Directed network motifs in Alzheimer's disease and mild cognitive impairment.

Directed network motifs are the building blocks of complex networks, such as human brain networks, and capture deep connectivity information that is not contained in standard network measures. In this paper we present the first application of directed network motifs in vivo to human brain networks, utilizing recently developed directed progression networks which are built upon rates of cortical thickness changes between brain regions. This is in contrast to previous studies which have relied on simulations and in vitro analysis of non-human brains. We show that frequencies of specific directed network motifs can be used to distinguish between patients with Alzheimer's disease (AD) and normal control (NC) subjects. Especially interesting from a clinical standpoint, these motif frequencies can also distinguish between subjects with mild cognitive impairment who remained stable over three years (MCI) and those who converted to AD (CONV). Furthermore, we find that the entropy of the distribution of directed network motifs increased from MCI to CONV to AD, implying that the distribution of pathology is more structured in MCI but becomes less so as it progresses to CONV and further to AD. Thus, directed network motifs frequencies and distributional properties provide new insights into the progression of Alzheimer's disease as well as new imaging markers for distinguishing between normal controls, stable mild cognitive impairment, MCI converters and Alzheimer's disease.

Stochastic geometric network models for groups of functional and structural connectomes.

Structural and functional connectomes are emerging as important instruments in the study of normal brain function and in the development of new biomarkers for a variety of brain disorders. In contrast to single-network studies that presently dominate the (non-connectome) network literature, connectome analyses typically examine groups of empirical networks and then compare these against standard (stochastic) network models. The current practice in connectome studies is to employ stochastic network models derived from social science and engineering contexts as the basis for the comparison. However, these are not necessarily best suited for the analysis of connectomes, which often contain groups of very closely related networks, such as occurs with a set of controls or a set of patients with a specific disorder. This paper studies important extensions of standard stochastic models that make them better adapted for analysis of connectomes, and develops new statistical fitting methodologies that account for inter-subject variations. The extensions explicitly incorporate geometric information about a network based on distances and inter/intra hemispherical asymmetries (to supplement ordinary degree-distribution information), and utilize a stochastic choice of network density levels (for fixed threshold networks) to better capture the variance in average connectivity among subjects. The new statistical tools introduced here allow one to compare groups of networks by matching both their average characteristics and the variations among them. A notable finding is that connectomes have high "smallworldness" beyond that arising from geometric and degree considerations alone.

Directed progression brain networks in Alzheimer's disease: properties and classification.

This article introduces a new approach in brain connectomics aimed at characterizing the temporal spread in the brain of pathologies like Alzheimer's disease (AD). The main instrument is the development of "directed progression networks" (DPNets), wherein one constructs directed edges between nodes based on (weakly) inferred directions of the temporal spreading of the pathology. This stands in contrast to many previously studied brain networks where edges represent correlations, physical connections, or functional progressions. In addition, this is one of a few studies showing the value of using directed networks in the study of AD. This article focuses on the construction of DPNets for AD using longitudinal cortical thickness measurements from magnetic resonance imaging data. The network properties are then characterized, providing new insights into AD progression, as well as novel markers for differentiating normal cognition (NC) and AD at the group level. It also demonstrates the important role of nodal variations for network classification (i.e., the significance of standard deviations, not just mean values of nodal properties). Finally, the DPNets are utilized to classify subjects based on their global network measures using a variety of data-mining methodologies. In contrast to most brain networks, these DPNets do not show high clustering and small-world properties.

Hierarchical networks, power laws, and neuronal avalanches.

We show that in networks with a hierarchical architecture, critical dynamical behaviors can emerge even when the underlying dynamical processes are not critical. This finding provides explicit insight into current studies of the brain's neuronal network showing power-law avalanches in neural recordings, and provides a theoretical justification of recent numerical findings. Our analysis shows how the hierarchical organization of a network can itself lead to power-law distributions of avalanche sizes and durations, scaling laws between anomalous exponents, and universal functions-even in the absence of self-organized criticality or critical points. This hierarchy-induced phenomenon is independent of, though can potentially operate in conjunction with, standard dynamical mechanisms for generating power laws.

Test-retest reliability of computational network measurements derived from the structural connectome of the human brain.

Structural magnetic resonance (MR) connectomics holds promise for the diagnosis, outcome prediction, and treatment monitoring of many common neurodevelopmental, psychiatric, and neurodegenerative disorders for which there is currently no clinical utility for MR imaging (MRI). Before computational network metrics from the human connectome can be applied in a clinical setting, their precision and their normative intersubject variation must be understood to guide the study design and the interpretation of longitudinal data. In this work, the reproducibility of commonly used graph theoretic measures is investigated, as applied to the structural connectome of healthy adult volunteers. Two datasets are examined, one consisting of 10 subjects scanned twice at one MRI facility and one consisting of five subjects scanned once each at two different facilities using the same imaging platform. Global graph metrics are calculated for unweighed and weighed connectomes, and two levels of granularity of the connectome are evaluated: one based on the 82-node cortical and subcortical parcellation from FreeSurfer and one based on an atlas-free parcellation of the gray-white matter boundary consisting of 1000 cortical nodes. The consistency of the unweighed and weighed edges and the module assignments are also computed for the 82-node connectomes. Overall, the results demonstrate good-to-excellent test-retest reliability for the entire connectome-processing pipeline, including the graph analytics, in both the intrasite and intersite datasets. These findings indicate that measurements of computational network metrics derived from the structural connectome have sufficient precision to be tested as potential biomarkers for diagnosis, prognosis, and monitoring of interventions in neurological and psychiatric diseases.

The structural connectome of the human brain in agenesis of the corpus callosum.

Adopting a network perspective, the structural connectome reveals the large-scale white matter connectivity of the human brain, yielding insights into cerebral organization otherwise inaccessible to researchers and clinicians. Connectomics has great potential for elucidating abnormal connectivity in congenital brain malformations, especially axonal pathfinding disorders. Agenesis of the corpus callosum (AgCC) is one of the most common brain malformations and can also be considered a prototypical genetic disorder of axonal guidance in humans. In this exploratory study, the structural connectome of AgCC is mapped and compared to that of the normal human brain. Multiple levels of granularity of the AgCC connectome are investigated, including summary network metrics, modularity analysis, and network consistency measures, with comparison to the normal structural connectome after simulated removal of all callosal connections ("virtual callostomy"). These investigations reveal four major findings. First, global connectivity is abnormally reduced in AgCC, but local connectivity is increased. Second, the network topology of AgCC is more variable than that of the normal human connectome, contradicting the predictions of the virtual callostomy model. Third, modularity analysis reveals that many of the tracts that comprise the structural core of the cerebral cortex have relatively weak connectivity in AgCC, especially the cingulate bundles bilaterally. Finally, virtual lesions of the Probst bundles in the AgCC connectome demonstrate that there is consistency across subjects in many of the connections generated by these ectopic white matter tracts, and that they are a mixture of cortical and subcortical fibers. These results go beyond prior diffusion tractography studies to provide a systems-level perspective on anomalous connectivity in AgCC. Furthermore, this work offers a proof of principle for the utility of the connectome framework in neurodevelopmental disorders.

Probabilistic convergence guarantees for type-II pulse-coupled oscillators.

We show that a large class of pulse-coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a classification scheme for type-II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These results suggest methods for the analysis of pulse-coupled oscillators, and provide insights into the balance of excitation and inhibition in the operation of biological type-II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.

Robust convergence in pulse-coupled oscillators with delays.

We show that for pulse-coupled oscillators a class of phase response curves with both excitation and inhibition exhibit robust convergence to synchrony on arbitrary aperiodic connected graphs with delays. We describe the basins of convergence and give explicit bounds on the convergence times. These results provide new and more robust methods for synchronization of sensor nets and also have biological implications.

Combinatorial games with a pass: a dynamical systems approach.

By treating combinatorial games as dynamical systems, we are able to address a longstanding open question in combinatorial game theory, namely, how the introduction of a "pass" move into a game affects its behavior. We consider two well known combinatorial games, 3-pile Nim and 3-row Chomp. In the case of Nim, we observe that the introduction of the pass dramatically alters the game's underlying structure, rendering it considerably more complex, while for Chomp, the pass move is found to have relatively minimal impact. We show how these results can be understood by recasting these games as dynamical systems describable by dynamical recursion relations. From these recursion relations, we are able to identify underlying structural connections between these "games with passes" and a recently introduced class of "generic (perturbed) games." This connection, together with a (non-rigorous) numerical stability analysis, allows one to understand and predict the effect of a pass on a game.

Construction and analysis of random networks with explosive percolation.

The existence of explosive phase transitions in random (Erdös Rényi-type) networks has been recently documented by Achlioptas, D'Souza, and Spencer [Science 323, 1453 (2009)] via simulations. In this Letter we describe the underlying mechanism behind these first-order phase transitions and develop tools that allow us to identify (and predict) when a random network will exhibit an explosive transition. Several interesting new models displaying explosive transitions are also presented.

Nonlinear dynamics in combinatorial games: renormalizing Chomp.

We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to transform a combinatorial game like Chomp into a type of dynamical system. Not only does this provide powerful insights into the game of Chomp (yielding a complete probabilistic description of optimal play in Chomp and an answer to a longstanding question about the nature of the winning opening move), but more generally, it offers a mathematical framework for exploring this unexpected relationship between combinatorial games and modern dynamical systems theory.

Behavior of coupled automata.

We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of "synchronization" between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles.

Immune modulation by ionizing radiation and its implications for cancer immunotherapy.

Ionizing radiation exhibits immunomodulatory properties, which could portend a future collaboration of cancer immunotherapy with radiation therapy. The danger model of immunity describes antigen-specific cellular immunity engendered by an inflammatory milieu. Dendritic cells (DCs) are attracted to this microenvironment, undergoing maturation after internalizing apoptotic and necrotic cellular debris. Mature DCs mediate antigen-specific cellular immunity via presentation of processed antigen to T cells. Administration of radiation has been utilized in vitro and in vivo to create an inflammatory setting, via induction of apoptosis, necrosis, cell surface molecules, and secretory molecules. Caspase-mediated cellular apoptosis is induced by radiation thro ugh multiple signaling pathways. Radiation upregulates expression of immunomodulatory surface molecules (MHC, costimulatory molecules, adhesion molecules, death receptors, heat shock proteins) and secretory molecules (cytokines, inflammatory mediators) in tumor, stromal, and vascular endothelial cells. Results of animal studies indicate possible radiation-mediated modulation of tumor antigen-specific immunity. Experimental data could indicate that the radiation-induced danger microenvironment engenders a DC-mediated antigen-specific immune response. Further enhancement of radiation-mediated inflammation and cell death can be achieved via administration of radiosensitizing pharmaceuticals. Radiation-mediated immune modulation currently remains unquantified and poorly understood. A major research effort will be required to elucidate mechanisms of action. With a thorough understanding of this phenomenon, we believe that ionizing radiation could be optimized for use with cancer vaccines and generate tumor antigen-specific cellular immunity.