Mary Kociuba^{1} and Daniel Rowe^{1,2}

^{1}Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, WI, United States, ^{2}Biophysics, Medical College of Wisconsin, Milwaukee, WI, United States

### Synopsis

**The
standard in fMRI is a magnitude-only statistical analysis of the
data, despite evidence of task related change in the phase time-series. This study demonstrates the increased sensitivity and specificity of implementing complex-valued
correlation models for low magnitude and phase contrast-to-noise ratio (CNR)
values in fMRI data sets.**### Purpose & Background

The
standard in fMRI is to discard the phase before the statistical analysis of the
data, despite evidence of task related change in the phase time-series. With a
real-valued isomorphism representation of Fourier reconstruction, correlation
is computed in the temporal frequency domain with complex-valued (CV)
time-series data, rather than with the standard of magnitude-only (MO) data. The
purpose of this study is to demonstrate the increased statistical power of
implementing complex-valued correlation models for low magnitude and phase
contrast-to-noise ratio (CNR) values in fMRI data sets.

### Methods

_{,}
To demonstrate the increased power of the
CV correlation over MO correlation in functional MRI studies, a MATLAB
simulation is run with a varying degree of magnitude contrast-to-noise ratio
(CNR_{ρ}) and phase
contrast-to-noise ratio (CNR_{ϕ}).
The SNR is defined as the mean magnitude signal over the standard deviation of
the noise in the time-series, SNR = *ρ/σ*.
For the CNR, the amplitude is defined as the difference between the baseline
signal and the task related change in the signal for the magnitude and phase
components of the time-series, *A*_{ρ} and
*A*_{ϕ}, so CNR_{ρ} = *A*_{ρ}/σ and CNR_{ϕ}
= *A*_{ϕ}/(σ/ρ). Since the standard
deviation of a phase-only time-series is *σ/ρ*,
the CNR_{ϕ} is proportional
to the SNR. Typically in fMRI
studies, the task related signal change in the magnitude corresponds to a 1-2%
signal change, and the phase CNR_{ϕ} is
approximately π/36 [1]. To compare MO
and CV correlations, two 96 × 96 surfaces are
generated with 600 time-points and standard normal random noise added to the
real and imaginary channels. As visualized in Fig. 1, each voxel has an SNR
between 0 and 50, and a task generated to represent a CNR_{ρ} between 0 and 1, and a CNR_{ϕ} between 0 and π/36. The MO and CV correlations are
computed between the two time-series in each surface with equivalent parameter
settings, so there is a 96 × 96 corresponding matrix
for MO and CV.

Note, the CV correlation
is computed in the Fourier frequency domain. The real-valued isomorphism representation of
the inverse Fourier reconstruction operator is denoted Ω_{T}[2]. So, the voxel time-series for voxel *j*,
*y*_{j}, is reconstructed from
the temporal frequencies,* y*_{j }= Ω_{T}*f*_{j}, and demeaned. Using the same
notation for voxel *k*, the spatial
covariance between voxels *j* and *k* is represented as cov(*y*_{j},*y*_{k} )=1/(2*n) y*_{j}^{T}*y*_{k} = 1/(2*n*)(Ω_{T}f_{j})^{T} (Ω_{T}f_{k})=
1/4(*f*_{j}^{T}f_{k})
. Note, the covariance
corresponds to the *jk*^{th}
entry in the voxel spatial covariance matrix, Σ, and is presented as
summation of the overlap of the real and imaginary components of temporal
frequencies. The diagonal matrix of spatial variances, *D*, is used to construct the spatial correlation,
*R* = *D*^{-1/2} Σ *D*^{-1/2}
. To compare the
correlations between the MO and CV models, the Fisher-*z* transform, *z,* is computed and plotted for each time-series correlation, *r*, as*
z* = ½ log((1+*r)* /(1-*r)*).

### Result

The simulation
demonstrates the power of CV correlations over MO correlations for low
magnitude contrast-to-noise time-series. Fig. 2 is the Fisher-

*z* transform for the MO and CV
correlations is computed for the surfaces generated with the parameters
described in Fig. 1. Including the phase in the CV correlation calculation
increases the sensitivity of the correlation value, as illustrated by comparing
the top left corner of the Fisher-

*z
*maps. The additional task related information of the phase time-series improves
the correlation calculation at low magnitude CNR values.

### Conclusion

The
simulation comparing decreasing CNR magnitude and phase values, illustrates the
statistical power of CV correlations with a specific advantage for low CNR fMRI
time-series. Including both magnitude and phase in spatial correlation
computations more accurately identifies correlated task-activated regions in
data sets with high noise variance. Identifying these correlations are
important to preserve the signal of interest, in order to reduce the influence
of processing or reconstruction induced correlations [3] on the statistical
analysis of fMRI data.

### Acknowledgements

No acknowledgement found.### References

1. Menon et al., MRM
2002. 2. Rowe, NeuroImage 2005. 3. Nencka et al., J. Neurosci. Meth. 2009. 4. Cordes et al., J. of
Am. NeuroRadiology 2000. 5. Cordes
et al., J. of Am. NeuroRadiology 2001.