Cameron Trapp^{1}, Kishore Vakamudi^{1}, and Stefan Posse^{1,2,3}

We analytically investigate the characteristics of a recently developed sliding window methodology designed for real time analysis of resting state connectivity. The suppression of various types of confounds is investigated both in this analytical framework and numerically. It is shown that this methodology not only acts as a high pass filter and denoiser, but behaves as a model free despiking and confound suppression tool.

Seed-based
resting-state fMRI analysis typically requires regression of movement
effects and signal fluctuations in CSF/white matter to minimize false
connectivity^{1,2}. In this study we present an analytical,
numerical, and experimental characterization of the recently
developed windowed seed based connectivity analysis (wSCA)
methodology^{3}. This technique utilizes a running mean of sliding
window correlations and has been shown to clean up resting state
connectivity maps *in vivo*.
We find that wSCA not only acts as a high pass filter and denoiser,
but also as a model free despiking tool, limiting the effects of
spurious correlations due to a variety of artifacts.

The following equations show correlations for the simple cases of spikes in a single time-course (Eq.1) and both time-courses (Eq.2):

$$r_{1spike} = \frac{N\sum xy - \sum x\sum y+n(Nsy_{i}-s\sum y)}{\sqrt{\bigg{(}N\sum x^{2}-\big{(}\sum x \big{)}^{2}+n(N-1)s^{2}+2ns(Nx_{i}-\sum x)\bigg{)}\bigg{(}N\sum y^{2}-\big{(}\sum y \big{)}^{2}\bigg{)}}} \quad [1]$$

$$r_{2spike} = \frac{N\sum xy-\sum x\sum y+n(N-1)s^{2}+ns(Nx_{i}+Ny_{i}-\sum x- \sum y)}{\sqrt{\bigg{(}N\sum x^{2}-\big{(}\sum x \big{)}^{2}+n(N-1)s^{2}+2ns(Nx_{i}-\sum x)\bigg{)}\bigg{(}N\sum y^{2}-\big{(}\sum y \big{)}^{2}+n(N-1)s^{2}+2ns(Ny_{i}-\sum y)\bigg{)}}} \quad [2]$$

where x and y are the time-courses being correlated, N is the number of time points, w is the window width, s is the spike amplitude, and n is the number of spikes. Equation 3 shows how wSCA modifies these correlations:

$$r_{wSCA}=\frac{1}{N-w+1}\sum_{1}^{N-2w+1}\sigma + \frac{n}{N-w+1}\sum_{1}^{w}r_{spike} \quad [3]$$

where σ is the correlation without confounds. Fig.1 shows how Eq.3 behaves analytically with varying w and s. For spike amplitude approaching zero, there is little difference between correlations with various window widths, including standard seed-based approaches (w=N). As spike strength increases, wSCA displays reduced changes in measured correlations.

In order to further characterize the behavior of this methodology, we take the limit of Eq.3 for large spikes in both time-courses, and small values of w/N:

$$\lim_{\frac{w}{N}\to 0;s\to\infty} r_{wSCA} = \sigma + \frac{nw}{N} \quad [4]$$

In these limits, the over/under-estimation of the correlation due to the spikes decreases linearly with window width and spike number. This behavior is further explored numerically in the following section.

Confound suppression was characterized for a 2-node network using the model in Fig.2. Random signals (variance=1, mean=0) were generated, filtered using a simple lowpass filter, and mixed in order to induce a known correlation. Noise and confounds were added to assess the viability of different methods for reproducing the original correlation. In order to assess the response of the wSCA to confounds, single spikes, randomly distributed multiple spikes (of varying quantity), offsets, and drifts were added to the signals and investigated at various window widths.

For *in vivo*
verification, multi-slab echo volumar imaging^{4} data
were collected in 7 healthy subjects (with informed consent) using a
3T Siemens Trio scanner (TR/TE 136/28 ms, 5 min scan time). Low
frequency (0.01 – 0.2 Hz) and high frequency data (0.5 – 3 Hz,
filtered using a high order FIR bandpass filter with 60 dB stopband
suppression) were analyzed using TurboFIRE (v5.14.5.1)^{5}
with standard preprocessing steps (motion correction, spatial
normalization, 8mm3
Gaussian spatial smoothing), and a 15s sliding window. Running means
of correlation values were computed across the windows.

Results

*Simulations:*
Suppression of all confound types increased with
decreasing window width (Fig.3). For small numbers of spikes
(n/N<0.01) and small window widths (w/N<0.23), we were able to
replicate the behavior predicted in Eq.4. As window width and spike
number increase, however, the effectiveness of this methodology was
limited, as confounds dominate a larger percentage of
windows, and the linearity breaks down (Fig.4).

*In Vivo:*
Results show this methodology removes false positive connectivity in
white/gray matter when mapping major resting state networks at both
low/high frequency (Fig.5). This methodology shows significant
improvements to inherently low signal^{6}, high frequency
correlation maps, allowing for consistent observation of auditory
networks across multiple subjects. Additionally, wSCA is shown to be
comparable and complementary to regression techniques in a concurrent
study^{7}.

As shown, wSCA
allows for computationally efficient high pass filtering, noise
reduction, and confound suppression. We have shown the benefits of this methodology that, while mathematically intricate, are
analytically and numerically quantifiable. While traditional
despiking methods^{8-10} provide stronger spike suppression,
they require the ability to reliably identify isolated signal events
and do not account for more complex confound waveforms. Therefore, wSCA is
particularly advantageous for mapping high frequency connectivity
where artifact characteristics are less well established.
Additionally, this approach is well suited to single subject and real
time applications, as it is computationally efficient and does not
require a-priori knowledge of signal characteristics. Current efforts
are focused on extending this methodology to probe the dynamics of
resting state connectivity, which are of interest for characterizing
neurological and psychiatric diseases.

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3-D
representation of how the analytical form of the wSCA methodology
(Eq.3) behaves for varying window width and spike strength for a
spike in a single time-course (top) and both time-courses (bottom).
There is little different between varying window widths for small
spikes, however, as spike strength increases, smaller windows
maintain the original correlation better than larger windows and
traditional seed based analysis (w=N).

Flow chart showing the data simulation and analysis pipeline. Random signals are initially generated (X1 and X2) with a mean of 0 and variance of 1. A simple autoregressive filter is applied if frequency dependence is of interest, and a mixing term is introduced to induce a correlation in the signals. Gaussian noise is added to both signals and the correlation is calculated for reference. Confounds of interest are introduced, and the correlation is once more calculated. Finally, the wSCA methodology is implemented to measure the 'meta-mean' correlation.

Numerical simulations on the effects of window width on the suppression of various confound types, including single spikes, multiple spikes, signal drifts, and a signal offset. The original correlation is shown in blue, while the confounded signal is shown in red. The shaded regions show the standard deviation over multiple iterations of the simulation. In general, smaller windows give a higher degree of confound suppression, however, the exact shape of the graphs depend on the type of confound and it's location within the time-course.

Numerical simulation dependence on the relative window width (left) and relative spike number (right) in comparison with the theoretical limits in Eq. 4. The behavior of the wSCA methodology with respect to relative window widths behaves roughly linearly in the regime where w/N < 0.23 and the slope is comparable to the theory. The behavior with respect to relative spike number behaves linearly below 1% spikes, however, the slope is slightly lower than theory.