Two fixed-point CICA algorithms were proposed in order to improve the stability of the OCICA method. CICA1 is based on a Newton-like gradient learning process; CICA2 is using a regular gradient descent approach. Algorithms for applying constraints to either the components or to the mixing time courses were derived especially for 2D source separations. No learning rate was used in both proposed algorithms. 1D randomly mixed synthetic data and 2D random noise contaminated synthetic data were used to evaluate the performance of the proposed algorithms with comparisons to the OCICA and standard ICA. The 1D dataset was used to validate the algorithms for separating randomly mixed signal. The 2D synthetic data were simulating fMRI data and were used to validate the proposed algorithms for source separation in the case of randomly generated noise; as well as their performance for source separation with either spatial constraints or temporal constraints. In both 1D and 2D source separation experiments, the proposed methods were demonstrated to be better than OCICA in terms of higher stability (converged for all experiments for both 1D and 2D source separations), higher source separation quality, and lower PI. All CICAs outperformed fICA for source separation with higher CC and lower PI due to the use of constraints. No performance difference was observed between CICA1 and CICA2. The proposed CICAs presented higher convergence rate than OCICA. Because fICA does not include any additional constraints, it showed the fastest convergence rate when converged. The ROC analysis showed that CICA yielded better sensitivity/specificity performance than GLM for brain activation detection. However, fICA was demonstrated to be even less effective than GLM for the pseudo-activation detection, which may partly due to the interference of noise added to the mixed signal. Another reason may be that FastICA does not yield the best separation results for fMRI data as compared to other ICA algorithms [7
]. The additive noise could contribute significant background fluctuations, and the sub-optimal fMRI data source separation could make ICA converge to a local optima, which subsequently contributes component split or cross-contaminations. Among the 50 times source separations, the fICA identified ICs had large background noise or contaminations from other sources as shown in , which markedly degraded the sensitivity and specificity performance for the activation detection. By attracting the separating weight to the correct direction in the first several iterations, CICA provides a practical solution to deal with both the noise interference and the local optima problem.
A detailed stability analysis was provided for the proposed CICA methods. It is worth to note that the stability analysis for the OCICA [21
] was based on an additional KKT condition λ > 0, which is not necessary to be true. However, the convergence of the OCICA is still satisfied if the stability analysis is based on the classical KKT condition and the convexity condition as described in this paper.
Random noise is usually not explicitly modeled in standard ICA since it can be treated as a regular IC if there is only one Gaussian noise source. However, noise added after source mixing could drastically degrade the source separation quality of regular ICA as demonstrated in . By contrast, CICA is much less sensitive to this additive noise as shown in . An interesting finding in the additive noise contaminated 2D source separation simulations is that better separation quality for IC can be achieved by using CICA with constraints for the ICs (the so-called spatial constraints in the 2D data separation simulations) and better separation quality for TC can be achieved by using CICA with constraints for the TCs (the so-called temporal constraints in the 2D data separation simulations), suggesting gaining further quality improvement for both ICs and the associated TCs using constraints for both of them if we do have the constraints.
Since the noise environment and source features might be quite different in real fMRI from what was simulated in the synthetic datasets, sensorimotor fMRI data were used to further validate the proposed methods for analyzing fMRI data. The proposed 2 CICA algorithms were able to converge at the first time when they were applied to each subject's data. As compared to OCICA, the proposed methods demonstrated increased sensitivity for detecting brain activations in response to the sensorimotor and visual task. As in the simulations, all CICAs demonstrated better activation detection sensitivity than fICA due to the use of constraints. Both CICA1 and CICA2 performed nearly the same for detecting both visual and sensorimotor activations. CICA including the proposed new CICAs and OCICA (when converged) yielded much higher sensitivity for sensorimotor brain activation detection than the GLM-based univariate method. The aforementioned ROC analysis results and the demonstrated superiority of CICA for the well-characterized sensorimotor task activation detection suggest that the proposed CICA methods can be employed as sensitive tools for brain activation in fMRI. One reason for this sensitivity gain is the multivariate data processing in ICA. As each input sample is treated as a single unit, ICA-based fMRI data analysis is searching the spatial patterns as an entity rather than pursuing a collection of discrete patterns separately at each voxel like in the univariate GLM. Because the spatially coherent activation patterns are more robust to noise interference than each single voxel if the noise is spatially independent which is a quite common assumption taken in the real MRI field [12
], assessing the patterns as a whole should then gain a sensitivity increase as compared to the way that assesses the patterns at each isolated voxel. However, standard ICA especially fICA might not be able to fully demonstrate this benefit as partly shown in . A right solution is then to incorporating prior information into the ICA learning process as in CICA.
The sensorimotor data analysis results () showed that all CICA methods and fICA yielded less sensitivity than GLM in left Insula in terms of lower peak t-value and small cluster size, which actually reflected a caveat of a multivariate fMRI data analysis method that it could induce a sensitivity loss to some focal brain activations if they are not coherent with the overall spatially connected activation pattern. Actually, we have identified in [31
] that the left insula was activated less coherently than the other three regions, resulting in a less sensitive detection using ICA and CICA. Nevertheless, combining prior information and the data-driven property of ICA, CICA yielded better sensitivity for brain activation detection in the target visual and motor cortex in a well characterized sensorimotor fMRI experiment. In reality, resting fMRI data analysis is one of the hottest ICA application in fMRI. Since several resting networks have been repeatedly identified in the literature, it is possible to build templates for those networks using CICA and retrospectively use them in CICA for general resting state analysis. However, either topic needs additional extensive future work and should be reported separately.
One issue of CICA is that adding constraints could possibly affect the blindness feature of ICA. Similar to [20
], this potential bias is minimized by quickly releasing the constraints, so that the constraints will be totally inactive after several iterations and the rest of iterations will be the same as those in standard ICA. From this point of view, adding constraints does not change the data-driven property of ICA. Including constraints does not force ICA to produce the desired ICs if there does not exist any separating weights or ICs close to the constraints, but CICA through the proposed CICA1, CICA2, or OCICA (when an appropriate learning rate is used) is guaranteed to converge if some local optima exist regardless of whether they are close to the constraints or not. All the simulations demonstrated that the proposed CICA even the OCICA (when converged) were capable to identify the sources when the corresponding constraints were chosen to be a binary pulse or box-car function or binary image mask which are correlated to but with large deviations from the latent sources.
Although the proposed methods do not use any learning rate, they are still using two other parameters, the Lagrangian multiplier μ and the scalar penalty parameter γ. Different settings could potentially give different convergence rate. But we did not observe significant difference when the initial value of μ was set to be any number between 0.1 to 5 and when two additional different updating rules: 0.1 × 5k−1
were used for γ. More details about these two parameters can be found in [3
For CICA itself, we noticed that there is another approach called semi-blind ICA [1
], which incorporates the prior information by adding an additional adjusting step to the mixing matrix to attract the separating weights to the desired directions. A possible issue is that appending an additional adjusting step is not guaranteed to keep the stability of the original ICA. The two factors for controlling the additional adjustment may not be easy to be tuned for different applications too. By contrast, the convergence of the proposed CICA methods was fully proven. As the focus of this paper was CICA method development, a direct comparison between CICA and semi-blind ICA would be beyond the scope of this paper and was not provided. We also noticed that a CICA algorithm very similar to CICA2 proposed in this paper has recently been published by Lin et al.
] using a Lagrangian similar to Eq. 17
. Although they did not provide a stability analysis for their method, they had demonstrated the superiority of their improved CICA (similar to CICA2 in this paper) to the improved OCICA proposed by the same group [19