et al., 2004). See Appendix A for a review of these
estimates.
To identify changes in the motor network between task and rest,
we contrasted the coherence maps, as well as the phase-delay
maps, across condition. Group level random-effects analyses were
231
performed on the entire volume with a threshold of P < .05, controlling for the family-wise error rate with a nonparametric permutation analysis (http://www.fil.ion.ucl.ac.uk/spm/snpm)
(Nichols and Hayasaka, 2003).
ROI analysis. To complement our whole-brain analysis and
demonstrate a more hypothesis-driven usage of coherence, we
selected several regions of interest (ROIs), defined a priori, based
on evidence from the literature and results from the univariate
analysis: they are located in the primary sensorimotor cortex (S1/
M1), premotor cortex (PM), and posterior parietal cortex (PPC).
For each ROI, an anatomical mask was defined in the subjects’
native space. The S1/M1 mask included cortex adjacent to the
central sulcus, extending anteriorly to the mid-line between the
central and precentral sulcus, posteriorly to the mid-line between
the central and post-central sulcus, and dorsally and ventrally to
include the primary motor hand area (Yousry et al., 1997). The
dorsal PM mask extended from the mid-line between the central
and precentral sulcus, anteriorly to the junction of the superior
frontal sulcus and the precentral sulcus. To exclude ventral
premotor regions, the PM mask included only cortex dorsal to
the inferior frontal sulcus (Picard and Strick, 2001). The PPC mask
included cortex in the superior parietal lobe posterior to and
inclusive of the post-central sulcus, and the cortex adjacent to and
inclusive of the intraparietal sulcus at the junction of the two sulci
(Simon et al., 2002). All masks were non-overlapping. Within each
masked region, ROIs (for both the right and left hemispheres) were
identified as the most significant voxel (based on univariate
statistics) and all surrounding significant voxels within a 6 mm
radius.
Results
Single subject example: coherence, phase-delay, and residual maps
For each subject and each condition, we calculated a
coherency magnitude map, phase-delay map, and residual map
using a reference seed within the SMA. Examples of these maps
are shown for a representative subject in Fig. 3. For the
coherence map (Figs. 3A, D), high values (shown in yellow)
represent regions that are functionally connected to the reference
region. For this representative subject, the coherence map during
the rest condition (Fig. 3A) reveals a network of regions highly
coupled with the SMA; these include predominantly the medial
frontal regions and bilateral M1, but also bilateral PMd and PPC.
During the task period (Fig. 3D), the network of regions highly
coupled to the SMA is similar to those in the rest network;
however, there are higher coherence values in the PPC along the
intraparietal sulcus.
The phase-delay maps indicate the relative latency between the
seed region and all other regions. On this map, blue indicates
regions that are active prior to the seed, and red indicates regions
that lag the seed. Note that the phase-delay map is based on a
bivariate measure and therefore cannot be used to specify an
ordering between two non-seed regions. Here, regions that have a
95% confidence interval of less than T 0.5 s are outlined in white.
These confidence intervals are derived from the RMSE values
displayed in the residual maps (Figs. 3C, F) (see Coherency section
for details). High RMSE values (shown in yellow) indicate that the
relationship between the reference voxel and the region is not
232
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
Fig. 3. Coherence, phase-delay, and residual maps are shown for an individual subject during rest and task. (A) The coherence map during the rest condition.
High coherence values are represented with warm colors, and low coherence values are represented in black. (B) The phase-delay map during the rest
condition. Red values indicate regions that lag the SMA, and blue values indicate regions that lead the SMA. (C) The residual map during the rest condition
displays the value of the root mean squared error (RMSE) of the linear delay estimate. High values are indicated in warm colors, and low values are indicated in
black. Residual values less than .03 correspond to a 95% confidence interval of less than T 0.5 s on the phase-delay map. These regions are outlined in white on
the phase-delay map. Panels (D), (E), and (F) Present the coherence, phase-delay, and residual maps, respectively, for a single subject during the task condition.
modeled well by a linear delay, and therefore the estimation of the
phase-delay is characterized by high variance. For this subject,
during the rest condition (Fig. 3B), the right M1, bilateral PMd,
right PPC, and precuneus (PreCu) lag the SMA, while the left M1
leads the SMA. In contrast, during the task condition, most regions
in the motor network, including bilateral M1, bilateral PMd, and
bilateral PPC lag the SMA, with a region in the left ventral PM
leading the seed.
During the task condition, it is also possible to obtain the trialaveraged response for each region. From the phase-delay map (Fig.
4A), we selected voxels in the lPM (2), lM1 (3), and lPPC (4) that
have increasing latency with respect to the SMA (1). The trialaveraged time-series for these regions (Fig. 4B) reflect these
delays. We also chose a region in the right ventral PM (5) with a
negative phase-delay (that is, region (5) leads region (1)). This lead
is also reflected in the trial-averaged time-series. These trialaveraged responses illustrate that the phase-delay measure is less
sensitive to differences in the shape of the response that may affect
other latency measures such as time-to-peak, or onset (see for
example time-series (3) and (5)), where the shape of the trialaveraged response deviates from a canonical hemodynamic
response; or time-series (2) and (3), which have similar peak
times yet consistently delayed time-courses over their duration).
Group analysis: map-wise
In order to determine whether the phase-delays were consistent
across subjects, we examined the group-averaged phase maps for
both the task and rest conditions (Fig. 5). During the rest condition,
regions that lag the SMA include the middle frontal gyrus, superior
frontal gyrus, precuneus, inferior parietal lobule, and right
precentral and post-central gyri (Fig. 5A). In contrast, during the
task condition, we find that regions with a high latency with
respect to the SMA were motor and premotor regions, as well as
the frontal eye fields. Additionally, we find that left prefrontal
regions lead the SMA in activity during task (Fig. 5B).
To quantify differences in the phase-delay between the task
period and the rest period, we contrasted the phase-delay maps,
using a nonparametric random-effects group analysis, with a
threshold of P < 05, corrected for multiple comparisons (FWE).
A nonparametric method is used here because the null distribution
of the phase-delay values is unknown. Regions that have a
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
233
significant difference in the relative latency with the SMA across
conditions are the left M1 ([ 44 18 64] MNI), the left middle
frontal gyrus ([ 26 16 62] MNI), and medial superior frontal
gyrus ([ 2 40 58] MNI). A change in latency suggests that the
timing of information flow differs from rest to task. Fig. 6 shows the
statistical map with a threshold of P < .005 (uncorrected) to present
the significant regions as well as other regions that follow the same
trend. The red indicates that the relative latency with SMA was
greater during task than during rest. These regions include the left
M1, left PMd, and bilateral PPC. Blue regions are those that have a
decreased latency with SMA during task compared to rest. These
regions include left middle frontal and superior frontal gyrus, the
paracentral lobule and the left superior parietal lobule.
Group analysis: ROIs
In order to investigate the interactions between the SMA and
specific nodes of the motor network, regions of interest (ROIs)
were selected and defined a priori. These ROIs were located in
Fig. 5. Group-averaged phase-delay maps during the rest condition and task
condition. The rest condition phase-delay maps (A) for the SMA seed
reveal regions which, averaged across subjects, lead (shown in blue) or lag
(shown in red) the seed. The task condition phase-delay maps are shown in
panel (B).
bilateral S1/M1, bilateral PMd, and bilateral PPC. The
coherency magnitude and phase-delay values for these regions
are shown in Fig. 7. While all ROIs have significant coherence
with SMA at rest, in all regions of interest, the coherency
magnitude was significantly greater during task than at rest. The
bar graph of phase-delays shows that delays also differ between
task and rest. Specifically, the latency from SMA to the left
PM, left M1, and bilateral PPC is significantly increased during
task compared to rest. The phase-delay bar graph also reveals a
trend from anterior to posterior regions during task. That is, the
latency from SMA increases from PM to M1 to PPC. The
group trial-averaged time-series for these ROIs, showing the
same trend, are plotted in Fig. 7B.
Discussion
Fig. 4. Phase-delay maps for single subject during task condition. This
phase-delay map shows how regions are temporally shifted with respect to
the reference region (SMA), identified by square 1. Regions shown in red
lag the reference region, and regions shown in blue lead the reference
region. All regions shown have a 95% confidence interval of less than T
0.5 s. The trial-averaged time-courses for several voxels in the motor
network are shown in the lower panel (B), with the SMA (1) shown in
black, the left PM (2) in burgundy, the left M1 (3) in red, and the left
parietal (4) in yellow. A right ventral PM (5) region is shown in blue.
We have presented a novel method for evaluating the temporal
relationship between functionally connected regions using phasedelay, a measure derived from the phase component of coherency.
This bivariate spectral measure, in conjunction with the coherence,
allows one to detect the linear relationship and to estimate the
linear delay between regions. We applied these methods to
functional imaging data acquired while subjects performed a motor
task and during fixation to investigate the temporal dynamics of the
same cortical motor network during task and at rest.
Phase-delay as a measure of network dynamics
The phase component of coherency was initially introduced to
the neuroimaging field as a method to measure temporal properties
234
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
Fig. 6. Group statistical maps. These group statistical maps are generated by contrasting the task and rest delay maps. Red/yellow regions are those that have an
increased latency (with respect to SMA) during the task period compared to the rest period. Blue/green regions have an increased latency during the rest period.
of highly localized regions for a periodic block design (Muller et
al., 2003). Here, we extend this method to investigate latencies
across a network of regions and to incorporate a variety of
experimental designs beyond periodic designs. While there have
been a number of other reported methods to measure the delay of
the fMRI response (Menon et al., 1998; Rajapakse et al., 1998;
Liao et al., 2002; Saad et al., 2003), phase-delay differs from these
methods in several important ways. First, the phase-spectrum
measures the overall linear delay between regions and is therefore
not dependent on a specific feature of the hemodynamic response
such as onset or time-to-peak. Because this measure takes into
account the entire time-series and not just a small portion of the
time-series, it may be less susceptible to noise (see also Saad et al.,
2003). Other methods depend upon a model of the hemodynamic
response function (HRF) (Rajapakse et al., 1998; Liao et al., 2002;
Saad et al., 2003). Since the HRF has been shown to be highly
variable across subjects and regions (Aguirre et al., 1998;
Handwerker et al., 2004), estimated delays would be dependant
Fig. 7. ROI analysis of coherency magnitude and delay. (A) These bar graphs show group coherency magnitude and delay values with the standard error bars
for each region of interest (ROI). The ROIs were selected within anatomical masks of M1, PM, and PPC (see Methods for details). The coherency magnitude
graph shows significant coherence in all ROIs and a significant increase in coherence during the task compared to rest for all ROIs, excluding the SMA. The
delay graph reveals a trend of increasing delay from anterior to posterior regions during the task period. Also note the significant delay between the SMA and
right motor regions, whereas no significant delay was found between the SMA and left motor regions except for the left PPC. The group trial-averaged timeseries for these voxels is plotted in panel (B) for the task condition.
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
on how well the model of the HRF fits the data. Furthermore,
because these methods depend on a model of the stimulus
paradigm, they cannot be used to measure latencies for conditions
where no specific timing information is available, such as natural
viewing epochs (Bartels and Zeki, 2004), delay periods during a
memory task (Postle et al., 2000), or rest periods (Biswal et al.,
1995).
Most importantly, however, the coherency method (coherence
and phase-spectrum) allows one to measure the latency between
functionally connected regions. The methods discussed above
measure latency with only an implicit understanding of connectivity.
Explicit measures of such temporal influences were recently
presented by Goebel et al. (2003) using Granger causality. Granger
causality allows one to measure influences between regions and to
determine whether influence is causal (that is, the past events of one
region predict the current event of another region) or instantaneous
(the past and present events of one region predict the current event in
another region). While this promising method incorporates both
linear and non-linear influences, there is no direct measure of the
latency between regions. Therefore, a second technique, such as the
methods described above, or the phase-delay method would be
necessary to quantify the temporal lead between regions with
directed influence as measured by Granger causality.
Network dynamics during task and rest
Using phase-delay, we measured the relative latency differences
between SMA and other motor regions during a bimanual sequence
task. During the motor task, we found latency differences between
SMA and M1 that were consistent with previous studies: Kansaku
et al. (1998) and Weilke et al. (2001) found latency differences of
470 ms and 700 ms, respectively, for an externally cued motor task.
In addition, we examined the latency differences between SMA
and other motor regions such as PM and PPC. We found that
activity in the SMA preceded all of the selected regions of interest
with an increasing latency from anterior to posterior. This early
activation suggests that the SMA may be involved in initiating and
coordinating subsequent motor processes across these regions.
We also examined latencies in the motor regions during the rest
condition. Functional networks during the resting or baseline state
have recently gained interest in the neuroimaging field. Biswal et
al. (1995) first identified a resting state network by measuring lowfrequency correlations of the fMRI signal between the bilateral
primary motor cortices. Following that study, resting state
correlations were also reported between homologous primary
motor, auditory, and visual cortices (Xiong et al., 1999; Cordes
et al., 2000; Lowe et al., 2000), as well as non-primary brain
regions such as the amygdala and hippocampus (Lowe et al.,
1998), and Wernicke’s area and Broca’s area (Hampson et al.,
2002). The high spatial and functional specificity of the correlated
regions suggests that resting-state correlations are not a result of
physiological artifacts such as breathing or cardiac pulsatility but
instead are coupled to neuronal activity and are driven by
spontaneous cognitive or sensory processes.
We examined the motor network during rest using the same
reference region and ROIs as for the task period. Our finding that
the SMA is functionally connected to M1 at rest corroborates
previous findings of correlations between M1 and SMA (Biswal et
al., 1995; Xiong et al., 1999; Cordes et al., 2000). Using phasedelay, we were also able to examine the relative latencies between
regions. To our knowledge, this study is the first to examine fMRI
235
latencies during rest. In contrast to the anterior to posterior trend in
the task period, an intra-hemispheric pattern dominated during the
rest period. M1 and PM regions ipsilateral to the reference region
had no significant phase-delay with respect to SMA, whereas M1
and PM contralateral were delayed by several hundred milliseconds. Our results suggest that during rest, there is a tighter
temporal coupling within hemisphere than across hemispheres.
By contrasting task and rest periods, we were able to identify
regions with significantly different latencies with respect to the
SMA during task compared to rest. A change in latency may
suggest a change in cognitive processing. For example, the relative
latency between two regions may be near zero in one condition and
significantly positive (or negative) in another condition, suggesting
that some serial cognitive processing between the regions may be
required during the second condition and not in the first. We
observed this type of interaction between the left M1 and PM and
the SMA. During the rest condition, there were no significant
differences in the relative latencies of these regions with the SMA;
however, during the task period, activity in the SMA preceded the
activity in these regions, suggesting that sequential processing was
present during the task period. In addition, because latency
measures can be either negative or positive, a significant difference
in latency may mean a change in the direction of information flow.
For example, the relative latency between two regions, A and B,
may be negative in one condition (B leads A) and positive in
another condition (A leads B). This type of interaction was present
in the left prefrontal cortex. From the group-averaged phase-delay
maps, we can see that activity in the SMA precedes activity in the
left PFC during the rest condition, but during the task condition,
activity in the left PFC leads the SMA.
Do relative fMRI latencies reflect hemodynamic delays or neural
delays?
There is still much discussion about whether the measured
fMRI delay between regions is due primarily to hemodynamic
differences or due to neural differences. While this method cannot
distinguish between the two, evidence from our data suggests that
the measured relative latencies are not solely governed by
hemodynamic delays but instead may partially reflect latency
differences of neural responses. First, we find that there are
different latencies during the task period and the rest period. If the
results were driven by hemodynamic latencies, then one would not
expect the latencies to change with task. Second, we find
differences in latency between homologous regions during task
and rest. For example, during the task period, the latency of the
right PPC was significantly shorter than the latency of the left PPC.
During the rest period, the latency of left M1 and PM was
significantly shorter than the latency of the right M1 and PM
regions. One would expect that the physiological parameters that
govern hemodynamic latencies would be comparable within
homologous regions or especially between neighboring regions
with common arterial supply or venous drainage.
Growing evidence from a number of studies has also suggested
that while absolute delays have high variance and a significant
hemodynamic contribution, relative latencies have a neural basis
(for review, see Formisano and Goebel, 2003). Within the primary
visual cortex, Menon et al. (1998) demonstrated that while the
absolute latency of the hemodynamic response in the visual cortex
was highly variable across subjects, the relative timing of the
difference of the hemodynamic response between the left and right
236
F.T. Sun et al. / NeuroImage 28 (2005) 227 – 237
visual cortex corresponded well to hemi-field stimulus offsets as
short as 125 ms. Using a dynamic susceptibility contrast (DSC),
Biswal et al. (2003) was able to decouple the hemodynamic delay
from the task-induced (neural and vasomotor) delay. He found that
during a bilateral finger-tapping task the hemodynamic delay was
over 2 s, while the task-induced delay was 0.7 s. Biswal et al. also
found that, during the task, the measured task-induced delay of
SMA preceded M1 by 560 ms. This is in agreement with our
latency measures between SMA and M1.
In summary, we have presented a novel method using the phase
component of coherency to measure inter-regional latencies of
fMRI data. Together, magnitude and phase components provide a
measure of the dynamic interactions within a network. An
important characteristic of these measures is that they do not
require a reference function or model of the hemodynamic response
and therefore can be used for a broad range of experimental designs.
In this study, we applied coherence and phase-delay to fMRI data
acquired from the motor network during rest and task periods. We
were thus able to examine the temporal dynamics of this network.
We found an anterior to posterior trend in the motor network during
task and an intra-hemispheric left to right trend during rest. These
results provide insight into the shift of information flow and
cognitive processes that occur within the same network during two
very different conditions. While much work remains, such as
resolving hemodynamic from neural latencies, we feel that this
method is a valuable addition to the currently limited set of tools for
investigating inter-regional dynamics.
Acknowledgments
We would like to thank Ben Inglis and Ajay Rao for technical
assistance. This work was supported by the National Institutes of
Health.
Appendix A
To estimate coherency, we first estimated the cross-spectral and
power-spectral densities using Welch’s modified periodogram
averaging method. An estimate of the cross-spectrum can be
obtained by multiplying the Fourier transform of x with the Fourier
transform of y. Since the variance of this estimate is dependent on
the power-spectrum and not on the length of the time-series, we
decrease the variance by averaging the cross-spectrum over several
segments of a shorter length. Therefore, we divide the time-series
into N segments of length T, where each segment is assumed to
approximate stationarity. Each segment is mean-centered and
windowed by a Hanning filter of length T. Windowing is applied
to reduce spectral leakage, which results from segmenting the timeseries. The reduced-variance estimate of the cross-spectrum is:
N
1 X
fˆ ðxyT Þ ðkÞ ¼
X ðT Þ ðkÞI Yn4ðT Þ ðkÞ
N n¼1 n
ð1Þ
where X n(T)(k) is the T-point discrete Fourier transform (DFT) of the
windowed and mean-centered nth segment of x. Likewise, the
power-spectrum estimate is:
N
1 X
fˆ ðxxT Þ ðkÞ ¼
jX ðT Þ ðkÞj2 :
N n¼1 n
ð2Þ
Using these estimates of the cross-spectrum and the powerspectrum, we can estimate coherency:
fˆ ðxyT Þ ðkÞ
R̂ ðxyT Þ ðkÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
R
fˆ ðxxT Þ ðkÞI fˆ ðyyT Þ ðkÞ
ð3Þ
The band-averaged coherence is estimated using band-averaged
estimates of the cross-spectrum and power-spectrum (Andrew and
Pfurtscheller, 1996):
j ~ fˆ xy ðkÞj2
Coh xy ðk̄Þ ¼
Ĉ
k
~ fˆ xx ðkÞI ~ fˆ yy ðkÞ
k
:
ð4Þ
k
We can also derive a statistical measure of the difference
between two coherence measures by using the arc-hyperbolic
tangent transformation, where:
tanh1 jRxy k¯ j tanh1 jRuv k¯ j
ð5Þ
is approximately normally distributed (Rosenberg et al., 1989).
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