The MI-clust Plug-in: Clustering independent components using mutual information
The infomax ICA algorithm tries to minimize mutual information between components and is greatly successful at this task. Since the activities of cortical and other EEG sources are not perfectly independent, after performing ICA a low level of dependence still exists between the returned maximally independent components( ICs). This residual mutual information can be used to infer relations between different ICs. Figure below shows the scalp maps 30 ICs returned for a sample dataset.
We can use the mi_pairs.m function to calculate the mutual information between the activities (or activations) of these first 30 ICs:
>> mutual_info = mi_pairs(EEG.icaact(1:30,:));
To visualize the resulting matrix, it is better to first remove the diagonal values (indicating the mutual information between each component and itself), which are quite higher than off-diagonal inter-IC pair values
>> mutual_info = mutual_info – diag(diag(mutual_info));
Now we can use a simple Matlab call to visualize this matrix:
>> figure; imagesc(mutual_info);
As we can see, there is some residual dependency between some of the component activities. For example, ICs 1-4 have a higher dependency on each other than on other components. The scalp maps of these ICs (below) show that they are all related to artifacts generated by subject eye movements, specifically eye blinks.
The MI_clust plug-in takes advantage of such residual mutual information to find subsets of maximally independent components whose activities are in some way related to each other. This makes it easier to differentiate various sources of the recorded EEG signal separated by ICA. To use this plug-in, we first need to perform ICA on our dataset. Then select Plot > Cluster dataset ICs
A pop-up dialog window will appear:
Here we need to input three values:
- Components to cluster: These are the indices of the ICs to cluster.
- Number of clusters: Number of different groups into which ICs are to be clustered. For example, we might want to keep at least one cluster per every type of EEG signal source (eye movements, noise subspace, cortical activities,...).
- Clustering method: Currently one can choose either "linkage" (which uses the Matlab linkage() function, using the 'ward' method) or "kmeans" (based on the Matlab kmeans() function) for clustering. "Linkage" seems to produce better results. An additional dendrogram of IC distance relationships is plotted if the "linkage" option is selected.
After receiving these inputs, the plug-in calculates mutual information between ICs using the mi_pairs.m function (this may take some time...). The function produces 2-D and 3-D IC "locations" whose distances from each other best represent the degree of dependence between each IC pair (closer ICs exhibiting more dependence). This uses the Matlab mdscale() function. These locations are clustered by the specified clustering function ("linkage" or "kmeans"). Several figures are plotted :
- Component Cluster Plots:
Each IC cluster is plotted in a separate figure. Figure below shows a cluster of vertical eye movement components:
Another cluster, plotted below, consists of likely brain sources:
The brightness of the background in these plots indicates the confidence level (returned by the Matlab silhouette() function) that each IC belongs to this cluster. The figure below shows an example in which three ICs on the top-right corner have a dark background, meaning they are not a strong part of this cluster. (Notice that their scalp maps are quite different as well). In such cases, increasing the number of clusters is recommended. The plotting function interpolates the silhouette() values to show a smooth background transition between components.
- The Silhouette Plot:
Silhouette() measures the degree of confidence that a certain IC belongs to the indicated cluster versus other clusters. It is measured by a value between -1 and +1. The plug-in plots silhouette() values for all the ICs that belong to the given cluster. From the figure below we can see that the high confidence (positive silhouette value) ICs in cluster 5 (eye component cluster) and low confidence (negative silhouette) for ICs in cluster 2 (above figure). This suggests increasing the number of clusters to provide additional groups for ICs with low silhouette.
- The 3-D Plot:
This plot shows a 3-D representation in which closer ICs have higher mutual information or activity dependence. Components are colored according to their respective cluster.
- The Dendrogram Plot:
If the "linkage" clustering method is selected, an IC dendrogram is plotted:
In this plot, IC are hierarchically joined together (in a binary manner) to form groups with high interdependency. For example, in the above figure, ICs 3 and 4 joint together to form a group that joins the group of ICs 1 and 2 at a higher level to form an \'eye component\' group. The height of the tree at each level is proportional to the distance (independence) between joined group at that level.