Reduced Model and Simulation of Neuron with Passive Dendritic Cable: An Eigenfunction Expansion Approach

The neuron models with passive dendritic cables are often used for detailed cortical network simulations (Protopapas et al., 1998; Suarez et al., 1995). For this, the compartment model based on finite volume or finite difference discretization was used. In this paper, we propose an eigenfunction expansion approach combined with singular perturbation and demonstrate that the proposed scheme can achieve an order of magnitude accuracy improvement with the same number of equations. Moreover, it is also shown that the proposed scheme converges much faster to attain a given accuracy. Hence, for a network simulation of the neurons with passive dendritic cables, the proposed scheme can be an attractive alternative to the compartment model, that leads to a low order model with much higher accuracy or that converges faster for a given accuracy.     

Model reduction using a posteriori analysis

Mathematical models in biology and physiology are often represented by large systems of non-linear ordinary differential equations. In many cases, an observed behaviour may be written as a linear functional of the solution of this system of equations. A technique is presented in this study for automatically identifying key terms in the system of equations that are responsible for a given linear functional of the solution. This technique is underpinned by ideas drawn from a posteriori error analysis. This concept has been used in finite element analysis to identify regions of the computational domain and components of the solution where a fine computational mesh should be used to ensure accuracy of the numerical solution. We use this concept to identify regions of the computational domain and components of the solution where accurate representation of the mathematical model is required for accuracy of the functional of interest. The technique presented is demonstrated by application to a model problem, and then to automatically deduce known results from a cell-level cardiac electrophysiology model.     

Numerical methods and parameter estimation of a structured population model with discrete events in the life history

We consider two numerical methods for the solution of a physiologically structured population (PSP) model with multiple life stages and discrete event reproduction. The model describes the dynamic behaviour of a predator-prey system consisting of rotifers predating on algae. The nitrate limited algal prey population is modelled unstructured and described by an ordinary differential equation (ODE). The formulation of the rotifer dynamics is based on a simple physiological model for their two life stages, the egg and the adult stage. An egg is produced when an energy buffer reaches a threshold value. The governing equations are coupled partial differential equations (PDE) with initial and boundary conditions. The population models together with the equation for the dynamics of the nutrient result in a chemostat model. Experimental data are used to estimate the model parameters. The results obtained with the explicit finite difference (FD) technique compare well with those of the Escalator Boxcar Train (EBT) method. This justifies the use of the fast FD method for the parameter estimation, a procedure which involves repeated solution of the model equations.     

Critical analysis of musculoskeletal modelling complexity in multibody biomechanical models of the upper limb

The inverse dynamics technique applied to musculoskeletal models, and supported by optimisation techniques, is used extensively to estimate muscle and joint reaction forces. However, the solutions of the redundant muscle force sharing problem are sensitive to the detail and modelling assumptions of the models used. This study presents four alternative biomechanical models of the upper limb with different levels of discretisation of muscles by bundles and muscle paths, and their consequences on the estimation of the muscle and joint reaction forces. The muscle force sharing problem is solved for the motions of abduction and anterior flexion, acquired using video imaging, through the minimisation of an objective function describing muscle metabolic energy consumption. While looking for the optimal solution, not only the equations of motion are satisfied but also the stability of the glenohumeral and scapulothoracic joints is preserved. The results show that a lower level of muscle discretisation provides worse estimations regarding the muscle forces. Moreover, the poor discretisation of muscles relevant to the joint in analysis limits the applicability of the biomechanical model. In this study, the biomechanical model of the upper limb describing the infraspinatus by a single bundle could not solve the complete motion of anterior flexion. Despite the small differences in the magnitude of the forces predicted by the biomechanical models with more complex muscular systems, in general, there are no significant variations in the muscular activity of equivalent muscles.     

A finite volume method solution for the bidomain equations and their application to modelling cardiac ischaemia

This paper presents an implementation of the finite volume method with the aim of studying subendocardial ischaemia during the ST segment. In this implementation, based on hexahedral finite volumes, each quadrilateral sub-face is split into two triangles to improve the accuracy of the numerical integration in complex geometries and when fibre rotation is included. The numerical method is validated against previously published solutions obtained from slab and cylindrical models of the left ventricle with subendocardial ischaemia and no fibre rotation. Epicardial potential distributions are then obtained for a half-ellipsoid model of the left ventricle. In this case it is shown that for isotropic cardiac tissue the degree of subendocardial ischaemia does not affect the epicardial potential distribution, which is consistent with previous findings from analytical studies in simpler geometries. The paper also considers the behaviour of various preconditioners for solving numerically the resulting system of algebraic equations resulting from the implementation of the finite volume method. It is observed that each geometry considered has its own optimal preconditioner.     

Numerical modelling of corneal biomechanical behaviour

Numerical modelling based on finite element analysis is used to represent the biomechanical behaviour of the cornea. The construction details of the model including the discretisation method, the mesh density, the thickness distribution, the topography idealisation, the boundary conditions and the material properties, are optimised to improve efficiency. Factors which are found to have a considerable effect on model accuracy are considered and those with effect below a certain low threshold are ignored to reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation of corneal trephinates while monitoring their behaviour. To illustrate the potential of the validated model in studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) is briefly described. In studying GAT, the model is able to accurately trace the behaviour of the cornea under tonometric pressure and monitor the gap closure and the progress of deformation to the point of applanation.     

Numerical modelling of corneal biomechanical behaviour

Numerical modelling based on finite element analysis is used to represent the biomechanical behaviour of the cornea. The construction details of the model including the discretisation method, the mesh density, the thickness distribution, the topography idealisation, the boundary conditions and the material properties, are optimised to improve efficiency. Factors which are found to have a considerable effect on model accuracy are considered and those with effect below a certain low threshold are ignored to reduce cost of analysis. The model is validated against laboratory tests involving pressure inflation of corneal trephinates while monitoring their behaviour. To illustrate the potential of the validated model in studying corneal biomechanics, its use in modelling Goldmann applanation tonometry (GAT) is briefly described. In studying GAT, the model is able to accurately trace the behaviour of the cornea under tonometric pressure and monitor the gap closure and the progress of deformation to the point of applanation.     

Electrical properties of frog skeletal muscle fibers interpreted with a mesh model of the tubular system.

This paper presents the construction, derivation, and test of a mesh model for the electrical properties of the transverse tubular system (T-system) in skeletal muscle. We model the irregular system of tubules as a random network of miniature transmission lines, using differential equations to describe the potential between the nodes and difference equations to describe the potential at the nodes. The solution to the equations can be accurately represented in several approximate forms with simple physical and graphical interpretations. All the parameters of the solution are specified by impedance and morphometric measurements. The effect of wide circumferential spacing between T-system openings is analyzed and the resulting restricted mesh model is shown to be approximated by a mesh with an access resistance. The continuous limit of the mesh model is shown to have the same form as the disk model of the T-system, but with a different expression for the tortuosity factor. The physical meaning of the tortuosity factor is examined, and a short derivation of the disk model is presented that gives results identical to the continuous limit of the mesh model. Both the mesh and restricted mesh models are compared with experimental data on the impedance of muscle fibers of the frog sartorius. The derived value for the resistivity of the lumen of the tubules is not too different from that of the bathing solution, the difference probably arising from the sensitivity of this value to errors in the morphometric measurements.     

A second-order high resolution finite difference scheme for a structured erythropoiesis model subject to malaria infection

We develop a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model describing the within-host dynamics of malaria infection. The model consists of two nonlinear partial differential equations coupled with three nonlinear ordinary differential equations. Convergence of the numerical method to the unique weak solution with bounded total variation is proved. Numerical simulations demonstrating the achievement of the designed accuracy are presented.     

A multiscale model for red blood cell mechanics

The objective of this article is the derivation of a continuum model for mechanics of red blood cells via multiscale analysis. On the microscopic level, we consider realistic discrete models in terms of energy functionals defined on networks/lattices. Using concepts of Γ-convergence, convergence results as well as explicit homogenisation formulae are derived. Based on a characterisation via energy functionals, appropriate macroscopic stress–strain relationships (constitutive equations) can be determined. Further, mechanical moduli of the derived macroscopic continuum model are directly related to microscopic moduli. As a test case we consider optical tweezers experiments, one of the most common experiments to study mechanical properties of cells. Our simulations of the derived continuum model are based on finite element methods and account explicitly for membrane mechanics and its coupling with bulk mechanics. Since the discretisation of the continuum model can be chosen freely, rather than it is given by the topology of the microscopic cytoskeletal network, the approach allows a significant reduction of computational efforts. Our approach is highly flexible and can be generalised to many other cell models, also including biochemical control.     

Thermoelastic modelling of the skin at finite deformations

The modelling and computation of the coupled thermal and mechanical response of human skin at finite deformations is considered. The model extends current thermal models to account for thermally- and mechanically-induced deformations. Details of the solution of the highly nonlinear system of governing equations using the finite element method are presented. A representative numerical example illustrates the importance of considering the coupled response for the problem of a rigid, hot indenter in contact with the skin.     

Assessment of the effect of mesh density on the material property discretisation within QCT based FE models: a practical example using the implanted proximal tibia

A three-dimensional, quantitative computed tomography based finite element model of a proximal implanted tibia was analysed in order to assess the effect of mesh density on material property discretisation and the resulting influence on the predicted stress distribution. The mesh was refined on the contact surfaces (matched meshes) with element sizes of 3, 2, 1.4, 1 and 0.8 mm. The same loading conditions were used in all models (bi-condylar load: 60% medial, 40% lateral). Significant variations were observed in the modulus distributions between the coarsest and finest mesh densities. Poor discretisation of the material properties also resulted in poor correlations of the stresses and risk ratios between the coarsest and finest meshes. Little difference in Young's modulus, von Mises stress and risk ratio distributions were observed between the three finest models; hence, it was concluded that for this particular case an element size of 1.4 mm on the contact surfaces was enough to properly describe the stiffness, stress and risk ratio distributions within the bone. Poor convergence of the material property distribution occurred when the element size was significantly larger than the pixel size of the source CT data. It was concluded that unless there is convergence in the Young's modulus distribution, convergence of the stress field or of other parameters of interest will not occur either.     

Iteroparous reproduction strategies and population dynamics

Asymptotic relationships between a class of continuous partial differential equation population models and a class of discrete matrix equations are derived for iteroparous populations. First, the governing equations are presented for the dynamics of an individual with juvenile and adult life stages. The organisms reproduce after maturation, as determined by the juvenile period, and at specific equidistant ages, which are determined by the iteroparous reproductive period. A discrete population matrix model is constructed that utilizes the reproductive information and a density-dependent mortality function. Mortality in the period between two reproductive events is assumed to be a continuous process where the death rate for the adults is a function of the number of adults and environmental conditions. The asymptotic dynamic behaviour of the discrete population model is related to the steady-state solution of the continuous-time formulation. Conclusions include that there can be a lack of convergence to the steady-state age distribution in discrete event reproduction models. The iteroparous vital ratio (the ratio between the maximal age and the reproductive period) is fundamental to determining this convergence. When the vital ratio is rational, an equivalent discrete-time model for the population can be derived whose asymptotic dynamics are periodic and when there are a finite number of founder cohorts, the number of cohorts remains finite. When the ratio is an irrational number, effectively there is convergence to the steady-state age distribution. With a finite number of founder cohorts, the number of cohorts becomes countably infinite. The matrix model is useful to clarify numerical results for population models with continuous densities as well as delta measure age distribution. The applicability in ecotoxicology of the population matrix model formulation for iteroparous populations is discussed.     

Synergism and antagonism of neurons caused by an electrical synapse

To investigate the role of electrical junctions in the nervous system, a model system consisting of two nearly identical neurons electrotonically coupled is studied. We assume that each neuron discharges a train of impulses or bursts either spontaneously or under constant stimulus via chemical synapses. It is known that not only an electric current but also chemical substances whose molecular weight is about 1000 can pass through the junction of an electrical synapse (gap junction). So, our model system is regarded as a set of non-linear oscillators coupled by diffusion, and it may be described by a system of ordinary differential equations. Neurons are excited constantly when they are stimulated by an electric current above the threshold level. Therefore, we expect Hopf bifurcation to occur at the critical magnitude of a stimulating electric current in the system of differential equations which describes the dynamics of a single neuron. Studying our model system according to the theory of Hopf bifurcation, we found regions of diffusion constants of the electrical junction which give two kinds of periodic solutions. One is the solution where two neurons oscillate in phase synchrony. The other is where two neurons oscillate 180° out of phase. In the case where one neuron is described by the BVP model, the following was found by computer simulation. When the initial difference between the phase of two neurons is small, the two neurons come to oscillate synchronously. If the initial difference is large, however, the two come to be excited alternately. The physiological implications of these results are discussed.Department of Behaviorology, Faculty of Human Sciences     

Stochastic modeling of aphid population growth with nonlinear, power-law dynamics

This paper develops a deterministic and a stochastic population size model based on power-law kinetics for the black-margined pecan aphid. The deterministic model in current use incorporates cumulative-size dependency, but its solution is symmetric. The analogous stochastic model incorporates the prolific reproductive capacity of the aphid. These models are generalized in this paper to include a delayed feedback mechanism for aphid death. Whereas the per capita aphid death rate in the current model is proportional to cumulative size, delayed feedback is implemented by assuming that the per capita rate is proportional to some power of cumulative size, leading to so-called power-law dynamics. The solution to the resulting differential equations model is a left-skewed abundance curve. Such skewness is characteristic of observed aphid data, and the generalized model fits data well. The assumed stochastic model is solved using Kolmogrov equations, and differential equations are given for low order cumulants. Moment closure approximations, which are simple to apply, are shown to give accurate predictions of the two endpoints of practical interest, namely (1) a point estimate of peak aphid count and (2) an interval estimate of final cumulative aphid count. The new models should be widely applicable to other aphid species, as they are based on three fundamental properties of aphid population biology.     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

Interrogating the effect of enzyme kinetics on metabolism using differentiable constraint-based models

Metabolic models are typically characterized by a large number of parameters. Traditionally, metabolic control analysis is applied to differential equation-based models to investigate the sensitivity of predictions to parameters. A corresponding theory for constraint-based models is lacking, due to their formulation as optimization problems. Here, we show that optimal solutions of optimization problems can be efficiently differentiated using constrained optimization duality and implicit differentiation. We use this to calculate the sensitivities of predicted reaction fluxes and enzyme concentrations to turnover numbers in an enzyme-constrained metabolic model of Escherichia coli. The sensitivities quantitatively identify rate limiting enzymes and are mathematically precise, unlike current finite difference based approaches used for sensitivity analysis. Further, efficient differentiation of constraint-based models unlocks the ability to use gradient information for parameter estimation. We demonstrate this by improving, genome-wide, the state-of-the-art turnover number estimates for E. coli. Finally, we show that this technique can be generalized to arbitrarily complex models. By differentiating the optimal solution of a model incorporating both thermodynamic and kinetic rate equations, the effect of metabolite concentrations on biomass growth can be elucidated. We benchmark these metabolite sensitivities against a large experimental gene knockdown study, and find good alignment between the predicted sensitivities and in vivo metabolome changes. In sum, we demonstrate several applications of differentiating optimal solutions of constraint-based metabolic models, and show how it connects to classic metabolic control analysis.     

A transversely isotropic hyperelastic constitutive model of the PDL. Analytical and computational aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A Transversely Isotropic Hyperelastic Constitutive Model of the PDL. Analytical and Computational Aspects

This study describes the development of a constitutive law for the modelling of the periodontal ligament (PDL) and its practical implementation into a commercial finite element code. The constitutive equations encompass the essential mechanical features of this biological soft tissue: non-linear behaviour, large deformations, anisotropy, distinct behaviour in tension and compression and the fibrous characteristics. The approach is based on the theory of continuum fibre-reinforced composites at finite strain where a compressible transversely isotropic hyperelastic strain energy function is defined. This strain energy density function is further split into volumetric and deviatoric contributions separating the bulk and shear responses of the material. Explicit expressions of the stress tensors in the material and spatial configurations are first established followed by original expressions of the elasticity tensors in the material and spatial configurations. As a simple application of the constitutive model, two finite element analyses simulating the mechanical behaviour of the PDL are performed. The results highlight the significance of integrating the fibrous architecture of the PDL as this feature is shown to be responsible for the complex strain distribution observed.     

A fully coupled porous media and channels flow approach for simulation of blood and bile flow through the liver lobules

Two dimensional, steady state, and incompressible blood and bile flows through the liver lobules are numerically simulated. Two different geometric models A and B are proposed to study the effects of lobule structure on the fluid flow behaviour. In Model A, the lobule tissue is represented as a hexagonal shape porous medium with a set of flow channels at its vertices accounting for the hepatic artery, portal and central veins along with bile ductules. Model B is a channelized porous medium constructed by adding a set of flow channels, representing the bile canaliculies and lobule sinusoids, to Model A. The bile and blood flow through the lobule is simulated by the finite element approach, based on the Darcy/Brinkman equations in the lobule tissue and the Navier-Stokes (or Stokes) equations in the flow channels. In Model B, a transmission factor on the boundaries of the bile canaliculies is introduced to connect the bile and blood flows. First, a single regular lobule is utilized to exhibit the fluid flow pattern through the liver lobule represented by proposed geometric models. Then, the model is extended to a group of liver lobules to demonstrate the flow through a liver slice represented by irregular lobules. Numerical results indicate that the Darcy and Brinkman equations provide nearly the same solutions for Model A and similar solutions with a little difference for Model B. It is shown that the existence of sinusoids and bile canaliculies inside the liver lobules has noticeable effects on its fluid flow pattern, in terms of pressure and velocity fields.