A method for representing and developing process models

Scientists investigate the dynamics of complex systems with quantitative models, employing them to synthesize knowledge, to explain observations, and to forecast future system behavior. Complete specification of systems is impossible, so models must be simplified abstractions. Thus, the art of modeling involves deciding which system elements to include and determining how they should be represented. We view modeling as search through a space of candidate models that is guided by model objectives, theoretical knowledge, and empirical data. In this contribution, we introduce a method for representing process-based models that facilitates the discovery of structures that explain observed behavior. This representation casts dynamic systems as interacting sets of processes that act on entities. Using this approach, a modeler first encodes relevant ecological knowledge into a library of generic entities and processes, then instantiates these theoretical components, and finally assembles candidate models from these elements. We illustrate this methodology with a model of the Ross Sea ecosystem.     

Predictive landscapes hidden beneath biological cellular automata

To celebrate Hans Frauenfelder’s achievements, we examine energy(-like) “landscapes” for complex living systems. Energy landscapes summarize all possible dynamics of some physical systems. Energy(-like) landscapes can explain some biomolecular processes, including gene expression and, as Frauenfelder showed, protein folding. But energy-like landscapes and existing frameworks like statistical mechanics seem impractical for describing many living systems. Difficulties stem from living systems being high dimensional, nonlinear, and governed by many, tightly coupled constituents that are noisy. The predominant modeling approach is devising differential equations that are tailored to each living system. This ad hoc approach faces the notorious “parameter problem”: models have numerous nonlinear, mathematical functions with unknown parameter values, even for describing just a few intracellular processes. One cannot measure many intracellular parameters or can only measure them as snapshots in time. Another modeling approach uses cellular automata to represent living systems as discrete dynamical systems with binary variables. Quantitative (Hamiltonian-based) rules can dictate cellular automata (e.g., Cellular Potts Model). But numerous biological features, in current practice, are qualitatively described rather than quantitatively (e.g., gene is (highly) expressed or not (highly) expressed). Cellular automata governed by verbal rules are useful representations for living systems and can mitigate the parameter problem. However, they can yield complex dynamics that are difficult to understand because the automata-governing rules are not quantitative and much of the existing mathematical tools and theorems apply to continuous but not discrete dynamical systems. Recent studies found ways to overcome this challenge. These studies either discovered or suggest an existence of predictive “landscapes” whose shapes are described by Lyapunov functions and yield “equations of motion” for a “pseudo-particle.” The pseudo-particle represents the entire cellular lattice and moves on the landscape, thereby giving a low-dimensional representation of the cellular automata dynamics. We outline this promising modeling strategy.

     

Aggregation error in nonlinear ecological models

Conditions are presented for accurately representing the dynamics of several components of a nonlinear ecological system by a single state variable. For mass-balance ecosystem models, only two conditions exist that permit such aggregation without introducing error into the model. Under the first condition, components can be lumped if their environmental losses (e.g. respiration or migration) are characterized by identical linear functions. Under the second condition, the components must maintain constant proportionality throughout their transient behavior. If these conditions are violated, it is not possible to aggregate without error. However, it is shown that aggregation error can still be minimized. The theorems developed here extend related results derived for the general theory of systems and have the advantage of generality, being applicable to nonlinear models of considerable complexity.     

Circumventing structural uncertainty: A Bayesian perspective on nonlinear forecasting for ecology

As a consequence of the complexity of ecosystems and context-dependence of species interactions, structural uncertainty is pervasive in ecological modeling. This is particularly problematic when ecological models are used to make conservation and management plans whose outcomes may depend strongly on model formulation. Nonlinear time series approaches allow us to circumvent this issue by using the observed dynamics of the system to guide policy development. However, these methods typically require long time series from stationary systems, which are rarely available in ecological settings. Here we present a Bayesian approach to nonlinear forecasting based on Gaussian processes that readily integrates information from several short time series and allows for nonstationary dynamics. We demonstrate the utility of our modeling methods on simulated from a wide range of ecological scenarios. We expect that these models will extend the range of ecological systems to which nonlinear forecasting methods can be usefully applied.     

Dynamical modeling of collective behavior from pigeon flight data: flock cohesion and dispersion

Several models of flocking have been promoted based on simulations with qualitatively naturalistic behavior. In this paper we provide the first direct application of computational modeling methods to infer flocking behavior from experimental field data. We show that this approach is able to infer general rules for interaction, or lack of interaction, among members of a flock or, more generally, any community. Using experimental field measurements of homing pigeons in flight we demonstrate the existence of a basic distance dependent attraction/repulsion relationship and show that this rule is sufficient to explain collective behavior observed in nature. Positional data of individuals over time are used as input data to a computational algorithm capable of building complex nonlinear functions that can represent the system behavior. Topological nearest neighbor interactions are considered to characterize the components within this model. The efficacy of this method is demonstrated with simulated noisy data generated from the classical (two dimensional) Vicsek model. When applied to experimental data from homing pigeon flights we show that the more complex three dimensional models are capable of simulating trajectories, as well as exhibiting realistic collective dynamics. The simulations of the reconstructed models are used to extract properties of the collective behavior in pigeons, and how it is affected by changing the initial conditions of the system. Our results demonstrate that this approach may be applied to construct models capable of simulating trajectories and collective dynamics using experimental field measurements of herd movement. From these models, the behavior of the individual agents (animals) may be inferred.     

Expert-Guided Generative Topographical Modeling with Visual to Parametric Interaction

Introduced by Bishop et al. in 1996, Generative Topographic Mapping (GTM) is a powerful nonlinear latent variable modeling approach for visualizing high-dimensional data. It has shown useful when typical linear methods fail. However, GTM still suffers from drawbacks. Its complex parameterization of data make GTM hard to fit and sensitive to slight changes in the model. For this reason, we extend GTM to a visual analytics framework so that users may guide the parameterization and assess the data from multiple GTM perspectives. Specifically, we develop the theory and methods for Visual to Parametric Interaction (V2PI) with data using GTM visualizations. The result is a dynamic version of GTM that fosters data exploration. We refer to the new version as V2PI-GTM. In this paper, we develop V2PI-GTM in stages and demonstrate its benefits within the context of a text mining case study.     

植被系统中植物与环境因子相互作用的动态模拟

本文在综合考虑现有植被发展动态模型的基础上提出包含植被,环境因子,人类或其它外力干扰三者间相互作用情况下的植被系统一般动态模型,并从控制论的角度阐述了植被与环境因子间的耦合,反馈作用。最后以线性逼近和微分拟合的方法求取模型的局部解,进而作出外延预测。实例分析表明,模型预测对实际观测有较好的跟踪性能。     

A new method of identifying a systems based on the minimal quadratic discrepancy criterion for biophysics problems

A wide range of biophysical systems are described by nonlinear dynamic models mathematically presented as a set of ordinary differential equations in the Cauchy explicit form: [formula: see text] Fij(X1(t),..,XN(t),t), (i = 1,...,N, j = 1,...,M), where Fij (X1(t), ..., XN(t), t) is a set of basis functions satisfying the Lipschitz condition. We investigate the problem of evaluation of model constants aij (the system identification) using experimental data about the time dependence of the dynamic parameters of the system Xi(t). A new method of system identification for the class of similar nonlinear dynamic models is proposed. It is shown that the problem of identifying an initial nonlinear model can be reduced to the solution of a system of linear equations for the matrix of the dynamic model constants [aj]i. It is proposed to determine the set of dynamic model constants aij using the criterion of minimal quadratic discrepancy for the time dependence of the set of dynamic parameters Xi(t). An important special case of the nonlinear model, the quadratic model, is considered. Test problems of identification using this method are presented for two nonlinear systems: the Van der Pol type multiparametric nonlinear oscillator and the strange attractor of Ressler, a widely known example of dynamic systems showing the stochastic behavior.     

Active Learning to Understand Infectious Disease Models and Improve Policy Making

Modeling plays a major role in policy making, especially for infectious disease interventions but such models can be complex and computationally intensive. A more systematic exploration is needed to gain a thorough systems understanding. We present an active learning approach based on machine learning techniques as iterative surrogate modeling and model-guided experimentation to systematically analyze both common and edge manifestations of complex model runs. Symbolic regression is used for nonlinear response surface modeling with automatic feature selection. First, we illustrate our approach using an individual-based model for influenza vaccination. After optimizing the parameter space, we observe an inverse relationship between vaccination coverage and cumulative attack rate reinforced by herd immunity. Second, we demonstrate the use of surrogate modeling techniques on input-response data from a deterministic dynamic model, which was designed to explore the cost-effectiveness of varicella-zoster virus vaccination. We use symbolic regression to handle high dimensionality and correlated inputs and to identify the most influential variables. Provided insight is used to focus research, reduce dimensionality and decrease decision uncertainty. We conclude that active learning is needed to fully understand complex systems behavior. Surrogate models can be readily explored at no computational expense, and can also be used as emulator to improve rapid policy making in various settings.     

A description of arterial wall mechanics using limiting chain extensibility constitutive models

Certain aspects of the mechanical response of arterial walls can be described using nonlinear elasticity theory. Uniaxial tests on vascular walls reveal nonlinear stress-strain behavior, with higher extensibility in the low stretch range and progressively lower extensibility with increasing stretch. This phenomenon is well known in the framework of rubber-like materials where it is called a strain-hardening or strain-stiffening effect. Constitutive models of incompressible hyperelasticity that take this into account include power-law models and limiting chain extensibility models. Our purpose in this paper is to bring to the attention of the biomechanics community some essential features of one such model of the latter type due to Gent. This model is compared with isotropic versions of biomechanical constitutive models by Takamizawa-Hayashi and Fung; the latter is a limiting version of a power-law material. Two particular problems are considered for which experimental data on arterial wall deformations are available. The first concerns small oscillations superposed on a large static stretch of a vertical string of arterial tissue. It is shown that the exponential model of Fung and the Gent model match well with the experimental data. The second problem is the extension of an internally pressurized circular cylindrical tube. It is shown that an inversion phenomenon observed experimentally for the human iliac artery can be described within a membrane theory by the Gent model whereas this cannot be described using the exponential model. The foregoing considerations are carried out for isotropic elastic materials in the absence of residual stress. Extensions to include anisotropy are also indicated.     

Extended Kalman Filter for Estimation of Parameters in Nonlinear State-Space Models of Biochemical Networks

It is system dynamics that determines the function of cells, tissues and organisms. To develop mathematical models and estimate their parameters are an essential issue for studying dynamic behaviors of biological systems which include metabolic networks, genetic regulatory networks and signal transduction pathways, under perturbation of external stimuli. In general, biological dynamic systems are partially observed. Therefore, a natural way to model dynamic biological systems is to employ nonlinear state-space equations. Although statistical methods for parameter estimation of linear models in biological dynamic systems have been developed intensively in the recent years, the estimation of both states and parameters of nonlinear dynamic systems remains a challenging task. In this report, we apply extended Kalman Filter (EKF) to the estimation of both states and parameters of nonlinear state-space models. To evaluate the performance of the EKF for parameter estimation, we apply the EKF to a simulation dataset and two real datasets: JAK-STAT signal transduction pathway and Ras/Raf/MEK/ERK signaling transduction pathways datasets. The preliminary results show that EKF can accurately estimate the parameters and predict states in nonlinear state-space equations for modeling dynamic biochemical networks.     

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.     

Feedback identification of continuous microbial growth systems

The fundamental problem of dynamic modeling of continuous culture systems for process control and optimization is addressed. Forcing a system to bifurcation via feedback control is a very promising method for model discrimination and identification. Dynamic information is obtained by using this technique, the dynamic behavior of the chemostat as predicted by unstructured models, the model with delay, and a structured model has been analyzed. The method exposes significant differences in the nonlinear dynamic structure of the various models and can be implemented to discriminate between various possible models for a continuous culture system.     

Reducing complexity in metabolic networks: making metabolic meshes manageable

The dynamics of complex systems can be effectively analyzed by judicious use of intrinsic time constants. Order of magnitude estimation based on time constants has been used successfully to examine the dynamic behavior of complicated processes. The main goal of this paper is to introduce this approach to the analysis of complex metabolic systems. Time constants and dynamic modes of motion are defined within the context of well-established linear algebra. The order of magnitude estimation is then introduced into the systemic framework. The main goals of the analysis are: to provide improved understanding of biochemical dynamics and their physiological significance, and to yield reduced dynamic models that are physiologically realistic but tractable for practical use.     

Next-Generation Individual-Based Models Integrate Biodiversity and Ecosystems: Yes We Can,and Yes We Must

Ecosystem and community ecology have evolved along different pathways, with little overlap. However, to meet societal demands for predicting changes in ecosystem services, the functional and structural view dominating these two branches of ecology, respectively, must be integrated. Biodiversity–ecosystem function research has addressed this integration for two decades, but full integration that makes predictions relevant to practical problems is still lacking. We argue that full integration requires going, in both branches, deeper by taking into account individual organisms and the evolutionary and physico-chemical principles that drive their behavior. Individual-based models are a major tool for this integration. They have matured by using individual-level mechanism to replace the demographic thinking which dominates classical theoretical ecology. Existing individual-based ecosystem models already have proven useful both for theory and application. Still, next-generation individual-based models will increasingly use standardized and re-usable submodels to represent behaviors and mechanisms such as growth, uptake of nutrients, foraging, and home range behavior. The strategy of pattern-oriented modeling then helps make such ecosystem models structurally realistic by developing theory for individual behaviors just detailed enough to reproduce and explain patterns observed at the system level. Next-generation ecosystem scientists should include the individual-based approach in their toolkit and focus on addressing real systems because theory development and solving applied problems go hand-in-hand in individual-based ecology.