Effects Of Symmetry On The Structural Controllability Of Neural Networks: A Perspective.

The controllability of a dynamical system or network describes whether a given set of control inputs can completely exert influence in order to drive the system towards a desired state. Structural controllability develops the canonical coupling structures in a network that lead to un-controllability, but does not account for the effects of explicit symmetries contained in a network. Recent work has made use of this framework to determine the minimum number and location of the optimal actuators necessary to completely control complex networks. In systems or networks with structural symmetries, group representation theory provides the mechanisms for how the symmetry contained in a network will influence its controllability, and thus affects the placement of these critical actuators, which is a topic of broad interest in science from ecological, biological and man-made networks to engineering systems and design.

Observability and Controllability of Nonlinear Networks: The Role of Symmetry.

Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.

Rainfall drives hydrocephalus in East Africa.

OBJECT: Hydrocephalus is one of the most common brain disorders in children throughout the world. The majority of infant hydrocephalus cases in East Africa appear to be postinfectious, related to preceding neonatal infections, and are thus preventable if the microbial origins and routes of infection can be characterized. In prior microbiological work, the authors noted evidence of seasonality in postinfectious hydrocephalus (PIH) cases. METHODS: The geographical address of 696 consecutive children with PIH who were treated over 6 years was fused with satellite rainfall data for the same time period. A comprehensive time series and spatiotemporal analysis of cases and rainfall was performed. RESULTS: Four infection-onset peaks were found to straddle the twice-yearly rainy season peaks, demonstrating that the infections occurred at intermediate levels of rainfall. CONCLUSIONS: The findings in this study reveal a previously unknown link between climate and a neurosurgical condition. Satellite-derived rainfall dynamics are an important factor in driving the infections that lead to PIH. Given prior microbial analysis, these findings point to the importance of environmental factors with respect to preventing the newborn infections that lead to PIH.

Observability of Neuronal Network Motifs.

We quantify observability in small (3 node) neuronal networks as a function of 1) the connection topology and symmetry, 2) the measured nodes, and 3) the nodal dynamics (linear and nonlinear). We find that typical observability metrics for 3 neuron motifs range over several orders of magnitude, depending upon topology, and for motifs containing symmetry the network observability decreases when observing from particularly confounded nodes. Nonlinearities in the nodal equations generally decrease the average network observability and full network information becomes available only in limited regions of the system phase space. Our findings demonstrate that such networks are partially observable, and suggest their potential efficacy in reconstructing network dynamics from limited measurement data. How well such strategies can be used to reconstruct and control network dynamics in experimental settings is a subject for future experimental work.

Data assimilation for heterogeneous networks: the consensus set.

Data assimilation in dynamical networks is intrinsically challenging. A method is introduced for the tracking of heterogeneous networks of oscillators or excitable cells in a nonstationary environment, using a homogeneous model network to expedite the accurate reconstruction of parameters and unobserved variables. An implementation using ensemble Kalman filtering to track the states of the heterogeneous network is demonstrated on simulated data and applied to a mammalian brain network experiment. The approach has broad applicability for the prediction and control of biological and physical networks.

Reconstructing the topology of sparsely connected dynamical networks.

Given a general physical network and measurements of node dynamics, methods are proposed for reconstructing the network topology. We focus on networks whose connections are sparse and where data are limited. Under these conditions, common in many biological networks, constrained optimization techniques based on the L1 vector norm are found to be superior for inference of the network connections.

Reconstruction of shared nonlinear dynamics in a network.

We study a general physical network consisting of a collection of response systems with complex nonlinear dynamics, influenced by a common driver. The goal is to reconstruct dynamics, regular or chaotic, that are common to all of the response systems, working from simultaneous time series measured at the responses systems only. A fundamental theorem is stated concerning the reconstruction of the common driver. An algorithm is developed, based on the theorem, to carry out the reconstruction, and is demonstrated with several examples.

Shadowing breakdown and large errors in dynamical simulations of physical systems.

Simulations play a crucial role in the modern study of physical systems. A major open question for long dynamical simulations of physical processes is the role of discretization and truncation errors in the outcome. A general mechanism is described that can cause extremely small noise inputs to result in errors in simulation statistics that are several orders of magnitude larger. A scaling law for the size of such errors in terms of the noise level and properties of the dynamics is given. This result brings into question trajectory averages that are computed for systems with particular dynamical behaviors, in particular, systems that exhibit fluctuating Lyapunov exponents or unstable dimension variability.