- Research
- Open Access

# Identifying directed links in large scale functional networks: application to brain fMRI

- Guillermo A Cecchi
^{1}Email author, - A Ravishankar Rao
^{1}, - Maria V Centeno
^{2}, - Marwan Baliki
^{2}, - A Vania Apkarian
^{2}and - Dante R Chialvo
^{2}

**8 (Suppl 1)**:S5

https://doi.org/10.1186/1471-2121-8-S1-S5

© Cecchi et al; licensee BioMed Central Ltd. 2007

**Published:**10 July 2007

## Abstract

### Background

Biological experiments increasingly yield data representing large ensembles of interacting variables, making the application of advanced analytical tools a forbidding task. We present a method to extract networks of correlated activity, specifically from functional MRI data, such that: (a) network nodes represent voxels, and (b) the network links can be directed or undirected, representing temporal relationships between the nodes. The method provides a snapshot of the ongoing dynamics of the brain without sacrificing resolution, as the analysis is tractable even for very large numbers of voxels.

### Results

We find that, based on topological properties of the networks, the method provides enough information about the dynamics to discriminate between subtly different brain states. Moreover, the statistical regularities previously reported are qualitatively preserved, i.e. the resulting networks display scale-free and small-world topologies.

### Conclusion

Our method expands previous approaches to render large scale functional networks, and creates the basis for an extensive and -due to the presence of mixtures of directed and undirected links- richer motif analysis of functional relationships.

## Keywords

- Degree Distribution
- Brain State
- Giant Component
- Topological Measure
- Functional Brain Network

## Background

A growing number of biological experiments are producing datasets consisting of large numbers of interacting variables, from genomics to neural networks to eco-systems, giving rise to the nascent field of systems biology. The eminent challenge of this discipline is how to simplify the analysis of these high dimensional dynamical systems while retaining their relevant features. In particular, despite the rich, complex dynamics of the brain, and the highly interconnected and non-linear nature of its information processing capabilities, the bulk of the literature on brain imaging involves slight variations on the main theme: the identification of the degree of correlation between the activation of a local brain area and external markers. A variety of attempts have been made trying to go beyond this paradigm, including ICA, Volterra kernels and supervised classification techniques [1–4]; however, the applicability of these multi-variate methods has been restricted to networks with only a few hundred vertices.

Recently, a different approach was introduced [5, 6], based on the analysis of the structure of pair-wise correlations between voxels in fMRI, embedding it in a graph structure whose nodes are the voxels and edges between them are defined by the covariance exceeding a threshold. With this relatively simple approach, we were able to demonstrate that the resulting networks have universal statistical topological properties, like scale-free connectivity and small-worldness [7], shared with other biological networks. Moreover, the spatial distribution of these networks is determined by the specific task the brain is engaged in, as opposed to capturing a task-independent network with little functional sensitivity. Similar ideas have been used to analyze and discriminate a variety of functional and dysfunctional brain states, and indeed have confirmed our initial results [8–10].

Another approach to capture the dynamics of complex systems is the causality analysis pioneered by Granger. The essence of this approach is to identify possible causal relationship between the variables of a system by analyzing how much the time course of one variable contributes to that of another one, based on an auto-regressive model. The method has been applied with success to neural data in the context of a few electro-physiological recordings, and to small numbers of brain areas represented by aggregates of several voxels [11, 12]. One aspect of Granger's method and its derivatives that makes it superior to a simple covariance analysis is that it takes care of the transitivity and symmetry of the covariance: if a variable *A* determines the dynamical behavior of variable *B*, and *B* in its turn drives *C*, the covariance will find symmetric "links" *A* ↔ *B*, *B* ↔ *C* and *A* ↔ *C*; similarly, if *A* drives *B* and *C* but the latter two do not influence each other, the covariance will still find B and C to be linked. Multivariate methods based on Granger's causality are able to break the symmetry and the transitivity, and therefore provide a more accurate description of the internal dynamics of the system. These methods have nevertheless several limitations arising from the need of a very large number of samples, as the auto-regressive model requires a matrix whose dimensions are (*M* × *T*)^{2}, where *M* is the number of brain areas, and *T* the number of time units looking into the past [13, 14].

In the present work, we try to circumvent both the limitations of covariance-based analysis and auto-regression-based causality analysis. We extend our previous findings by attempting to capture a larger signature of the dynamics of the brain by including directionality in the edges of the network, based on the concept of delayed covariance.

In our previous work, we defined a functional network by considering all functional voxels {*v*_{
i
}} as possible nodes; their covariance determines whether a binary functional link (or edge) exists between them: *c*_{
ij
}= ⟨(*v*_{
i
}(*t*) - $\overline{v}$_{
i
})(*v*_{
j
}(*t*) - $\overline{v}$_{
j
})${\sigma}_{i}^{-1}{\sigma}_{j}^{-1}$, where $\overline{v}$_{
i
}= ⟨(*v*_{
i
}(*t*)⟩_{
t
}and ${\sigma}_{i}^{2}$ = ⟨(*v*_{
i
}(*t*) - $\overline{v}$_{
i
})^{2}⟩, such that if the correlation between *i* and *j* exceeds a threshold, a functional link is considered, and none otherwise: **if** *c*_{
ij
}> *C*_{
T
}**then** *d*_{
ij
}= 1, **else** *d*_{
ij
}= 0. We extend here this approach by considering the delayed or lagged covariance: *c*_{
ij
}(*τ*) = ⟨(*v*_{
i
}(*t* - *τ*)- $\overline{v}$_{
i
})(*v*_{
j
}(*t*) - $\overline{v}$_{
j
})${\sigma}_{i}^{-1}{\sigma}_{j}^{-1}$. We reason as follows: if there is a significant peak of the covariance between *i* and *j* at zero lag, then there is a potential binary symmetric link between them, as before. However, if the significant peak is not at zero lag, then we will consider that the preceding voxel, and only it, has a directed link pointing to the succeeding one. That is, **if** *c*_{
ij
}(*τ* = 0) > *C*_{
T
}**then** *d*_{
ij
}= *d*_{
ji
}= 1; **else if** *c*_{
ij
}(*τ* > 0) > *C*_{
T
}**then** *d*_{
ij
}= 1 **and** *d*_{
ji
}= 0; **else** *d*_{
ij
}= 0.

In other words, two voxels whose activity is highly correlated and simultaneous are considered to be symmetrically linked; a voxel that is highly correlated with the future of another one will be considered as a "source", and the latter as a "sink". This approach clearly can break the symmetry of the covariance, but as described cannot deal with the problem of the transitivity described above. Taking into consideration the relatively poor temporal resolution of fMRI, we reasoned that for the time being we could only in earnest tackle one of the confounding sources of undirected links, namely the explanation of a zero-lag covariance (i.e. undirected link) between two voxels by the presence of a common source, of which they are both targets. That is, after identifying all sources and sinks, every potential undirected link that can be explained by a common source is removed. We also considered possible reductions of triangulations of directed links (as in *A* → *B*, *B* → *C*, *A* → *C*), but we did not find this approach to have a significant impact, presumably due to the low temporal resolution of the signal (data not shown, see the Methods section).

## Results

We analyzed a dataset discussed in the literature [6], consisting of subjects undertaking a simple self-paced finger-tapping task, for the purpose of exploring the potential of our method. The task is composed of three variants, which were designed to be relatively similar, in the hope that the method could discriminate subtle differences in the patterns of brain activation. The variation between the tasks is based on the cue utilized to signal the start and stop of the finger-tapping. In the first task, a small featureless visual cue presented in the center of the screen; in the second task the cue is a short auditory tone; in the third one a visual cue similar to the first one is presented, the only difference being its larger size.

After standard functional pre-processing (see Methods), and delayed covariance analysis as explained above, the resulting networks were studied using the tools of statistical network theory. The first observation is that most links are undirected, comprising on average of 50% to 60% of total links; this is compatible with the notion that most neural interactions result from fast, local and presumably symmetric connections, whose subtle dynamics are for the most part beyond the present reach of functional MRI. However, directed links account for a significant number of the observed correlations between voxels, suggesting that our approach can indeed be fruitful in terms of capturing ongoing dynamics.

_{ a }> $\mathcal{L}$

_{ sv }and $\mathcal{L}$

_{ a }> $\mathcal{L}$

_{ lv }) yields a p-value of 0.017. This indicates that the subtle differences in activation elicited by the tasks have a measurable effect in the overall structure of correlations and flow of information of the networks. This tendency was probably amplified by the fact that only the

*giant*component of the network was considered for the purpose of measuring the average minimal path. The giant component is defined as the largest connected sub-network of a graph; we observed that in all cases this component was at least one order of magnitude larger in size that the runner-ups. In other words, we targeted a

*core*of correlated activity and disregarded much smaller sub-networks that could otherwise be relevant. It is also worth mentioning that the directional nature of the links enters explicitly the computation of the mean path, as a link

*A*→

*B*means that there is no direct access from

*B*to

*A*.

## Discussion

Topological regularities of functional brain networks have been described before in the literature, including our own work, but they were based on a narrower window on the properties of the underlying dynamical system (i.e. zero-lag correlation), and could not provide for discernibility of subtly different brain states. The finding that hybrid networks with directed and undirected links can be discriminated based on global topological measures is very relevant to theoretical approaches to brain function, as it is a formalization of the collective properties of complex systems.

The other novel, and certainly unexpected, result is the one summarized in Figure 3: the nodes of the networks tend to be either in- or out-hubs of directed links. The interpretation of this finding is not straightforward; at the neuronal level, we would expect a more balanced relation between "driving" and "driven" connections subtending hypothetical closed circuits across the brain during the completion of a task. However, this interpretation cannot proceed at the level of resolution that the technique provides. In purely speculative terms, this finding points to a role for very large neuronal patches, possibly entire functional areas, in driving other large patches at a relatively slow time scale, compatible with that of the task. Under this preliminary interpretation, the observed in-hub/out-hub exclusion would be an emergent feature of neuronal ensembles, amplified by the highly non-linear (i.e. binary) process of rendering directed links. We are currently studying different alternatives to clarify these ideas, most prominently ensemble modeling.

## Conclusion

Building on our previous work on graph theoretic analysis of functional magnetic resonance imaging data, we have introduced a novel method to capture more of the complex dynamical interactions of brain networks. We find that this approach yields networks with similar topological properties as those described previously, i.e. they display scale-free connectivity, non-hierarchical architecture and small-world topology, even when considering only the giant component for the analysis and strictly enforcing directionality in the computation of the average mean path. However, the topological information contained in directed and undirected links is enough to reveal subtle brain state differences, consisting of a very short auditory or visual cue to trigger a relatively much longer finger-tapping motor sequence. Initial results suggest that these topological differences are concurrent with distinct patterns in the spatial distribution of hubs of directed links, i.e. the location of in- and out-hubs. These findings point in the direction of a functional dissection based on the density and architecture of directed connections, and more specifically on the spatial patterns of topological motifs, similar to approaches already advanced in the study of genetic networks [15, 16]. This is the subject of current investigation in our group.

Finally, we conclude that the computational feasibility of the approach (see Methods), even when dealing with 10,000 to 20,000 independent variables, renders it applicable to other similarly large biological networks, like gene-expression patterns generated by cDNA microarrays [17].

## Methods

*p*=

*GC/N*that peaks at 1/2,

*I*

_{ GC }=

*p*log

_{2}

*p*+ (1 -

*p*) log

_{2}(1 -

*p*), averaged over networks. Observe again that

*I*

_{ GC }is close to the maximum of 1 for a range of thresholds that includes the one we selected. In all cases, the threshold for non-zero-lag was set 0.05 below that for zero-lag, for the reasons explained above.

The weeding of confounding undirected or neutral links was implemented as follows: if two nodes have an undirected link, *A* ↔ *B*, but they have a common directed source, *A* → *B* and *A* → *C*, then the undirected link is removed. We considered directed triangulations with commensurable time delays: *A* → *B*, *B* → *C*, *A* → *C* and Δ*τ*(*A*, *B*) + Δ*τ*(*B*, *C*) = Δ*τ*(*A*, *C*), such that the links *B* → *C* or *A* → *C* could be confounds. The statistical significance of these configurations was very low, and therefore we decided not to include them in the present analysis. The average minimal path or geodesic of a graph is defined as the average over all nodes of the shortest distance from one node to every other $\mathcal{L}$ = ⟨$\mathcal{L}$_{
g
}(*i*, *j*)⟩, and it has a simple interpretation as the "ease" of navigation of the network. The p-value for the data in Fig. 2C was computed as *p* = ∏(6, 5)*q*^{5}(1 - *q*) + ∏(6, 6)*q*^{6}, where ∏(*a*, *b*) is the combinatorial number *a*!/*b*!(*a* - *b*)! and *q* is the probability for the normalized mean path of the auditory cue network to be bigger than that of the small and large visual cues, for the null hypothesis: *q* = *P*($\mathcal{L}$_{
a
}> $\mathcal{L}$_{
sv
}, $\mathcal{L}$_{
a
}> $\mathcal{L}$_{
lv
}) = 1/3. The clustering is defined as the average of the ratio, for each node, between how many of its neighbors share connections amongst them, and the total possible number of connections $\mathcal{C}$ = ⟨*N*_{
i
}/∏(*k*_{
i
}, 2)⟩, where *k*_{
i
}is the number of neighbors of node *i*, and *N*_{
i
}is the number of connections between the neighbors. This topological measure can also be interpreted as the density of triangulations of the network, as reflecting the presence of local structures. The theory developed by Erdös establishes that the geodesic path and clustering of a random network with *N* vertices and average connectivity *k* are, respectively: $\mathcal{L}$ = log(*N*)/log(*k*), $\mathcal{C}$ = *k*/*N*. The alternative null hypothesis is a regular lattice. The mean geodesic path and clustering for the regular 1d-lattice in the original model proposed by Strogatz and Watts can also be computed as $\mathcal{L}$ = *N*/2*k* and $\mathcal{C}$ = 3(*k* - 1)/4(*k* - 2). Erdös random networks are constructed so that an edge can be attached to any two nodes with equal a priori probability, and display a binomial degree distribution, low clustering and small mean path. For the purpose of our analysis, we compared the functional networks with Erdös networks with the same number of nodes and edges. The covariance in Fig. 3 is defined as *c*_{
xy
}= ⟨(*x* - $\overline{x}$)(*y* - $\overline{y}$)${\sigma}_{x}^{-1}{\sigma}_{y}^{-1}$. Pre-processing of the functional data was implemented as in Ref. [6]: right-handed human subjects were studied using a Siemens-Trio 3.0 Tesla imaging system using a birdcage radio-frequency head coil. Blood oxygenation level-dependent single-shot echo-planar T2-weighted imaging was obtained using scan repeat time of 3000 ms, echo time of 30 ms, flip angle 90°, and field 256 mm. The data were preprocessed for motion correction and spatial smoothing using the FSL package [19]. The link density maps were computed based on the degree of each node, and were then, for the purpose of comparison, normalized to the anatomy of the subject using the registration tool in FSL.

## Declarations

### Acknowledgements

G.A.C. would like to acknowledge useful discussions with G. Stolovitzky and J. Kozloski. Experimental work funded through NIH-NINDS grants 42660 and 35115.

This article has been published as part of *BMC Cell Biology* Volume 8 Supplement 1, 2007: 2006 International Workshop on Multiscale Biological Imaging, Data Mining and Informatics. The full contents of the supplement are available online at http://www.biomedcentral.com/1471-2121/8?issue=S1

## Authors’ Affiliations

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## Copyright

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