^{1}, Danny Grossman

^{1}, Vladislav Volman

^{1}, Mark Shein

^{1}, John Hunter

^{2}, Vernon L. Towle

^{2}and Eshel Ben-Jacob

^{3,a)}

### 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.

We propose a new mathematical conceptualization for analyzing the complex activity of information-based, task-performing biological networks [e.g., gene networks, the immune system and the central nervous system (CNS)]. Our approach is guided by the “whole in every part” nature of holograms and the “simple complexity” assumption about the nature of the biological system dynamics—the idea that simple motives are hidden in the observed behavior. The mathematical basis for our approach is a novel functional holography (FH) analysis in which the complex activity of biological networks is unfolded in the space of functional correlations between the activities of the network constituents. Using dimension reduction algorithms, a connectivity diagram is generated in the three-dimensional space that captures the maximal relevant information. Temporal (causal) information is superimposed on the resulted diagram by coloring the elements locations according to the temporal ordering of their activities. By this analysis, the existence of hidden causal manifolds with simple yet characteristic geometrical and topological features in the complex biological activity can be deciphered. We propose that our findings hint that the functional holographyanalysis is consistent with a new holographic principle by which biological networks regulate their complex activity to perform information processing in the space of functional correlations.

The method presented here has evolved from the joint study with Dr. Ronen Segev, Eyal Hulata, and Yoash Shapira (Ref. 8). Much insight has been gained from analyzing the structure of artificial and neural networks in collaboration with Pablo Blinder and Dr. Danny Baranes. One of us (E.B-J.) is most thankful to Professor Steven Schiff for many illuminating conversations, guidance into the foundations and the literature about epilepsy and constructive comments and advices during the development of this research. We thank Professor Robert Benzi, Professor Eytan Domany, Professor Sir Sam Edwards, Professor Herbert Levine, and Professor Itamar Procaccia for constructive comments about the mathematical basis of the method. Preleminary analyses of fMRI measurements are done in collaboration with Dr. Talma Handler and Dr. Yaniv Asaf. This research was partially supported by a grant from the Israel Science Foundation, the Maguy-Glass chair in Physics of Complex Systems, and NSF Grant No. PHY99-07949. One of the authors (E.B-J.) thanks the KITP at University of California Santa Barbara, the Weitzman Institute and the Center for Theoretical and Biological Physics for hospitality during various stages of this research.

I. INTRODUCTION

II. THE FUNCTIONAL HOLOGRAPHYANALYSIS METHOD

A. Common analyses of correlation matrices

B. Collective normalization—the affinity transformation

C. Retrieval of lost information and the holographicnetwork

D. Inclusion of temporal information

E. Interpretations—contraction and expansion of features

III. ANALYZING THE ACTIVITY OF NEURAL NETWORKS

A. Cultured neural networks

B. Modeled neural networks

IV. ANALYZING ECoG RECORDED HUMAN BRAIN ACTIVITY

V. EXTENSIONS OF THE ANALYSIS

A. Topological characterization of the manifolds

B. Holographic comparison and superposition of networks activity

C. Holographic zooming

VI. CONCLUDING REMARKS AND LOOKING AHEAD

## Figures

(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.

(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.

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.

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.

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.

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.

(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.

(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.

(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.

(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.

(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.

(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.

(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).

(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).

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.

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.

(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.

(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.

(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.

(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.

(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.

(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.

(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.

(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.

(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.

(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.

(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).

(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).

(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 } .

(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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