Robust steady states in ecosystems with symmetries.

Steady states of dynamical systems, whether stable or unstable, are critical for understanding future evolution. Robust steady states, ones that persist under small changes in the model parameters, are desired when modelling ecological systems, where it is common for accurate and detailed information on functional form and parameters to be unavailable. Previous work by Jahedi et al. [Robustness of solutions of almost every system of equations, SIAM J. Appl. Math. 82(5) (2022), pp. 1791-1807; Structured systems of nonlinear equations, SIAM J. Appl. Math. 83(4) (2023), pp. 1696-1716.] has established criteria to imply the prevalence of robust steady states for systems with minimal predetermined structure, including conventional structured systems. We review that work and extend it by allowing symmetries in the system structure, which present added obstructions to robustness.

Stabilizing the return to normal behavior in an epidemic.

Predicting the interplay between infectious disease and behavior has been an intractable problem because behavioral response is so varied. We introduce a general framework for feedback between incidence and behavior for an infectious disease. By identifying stable equilibria, we provide policy end-states that are self-managing and self-maintaining. We prove mathematically the existence of two new endemic equilibria depending on the vaccination rate: one in the presence of low vaccination but with reduced societal activity (the "new normal"), and one with return to normal activity but with vaccination rate below that required for disease elimination. This framework allows us to anticipate the long-term consequence of an emerging disease and design a vaccination response that optimizes public health and limits societal consequences.

IDENTIFIABILITY OF INFECTION MODEL PARAMETERS EARLY IN AN EPIDEMIC.

It is known that the parameters in the deterministic and stochastic SEIR epidemic models are structurally identifiable. For example, from knowledge of the infected population time series I(t) during the entire epidemic, the parameters can be successfully estimated. In this article we observe that estimation will fail in practice if only infected case data during the early part of the epidemic (prepeak) is available. This fact can be explained using a well-known phenomenon called dynamical compensation. We use this concept to derive an unidentifiability manifold in the parameter space of SEIR that consists of parameters indistinguishable from I(t) early in the epidemic. Thus, identifiability depends on the extent of the system trajectory that is available for observation. Although the existence of the unidentifiability manifold obstructs the ability to exactly determine the parameters, we suggest that it may be useful for uncertainty quantification purposes. A variant of SEIR recently proposed for COVID-19 modeling is also analyzed, and an analogous unidentifiability surface is derived.

A Generalized Unscented Transformation for Probability Distributions.

The unscented transform uses a weighted set of samples called sigma points to propagate the means and covariances of nonlinear transformations of random variables. However, unscented transforms developed using either the Gaussian assumption or a minimum set of sigma points typically fall short when the random variable is not Gaussian distributed and the nonlinearities are substantial. In this paper, we develop the generalized unscented transform (GenUT), which uses 2n+1 sigma points to accurately capture up to the diagonal components of the skewness and kurtosis tensors of most probability distributions. Constraints can be analytically enforced on the sigma points while guaranteeing at least second-order accuracy. The GenUT uses the same number of sigma points as the original unscented transform while also being applicable to non-Gaussian distributions, including the assimilation of observations in the modeling of infectious diseases such as coronavirus (SARS-CoV-2) causing COVID-19.

Correcting observation model error in data assimilation.

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications, the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models and on a single-column radiative transfer model with multicloud parameterization.

Tracking intracellular dynamics through extracellular measurements.

Extracellular recordings of neuronal cells are frequently a part of in vitro and in vivo experimental studies as a means of monitoring network-level dynamics. Their connections to intracellular dynamics are not well understood. Single-unit recordings are a more direct way to measure intracellular dynamics, but are typically difficult and expensive. On the other hand, simple differential equations models exist for single neurons. In this article, we apply a recent advance in data assimilation theory, designed to correct bias in general observation functions, toward the reconstruction of model-based intracellular dynamics from extracellular recordings.

Limits on reconstruction of dynamics in networks.

An observability condition number is defined for physical systems modeled by network dynamics. Assuming that the dynamical equations of the network are known and a noisy trajectory is observed at a subset of the nodes, we calculate the expected distance to the nearest correct trajectory as a function of the observation noise level and discuss how it varies over the unobserved nodes of the network. When the condition number is sufficiently large, reconstructing the trajectory from observations from the subset will be infeasible. This knowledge can be used to choose an optimal subset from which to observe a network.

Changes in Ugandan Climate Rainfall at the Village and Forest Level.

In 2013, the US National Oceanographic and Atmospheric Administration (NOAA) refined the historical rainfall estimates over the African Continent and produced the African Rainfall Climate version 2.0 (ARC2) estimator. ARC2 offers a nearly complete record of daily rainfall estimates since 1983 at 0.1° × 0.1° resolution. Despite short-term anomalies, we identify an overall decrease in average rainfall of about 12% during the past 34 years in Uganda. Spatiotemporally, these decreases are greatest in agricultural regions of central and western Uganda, but similar rainfall decreases are also reflected in the gorilla habitat within the Bwindi Forest in Southwest Uganda. The findings carry significant implications for agriculture production, food security, wildlife habitat, and economic impact at the community and societal level.

Developing and Applying the Propensity Score to Make Causal Inferences: Variable Selection and Stratification.

This Monte Carlo simulation examined the effects of variable selection (combinations of confounders with four patterns of relationships to outcome and assignment to treatment) and number of strata (5, 10, or 20) in propensity score analyses. The focus was on how the variations affected the average effect size compared to quasi-assignment without adjustment for bias. Results indicate that if a propensity score model does not include variables strongly related to both outcome and assignment, not only will bias not decrease, but it may possibly increase. Furthermore, models that include a variable highly related to assignment to treatment but do not also include a variable highly related to the outcome could increase bias. In regards to the number of strata, results varied depending on the propensity score model and sample size. In 75% of the models that resulted in a significant reduction in bias, quintiles outperformed the other stratification schemes. In fact, the richer that the propensity score model was (i.e., including multiple covariates of varying relationships to the outcome and to assignment to treatment), the more likely that the model required fewer strata to balance the covariates. In models without that same richness, additional strata were necessary. Finally, the study suggests that when developing a rich propensity score model with stratification, it is crucial to examine the strata for overlap.

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.

Predicting chaotic time series with a partial model.

Methods for forecasting time series are a critical aspect of the understanding and control of complex networks. When the model of the network is unknown, nonparametric methods for prediction have been developed, based on concepts of attractor reconstruction pioneered by Takens and others. In this Rapid Communication we consider how to make use of a subset of the system equations, if they are known, to improve the predictive capability of forecasting methods. A counterintuitive implication of the results is that knowledge of the evolution equation of even one variable, if known, can improve forecasting of all variables. The method is illustrated on data from the Lorenz attractor and from a small network with chaotic dynamics.

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.

Real-time tracking of neuronal network structure using data assimilation.

A nonlinear data assimilation technique is applied to determine and track effective connections between ensembles of cultured spinal cord neurons measured with multielectrode arrays. The method is statistical, depending only on confidence intervals, and requiring no form of arbitrary thresholding. In addition, the method updates connection strengths sequentially, enabling real-time tracking of nonstationary networks. The ensemble Kalman filter is used with a generic spiking neuron model to estimate connection strengths as well as other system parameters to deal with model mismatch. The method is validated on noisy synthetic data from Hodgkin-Huxley model neurons before being used to find network connections in the neural culture recordings.

Detecting connectivity changes in neuronal networks.

We develop a method from semiparametric statistics (Cox, 1972) for the purpose of tracking links and connection strengths over time in a neuronal network from spike train data. We consider application of the method as implemented in Masud and Borisyuk (2011), and evaluate its use on data generated independently of the Cox model hypothesis, in particular from a spiking model of Izhikevich in four different dynamical regimes. Then, we show how the Cox method can be used to determine statistically significant changes in network connectivity over time. Our methodology is demonstrated using spike trains from multi-electrode array measurements of networks of cultured mammalian spinal cord cells.

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.

Chaotic itinerancy based on attractors of one-dimensional maps.

A general methodology is described for constructing systems that have a slowly converging Lyapunov exponent near zero, based on one-dimensional maps with chaotic attractors. In certain parameter ranges, these relatively simple systems display the properties of intermittent dynamics known as chaotic itinerancy. We show that in addition to the local sensitivity characteristic of chaotic dynamics, these itinerant systems display a global sensitivity, in the sense that fine-scale additive noise may significantly change the natural measure on the large scale.