The impact of regularization parameter to the point-spread function of minimum-norm inverse

 

The minimum-norm estimate (MNE) and noise-normalized MNE provide analytic solutions to map MEG scalp recordings into currents on the cortical surface. In the implementation of MNE regularization is employed to stabilize the solution. The regularization parameter characterizes the relative weights of data and current prior error terms in the cost function to be minimized. One choice of regularization parameter is the inverse of the SNR in the data. We study the point-spread function (PSF) of the MNE linear inverse operators as a function of the regularization parameter with different cortical orientation constraints with SNR varying between 0.1 and 100.0. The PSFs with different orientation constraints have a similar spatial distributions. MNE inverse operators have a larger aPSF (>20 mm) at insula region. Noise-normalized MNE has a smaller PSF than MNE at insula but a larger PSF than MNE at lateral aspect of temporal lobe. The PSFs do not change significantly if SNR >1. Both MNE and noise-normalized MNE have wide PSFs around thalamus and medial aspect of temporal lobe (>20 mm). Dynamic estiamtion of SNR is also tested using L-curve, Generalized Cross-Validation and whitened measurements on evoked somatosensory data to reveal dynamic variation of SNR. All three automatic regularization methods report SNR above 1.0. This indicates the linear time-invariant MNE inverse suffices to provide stable PSF and localization accuracy in the experiment with dynamic SNR variation.

 
Distributed source modeling is used widely to estimate the spatial distribution of neuronal activities based on the extracranially recorded MEG data. Among the available approaches the minimum-norm estimate (MNE) (Hamalainen and Ilmoniemi 1984) , which assumes the minimum L-2 norm dipole current power across the whole brain, has been adopted most widely. Noise-normalized MNE (Dale, Liu et al. 2000) was also proposed to dynamically estimate the statistical significance of neuronal activities relative to the baseline. The derivation of MNE and noise-normalized MNE inverse is formulated within the Tikhonov regularization framework (Tikhonov and Arsenin 1977) , which uses regularization to balance data errors and model errors (Liu, Dale et al. 2002) . It has been suggested that the inverse of signal-to-noise ratio (SNR) can be use as the regularization parameter (Dale and Sereno 1993) . Since the SNR in MEG experiments varies dynamically, and both MNE and noise-normalized MNE are linear time-invariant regularized operators, assessment of the regularization parameter's impact on the PSF is desired for more better localization of MEG data using MNE and noise-normalized MNE and to understand the consequences of time dependent regularization.
 

Assuming that the genesis of extracranial neuromagnetic fields and electric potentials due to postsynaptic currents is an instantaneous process of linear transformation, we can relate  the recorded MEG/EEG signal and neuronal activities using integral equations. In the discrete format, we have the following matrix equations. y(t )=Ax(t)+n. y(t) indicates the instantaneous MEG measurements at time t with  p sensors. x(t) indicates the neuronal currents. A is the "forward matrix" mapping the intracranial neuronal currents to extracranial MEG/EEG signals. n is the the contaminating noise. Due to the ill-poseness of the linear inverse, the assumption that the neuronal currents over the whole brain are of minimum power (in the L-2 norm sense) has to be made, yielding the minimum-norm estimate (MNE)  with cost function. x(t)=argmin{||y(t)-Ax(t)||2+l||x(t)||2}. ||.||2 represents the L-2 norm. And l is the regularization parameter. Using the linear inverse operator we can estimate the source currents by x(t) = W y(t). W is the "inverse operator" which is calculated as maximal a posteriori (MAP) estimate Wmne = RAT(ARAT+lC)-1. Here R is the source covariance matrix and C is the noise covariance matrix. We can derive the noise-normalized MNE Wspm = Wmne/sqrt (Wmne C WmneT),  thus calculating a statistical parametric map of the neuronal activity compared to baseline.

We can also modify A and R in order to use free orientation (FO), strict cortical orientation constraint (SOC) and loose cortical orientation constraint (LOC). SOC requires the estimated dipoles to be perpendicular to the cortical surface, and LOC allows for additional tangential source components in local cortex. The regularization parameter was proposed to be the inverse of the SNR of the inverse as l =Tr(C)/Tr(ARAT)/SNR

To estimate the regularization parameter, we compared three automatic regularization techniques, the L-curve (Hansen 1998) , the Generalized Cross-Validateion (GCV) (Golub, Heath et al. 1979) , and additionally we propose to estimate the inverse of signal-to-noise ratio to trade-off the two cost function terms. In the latter method the SNR can be derived from the whitened measurements, yw(t) = sqrt(S)-1UTy(t) as SNR(t) = (yw(t)T yw(t))/p. Here U and S are the left singular vectors and singular values of the noise covariance C. We used the averaged point spread function(aPSF) to assess the regularization parameter's impact on the MNE. At each source location r, the resolution matrix S(r) = W(r) A and  a distance vector from all source locations in the brain relative to r  were calculated. The aPSF is then calculated by the center of mass of distance vector using S(r) as weightings.We varied the SNR from 0.1 to 100 parametrically and we also used evoked somatosensory data to evaluate dynamic range of SNR in actual measurements. An MEG experiment with the right median nerve stimulation was conducted with 0.5-ms constant current pulses and with amplitude clearly above the motor threshold. A 306-channel MEG system (Elekta Neuromag Oy) was used to record the neuromagnetic responses.

The inter-stimulus-interval between current pulse was 4 seconds. The measurement bandwidth was 0.03 to 250 Hz and the data were digitized at 1004 Hz and 100 responses were averaged.
 
aPSF Distributions of MNE and noise-normalized MNE using free orientation at SNR between 0.1 and 100. Light gray indicates gyri and dark gray indicates sulci.
 
aPSF of MNE and noise-normalized MNE using free orientation (FO), loose cortical orientation constraint (LOC) and strict cortical orientation constraint (SOC).
 
 Dynamic estimation of SNR using whitened MEG measurements, L-curve and GCV on median nerve stimulation experiment
 
In this study, we used aPSF to quantify the pointspread function of linear inverse operators depending on the regularization parameter. We compared the results between MNE and noise-normalized MNE. Ideally the PSF should be small for high localization accuracy. Several strategies have been proposed to optimize the aPSF, and optimal combination of MEG and fMRI was studied based on aPSF analysis (Liu, Belliveau et al. 1998) . Here we showed that the distribution of PSF varied with different regularization parameters and SNRs. The distributions of PSF values are different in MNE and noise-normalized MNE, and they vary with different orientation constraints. Nevertheless, the asymptotic aPSF values indicate the validity of using static regularization parameter in the experiments with time-varying SNR when the SNR exceeds 1. Finally, we validated the estimated SNR based on whitened measurements by L-curve and GCV in somatosensory evoked field MEG experiment to show that the SNR remains larger than 1 during the first 250 ms after median nerve stimulation. This concludes that a static regularization parameter can be applied to both MNE and noise-normalized MNE when the SNR exceeds 1 for stable localization performance.

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