Modeling of dynamical systems through deep learning

This review presents a modern perspective on dynamical systems in the context of current goals and open challenges. In particular, our review focuses on the key challenges of discovering dynamics from data and finding data-driven representations that make nonlinear systems amenable to linear analysis. We explore various challenges in modern dynamical systems, along with emerging techniques in data science and machine learning to tackle them. The two chief challenges are (1) nonlinear dynamics and (2) unknown or partially known dynamics. Machine learning is providing new and powerful techniques for both challenges. Dimensionality reduction methods are used for projecting dynamical methods in reduced form, and these methods perform computational efficiency on real-world data. Data-driven models drive to discover the governing equations and give laws of physics. The identification of dynamical systems through deep learning techniques succeeds in inferring physical systems. Machine learning provides advanced new and powerful algorithms for nonlinear dynamics. Advanced deep learning methods like autoencoders, recurrent neural networks, convolutional neural networks, and reinforcement learning are used in modeling of dynamical systems.     

Identification of dynamic mass-action biochemical reaction networks using sparse Bayesian methods

Identifying the reactions that govern a dynamical biological system is a crucial but challenging task in systems biology. In this work, we present a data-driven method to infer the underlying biochemical reaction system governing a set of observed species concentrations over time. We formulate the problem as a regression over a large, but limited, mass-action constrained reaction space and utilize sparse Bayesian inference via the regularized horseshoe prior to produce robust, interpretable biochemical reaction networks, along with uncertainty estimates of parameters. The resulting systems of chemical reactions and posteriors inform the biologist of potentially several reaction systems that can be further investigated. We demonstrate the method on two examples of recovering the dynamics of an unknown reaction system, to illustrate the benefits of improved accuracy and information obtained.     

Creative dynamics approach to neural intelligence

The thrust of this paper is to introduce and discuss a substantially new type of dynamical system for modelling biological behavior. The approach was motivated by an attempt to remove one of the most fundamental limitations of artificial neural networks — their rigid behavior compared with even simplest biological systems. This approach exploits a novel paradigm in nonlinear dynamics based upon the concept of terminal attractors and repellers. It was demonstrated that non-Lipschitzian dynamics based upon the failure of Lipschitz condition exhibits a new qualitative effect — a multi-choice response to periodic external excitations. Based upon this property, a substantially new class of dynamical systems — the unpredictable systems — was introduced and analyzed. These systems are represented in the form of coupled activation and learning dynamical equations whose ability to be spontaneously activated is based upon two pathological characteristics. Firstly, such systems have zero Jacobian. As a result of that, they have an infinite number of equilibrium points which occupy curves, surfaces or hypersurfaces. Secondly, at all these equilibrium points, the Lipschitz conditions fails, so the equilibrium points become terminal attractors or repellers depending upon the sign of the periodic excitation. Both of these pathological characteristics result in multi-choice response of unpredictable dynamical systems. It has been shown that the unpredictable systems can be controlled by sign strings which uniquely define the system behaviors by specifying the direction of the motions in the critical points. By changing the combinations of signs in the code strings the system can reproduce any prescribed behavior to a prescribed accuracy. That is why the unpredictable systems driven by sign strings are extremely flexible and are highly adaptable to environmental changes. It was also shown that such systems can serve as a powerful tool for temporal pattern memories and complex pattern recognition. It has been demonstrated that new architecture of neural networks based upon non-Lipschitzian dynamics can be utilized for modelling more complex patterns of behavior which can be associated with phenomenological models of creativity and neural intelligence.     

Integrating data mining and transmission theory in the ecology of infectious diseases

Our understanding of ecological processes is built on patterns inferred from data. Applying modern analytical tools such as machine learning to increasingly high dimensional data offers the potential to expand our perspectives on these processes, shedding new light on complex ecological phenomena such as pathogen transmission in wild populations. Here, we propose a novel approach that combines data mining with theoretical models of disease dynamics. Using rodents as an example, we incorporate statistical differences in the life history features of zoonotic reservoir hosts into pathogen transmission models, enabling us to bound the range of dynamical phenomena associated with hosts, based on their traits. We then test for associations between equilibrium prevalence, a key epidemiological metric and data on human outbreaks of rodent‐borne zoonoses, identifying matches between empirical evidence and theoretical predictions of transmission dynamics. We show how this framework can be generalized to other systems through a rubric of disease models and parameters that can be derived from empirical data. By linking life history components directly to their effects on disease dynamics, our mining‐modelling approach integrates machine learning and theoretical models to explore mechanisms in the macroecology of pathogen transmission and their consequences for spillover infection to humans.     

Forecasting and Uncertainty Quantification Using a Hybrid of Mechanistic and Non-mechanistic Models for an Age-Structured Population Model

In this paper, we present a new method for the prediction and uncertainty quantification of data-driven multivariate systems. Traditionally, either mechanistic or non-mechanistic modeling methodologies have been used for prediction; however, it is uncommon for the two to be incorporated together. We compare the forecast accuracy of mechanistic modeling, using Bayesian inference, a non-mechanistic modeling approach based on state space reconstruction, and a novel hybrid methodology composed of the two for an age-structured population data set. The data come from cannibalistic flour beetles, in which it is observed that the adults preying on the eggs and pupae result in non-equilibrium population dynamics. Uncertainty quantification methods for the hybrid models are outlined and illustrated for these data. We perform an analysis of the results from Bayesian inference for the mechanistic model and hybrid models to suggest reasons why hybrid modeling methodology may enable more accurate forecasts of multivariate systems than traditional approaches.     

Practical limits for reverse engineering of dynamical systems: a statistical analysis of sensitivity and parameter inferability in systems biology models

The size and complexity of cellular systems make building predictive models an extremely difficult task. In principle dynamical time-course data can be used to elucidate the structure of the underlying molecular mechanisms, but a central and recurring problem is that many and very different models can be fitted to experimental data, especially when the latter are limited and subject to noise. Even given a model, estimating its parameters remains challenging in real-world systems. Here we present a comprehensive analysis of 180 systems biology models, which allows us to classify the parameters with respect to their contribution to the overall dynamical behaviour of the different systems. Our results reveal candidate elements of control in biochemical pathways that differentially contribute to dynamics. We introduce sensitivity profiles that concisely characterize parameter sensitivity and demonstrate how this can be connected to variability in data. Systematically linking data and model sloppiness allows us to extract features of dynamical systems that determine how well parameters can be estimated from time-course measurements, and associates the extent of data required for parameter inference with the model structure, and also with the global dynamical state of the system. The comprehensive analysis of so many systems biology models reaffirms the inability to estimate precisely most model or kinetic parameters as a generic feature of dynamical systems, and provides safe guidelines for performing better inferences and model predictions in the context of reverse engineering of mathematical models for biological systems.     

Transfer Functions for Protein Signal Transduction: Application to a Model of Striatal Neural Plasticity

We present a novel formulation for biochemical reaction networks in the context of protein signal transduction. The model consists of input-output transfer functions, which are derived from differential equations, using stable equilibria. We select a set of “source” species, which are interpreted as input signals. Signals are transmitted to all other species in the system (the “target” species) with a specific delay and with a specific transmission strength. The delay is computed as the maximal reaction time until a stable equilibrium for the target species is reached, in the context of all other reactions in the system. The transmission strength is the concentration change of the target species. The computed input-output transfer functions can be stored in a matrix, fitted with parameters, and even recalled to build dynamical models on the basis of state changes. By separating the temporal and the magnitudinal domain we can greatly simplify the computational model, circumventing typical problems of complex dynamical systems. The transfer function transformation of biochemical reaction systems can be applied to mass-action kinetic models of signal transduction. The paper shows that this approach yields significant novel insights while remaining a fully testable and executable dynamical model for signal transduction. In particular we can deconstruct the complex system into local transfer functions between individual species. As an example, we examine modularity and signal integration using a published model of striatal neural plasticity. The modularizations that emerge correspond to a known biological distinction between calcium-dependent and cAMP-dependent pathways. Remarkably, we found that overall interconnectedness depends on the magnitude of inputs, with higher connectivity at low input concentrations and significant modularization at moderate to high input concentrations. This general result, which directly follows from the properties of individual transfer functions, contradicts notions of ubiquitous complexity by showing input-dependent signal transmission inactivation.     

Automated Design of Complex Dynamic Systems

Several fields of study are concerned with uniting the concept of computation with that of the design of physical systems. For example, a recent trend in robotics is to design robots in such a way that they require a minimal control effort. Another example is found in the domain of photonics, where recent efforts try to benefit directly from the complex nonlinear dynamics to achieve more efficient signal processing. The underlying goal of these and similar research efforts is to internalize a large part of the necessary computations within the physical system itself by exploiting its inherent non-linear dynamics. This, however, often requires the optimization of large numbers of system parameters, related to both the system''s structure as well as its material properties. In addition, many of these parameters are subject to fabrication variability or to variations through time. In this paper we apply a machine learning algorithm to optimize physical dynamic systems. We show that such algorithms, which are normally applied on abstract computational entities, can be extended to the field of differential equations and used to optimize an associated set of parameters which determine their behavior. We show that machine learning training methodologies are highly useful in designing robust systems, and we provide a set of both simple and complex examples using models of physical dynamical systems. Interestingly, the derived optimization method is intimately related to direct collocation a method known in the field of optimal control. Our work suggests that the application domains of both machine learning and optimal control have a largely unexplored overlapping area which envelopes a novel design methodology of smart and highly complex physical systems.     

Information-Theoretic Analysis of the Dynamics of an Executable Biological Model

To facilitate analysis and understanding of biological systems, large-scale data are often integrated into models using a variety of mathematical and computational approaches. Such models describe the dynamics of the biological system and can be used to study the changes in the state of the system over time. For many model classes, such as discrete or continuous dynamical systems, there exist appropriate frameworks and tools for analyzing system dynamics. However, the heterogeneous information that encodes and bridges molecular and cellular dynamics, inherent to fine-grained molecular simulation models, presents significant challenges to the study of system dynamics. In this paper, we present an algorithmic information theory based approach for the analysis and interpretation of the dynamics of such executable models of biological systems. We apply a normalized compression distance (NCD) analysis to the state representations of a model that simulates the immune decision making and immune cell behavior. We show that this analysis successfully captures the essential information in the dynamics of the system, which results from a variety of events including proliferation, differentiation, or perturbations such as gene knock-outs. We demonstrate that this approach can be used for the analysis of executable models, regardless of the modeling framework, and for making experimentally quantifiable predictions.     

Learning and evolution in bacterial taxis: an operational amplifier circuit modeling the computational dynamics of the prokaryotic 'two component system' protein network

Adaptive behavior in unicellular organisms (i.e., bacteria) depends on highly organized networks of proteins governing purposefully the myriad of molecular processes occurring within the cellular system. For instance, bacteria are able to explore the environment within which they develop by utilizing the motility of their flagellar system as well as a sophisticated biochemical navigation system that samples the environmental conditions surrounding the cell, searching for nutrients or moving away from toxic substances or dangerous physical conditions. In this paper we discuss how proteins of the intervening signal transduction network could be modeled as artificial neurons, simulating the dynamical aspects of the bacterial taxis. The model is based on the assumption that, in some important aspects, proteins can be considered as processing elements or McCulloch-Pitts artificial neurons that transfer and process information from the bacterium's membrane surface to the flagellar motor. This simulation of bacterial taxis has been carried out on a hardware realization of a McCulloch-Pitts artificial neuron using an operational amplifier. Based on the behavior of the operational amplifier we produce a model of the interaction between CheY and FliM, elements of the prokaryotic two component system controlling chemotaxis, as well as a simulation of learning and evolution processes in bacterial taxis. On the one side, our simulation results indicate that, computationally, these protein 'switches' are similar to McCulloch-Pitts artificial neurons, suggesting a bridge between evolution and learning in dynamical systems at cellular and molecular levels and the evolutive hardware approach. On the other side, important protein 'tactilizing' properties are not tapped by the model, and this suggests further complexity steps to explore in the approach to biological molecular computing.     

Adding Detritus to a Nutrient-Phytoplankton-Zooplankton Model:A Dynamical-Systems Approach

The dynamics of two plankton population models are investigatedto examine sensitivities to model complexity and to parametervalues. The models simulate concentrations of nutrients, phytoplankton,zooplankton and detritus in the oceanic mixed layer. In Model1, zooplankton can graze only upon phytoplankton, whereas inModel 2 they can graze upon phytoplankton and detritus. Bothfeeding strategies are employed by zooplankton in the ocean,and both are features of models in the literature. Each modelhere consists of four coupled ordinary differential equations,and can exhibit unforced oscillations (limit cycles) of thefour concentrations. By constructing diagrams that show howsteady states and oscillations persist as each parameter isvaried, a general picture of the dynamics of each model is builtup. The addition of the detritus pool to an earlier nutrient–phytoplankton–zooplanktonmodel appears to have little influence on the dynamics whenthe zooplankton cannot graze upon the detritus (Model 1), butif the zooplankton can graze upon the detritus (Model 2), thenthe dynamics are affected in a significant way. These results,obtained using the theory of dynamical systems, enhance ourknowledge of the factors governing the dynamics of planktonpopulation models.     

A Lyapunov function for piecewise-independent differential equations: stability of the ideal free distribution in two patch environments

In this article we construct Lyapunov functions for models described by piecewise-continuous and independent differential equations. Because these models are described by discontinuous differential equations, the theory of Lyapunov functions for smooth dynamical systems is not applicable. Instead, we use a geometrical approach to construct a Lyapunov function. Then we apply the general approach to analyze population dynamics describing exploitative competition of two species in a two-patch environment. We prove that for any biologically meaningful parameter combination the model has a globally stable equilibrium and we analyze this equilibrium with respect to parameters.      

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.

     

Solitary wave and spatially locked solitary pattern in a chemical reaction system

We investigate the behavior of a one-dimensional two component dynamical system. The dynamical equations are obtained by extracting an essence out of equations which describe the behavior of a biochemical reaction catalyzed by an allosteric protein. The obtained dynamical equations are similar to van der Pol equations. The dynamical equations are solved numerically. In the continuous system, a solitary wave is found to occur in certain ranges of the parameter space. The condition of occurrence of the solitary wave is investigated. The solitary wave can be induced by various initial perturbations, including rectangular ones with space-wise length longer than a certain critical value. The property of the solitary wave is similar to that of the impulses in nervous systems. In the discrete system, a spatially locked solitary pattern is found to occur in certain ranges of the parameter space.     

Heterogeneous animal group models and their group-level alignment dynamics: an equation-free approach

We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.     

On learning dynamics underlying the evolution of learning rules

In order to understand the development of non-genetically encoded actions during an animal’s lifespan, it is necessary to analyze the dynamics and evolution of learning rules producing behavior. Owing to the intrinsic stochastic and frequency-dependent nature of learning dynamics, these rules are often studied in evolutionary biology via agent-based computer simulations. In this paper, we show that stochastic approximation theory can help to qualitatively understand learning dynamics and formulate analytical models for the evolution of learning rules. We consider a population of individuals repeatedly interacting during their lifespan, and where the stage game faced by the individuals fluctuates according to an environmental stochastic process. Individuals adjust their behavioral actions according to learning rules belonging to the class of experience-weighted attraction learning mechanisms, which includes standard reinforcement and Bayesian learning as special cases. We use stochastic approximation theory in order to derive differential equations governing action play probabilities, which turn out to have qualitative features of mutator-selection equations. We then perform agent-based simulations to find the conditions where the deterministic approximation is closest to the original stochastic learning process for standard 2-action 2-player fluctuating games, where interaction between learning rules and preference reversal may occur. Finally, we analyze a simplified model for the evolution of learning in a producer–scrounger game, which shows that the exploration rate can interact in a non-intuitive way with other features of co-evolving learning rules. Overall, our analyses illustrate the usefulness of applying stochastic approximation theory in the study of animal learning.     

Multivariate recurrence plots for visualizing and quantifying the dynamics of spatially extended ecosystems

Few methods for quantifying the dynamics of temporal processes are readily applicable to spatially extended systems when equations governing the motion are unknown. The objective of this paper is to illustrate how the MRP-RQA (multivariate recurrence plot-recurrence quantification analysis) approach may serve to characterize ecosystems driven by both deterministic and stochastic forces. The strength of the MRP-RQA approach resides in its independence from constraining assumptions regarding outliers, noise, stationarity and transients. Its utility is demonstrated by means of two spatiotemporal series (summer and spring datasets) of light intensity variations in an old growth forest ecosystem. Results revealed qualitative differences in homogeneity, transiency, and drift typologies between the MRPs derived from each dataset. RQA estimates of determinism and Kolmogorov entropy supported the idea that mixed chaotic–stochastic dynamics may be common in mesoscale forest habitats. Advantages and inconveniences of the MRP-RQA approach are also discussed in the more general context of monitoring ecosystems.     

On the use of qualitative reasoning to simulate and identify metabolic pathways

MOTIVATION: Perhaps the greatest challenge of modern biology is to develop accurate in silico models of cells. To do this we require computational formalisms for both simulation (how according to the model the state of the cell evolves over time) and identification (learning a model cell from observation of states). We propose the use of qualitative reasoning (QR) as a unified formalism for both tasks. The two most commonly used alternative methods of modelling biochemical pathways are ordinary differential equations (ODEs), and logical/graph-based (LG) models. RESULTS: The QR formalism we use is an abstraction of ODEs. It enables the behaviour of many ODEs, with different functional forms and parameters, to be captured in a single QR model. QR has the advantage over LG models of explicitly including dynamics. To simulate biochemical pathways we have developed 'enzyme' and 'metabolite' QR building blocks that fit together to form models. These models are finite, directly executable, easy to interpret and robust. To identify QR models we have developed heuristic chemoinformatics graph analysis and machine learning procedures. The graph analysis procedure is a series of constraints and heuristics that limit the number of ways metabolites can combine to form pathways. The machine learning procedure is generate-and-test inductive logic programming. We illustrate the use of QR for modelling and simulation using the example of glycolysis. AVAILABILITY: All data and programs used are available on request.