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Edited by: Alard Roebroeck, Maastricht University, Netherlands

Reviewed by: Benito de Celis Alonso, Meritorious Autonomous University of Puebla, Mexico; Miguel Castelo-Branco, Coimbra Institute for Biomedical Imaging and Translational Research, Portugal

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this work fMRI BOLD datasets are shown to contain slice-dependent non-stationarities. A model containing slice-dependent, non-stationary signal power is proposed to address time-varying signal power during BOLD data acquisition. The impact of non-stationary power on functional MRI connectivity is analytically derived, establishing that pairwise connectivity estimates are scaled by a function of the time-varying signal power, with magnitude upper bound by 1, and that the variance of sample correlation is increased, thereby inducing spurious connectivity. Consequently, we make the observation that time-varying power during acquisition of BOLD timeseries has the propensity to diminish connectivity estimates. To ameliorate the impact of non-stationary signal power, a simple correction for slice-dependent non-stationarity is proposed. Our correction is analytically shown to restore both signal stationarity and, subsequently, the integrity of connectivity estimates. Theoretical results are corroborated with empirical evidence demonstrating the utility of our correction. In addition, slice-dependent non-stationary variance is experimentally determined to be optimally characterized by an inverse Gamma distribution. The resulting distribution of a voxel's signal intensity is analytically derived to be a generalized Student's-

Functional MRI (fMRI) connectivity analysis utilizes blood oxygen level dependent (BOLD) data to determine how spatially remote brain regions cooperate when a subject is completing a task or is simply at rest. BOLD data is noisy, being impacted by both equipment-related noise, such as coil interference (Bandettini et al., _{0} fluctuations exhibit slice dependence due to factors such as respiration effects and cardiac effects (de Moortele et al.,

Connectivity analyses in fMRI frequently employ a linear Gaussian model (LGM) to estimate dependence between a seed timeseries and brain voxels (Goebel et al.,

Non-stationarity of BOLD timeseries has been modeled in an effort to resolve the loss of integrity associated with a stationarity assumption imposed on non-stationary data. Lund et al. (

The necessity for modeling non-stationarity of BOLD timeseries has often been argued with visual evidence (Calhoun et al.,

More recently, there has been increased interest in examining the non-stationarity of functional connectivity in resting state fMRI, such as Hindriks et al. (

In this paper, we correct fMRI connectivity estimates for non-stationarity induced by time-varying signal power during BOLD data acquisition. Experimental fMRI data is empirically shown to contain time-varying slice variance that is optimally characterized by an inverse gamma distribution. The resulting distribution of voxel intensity values is analytically determined to be a generalization of Student's-

This paper is organized as follows. Theoretical results are detailed in section 3.1, incorporating a description of the non-stationary signal model, with a derivation for the distribution of voxel intensity for signals with time-varying variance (subsection 3.1.1), and the consequent expression for correlation between non-stationary timeseries with proposed correction (subsection 3.1.2). Empirical and experimental methods are described in section 2, followed by a description of results in section 3 and discussion in section 4.

We examine the impact of non-stationary signal power multiplicatively applied to underlying stationary signals during acquisition. In order to avoid confusion, we introduce a nomenclature to describe the multiple sources of signal variance. Underlying stationary timeseries are associated with a constant variance, which will be referred to as stationary signal variance. Time-varying signal power, modeled as a slice-dependent, non-stationary variance, will be referred to as slice dependent non-stationary variance or, equivalently, as time-varying signal power. Sample variance of voxel intensities within a slice will be referred to as sample slice variance and, similarly, sample variance of voxels within a specific tissue type will be referred to as sample tissue variance. Finally, variance of sample correlation will be explicitly identified as such.

MATLAB was used to generate simulation datasets of 2, 000 timeseries pairs. Denote the constituent timeseries in each pair by _{x} and μ_{y}, respectively, and drawn from the range [−100, 100]; and stationary signal variance of _{x} and σ_{y}, respectively, both randomly selected from the range (0, 10]. For each dataset, a slice dependent non-stationary variance was randomly generated for timeseries in slice

Slice dependent non-stationary variance was estimated for each slice by calculating sample variance spatially, across the 2, 000

In order to determine the impact of both time-varying signal power, and our subsequent correction, on autocorrelated signals, we followed the same technique as for the white signals above, but generated the signals using a vector autoregression (VAR) model. In this case,

where ν_{x} and ν_{y} allow for timeseries with non-zero mean, _{p}

We generated random VAR models using randomly selected model order, _{x} and ν_{y} randomly set so that μ_{x} and μ_{y} are in the range [−100, 100], where _{p} matrices were specified via random generation of the poles of each VAR(

We followed the same precision correction as for the white signals, in weighting the timeseries with time-varying variance according to whether the signal was allocated to slice

Two distinct datasets were collected. For the first dataset, two healthy subjects were scanned on a 3T Siemens Tim TRIO MRI scanner using two different procedures: (1) resting state BOLD echo planar imaging, and (2) a motor task performance BOLD EPI. For both procedures, we collected 219 volumes for each participant (repetition time = 1, 600 ms; echo time = 20 ms; flip angle = 90°; 24 trans-axial slices, each 5.5mm thick, matrix = 64 × 64, field of view [FOV] = 200 × 200 mm^{2}; acquisition voxel size=3.125 × 3.125 × 5.5 mm^{3}). The motor task was a block design containing alternating periods of 30s each. During the active period, the subjects were instructed to press either the left or right button, according to an arrow displayed on the screen that changed randomly every second. During the baseline period, subjects were instructed to focus on a crosshair at the screen's center point. This dataset will be referred to as dataset 1.

For the second dataset three additional subjects were scanned on a 3T Siemens Tim TRIO MRI scanner using resting state BOLD echo planar imaging. We collected 240 volumes for each subject (repetition time = 2, 000 ms; echo time = 20 ms; flip angle = 90°; 91 trans-axial slices, each 5 mm thick, matrix = 91 × 109, field of view [FOV] = 200 × 200 mm^{2}; acquisition voxel size=3.125 × 3.125 × 5.5 mm^{3}). This dataset will be referred to as dataset 2.

All images were motion corrected and smoothed using a 2D isotropic Gaussian 6mm kernel, applied using MATLAB. A conventional general linear model (GLM) activation analysis (Friston et al.,

For each subject, slice dependent non-stationary variance was estimated by calculating the sample slice variance of voxel intensities in each slice at each time point. The best-fit distribution for the sample slice variance of each slice was determined by calculating the maximum likelihood estimates (MLEs) for distribution parameters and identifying the distribution that attained the minimum negative log-likelihood amongst the set of Weibull, Gaussian, Gamma, Inverse Gamma, Student's-t, Exponential, Log-normal, Laplace, and Rayleigh distributions. The Wilcoxon signed-rank test was used to determine if sample slice variances of different slices were drawn from distributions with equal mean (Wilcoxon,

To establish whether tissue type is a significant determinant of slice dependent variance, tissue masks were generated for each subject using the FSL software suite. For each image volume, the sample tissue variance across all voxels associated with a specific tissue type was calculated, yielding a timeseries of volume dependent variance for each tissue, including gray matter, white matter, and cerebrospinal fluid (CSF). Given that sample tissue variance and sample slice variance were necessarily generated from a common set of voxels, to establish whether tissue type is a significant determinant of slice-dependent variance, whole-volume variance was calculated at each time point and regressed out from both the tissue variance timeseries and the slice variance timeseries. The resulting tissue variance timeseries were correlated with slice variance timeseries, and tested for significance using a Student's

Corrected voxel timeseries were formed by weighting voxel samples using the slice dependent variance, as an estimate of time-varying signal power. Seed-voxel correlation maps were generated with a RMC seed for datasets with and without applying the correction for time-varying signal power. Correlation estimates were significance tested using Fisher's z-transformation (Fisher,

In the acquisition of EPI BOLD data, two-dimensional slices are usually obtained serially, on a millisecond time scale. This can result in changes in the equipment and subject environment, ultimately modifying signal characteristics between slices, resulting in non-stationarity of voxel timeseries (Calhoun et al.,

fMRI analyses typically assume stationarity of voxel timeseries and therefore it is desirable to determine a correction to restore stationarity. We propose a model of non-stationarity for resting state BOLD data in which each two-dimensional slice variance is characterized by a distinct time-varying signal power. The distribution of voxel intensity is analytically derived for this model and the impact of the non-stationarity on correlation is determined. Crucially, a simple correction is proposed that is analytically shown to restore stationarity and correct correlation values so that they reflect the underlying linear dependence between voxel timeseries.

We begin by proposing a model for the measured, non-stationary signal in a voxel

where _{m, t} is a time-varying multiplicative variance process associated with slice

Here α_{m} and β_{m} are slice specific shape and scale parameters, respectively. Consequently, slice dependent non-stationary variance has a mean of _{m} > 0, α_{m} > 2.

We now determine the distribution of non-stationary voxel intensity, _{m, t} is unknown (Equation 2). The resting state stationary component,

As _{m, t} are independent random variables, at each time point a voxel in slice

The distribution of voxel intensity as derived in

The voxel intensity distribution (Equation 6), is an instance of the generalized Student's-_{t} = 0, ν_{t} = 2α, and

Correlation-based measures assume stationarity of timeseries and are known to be susceptible to spurious significance if the stationarity assumption is violated (Granger and Newbold,

where

and by the Cauchy-Schwarz inequality (Cauchy,

Consequently, time-varying signal weights necessarily reduce the magnitude of correlation between non-stationary signals, where the extent of the reduction depends on the similarity of the time-varying weight processes. It is important to note that it is feasible for the time-varying processes to modify the sign of the correlation in the event that the weights are drawn from the set of real numbers. In such a case, correlation may appear as anti-correlation and vice-versa.

The variance of sample correlation was found in

Since κ ≤ 1, the variance of sample correlation between non-stationary timeseries will be larger than that of stationary signals. Consequently, estimates of correlation will be more prone to spurious significance.

Several fMRI connectivity methods, such as correlation-based measures (Friston et al.,

Let

where _{m}, such that

Observe that the corrected voxel timeseries is now stationary, but with a variance that differs by a constant factor from that of the underlying stationary signal variance component,

Correlation between corrected timeseries is given by

where the first line is a direct result of the definition of corrected timeseries in Equation (12). This demonstrates that correlation between timeseries with time-varying signal power, corrected for this source of non-stationarity, is equivalent to correlation between the underlying stationary voxel timeseries. The impact of the non-stationary signal weighting has been removed and the intrinsic linear dependence has been recovered, as required.

Simulated datasets were used to examine the impact of spatially dependent non-stationary variance on sample correlation, as well as the utility of the timeseries correction proposed in Equation (12). Pairs of correlated, stationary timeseries were generated using the methods described in section 2.1, and sample correlation calculated between each pair. A histogram of the correlation estimates is shown in

The non-stationary timeseries were corrected by an estimate of slice variance at each time point, according to the simple correction proposed in Equation (12). Sample correlation was calculated between corrected timeseries pairs (

The impact of the autocorrelation of fMRI timeseries on our proposed correction was examined. Pairs of stationary VAR timeseries were generated according to Equation (1), and one timeseries within each pair allocated to slice

As shown in

For each subject, and for all slices within each subject's experimental dataset, the inverse gamma distribution best characterized the distribution of sample slice variance estimates of voxel intensities within the slice, as shown in

Comparison of MLE distribution fits to sample slice variance. Goodness of fit was evaluated using negative log likehood. The optimal fit is identified by a solid circle.

Distribution of sample slice variance of voxel intensities within each slice. Distribution of sample slice variance values within each slice (dashed) and MLE inverse gamma distribution fit to slice variance (solid), across slices.

The probability distribution of Gaussian intensity values with an inverse gamma, non-stationary variance was analytically derived to be a generalization of a Student's-

Three parameter Student's-

Our theoretical results in section 3.1.1 were derived using a model of slice dependent, time-varying variance. This model differs to that of Diedrichsen and Shadmehr (

A more formal examination of dissimilarity was performed using the Wilcoxin rank-sum test to assess the equality of sample slice variance distributions. The test was applied to each slice pair;

Test of distribution dependence between sample slice variance processes. The Wilcoxon signed-rank test was applied to each slice pair for each subject to identify dependence between sample slice variance distributions.

To determine if the slice dependence of the non-stationary weights was induced by differing ratios of tissue types in each slice, the linear association between slice time-varying weights, and each sample tissue variance timeseries, was examined. For all subjects, <10% of slices showed significant correlation with the sample tissue variance timeseries, and none of the inverse gamma parameters were significantly associated with the proportion of tissue type within each slice.

Stationarity of slices was evaluated by applying the augmented Dickey-Fuller (ADF) stationarity test to each slice. For all subjects, more than 70% of the sample slice variance timeseries were found to be non-stationary. None of the inverse gamma parameters were significantly associated with the proportion of tissue type within each slice.

Non-stationary slice variance is problematic when methods that assume stationarity are employed to estimate brain connectivity. The theoretical results detailed in section 3.1.1 model non-stationarity derived from temporal changes in signal power. The impact of non-stationarity on correlation was analytically established in section 3.1.2, in which it was shown that a non-stationary weighting alters the expected value of sample correlation, which is a critical result for fMRI connectivity research. In this section, the theoretical results are corroborated with empirical evidence from both simulated and experimental data.

The simple timeseries correction (Equation 12), was applied to the experimental resting state dataset introduced in section 2.2 to examine its ability to restore stationarity of fMRI data. The weighting of correlation anticipated by the theory as a consequence of the non-stationary signal power (Equation 8), was found to be in close agreement with those found experimentally. For example, for voxels co-located with the RMC, the weighting anticipated by the sample slice variance was 0.8 for subject 1, while experimentally the change was 0.81. Furthermore, the scatterplot between experimental correlation estimates before and after correction,

Scatterplot of correlation estimates between the RMC and brain voxels (slice 19), before correcting for non-stationarity, against correlation estimates acquired after applying the correction to restore stationarity. A line of best fit of pre-correction correlation, to post-correction correlation, values is shown (dashed), as well as the identity line denoting no change, to aid comparison (solid).

The utility of our correction is demonstrated in

Slice maps of significant RMC-seed voxel correlation for uncorrected timeseries, and timeseries corrected for non-stationary signal power, tested using Fisher's z-transformation.

Connectivity analyses of resting state fMRI data typically require stationarity of timeseries. However, this assumption is violated by non-stationarity derived from noise sources such as head movement, or by variable signal power originating from sources such as inhomogeneous RF amplification (Tanase et al.,

We propose a model of non-stationarity motivated by time-varying signal power during acquisition. Since voxels within a slice are acquired simultaneously in 2D EPI acquisitions, we examined sample slice variance, describing temporal changes in the variance of voxel intensities within a slice. We established empirically that sample slice variance is non-stationary and can be well modeled by an inverse gamma distribution, in agreement with Hansen et al. (

While signal non-stationarity can come from many potential sources, each impacting a different spatial extent, we chose to focus on slice specific variance due to striking observations of this in experimental results. Conversely, a voxelwise model of non-stationarity is difficult to substantiate and implement due to the need for sample variance to be calculated from multiple samples over space or time. Further, there is an infinite number of ways to separate the voxel's signal variance into spatial and temporal origins, that in the absence of prior or secondary information renders the approach intractable.

Our slice-specific method does not preclude the removal of regionally specific non-stationarity as a subsequent pre-processing step. The combination of the two would be confounded, as a given brain region may cross multiple slices, and would then contain voxels acquired with varying levels of signal power that should be removed first. The secondary analysis and removal of regional non-stationarity is beyond the scope of this paper, and research we intend to pursue in the future.

In establishing that the non-stationarity of voxel timeseries has a spatial dependence, it was shown that the time-varying weights for each slice are drawn from statistically significant distributions. We further established that there does not appear to be evidence for tissue specificity. Given that the slice dependence is dispersed across all tissue types, scaling timeseries by sample slice variance will not destroy dynamic connectivity. Consequently, additional methods can be employed to model time-varying connectivity.

Non-stationary signal power applied to voxel signals during acquisition has the propensity to change the distribution of the intensity values (Granger and Newbold,

Having modeled the non-stationary distribution of fMRI voxel intensities, we subsequently considered the impact of non-stationary slice variance in the context of connectivity analyses, obtaining an analytic expression for correlation between non-stationary signals. Sample correlation between non-stationary timeseries was shown to derive from correlation between the underlying stationary signals, scaled by a function of the time-varying processes. Importantly, the scale factor was shown to have an absolute value less than unity. Consequently, slice dependent non-stationary variance processes diminish the strength of linear association between voxels. Furthermore, in the case where time-varying weights are drawn from the set of real numbers, the non-stationary process has the propensity to change the sign of association so that a positive correlation can become negative and vice-versa.

We derived the variance of sample correlation estimates between non-stationary timeseries. We demonstrated that time-varying power increases the variance. The theoretical results were corroborated with empirical results, which demonstrated a discernible change in the variance of sample correlation between non-stationary timeseries. This is significant as, if not accounted for, it will result in an increase in the incidence of spurious correlation. This may explain the large amount of correlation found in some fMRI datasets before correction, and the reduced amount of activation following correcting for time-varying signal power.

While this paper considers non-stationary signal power in resting state data, the result can equally be applied to activation data. The correction procedure has the propensity to change the signal means, however relative differences between signal means both spatially, in different brain regions, and temporally, will be preserved, and the linear relationship between the signal and the task will be preserved as the signal mean is an offset factor. This is the subject of our current work, and our preliminary testing confirms this supposition.

To address the impact of non-stationarity signal power on fMRI connectivity, a simple precision correction was proposed, which was analytically demonstrated to restore signal stationarity. The correction generated stationary signals that differ in value from the underlying stationary signals by a constant scale factor, which is inconsequential for correlation since it is insensitive to scale. The efficacy of our correction was validated using simulation data; after applying our proposed correction the mean value of sample correlation coincided with the underlying correlation between the stationary signals. Experimental data showed reduced outlier connectivity and the number of significant connections reduced in accordance with the smaller variance of stationary signals. Furthermore, the change in mean correlation agreed with that anticipated by the analytically derived expression for correlation between non-stationary signals. Note that removal of time-varying signal power will increase estimates of underlying linear connectivity since the time-varying weights diminish sample correlation estimates between timeseries.

The scope of current work has been to identify slice dependent non-stationary variance and propose a statistical correction for its presence in fMRI connectivity analyses. The 2D EPI datasets examined were both acquired in the standard axial slice direction. In order to probe the physical origins of non-stationarity, it will be of interest in future work to alter the slice direction to coronal, sagittal, or oblique slices. With B0 inhomogeneities more concentrated at the base of the frontal lobe, this is expected to change the signal power distributions when distributed across slices rather than confined within certain slices. Investigation into slice ordering, contiguous vs interleaved, and a comparison with 3D acquisitions are all potential ways in which the origins of the non-stationarity can be scrutinized. Further, it is of interest to explore Simultaneous Multi-Slice fMRI approaches that are increasingly popular for the increased temporal resolution they offer. The concurrent acquisition of multiple slices and un-aliasing of the slices via the receive array coils will likely render this a challenging statistical inference problem, but as with the current work, this is vital for the statistical integrity of the resultant brain connectivity maps.

Linear fMRI connectivity methods typically require stationarity of timeseries. We have demonstrated that resting state fMRI data do not meet this requirement but rather have time-varying, slice-dependent, variance that is well-modeled by an inverse gamma distribution. The resulting voxel intensity distribution under this model is a generalization of a Student's-

Requests to access these datasets should be directed to

All human imaging was conducted with the approval of the University of Melbourne Human Research Ethics Committee, and volunteers gave an informed consent prior to the experiment. The patients/participants provided their written informed consent to participate in this study.

LJ made the initial proposal to model variance using a time-varying inverse Gamma distribution. LJ and DG provided supervision and guidance on maths and data processing techniques. CD did the maths derivations and primarily implemented the Matlab coding and fMRI data processing. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The content of this manuscript has been published as part of the thesis of Davey (

The Supplementary Material for this article can be found online at:

autocorrelation function

Akaike's information criterion

autoregressive

autoregressive exogenous input

Bayesian information criterion

blood oxygen level dependent

conditional directed Granger causality

conditional Geweke's Granger causality

conditional instantaneous Granger causality

cerebrospinal fluid

dynamic causal modeling

directed Granger causality

electroencephalography

echo-planar imaging

finite impulse response

Functional MRI

false discovery rate

final prediction error

family-wise error rate

Granger causality

Geweke's Granger causality

general linear model

goodness of fit

Hannan and Quinn's information criterion

hemodynamic response function

independent and identically distributed

independent component analysis

instantaneous Granger causality

infinite impulse response

local field potential

linear Gaussian model

linear structural relations

left hemisphere primary motor cortex

least squares estimation

matrix laboratory

multiple correlation

minimum description length

magnetoencephalography

maximum likelihood estimation

magnetic resonance imaging

order selection criteria

partial autocorrelation function

partial correlation

principal component analysis

positron emission tomography

probability density function

partial directed Granger causality

positron emission tomography

radio-frequency

right hemisphere primary motor cortex

receiver operating characteristic

region of interest

residual sum of squares

structural equation modeling

spatial ICA

structural MRI

signal to noise ratio

singular value decomposition

temporal ICA

repetition time

temporal SNR

vector autoregressive

white Gaussian noise.