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The brain is a complex system comprising a myriad of interacting elements, posing significant challenges in understanding its structure, function, and dynamics. Network science has emerged as a powerful tool for studying such intricate systems, offering a framework for integrating multiscale data and complexity. Here, we discuss the application of network science in the study of the brain, addressing topics such as network models and metrics, the connectome, and the role of dynamics in neural networks. We explore the challenges and opportunities in integrating multiple data streams for understanding the neural transitions from development to healthy function to disease, and discuss the potential for collaboration between network science and neuroscience communities. We underscore the importance of fostering interdisciplinary opportunities through funding initiatives, workshops, and conferences, as well as supporting students and postdoctoral fellows with interests in both disciplines. By uniting the network science and neuroscience communities, we can develop novel network-based methods tailored to neural circuits, paving the way towards a deeper understanding of the brain and its functions., Comment: 19 pages, 1 figure, 1 box

The standard approach to modeling the human brain as a complex system is with a network, where the basic unit of interaction is a pairwise link between two brain regions. While powerful, this approach is limited by the inability to assess higher-order interactions involving three or more elements directly. In this work, we present a method for capturing higher-order dependencies in discrete data based on partial entropy decomposition (PED). Our approach decomposes the joint entropy of the whole system into a set of strictly non-negative partial entropy atoms that describe the redundant, unique, and synergistic interactions that compose the system's structure. We begin by showing how the PED can provide insights into the mathematical structure of both the FC network itself, as well as established measures of higher-order dependency such as the O-information. When applied to resting state fMRI data, we find robust evidence of higher-order synergies that are largely invisible to standard functional connectivity analyses. This synergistic structure distinct from structural features based on redundancy that have previously dominated FC analyses. Our approach can also be localized in time, allowing a frame-by-frame analysis of how the distributions of redundancies and synergies change over the course of a recording. We find that different ensembles of regions can transiently change from being redundancy-dominated to synergy-dominated, and that the temporal pattern is structured in time. These results provide strong evidence that there exists a large space of unexplored structures in human brain data that have been largely missed by a focus on bivariate network connectivity models. This synergistic "shadow structures" is dynamic in time and, likely will illuminate new and interesting links between brain and behavior.

One of the most well-established tools for modeling the brain as a complex system is the functional connectivity network, which examines the correlations between pairs of interacting brain regions. While powerful, the network model is limited by the restriction that only pairwise dependencies are visible and potentially higher-order structures are missed. In this work, we explore how multivariate information theory can reveal higher-order, synergistic dependencies in the human brain. Using the O-information, a measure of whether the structure of a system is redundancy- or synergy-dominated, we show that synergistic subsystems are widespread in the human brain. We provide a mathematical analysis of the O-information to locate it within a larger taxonomy of multivariate complexity measures. We also show the O-information is related to a previously established measure, the Tononi-Sporns-Edelman complexity, and can be understood as an expected difference in integration between system scales. Highly synergistic subsystems typically sit between canonical functional networks, and may serve to integrate those networks. We then use simulated annealing to find maximally synergistic subsystems, finding that such systems typically comprise $\approx$10 brain regions, also recruited from multiple canonical brain systems. Though ubiquitous, highly synergistic subsystems are invisible when considering pairwise functional connectivity, suggesting that higher-order dependencies form a kind of ``shadow structure" that has been unrecognized by established network-based analyses. We assert that higher-order interactions in the brain represent a vast and under-explored space that, made accessible with tools of multivariate information theory, may offer novel scientific insights., Comment: 13 pages, 5 figures, 74 references

Network models describe the brain as sets of nodes and edges that represent its distributed organization. So far, most discoveries in network neuroscience have prioritized insights that highlight distinct groupings and specialized functional contributions of network nodes. Importantly, these functional contributions are determined and expressed by the web of their interrelationships, formed by network edges. Here, we underscore the important contributions made by brain network edges for understanding distributed brain organization. Different types of edges represent different types of relationships, including connectivity and similarity among nodes. Adopting a specific definition of edges can fundamentally alter how we analyze and interpret a brain network. Furthermore, edges can associate into collectives and higher-order arrangements, describe time series, and form edge communities that provide insights into brain network topology complementary to the traditional node-centric perspective. Focusing on the edges, and the higher-order or dynamic information they can provide, discloses previously underappreciated aspects of structural and functional network organization., Comment: 35 pages, 4 figures

Are there parallels between the furthest reaches of our universe, and the foundations of thought, awareness, perception, and emotion? What are the connections between the webs and structures that define both? What are the differences? "As Above As Below" was an exhibition that examined these questions. It consisted of six artworks, each of them the product of a collaboration that included at least one artist, astrophysicist, and neuroscientist. The installations explored new parallels between intergalactic and neuronal networks through media such as digital projection, virtual reality, and interactive multimedia, and served to illustrate diverse collaboration practices and ways to communicate across very different fields., Comment: In SciArt Magazine, https://www.sciartmagazine.com/collaboration-as-above-as-below.html ; exhibition video https://youtu.be/RM3gDjb3phU ; panel discussion https://youtu.be/qNhtyvPz8yw ; project website https://asaboveasbelow.com/

We propose a information theoretical framework to capture transition and information costs of network navigation models. Based on the minimum description length principle and the Markov decision process, we demonstrate that efficient global navigation can be achieved with only partial information. Additionally, we derived a scalable algorithm for optimal solutions under certain conditions. The proposed algorithm can be interpreted as a dynamical process on network, making it a useful tool for analysing and understanding navigation strategies on real world networks., Comment: To appear in the Proceedings of the 2020 IEEE International Conference on Networking, Sensing and Control (ICNSC2020)

Communication of signals among nodes in a complex network poses fundamental problems of efficiency and cost. Routing of messages along shortest paths requires global information about the topology, while spreading by diffusion, which operates according to local topological features, is informationally "cheap" but inefficient. We introduce a stochastic model for network communication that combines varying amounts of local and global information about the network topology. The model generates a continuous spectrum of dynamics that converge onto shortest-path and random-walk (diffusion) communication processes at the limiting extremes. We implement the model on two cohorts of human connectome networks and investigate the effects of varying amounts of local and global information on the network's communication cost. We identify routing strategies that approach a (highly efficient) shortest-path communication process with a relatively small amount of global information. Moreover, we show that the cost of routing messages from and to hub nodes varies as a function of the amount of global information driving the system's dynamics. Finally, we implement the model to identify individual subject differences from a communication dynamics point of view. The present framework departs from the classical shortest paths vs. diffusion dichotomy, suggesting instead that brain networks may exhibit different types of communication dynamics depending on varying functional demands and the availability of resources.

Although there is increasing evidence of criticality in the brain, the processes that guide neuronal networks to reach or maintain criticality remain unclear. The present research examines the role of neuronal gain plasticity in time-series of simulated neuronal networks composed of integrate-and-fire stochastic spiking neurons, and the utility of fractal methods in assessing network criticality. Simulated time-series were derived from a network model of fully connected discrete-time stochastic excitable neurons. Monofractal and multifractal analyses were applied to neuronal gain time-series. Fractal scaling was greatest in networks with a mid-range of neuronal plasticity, versus extremely high or low levels of plasticity. Peak fractal scaling corresponded closely to additional indices of criticality, including average branching ratio. Networks exhibited multifractal structure, or multiple scaling relationships. Multifractal spectra around peak criticality exhibited elongated right tails, suggesting that the fractal structure is relatively insensitive to high-amplitude local fluctuations. Networks near critical states exhibited mid-range multifractal spectra width and tail length, which is consistent with literature suggesting that networks poised at quasi-critical states must be stable enough to maintain organization but unstable enough to be adaptable. Lastly, fractal analyses may offer additional information about critical state dynamics of networks by indicating scales of influence as networks approach critical states., Comment: 11 pages, 3 subfigures divided into 2 figures

Networks often exhibit structure at disparate scales. We propose a method for identifying community structure at different scales based on multiresolution modularity and consensus clustering. Our contribution consists of two parts. First, we propose a strategy for sampling the entire range of possible resolutions for the multiresolution modularity quality function. Our approach is directly based on the properties of modularity and, in particular, provides a natural way of avoiding the need to increase the resolution parameter by several orders of magnitude to break a few remaining small communities, necessitating the introduction of ad-hoc limits to the resolution range with standard sampling approaches. Second, we propose a hierarchical consensus clustering procedure, based on a modified modularity, that allows one to construct a hierarchical consensus structure given a set of input partitions. While here we are interested in its application to partitions sampled using multiresolution modularity, this consensus clustering procedure can be applied to the output of any clustering algorithm. As such, we see many potential applications of the individual parts of our multiresolution consensus clustering procedure in addition to using the procedure itself to identify hierarchical structure in networks., Comment: 19 pages, 9 figures, code available at https://github.com/LJeub/HierarchicalConsensus

Stochastic resonance is a phenomenon in which noise enhances the response of a system to an input signal. The brain is an example of a system that has to detect and transmit signals in a noisy environment, suggesting that it is a good candidate to take advantage of SR. In this work, we aim to identify the optimal levels of noise that promote signal transmission through a simple network model of the human brain. Specifically, using a dynamic model implemented on an anatomical brain network (connectome), we investigate the similarity between an input signal and a signal that has traveled across the network while the system is subject to different noise levels. We find that non-zero levels of noise enhance the similarity between the input signal and the signal that has traveled through the system. The optimal noise level is not unique; rather, there is a set of parameter values at which the information is transmitted with greater precision, this set corresponds to the parameter values that place the system in a critical regime. The multiplicity of critical points in our model allows it to adapt to different noise situations and remain at criticality.

We investigate the relationship of resting-state fMRI functional connectivity estimated over long periods of time with time-varying functional connectivity estimated over shorter time intervals. We show that using Pearson's correlation to estimate functional connectivity implies that the range of fluctuations of functional connections over short time scales is subject to statistical constraints imposed by their connectivity strength over longer scales. We present a method for estimating time-varying functional connectivity that is designed to mitigate this issue and allows us to identify episodes where functional connections are unexpectedly strong or weak. We apply this method to data recorded from $N=80$ participants, and show that the number of unexpectedly strong/weak connections fluctuates over time, and that these variations coincide with intermittent periods of high and low modularity in time-varying functional connectivity. We also find that during periods of relative quiescence regions associated with default mode network tend to join communities with attentional, control, and primary sensory systems. In contrast, during periods where many connections are unexpectedly strong/weak, default mode regions dissociate and form distinct modules. Finally, we go on to show that, while all functional connections can at times manifest stronger (more positively correlated) or weaker (more negatively correlated) than expected, a small number of connections, mostly within the visual and somatomotor networks, do so a disproportional number of times. Our statistical approach allows the detection of functional connections that fluctuate more or less than expected based on their long-time averages and may be of use in future studies characterizing the spatio-temporal patterns of time-varying functional connectivity, Comment: 47 Pages, 8 Figures, 4 Supplementary Figures

Modularity is an important topological attribute for functional brain networks. Recent studies have reported that modularity of functional networks varies not only across individuals being related to demographics and cognitive performance, but also within individuals co-occurring with fluctuations in network properties of functional connectivity, estimated over short time intervals. However, characteristics of these time-resolved functional networks during periods of high and low modularity have remained largely unexplored. In this study we investigate spatiotemporal properties of time-resolved networks in the high and low modularity periods during rest, with a particular focus on their spatial connectivity patterns, temporal homogeneity and test-retest reliability. We show that spatial connectivity patterns of time-resolved networks in the high and low modularity periods are represented by increased and decreased dissociation of the default mode network module from task-positive network modules, respectively. We also find that the instances of time-resolved functional connectivity sampled from within the high (low) modularity period are relatively homogeneous (heterogeneous) over time, indicating that during the low modularity period the default mode network interacts with other networks in a variable manner. We confirmed that the occurrence of the high and low modularity periods varies across individuals with moderate inter-session test-retest reliability and that it is correlated with previously-reported individual differences in the modularity of functional connectivity estimated over longer timescales. Our findings illustrate how time-resolved functional networks are spatiotemporally organized during periods of high and low modularity, allowing one to trace individual differences in long-timescale modularity to the variable occurrence of network configurations at shorter timescales., Comment: Reorganized the paper; to appear in NeuroImage; arXiv abstract shortened to fit within character limits

The human brain is a complex network of interconnected brain regions organized into functional modules with distinct roles in cognition and behavior. An important question concerns the persistence and stability of these modules over the human lifespan. Here we use graph-theoretic analysis to algorithmically uncover the brain's intrinsic modular organization across multiple spatial scales ranging from small communities comprised of only a few brain regions to large communities made up of many regions. We find that at coarse scales modules become progressively more segregated, while at finer scales segregation decreases. Module composition also exhibits scale-specific and age-dependent changes. At coarse scales, the module assignments of regions normally associated with control, default mode, attention, and visual networks are highly flexible. At fine scales the most flexible regions are associated with the default mode network. Finally, we show that, with age, some regions in the default mode network, specifically retrosplenial cortex, maintain a greater proportion of functional connections to their own module, while regions associated with somatomotor and saliency/ventral attention networks distribute their links more evenly across modules., Comment: 56 pages, 7 figures, 6 supplemental figures

The human connectome represents a network map of the brain's wiring diagram and the pattern into which its connections are organized is thought to play an important role in cognitive function. The generative rules that shape the topology of the human connectome remain incompletely understood. Earlier work in model organisms has suggested that wiring rules based on geometric relationships (distance) can account for many but likely not all topological features. Here we systematically explore a family of generative models of the human connectome that yield synthetic networks designed according to different wiring rules combining geometric and a broad range of topological factors. We find that a combination of geometric constraints with a homophilic attachment mechanism can create synthetic networks that closely match many topological characteristics of individual human connectomes, including features that were not included in the optimization of the generative model itself. We use these models to investigate a lifespan dataset and show that, with age, the model parameters undergo progressive changes, suggesting a rebalancing of the generative factors underlying the connectome across the lifespan., Comment: 38 pages, 5 figures + 19 supplemental figures, 1 table