Accurate modeling of temporal correlations in rapidly sampled fMRI time series

Abstract

Rapid imaging techniques are increasingly used in functional MRI studies because they allow a greater number of samples to be acquired per unit time, thereby increasing statistical power. However, temporal correlations limit the increase in functional sensitivity and must be accurately accounted for to control the false-positive rate. A common approach to accounting for temporal correlations is to whiten the data prior to estimating fMRI model parameters. Models of white noise plus a first-order autoregressive process have proven sufficient for conventional imaging studies, but more elaborate models are required for rapidly sampled data. Here we show that when the "FAST" model implemented in SPM is used with a well-controlled number of parameters, it can successfully prewhiten 80% of grey matter voxels even with volume repetition times as short as 0.35 s. We further show that the temporal signal-to-noise ratio (tSNR), which has conventionally been used to assess the relative functional sensitivity of competing imaging approaches, can be augmented to account for the temporal correlations in the time series. This amounts to computing the t-score testing for the mean signal. We show in a visual perception task that unlike the tSNR weighted by the number of samples, the t-score measure is directly related to the t-score testing for activation when the temporal correlations are correctly modeled. This score affords a more accurate means of evaluating the functional sensitivity of different data acquisition options.

Keywords: accelerated acquisitions; functional MRI; functional sensitivity; prewhitening; temporal correlation.

Figure 1

Covariance components of the FAST model with p = 9 [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 2

Proportion of voxels showing temporal correlations in the residuals of the GLM (without (a) and with (b) physiological regressors included in the design matrix). Each data point is the average across all the participants. The Ljung‐Box Q test is performed with 11 different models for temporal correlations including the conventional AR(1)+ white noise model (AR), no temporal correlations (No), and the FAST model with p varying from 1 to 9. The significance level is defined as p < .05 after false discovery rate correction [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 3

Power spectra of the residuals after fitting the GLM on the four time‐series (TR = 2.8, 1.4, 0.7, and 0.35 s) acquired on one exemplar participant. The design matrix did (bottom line) or did not (top line) include the physiological regressors. Two different models for temporal correlations were tested: AR(1) + white noise (blue curve) and FAST with 18 components (red curve) and compared to the residuals obtained without prewhitening (“No Model,” yellow curve) [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 4

Relationship between the standard precision (inverse standard error) of the parameter of the constant term and the number of samples. Temporal correlations were modeled with 9, 18, and 27 components for TR = 1.4 s; with 12, 18, and 27 components for TR = 0.7 s; and with 15, 18, and 27 components for TR = 0.35 s. The coefficient of determination R ² of the linear regression averaged across participants is indicated for each TR and model. Each color represents one participant [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 5

The resulting free energy for each model (abscissae), each participant (colors), and each TR: (a) 2.8 s; (b) 1.4 s; (c) 0.7 s; (d) 0.35s [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 6

(a) Simulated relationship between the t‐score testing for the task and the weighted tSNR (tSNR_w, blue cross) or the t‐score testing for the mean signal (t ₀, red circle). By increasing the number of samples while decreasing the sampling interval, the weighted tSNR overestimates the increase in functional sensitivity, whereas the t‐score testing for the mean is directly proportional to the functional sensitivity. Indeed the t‐score testing for the mean signal and the weighted tSNR are not proportional (b), instead the t‐score testing for the mean tends to increase less rapidly than the weighted tSNR as the sampling interval decreases [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 7

Relationship between the 10% highest t‐scores testing for the task (scenes vs object) in V1 and both the t‐score testing for the mean signal, $t_0$ (a) and the weighted tSNR, $\mathrm{t}\mathrm{S}\mathrm{N}{\mathrm{R}}_{\mathrm{w}}$ (b) averaged across V1. Each data point is the median across participants. The vertical and horizontal bars illustrate the first and third interquartiles across participants. (c) The relationship between the t‐score testing for the mean signal and the weighted tSNR. The optimal model determined by the Ljung‐Box Q test is used for the prewhitening to remove temporal correlations in the time series [Color figure can be viewed at http://wileyonlinelibrary.com]

Figure 8

Variation of the 10% highest t‐scores testing for the task in V1, the averaged weighted tSNR ( $\mathrm{t}\mathrm{S}\mathrm{N}{\mathrm{R}}_{\mathrm{w}})$ across V1, and the averaged t‐score testing for the mean signal ( $t_0)$ across V1 with respect to the sampling interval (i.e., volume TR), the number of samples, and the model used to account for temporal correlations. Box plots represent the median and the interquartile range of the metrics across participants. Below the graphs, the median (interquartile range) of the percent difference between models are indicated: AR(1) + white noise with respect to the optimal model (pink line) and the FAST model with 18 components with respect to the optimal model (green line) [Color figure can be viewed at http://wileyonlinelibrary.com]