A combined neuronal and mechanical model of fish swimming

A simulated neural network has been connected to a simulated mechanical environment. The network is based on a model of the spinal central pattern generator producing rhythmic swimming movements in the lamprey and the model is similar to that used in earlier simulations of fictive swimming. Here, the network has been extended with a model of how motoneuron activity is transformed via the muscles to mechanical forces. Further, these forces are used in a two-dimensional mechanical model including interaction with the surrounding water, giving the movements of the different parts of the body. Finally, these movements are fed back through stretch receptors interacting with the central pattern generator. The combined model provides a platform for various simulation experiments relating the currently known neural properties and connectivity to the behavior of the animalin vivo. By varying a small set of parameters, corresponding to brainstem input to the spinal network, a variety of basic locomotor behaviors, like swimming at different speeds and turning can be produced. This paper describes the combined model and its basic properties.     

Selection of evolutionary models for phylogenetic hypothesis testing using parametric methods

Recent molecular studies have incorporated the parametric bootstrap method to test a priori hypotheses when the results of molecular based phylogenies are in conflict with these hypotheses. The parametric bootstrap requires the specification of a particular substitutional model, the parameters of which will be used to generate simulated, replicate DNA sequence data sets. It has been both suggested that, (a) the method appears robust to changes in the model of evolution, and alternatively that, (b) as realistic model of DNA substitution as possible should be used to avoid false rejection of a null hypothesis. Here we empirically evaluate the effect of suboptimal substitution models when testing hypotheses of monophyly with the parametric bootstrap using data sets of mtDNA cytochrome oxidase I and II (COI and COII) sequences for Macaronesian Calathus beetles, and mitochondrial 16S rDNA and nuclear ITS2 sequences for European Timarcha beetles. Whether a particular hypothesis of monophyly is rejected or accepted appears to be highly dependent on whether the nucleotide substitution model being used is optimal. It appears that a parameter rich model is either equally or less likely to reject a hypothesis of monophyly where the optimal model is unknown. A comparison of the performance of the Kishino–Hasegawa (KH) test shows it is not as severely affected by the use of suboptimal models, and overall it appears to be a less conservative method with a higher rate of failure to reject null hypotheses.     

Parametric and non-parametric modeling of short-term synaptic plasticity. Part II: Experimental study

This paper presents a synergistic parametric and non-parametric modeling study of short-term plasticity (STP) in the Schaffer collateral to hippocampal CA1 pyramidal neuron (SC) synapse. Parametric models in the form of sets of differential and algebraic equations have been proposed on the basis of the current understanding of biological mechanisms active within the system. Non-parametric Poisson–Volterra models are obtained herein from broadband experimental input–output data. The non-parametric model is shown to provide better prediction of the experimental output than a parametric model with a single set of facilitation/depression (FD) process. The parametric model is then validated in terms of its input–output transformational properties using the non-parametric model since the latter constitutes a canonical and more complete representation of the synaptic nonlinear dynamics. Furthermore, discrepancies between the experimentally-derived non-parametric model and the equivalent non-parametric model of the parametric model suggest the presence of multiple FD processes in the SC synapses. Inclusion of an additional set of FD process in the parametric model makes it replicate better the characteristics of the experimentally-derived non-parametric model. This improved parametric model in turn provides the requisite biological interpretability that the non-parametric model lacks.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Neuronal Network Models of Phase Separation Between Limb CPGs of Digging Sand Crabs

Ordinary differential equations are used to model a peculiar motor behaviour in the anomuran decapod crustacean Emerita analoga. Little is known about the neural circuitry that permits E. analoga to control the phase relationships between movements of the fourth legs and pair of uropods as it digs into sand, so mathematical models might aid in identifying features of the neural structures involved. The geometric arrangement of segmental ganglia controlling the movements of each limb provides an intuitive framework for modelling. Specifically, due to the rhythmic nature of movement, the network controlling the fourth legs and uropods is viewed as three coupled identical oscillators, one dedicated to the control of each fourth leg and one for the pair of uropods, which always move in bilateral synchrony. Systems of Morris–Lecar equations describe the voltage and ion channel dynamics of neurons. Each central pattern generator for a limb is first modelled as a single neuron and then, more realistically as a multi-neuron oscillator. This process results in high-dimensional systems of equations that are difficult to analyse. In either case, reduction to phase equations by averaging yields a two-dimensional system of equations where variables describe only each oscillator’s phase along its limit cycle. The behaviour observed in the reduced equations approximates that of the original system. Results suggest that the phase response function in the two dimensional system, together with minimal input from asymmetric bilateral coupling parameters, is sufficient to account for the observed behaviour.     

The Mechanics of Metastasis: Insights from a Computational Model

Although it may seem obvious that mechanical forces are required to drive metastatic cell movements, understanding of the mechanical aspects of metastasis has lagged far behind genetic and biochemical knowledge. The goal of this study is to learn about the mechanics of metastasis using a cell-based finite element model that proved useful for advancing knowledge about the forces that drive embryonic cell and tissue movements. Metastasis, the predominant cause of cancer-related deaths, involves a series of mechanical events in which one or more cells dissociate from a primary tumour, migrate through normal tissue, traverse in and out of a multi-layer circulatory system vessel and resettle. The present work focuses on the dissemination steps, from dissociation to circulation. The model shows that certain surface tension relationships must be satisfied for cancerous cells to dissociate from a primary tumour and that these equations are analogous to those that govern dissociation of embryonic cells. For a dissociated cell to then migrate by invadopodium extension and contraction and exhibit the shapes seen in experiments, the invadopodium must generate a contraction equal to approximately twice that produced by the interfacial tension associated with surrounding cells. Intravasation through the wall of a vessel is governed by relationships akin to those in the previous two steps, while release from the vessel wall is governed by equations that involve surface and interfacial tensions. The model raises a number of potential research questions. It also identifies how specific mechanical properties and the sub-cellular structural components that give rise to them might be changed so as to thwart particular metastatic steps and thereby block the spread of cancer.     

Parametric Sensitivity Analysis of Oscillatory Delay Systems with an Application to Gene Regulation

A parametric sensitivity analysis for periodic solutions of delay-differential equations is developed. Because phase shifts cause the sensitivity coefficients of a periodic orbit to diverge, we focus on sensitivities of the extrema, from which amplitude sensitivities are computed, and of the period. Delay-differential equations are often used to model gene expression networks. In these models, the parametric sensitivities of a particular genotype define the local geometry of the evolutionary landscape. Thus, sensitivities can be used to investigate directions of gradual evolutionary change. An oscillatory protein synthesis model whose properties are modulated by RNA interference is used as an example. This model consists of a set of coupled delay-differential equations involving three delays. Sensitivity analyses are carried out at several operating points. Comments on the evolutionary implications of the results are offered.     

Mechanics of the limbs of the walrus (Odobenus rosmarus) and the california sea lion (Zalophus californianus)

Moment arms and cross-sectional areas of muscles from the forelimbs and hind limbs of Odobenus and Zalophus were used to estimate relative torques associated with several locomotor movements. Odobenus uses predominantly the hind limbs for aquatic propulsion, while Zalophus only uses the forelimbs. Torques contributing to locomotor movements used during the power and recovery strokes were substantially greater for the hind limbs of Odobenus and for the forelimbs of Zalophus. Apparently, the lineages of these two genera shared an aquatic ancestor during the Miocene; therefore, they also shared a common method of aquatic propulsion. This common method of aquatic propulsion could have been either forelimb, hind limb, or both. Since the Miocene then, the method of aquatic propulsion and the mechanical characteristics of the limbs have diverged for at least one of these two lineages. A model is advanced to explain key mechanical differences between Odobenus and Zalophus that enable Odobenus to maneuver effectively in a head-down posture on the ocean floor.     

Parametric energy conversion--a possible, universal approach to bioenergetics in biological structures.

It is shown that the mechanism of parametric energy conversion—a non-linear phenomenon which is known to occur in all branches of physics—may play a fundamental role in energy conversion in biological structures. Parametric energy conversion means pumping of energy through the variation of an energy storing quantity (a parameter). In biological systems the energy storing parameter is the membrane itself, the structure and composition of which is varied by proceeding structure bound biochemical reactions. The principle of parametric energy conversion is introduced into a molecular kinetical model and three coupled differential equations are derived, which interconnect chemical, electrical and mechanical energy in biological structures. It is shown that they describe parametric pumping of energy. It is a particular mechanism, which is also found in the physical phenomenon of Bethenod. The mechanism is tested with the derivation and explanation of various important bioenergetical functions as special cases of parametric energy conversion, of ATP synthesis, the pumping of ions and molecules during active transport, the excitability of nerve membranes and the dynamics of oscillatory muscles. A new interpretation of the connection of structure and function in striated muscles is also derived and signal transformation in receptors discussed. It is suggested that parametric energy conversion may be the uniform basis of energy conversion in biological structures and that the path of bioenergetic evolution might have essentially followed the line marked by the characteristic properties of this flexible mechanism. The parametric hypothesis offers an elegant ordering scheme and reasonable explanation for evolution and function of a large variety of important bioenergetic mechanisms. In order to handle the intricate mechanism properly it would be necessary to give up the conventional, intuitive way of formulating and understanding biochemical mechanisms and to develop a new dimension of chemical thinking.     

A neural network model for limb trajectory formation

This paper deals with the problem of representing and generating unconstrained aiming movements of a limb by means of a neural network architecture. The network produced time trajectories of a limb from a starting posture toward targets specified by sensory stimuli. Thus the network performed a sensory-motor transformation. The experimenters trained the network using a bell-shaped velocity profile on the trajectories. This type of profile is characteristic of most movements performed by biological systems. We investigated the generalization capabilities of the network as well as its internal organization. Experiments performed during learning and on the trained network showed that: (i) the task could be learned by a three-layer sequential network; (ii) the network successfully generalized in trajectory space and adjusted the velocity profiles properly; (iii) the same task could not be learned by a linear network; (iv) after learning, the internal connections became organized into inhibitory and excitatory zones and encoded the main features of the training set; (v) the model was robust to noise on the input signals; (vi) the network exhibited attractor-dynamics properties; (vii) the network was able to solve the motorequivalence problem. A key feature of this work is the fact that the neural network was coupled to a mechanical model of a limb in which muscles are represented as springs. With this representation the model solved the problem of motor redundancy.     

Numerical simulation of load-induced bone structural remodelling using stress-limit criterion

A simple and efficient numerical method for predicting the remodelling of adaptive materials and structures under applied loading was presented and implemented within a finite element framework. The model uses the trajectorial architecture theory of optimisation to predict the remodelling of material microstructure and structural organisation under mechanical loading. We used the proposed model to calculate the density distribution of proximal femur in the frontal plane. The loading considered was the hip joint contact forces and muscular forces at the attachment sites of the muscles to the bone. These forces were estimated from a separate finite element calculation using a heterogeneous three-dimensional model of the proximal femur. The density distributions obtained by this procedure has a qualitative similarity with in vivo observations. Solutions displayed the characteristic high-density channels that are evident in the Dual X-ray Absorptiometry scan. There is also evidence of the intramedullary canal, as well as low-density regions in the femoral neck. Several parametric studies were carried out to highlight the advantages of the proposed method, which includes fast convergence and low-computational cost. The potential applications of the proposed method in predicting bone structural remodelling in cancer are also briefly discussed.     

Model of mechanical alternans in the mammalian myocardium

A model is proposed to elucidate the cause and mechanism of mechanical alternans in cardiac muscle in terms of discrete calcium movements. Mechanical alternans, the cause of which lies within the borders of excitation-contraction-coupling (ECC), is analyzed. In this case, the "input" of the ECC system (the action potentials and intervals) is constant while the "output" (contractile force) oscillates between two constant values, indicating that the system has a "memory" with two "internal states". It is proposed that these two "states" are associated with a part of the sarcoplasmic reticulum ("releasable terminal") containing the readily releasable calcium. A mechanism of "calcium-concentration-dependent threshold" is suggested to govern the "release function", i.e. the release of calcium from the "releasable terminal" to the myofilaments. The "release function" is analyzed in both the linear and the non-linear cases and its implication on the initiation of sustained and transient mechanical alternans are described. The dependence of mechanical alternans on a disturbance is also explained. The model response resembles the experimental observations of mechanical alternans in mammalian myocardium in the following manners: abrupt transition from low to high heart rates, slow progressive acceleration of rate, variations in persistence at subthreshold rates, effect of premature and delayed beat following the small and large beats, restitution curves, and transient mechanical alternans initiated by a delayed beat.     

Modelling HIV immune response and validation with clinical data

A system of ordinary differential equations is formulated to describe the pathogenesis of HIV infection, wherein certain features that have been shown to be important by recent experimental research are incorporated in the model. These include the role of CD4+memory cells that serve as a major reservoir of latently infected cells, a critical role for T-helper cells in the generation of CD8 memory cells capable of efficient recall response, and stimulation by antigens other than HIV. A stability analysis illustrates the capability of this model in admitting multiple locally asymptotically stable (locally a.s.) off-treatment equilibria. We show that this more biologically detailed model can exhibit the phenomenon of transient viremia experienced by some patients on therapy with viral load levels suppressed below the detection limit. We also show that the loss of CD4+T-cell help in the generation of CD8+memory cells leads to larger peak values for the viral load during transient viremia. Censored clinical data is used to obtain parameter estimates. We demonstrate that using a reduced set of 16 free parameters, obtained by fixing some parameters at their population averages, the model provides reasonable fits to the patient data and, moreover, that it exhibits good predictive capability. We further show that parameter values obtained for most clinical patients do not admit multiple locally a.s off-treatment equilibria. This suggests that treatment to move from a high viral load equilibrium state to an equilibrium state with a lower (or zero) viral load is not possible for these patients.     

Suggestions for a mathematical model of a pathological neuron with spontaneous repetitive discharges

A formal mathematical model is proposed for a spontaneously repetitively firing neuron. It is based on the assumption that an excitatory and inhibitory substance, possibly different from those involved in synaptic transmissions, is formed in the soma of everynormal neuron. Furthermore, the decay of the substances is ascribed to their combination with some other substances, present in healthy individuals. A generalized two factor system of differential equations is used. It is shown that when the normally present substances are absent, possibly due to genetic defects so that the decay constants become zero, the equations lead to undamped sinusoidal solutions of the difference between excitatory and inhibitory factors, thus producing a trulyspontaneous repetitive discharge, in the absence of external currents or other stimulation. It is suggested that convulsants may act by destroying the substances present in healthy individuals. It is further suggested that by administering to epileptics those substances, which are present in normal healthy persons, perhaps by using brain extracts fromhealthy higher animals which sometimes suffer from epilepsy, an actual cure rather than symptomatic treatment by anticonvulsants may be obtained.     

Parametric-equation representation of biconcave erythrocytes

The representation of the shape of a biconcave erythrocyte by a set of three parametric equations was achieved by using the expressions that transform the curvilinear coordinates from the disc-cyclide coordinate system [denoted J2R; Moon and Spencer (1988), Field Theory Handbook, Springer-Verlag, Berlin] to Cartesian coordinates. The equations are products of elliptic functions, so the challenge was to relate the three major ’shape-defining’ measurements of the human erythrocyte in Cartesian coordinates to three parameters in the new curvilinear coordinates, to give a realistic representation of the shape of the membrane-surface. The relationships between the coefficients of the Cartesian degree-4 surface that describes the discocyte and the coordinate transformation equations were derived with the aid of Mathematica; and the membrane-surface of the cell was drawn using the ParametricPlot3D function in this ‘package’. By having the erythrocyte shape expressed in its new form it is readily amenable to further transformations that might be used to model those changes in shape that are seen when the cells are immersed in media of various osmolalities, or when they change metabolic ’states’. On the other hand, the relationship between the coefficients of the Cartesian expression for the disc-cyclide surface is relevant to image analysis of erythrocytes, as determined by physical methods that rely on Cartesian imaging ’slices’. These methods include confocal microscopy and various nuclear magnetic resonance microimaging procedures.     

A biomechanical hypothesis explaining upstream movements by the freshwater snail Elimia

1. Many taxa of freshwater invertebrates show active upstream movements, particularly the snails. Hypotheses explaining this behaviour invoke the search for food or space, compensation for drift, avoidance of predation and hydrodynamic effects. The pervasiveness of upstream movements among remote lineages of snails (two subclasses, three orders, 10 families), however, suggests that snails may move upstream for mechanical rather than adaptive reasons.
2. It is proposed that upstream movements by snails are a function of torque on the snail's foot generated by hydrodynamic drag on the shell. When subject to high broadside drag-forces on their shells, snails are able to reduce torque and stabilize orientation only by directing their anterior aspect upstream.
3. Movements of the freshwater pleurocerid snail Elimia were studied by following marked free-ranging individuals in six streams in Alabama, USA (four species, eight populations).
4. Populations showed either no net movement (two streams) or significant upstream movements ranging to a mean of ≈40 m over a 3-month period (four streams). Movement patterns were stream specific rather than species or population specific. Within populations showing significant upstream movements, snails with shell lengths ≤10 mm showed little net movement. Larger snails showed movements from 0 to >200 m upstream.
5. A torque-constrained random walk model was used to perform a post hoc test of the hypothesis that upstream movements were a function of torque on the snail's foot generated by hydrodynamic drag on the shell. The model predicted upstream and size-dependent movement patterns that approximated those observed for snails in the field.     

The Cardiocoil stent-artery interaction

An analytical approach for the mechanical interaction of the self-expanding Cardiocoil stent with the stenosed artery is presented. The damage factor as the contact stress at the stent-artery interface is determined. The stent is considered as an elastic helical rod having a nonlinear pressure-displacement dependence, while the artery is modeled by an elastic cylindrical shell. An influence of a moderate relative thickness of the shell is estimated. The equations for both the stent and the artery are presented in the stent-associated helical coordinates. The computational efficiency of the model enabled to carry out a parametric study of the damage factor. Comparative examinations are conducted for the stents made of the helical rods with circular and rectangular cross sections. It was found, in particular, that, under same other conditions, the damage factor for the stent with a circular cross section may be two times larger than that for a rectangular one.     

Soft Tissue Modelling of Cardiac Fibres for Use in Coupled Mechano-Electric Simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical—either in terms of memory or time—to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations.     

How a simple adaptive foraging strategy can lead to emergent home ranges and increased food intake

Animals often alternate between searching for food locally and moving over larger distances depending on the amount of food they find. This ability to switch between movement modes can have large implications on the fate of individuals and populations, and a mechanism that allows animals to find the optimal balance between alternative movement strategies is therefore selectively advantageous. Recent theory suggests that animals are capable of switching movement mode depending on heterogeneities in the landscape, and that different modes may predominate at different temporal scales. Here we develop a conceptual model that enables animals to use either an area‐concentrated food search behavior or undirected random movements. The model builds on the animals’ ability to remember the profitability and location of previously visited areas. In contrast to classical optimal foraging models, our model does not assume food to be distributed in large, well‐defined patches, and our focus is on animal movement rather than on how animals choose between foraging patches with known locations and value. After parameterizing the fine‐scale movements to resemble those of the harbor porpoise Phocoena phocoena we investigate whether the model is capable of producing emergent home ranges and use pattern‐oriented modeling to evaluate whether it can reproduce the large‐scale movement patterns observed for porpoises in nature. Finally we investigate whether the model enables animals to forage optimally. We found that the model was indeed able to produce either stable home ranges or movement patterns that resembled those of real porpoises. It enabled animals to maximize their food intake when fine‐tuning the memory parameters that controlled the relative contribution of area concentrated and random movements.