Evaluating functional network inference using simulations of complex biological systems.

MOTIVATION: Although many network inference algorithms have been presented in the bioinformatics literature, no suitable approach has been formulated for evaluating their effectiveness at recovering models of complex biological systems from limited data. To overcome this limitation, we propose an approach to evaluate network inference algorithms according to their ability to recover a complex functional network from biologically reasonable simulated data. RESULTS: We designed a simulator to generate data representing a complex biological system at multiple levels of organization: behaviour, neural anatomy, brain electrophysiology, and gene expression of songbirds. About 90% of the simulated variables are unregulated by other variables in the system and are included simply as distracters. We sampled the simulated data at intervals as one would sample from a biological system in practice, and then used the sampled data to evaluate the effectiveness of an algorithm we developed for functional network inference. We found that our algorithm is highly effective at recovering the functional network structure of the simulated system-including the irrelevance of unregulated variables-from sampled data alone. To assess the reproducibility of these results, we tested our inference algorithm on 50 separately simulated sets of data and it consistently recovered almost perfectly the complex functional network structure underlying the simulated data. To our knowledge, this is the first approach for evaluating the effectiveness of functional network inference algorithms at recovering models from limited data. Our simulation approach also enables researchers a priori to design experiments and data-collection protocols that are amenable to functional network inference.

Emergence of a small-world functional network in cultured neurons

The functional networks of cultured neurons exhibit complex network properties similar to those found in vivo. Starting from random seeding, cultures undergo significant reorganization during the initial period in vitro, yet despite providing an ideal platform for observing developmental changes in neuronal connectivity, little is known about how a complex functional network evolves from isolated neurons. In the present study, evolution of functional connectivity was estimated from correlations of spontaneous activity. Network properties were quantified using complex measures from graph theory and used to compare cultures at different stages of development during the first 5 weeks in vitro. Networks obtained from young cultures (14 days in vitro) exhibited a random topology, which evolved to a small-world topology during maturation. The topology change was accompanied by an increased presence of highly connected areas (hubs) and network efficiency increased with age. The small-world topology balances integration of network areas with segregation of specialized processing units. The emergence of such network structure in cultured neurons, despite a lack of external input, points to complex intrinsic biological mechanisms. Moreover, the functional network of cultures at mature ages is efficient and highly suited to complex processing tasks.

Parameter-free extraction of the functional network topology from intracranial EEG recordings: a time-resolved study of graph properties in focal onset seizures

Rationale: Focal onset epileptic seizures are due to abnormal interactions between distributed brain areas. By estimating the cross-correlation matrix of multi-site intra-cerebral EEG recordings (iEEG), one can quantify these interactions. To assess the topology of the underlying functional network, the binary connectivity matrix has to be derived from the cross-correlation matrix by use of a threshold. Classically, a unique threshold is used that constrains the topology [1]. Our method aims to set the threshold in a data-driven way by separating genuine from random cross-correlation. We compare our approach to the fixed threshold method and study the dynamics of the functional topology. Methods: We investigate the iEEG of patients suffering from focal onset seizures who underwent evaluation for the possibility of surgery. The equal-time cross-correlation matrices are evaluated using a sliding time window. We then compare 3 approaches assessing the corresponding binary networks. For each time window: * Our parameter-free method derives from the cross-correlation strength matrix (CCS)[2]. It aims at disentangling genuine from random correlations (due to finite length and varying frequency content of the signals). In practice, a threshold is evaluated for each pair of channels independently, in a data-driven way. * The fixed mean degree (FMD) uses a unique threshold on the whole connectivity matrix so as to ensure a user defined mean degree. * The varying mean degree (VMD) uses the mean degree of the CCS network to set a unique threshold for the entire connectivity matrix. * Finally, the connectivity (c), connectedness (given by k, the number of disconnected sub-networks), mean global and local efficiencies (Eg, El, resp.) are computed from FMD, CCS, VMD, and their corresponding random and lattice networks. Results: Compared to FMD and VMD, CCS networks present: *topologies that are different in terms of c, k, Eg and El. *from the pre-ictal to the ictal and then post-ictal period, topological features time courses that are more stable within a period, and more contrasted from one period to the next. For CCS, pre-ictal connectivity is low, increases to a high level during the seizure, then decreases at offset. k shows a ‘‘U-curve’’ underlining the synchronization of all electrodes during the seizure. Eg and El time courses fluctuate between the corresponding random and lattice networks values in a reproducible manner. Conclusions: The definition of a data-driven threshold provides new insights into the topology of the epileptic functional networks.

Optimizing Functional Network Representation of Multivariate Time Series

By combining complex network theory and data mining techniques, we provide objective criteria for optimization of the functional network representation of generic multivariate time series. In particular, we propose a method for the principled selection of the threshold value for functional network reconstruction from raw data, and for proper identification of the network's indicators that unveil the most discriminative information on the system for classification purposes. We illustrate our method by analysing networks of functional brain activity of healthy subjects, and patients suffering from Mild Cognitive Impairment, an intermediate stage between the expected cognitive decline of normal aging and the more pronounced decline of dementia. We discuss extensions of the scope of the proposed methodology to network engineering purposes, and to other data mining tasks.

Altered brain rhythms and functional network disruptions involved in patients with generalized fixation-off epilepsy

Fixation-off sensitivity (FOS) denotes the forms of epilepsy elicited by elimination of fixation. FOS-IGE patients are rare cases [1]. In a previous work [2] we showed that two FOS-IGE patients had different altered EEG rhythms when closing eyes; only beta band was altered in patient 1 while theta, alpha and beta were altered in patient 2. In the present work, we explain the relationship between the altered brain rhythms in these patients and the disruption in functional brain networks.

Heritability of the network architecture of intrinsic brain functional connectivity

The brain's functional network exhibits many features facilitating functional specialization, integration, and robustness to attack. Using graph theory to characterize brain networks, studies demonstrate their small-world, modular, and "rich-club" properties, with deviations reported in many common neuropathological conditions. Here we estimate the heritability of five widely used graph theoretical metrics (mean clustering coefficient (γ), modularity (Q), rich-club coefficient (ϕnorm), global efficiency (λ), small-worldness (σ)) over a range of connection densities (k=5-25%) in a large cohort of twins (N=592, 84 MZ and 89 DZ twin pairs, 246 single twins, age 23±2.5). We also considered the effects of global signal regression (GSR). We found that the graph metrics were moderately influenced by genetic factors h2 (γ=47-59%, Q=38-59%, ϕnorm=0-29%, λ=52-64%, σ=51-59%) at lower connection densities (≤15%), and when global signal regression was implemented, heritability estimates decreased substantially h2 (γ=0-26%, Q=0-28%, ϕnorm=0%, λ=23-30%, σ=0-27%). Distinct network features were phenotypically correlated (|r|=0.15-0.81), and γ, Q, and λ were found to be influenced by overlapping genetic factors. Our findings suggest that these metrics may be potential endophenotypes for psychiatric disease and suitable for genetic association studies, but that genetic effects must be interpreted with respect to methodological choices.

Hippocampal functional connectivity reflects verbal episodic memory network integrity

Using functional magnetic resonance imaging during a verbal memory task, we investigated correlations of signal fluctuations within the hippocampus and ipsilateral frontal as well as temporal areas in temporal lobe epilepsy patients. Declarative memory abilities were additionally examined before and after temporal lobe epilepsy surgery. A significant difference exists in functional connectivity between patients whose mnemonic functions deteriorated and those who remained stable or improved. Univariate analyses showed significantly higher preoperative coupling between the hippocampus and Brodmann area 22 for the group that decreased in verbal learning. We suggest greater coupling to reflect higher functional network integrity. Postoperatively reduced learning ability in patients with higher preoperative coupling underlines the importance of hippocampal interaction with cortical areas for successful memory formation.

Functional brain network efficiency predicts intelligence

The neuronal causes of individual differences in mental abilities such as intelligence are complex and profoundly important. Understanding these abilities has the potential to facilitate their enhancement. The purpose of this study was to identify the functional brain network characteristics and their relation to psychometric intelligence. In particular, we examined whether the functional network exhibits efficient small-world network attributes (high clustering and short path length) and whether these small-world network parameters are associated with intellectual performance. High-density resting state electroencephalography (EEG) was recorded in 74 healthy subjects to analyze graph-theoretical functional network characteristics at an intracortical level. Ravens advanced progressive matrices were used to assess intelligence. We found that the clustering coefficient and path length of the functional network are strongly related to intelligence. Thus, the more intelligent the subjects are the more the functional brain network resembles a small-world network. We further identified the parietal cortex as a main hub of this resting state network as indicated by increased degree centrality that is associated with higher intelligence. Taken together, this is the first study that substantiates the neural efficiency hypothesis as well as the Parieto-Frontal Integration Theory (P-FIT) of intelligence in the context of functional brain network characteristics. These theories are currently the most established intelligence theories in neuroscience. Our findings revealed robust evidence of an efficiently organized resting state functional brain network for highly productive cognitions.

Identification of pre-spike network in patients with mesial temporal lobe epilepsy

Background: Seizures and interictal spikes in mesial temporal lobe epilepsy (MTLE) affect a network of brain regions rather than a single epileptic focus. Simultaneous electroencephalography and functional magnetic resonance imaging (EEG-fMRI) studies have demonstrated a functional network in which hemodynamic changes are time-locked to spikes. However, whether this reflects the propagation of neuronal activity from a focus, or conversely the activation of a network linked to spike generation remains unknown. The functional connectivity (FC) changes prior to spikes may provide information about the connectivity changes that lead to the generation of spikes. We used EEG-fMRI to investigate FC changes immediately prior to the appearance of interictal spikes on EEG in patients with MTLE. Methods/principal findings: Fifteen patients with MTLE underwent continuous EEG-fMRI during rest. Spikes were identified on EEG and three 10 s epochs were defined relative to spike onset: spike (0–10 s), pre-spike (−10 to 0 s), and rest (−20 to −10 s, with no previous spikes in the preceding 45s). Significant spike-related activation in the hippocampus ipsilateral to the seizure focus was found compared to the pre-spike and rest epochs. The peak voxel within the hippocampus ipsilateral to the seizure focus was used as a seed region for FC analysis in the three conditions. A significant change in FC patterns was observed before the appearance of electrographic spikes. Specifically, there was significant loss of coherence between both hippocampi during the pre-spike period compared to spike and rest states. Conclusion/significance: In keeping with previous findings of abnormal inter-hemispheric hippocampal connectivity in MTLE, our findings specifically link reduced connectivity to the period immediately before spikes. This brief decoupling is consistent with a deficit in mutual (inter-hemispheric) hippocampal inhibition that may predispose to spike generation.

Evaluation of predicted network modules in yeast metabolism using NMR-based metabolite profiling

Genome-scale metabolic models promise important insights into cell function. However, the definition of pathways and functional network modules within these models, and in the biochemical literature in general, is often based on intuitive reasoning. Although mathematical methods have been proposed to identify modules, which are defined as groups of reactions with correlated fluxes, there is a need for experimental verification. We show here that multivariate statistical analysis of the NMR-derived intra- and extracellular metabolite profiles of single-gene deletion mutants in specific metabolic pathways in the yeast Saccharomyces cerevisiae identified outliers whose profiles were markedly different from those of the other mutants in their respective pathways. Application of flux coupling analysis to a metabolic model of this yeast showed that the deleted gene in an outlying mutant encoded an enzyme that was not part of the same functional network module as the other enzymes in the pathway. We suggest that metabolomic methods such as this, which do not require any knowledge of how a gene deletion might perturb the metabolic network, provide an empirical method for validating and ultimately refining the predicted network structure.

Network mechanisms of intentional learning

The ability to learn new tasks rapidly is a prominent characteristic of human behaviour. This ability relies on flex- ible cognitive systems that adapt in order to encode temporary programs for processing non-automated tasks. Previous functional imaging studies have revealed distinct roles for the lateral frontal cortices (LFCs) and the ven- tral striatum in intentional learning processes. However, the human LFCs are complex; they house multiple dis- tinct sub-regions, each of which co-activates with a different functional network. It remains unclear how these LFC networks differ in their functions and how they coordinate with each other, and the ventral striatum, to support intentional learning. Here, we apply a suite of fMRI connectivity methods to determine how LFC networks activate and interact at different stages of two novel tasks, in which arbitrary stimulus-response rules are learnt either from explicit instruction or by trial-and-error. We report that the networks activate en masse and in synchrony when novel rules are being learnt from instruction. However, these networks are not homogeneous in their functions; instead, the directed connectivities between them vary asymmetrically across the learning timecourse and they disengage from the task sequentially along a rostro-caudal axis. Furthermore, when negative feedback indicates the need to switch to alternative stimulus–response rules, there is additional input to the LFC networks from the ventral striatum. These results support the hypotheses that LFC networks interact as a hierarchical system during intentional learning and that signals from the ventral striatum have a driving influence on this system when the internal program for processing the task is updated.

Dynamic Network Mechanisms of Relational Integration

A prominent hypothesis states that specialized neural modules within the human lateral frontopolar cortices (LFPCs) support “relational integration” (RI), the solving of complex problems using inter-related rules. However, it has been proposed that LFPC activity during RI could reflect the recruitment of additional “domain-general” resources when processing more difficult problems in general as opposed to RI specifi- cally. Moreover, theoretical research with computational models has demonstrated that RI may be supported by dynamic processes that occur throughout distributed networks of brain regions as opposed to within a discrete computational module. Here, we present fMRI findings from a novel deductive reasoning paradigm that controls for general difficulty while manipulating RI demands. In accordance with the domain- general perspective, we observe an increase in frontoparietal activation during challenging problems in general as opposed to RI specifically. Nonetheless, when examining frontoparietal activity using analyses of phase synchrony and psychophysiological interactions, we observe increased network connectivity during RI alone. Moreover, dynamic causal modeling with Bayesian model selection identifies the LFPC as the effective connectivity source. Based on these results, we propose that during RI an increase in network connectivity and a decrease in network metastability allows rules that are coded throughout working memory systems to be dynamically bound. This change in connectivity state is top-down propagated via a hierarchical system of domain-general networks with the LFPC at the apex. In this manner, the functional network perspective reconciles key propositions of the globalist, modular, and computational accounts of RI within a single unified framework.

Multiple Pathways Analysis of Brain Functional Networks from EEG Signals: An Application to Real Data

In the present study, we propose a theoretical graph procedure to investigate multiple pathways in brain functional networks. By taking into account all the possible paths consisting of h links between the nodes pairs of the network, we measured the global network redundancy R (h) as the number of parallel paths and the global network permeability P (h) as the probability to get connected. We used this procedure to investigate the structural and dynamical changes in the cortical networks estimated from a dataset of high-resolution EEG signals in a group of spinal cord injured (SCI) patients during the attempt of foot movement. In the light of a statistical contrast with a healthy population, the permeability index P (h) of the SCI networks increased significantly (P < 0.01) in the Theta frequency band (3-6 Hz) for distances h ranging from 2 to 4. On the contrary, no significant differences were found between the two populations for the redundancy index R (h) . The most significant changes in the brain functional network of SCI patients occurred mainly in the lower spectral contents. These changes were related to an improved propagation of communication between the closest cortical areas rather than to a different level of redundancy. This evidence strengthens the hypothesis of the need for a higher functional interaction among the closest ROIs as a mechanism to compensate the lack of feedback from the peripheral nerves to the sensomotor areas.

Epileptic seizures as condensed sleep: an analysis of network dynamics from electroencephalogram signals

Both deepening sleep and evolving epileptic seizures are associated with increasing slow-wave activity. Larger-scale functional networks derived from electroencephalogram indicate that in both transitions dramatic changes of communication between brain areas occur. During seizures these changes seem to be 'condensed', because they evolve more rapidly than during deepening sleep. Here we set out to assess quantitatively functional network dynamics derived from electroencephalogram signals during seizures and normal sleep. Functional networks were derived from electroencephalogram signals from wakefulness, light and deep sleep of 12 volunteers, and from pre-seizure, seizure and post-seizure time periods of 10 patients suffering from focal onset pharmaco-resistant epilepsy. Nodes of the functional network represented electrical signals recorded by single electrodes and were linked if there was non-random cross-correlation between the two corresponding electroencephalogram signals. Network dynamics were then characterized by the evolution of global efficiency, which measures ease of information transmission. Global efficiency was compared with relative delta power. Global efficiency significantly decreased both between light and deep sleep, and between pre-seizure, seizure and post-seizure time periods. The decrease of global efficiency was due to a loss of functional links. While global efficiency decreased significantly, relative delta power increased except between the time periods wakefulness and light sleep, and pre-seizure and seizure. Our results demonstrate that both epileptic seizures and deepening sleep are characterized by dramatic fragmentation of larger-scale functional networks, and further support the similarities between sleep and seizures.