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  <cit> ; 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  <cit> , 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  <cit> , 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  <cit> .
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  <cit> .
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  <dig>  where m is the number of brain areas, and t the number of time units looking into the past  <cit> .
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 {vi} as possible nodes; their covariance determines whether a binary functional link  exists between them: cij = ⟨ - v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i) - v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@j)σi−1σj−1
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaeyoei0iaegymaedaaogae83wdm3aa0baasqaaiabdqgaqbqaaiabgkhitiabigdaxaaaaaa@370a@, where v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i = ⟨⟩t and σi2
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaegomaidaaaaa@30f0@ = ⟨ - v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i)2⟩, such that if the correlation between i and j exceeds a threshold, a functional link is considered, and none otherwise: if cij > ct then dij =  <dig>  else dij =  <dig>  we extend here this approach by considering the delayed or lagged covariance: cij = ⟨- v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i) - v¯
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@j)σi−1σj−1
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaeyoei0iaegymaedaaogae83wdm3aa0baasqaaiabdqgaqbqaaiabgkhitiabigdaxaaaaaa@370a@.
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 cij > ct then dij = dji = 1; else if cij > ct then dij =  <dig> and dji = 0; else dij =  <dig>
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  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 , but we did not find this approach to have a significant impact, presumably due to the low temporal resolution of the signal .
RESULTS
we analyzed a dataset discussed in the literature  <cit> , 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 , 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.
although directed links have less statistical power than undirected ones, their degree distribution still shows a power-law behavior.
this is exemplified in fig.
1a, where the histogram of source  and sink  degrees, i.e.
out- and in-directed links, averaged over all subjects and tasks, is represented.
observe that the distribution is very similar for both classes of links, and approaches a power-law with an exponent of 3/ <dig>  which is even more evident when all links are considered for each node .
this result is in line with our previous findings, as it reinforces the notion of information hubs dominating the flow of information in the brain.
the effect of removing some of the undirected edges is to reduce the scaling from an exponent near  <dig> as estimated with our previous approach.
figure 1b depicts the result of analyzing the small-world topology of the functional networks.
the panel represents the clustering vs. average minimal path for all the studied cases , the equivalent random networks , and the equivalent regular lattices .
observe that the clustering of the functional networks is several orders of magnitude larger than that of the equivalent random networks, and comparable to that of the equivalent regular lattices.
the average minimal path of the functional networks, while larger than that of the random networks, is still smaller than that of the equivalent regular lattices.
in comparison with our previous study, the main topological features are preserved: for the same threshold , there is an increase from  <dig>  to  <dig>  in the average path due to the removal of undirected links explained by a common source; interestingly, the clustering coefficient increases from  <dig>  to  <dig> , presumably due to triangulations that could not be detected before.
similarly, a comparison with our previous finding regarding the assortative nature of functional brain networks shows that indeed this property does not change when directed links are included.
figure 1c represents the total degree of each node vs. the average degree of the first neighbors, i.e.
nodes reachable through in-, out- and undirected edges.
again, we find a correlation between these two degrees, in contrast with all other biological networks investigated to date, which implies a lack of hierarchical organization in the networks: a hierarchical network would display a negative correlation, such that nodes with high degree  tend to be connected with low degree nodes .
a more detailed study, analyzing differences between in- and out-hubs, will be reported in future publications.
interestingly, the networks hold enough information about the dynamics of brain states so that even a global measure of their properties can discriminate between tasks.
figure  <dig> shows the result of comparing different topological measures of the three resulting networks  for  <dig> different subjects.
the first observation in this regard is that the total number of nodes identified by the method  is not a good indicator of the identity of the task, or of the subject for that matter; this is to be expected, as the number of nodes depends dramatically on the chosen threshold.
the average degree, or connectivity , does not seem to provide much information either; this is clearly a very local measure of the structure of the network, for the most part unrelated to the flow of information.
we observed a tendency towards discriminating between the tasks when we computed the average minimal path of the networks , defined as the minimal number of links needed to reach a node from another one, averaged over all nodes.
this tendency, though, is not consistent across subjects, as the auditory cue task is in some cases large and small in others.
however, we were able to improve this measurement by considering the normalized mean path, as the mean path of the network over that of the equivalent erdös random network, displayed in panel 2c .
observe that for  <dig> out of the  <dig> subjects, the auditory task is consistently larger  that the equivalent visual cue tasks.
non-parametric statistical analysis on the rank of the normalized mean path  yields a p-value of  <dig> .
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.
a remarkable regularity displayed by these networks is the tendency for nodes to be mostly "sources"  or "sinks" .
that is, nodes with a large number of out-links tend to have relatively few in-edges, and vice versa, although, interestingly, this is not a strictly enforced rule.
moreover, in-hubs tend to have relatively few undirected links, whereas out-hubs tend to be also undirected hubs.
this seems to be counter-intuitive at face value, as one may naïvely think that the hubs are balanced; however, they need not be so, as one would expect in, for instance, tracffic hubs.
in other words, there are no conserved quantities at the hub level to be balanced.
these results are summarized in fig.
<dig>  the tendency for nodes to be either in- or out-hubs is particularly clear in panel d, where the maximum between the in-degree and the out-degree is plotted against the absolute value of their difference, showing a strong correlation.
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 , 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.
however, to move beyond a simply phenomenological description of such systems, we are compelled to bridge the gap between emergent global behavior and local functional properties.
one possibility is to interpret the results depicted in fig.
2c in terms of the spatial organization of the networks.
to do so, and given the limitations of representing the full complexity of the networks, we resorted to a simplified approach that captures only part of this complexity but serves as a possible guide for reasoning.
figure  <dig> represents the networks by link density maps, where the nodes are depicted in their original spatial location, and the links are mapped using a color-coded representation of the degree .
the degree of undirected links is shown for the same subject in the small visual task in panel a, and the auditory cue task in panel b. observe that the density maps are very similar in both conditions.
the out-degree, i.e.
sources of directed links, is depicted in panels c for the visual cue task and panel d for auditory.
in this case, the differences between the maps are much more compelling; however, it is to be expected that precisely in a class of tasks where the varying factor is a short cue, the main source of functional differences will be "sources of information", in a manner of speak.
a thorough analysis of the regional discernibility of the method is beyond the scope of the present publication, focused on topological properties of the networks.
this would require a systematic topographic comparison of topological features  across subjects and tasks, beyond the scope of this manuscript.
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  process of rendering directed links.
we are currently studying different alternatives to clarify these ideas, most prominently ensemble modeling.
CONCLUSIONS
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  <cit> .
this is the subject of current investigation in our group.
finally, we conclude that the computational feasibility of the approach , even when dealing with  <dig>  to  <dig>  independent variables, renders it applicable to other similarly large biological networks, like gene-expression patterns generated by cdna microarrays  <cit> .

 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; 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, 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, 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. 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. 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 where m is the number of brain areas, and t the number of time units looking into the past. 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 {vi} as possible nodes; their covariance determines whether a binary functional link exists between them: cij = ⟨ - v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i) - v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@j)σi−1σj−1 mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaeyoei0iaegymaedaaogae83wdm3aa0baasqaaiabdqgaqbqaaiabgkhitiabigdaxaaaaaa@370a@, where v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i = ⟨⟩t and σi2 mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaegomaidaaaaa@30f0@ = ⟨ - v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i)2⟩, such that if the correlation between i and j exceeds a threshold, a functional link is considered, and none otherwise: if cij > ct then dij = else dij = we extend here this approach by considering the delayed or lagged covariance: cij = ⟨- v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@i) - v¯ mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacuwg2bgdgaqeaaaa@2e39@j)σi−1σj−1 mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaaiigacqwfdpwcdaqhaawcbagaemyaakgabagaeyoei0iaegymaedaaogae83wdm3aa0baasqaaiabdqgaqbqaaiabgkhitiabigdaxaaaaaa@370a@. 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 cij > ct then dij = dji = 1; else if cij > ct then dij = and dji = 0; else dij = 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 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, but we did not find this approach to have a significant impact, presumably due to the low temporal resolution of the signal.  we analyzed a dataset discussed in the literature, 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, 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. although directed links have less statistical power than undirected ones, their degree distribution still shows a power-law behavior. this is exemplified in fig. 1a, where the histogram of source and sink degrees, i.e. out- and in-directed links, averaged over all subjects and tasks, is represented. observe that the distribution is very similar for both classes of links, and approaches a power-law with an exponent of 3/ which is even more evident when all links are considered for each node. this result is in line with our previous findings, as it reinforces the notion of information hubs dominating the flow of information in the brain. the effect of removing some of the undirected edges is to reduce the scaling from an exponent near as estimated with our previous approach. figure 1b depicts the result of analyzing the small-world topology of the functional networks. the panel represents the clustering vs. average minimal path for all the studied cases, the equivalent random networks, and the equivalent regular lattices. observe that the clustering of the functional networks is several orders of magnitude larger than that of the equivalent random networks, and comparable to that of the equivalent regular lattices. the average minimal path of the functional networks, while larger than that of the random networks, is still smaller than that of the equivalent regular lattices. in comparison with our previous study, the main topological features are preserved: for the same threshold, there is an increase from to in the average path due to the removal of undirected links explained by a common source; interestingly, the clustering coefficient increases from to, presumably due to triangulations that could not be detected before. similarly, a comparison with our previous finding regarding the assortative nature of functional brain networks shows that indeed this property does not change when directed links are included. figure 1c represents the total degree of each node vs. the average degree of the first neighbors, i.e. nodes reachable through in-, out- and undirected edges. again, we find a correlation between these two degrees, in contrast with all other biological networks investigated to date, which implies a lack of hierarchical organization in the networks: a hierarchical network would display a negative correlation, such that nodes with high degree tend to be connected with low degree nodes. a more detailed study, analyzing differences between in- and out-hubs, will be reported in future publications. interestingly, the networks hold enough information about the dynamics of brain states so that even a global measure of their properties can discriminate between tasks. figure shows the result of comparing different topological measures of the three resulting networks for different subjects. the first observation in this regard is that the total number of nodes identified by the method is not a good indicator of the identity of the task, or of the subject for that matter; this is to be expected, as the number of nodes depends dramatically on the chosen threshold. the average degree, or connectivity, does not seem to provide much information either; this is clearly a very local measure of the structure of the network, for the most part unrelated to the flow of information. we observed a tendency towards discriminating between the tasks when we computed the average minimal path of the networks, defined as the minimal number of links needed to reach a node from another one, averaged over all nodes. this tendency, though, is not consistent across subjects, as the auditory cue task is in some cases large and small in others. however, we were able to improve this measurement by considering the normalized mean path, as the mean path of the network over that of the equivalent erdös random network, displayed in panel 2c. observe that for out of the subjects, the auditory task is consistently larger that the equivalent visual cue tasks. non-parametric statistical analysis on the rank of the normalized mean path yields a p-value of. 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. a remarkable regularity displayed by these networks is the tendency for nodes to be mostly "sources" or "sinks". that is, nodes with a large number of out-links tend to have relatively few in-edges, and vice versa, although, interestingly, this is not a strictly enforced rule. moreover, in-hubs tend to have relatively few undirected links, whereas out-hubs tend to be also undirected hubs. this seems to be counter-intuitive at face value, as one may naïvely think that the hubs are balanced; however, they need not be so, as one would expect in, for instance, tracffic hubs. in other words, there are no conserved quantities at the hub level to be balanced. these results are summarized in fig. the tendency for nodes to be either in- or out-hubs is particularly clear in panel d, where the maximum between the in-degree and the out-degree is plotted against the absolute value of their difference, showing a strong correlation. 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, 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. however, to move beyond a simply phenomenological description of such systems, we are compelled to bridge the gap between emergent global behavior and local functional properties. one possibility is to interpret the results depicted in fig. 2c in terms of the spatial organization of the networks. to do so, and given the limitations of representing the full complexity of the networks, we resorted to a simplified approach that captures only part of this complexity but serves as a possible guide for reasoning. figure represents the networks by link density maps, where the nodes are depicted in their original spatial location, and the links are mapped using a color-coded representation of the degree. the degree of undirected links is shown for the same subject in the small visual task in panel a, and the auditory cue task in panel b. observe that the density maps are very similar in both conditions. the out-degree, i.e. sources of directed links, is depicted in panels c for the visual cue task and panel d for auditory. in this case, the differences between the maps are much more compelling; however, it is to be expected that precisely in a class of tasks where the varying factor is a short cue, the main source of functional differences will be "sources of information", in a manner of speak. a thorough analysis of the regional discernibility of the method is beyond the scope of the present publication, focused on topological properties of the networks. this would require a systematic topographic comparison of topological features across subjects and tasks, beyond the scope of this manuscript. 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 process of rendering directed links. we are currently studying different alternatives to clarify these ideas, most prominently ensemble modeling.  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. this is the subject of current investigation in our group. finally, we conclude that the computational feasibility of the approach, even when dealing with to independent variables, renders it applicable to other similarly large biological networks, like gene-expression patterns generated by cdna microarrays. 
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. 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. 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. biological experiments increasingly yield data representing large ensembles of interacting variables, making the application of advanced analytical tools a forbidding task.
 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: network nodes represent voxels, and 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.  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.  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. 7– september international workshop on multiscale biological imaging, data mining and informatics santa barbara, ca, usa 