Figure 4. fMRI BOLD signal as a function of neural response.

(A) Five pairs of BOLD response amplitudes evoked in V1-V3 with the single- and double-sided stimulations, each with two stimulus durations, 6-s (left column) and 1-s (right column). If the neural response to a single-sided stimulus is Z_i, then the neural response to the corresponding double-sided stimulus will be 2Z_i, given our empirical determinations of co-localization and independence of the neuronal populations in an achiasmic visual cortex. (B) The BOLD vs. neural response (BvZ) functions for V1-V3 as inferred by the stitching procedure for the two stimulus durations. The inferred functions can be well fitted with power-law functions (i.e. straight lines in log-log coordinates). These functions are nonlinear, with a log-log slope significantly shallower than unity (the background gray lines). (C) The exponents (γ) of the power-law fit of the BvZ functions for V1-V3. Error bars denote 95% CI. The red line indicates γ = 0.5. γ estimated from V2 and V3 (γ ~ 0.5) were not significantly different, while that obtained from V1 was biased upward, due to a violation of the co-localization assumption (see Discussion) required for inferring the BvZ function using the summation experiment. We thus inferred the (true) BvZ function of V1-V3 using the average γ estimated from V2 and V3 only.

DOI: http://dx.doi.org/10.7554/eLife.09600.011

Figure 4—figure supplement 1. Alternative derivation of the BvZ function.

(A) BOLD amplitudes evoked with a double-sided stimulus were linearly proportional to that evoked with the corresponding single-sided stimulus of the same contrast. There were 5 runs per data point for the 6-s condition and 9 runs for the 1-s condition. For each of the visual areas V1-V3, a linear function provides a good fit the data points [6 s: R²≥0.90; 1 s: R²≥0.85], and the intercepts are not significantly different from zero [nested model comparison for each visual area; 6 s: F(1,3)<0.37, p>0.59; 1 s: F(1,3)<0.27, p>0.64]. (B) Equivalently, contrast had no significant effect on the ratio of BOLD amplitude of a double-sided condition to that of the corresponding single-sided condition [nested model comparison for each visual area, 6 s: F(1,3)<0.53, p>0.52; 1 s: F(1,3)<0.16, p>0.72]. Performing a one-way ANOVA on the BOLD ratios obtained from individual runs also failed to find any significant effect of contrast in any of the visual areas [6 s: F(4,20)<0.37, p>0.8; 1 s: F(4,40)<1.76 p>0.15]. Assuming that the underlying neural responses to the double-sided stimuli is twice those of the corresponding single-sided stimuli, these results imply that the relationship between neural and fMRI BOLD responses follows power law: B = kZ^γ, where γ can be determined from the ratio of BOLD amplitude evoked by the double-sided stimuli to that evoked by the single-sided ones: γ = log₂(B_2,i/B_1,i). The estimated values of γ from the BOLD ratios are 0.75 (V1), 0.54 (V2), 0.53 (V3) from the 6 s presentation and 0.55 (V1), 0.48 (V2), 0.44 (V3) from the 1 s presentation. These values are very close to those estimated using the stitching procedure (Figure 4C).

DOI: http://dx.doi.org/10.7554/eLife.09600.012

Figure 4—figure supplement 2. Robustness of BOLD summation results.

(A) BOLD amplitudes of every voxel in each ROI (V1-V3) as evolved by the two versions of the single-sided stimulus (Stim A and B of Figure 3) in the 6-s experiment. The voxels responded approximately equally to both versions of the stimulus, indicating that any imperfect homotopy and/or gaze control did not significantly affect the stimuli's ability to equally co-activate the voxels. Red dots represent voxels in the ROIs that were strongly responsive (uncorrected p<0.001) to both versions of the stimulus. Reanalyzing the data using only these strongly responsive voxels yielded essentially the same results as in Figure 4 when all the voxels with the ROIs were used (right vs. left column, respectively, of B–D.

DOI: http://dx.doi.org/10.7554/eLife.09600.013