Uncovering functional signature in neural systems via random matrix theory

Reduced Plasticity in Coupling Strength in the Aging SCN Clock as Revealed by Kuramoto Modeling

The mammalian circadian clock is located in the suprachiasmatic nucleus (SCN) and consists of a network of coupled neurons, which are entrained to the environmental light-dark cycle. The phase coherence of the neurons is plastic and driven by the duration of daylight. With aging, the capacity to behaviorally adapt to seasonal changes in photoperiod reduces. The mechanisms underlying photoperiodic adaptation are largely unknown, but are important to unravel for the development of novel interventions to improve the quality of life of the elderly. We analyzed the phase coherence of single-cell PERIOD2::LUCIFERASE (PER2::LUC) expression rhythms in the SCN of young and old mice entrained to either long or short photoperiod. The phase coherence was used as input to a 2-community noisy Kuramoto model to estimate the coupling strength between and within neuronal subpopulations. The model revealed a correlation between coupling strength and photoperiod-induced changes in the phase relationship among neurons, suggesting a functional link. We found that the SCN of young mice adapts in coupling strength over a large range, with weak coupling in long photoperiod (LP) and strong coupling in short photoperiod (SP). In aged mice, we also found weak coupling in LP, but a reduced capacity to reach strong coupling in SP. The inability to respond with an increase in coupling strength suggests that manipulation of photoperiod is not a suitable strategy to enhance clock function with aging. We conclude that the inability of aged mice to reach strong coupling contributes to deficits in behavioral adaptation to seasonal changes in photoperiod.

Network Structure of the Master Clock Is Important for Its Primary Function

A master clock located in the suprachiasmatic nucleus (SCN) regulates the circadian rhythm of physiological and behavioral activities in mammals. The SCN has two main functions in the regulation: an endogenous clock produces the endogenous rhythmic signal in body rhythms, and a calibrator synchronizes the body rhythms to the external light-dark cycle. These two functions have been determined to depend on either the dynamic behaviors of individual neurons or the whole SCN neuronal network. In this review, we first introduce possible network structures for the SCN, as revealed by time series analysis from real experimental data. It was found that the SCN network is heterogeneous and sparse, that is, the average shortest path length is very short, some nodes are hubs with large node degrees but most nodes have small node degrees, and the average node degree of the network is small. Secondly, the effects of the SCN network structure on the SCN function are reviewed based on mathematical models of the SCN network. It was found that robust rhythms with large amplitudes, a high synchronization between SCN neurons and a large entrainment ability exists mainly in small-world and scale-free type networks, but not other types. We conclude that the SCN most probably is an efficient small-world type or scale-free type network, which drives SCN function.

Fluctuation spectra of large random dynamical systems reveal hidden structure in ecological networks

Understanding the relationship between complexity and stability in large dynamical systems-such as ecosystems-remains a key open question in complexity theory which has inspired a rich body of work developed over more than fifty years. The vast majority of this theory addresses asymptotic linear stability around equilibrium points, but the idea of 'stability' in fact has other uses in the empirical ecological literature. The important notion of 'temporal stability' describes the character of fluctuations in population dynamics, driven by intrinsic or extrinsic noise. Here we apply tools from random matrix theory to the problem of temporal stability, deriving analytical predictions for the fluctuation spectra of complex ecological networks. We show that different network structures leave distinct signatures in the spectrum of fluctuations, and demonstrate the application of our theory to the analysis of ecological time-series data of plankton abundances.

From Univariate to Multivariate Coupling Between Continuous Signals and Point Processes: A Mathematical Framework

Time series data sets often contain heterogeneous signals, composed of both continuously changing quantities and discretely occurring events. The coupling between these measurements may provide insights into key underlying mechanisms of the systems under study. To better extract this information, we investigate the asymptotic statistical properties of coupling measures between continuous signals and point processes. We first introduce martingale stochastic integration theory as a mathematical model for a family of statistical quantities that include the phase locking value, a classical coupling measure to characterize complex dynamics. Based on the martingale central limit theorem, we can then derive the asymptotic gaussian distribution of estimates of such coupling measure that can be exploited for statistical testing. Second, based on multivariate extensions of this result and random matrix theory, we establish a principled way to analyze the low-rank coupling between a large number of point processes and continuous signals. For a null hypothesis of no coupling, we establish sufficient conditions for the empirical distribution of squared singular values of the matrix to converge, as the number of measured signals increases, to the well-known Marchenko-Pastur (MP) law, and the largest squared singular value converges to the upper end of the MP support. This justifies a simple thresholding approach to assess the significance of multivariate coupling. Finally, we illustrate with simulations the relevance of our univariate and multivariate results in the context of neural time series, addressing how to reliably quantify the interplay between multichannel local field potential signals and the spiking activity of a large population of neurons.

Two-community noisy Kuramoto model with general interaction strengths. I

We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. By developing a geometric interpretation of the self-consistency equations, we are able to separate the parameter space into ten regions in which we identify the maximum number of solutions in the steady state. Furthermore, we prove that in the steady state, only the angles 0 and π are possible between the average phases of the two communities and derive the solution boundary for the unsynchronized solution. Last, we identify the equivalence class relation in the parameter space corresponding to the symmetrically synchronized solution.

Aging Affects the Capacity of Photoperiodic Adaptation Downstream from the Central Molecular Clock

Aging impairs circadian clock function, leading to disrupted sleep-wake patterns and a reduced capability to adapt to changes in environmental light conditions. This makes shift work or the changing of time zones challenging for the elderly and, importantly, is associated with the development of age-related diseases. However, it is unclear what levels of the clock machinery are affected by aging, which is relevant for the development of targeted interventions. We found that naturally aged mice of >24 months had a reduced rhythm amplitude in behavior compared with young controls (3-6 months). Moreover, the old animals had a strongly reduced ability to adapt to short photoperiods. Recording PER2::LUC protein expression in the suprachiasmatic nucleus revealed no impairment of the rhythms in PER2 protein under the 3 different photoperiods tested (LD: 8:16, 12:12, and 16:8). Thus, we observed a discrepancy between the behavioral phenotype and the molecular clock, and we conclude that the aging-related deficits emerge downstream of the core molecular clock. Since it is known that aging affects several intracellular and membrane components of the central clock cells, it is likely that an impairment of the interaction between the molecular clock and these components is contributing to the deficits in photoperiod adaptation.

Determining the Hierarchical Architecture of the Human Brain Using Subject-Level Clustering of Functional Networks

Optimal integration and segregation of neuronal connections are necessary for efficient large-scale network communication between distributed cortical regions while allowing for modular specialization. This dynamic in the cortex is enabled at the network mesoscale by the organization of nodes into communities. Previous in vivo efforts to map the mesoscale architecture in humans had several limitations. Here we characterize a consensus multiscale community organization of the functional cortical network. We derive this consensus from the clustering of subject-level networks. We applied this analysis to magnetic resonance imaging data from 1003 healthy individuals part of the Human Connectome Project. The hierarchical atlas and code will be made publicly available for future investigators.

Structural Entropy: Monitoring Correlation-Based Networks Over Time With Application To Financial Markets

The concept of "Structural Diversity" of a network refers to the level of dissimilarity between the various agents acting in the system, and it is typically interpreted as the number of connected components in the network. This key property of networks has been studied in multiple settings, including diffusion of ideas in social networks and functional diversity of regions in brain networks. Here, we propose a new measure, "Structural Entropy", as a revised interpretation to "Structural Diversity". The proposed measure relies on the finer-grained network communities (in contrast to the network's connected components), and takes into consideration both the number of communities and their sizes, generating a single representative value. We then propose an approach for monitoring the structure of correlation-based networks over time, which relies on the newly suggested measure. Finally, we illustrate the usefulness of the new approach, by applying it to the particular case of emergent organization of financial markets. This provides us a way to explore their underlying structural changes, revealing a remarkably high linear correlation between the new measure and the volatility of the assets' prices over time.