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Functional holography analysis: Simplifying the complexity of dynamical networks
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color) Correlation matrix of synthetically produced signals. Left (a), synthetic signals that include three groups—the first subgroup of nine signals (signals 7,8,10,11,13,15,17,22,24) was generated by harmonic signals with the same periodicity, a phase shift of about and added noise. The second subgroup (signals 3,5,6,14) is another set of harmonic signals, with a different frequency. The other signals just have pure noise with no correlations. Right top (b), the corresponding similarity matrix—the correlation matrix in this case that was computed using the Pearson’s correlations (Appendix A). Right bottom (c), the sorted correlation matrix using the dendrogramed clustering algorithm (Appendix A). In this matrix the two subgroups form distinct clusters.

Image of FIG. 2.
FIG. 2.

Illustration of dimension reduction using the PCA algorithm. We use the PCA algorithm for dimension reduction from the 25-dimension space of correlations for the example shown in Fig. 1 into a three-dimension space. In (a) we show projection of the entire matrix. In (b) we show the same process after taking the correlation matrix for only the nine components that belong to the large subgroup that is captured by reordering of the correlation matrix [Fig. 1(c) ] and is also identified by the clustering algorithm.

Image of FIG. 3.
FIG. 3.

The affinity transformation. We show two examples of the affinity transformation for the correlation matrix presented in Fig. 1 . In (a) the normalization is performed using the correlation distances and in (b) the metacorrelation metric is used.

Image of FIG. 4.
FIG. 4.

(Color) Construction of the holographic network. In (a) and (b) we show the holographic network in the 3D principal space that is computed for the affinity matrix for the nine periodic signals described in Figs. 1 and 2 . In (a) we link the pairs that have level of correlation above 0.5. In (b) the threshold level is 0.1.

Image of FIG. 5.
FIG. 5.

(Color) Inclusion of temporal information. We show the inclusion of the causal information for the holographic network shown in Figs. 3 and 4 . The activity propagation is added by coloring the nodes location according to the relative phases or time lag between them. Blue is for early times (negative phases) and red for late times (positive phases). Note that adding this information helps to reveal the phase shifts imposed in the generation of the signals.

Image of FIG. 6.
FIG. 6.

(Color) The expansion effect of the affinity transformation. Top left (a) is a synthetic correlation matrix that has two groups. Top right (b) is the corresponding affinity matrix. The pictures below (c), (d) show the projection of the matrices on the corresponding 3D space of principal PCA vectors (for the correlation matrix on the left and for the affinity matrix on the right). The red lines show the links with the highest correlations. Note that for both cases the two subgroups form distinct clusters. However, for the affinity matrix we see that each of the clusters is “stretched” in the plane that is perpendicular to the line connecting the two clusters. This effect helps to reveal internal information of each of the subgroups while keeping the clusters apart.

Image of FIG. 7.
FIG. 7.

(Color) The contraction effect of the affinity transformation. In this case the correlation matrix was constructed to have two subgroups but with two “bridging channels” that is two channels that have relatively high correlations with both subgroups. The affinity transformation helps to identify the special functional role of the bridging channels as they are located together and with some distance from the other channels of the small subgroup. The correlation matrix is shown in (a) and its corresponding affinity matrix is shown in (b).

Image of FIG. 8.
FIG. 8.

The recorded activity of cultured neural network. Top (a), formation of SBEs in the recorded activity of cultured networks. The time axis is divided into bins. Each row is a binary bar-code representation of the activity of an individual neuron, i.e., the bars mark detection of spikes. Bottom left (b), zoom into the synchronized bursting event. During the SBE, each neuron has its specific spiking profile. Bottom right (c), the averaged spikes density (firing rate) of the neurons during the SBEs.

Image of FIG. 9.
FIG. 9.

(Color) Applying the functional holography analysis on the recorded activity of cultured neural network. Top, the interneuron correlation matrix. Middle, the activity connectivity network in real space. Bottom, the causal manifold. As can be seen the holographic network has a relatively simple geometry and topology. Moreover, the activity propagates along the backbone of the resulted manifold.

Image of FIG. 10.
FIG. 10.

(Color) The internal dynamics of a simulated neural network with uniform synaptic connectivity. We show the results for 50 neurons from a model network of 50 cells. In (a) we show the sorted interneuron correlation matrix for the model network. Note that the structure of the matrix is blocklike, with larger block corresponding to the excitatory neurons, and smaller block—to the inhibitory ones. The connectivity diagram in real space is shown in (b). The inhibitory neurons in the model are marked by a red circle and excitatory ones by a blue circle.

Image of FIG. 11.
FIG. 11.

(Color) The FH manifold of the uniform network model. We show the corresponding holographic network for all the 50 neurons in the modeled network for two different angles of view. There is a clear separation into two clusters. The large one composed of the excitatory neurons (colored blue) and the smaller one (colored red) is composed of the inhibitory ones.

Image of FIG. 12.
FIG. 12.

(Color) Analyzing the activity of structured modeled network. (a) The ordered correlation matrix for the structured model network. (b) The FH manifold for the activity of the model network. Note that the holographic network has a relatively simple geometry and topology that resembles that of the real network. Another common feature is that, the activity propagates along the backbone of the resulted manifold.

Image of FIG. 13.
FIG. 13.
Image of FIG. 14.
FIG. 14.

(Color) Holographic networks of recorded brain activity. The holographic networks are for the ECoG recorded human brain activity for the inter-Ictal and Ictal activities shown in Fig. 13 —inter-Ictal (a) and Ictal (b). The pictures show the manifolds from different angles of view. In the analysis we included only electrodes whose correlations with the other electrodes are above noise level. Note, that the locations change their functional role during seizure (Ictal) relative to those during the inter-Ictal durations.

Image of FIG. 15.
FIG. 15.

(Color) The topological characters of the manifolds. We show the interpolating surfaces between the nodes for two cases of inter-gene correlations extracted from Microarrays experiments (the details will be provided elsewhere).

Image of FIG. 16.
FIG. 16.

(Color) Holographic superposition. Large cultured networks exhibit distinct modes of dynamical behavior. This phenomenon is manifested by the observation that the sequence of SBEs is composed of distinct subgroups of SBEs, each with its own characteristic spatio-temporal pattern of activity and interneuron similarity matrix (Refs. 4 and 6 ). The top pictures (a),(b) show two manifolds, each for a specific mode of the network activity—a specific subgroup of SBEs. For clarity, we present the manifolds by the 2D interpolating surfaces. The bottom pictures show the holographic superposition of these two subgroups of SBEs. As described in Ref. 6 , the similarity matrix for each subgroup is projected on a joint 3D PCA space. That is a space whose axes are the leading PCA vectors of the combined affinity matrix. As seen, each subgroup generates its own manifold (each has a different gray level). For clarification, for each case we used the 2D manifold (extrapolation of a curved surface between the nodes). It is also quite transparent that the manifold is intermingled yet perpendicular. We emphasize that each manifold is composed of all the recorded neurons. The holographic superposition makes it possible to enhance the fact that each neuron has different similarities for each subgroup of SBEs. The results shown are for the network analyzed in Refs. 4 and 6 .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Functional holography analysis: Simplifying the complexity of dynamical networks