**2. Methods and applications**

#### **2.1 FH analysis**

The FH method was developed from the perspective of cultured networks (Segev et al., 2004). Yet it can be applied to essentially any type of neural signal from the analysis of slices, to cortex-recorded activities, ranging from electro- or magneto-encephalographic (EEG and MEG respectively) to functional magnetic resonance imaging (fMRI) signals. Moreover, it makes it possible to place the recordings from all of these levels within the same presentation schema for comparative studies.

The FH approach allows identifying additional motifs embedded in the inter-neuron correlation matrices— analogous to the inter-location coherence matrices evaluated for ECoG recordings of brain activity (Milton & Jung, 2002; Towle et al., 1999) that are not transparent in the real space connectivity networks. The correlation matrix is represented in a higher dimensional space of functional correlations, or correlation affinities.

The FH method involves the following steps:


#### **2.1.1 Correlation matrices**

The first stage in the FH analysis is computation of the signals correlation matrices – the matrices of correlations between the dynamical responses of all pairs of signals. We used the Pearson formula (Pearson, 1901) to calculate the correlation *C i* , *j* between signals (i) and (j):

$$C(i,j) = \frac{\sum\_{k=1}^{T} \left( X(i,k) - \mu(i) \right) \left( X(j,k) - \mu(j) \right)}{\sigma(i)\sigma(j)} \tag{1}$$

where *X i* and *X j* are the recorded time signals (i) and (j), with corresponding means *i* , *j* and standard deviations *i* , *j* .

For N signals, the pair–wise correlations define a symmetric NxN correlation matrix. In order to reveal subgroups in the correlation matrix, we make use of the commonly used dendrogram clustering algorithm (Dubes & Jain, 1980). This algorithm reorders the

signals, and will finally conclude by discussing the scope of the FH method in the context of

The FH method was developed from the perspective of cultured networks (Segev et al., 2004). Yet it can be applied to essentially any type of neural signal from the analysis of slices, to cortex-recorded activities, ranging from electro- or magneto-encephalographic (EEG and MEG respectively) to functional magnetic resonance imaging (fMRI) signals. Moreover, it makes it possible to place the recordings from all of these levels within the same

The FH approach allows identifying additional motifs embedded in the inter-neuron correlation matrices— analogous to the inter-location coherence matrices evaluated for ECoG recordings of brain activity (Milton & Jung, 2002; Towle et al., 1999) that are not transparent in the real space connectivity networks. The correlation matrix is represented in

5. Retrieval from the correlation matrix of the information lost in the dimension reduction. 6. Inclusion of temporal causal information describing the activity propagation in the

The first stage in the FH analysis is computation of the signals correlation matrices – the matrices of correlations between the dynamical responses of all pairs of signals. We used the Pearson formula (Pearson, 1901) to calculate the correlation *C i* , *j* between signals (i)

 

(1)

*Xik i X j k j*

, ,

 

where *X i* and *X j* are the recorded time signals (i) and (j), with corresponding means

 *i* , *j* . For N signals, the pair–wise correlations define a symmetric NxN correlation matrix. In order to reveal subgroups in the correlation matrix, we make use of the commonly used dendrogram clustering algorithm (Dubes & Jain, 1980). This algorithm reorders the

a higher dimensional space of functional correlations, or correlation affinities.

brain imaging data analysis.

**2.1 FH analysis** 

**2. Methods and applications** 

presentation schema for comparative studies.

The FH method involves the following steps:

7. Holographic zooming and comparison.

*i* , *j* and standard deviations

4. Dimension reduction.

**2.1.1 Correlation matrices** 

network.

and (j):

 

1. Evaluation of the similarity matrix between components. 2. Clustering by sorting or reordering of the similarity matrix.

3. Construction of a matrix of functional correlations.

,

1

*Cij i j*

*T*

*k*

  correlation matrix such that highly correlated signals are closely located. This is performed using the correlation distance D(i,j) between signals (i) and (j), which is the Euclidean distance between the rows i and j in the correlation matrix (the vectors of correlations of each one of the signals with all other ones)

$$D\left(i,j\right) = \left\|\vec{\mathcal{C}}\left(i\right) - \vec{\mathcal{C}}\left(j\right)\right\| = \sqrt{\sum\_{k=1}^{N} \left(\mathcal{C}\left(i,k\right) - \mathcal{C}\left(j,k\right)\right)^{2}}\tag{2}$$

where is the correlation vector between signal (i) and all other signals. Next, the algorithm reorders the correlation matrix by sorting it according to the hierarchical tree of correlation distances. In such a way we produce a real metric that satisfies the triangle inequality. In Fig. 1 we illustrate the analysis with a simple example. We generate 25 signals to imitate a multichannel recording of the activity of a network of 25 components. The signals (Fig. 1a) include two subgroups of periodic signals with higher correlations and a group of random signals. In Fig. 1b we show the corresponding correlation matrix computed using the Pearson correlations. Applying the dendrogram clustering algorithm (Fig. 1c) on the correlation matrix, the subgroups are delineated in the resulting sorted (reordered) matrix (Fig. 1d). The correlation matrix can be associated with the *correlation space*, i.e. the N-1 dimensional space of correlations (Baruchi et al., 2006; Baruchi et al., 2004). 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. *C i*

#### **2.1.2 Collective normalization**

The next step of the analysis is designed to capture mutual or relative effects between several signals. 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; Baruchi et al., 2004). 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. The affinity collective normalized matrix is the product of the correlation matrix and the meta-correlation matrix as defined in Eq. 4.

$$\text{MC}\{i,j\} = \frac{\sum\_{k \neq i,j}^{N} \{\text{C}(i,k) - \mu \text{c}(i)\} \{\text{C}(j,k) - \mu \text{c}(j)\}}{\sqrt{\left\{\hat{\text{C}}(i)^{2}\right\} \cdot \left\{\hat{\text{C}}(j)^{2}\right\}}} \tag{3}$$

$$A\{i,j\} = \sqrt{\text{C}\{i,j\} \cdot \text{MC}\{i,j\}} \tag{4}$$

This MC matrix is calculated on the reshuffled rows of the matrix in such a way that all the elements between the signals (i) and (j) themselves are not included in the calculation. We note that the affinity transformation is performed after rescaling the range of the correlations to [0,1].

Functional Holography and Cliques in Brain Activation Patterns 107

Each node is placed in this space according to its three eigenvalues for the three leading principal vectors. 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; Baruchi et al., 2004; Madi et al., 2008). The result is a holographic

Fig. 2. Holographic representation of the synthetically produced signals from Fig.1 in the 3D space. The axes are the three leading principal PCA vectors of the correlation matrix. Each node is located in this space according to its eigenvalues corresponding to the leading principal vectors. All the nodes with correlations above 0.8 are linked by lines color coded according to the correlations (represented in the colorbar), creating the holographic

Often, one is interested in more details about a part of the manifold. Details cannot be extracted simply by rescaling of the axes as done, for example, when a part of a picture is magnified. The idea of the *holographic zooming* is to take advantage of the collective normalization in the following way: 1) Identifying the part of the manifold to be magnified; 2) isolating the *subsimilarity* matrix for the cluster; 3) performing a second iteration on this matrix, i.e., the affinity transformation, dimension reduction and construction of a manifold

An essential, though often neglected aspect of brain activity is represented by its temporal dimension. The similarity matrices, the cornerstone of the FH method exposed so far, do not include essential information about the temporal propagation of activity across the components. When available, this information can be presented in temporal ordering matrices the generic *Ti,j* element of which describes the relative timing or phase difference between the activity of components *i* and *j*. Various methods can be used to evaluate the temporal ordering matrices. Recently, a new notion—the *temporal center of mass*, or temporal

manifold.

(see Fig. 3).

**2.1.4 Holographic zooming** 

**2.1.5 Inclusion of temporal information** 

representation (Fig. 2) of a network (or manifold) of linked nodes in the PCA space.

Fig. 1. Correlation matrix of synthetically produced signals. (a) Synthetic signals that include three groups—the first subgroup of nine signals (color coded in magenta) was generated by harmonic signals with the same periodicity, a phase shift of about 2π/10 and added noise. The second subgroup signals (color coded in green), is another set of harmonic signals, with a different frequency. The other signals just have pure noise with no correlations. (b) The corresponding correlation matrix that was computed using the Pearson's correlations. (c) The dendrogram tree. The vertical axis is the correlation distance between the signals (the Euclidian distance between the vectors of correlations of each signal with all the others, or the row in the correlation matrix that corresponds to the signal). Longer/shorter distances correspond to lower/higher correlations. (d) The sorted correlation matrix using the dendrogramed clustering algorithm. In this matrix the two subgroups form distinct clusters.

#### **2.1.3 Dimension reduction and construction of the holographic networks**

To search for hidden functional motifs of brain activity induced by the execution of a given task, dimension reduction of the correlation matrices is performed. 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; Baruchi et al., 2004) the axes of which are the three leading PCA principal vectors.

Fig. 1. Correlation matrix of synthetically produced signals. (a) Synthetic signals that include three groups—the first subgroup of nine signals (color coded in magenta) was generated by harmonic signals with the same periodicity, a phase shift of about 2π/10 and added noise. The second subgroup signals (color coded in green), is another set of harmonic signals, with a different frequency. The other signals just have pure noise with no correlations. (b) The corresponding correlation matrix that was computed using the Pearson's correlations. (c) The dendrogram tree. The vertical axis is the correlation distance between the signals (the Euclidian distance between the vectors of correlations of each signal with all the others, or the row in the correlation matrix that corresponds to the signal). Longer/shorter distances correspond to lower/higher correlations. (d) The sorted correlation matrix using the

dendrogramed clustering algorithm. In this matrix the two subgroups form distinct clusters.

To search for hidden functional motifs of brain activity induced by the execution of a given task, dimension reduction of the correlation matrices is performed. 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; Baruchi et al., 2004) the axes of which are the three leading PCA principal vectors.

**2.1.3 Dimension reduction and construction of the holographic networks** 

Each node is placed in this space according to its three eigenvalues for the three leading principal vectors. 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; Baruchi et al., 2004; Madi et al., 2008). The result is a holographic representation (Fig. 2) of a network (or manifold) of linked nodes in the PCA space.

Fig. 2. Holographic representation of the synthetically produced signals from Fig.1 in the 3D space. The axes are the three leading principal PCA vectors of the correlation matrix. Each node is located in this space according to its eigenvalues corresponding to the leading principal vectors. All the nodes with correlations above 0.8 are linked by lines color coded according to the correlations (represented in the colorbar), creating the holographic manifold.

#### **2.1.4 Holographic zooming**

Often, one is interested in more details about a part of the manifold. Details cannot be extracted simply by rescaling of the axes as done, for example, when a part of a picture is magnified. The idea of the *holographic zooming* is to take advantage of the collective normalization in the following way: 1) Identifying the part of the manifold to be magnified; 2) isolating the *subsimilarity* matrix for the cluster; 3) performing a second iteration on this matrix, i.e., the affinity transformation, dimension reduction and construction of a manifold (see Fig. 3).

#### **2.1.5 Inclusion of temporal information**

An essential, though often neglected aspect of brain activity is represented by its temporal dimension. The similarity matrices, the cornerstone of the FH method exposed so far, do not include essential information about the temporal propagation of activity across the components. When available, this information can be presented in temporal ordering matrices the generic *Ti,j* element of which describes the relative timing or phase difference between the activity of components *i* and *j*. Various methods can be used to evaluate the temporal ordering matrices. Recently, a new notion—the *temporal center of mass*, or temporal

Functional Holography and Cliques in Brain Activation Patterns 109

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

A versatile method of data analysis should also come with the ability to quantitatively assess difference between experimental conditions. Clustering algorithms are often used for comparison between the activities of different networks, e.g., gene expression in different groups of patients, or between two modes of behavior of the same network, e.g., during and between epileptic seizures of the same patient. We propose the following holographic comparison between networks: 1) Compute the PCA leading eigenvectors of the affinity matrix for each network. 2) Project the affinity matrix of each network on the leading eigenvectors of the other one. Clearly, this approach can also be used for comparison between different modes of activity of the same networks, like the above-mentioned case of brain activity in between and during seizure, or different clusters identified in a given matrix. Once the clusters are identified, the similarity matrix for each is isolated from the combined matrix and the above two stages are applied. The holographic superposition is designed as additional method for comparison between different modes of activity of the same network. The idea is similar to the holographic comparison; however, the projection is on the mutual PCA leading eigenvectors, i.e. the leading eigenvectors of a combined matrix

Once clusters of functional brain activity are singled out, it is often useful to describe them in a quantitative fashion. This in particular enables to compare different clusters within and

Entropy has been used in statistics and information theory to develop measures of the information content of signals (Shannon, 1948). However, entropy can also be used to measure the amount of information or variance embedded in a cluster, and to quantify the deviation of the cluster's eigenvalue distribution from a uniform one (Alter et al., 2000). This

**2.1.6 Holographic comparison and superposition of networks activity** 

generation of the signals.

that includes the different modes.

across subjects.

**2.1.7 Quantifying cluster information: Cluster entropy** 

location was introduced in the context of cultured networks (Segev et al., 2004), but works equally well for other fast continuous neurophysiological signals, ECoG and EEG, and though with minor modifications even for relatively slow signals, viz. fMRI.

Fig. 3. Holographic zooming. The FH algorithm conducted for each cluster separately; (a) cluster 1 (b) cluster 2 (c) noise signals nodes. All the nodes are linked by lines color coded according to the correlations (represented in the colorbar), creating the holographic manifold. Note that the two clusters are highly correlated whereas the noise group has no high or low correlations between them. The FH diagrams in (d,e and f) represent the same diagrams as in (a,b and c) from a different point of view while all nodes with correlations above 0.8 are linked.

The idea is to regard the activity density of each node *i* as a temporal weight function so that it's temporal center of mass, Tin, during a synchronized bursting event (SBEs), i.e. a time segment in which most of the recorded neurons exhibit rapid firing is given by *Ti*

$$\mathbf{T\_i^n} = \frac{\int (\mathbf{t} - \mathbf{T\_n}) \mathbf{D\_t^n (t - T\_n)} \mathbf{dt}}{\mathbf{D\_t^n (t - T\_n)} \mathbf{dt}} \tag{5}$$

where the integral is over the time window of the SBE, and *Tn* marks the temporal location of the nth SBE, which is the combined "center of mass" of all the neurons. The temporal center of mass of each neuron can vary between the different SBEs. Therefore we define the relative timing of a neuron i to be �� � 〈�� �〉� the average of the sequence of SBEs. Similarly, we define the temporal ordering matrix as follows:

$$T\_{l,f} = \langle T\_l^n - T\_f^n \rangle \tag{6}$$

Interestingly, when the temporal information is superimposed on the 3-D space of leading PCA eigenvectors, the activity propagates along the manifold in an orderly fashion from one end to the other (Fig. 4). For this reason, it is proposed to view the resulted manifold, which includes the temporal information as a causal manifold.

location was introduced in the context of cultured networks (Segev et al., 2004), but works equally well for other fast continuous neurophysiological signals, ECoG and EEG, and

Fig. 3. Holographic zooming. The FH algorithm conducted for each cluster separately; (a) cluster 1 (b) cluster 2 (c) noise signals nodes. All the nodes are linked by lines color coded

The idea is to regard the activity density of each node *i* as a temporal weight function so that it's temporal center of mass, Tin, during a synchronized bursting event (SBEs), i.e. a time

���������

��������� (5)

�〉� the average of the sequence of SBEs. Similarly,

�〉 (6)

according to the correlations (represented in the colorbar), creating the holographic manifold. Note that the two clusters are highly correlated whereas the noise group has no high or low correlations between them. The FH diagrams in (d,e and f) represent the same diagrams as in (a,b and c) from a different point of view while all nodes with correlations

segment in which most of the recorded neurons exhibit rapid firing is given by *Ti*

���� � 〈��

� � ���������

��

where the integral is over the time window of the SBE, and *Tn* marks the temporal location of the nth SBE, which is the combined "center of mass" of all the neurons. The temporal center of mass of each neuron can vary between the different SBEs. Therefore we define the

� � ��

Interestingly, when the temporal information is superimposed on the 3-D space of leading PCA eigenvectors, the activity propagates along the manifold in an orderly fashion from one end to the other (Fig. 4). For this reason, it is proposed to view the resulted manifold, which

T�

above 0.8 are linked.

relative timing of a neuron i to be �� � 〈��

we define the temporal ordering matrix as follows:

includes the temporal information as a causal manifold.

though with minor modifications even for relatively slow signals, viz. fMRI.

Fig. 4. Inclusion of temporal information. (a) and (b) show the inclusion of the causal information for the holographic network shown in Fig. 3a,b respectively. The activity propagation is added by coloring the nodes location according to the relative phases or time lag between them. Blue is for early time (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.

#### **2.1.6 Holographic comparison and superposition of networks activity**

A versatile method of data analysis should also come with the ability to quantitatively assess difference between experimental conditions. Clustering algorithms are often used for comparison between the activities of different networks, e.g., gene expression in different groups of patients, or between two modes of behavior of the same network, e.g., during and between epileptic seizures of the same patient. We propose the following holographic comparison between networks: 1) Compute the PCA leading eigenvectors of the affinity matrix for each network. 2) Project the affinity matrix of each network on the leading eigenvectors of the other one. Clearly, this approach can also be used for comparison between different modes of activity of the same networks, like the above-mentioned case of brain activity in between and during seizure, or different clusters identified in a given matrix. Once the clusters are identified, the similarity matrix for each is isolated from the combined matrix and the above two stages are applied. The holographic superposition is designed as additional method for comparison between different modes of activity of the same network. The idea is similar to the holographic comparison; however, the projection is on the mutual PCA leading eigenvectors, i.e. the leading eigenvectors of a combined matrix that includes the different modes.

#### **2.1.7 Quantifying cluster information: Cluster entropy**

Once clusters of functional brain activity are singled out, it is often useful to describe them in a quantitative fashion. This in particular enables to compare different clusters within and across subjects.

Entropy has been used in statistics and information theory to develop measures of the information content of signals (Shannon, 1948). However, entropy can also be used to measure the amount of information or variance embedded in a cluster, and to quantify the deviation of the cluster's eigenvalue distribution from a uniform one (Alter et al., 2000). This

Functional Holography and Cliques in Brain Activation Patterns 111

The MST method requires assigning weights to the links between the nodes. The weights are assigned according to the affinity matrix. A commonly used distance transformation is the ultrametric distance, suggested by Rammal et al. (1986) (Rammal et al., 1986) and Mantegna et al. (2000) (Mantegna & Stanley, 2000). Using this distance, the pairwise correlation coefficient for each pair of nodes is translated into the distance between those

where *C(i,j)* is the correlation coefficient between nodes *i* and *j*. This distance metric satisfies the ultrametric inequality, (I) *UD(i,j)* = 0 if and only if *i* = *j*, (II) *UD(i,j)* = *UD(j,i)*, (III) *UD(i,j)* 

*UD i j C i j* , 21 ,

The result of the ultrametric transformation is that strong positive correlations are translated into short distances, and strong negative correlations are translated into long distances. For the case of perfect positive correlation, i.e. *C(i,j)* = 1, the distance is 0; for the case of no correlation, i.e. *C(i,j)* = 0.5, the distance is 1; and for the case of perfect negative correlation, i.e. *C(i,j)* = 0, the distance is √2. The ultrametric distance matrix describes the complete network´s topological structure that yields no significant information (West, 2001). To construct the MST, the Kruskal algorithm (Kruskal, 1956) can be applied. This algorithm is considered greedy, as it runs in polynomial time (this problem however is not particularly severe if the networks have about 300 nodes, which renders the problem computable), and in each phase some local optimum is chosen. Fig. 5 demonstrates the use of the Kruskal

Fig. 5. An example of how the Kruskal algorithm can be used in order to find the minimal spanning tree from the complete graph. (a) Original graph. The numbers near the links indicate their weight. AD and CE are the shortest links, with length 5, and AD highlighted to indicate that it has been arbitrarily chosen. (b) CE is now the shortest link with length 5,

which does not form a cycle, so it is highlighted as the second link. (c) The next

(9)

two nodes. The ultrametric distance is *UD(i,j)* given by

algorithm to find the MST for a complete graph.

≤ Max{*UD(i,k)*,*UD(k,j)*}.

idea has been used in the context of biological systems (Varshavsky et al., 2007; Varshavsky et al., 2006) and economic systems (Shapira et al., 2009). The eigenvalue entropy is defined as

$$S = -\frac{1}{\log(N)} \sum\_{i=1}^{N} \Omega(i) \log\left[\varOmega(i)\right] \tag{7}$$

where Ω is given by,

$$\Omega(i) = \frac{\mathbb{A}(i)^2}{\sum\_{i=1}^{N} \mathbb{A}(i)^2} \tag{8}$$

N the number of signals, and denotes the matrix eigenvalues. S ranges from 0 to 1. Note that is a normalization factor ensuring that S reaches its maximum (S=1) for a uniform eigenvalue distribution (i.e. random correlations matrix). *i* 1 /log*N*

#### **2.1.8 Extracting topological information: MST analysis**

The correlation matrix of the system creates a topological network structure, where the links between nodes are the pair-wise correlations, and the correlation coefficients as the weights of these links. Valuable information can be extracted from the topological properties of the network, over and above the mere localization in the brain volume. To extract this information, graph theoretical techniques can be applied to the data.

The graph induced by the correlation matrix is complete and therefore difficult to interpret per se. Extracting meaningful information from this complete graph involves providing a more compact description of the graph and analyzing its topological properties (e.g. (Newman, 2003; Tumminello et al., 2007). A graph and its connectivity can be synthetically described by its minimum spanning tree (MST) (West, 2001), i.e. a connected, undirected graph composed of subgroups of edges with the following two properties: I) The tree spans the graph, i.e. connects all the nodes of the graph. The number of links retained is (n − 1) for a network of (n) nodes. II) The sum of the edges' weights is minimal out of all possible spanning trees. The MST creates a subgraph without loops, maintains the connection of all nodes, using only the links with minimal weight.

The topological structure of the constructed tree creates a new visualization of the complex system, which allows visually tracking clusters of nodes, as well as structural similarities and differences of the system under different conditions. Other graph properties, such as node degree, node centrality and betweenness can be used to extract information from the tree (Newman, 2003; Tumminello et al., 2007; West, 2001).

The MST can be constructed based of the correlation matrix (i.e. the correlation based system) obtained by the FH algorithm. Plotting the MST upon the PCA affinity space and on the anatomical slice image enables us to monitor the dynamical changes of the selected voxels and the tree they create (their connections) over the entire time course of the experiment.

idea has been used in the context of biological systems (Varshavsky et al., 2007; Varshavsky et al., 2006) and economic systems (Shapira et al., 2009). The eigenvalue entropy is defined

> 1 <sup>1</sup> log log *N*

*i*

for a uniform eigenvalue distribution (i.e. random correlations matrix).

*i*

information, graph theoretical techniques can be applied to the data.

**2.1.8 Extracting topological information: MST analysis** 

nodes, using only the links with minimal weight.

experiment.

tree (Newman, 2003; Tumminello et al., 2007; West, 2001).

*i*

N the number of signals, and denotes the matrix eigenvalues. S ranges from 0 to 1. Note that is a normalization factor ensuring that S reaches its maximum (S=1)

The correlation matrix of the system creates a topological network structure, where the links between nodes are the pair-wise correlations, and the correlation coefficients as the weights of these links. Valuable information can be extracted from the topological properties of the network, over and above the mere localization in the brain volume. To extract this

The graph induced by the correlation matrix is complete and therefore difficult to interpret per se. Extracting meaningful information from this complete graph involves providing a more compact description of the graph and analyzing its topological properties (e.g. (Newman, 2003; Tumminello et al., 2007). A graph and its connectivity can be synthetically described by its minimum spanning tree (MST) (West, 2001), i.e. a connected, undirected graph composed of subgroups of edges with the following two properties: I) The tree spans the graph, i.e. connects all the nodes of the graph. The number of links retained is (n − 1) for a network of (n) nodes. II) The sum of the edges' weights is minimal out of all possible spanning trees. The MST creates a subgraph without loops, maintains the connection of all

The topological structure of the constructed tree creates a new visualization of the complex system, which allows visually tracking clusters of nodes, as well as structural similarities and differences of the system under different conditions. Other graph properties, such as node degree, node centrality and betweenness can be used to extract information from the

The MST can be constructed based of the correlation matrix (i.e. the correlation based system) obtained by the FH algorithm. Plotting the MST upon the PCA affinity space and on the anatomical slice image enables us to monitor the dynamical changes of the selected voxels and the tree they create (their connections) over the entire time course of the

*N*

1

*i*

2

2

*i*

(7)

(8)

*i S ii N*

as

where Ω is given by,

1 /log*N*

The MST method requires assigning weights to the links between the nodes. The weights are assigned according to the affinity matrix. A commonly used distance transformation is the ultrametric distance, suggested by Rammal et al. (1986) (Rammal et al., 1986) and Mantegna et al. (2000) (Mantegna & Stanley, 2000). Using this distance, the pairwise correlation coefficient for each pair of nodes is translated into the distance between those two nodes. The ultrametric distance is *UD(i,j)* given by

$$\text{LID}\left(i,j\right) = \sqrt{2 \cdot \left(1 - \text{C}\left(i,j\right)\right)}\tag{9}$$

where *C(i,j)* is the correlation coefficient between nodes *i* and *j*. This distance metric satisfies the ultrametric inequality, (I) *UD(i,j)* = 0 if and only if *i* = *j*, (II) *UD(i,j)* = *UD(j,i)*, (III) *UD(i,j)*  ≤ Max{*UD(i,k)*,*UD(k,j)*}.

The result of the ultrametric transformation is that strong positive correlations are translated into short distances, and strong negative correlations are translated into long distances. For the case of perfect positive correlation, i.e. *C(i,j)* = 1, the distance is 0; for the case of no correlation, i.e. *C(i,j)* = 0.5, the distance is 1; and for the case of perfect negative correlation, i.e. *C(i,j)* = 0, the distance is √2. The ultrametric distance matrix describes the complete network´s topological structure that yields no significant information (West, 2001). To construct the MST, the Kruskal algorithm (Kruskal, 1956) can be applied. This algorithm is considered greedy, as it runs in polynomial time (this problem however is not particularly severe if the networks have about 300 nodes, which renders the problem computable), and in each phase some local optimum is chosen. Fig. 5 demonstrates the use of the Kruskal algorithm to find the MST for a complete graph.

Fig. 5. An example of how the Kruskal algorithm can be used in order to find the minimal spanning tree from the complete graph. (a) Original graph. The numbers near the links indicate their weight. AD and CE are the shortest links, with length 5, and AD highlighted to indicate that it has been arbitrarily chosen. (b) CE is now the shortest link with length 5, which does not form a cycle, so it is highlighted as the second link. (c) The next

Functional Holography and Cliques in Brain Activation Patterns 113

Lee et al. (Lee et al., 2006). This measure is based on the information metric d(X,Y), which quantifies the conditional entropies (or the difference) between two information sources,

where H(X|Y) and H(Y|X) are the conditional entropies between sources X and Y. This metric satisfies the triangle inequality. The conditional entropy H(X|Y) denotes the amount of information that is obtained by measuring an information source Y with the knowledge of a different source X. The information gain can be approximated by the information change between two different sources. The source X is transformed into the Y source space by the transformation Y=f(X). The average information change by the transformation is

> 1 <sup>1</sup> ln *N*

*H*

where N is the number of elements of an information source.

*DY X*

dissimilarity between two MSTs the metric distance is given by,

can then be delineated using the dendrogram clustering algorithm.

**2.2.1 Analyzing ECoG recorded human brain activity** 

in order to apply it between two MSTs,

*i i*

*N x* 

The divergence rate can be defined as an approximation of the average information change

Where *D(X)(i)* is the sum of all distances taken from a reference node *i* to the neighboring nodes in the XMST, and *D(X|Y)(i)* is the sum of all distances taken from the node *i* to the nodes in the YMST. Given the distances between node *i* to its neighbors in the XMST, *D(Y|X)(i)* evaluates how much the distance changed in the YMST. Note that the number of neighboring nodes for every reference node is different. This measure evaluates how much information is needed on average to explain YMST, given XMST. Then, to quantify the

If the MSTs are identical, *D(X,Y)* = 0, otherwise *D(X,Y)* > 0. Subgroups in the metric distance

The occurrence of epilepsy is rising and is estimated to affect, at some level, 1%–2% of the world population (Towle et al., 2002). Due to availability of many antiepileptic drugs, approximately 80% of all epileptic patients can be kept seizure free. But for the remaining 20%, the only cure is surgical resection of the seizure focus (Chkhenkeli et al., 1998; Doyle & Spencer, 1998). One of the most challenging tasks facing epileptologists is precise identification of brain areas to be removed so that the problem can be cured with minimal damage and side effects. Often, the precise location of the epileptogenic region remains

1 <sup>|</sup> <sup>1</sup> | log

10

*N DX*

*<sup>N</sup> <sup>i</sup> i i*

*DY X*

*i*

*f x*

defined as

**2.2 Applications** 

*d XY H X Y H Y X* , || , (10)

, (11)

. (12)

*DXY DY X DX Y* ,|| (13)

link, DF with length 6, is highlighted using the same method. (d) The next-shortest links are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The link BD cannot be chosen as it would form a cycle (ABD) and has therefore been marked in red. (e) The process continues to highlight the next-smallest link, BE with length 7. Many more links are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD. (f) Finally, the process finishes with the link EG, of length 9, and the minimum spanning tree is found.

The MST analysis can be used in combination with FH analysis. The FH analysis performs the aforementioned PCA dimension reduction algorithm creating a visualization of this complex network. The nodes in the reduced 3D space are then linked according to the MST connections, with lines color coded according to their original correlations. These lines create a topological MST manifold upon the 3D correlations space (see figs. 11 and 14 in section 2.2.2). This MST manifold is displayed on the brain slice image in the same way as that of the FH, thus providing information about connectivity on the real space (see figs. 12 and 15 in section 2.2.2).

Fig. 6. The MST representation of the synthetically produced signals from Fig.1. The magenta and green colors represent clusters 1 and 2 respectively, and the cyan color represents the noise signals.

#### **2.1.9 Dissimilarity measure for MSTs**

The MST can conveniently be used to quantify similarities between different networks of brain activity. This can be done by resorting to the divergence rate measure developed by

The MST analysis can be used in combination with FH analysis. The FH analysis performs the aforementioned PCA dimension reduction algorithm creating a visualization of this complex network. The nodes in the reduced 3D space are then linked according to the MST connections, with lines color coded according to their original correlations. These lines create a topological MST manifold upon the 3D correlations space (see figs. 11 and 14 in section 2.2.2). This MST manifold is displayed on the brain slice image in the same way as that of the FH, thus providing

link, DF with length 6, is highlighted using the same method. (d) The next-shortest links are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The link BD cannot be chosen as it would form a cycle (ABD) and has therefore been marked in red. (e) The process continues to highlight the next-smallest link, BE with length 7. Many more links are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD. (f) Finally, the process

finishes with the link EG, of length 9, and the minimum spanning tree is found.

information about connectivity on the real space (see figs. 12 and 15 in section 2.2.2).

Fig. 6. The MST representation of the synthetically produced signals from Fig.1. The magenta and green colors represent clusters 1 and 2 respectively, and the cyan color

The MST can conveniently be used to quantify similarities between different networks of brain activity. This can be done by resorting to the divergence rate measure developed by

represents the noise signals.

**2.1.9 Dissimilarity measure for MSTs** 

Lee et al. (Lee et al., 2006). This measure is based on the information metric d(X,Y), which quantifies the conditional entropies (or the difference) between two information sources,

$$d\left(X,Y\right) = H\left(X\mid Y\right) + H\left(Y\mid X\right)\_{\text{red}} \tag{10}$$

where H(X|Y) and H(Y|X) are the conditional entropies between sources X and Y. This metric satisfies the triangle inequality. The conditional entropy H(X|Y) denotes the amount of information that is obtained by measuring an information source Y with the knowledge of a different source X. The information gain can be approximated by the information change between two different sources. The source X is transformed into the Y source space by the transformation Y=f(X). The average information change by the transformation is defined as

$$\overline{\Delta H} = \frac{1}{N} \sum\_{i=1}^{N} \ln \left| \frac{\Delta f\left(\mathbf{x}\_{i}\right)}{\Delta \mathbf{x}\_{i}} \right|,\tag{11}$$

where N is the number of elements of an information source.

The divergence rate can be defined as an approximation of the average information change in order to apply it between two MSTs,

$$D\left(Y \mid X\right) = \frac{1}{N} \sum\_{i=1}^{N} \log\_{10} \left| \frac{D\left(Y \mid X\right)\_{(i)}}{D\left(X\right)\_{(i)}} \right| \,. \tag{12}$$

Where *D(X)(i)* is the sum of all distances taken from a reference node *i* to the neighboring nodes in the XMST, and *D(X|Y)(i)* is the sum of all distances taken from the node *i* to the nodes in the YMST. Given the distances between node *i* to its neighbors in the XMST, *D(Y|X)(i)* evaluates how much the distance changed in the YMST. Note that the number of neighboring nodes for every reference node is different. This measure evaluates how much information is needed on average to explain YMST, given XMST. Then, to quantify the dissimilarity between two MSTs the metric distance is given by,

$$D(X,Y) = D(Y \mid X) + D(X \mid Y) \tag{13}$$

If the MSTs are identical, *D(X,Y)* = 0, otherwise *D(X,Y)* > 0. Subgroups in the metric distance can then be delineated using the dendrogram clustering algorithm.

#### **2.2 Applications**

#### **2.2.1 Analyzing ECoG recorded human brain activity**

The occurrence of epilepsy is rising and is estimated to affect, at some level, 1%–2% of the world population (Towle et al., 2002). Due to availability of many antiepileptic drugs, approximately 80% of all epileptic patients can be kept seizure free. But for the remaining 20%, the only cure is surgical resection of the seizure focus (Chkhenkeli et al., 1998; Doyle & Spencer, 1998). One of the most challenging tasks facing epileptologists is precise identification of brain areas to be removed so that the problem can be cured with minimal damage and side effects. Often, the precise location of the epileptogenic region remains

Functional Holography and Cliques in Brain Activation Patterns 115

Notably, the manifold of the inter-ictal activity has a very simple topology of almost circular horseshoe like part and another subgroup perpendicular to its plane and position at the center of the horseshoe. Although the new manifold has as expected a more complex topology, it retains some of the features of the one associated with inter-Ictal activity, when viewed from specific angle. Preliminary analyses also indicate that causal features are captured when the temporal (i.e., phase coherences) information is imposed on the manifolds. These results bear the promise that functional holography might become a

When applied to fMRI data, FH is an effective clustering method, capable of capturing system level networks using voxel-voxel correlation matrices (Jacob et al., 2010). Here we show how the algorithm using a dendrogram clustering method combined with a standard deviation (STD) filter can effectively be used to identify and extract voxel clusters. Subjects were instructed to clench and open either their left or right hand, according to an auditory cue. The paradigm consisted of 11 blocks of 114 volumes. Each consisted of a resting period with cross xation (6–15 s), an auditory instruction period regarding hand movement (right or left; 3 s), and a period of hand movement execution (15 s). The blocks were presented in a constant order across subjects with regard to which hand to move. Two types of sequences were examined: *repetitive* (two consecutive movements of the same hand) and *alternating*

Even for this simple hand clenching motor task, the FH analysis conducted for a single block revealed interesting motifs. For example, unilateral hand movement yielded two clusters, one located in the contralateral primary motor cortex showing increased signal (i.e. activation), and the other one in the ipsilateral homologue region, showing reduced signal (i.e. deactivation) (Fig. 8). Inspection of repetitive vs. alternating hand movements suggested that this pattern could be indicative of an inhibition mechanism of the ipsilateral hemisphere. In addition, a single-block level analysis, using only 12 time points corresponding to a 36 second recording session, was enough to determine which hand was moved by the subject, while other methods required the entire experiment time course. Moreover, cluster quantification based on eigenvalue entropy showed lower entropy for the motor-dominant hemisphere clusters. This lower entropy demonstrates less variability in the cluster's correlations, suggesting a higher

The MST was extracted for each subject and each block. The divergence rate measure was used for quantification of the structural similarities and differences of the system under the two different conditions of right or left hand movement. Fig. 9 displays examples of the dendrograms constructed from the divergence rate measure. Each dendrogram represents the distance, i.e. the divergence rate measure or similarity between all pairwise MSTs of the experiment's different blocks. Half of the subjects exhibited good separation between right and

(d) show the corresponding dendrogram correlation matrices. (e) and (f) show the corresponding FH manifolds in the PCA affinity space. 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-

valuable diagnostic procedure in the treatment of intractable epilepsy.

**2.2.2 Analyzing fMRI recorded human brain activity** 

modular organization in specific motor dominant hemisphere.

Ictal durations.

hand movements.

uncertain after obtaining conventional, noninvasive measurements such as electroencephalogram (EEG) and magnetoencephalogram (MEG) cannot provide sufficient information because of the relatively low spatial resolution of these methods. In these cases, the activity is directly recorded by the electrocorticography (ECoG) procedure in which the recording electrodes are placed directly on the brain surface.

Here we illustrate how the FH method can be applied to reveal the existence of hidden causal manifolds in the electrical brain activity of epileptic patients with implanted electrodes. We note that the method can also be applied to experimental seizure studies that have gained much attention (Ben-Jacob et al., 2007 ). Typical results are presented in Fig. 7.

Fig. 7. Holographic networks of recorded brain activity. The holographic networks are for the ECoG recorded human brain activity for the inter-Ictal and Ictal activities. (a) and (b) show the connectivity diagram for the inter-Ictal and Ictal respectively, constructed upon the set of electrodes placed on the surface of the brain (the frontal lobe in this case). (c) and

uncertain after obtaining conventional, noninvasive measurements such as electroencephalogram (EEG) and magnetoencephalogram (MEG) cannot provide sufficient information because of the relatively low spatial resolution of these methods. In these cases, the activity is directly recorded by the electrocorticography (ECoG) procedure in which the

Here we illustrate how the FH method can be applied to reveal the existence of hidden causal manifolds in the electrical brain activity of epileptic patients with implanted electrodes. We note that the method can also be applied to experimental seizure studies that have gained much attention (Ben-Jacob et al., 2007 ). Typical results are presented in Fig. 7.

Fig. 7. Holographic networks of recorded brain activity. The holographic networks are for the ECoG recorded human brain activity for the inter-Ictal and Ictal activities. (a) and (b) show the connectivity diagram for the inter-Ictal and Ictal respectively, constructed upon the set of electrodes placed on the surface of the brain (the frontal lobe in this case). (c) and

recording electrodes are placed directly on the brain surface.

(d) show the corresponding dendrogram correlation matrices. (e) and (f) show the corresponding FH manifolds in the PCA affinity space. 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.

Notably, the manifold of the inter-ictal activity has a very simple topology of almost circular horseshoe like part and another subgroup perpendicular to its plane and position at the center of the horseshoe. Although the new manifold has as expected a more complex topology, it retains some of the features of the one associated with inter-Ictal activity, when viewed from specific angle. Preliminary analyses also indicate that causal features are captured when the temporal (i.e., phase coherences) information is imposed on the manifolds. These results bear the promise that functional holography might become a valuable diagnostic procedure in the treatment of intractable epilepsy.

#### **2.2.2 Analyzing fMRI recorded human brain activity**

When applied to fMRI data, FH is an effective clustering method, capable of capturing system level networks using voxel-voxel correlation matrices (Jacob et al., 2010). Here we show how the algorithm using a dendrogram clustering method combined with a standard deviation (STD) filter can effectively be used to identify and extract voxel clusters. Subjects were instructed to clench and open either their left or right hand, according to an auditory cue. The paradigm consisted of 11 blocks of 114 volumes. Each consisted of a resting period with cross xation (6–15 s), an auditory instruction period regarding hand movement (right or left; 3 s), and a period of hand movement execution (15 s). The blocks were presented in a constant order across subjects with regard to which hand to move. Two types of sequences were examined: *repetitive* (two consecutive movements of the same hand) and *alternating* hand movements.

Even for this simple hand clenching motor task, the FH analysis conducted for a single block revealed interesting motifs. For example, unilateral hand movement yielded two clusters, one located in the contralateral primary motor cortex showing increased signal (i.e. activation), and the other one in the ipsilateral homologue region, showing reduced signal (i.e. deactivation) (Fig. 8). Inspection of repetitive vs. alternating hand movements suggested that this pattern could be indicative of an inhibition mechanism of the ipsilateral hemisphere. In addition, a single-block level analysis, using only 12 time points corresponding to a 36 second recording session, was enough to determine which hand was moved by the subject, while other methods required the entire experiment time course. Moreover, cluster quantification based on eigenvalue entropy showed lower entropy for the motor-dominant hemisphere clusters. This lower entropy demonstrates less variability in the cluster's correlations, suggesting a higher modular organization in specific motor dominant hemisphere.

The MST was extracted for each subject and each block. The divergence rate measure was used for quantification of the structural similarities and differences of the system under the two different conditions of right or left hand movement. Fig. 9 displays examples of the dendrograms constructed from the divergence rate measure. Each dendrogram represents the distance, i.e. the divergence rate measure or similarity between all pairwise MSTs of the experiment's different blocks. Half of the subjects exhibited good separation between right and

Functional Holography and Cliques in Brain Activation Patterns 117

right and left hand movements. Figs.11-15 depicts the MSTs on the FH correlation PCA space, and on the anatomical brain slice image of a single subject that showed a good separation between right and left hand movements (Figs.11-12). A single subject who did not yield a good separation is also shown (Figs.13-15). Looking at the dynamic changes of the MSTs along the experiment time course in the anatomical slice image it becomes visible that blocks of sequences of repetitive hand movements resulted in numerous connections between the clusters as opposed to the alternating hand movements' blocks. The graph in Fig. 16 displays the average Z score of the number of connections between the clusters for each block across subjects (N=15). This demonstrates that these two kinds of sequences can

Fig. 9. Dendrograms constructed from the divergence rate measure for two different subjects. Each dendrogram represents the distances i.e. the divergence rate measure between all pair wise MSTs. The dendrogram clusters all the MSTs that show similar structure. The subjects were divided into two groups; (a) An example of a subject who's MSTs had much similarity showing clusters separating between right and left hand movement MSTs. (b) An example of a subject that showed no distinct clusters for MSTs

Fig. 10. Example of the MSTs constructed for a single left-handed male subject that shows a good separation in the divergence rate dendrogram. Displaying the MSTs for all the experiment's ten blocks i.e. right hand (RH) and left hand (LH) movements. The MST color

also be differentiated by the MST visualization.

associated with right and left hand movements.

left hand movement MSTs (e.g. Fig. 9a), displaying clusters of 3-4 same hand movement trees. The other half of the subjects did not yield such good separation (e.g. Fig. 9b). Fig. 10 depicts the MSTs for a single subject that showed good separation. To elucidate the functional and structural meaning of the MST, the tree's nodes were colored according to their location in the brain; the red and blue colors represent voxels in the right and left hemisphere respectively, while yellow represents the midline SMA region. All the resulting MSTs showed two distinct clusters, one dominated by the right (red) and one by the left hemisphere (blue). The interesting result is that this representation highlights for all blocks a few red voxels in the blue cluster and blue ones in the red cluster, these voxels are the same in every block.

Fig. 8. FH applied on fMRI data. Presented here is an example of the results of a righthanded single subject, for a right hand movement block (12 TRs). (a) The correlation matrix shows a pattern of two dominant distinguished clusters. (b) The magenta cluster averaged BOLD signal demonstrated clear activation, whereas the green cluster showed deactivation. (c) The magenta activation cluster was located at the left hemisphere in the M1 region and, as expected. The second green cluster was located in the M1 region of the ipsihemisphere. (d) and (e) show the holographic presentations of the voxels or the holographic networks while the voxels with correlations above 0.8 are linked in (d) and voxels with correlations below -0.4 are linked in (e). (f) Displays the corresponding holographic networks on the brain slice image for correlations with a specific range of 0.98-0.99 and (-0.9)–(-1.0).

To further investigate whether the two groups of subjects as obtained by the divergence rate measure, differed in terms of topological structure of their functional networks of correlated activity, the MST was used in combination with the FH visualization. The MST constructed for every single block shows a good separation between the hemispheres. For half of the subjects, the divergence rate measure allowed to partition the MSTs into two clusters of

left hand movement MSTs (e.g. Fig. 9a), displaying clusters of 3-4 same hand movement trees. The other half of the subjects did not yield such good separation (e.g. Fig. 9b). Fig. 10 depicts the MSTs for a single subject that showed good separation. To elucidate the functional and structural meaning of the MST, the tree's nodes were colored according to their location in the brain; the red and blue colors represent voxels in the right and left hemisphere respectively, while yellow represents the midline SMA region. All the resulting MSTs showed two distinct clusters, one dominated by the right (red) and one by the left hemisphere (blue). The interesting result is that this representation highlights for all blocks a few red voxels in the blue

cluster and blue ones in the red cluster, these voxels are the same in every block.

Fig. 8. FH applied on fMRI data. Presented here is an example of the results of a righthanded single subject, for a right hand movement block (12 TRs). (a) The correlation matrix shows a pattern of two dominant distinguished clusters. (b) The magenta cluster averaged BOLD signal demonstrated clear activation, whereas the green cluster showed deactivation. (c) The magenta activation cluster was located at the left hemisphere in the M1 region and, as expected. The second green cluster was located in the M1 region of the ipsihemisphere. (d) and (e) show the holographic presentations of the voxels or the holographic networks while the voxels with correlations above 0.8 are linked in (d) and voxels with correlations below -0.4 are linked in (e). (f) Displays the corresponding holographic networks on the brain slice image for correlations with a specific range of 0.98-0.99 and (-0.9)–(-1.0).

To further investigate whether the two groups of subjects as obtained by the divergence rate measure, differed in terms of topological structure of their functional networks of correlated activity, the MST was used in combination with the FH visualization. The MST constructed for every single block shows a good separation between the hemispheres. For half of the subjects, the divergence rate measure allowed to partition the MSTs into two clusters of right and left hand movements. Figs.11-15 depicts the MSTs on the FH correlation PCA space, and on the anatomical brain slice image of a single subject that showed a good separation between right and left hand movements (Figs.11-12). A single subject who did not yield a good separation is also shown (Figs.13-15). Looking at the dynamic changes of the MSTs along the experiment time course in the anatomical slice image it becomes visible that blocks of sequences of repetitive hand movements resulted in numerous connections between the clusters as opposed to the alternating hand movements' blocks. The graph in Fig. 16 displays the average Z score of the number of connections between the clusters for each block across subjects (N=15). This demonstrates that these two kinds of sequences can also be differentiated by the MST visualization.

Fig. 9. Dendrograms constructed from the divergence rate measure for two different subjects. Each dendrogram represents the distances i.e. the divergence rate measure between all pair wise MSTs. The dendrogram clusters all the MSTs that show similar structure. The subjects were divided into two groups; (a) An example of a subject who's MSTs had much similarity showing clusters separating between right and left hand movement MSTs. (b) An example of a subject that showed no distinct clusters for MSTs associated with right and left hand movements.

Fig. 10. Example of the MSTs constructed for a single left-handed male subject that shows a good separation in the divergence rate dendrogram. Displaying the MSTs for all the experiment's ten blocks i.e. right hand (RH) and left hand (LH) movements. The MST color

Functional Holography and Cliques in Brain Activation Patterns 119

Fig. 12. The same MSTs from figs. 10 and 11 constructed upon the brain slice EPI image. This presentation shows that according to the MSTs the two hemispheres are highly coupled

Fig. 13. Example of the MSTs constructed for a single left-handed subject that does not show a good separation in the divergence rate dendrogram. Displaying the MSTs for all ten blocks i.e. right hand (RH) and left hand (LH) movements. The red and blue colors represent the

(with positive correlation and with negative correlation) in the motor task.

coded according to the brain anatomic structure. The red and blue colors represent the right and left hemisphere voxels respectively, and the yellow color represents the SMA area. The MSTs give a good separation between the right and left hemisphere. Note that in every MST there are few red voxles in the blue cluster and one blue voxle in the red cluster, since all the voxles are labeled a closer inspection shows that these outlier voxels are the exact same voxles in every MST.

Fig. 11. The MSTs from fig. 10 constructed on the FH affinity PCA space. Displaying the MSTs of a subject that resulted in a good separation between right and left hand MSTs, showing high similarity between right hand block MSTs, and between left hand block MSTs. Each 3D graph represent the same voxels (color coded according to their original cluster with magenta and green), for each experiment block. These voxels are presented in their new location in the correlation space and connected according to the block's MST with lines color coded according to their correlation coefficient. In this visualization it is hard to detect the similarities and dissimilarities between the tree's structures. However it becomes visible in this case, that right hand movement's MSTs yielded a better separation of the clusters as opposed to left hand movements.

coded according to the brain anatomic structure. The red and blue colors represent the right and left hemisphere voxels respectively, and the yellow color represents the SMA area. The MSTs give a good separation between the right and left hemisphere. Note that in every MST there are few red voxles in the blue cluster and one blue voxle in the red cluster, since all the voxles are labeled a closer inspection shows that these outlier voxels are the exact same

Fig. 11. The MSTs from fig. 10 constructed on the FH affinity PCA space. Displaying the MSTs of a subject that resulted in a good separation between right and left hand MSTs, showing high similarity between right hand block MSTs, and between left hand block MSTs. Each 3D graph represent the same voxels (color coded according to their original cluster with magenta and green), for each experiment block. These voxels are presented in their new location in the correlation space and connected according to the block's MST with lines color coded according to their correlation coefficient. In this visualization it is hard to detect the similarities and dissimilarities between the tree's structures. However it becomes visible in this case, that right hand movement's MSTs yielded a better separation of the clusters as

voxles in every MST.

opposed to left hand movements.

Fig. 12. The same MSTs from figs. 10 and 11 constructed upon the brain slice EPI image. This presentation shows that according to the MSTs the two hemispheres are highly coupled (with positive correlation and with negative correlation) in the motor task.

Fig. 13. Example of the MSTs constructed for a single left-handed subject that does not show a good separation in the divergence rate dendrogram. Displaying the MSTs for all ten blocks i.e. right hand (RH) and left hand (LH) movements. The red and blue colors represent the

Functional Holography and Cliques in Brain Activation Patterns 121

Fig. 15. The same MSTs from figs. 13 and 14 constructed upon the brain slice image. In this presentation, the two hemispheres seem to be highly negatively connected within the motor

The topological structure of the constructed tree allows visual tracking clusters of nodes, and the variations undergone by the system as it faces different experimental conditions. This visualization is of paramount importance when dealing with highly complex systems, and is particularly helpful in the identification of clusters and their hierarchies. Thus two different clusters, each dominated by one hemisphere (figs. 10 and 13) or "outlier" voxels, which may have an important part in inter-hemispheric communication, could be

Finally we point out one drawback of the MST method. When applied on a correlationbased system the MST uses the shortest distances. This induces a bias to positive correlations, while anti-correlations are overlooked. Further analysis of the data treating positive and the negative correlations on a par level (e.g. using the absolute values of the

Overall, with this extremely simple example, we have illustrated how analyzing a very basic topological network property of networks of correlated activity associated with different cognitive conditions can reveal, in a rather parsimonious way, it's most important connections, suggesting the potential of this type of analysis in dealing with more

task blocks.

highlighted.

correlation matrix) may be recommended.

challenging and rich data.

right and left hemisphere voxels respectively, and the yellow color represents the SMA area. Although this subject does not yield a good separation between the right and left hand MSTs his MSTs do show a good separation between the right and left hemispheres.

Fig. 14. The MSTs from fig. 13 constructed on the FH affinity PCA space. Demonstrating the MSTs of a subject who displayed no similarities or dissimilarities between right and left hand block MSTs in the divergence rate dendrogram tree.

right and left hemisphere voxels respectively, and the yellow color represents the SMA area. Although this subject does not yield a good separation between the right and left hand MSTs his MSTs do show a good separation between the right and left hemispheres.

Fig. 14. The MSTs from fig. 13 constructed on the FH affinity PCA space. Demonstrating the MSTs of a subject who displayed no similarities or dissimilarities between right and left

hand block MSTs in the divergence rate dendrogram tree.

Fig. 15. The same MSTs from figs. 13 and 14 constructed upon the brain slice image. In this presentation, the two hemispheres seem to be highly negatively connected within the motor task blocks.

The topological structure of the constructed tree allows visual tracking clusters of nodes, and the variations undergone by the system as it faces different experimental conditions. This visualization is of paramount importance when dealing with highly complex systems, and is particularly helpful in the identification of clusters and their hierarchies. Thus two different clusters, each dominated by one hemisphere (figs. 10 and 13) or "outlier" voxels, which may have an important part in inter-hemispheric communication, could be highlighted.

Finally we point out one drawback of the MST method. When applied on a correlationbased system the MST uses the shortest distances. This induces a bias to positive correlations, while anti-correlations are overlooked. Further analysis of the data treating positive and the negative correlations on a par level (e.g. using the absolute values of the correlation matrix) may be recommended.

Overall, with this extremely simple example, we have illustrated how analyzing a very basic topological network property of networks of correlated activity associated with different cognitive conditions can reveal, in a rather parsimonious way, it's most important connections, suggesting the potential of this type of analysis in dealing with more challenging and rich data.

Functional Holography and Cliques in Brain Activation Patterns 123

The method also provides an effective visualization of the system, of critical importance when dealing with highly complex systems, and is particularly helpful in the identification of clusters and their hierarchies. Even more important though is the ability of the FH analysis to reveal subtle, system-level dynamical features that are hard to detect through other methods, even at the single subject level. And that might be overlooked due to prior assumptions by hypothesis-driven methods. In fact, the FH method can capture sensitive hemodynamic variations at the single block level, without further need for averaging or for contrasts between experimental conditions. In addition the method requires far less time point to localize activations than other clustering methods, viz. ICA (Bell & Sejnowski, 1995), FCA (Windischberger et al., 2003) or TCA (Zhao et al., 2007), suggesting that the FH method may play a prominent role in the development of classification algorithms for blind

It is important to portray the FH method not only as an alternative but also as a valuable complement to existing methods. For instance, its dimension reduction step could be carried out using a variety of clustering techniques. Perhaps even more cogently, there is a clear complementarily between network theory and the FH method. The application of the former that we presented, i.e. the MST, clearly represents but one out of the many possible applications. To the extremely vast field of issues that network theory allows to address in a versatile but quantitatively rigorous and qualitatively explicit way, the FH method adds a compact representation in an auxiliary field that makes functional networks more explicit,

A distinctive quality of the FH method is represented by its versatility. While originally developed for cultured neural networks, the method can be applied to the analysis of essentially any type of signal, including the main tools for system-level neuroimaging, viz. EEG/MEG and fMRI. Although fewer examples of application to the latter are and further investigation of the method on different (viz. event-related) designs is needed, the proposed method shows great potential even for fMRI data in differentiating experimental conditions particularly when the corresponding signals are separated (Jacob et al., 2010). Since the outcome of the analysis is a holographic presentation in an abstract reduced space, it represents an ideal tool for multi-modal analysis of data from experiments combining EEG´s temporal precision with fMRI´s spatial one. Finally, the principles and implementation of the FH analysis are relatively simple and straightforward; taken together with the methods efficiency in delineating and tracking the time-varying unfolding of fine details of clustered activity at different spatial scales, it may represent a tool of election for brain scientists and

We are most thankful to Asaf Madi, Dror Kenett, Amir Rapson, Michal Kafri and Keren Rosenberg for fruitful discussions. This research has been supported in part by the Israel-US Binational Science Foundation – 2005385 (EBJ, TH), the Maguy-Glass chair in physics of complex systems (EBJ), the Tauber Family Funds and the Italy-Israel program in System Level Network Neuroscience at Tel Aviv University (EBJ), the National Science Foundation Grants PHY- 0216576 and 0225630 at UCSD (EBJ), the Israel Science Foundation - 1747/07

(TH) and by the U.S Department Of Defense W81XWH-11-2-0008 (TH).

identification of different conditions in extremely short time series.

as it divorces them from the anatomical space in which they live.

for clinical neurologists alike.

**4. Acknowledgment** 

Fig. 16. The averaged Z score of number of links in the MST connecting the two clusters across subjects (N=15) for each experiment block. The two repetitive hand movements (i.e. blocks 2 and 8) resulted in higher average than the rest of the blocks. A statistical Z-test was calculated for each block with the null hypothesis that the scores in each block are a random sample from a normal distribution with mean zero. According to this test the two repetitive hand movements were found significant (p=2x10-4 and p=6.9x10-10 for blocks 2 and 8 respectively).
