**1. Introduction**

100 Advances in Brain Imaging

[117]Cope, M. and D.T. Delpy, *System for long-term measurement of cerebral blood and tissue* 

[118]Obrig, H. and A. Villringer, *Beyond the visible--imaging the human brain with light.* J Cereb

[119]Chance, B., et al., *Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain.* Proc Natl Acad Sci U S A, 1988. 85(14): p. 4971-5. [120]Fantini, S., et al., *Non-invasive optical monitoring of the newborn piglet brain using* 

[121]Tsuchiya, Y., *Photon path distribution and optical responses of turbid media: theoretical* 

[122]Firbank, M., et al., *Experimental and theoretical comparison of NIR spectroscopy measurements of cerebral hemoglobin changes.* J Appl Physiol, 1998. 85(5): p. 1915-21. [123]van der Zee, P., et al., *Experimentally measured optical pathlengths for the adult head, calf* 

[124]Hoshi, Y. and S.J. Chen, *Regional cerebral blood flow changes associated with emotions in* 

[125]Adcock, L.M., et al., *Neonatal intensive care applications of near-infrared spectroscopy.* Clin

[126]Hopton, P., T.S. Walsh, and A. Lee, *CBF in adults using near infrared spectroscopy (NIRS):* 

[127]Sokol, D.K., et al., *Near infrared spectroscopy (NIRS) distinguishes seizure types.* Seizure,

[128]Soul, J.S. and A.J. du Plessis, *New technologies in pediatric neurology. Near-infrared* 

[129]Boas, D.A., et al., *The accuracy of near infrared spectroscopy and imaging during focal* 

[130]Hess, A., et al., *New insights into the hemodynamic blood oxygenation level-dependent* 

[131]Pouratian, N., et al., *Spatial/temporal correlation of BOLD and optical intrinsic signals in* 

[132]Paley, M., et al., *Design and initial evaluation of a low-cost 3-Tesla research system for* 

*response through combination of functional magnetic resonance imaging and optical* 

*combined optical and functional MR imaging with interventional capability.* J Magn

*potential for bedside measurement?* Br J Anaesth, 1996. 77(1): p. 131.

*changes in cerebral hemodynamics.* Neuroimage, 2001. 13(1): p. 76-90.

*recording in gerbil barrel cortex.* J Neurosci, 2000. 20(9): p. 3328-38.

*spectroscopy.* Semin Pediatr Neurol, 1999. 6(2): p. 101-10.

*humans.* Magn Reson Med, 2002. 47(4): p. 766-76.

Reson Imaging, 2001. 13(1): p. 87-92.

Comput, 1988. 26(3): p. 289-94.

1543-63.

84.

Blood Flow Metab, 2003. 23(1): p. 1-18.

Exp Med Biol, 1992. 316: p. 143-53.

Perinatol, 1999. 26(4): p. 893-903, ix.

2000. 9(5): p. 323-7.

*children.* Pediatr Neurol, 2002. 27(4): p. 275-81.

*oxygenation on newborn infants by near infra-red transillumination.* Med Biol Eng

*continuous-wave and frequency-domain spectroscopy.* Phys Med Biol, 1999. 44(6): p.

*analysis based on the microscopic Beer-Lambert law.* Phys Med Biol, 2001. 46(8): p. 2067-

*and forearm and the head of the newborn infant as a function of inter optode spacing.* Adv

The brain is a complex spatially extended biological system, where a great number of neurons (~1011) interact to carry out extremely sophisticated tasks. Alongside a wellestablished tradition of studies of single neuron activity, a wealth of neuroimaging techniques has been developed where brain activity at various spatial scales is observed in terms of multichannel recordings of the dynamics of its components.

Early neuroimaging studies of brain activity mainly focused on the functional specialization of segregated brain modules. The main concern of these studies was that of finding which brain areas change their activity as subjects carry out well-controlled tasks. A robust statistical underpinning for the quantitative analysis of results was offered by the general linear model and Gaussian field theory (Worsley & Friston, 1995), which allowed delineating a collection of significant cortical *activations* and *deactivations* associated with the execution of these tasks. From a computational point of view, this general univariate framework treated the brain as a collection of independent brain regions.

While the brain developed largely segregated modules, communication between and within these modules is essential to the transfer and processing of information. Accordingly, neuroimaging studies started incorporating the idea that the neural activity associated with the execution of given cognitive tasks is indeed diffuse, and that the influence that one brain region exerts over the others cannot be neglected. As a consequence, over the past few years, the neuroimaging literature has seen a shift towards a focus on measures of functional integration of brain activity. Many methods were developed to estimate functional and effective connectivity (Friston, 1994). These methods were designed to investigate how a

Functional Holography and Cliques in Brain Activation Patterns 103

been convincingly shown that brain anatomical networks have characteristically smallworld properties of dense or clustered local connectivity with relatively few long-range connections (Sporns et al., 2004). Similarly, human brain functional networks associated with the execution of cognitive tasks have also been associated with fractal small-world architecture (Achard et al., 2006; Bassett et al., 2006; Eguíluz et al., 2005; Salvador et al., 2005), which support efficient parallel information transfer at relatively low costs and is differently impaired by normal aging and pharmacological manipulations (Achard & Bullmore, 2007; Bassett et al., 2009). Furthermore, specific neuroanatomical connectivity patterns are univocally associated with given functional complexity levels, and networks capable of producing highly complex functional dynamics share common structural motifs (see e.g. (Sporns et al., 2000, 2002)). Finally, simulations showed that brain dynamics exhibits a modular hierarchical organization, where clusters coincide with the topological

Arguably graph theory's greatest strengths is that it has made possible to address a whole range of new research questions, far exceeding the original main one addressed by neuroimaging, of localizing brain activity, particularly issues related to *how* the brain organizes its activity as it carries out tasks of arbitrary complexity. A relative limitation of graph theoretical applications, in their current form, to neuroimaging is that both computations and visualization of functional brain networks are performed based on the Euclidean coordinates of observed activity. However, it has long been known that there is no straightforward correspondence between spatial and functional proximity between brain regions, so that regions that are contiguous to each other can in fact be involved in the execution of completely different tasks. It is then of great interest to be able to represent the topology of the functional space, and ultimately to delineate the correspondence between

Here we propose a new method, Functional Holography (FH), designed to describe the information content of a network as it functions as a whole unit. The term used for the familiar holograms indicates that the photographic plates can capture the whole information about the 3D image. The FH method can overcome the main limitations of previous methods by visualizing networks of correlated activity in an auxiliary space of correlations

1. To overcome the limitations of existing methods taking into account only a fraction of

2. To identify underlying functional motives embedded in complex spatio-temporal

3. To identify functional subgroups functional clusters and to reveal the causal relations

4. To relate the observed temporal ordering activity propagation to underlying causal

5. To be able to compare the activity of two different networks or different modes of

In the remainder of this chapter, we will first illustrate the mathematical procedure of the method; we will then show some applications of the method to various neurophysiological

community structure of anatomical networks (Zhou et al., 2006).

and linking the components according to similarities between them.

motives propagation of information and causal connectivity.

anatomical and functional spaces.

The main objectives of the FH method are:

the network components.

behavior of the same network.

behavior.

between them.

generally rather small set of brain areas interact, and how different experimental manipulations may affect their mutual relationships (Friston et al., 1997). More or less coarse-grained brain regions are identified with the nodes of a network, while some metric of brain coupling between these regions is identified with an edge between these nodes. Prominent among these methods are *data-driven* methods such as Independent component analysis (ICA) (McKeown et al., 1998), Fuzzy Clustering Analysis (FCA) (Windischberger et al., 2003), Temporal Clustering Analysis (TCA) (Zhao et al., 2007), and autoregressive models such as the Granger Causal Mapping (GCM) (Goebel et al., 2003) and *model-driven* dynamical models expressed in dynamic causal modeling (DCM) (Friston et al., 2003). The former set of methods started exploiting the inherent multivariate and stochastic nature of fMRI data. Model-driven approaches, on the other hand, used causal influences among neural sources to produce an explicit computational model generating the observed signal. This method improved on early methods by incorporating an explicit temporal component into effective connectivity estimation (Penny et al., 2004). The main merits of these methods were that of making explicit the spatially non-local nature of task-related brain activity, and of adding to it a (rather coarse) temporal dimension. However, these methods are typically limited in the number of regions they can incorporate. Furthermore, while these methods incorporate the idea that correlations among neuronal assemblies play an important role in brain activity (Segev et al., 2004), no clear distinction between information processing and information transfer is made, and the output is essentially a flow-chart of communication between nodes. As a consequence, the meaningfulness of the networks that are delineated boils down to the combined functional properties attributed to the segregated brain regions that are identified with the network nodes, but it is unrelated to some general property of the network *per se*. This in turn implies, among other things, that no clear relationship exists between brain anatomy, the structure of functional networks of brain activity and the dynamics taking place on them.

While single region activity can be characterized in a straightforward way through timevarying profiles of amplitudes of some aspect of brain activity, network activity needs appropriate non-trivial observables to be defined. Graph theory (Boccaletti et al., 2006) offers a convenient and flexible way to analyze topological properties of systems with a network organization (Bullmore et al., 2009). Most importantly for neuroscientists, graph theory can be used to understand the complex relationship between structure, dynamics and function in the brain. Graph theory shows that the topology of structural networks influences the dynamical processes (namely synchronization) taking place on them (Boccaletti et al., 2006). For instance, small-world properties of dense or clustered local connectivity with relatively few long-range connections confer distinctive dynamical and functional properties: in addition to optimizing information processing (Strogatz, 2001), facilitating synchronization (Bucolo et al., 2003), ensuring rapid response and emergence of coherent oscillations (Lago-Fernández et al., 2000), and conferring resilience against pathological attack, small-world architecture has been shown to provide an optimum trade-off between efficiency and wiring costs, conferring high local and global efficiencies for relatively low connection costs (Latora & Marchiori, 2001).

Recently, an increasing number of neuroimaging studies using graph theoretical tools have started showing that the brain developed in such a way that a clear correspondence exists between anatomical network topology and dynamical processes taking place on it. It has

generally rather small set of brain areas interact, and how different experimental manipulations may affect their mutual relationships (Friston et al., 1997). More or less coarse-grained brain regions are identified with the nodes of a network, while some metric of brain coupling between these regions is identified with an edge between these nodes. Prominent among these methods are *data-driven* methods such as Independent component analysis (ICA) (McKeown et al., 1998), Fuzzy Clustering Analysis (FCA) (Windischberger et al., 2003), Temporal Clustering Analysis (TCA) (Zhao et al., 2007), and autoregressive models such as the Granger Causal Mapping (GCM) (Goebel et al., 2003) and *model-driven* dynamical models expressed in dynamic causal modeling (DCM) (Friston et al., 2003). The former set of methods started exploiting the inherent multivariate and stochastic nature of fMRI data. Model-driven approaches, on the other hand, used causal influences among neural sources to produce an explicit computational model generating the observed signal. This method improved on early methods by incorporating an explicit temporal component into effective connectivity estimation (Penny et al., 2004). The main merits of these methods were that of making explicit the spatially non-local nature of task-related brain activity, and of adding to it a (rather coarse) temporal dimension. However, these methods are typically limited in the number of regions they can incorporate. Furthermore, while these methods incorporate the idea that correlations among neuronal assemblies play an important role in brain activity (Segev et al., 2004), no clear distinction between information processing and information transfer is made, and the output is essentially a flow-chart of communication between nodes. As a consequence, the meaningfulness of the networks that are delineated boils down to the combined functional properties attributed to the segregated brain regions that are identified with the network nodes, but it is unrelated to some general property of the network *per se*. This in turn implies, among other things, that no clear relationship exists between brain anatomy, the structure of functional networks of brain activity and the

While single region activity can be characterized in a straightforward way through timevarying profiles of amplitudes of some aspect of brain activity, network activity needs appropriate non-trivial observables to be defined. Graph theory (Boccaletti et al., 2006) offers a convenient and flexible way to analyze topological properties of systems with a network organization (Bullmore et al., 2009). Most importantly for neuroscientists, graph theory can be used to understand the complex relationship between structure, dynamics and function in the brain. Graph theory shows that the topology of structural networks influences the dynamical processes (namely synchronization) taking place on them (Boccaletti et al., 2006). For instance, small-world properties of dense or clustered local connectivity with relatively few long-range connections confer distinctive dynamical and functional properties: in addition to optimizing information processing (Strogatz, 2001), facilitating synchronization (Bucolo et al., 2003), ensuring rapid response and emergence of coherent oscillations (Lago-Fernández et al., 2000), and conferring resilience against pathological attack, small-world architecture has been shown to provide an optimum trade-off between efficiency and wiring costs, conferring high local and global efficiencies for relatively low connection costs (Latora

Recently, an increasing number of neuroimaging studies using graph theoretical tools have started showing that the brain developed in such a way that a clear correspondence exists between anatomical network topology and dynamical processes taking place on it. It has

dynamics taking place on them.

& Marchiori, 2001).

been convincingly shown that brain anatomical networks have characteristically smallworld properties of dense or clustered local connectivity with relatively few long-range connections (Sporns et al., 2004). Similarly, human brain functional networks associated with the execution of cognitive tasks have also been associated with fractal small-world architecture (Achard et al., 2006; Bassett et al., 2006; Eguíluz et al., 2005; Salvador et al., 2005), which support efficient parallel information transfer at relatively low costs and is differently impaired by normal aging and pharmacological manipulations (Achard & Bullmore, 2007; Bassett et al., 2009). Furthermore, specific neuroanatomical connectivity patterns are univocally associated with given functional complexity levels, and networks capable of producing highly complex functional dynamics share common structural motifs (see e.g. (Sporns et al., 2000, 2002)). Finally, simulations showed that brain dynamics exhibits a modular hierarchical organization, where clusters coincide with the topological community structure of anatomical networks (Zhou et al., 2006).

Arguably graph theory's greatest strengths is that it has made possible to address a whole range of new research questions, far exceeding the original main one addressed by neuroimaging, of localizing brain activity, particularly issues related to *how* the brain organizes its activity as it carries out tasks of arbitrary complexity. A relative limitation of graph theoretical applications, in their current form, to neuroimaging is that both computations and visualization of functional brain networks are performed based on the Euclidean coordinates of observed activity. However, it has long been known that there is no straightforward correspondence between spatial and functional proximity between brain regions, so that regions that are contiguous to each other can in fact be involved in the execution of completely different tasks. It is then of great interest to be able to represent the topology of the functional space, and ultimately to delineate the correspondence between anatomical and functional spaces.

Here we propose a new method, Functional Holography (FH), designed to describe the information content of a network as it functions as a whole unit. The term used for the familiar holograms indicates that the photographic plates can capture the whole information about the 3D image. The FH method can overcome the main limitations of previous methods by visualizing networks of correlated activity in an auxiliary space of correlations and linking the components according to similarities between them.

The main objectives of the FH method are:


In the remainder of this chapter, we will first illustrate the mathematical procedure of the method; we will then show some applications of the method to various neurophysiological

Functional Holography and Cliques in Brain Activation Patterns 105

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

> <sup>2</sup> 1

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

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.

, ˆ ˆ

 

*A i* ,,, *j C i j MC i j* (4)

(3)

*Cik ci C j k c j*

, ,

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

2 2

*Ci Cj*

*k Dij Ci C j Cik C jk*

(2)

, , , *N*

each one of the signals with all other ones)

*C i*

eigenvectors do not create an orthogonal mathematical space.

*MC i j*

,

*k ij*

*N*

**2.1.2 Collective normalization** 

correlations to [0,1].

signals, and will finally conclude by discussing the scope of the FH method in the context of brain imaging data analysis.
