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.     

A system identification analysis of neural adaptation dynamics and nonlinear responses in the local reflex control of locust hind limbs

Nonlinear type system identification models coupled with white noise stimulation provide an experimentally convenient and quick way to investigate the often complex and nonlinear interactions between the mechanical and neural elements of reflex limb control systems. Previous steady state analysis has allowed the neurons in such systems to be categorised by their sensitivity to position, velocity or acceleration (dynamics) and has improved our understanding of network function. These neurons, however, are known to adapt their output amplitude or spike firing rate during repetitive stimulation and this transient response may be more important than the steady state response for reflex control. In the current study previously used system identification methods are developed and applied to investigate both steady state and transient dynamic and nonlinear changes in the neural circuit responsible for controlling reflex movements of the locust hind limbs. Through the use of a parsimonious model structure and Monte Carlo simulations we conclude that key system dynamics remain relatively unchanged during repetitive stimulation while output amplitude adaptation is occurring. Whilst some evidence of a significant change was found in parts of the systems nonlinear response, the effect was small and probably of little physiological relevance. Analysis using biologically more realistic stimulation reinforces this conclusion.     

In silico analysis of lactic acid secretion metabolism through the top-down approach: Effect of grouping in enzyme kinetics

A top-down approach is known to be a useful and effective technique for the design and analysis of metabolic systems. In this study, we have constructed a grouped metabolic network forLactococcus lactis under aerobic conditions using grouped enzyme kinetics. To test the usefulness of grouping work, a non-grouped system and grouped systems were compared quantitatively with each other. Here, grouped systems were designed as two groups according to the extent of grouping. The overall simulated flux values in grouped and non-grouped models had pretty similar distribution trends, but the details on flux ratio at the pyruvate branch point showed a little difference. This result indicates that our grouping technique can be used as a good model for complicated metabolic networks, however, for detailed analysis of metabolic network, a more robust mechanism should be considered. In addition to the data for the pyruvate branch point analysis, some major flux control coefficients were obtained in this research.     

Spontaneous switching of frequency-locking by periodic stimulus in oscillators of plasmodium of the true slime mold

Microfabrication technique was used to construct a model system with a living cell of plasmodium of the true slime mold, Physarum polycephalum, a living coupled oscillator system. Its parameters can be systematically controlled as in computer simulations, so that results are directly comparable to those of general mathematical models. As the first step, we investigated responses in oscillatory cells, the oscillators of the plasmodium, to periodic stimuli by temperature changes to elucidate characteristics of the cells as nonlinear systems whose internal dynamics are unknown because of their complexity. We observed that the forced oscillator of the plasmodium show 1:1, 2:1, 3:1 frequency locking inside so-called Arnold tongues regions as well as in other nonlinear systems such as chemical systems and other biological systems. In addition, we found spontaneous switching behavior from certain frequency locking states to other states, even under certain fixed parameters. This technique can be applied to more complex systems with multiple elements, such as coupled oscillator systems, and would be useful to investigate complicated phenomena in biological systems such as information processing.     

A method for constructing data-based models of spiking neurons using a dynamic linear-static nonlinear cascade

This paper describes a general nonlinear dynamical model for neural system identification. It describes an algorithm for fitting a simple form of the model to spike train data, and reports on this algorithm's performance in identifying the structure and parameters of simulated neurons. The central element of the model is a Wiener-Bose dynamic nonlinearity that ensures that the model is able to approximate the behaviour of an arbitrary nonlinear dynamical system. Nonlinearities associated with spike generation and transmission are treated by placing the Wiener-Bose system in cascade with pulse frequency modulators and demodulators, and the static nonlinearity at the output of the Wiener-Bose system is decomposed into a rectifier and a multinomial. This simplifies the model without reducing its generality for neuronal system identification. Model elements can be characterised using standard methods of dynamical systems analysis, and the model has a simple form that can be implemented and simulated efficiently. This model bears a structural resemblance to real neurons; it may be regarded as a connectionist neuron that has been generalized in a realistic way to enable it to mimic the behaviour of an arbitrary nonlinear system, or conversely as a general nonlinear model that has been constrained to make it easy to fit to spike train data. Tests with simulated data show that the identification algorithm can accurately estimate the structure and parameters of neuron-like nonlinear dynamical systems using data sets containing only a few hundred spikes.     

Bilinear state space systems for nonlinear dynamical modelling

Summary We discuss the identification of multiple input, multiple output, discrete-time bilinear state space systems. We consider two identification problems. In the first case, the input to the system is a measurable white noise sequence. We show that it is possible to identify the system by solving a nonlinear optimization problem. The number of parameters in this optimization problem can be reduced by exploiting the principle of separable least squares. A subspace-based algorithm can be used to generate initial estimates for this nonlinear identification procedure. In the second case, the input to the system is not measurable. This makes it a much more difficult identification problem than the case with known inputs. At present, we can only solve this problem for a certain class of single input, single output bilinear state space systems, namely bilinear systems in phase variable form.     

Modeling Metal Flow Systems

Substance flow analysis (SFA) is a frequently used industrial ecology technique for studying societal metal flows, but it is limited in its ability to inform us about future developments in metal flow patterns and how we can affect them. Equation‐based simulation modeling techniques, such as dynamic SFA and system dynamics, can usefully complement static SFA studies in this respect, but they are also restricted in several ways. The objective of this article is to demonstrate the ability of agent‐based modeling to overcome these limitations and its usefulness as a tool for studying societal metal flow systems. The body of the article summarizes the parallel implementation of two models—an agent‐based model and a system dynamics model—both addressing the following research question: What conditions foster the development of a closed‐loop flow network for metals in mobile phones? The results from in silico experimentation with these models highlight three important differences between agent‐based modeling (ABM) and equation‐based modeling (EBM) techniques. An analysis of how these differences affected the insights that could be extracted from the constructed models points to several key advantages of ABM in the study of metal flow systems. In particular, this analysis suggests that a key advantage of the ABM technique is its flexibility to enable the representation of societal metal flow systems in a more native manner. This added flexibility endows modelers with enhanced leverage to identify options for steering metal flows and opens new opportunities for using the metaphor of an ecosystem to understand metal flow systems more fully.     

On-line identification of sensory systems using pseudorandom binary noise perturbations.

A technique of on-line identification of linear system characteristics from sensory systems with spike train or analog voltage outputs was developed and applied to the semicircular canal. A pseudorandom binary white noise input was cross-correlated with the system's output to produce estimates of linear system unit impulse responses (UIRs), which were then corrected for response errors of the input transducers. The effects of variability in the system response characteristics and sensitivity were studied by employing the technique with known linear analog circuits. First-order unit afferent responses from the guitarfish horizontal semicircular canal were cross-correlated with white noise rotational acceleration inputs to produce non-parametric UIR models. In addition, the UIRs were fitted by nonlinear regression to truncated exponential series to produce parametric models in the form of low-order linear system equations. The experimental responses to the white noise input were then compared with those predicted from the UIR models linear convolution, and the differences were expressed as a percent mean-square-error (%MSE). The average difference found from a population of 62 semicircular canal afferents was relatively low mean and standard deviation of 10.2 +/- 5.9 SD%MSE, respectively. This suggests that relatively accurate inferences can be made concerning the physiology of the semicircular canal from the linear characteristics of afferent responses.     

A ternary logic model for recurrent neuromime networks with delay

 In contrast to popular recurrent artificial neural network (RANN) models, biological neural networks have unsymmetric structures and incorporate significant delays as a result of axonal propagation. Consequently, biologically inspired neural network models are more accurately described by nonlinear differential-delay equations rather than nonlinear ordinary differential equations (ODEs), and the standard techniques for studying the dynamics of RANNs are wholly inadequate for these models. This paper develops a ternary-logic based method for analyzing these networks. Key to the technique is the realization that a nonzero delay produces a bounded stability region. This result significantly simplifies the construction of sufficient conditions for characterizing the network equilibria. If the network gain is large enough, each equilibrium can be classified as either asymptotically stable or unstable. To illustrate the analysis technique, the swim central pattern generator (CPG) of the sea slug Tritonia diomedea is examined. For wide range of reasonable parameter values, the ternary analysis shows that none of the network equilibria are stable, and thus the network must oscillate. The results show that complex synaptic dynamics are not necessary for pattern generation. Received: 15 June 1994/Accepted in revised form: 10 February 1995     

Minimum-order Wiener modelling of spike-output systems

Systems that generate spike outputs in response to continuous inputs abound in neurophysiology. The study of their dynamics with the use of systems analysis methods has been complicated by the difference in modality of the input and output signals. When the problem is placed in the framework of Wiener's theory in discrete time, an infinite functional series is required for the formal representation of the input-output relation. This has given rise to the belief that a large number of Wiener functionals is needed in practice before a model of reasonable accuracy can be obtained. In this paper, we introduce the concept of minimum-order Wiener models for spike-output systems, and we show that a low-order Wiener model is adequate in many cases for predicting fully the timing of the output spikes.     

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ₁-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.     

Nomograms Aid Interpretation of Complex Regression Models

ABSTRACT Ecologists often develop complex regression models that include multiple categorical and continuous variables, interactions among predictors, and nonlinear relationships between the response and predictor variables. Nomograms, which are graphical devices for presenting mathematical functions and calculating output values, can aid biologists in interpreting and presenting these complex models. To illustrate benefits of nomograms, we developed a logistic regression model of elk (Cervus elaphus) resource selection. With this model, we demonstrated how a nomogram helps scientists and managers interpret interactions among variables, compare the relative biological importance of variables, and examine predicted shapes of relationships (e.g., linear vs. nonlinear) between response and predictor variables. Although our example focused on logistic regression, nomograms are equally useful for other linear and nonlinear models. Regardless of the approach used for model development, nomograms and other graphical summaries can help scientists and managers develop, interpret, and apply statistical models.     

Use of in vitro gas production models in ruminal kinetics.

Physiological systems models for ruminant animals are used to predict the extent of ruminal carbohydrate digestion, based on rates of intake, digestion, and passage to the lower tract. Digestion of feed carbohydrates is described in these models by a first-order rate constant. Recently, an in vitro gas production technique has been developed to determine the digestion kinetics in batch fermentation, and nonlinear mathematical models have been fitted to the cumulative gas production data from these experiments. In this paper, we present an analysis that converts these gas production models to an effective first-order rate constant that can be used directly in rumen systems models. The analysis considers the digestion of an incremental mass of substrate entering the rumen. The occurrence of passage is represented probabilistically, and integration through time gives the total mass of substrate and total rate of digestion in the rumen. To demonstrate the analysis, several gas production models are fitted to a sample data set for corn silage, and the effective first-order rate constants are calculated. The rate constants for digestion depend on ruminal passage rate, an interaction that arises from the nonlinearity of the gas production models.     

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.     

Neurobiological models of two-choice decision making can be reduced to a one-dimensional nonlinear diffusion equation

The response behaviors in many two-alternative choice tasks are well described by so-called sequential sampling models. In these models, the evidence for each one of the two alternatives accumulates over time until it reaches a threshold, at which point a response is made. At the neurophysiological level, single neuron data recorded while monkeys are engaged in two-alternative choice tasks are well described by winner-take-all network models in which the two choices are represented in the firing rates of separate populations of neurons. Here, we show that such nonlinear network models can generally be reduced to a one-dimensional nonlinear diffusion equation, which bears functional resemblance to standard sequential sampling models of behavior. This reduction gives the functional dependence of performance and reaction-times on external inputs in the original system, irrespective of the system details. What is more, the nonlinear diffusion equation can provide excellent fits to behavioral data from two-choice decision making tasks by varying these external inputs. This suggests that changes in behavior under various experimental conditions, e.g. changes in stimulus coherence or response deadline, are driven by internal modulation of afferent inputs to putative decision making circuits in the brain. For certain model systems one can analytically derive the nonlinear diffusion equation, thereby mapping the original system parameters onto the diffusion equation coefficients. Here, we illustrate this with three model systems including coupled rate equations and a network of spiking neurons.     

From Markovian to pairwise epidemic models and the performance of moment closure approximations

Many if not all models of disease transmission on networks can be linked to the exact state-based Markovian formulation. However the large number of equations for any system of realistic size limits their applicability to small populations. As a result, most modelling work relies on simulation and pairwise models. In this paper, for a simple SIS dynamics on an arbitrary network, we formalise the link between a well known pairwise model and the exact Markovian formulation. This involves the rigorous derivation of the exact ODE model at the level of pairs in terms of the expected number of pairs and triples. The exact system is then closed using two different closures, one well established and one that has been recently proposed. A new interpretation of both closures is presented, which explains several of their previously observed properties. The closed dynamical systems are solved numerically and the results are compared to output from individual-based stochastic simulations. This is done for a range of networks with the same average degree and clustering coefficient but generated using different algorithms. It is shown that the ability of the pairwise system to accurately model an epidemic is fundamentally dependent on the underlying large-scale network structure. We show that the existing pairwise models are a good fit for certain types of network but have to be used with caution as higher-order network structures may compromise their effectiveness.     

Deterministic chaos and the first positive Lyapunov exponent: a nonlinear analysis of the human electroencephalogram during sleep

Under selected conditions, nonlinear dynamical systems, which can be described by deterministic models, are able to generate so-called deterministic chaos. In this case the dynamics show a sensitive dependence on initial conditions, which means that different states of a system, being arbitrarily close initially, will become macroscopically separated for sufficiently long times. In this sense, the unpredictability of the EEG might be a basic phenomenon of its chaotic character. Recent investigations of the dimensionality of EEG attractors in phase space have led to the assumption that the EEG can be regarded as a deterministic process which should not be mistaken for simple noise. The calculation of dimensionality estimates the degrees of freedom of a signal. Nevertheless, it is difficult to decide from this kind of analysis whether a process is quasiperiodic or chaotic. Therefore, we performed a new analysis by calculating the first positive Lyapunov exponent L ₁ from sleep EEG data. Lyapunov exponents measure the mean exponential expansion or contraction of a flow in phase space. L ₁ is zero for periodic as well as quasiperiodic processes, but positive in the case of chaotic processes expressing the sensitive dependence on initial conditions. We calculated L ₁ for sleep EEG segments of 15 healthy men corresponding to the sleep stages I, II, III, IV, and REM (according to Rechtschaffen and Kales). Our investigations support the assumption that EEG signals are neither quasiperiodic waves nor a simple noise. Moreover, we found statistically significant differences between the values of L ₁ for different sleep stages. All together, this kind of analysis yields a useful extension of the characterization of EEG signals in terms of nonlinear dynamical system theory.     

Development and application of white-noise modeling techniques for studies of insect visual nervous system

The nonlinear system identification technique through white-noise stimulation is extended to multi-input, -output systems with consideration given to applications in the functional study of the nervous system. The applicability of the method is discussed in general and in particular for the motion detection neuronal system of the fly. Two series of experiments are performed; one with moving striped-pattern stimuli and the other with spot stimuli of fluctuating intensity. In both cases nonlinear dynamic models are derived which describe the system with considerable accuracy over the frequency range of 0.2–50 Hz and a dynamic amplitude range of about 40-1. These models are able to predict accurately all the discrete experiments so far performed on this system for which the models are applicable. The differences in dynamic characteristics between the corresponding system of the Musca and Phoenicia families of flies are minor except for a difference in latencies and if the difference in geometry of their faceted eyes is taken into account. The large field response of the motion detection unit is a linear weighted summation of all the smaller field highly nonlinear subsystems of which the large field is comprised.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.