Neurodegenerative disease disrupts functional network structure. It is therefore of great interest to study the change of functional network architecture during disease progression. Independent component analysis (ICA) is a common approach to identify hidden functional network architecture from BOLD-fMRI images. Current ICA implementations have been shown to optimise for sparsity rather than for statistical independence between components.$$$^{1,2}$$$ We describe an efficient ICA implementation that scales linearly with the amount of voxels, optimises directly for sparsity and allows for overcomplete basis representation.

ICA is a blind source separation technique that finds components which are maximal statistically independent to each other. However, the two most common ICA implementations for ICA in BOLD-fMRI (InfoMax and FastICA) have been shown to select sparse rather than statistically independent components.$$$^{1,2}$$$ It is therefore of great interest to find optimisation strategies that directly optimise for sparsity rather than independence. The common ICA optimisation problem is formulated as follows $$\underset{x}{\text{min}} \sum_{i=1}^{m}\sum_{j=1}^{k}g(W_jx^{(i)}),\quad\text{subject to}\quad WW^T=\text{Id}$$ where g is a nonlinear convex function, W is the weight matrix $$$W\in\mathbb R^{k\times n}$$$, $$$k$$$ is the number of components and $$$n$$$ the number of voxels. The orthonormality constrain $$$WW^T=\text{Id}$$$ prevents the bases of $$$W$$$ from becoming degenerate. We replace the hard orthogonality constrain with a sparsity constrain as proposed in Le et al.$$$^3$$$ resulting in: $$ \underset{x}{\text{min}} \frac{\lambda}{m}\sum_{i=1}^{m}\|W^T Wx^{(i)} - x^{(i)}\|_{2}^2 + \sum_{i=1}^{m}\sum_{j=1}^{k}g(W_jx^{(i)})$$ Optimising this problem formulation in a neural network setting as shown in Figure 1 circumvents the computation of the expensive covariance matrix $$$W^TW$$$ and therefore scales easily to very high dimensional BOLD-fMRI data. Our cost function comprises the L2-norm between $$$\tilde{x}$$$ and $$$x$$$ and a L1 regularisation term. We then apply the backpropagation algorithm, computing the partial derivatives of the cost function.

We applied our ICA implementation to a recently published resting-state fMRI open-source cohort of 88 scans $$$^4$$$, comprising 22 subjects with 4 scans respectively, average age 23 years. The fMRI scans were acquired on a Siemens MAGNETOM 7T (TR 3000ms, TE 17ms, flip angle 70$$$^{\circ}$$$, voxel size 1.5mm isotropic, field of view (FoV) 192 mm$$$^2$$$, 70 oblique transverse slices, slice order interleaved). We excluded 4 scans due to an acquisition error. All fMRI scans were motion corrected, highpass-filtered (0.01Hz) and spatial-smoothed with a 6mm FWHM Gaussian filter. Scans were temporally demeaned and variance normalized. To compute a group ICA decomposition, subjects were mapped into a group space and the MIGP algorithm $$$^5$$$ was used to reduce the cohort to 599 spatial eigenvectors. Our ICA implementation was applied to the 347242 voxels x 599 spatial eigenvector matrix, resulting in 40 estimated sparse components. Individual components are described by their distribution of values. To identify voxels significantly contributing to a component, voxel values are scaled to a z-score. We identified several common resting-state networks$$$^6$$$ such as the default mode, sensory-motor, visual and fronto-parietal network as depicted in Figure 2,3,4.

BOLD-fMRI image acquisition has seen an enormous leap in spatial and temporal image resolution. Data initiatives such as the Human Connectome Project, Alzheimers Disease Neuroimaging Initiative as well as 7 Tesla BOLD-fMRI scans produce massive amounts of data that already need several terabytes to store. With this increasing scale of data, it is a necessity to develop algorithms that are able to identify recurrent patterns and produce intelligent data compressions. Our ICA implementation is one of a number of blind source separation approaches that rely on mathematical concepts rather than biological prior information. Although this makes it difficult to directly compare the results of such approaches between one another, we have provided evidence that we are able to find similar functional networks as those previously found.$$$^6$$$ Common ICA implementations were intended to optimise for statistical independence but produce sparse rather than independent components. Our ICA implementation directly optimises for sparsity in a very efficient way using a neural network optimisation scheme. This efficiency will be a crucial feature when applying this analysis techniques to large neuroimaging studies.

The flexibility of deep learning models make them an interesting tool for high-dimensional BOLD-fMRI image analysis. Our neural network optimisation scheme allows the efficient learning of sparse components with the backpropagation algorithm while avoiding the computation of a large, possibly intractable, spatial covariance matrix. Our framework furthermore provides the flexibility of learning overcomplete basis representation which might be of use for very short BOLD-fMRI acquisitions. Our future work will enhance our proposed ICA implementation by using additional hidden layers in the hope of learning interesting features from large cohort studies that might help elucidate the subtle differences between normal and abnormal brain networks.