Slow oscillations in blood pressure via a nonlinear feedback model

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.     

Force feasible set prediction with artificial neural network and musculoskeletal model

Abstract

Developing tools to predict the force capabilities of the human limbs through the Force Feasible Set (FFS) may be of great interest for robotic assisted rehabilitation and digital human modelling for ergonomics. Indeed, it could help to refine rehabilitation programs for active participation during exercise therapy and to prevent musculoskeletal disorders. In this framework, the purpose of this study is to use artificial neural networks (ANN) to predict the FFS of the upper-limb based on joint centre Cartesian positions and anthropometric data. Seventeen right upper-limb musculoskeletal models based on individual anthropometric data are created. For each musculoskeletal model, the FFS is computed for 8428 different postures. For any combination of force direction and joint positions, ANNs can predict the FFS with high values of coefficient of determination (R²?>?0.89) between the true and predicted data.     

A dynamic model of a two-synapse feedback loop in the vertebrate retina

The theoretical properties of synapses such as those in the retina which operate on graded potentials are developed using work on tetrodotoxin-treated synapses as a basis. A linearized model of a two-synapse negative feedback loop analogous to the bipolaramacrine feedback loop in the retina possesses a frequency response which developes an increasingly prominent resonance peak at higher input levels and under some circumstances shows instability. Psychophysical studies have shown that the visual system also exhibits this behaviour suggestive of progressive underdamping in a harmonic oscillator. Evidence in favor of the hypothesis that resonance originates in the loop is presented, the conclusions being that the loop functions to tune the retina to a range of temporal frequencies.Symbol Table V millivolts depolarisation relative to resting membrane potential - V _n , V _out pre-synaptic, post-synaptic depolarisation respectively - V _e , V _i reversal potential or e.m.f. of post-synaptic battery of excitatory, inhibitory synapses respectively - V _out (max) maximal post-synaptic depolarisation defined by Eq.(10c) - V ₀ input depolarisation for feedback loop - depolarisation potential normalised with respect to V _out(max) - I milliamperes of depolarising current - I _s post-synaptic membrane current - I _c cable current - I ₀ input depolarising current for feedback loop - I _max maximal physiological value for I ₀ =V _e ·G ₀ - i depolarising current normalised with respect to I_max - e reversal potential normalised with respect to V _e - r _i specific resistivity of internal medium - R _m membrane resistance - C _m membrane capacitance - cable space constant = R _m /2R _i - g ₀ characteristic cable conductance = 2/R _m ·R _i - G conductance of post-synaptic membrane - G _s maximal post-synaptic membrane conductance - g fraction of receptors occupied by transmitter = G/G _s - r the ratio G _s/G ₀ - membrane time constant = R _m·C_m - ₁ time constant of transmitter release in response to presynaptic depolarisation [Eq. (6)] - ₂ time constant of decay of g [Eq. (7a)] - _{2 2}·[1+k·exp(b·v _in)]^–1 - k equilibrium constant for transmitter-receptor interaction [Eq. (7a)] - b constant determining increase in rate of transmitter release with pre-synaptic depolarisation [Eq. (6)] - c concentration of transmitter in synaptic cleft normalised with respect to resting concentration - H _jk (s) linearised transfer function for synaptic transmission from neurone j to neurone k - G(s) H₁₂(s) - H(s) -H₂₁(s) - F(s) linearised closed-loop transfer function - _x 2 times spatial frequency of counterphase grating pattern - the ratio (1+s)/(_x)₂ - a the product (1+r)·k - d density of bipolars per unit area     

Learning combined feedback and feedforward control of a musculoskeletal system

The goal of this paper is the learning of neuromuscular control, given the following necessary conditions: (1) time delays in the control loop, (2) non-linear muscle characteristics, (3) learning of feedforward and feedback control, (4) possibility of feedback gain modulation during a task. A control system and learning methodology that satisfy those conditions is given. The control system contains a neural network, comprising both feedforward and feedback control. The learning method is backpropagation through time with an explicit sensitivity model. Results will be given for a one degree of freedom arm with two muscles. Good control results are achieved which compare well with experimental data. Analysis of the controller shows that significant differences in controller characteristics are found if the loop delays are neglected. During a control task the system shows feedback gain modulation, similar to experimentally found reflex gain modulation during rapid voluntary contraction. If only limited feedback information is available to the controller the system learns to co-contract the antagonistic muscle pair. In this way joint stiffness increases and stable control is more easily maintained. Received: 7 November 1995 / Accepted in revised form: 13 February 1996     

A nonlinear dynamic model of DNA with a sequence-dependent stacking term

No simple model exists that accurately describes the melting behavior and breathing dynamics of double-stranded DNA as a function of nucleotide sequence. This is especially true for homogenous and periodic DNA sequences, which exhibit large deviations in melting temperature from predictions made by additive thermodynamic contributions. Currently, no method exists for analysis of the DNA breathing dynamics of repeats and of highly G/C- or A/T-rich regions, even though such sequences are widespread in vertebrate genomes. Here, we extend the nonlinear Peyrard–Bishop–Dauxois (PBD) model of DNA to include a sequence-dependent stacking term, resulting in a model that can accurately describe the melting behavior of homogenous and periodic sequences. We collect melting data for several DNA oligos, and apply Monte Carlo simulations to establish force constants for the 10 dinucleotide steps (CG, CA, GC, AT, AG, AA, AC, TA, GG, TC). The experiments and numerical simulations confirm that the GG/CC dinucleotide stacking is remarkably unstable, compared with the stacking in GC/CG and CG/GC dinucleotide steps. The extended PBD model will facilitate thermodynamic and dynamic simulations of important genomic regions such as CpG islands and disease-related repeats.     

Role of feedback inhibition in stabilizing the classical operon

The Goodwin equations for a repressible operon (Goodwin, 1965) are modified (1) to describe a time lag between genetic regulation and appearance of functional enzyme, (2) to describe consumption of endproduct in protein synthesis, and (3) to describe feedback inhibition of enzyme activity. The stability of the modified equations is determined by a method outlined in the appendix which treats a class of negative feedback systems with time delays. With parameters estimated from experimental data on the tryptophan operon of Escherichia coli, we conclude that the operon becomes unstable as normal feedback inhibition is lost. Numerical solution of the modified equations shows that an example with a partial loss of feedback inhibition can have a period of oscillation less than the cell generation time, and the numerical solutions are shown to be in qualitative agreement with experiments showing oscillations in tryptophan operon expression.     

Stabilizing feedback of a nonlinear process involving uncertain data

A nonlinear control system describing the process of continuous methane fermentation is considered. Assuming that the parameters of the model are not exactly known but bounded within intervals, the set of optimal static points according to a practical criterion is computed. A continuous feedback control is proposed, which asymptotically stabilizes the dynamic system towards this set. Numerical results are also reported.     

A comprehensive and volumetric musculoskeletal model for the dynamic simulation of the shoulder function

We present a volumetric and extensive finite element model of the shoulder usable in the context of inverse control, in which the scapula is left unconstrained on the ribcage. Such a model allows for exploring various shoulder movements, which are essential for making patient-specific decisions. The proposed model consists of 23 volumetric muscles parts modelled using the finite element method. The glenohumeral, acromioclavicular and sternoclavicular joints are modelled with soft ball-socket constraints. The musculoskeletal model can be controlled by a tracking-based algorithm, finding the excitations values in the muscles needed to follow some target points. The moment arms obtained during abduction and rotation are compared with the literature, which includes results from cadaveric data and a fine FE model of the rotator cuff and the deltoid. We simulated the paralysis of serratus anterior, a main reason of scapular winging, and compared it with its physiological counterpart. A deficiency in the range of motion as well as a reduction in upward rotation were observed, which both corroborate clinical observations. This is one of the most comprehensive model of the shoulder, which can be used to study complex pathologies of the shoulder and their impact on functional outcome such as range-of-motion.     

Benchmarking of dynamic simulation predictions in two software platforms using an upper limb musculoskeletal model

Several opensource or commercially available software platforms are widely used to develop dynamic simulations of movement. While computational approaches are conceptually similar across platforms, technical differences in implementation may influence output. We present a new upper limb dynamic model as a tool to evaluate potential differences in predictive behavior between platforms. We evaluated to what extent differences in technical implementations in popular simulation software environments result in differences in kinematic predictions for single and multijoint movements using EMG- and optimization-based approaches for deriving control signals. We illustrate the benchmarking comparison using SIMM–Dynamics Pipeline–SD/Fast and OpenSim platforms. The most substantial divergence results from differences in muscle model and actuator paths. This model is a valuable resource and is available for download by other researchers. The model, data, and simulation results presented here can be used by future researchers to benchmark other software platforms and software upgrades for these two platforms.     

A nonlinear mixed-effect mixture model for functional mapping of dynamic traits

Functional mapping has emerged as a next-generation statistical tool for mapping quantitative trait loci (QTL) that affect complex dynamic traits. In this article, we incorporated the idea of nonlinear mixed-effect (NLME) models into the mixture-based framework of functional mapping, aimed to generalize the spectrum of applications for functional mapping. NLME-based functional mapping, implemented with the linearization algorithm based on the first-order Taylor expansion, can provide reasonable estimates of QTL genotypic-specific curve parameters (fixed effect) and the between-individual variation of these parameters (random effect). Results from simulation studies suggest that the NLME-based model is more general than traditional functional mapping. The new model can be useful for the identification of the ontogenetic patterns of QTL genetic effects during time course.     

A technical method using musculoskeletal model to analyse dynamic properties of muscles during human movement

An effective way to avoid invading or injuring the subjects is to use the musculoskeletal model when studying the dynamic properties of muscles in vivo. So, we put forward a joint coordinate system-based method, which mainly focuses on the coordinate's transformation of corresponding muscle attachment points, respectively, in the model and the subject in order to reproduce the movement of the subject on the model. As muscle moment arm is usually used to evaluate the dynamic properties of muscles, the moment arms in elbow flexion for each of the major muscles crossing the elbow in 50 healthy subjects (25 males and 25 females), ranging in height from 1.50 to 1.80 m (mean 1.6542 m) are calculated and compared with the measured data obtained from anatomical studies reported in the literature. The trends of the value basically coincide with each other. So, this novel method can be valid.     

A technical method using musculoskeletal model to analyse dynamic properties of muscles during human movement

An effective way to avoid invading or injuring the subjects is to use the musculoskeletal model when studying the dynamic properties of muscles in vivo. So, we put forward a joint coordinate system-based method, which mainly focuses on the coordinate's transformation of corresponding muscle attachment points, respectively, in the model and the subject in order to reproduce the movement of the subject on the model. As muscle moment arm is usually used to evaluate the dynamic properties of muscles, the moment arms in elbow flexion for each of the major muscles crossing the elbow in 50 healthy subjects (25 males and 25 females), ranging in height from 1.50 to 1.80 m (mean 1.6542 m) are calculated and compared with the measured data obtained from anatomical studies reported in the literature. The trends of the value basically coincide with each other. So, this novel method can be valid.     

On a genetic model of intraspecific competition and stabilizing selection

A genetic model is investigated in which two recombining loci determine the genotypic value of a quantitative trait additively. Two opposing evolutionary forces are assumed to act: stabilizing selection on the trait, which favors genotypes with an intermediate phenotype, and intraspecific competition mediated by that trait, which favors genotypes whose effect on the trait deviates most from that of the prevailing genotypes. Accordingly, fitnesses of genotypes have a frequency-independent component describing stabilizing selection and a frequency- and density-dependent component modeling competition. We study how the underlying genetics, in particular recombination rate and relative magnitude of allelic effects, interact with the conflicting selective forces and derive the resulting, surprisingly complex equilibrium patterns. We also investigate the conditions under which disruptive selection on the phenotypes can be observed and examine how much genetic variation can be maintained in such a model. We discovered a number of unexpected phenomena. For instance, we found that with little recombination the degree of stably maintained polymorphism and the equilibrium genetic variance can decrease as the strength of competition increases relative to the strength of stabilizing selection. In addition, we found that mean fitness at the stable equilibria is usually much lower than the maximum possible mean fitness and often even lower than the fitness at other, unstable equilibria. Thus, the evolutionary dynamics in this system are almost always nonadaptive.     

The two-locus model of Gaussian stabilizing selection

We study the equilibrium structure of a well-known two-locus model in which two diallelic loci contribute additively to a quantitative trait that is under Gaussian stabilizing selection. The population is assumed to be infinitely large, randomly mating, and having discrete generations. The two loci may have arbitrary effects on the trait, the strength of selection and the recombination rate may also be arbitrary. We find that 16 different equilibrium patterns exist, having up to 11 equilibria; up to seven interior equilibria may coexist, and up to four interior equilibria, three in negative and one in positive linkage disequilibrium, may be simultaneously stable. Also, two monomorphic and two fully polymorphic equilibria may be simultaneously stable. Therefore, the result of evolution may be highly sensitive to perturbations in the initial conditions or in the underlying genetic parameters. For the special case of equal effects, global stability results are proved. In the general case, we rely in part on numerical computations. The results are compared with previous analyses of the special case of extremely strong selection, of an approximate model that assumes linkage equilibrium, and of the much simpler quadratic optimum model.     

Delayed feedback control for a chemostat model

In this paper, we discuss asymptotic properties and numerical simulations of a chemostat model with delayed feedback control. A chemostat model with two organisms can be made coexistent by feedback control of the dilution rate which depends affinely on the concentrations of two organisms [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48]. Then the coexistence takes its simplest form; the equilibrium point in the non-negative orthant is globally asymptotically stable. We show that stability of the equilibrium point is changed by 'time-delay' caused in controlling the dilution rate after measuring the concentrations of two organisms.     

Computation of steady flow in a two-dimensional arterial model

The Navier-Stokes equations are solved numerically for steady flow through a double-branched two-dimensional section of a three-dimensional model of a canine aorta for which experimental data is available. The numerical scheme involves transforming the physical coordinates to a curvilinear boundary-fitted coordinate system and performing finite-difference computations in the transformed system. Shear stress at the wall is calculated for a Reynolds number of a 1,000 with branch-to-main aortic flow rate ratios as a parameter. The results are compared with the aforementioned experimental data and show reasonable qualitative agreement.