Optimal population stabilization and control using the Leslie matrix model

We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class ofshifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained opitmization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.     

A multi-phase optimal control technique for the simulation of a human vertical jump

A multi-phase optimal control technique is presented that can be used to solve dynamic optimization problems involving musculoskeletal systems. The biomechanical model consists of a set of differential equations describing the dynamics of the multi-body system and the generation of the dynamic forces of the human muscles. Within the optimization technique, subintervals can be defined in which the differential equations are continuous. At the boundaries the dimension of the state- and control vector as well as the dimension of the right-hand side may change. The problem is solved by a multiple shooting approach which converts the problem into a non-linear program. The method is applied to simulate a human jump movement.     

A method for solution of the multi-objective inverse problems under uncertainty

We describe a method to solve multi-objective inverse problems under uncertainty. The method was tested on non-linear models of dynamic series and population dynamics, as well as on the spatiotemporal model of gene expression in terms of non-linear differential equations. We consider how to identify model parameters when experimental data contain additive noise and measurements are performed in discrete time points. We formulate the multi-objective problem of optimization under uncertainty. In addition to a criterion of least squares difference we applied a criterion which is based on the integral of trajectories of the system spatiotemporal dynamics, as well as a heuristic criterion CHAOS based on the decision tree method. The optimization problem is formulated using a fuzzy statement and is constrained by penalty functions based on the normalized membership functions of a fuzzy set of model solutions. This allows us to reconstruct the expression pattern of hairy gene in Drosophila even-skipped mutants that is in good agreement with experimental data. The reproducibility of obtained results is confirmed by solution of inverse problems using different global optimization methods with heuristic strategies.     

Persistence of Tb in possums: Insights from a spatially explicit, stochastic model

The majority of models for predicting the dynamics of bovine Tb in brushtail possums in New Zealand owe much to the pioneering work of the late Nigel Barlow. These non-spatial, deterministic models subsumed local disease dynamics by using a heterogeneous mixing term that assumed Tb was confined to a fraction of the population (in patches). However, the underlying mechanism(s) that could result in this heterogeneity of infection risk were obscure. We present a new individual-based, spatial, stochastic simulation model of Tb in possums that provides an explicit mechanism for simulating heterogeneous risk of infection based on a model of individual home range utilisation and disease susceptibility. The manipulation of parameters governing individual utilisation of space also means that processes such as non-linear contact structure can be handled naturally. We use the model to predict the persistence of Tb in possums under scenarios currently implemented for the management of bovine Tb in wildlife and determine conditions under which Tb might be predicted to persist despite control efforts.     

Role of the autonomic nervous system in generating non-linear dynamics in short-term heart period variability.

We evaluated the role played by the autonomic nervous system in producing non-linear dynamics in short heart period variability (HPV) series recorded in healthy young humans. Non-linear dynamics are detected using an index of predictability based on a local non-linear predictor and a surrogate data approach. Different types of surrogates are utilized: (i) phase-randomized Fourier-transform based (FT) data; (ii) amplitude-adjusted FT (AAFT) data; and (iii) iteratively refined AAFT (IAAFT) data of two types (IAAFT-1 and IAAFT-2). The approach was applied to experimental protocols activating or blocking the sympathetic or parasympathetic branches of the autonomic nervous system or periodically perturbing cardiovascular control via paced respiration at different breathing rates. We found that short-term HPV was mostly linear at rest. Experimental protocols activating the sympathetic or parasympathetic nervous system did not produce non-linear dynamics. In contrast, paced respiration, especially at slow breathing rates, elicited significantly non-linear dynamics. Therefore, in short-term HPV ( approximately 300 beats) the use of non-linear models is not supported by the data, except under conditions whereby the subject is constrained to a slow respiratory rate.     

A control model of human tongue movements in speech

Tongue movements during speech production have been investigated by means of a simple yet realistic biomechanical model, based on a finite elements modeling of soft tissues, in the framework of the equilibrium point hypothesis (-model) of motor control. In particular, the model has been applied to the estimation of the “central” control commands issued to the muscles, for a data set of mid-sagittal digitized tracings of vocal tract shape, r ecorded by means of low-intensity X-ray cineradiographies during speech. In spite of the highly non-linear mapping between the shape of the oral cavity and its acoustic consequences, the organization of control commands preserves the peculiar spatial organization of vowel phonemes in acoustic space. A factor analysis of control commands, which have been decomposed into independent or “orthogonal” muscle groups, has shown that, in spite of the great mobility of the tongue and the highly complex arrangement of tongue muscles, its movements can be explained in terms of the activation of a small number of independent muscle groups, each corresponding to an elementary or “primitive” movement. These results are consistent with the hypothesis that the tongue is controlled by a small number of independent “articulators”, for which a precise biomechanical substrate is provided. The influence of the effect of jaw and hyoid movements on tongue equilibrium has also bee n evaluated, suggesting that the bony structures cannot be considered as a moving frame of reference, but, indeed, there may be a substantial interaction between them and the tongue, that may only be accounted for by a “global” model. The reported results also define a simple control model for the tongue and, in analogy with similar modelling studies, they suggest that, because of the peculiar geometrical arrangement of tongue muscles, the central nervous system (CNS) may not need a de tailed representation of tongue mechanics but rather may make use of a relatively small number of muscle synergies, that are invariant over the whole space of tongue configurations. Received: 27 August 1996 / Accepted in revised form: 25 February 1997     

Balance of a multijoint biomechanical system in natural and artificial environments: a simulation model

The paper is devoted to a neurobionic simulation model for controlling balance in a biomechanical pendulum. The model is realized by a complex of fuzzy regulators and an artificial neural network. Fuzzy regulators are used for simulating the physiological characteristics of the motor system and the functions of the sensory systems. The second level of control is the central integrator. It is realized as an artificial neural network (ANN), which simulates a real process of analysis and synthesis of afferent signals, formation of the model of action, etc.Equilibrium control in a multijoint biomechanical object is a specific example of a self-developing multilevel system of movement control. In the course of elaboration of the model and further examination of its behavior we have received model results which revealed correspondence with the results demonstrated by real subjects in stabilographic tests performed after long-term space flights. We concluded that the model permits us to simulate the peculiarities of human movement control and can be used for creating individual plans of recovery and rehabilitation of patients after long-term motionless or learning movement control in unknown environments.     

Response linearity determined by recruitment strategy in detailed model of nictitating membrane control

Many models of eyeblink conditioning assume that there is a simple linear relationship between the firing patterns of neurons in the interpositus nucleus and the time course of the conditioned response (CR). However, the complexities of muscle behaviour and plant dynamics call this assumption into question. We investigated the issue by implementing the most detailed model available of the rabbit nictitating membrane response (Bartha and Thompson in Biol Cybern 68:135-143, 1992a and in Biol Cybern 68:145-154, 1992b), in which each motor unit of the retractor bulbi muscle is represented by a Hill-type model, driven by a non-linear activation mechanism designed to reproduce the isometric force measurements of Lennerstrand (J Physiol 236:43-55, 1974). Globe retraction and NM extension are modelled as linked second order systems. We derived versions of the model that used a consistent set of SI units, were based on a physically realisable version of calcium kinetics, and used simulated muscle cross-bridges to produce force. All versions showed similar non-linear responses to two basic control strategies. (1) Rate-coding with no recruitment gave a sigmoidal relation between control signal and amplitude of CR, reflecting the measured relation between isometric muscle force and stimulation frequency. (2) Recruitment of similar strength motor units with no rate coding gave a sublinear relation between control signal and amplitude of CR, reflecting the increase in muscle stiffness produced by recruitment. However, the system response could be linearised by either a suitable combination of rate-coding and recruitment, or by simple recruitment of motor units in order of (exponentially) increasing strength. These plausible control strategies, either alone or in combination, would in effect present the cerebellum with the simplified virtual plant that is assumed in many models of eyeblink conditioning. Future work is therefore needed to determine the extent to which motor neuron firing is in fact linearly related to the nictitating membrane response.     

State-space model combining local camera data and regional administration data reveals population dynamics of wild boar

Recent increases in wildlife cause negative impacts on humans through both economic and ecological damage, as well as the spread of pathogens. Understanding the population dynamics of wildlife is crucial to develop effective management strategies. However, it is difficult to estimate accurate and precise population size over large spatial and temporal scales because of the limited data availability. We addressed these issues by first fitting a random encounter and staying time (REST) model based on camera trap data to construct an informative prior distribution for a capture rate parameter in a harvest-based Bayesian state-space model. We constructed a Bayesian state-space model that integrated administration data on the number of captured wild boar with the prior distribution of capture efficiency estimated by camera trap data. The model with informative prior distribution from the REST model successfully estimated population dynamics, whereas the model using only the administration data did not, owing to a lack of parameter convergence. We identified areas where (1) wild boars exhibit a high potential population growth rate and a high carrying capacity, (2) current trapping efforts are effectively suppressing local populations, and (3) trapping reinforcement is required to control populations in the whole region. The model could be used to predict future trends in populations under the assumptions of ongoing trapping pressure. This will help identify spatially explicit trapping efforts to achieve target population levels.     

State reduction for network intervention in probabilistic Boolean networks

MOTIVATION: A key goal of studying biological systems is to design therapeutic intervention strategies. Probabilistic Boolean networks (PBNs) constitute a mathematical model which enables modeling, predicting and intervening in their long-run behavior using Markov chain theory. The long-run dynamics of a PBN, as represented by its steady-state distribution (SSD), can guide the design of effective intervention strategies for the modeled systems. A major obstacle for its application is the large state space of the underlying Markov chain, which poses a serious computational challenge. Hence, it is critical to reduce the model complexity of PBNs for practical applications. RESULTS: We propose a strategy to reduce the state space of the underlying Markov chain of a PBN based on a criterion that the reduction least distorts the proportional change of stationary masses for critical states, for instance, the network attractors. In comparison to previous reduction methods, we reduce the state space directly, without deleting genes. We then derive stationary control policies on the reduced network that can be naturally induced back to the original network. Computational experiments study the effects of the reduction on model complexity and the performance of designed control policies which is measured by the shift of stationary mass away from undesirable states, those associated with undesirable phenotypes. We consider randomly generated networks as well as a 17-gene gastrointestinal cancer network, which, if not reduced, has a 2(17) × 2(17) transition probability matrix. Such a dimension is too large for direct application of many previously proposed PBN intervention strategies.     

Adaptive dynamic programming as a theory of sensorimotor control

Many characteristics of sensorimotor control can be explained by models based on optimization and optimal control theories. However, most of the previous models assume that the central nervous system has access to the precise knowledge of the sensorimotor system and its interacting environment. This viewpoint is difficult to be justified theoretically and has not been convincingly validated by experiments. To address this problem, this paper presents a new computational mechanism for sensorimotor control from a perspective of adaptive dynamic programming (ADP), which shares some features of reinforcement learning. The ADP-based model for sensorimotor control suggests that a command signal for the human movement is derived directly from the real-time sensory data, without the need to identify the system dynamics. An iterative learning scheme based on the proposed ADP theory is developed, along with rigorous convergence analysis. Interestingly, the computational model as advocated here is able to reproduce the motor learning behavior observed in experiments where a divergent force field or velocity-dependent force field was present. In addition, this modeling strategy provides a clear way to perform stability analysis of the overall system. Hence, we conjecture that human sensorimotor systems use an ADP-type mechanism to control movements and to achieve successful adaptation to uncertainties present in the environment.     

Path-finding in real and simulated rats: assessing the influence of path characteristics on navigation learning

A large body of experimental evidence suggests that the hippocampal place field system is involved in reward based navigation learning in rodents. Reinforcement learning (RL) mechanisms have been used to model this, associating the state space in an RL-algorithm to the place-field map in a rat. The convergence properties of RL-algorithms are affected by the exploration patterns of the learner. Therefore, we first analyzed the path characteristics of freely exploring rats in a test arena. We found that straight path segments with mean length 23 cm up to a maximal length of 80 cm take up a significant proportion of the total paths. Thus, rat paths are biased as compared to random exploration. Next we designed a RL system that reproduces these specific path characteristics. Our model arena is covered by overlapping, probabilistically firing place fields (PF) of realistic size and coverage. Because convergence of RL-algorithms is also influenced by the state space characteristics, different PF-sizes and densities, leading to a different degree of overlap, were also investigated. The model rat learns finding a reward opposite to its starting point. We observed that the combination of biased straight exploration, overlapping coverage and probabilistic firing will strongly impair the convergence of learning. When the degree of randomness in the exploration is increased, convergence improves, but the distribution of straight path segments becomes unrealistic and paths become 'wiggly'. To mend this situation without affecting the path characteristic two additional mechanisms are implemented: a gradual drop of the learned weights (weight decay) and path length limitation, which prevents learning if the reward is not found after some expected time. Both mechanisms limit the memory of the system and thereby counteract effects of getting trapped on a wrong path. When using these strategies individually divergent cases get substantially reduced and for some parameter settings no divergence was found anymore at all. Using weight decay and path length limitation at the same time, convergence is not much improved but instead time to convergence increases as the memory limiting effect is getting too strong. The degree of improvement relies also on the size and degree of overlap (coverage density) in the place field system. The used combination of these two parameters leads to a trade-off between convergence and speed to convergence. Thus, this study suggests that the role of the PF-system in navigation learning cannot be considered independently from the animals' exploration pattern.     

Search and navigation in dynamic environments – from individual behaviors to population distributions

Animal movement receives widespread attention within ecology and behavior. However, much research is restricted within isolated sub-disciplines focusing on single phenomena such as navigation (e.g. homing behavior), search strategies (e.g. Levy flights) or theoretical considerations of optimal population dispersion (e.g. ideal free distribution). To help synthesize existing research, we outline a unifying conceptual framework that integrates individual-level behaviors and population-level spatial distributions with respect to spatio-temporal resource dynamics. We distinguish among (1) non-oriented movements based on diffusion and kinesis in response to proximate stimuli, (2) oriented movements utilizing perceptual cues of distant targets, and (3) memory mechanisms that assume prior knowledge of a target's location. Species' use of these mechanisms depends on life-history traits and resource dynamics, which together shape population-level patterns. Resources with little spatial variability should facilitate sedentary ranges, whereas resources with predictable seasonal variation in spatial distributions should generate migratory patterns. A third pattern, 'nomadism', should emerge when resource distributions are unpredictable in both space and time. We summarize recent advances in analyses of animal trajectories and outline three major components on which future studies should focus: (1) integration across alternative movement mechanisms involving links between state variables and specific mechanisms, (2) consideration of dynamics in resource landscapes or environments that include resource gradients in predictability, variability, scale, and abundance, and finally (3) quantitative methods to distinguish among population distributions. We suggest that combining techniques such as evolutionary programming and pattern oriented modeling will help to build strong links between underlying movement mechanisms and broad-scale population distributions.     

Computational investigation of the dynamic control of cAMP signaling by PDE4 isoform types

Cyclic adenosine monophosphate (cAMP) is a generic signaling molecule that, through precise control of its signaling dynamics, exerts distinct cellular effects. Consequently, aberrant cAMP signaling can have detrimental effects. Phosphodiesterase 4 (PDE4) enzymes profoundly control cAMP signaling and comprise different isoform types wherein enzymatic activity is modulated by differential feedback mechanisms. Because these feedback dynamics are non-linear and occur coincidentally, their effects are difficult to examine experimentally but can be well simulated computationally. Through understanding the role of PDE4 isoform types in regulating cAMP signaling, PDE4-targeted therapeutic strategies can be better specified. Here, we established a computational model to study how feedback mechanisms on different PDE4 isoform types lead to dynamic, isoform-specific control of cAMP signaling. Ordinary differential equations describing cAMP dynamics were implemented in the VirtualCell environment. Simulations indicated that long PDE4 isoforms exert the most profound control on oscillatory cAMP signaling, as opposed to the PDE4-mediated control of single cAMP input pulses. Moreover, elevating cAMP levels or decreasing PDE4 levels revealed different effects on downstream signaling. Together these results underline that cAMP signaling is distinctly regulated by different PDE4 isoform types and that this isoform specificity should be considered in both computational and experimental follow-up studies to better define PDE4 enzymes as therapeutic targets in diseases in which cAMP signaling is aberrant.     

A general modeling framework for describing spatially structured population dynamics

Variation in movement across time and space fundamentally shapes the abundance and distribution of populations. Although a variety of approaches model structured population dynamics, they are limited to specific types of spatially structured populations and lack a unifying framework. Here, we propose a unified network‐based framework sufficiently novel in its flexibility to capture a wide variety of spatiotemporal processes including metapopulations and a range of migratory patterns. It can accommodate different kinds of age structures, forms of population growth, dispersal, nomadism and migration, and alternative life‐history strategies. Our objective was to link three general elements common to all spatially structured populations (space, time and movement) under a single mathematical framework. To do this, we adopt a network modeling approach. The spatial structure of a population is represented by a weighted and directed network. Each node and each edge has a set of attributes which vary through time. The dynamics of our network‐based population is modeled with discrete time steps. Using both theoretical and real‐world examples, we show how common elements recur across species with disparate movement strategies and how they can be combined under a unified mathematical framework. We illustrate how metapopulations, various migratory patterns, and nomadism can be represented with this modeling approach. We also apply our network‐based framework to four organisms spanning a wide range of life histories, movement patterns, and carrying capacities. General computer code to implement our framework is provided, which can be applied to almost any spatially structured population. This framework contributes to our theoretical understanding of population dynamics and has practical management applications, including understanding the impact of perturbations on population size, distribution, and movement patterns. By working within a common framework, there is less chance that comparative analyses are colored by model details rather than general principles.     

Dynamic finite element modeling of poroviscoelastic soft tissue

Clinical evidences relative to biomechanical factors have demonstrated their important contribution to the behaviour of soft tissues. Finite element (FE) analysis is used to study the mechanical behaviour of soft tissue because it can provide numerical solutions to problems that are intractable to analytic solutions. This study focuses on the development of a FE model of a poroelastic biological tissue, which incorporates the viscoelastic material behaviour, finite deformation and inertial effect. The FE formulation is based on the weak form derived from the governing equation, and Newmark-beta method as well as Newton's method is incorporated into the implicit non-linear solutions. One-dimensional analytical solutions were used to verify the theoretical formulation and the numerical implementation of the proposed model. This study was further extended to analyze two-dimensional biomechanical models and the results clearly demonstrate the importance of including finite deformation, viscoelasticity and inertial effects.     

A model of fluid-biofilm interaction using a Burger material law

A two-dimensional finite element model of the biofilm response to flow was developed. The numerical code sequentially coupled the fluid dynamics of turbulent, incompressible flow with the mechanical response of a single hemispherical biofilm cluster (approximately 100 microm) attached to the flow boundary. A non-linear Burger material law was used to represent the viscoelastic response of a representative microbial biofilm. This constitutive law was incorporated into the numerical model as a Prony series representation of the biofilm's relaxation modulus. Model simulations illuminated interesting details of this fluid-structure interaction. Simulations revealed that softer biofilms (characterized by lower elastic moduli) were highly susceptible to lift forces and consequently were subject to even greater drag forces found higher in the velocity field. A bimodal deformation path due to the two Burger relaxation times was also observed in several simulations. This suggested that interfacial biofilm may be most susceptible to hydrodynamically induced detachment during the initial relaxation time. This result may prove useful in developing removal strategies. Additionally, plots of lift versus drag suggested that the deformation paths taken by viscoelastic biofilms are largely insensitive to specific material coefficients. Softer biofilms merely seem to follow the same path (as a stiffer biofilm) at a faster rate. These relationships may be useful in estimating the hydrodynamic forces acting on an attached biofilm based on changes in scale and cataloged material properties.     

Modeling the physiological glucose-insulin dynamic system on diabetics

A novel mathematical model is presented to describe the dynamic behavior of plasma glucose and insulin on diabetic subjects. Though various models have been proposed to simulate the short-term (a variety of intravenous glucose or insulin injection) glucose-insulin dynamics, it is intended to construct a modified delay differential equations (DDEs) model based on the human glucose-insulin metabolic system. Five specific adjustable parameters inside the model are defined as the factors of the major physiological functions. Then several clinical data sets (56 subjects) which includes the information of food ingestion and exogenous insulin injection are verified and the model could practically reflect the dynamics and oscillation behavior on diabetic subjects by varying the adjustable parameters. Moreover, the corresponding parameters are fairly helpful to identify the patient's conditions of major physiological functions. This generic glucose-insulin dynamic model can be expected to develop such advanced therapy strategies for diabetics in the future.