## 4 Discussion  
Our results showed that BHs induced significant hemodynamic responses in all three channels including the 1 cm channel, which mainly interrogated the scalp and partially the skull. Hemodynamic responses measured by the 3 cm channel were quite similar to those measured at 1 cm. In Figures 4 and 5 only HHb and rCCO signals measured at 4 cm showed different responses to those of the extracerebral tissue. The clearest difference from the scalp response was observed at 4 cm in the rCCO response (Figures 4 and 5). No rCCO responses were observed at 1 cm and 3 cm using the homogeneous tissue model. Clear HHb-, tHB-, and rCCO-specific cerebral responses seen in Figure 7 and Table 2 indicate that the dynamics of changes measured at 3 cm and 4 cm channels were different from that measured at 1 cm, and this difference can be attributed to the differences between the brain and scalp responses to BHs. Note that hemodynamic responses at 3 cm and 4 cm shown in Figure 7 and Table 2 had significantly smaller magnitudes than those shown in Figures 4 and 5 and in Table 1. However, rCCO responses in both Figures 4 and 7 had their magnitudes close to 0.1 uM. This result supports the conclusion that we succeeded in decoupling rCCO changes from changes in other chromophores, and that the measured rCCO changes occurred in the brain. We would like to underline that in both homogeneous and two-layer algorithms we used the most conservative treatments of rCCO changes such as thresholding by the quantitative improvement in the goodness of fit, which suppressed spurious rCCO changes. The time-courses of cerebral responses to BHs in healthy adult humans were studied in detail using blood-oxygen-level-dependent (BOLD) fMRI in [25,26]. The quantitative relationship between the BOLD changes measured by fMRI and hemodynamic signals  
Figure 7: Cross-subject averaged changes in [HbO2], [HHb], [tHB], and [rCCO] calculated from the time-domain regression residuals using stepwise linear regression in spectral domain. measured by NIRS was investigated in [35]. Figure 8 (adopted from [25]) shows averaged BOLD signal time-courses with a 15 s BH after the full expiration on the areas supplied by the middle cerebral artery (MCA, red), anterior cerebral artery (ACA, blue) and posterior cerebral artery (PCA, black). Since our hNIRS probe was positioned on the left side of the forehead near the midline, it could interrogate cortical regions supplied by both ACA and MCA. The main difference between ACA and MCA regional responses was the early rise of the BOLD signal response in the ACA region and a 10 s delay in the MCA region. In our results the initial dynamics (during the first 15 s of BH) of cerebral HbO\({}_{2}\) and HHb responses rather corresponded to the MCA type of the BOLD response [25,33]. A delayed rCCO response after the regression (Figure 7) corresponded to the delayed ACA response. Such delayed cerebral responses were most different from the scalp hemodynamic responses to BHs measured at 1 cm (Figure 4), which were also very significant. Another significant difference between the long- and short-distance channels was the longer peak times of HHb responses in the short-distance channel (longer than BH duration, Table 1). The peak times of responses to BHs after expiration measured by hNIRS at 4 cm were close to those measured by BOLD fMRI in [26] (see Figure 2 in [26]). Our results show that without using a short-distance channel to remove biasing by the scalp, the cerebral responses to BHs were clearest in the rCCO time-course at 4 cm. Some features different from the scalp responses could also be seen in the HHb time-course at 4 cm. At 3 cm, some features of the cerebral response could be poorly recognized in the HHb time-course. The HbO\({}_{2}\) time-course did not show clear cerebral responses to BH at 3 cm or at 4 cm, in particular due to the high inter-subject variability. However, measurement of rCCO changes required hyperspectral or many-wavelength multispectral technology.