Functional holography analysis: Simplifying the complexity of dynamical networks

Abstract

We present a novel functional holography (FH) analysis devised to study the dynamics of task-performing dynamical networks. The latter term refers to networks composed of dynamical systems or elements, like gene networks or neural networks. The new approach is based on the realization that task-performing networks follow some underlying principles that are reflected in their activity. Therefore, the analysis is designed to decipher the existence of simple causal motives that are expected to be embedded in the observed complex activity of the networks under study. First we evaluate the matrix of similarities (correlations) between the activities of the network's components. We then perform collective normalization of the similarities (or affinity transformation) to construct a matrix of functional correlations. Using dimension reduction algorithms on the affinity matrix, the matrix is projected onto a principal three-dimensional space of the leading eigenvectors computed by the algorithm. To retrieve back information that is lost in the dimension reduction, we connect the nodes by colored lines that represent the level of the similarities to construct a holographic network in the principal space. Next we calculate the activity propagation in the network (temporal ordering) using different methods like temporal center of mass and cross correlations. The causal information is superimposed on the holographic network by coloring the nodes locations according to the temporal ordering of their activities. First, we illustrate the analysis for simple, artificially constructed examples. Then we demonstrate that by applying the FH analysis to modeled and real neural networks as well as recorded brain activity, hidden causal manifolds with simple yet characteristic geometrical and topological features are deciphered in the complex activity. The term "functional holography" is used to indicate that the goal of the analysis is to extract the maximum amount of functional information about the dynamical network as a whole unit.

Organization of the autoantibody repertoire in healthy newborns and adults revealed by system level informatics of antigen microarray data

“…We found that reduction to three dimensions extracts most of the relevant information (Ͼ85% of the information as described in section II of SI Text and in Fig. S1), similar to what was found for brain activity data (34) and gene expression data (35). Therefore, we present the immune response information in a 3-dimensional PCA space, whose axes are the three leading principal vectors computed by the PCA algorithm.…”

Functional Holography and Cliques in Brain Activation Patterns

“…The correlation matrix can be associated with the correlation space, i.e. the N-1 dimensional space of correlations (Baruchi et al, 2006;. We note that the correlation space does not represent a real space in the sense that the eigenvectors do not create an orthogonal mathematical space.…”

“…Reduction to three dimensions (projection on the three leading principal components) typically extracts most of the relevant information (above 85%), (Baruchi et al, 2006;Madi et al, 2008). To retrieve the information lost as a result of dimension reduction, we link each pair of nodes by lines color coded according to the correlations between them (Baruchi et al, 2006;Madi et al, 2008).…”

“…Principal component analysis (PCA) a standard dimension reduction algorithm can be used to extract the maximal relevant information embedded in the signal correlation matrices. The relevant information can then be presented in a 3-dimensional principal component space (Baruchi et al, 2006;) the axes of which are the three leading PCA principal vectors.…”

“…A collective normalization of the correlations (cross-correlation) is performed and an affinity matrix is computed. The affinity transformation represents a collective property of all channels, and can help capturing hidden collective motifs related to functional connectivity in the network behaviors (Baruchi et al, 2006;. The affinity matrix is calculated using the meta-correlation matrix MC(i,j), which is the Pearson's correlation between the rows of the reordered correlation matrix of any two components (i) and (j) as described in Eq.3.…”