## 2 Materials and Methods
### Data Processing
This model assumes that at any moment of time, \(t\), the absorbance measured at 1 cm and at the wavelength \(\lambda\) can be related to the changes of the absorption coefficient of the scalp only \(\Delta\mu_{a,s}(\lambda,t)\):  
\[\Delta OD_{\text{1 cm}}(\lambda,t)=L_{s\text{ 1 cm}}\Delta\mu_{a,s}\;(\lambda,t), \tag{4}\]  
where \(L_{s\text{ 1 cm}}\) is the optical pathlength in the scalp at 1 cm, and the absorbance at 3 cm and 4 cm could be expressed as  
\[\Delta OD_{\text{3 cm}}(\lambda,t)=L_{s\text{ 3 cm}}\Delta\mu_{a,s}(\lambda,t)+L_{c\text{ 3 cm}}\Delta\mu_{a,c}(\lambda,t) \tag{5}\]  
\[\Delta OD_{\text{4 cm}}(\lambda,t)=L_{s\text{ 4 cm}}\Delta\mu_{a,s}(\lambda,t)+L_{c\text{ 4 cm}}\Delta\mu_{a,c}(\lambda,t) \tag{6}\]  
where \(\Delta\mu_{a,c}(\lambda,t)\) is the change in the cerebral absorption coefficient, \(L_{s\text{ 3 cm}}\) and \(L_{s\text{ 4 cm}}\) are the scalp partial pathlength at 3 cm and 4 cm, respectively, and \(L_{c\text{ 3 cm}}\) and \(L_{c\text{ 4 cm}}\) are the cerebral partial pathlength at 3 cm and 4 cm, respectively. Since the pathlengths in Equations (4)-(6) were specific for every individual and unknown, \(\Delta\mu_{a,s}\) could not be algebraically excluded to find \(\Delta\mu_{a,c}\). However, since Equations (4) and (5) were linear with respect to the common time-dependent \(\Delta\mu_{a,s}(\lambda,t)\) and \(\Delta\mu_{a,c}(\lambda,t)\), for every wavelength,  
Figure 2: Absorption spectra of HHb, HbO\({}_{2}\), H2O, and rCCO. \(\lambda\), one could exclude the time-course common with \(\Delta\mu_{a,c}(\lambda,t)\) from \(\Delta OD_{\text{3 cm}}(\lambda,t)\) and \(\Delta OD_{\text{4 cm}}(\lambda,t)\). Indeed, one could rewrite Equations (5) and (6) in the form  
\[\Delta OD_{\text{3 cm}}(\lambda,t)=L_{c\text{ 3 cm}}\delta\mu_{a,c}(\lambda,t)+(L_{c \text{ 3 cm}}+L_{\text{s 3 cm}})\Big{(}L_{\text{(s 1 cm)}}\Big{)}^{-1}\Delta OD_{\text{1 cm}}(\lambda,t) \tag{7}\]  
\[\Delta OD_{\text{4 cm}}(\lambda,t)=L_{c\text{ 4 cm}}\delta\mu_{a,c}(\lambda,t)+(L_{c \text{ 4 cm}}+L_{\text{s 4 cm}})\Big{(}L_{\text{(s 1 cm)}}\Big{)}^{-1}\Delta OD_{\text{1 cm}}(\lambda,t) \tag{8}\]  
where  
\[\delta\mu_{a,c}(\lambda,t)=\Delta\mu_{a,c}(\lambda,t)-\Delta\mu_{a,s}(\lambda,t) \tag{9}\]  
represents the difference between the cerebral and scalp responses to BH. Note that Equations (7) and (8) are linear with respect to \(\Delta OD_{\text{1 cm}}(\lambda,t)\) and \(\delta\mu_{a,c}(\lambda,t)\), and also we expect that the latter functions of time and wavelength will be uncorrelated in the time domain. Therefore the time-domain linear regression coefficients between \(\Delta OD_{\text{1 cm}}(\lambda,t)\) and \(\Delta OD_{\text{3 cm}}(\lambda,t)\) and between \(\Delta OD_{\text{1 cm}}(\lambda,t)\) and \(\Delta OD_{\text{4 cm}}(\lambda,t)\) should be close to \(\beta_{\text{1,3 cm}}\approx(L_{c\text{ 3 cm}}+L_{\text{s 3 cm}})\Big{(}L_{\text{(s 1 cm)}}\Big{)}^{-1}\) and \(\beta_{\text{1,4 cm}}\approx(L_{c\text{ 4 cm}}+L_{\text{s 4 cm}})\Big{(}L_{\text{(s 1 cm)}}\Big{)}^{-1}\), respectively. Then, by subtracting \(\beta_{\text{1,3 cm}}(\lambda)\Delta OD_{\text{1 cm}}(\lambda,t)\) and \(\beta_{\text{1,4 cm}}(\lambda)\Delta OD_{\text{1 cm}}(\lambda,t)\) from Equations (7) and (8) we obtain:  
\[\Delta\mu_{a}(\lambda,t)=\Delta[\text{Hb}](t)\varepsilon(\lambda)_{\text{Hb} }+\Delta[\text{HbO}_{2}](t)\varepsilon(\lambda)_{\text{HbO}_{2}}+\Delta[\text {CCO}](t)\varepsilon(\lambda)_{\text{ccO}}, \tag{2}\]0  
Note that at all wavelengths the true cerebral response to BH could have a component close to the scalp response. Therefore, \(\delta OD_{\text{3,4 cm}}(\lambda,t)\) represent not the pure time-course of the cerebral absorption coefficient, but rather the difference between the cerebral and scalp absorption time-courses.