A three-layer continuous model of porous media to describe the first phase of skin irritation

Mechanical skin irritation induces vasodilation on the line of scratch and in the neighboring zone. In order to model the effect of an irritation on the microcirculation, the vascular network has been described using a three-layer model. The first and last layer, considered as horizontal two-dimensional porous media, describe irrigation and drainage of the system, respectively. The intermediate layer, described by means of a lumped parameter method, does not permit horizontal fluxes. Hierarchical fluxes are directed from the first to the second layer and then towards the drainage layer in order to take into account physiological flow direction. Irritation is modeled by changing compliance of vessels situated at the entrance of the micro-circulation. The model permits to investigate the influence of change in compliance on flow and pressure behavior at microscopic and macroscopic level.     

Factors influencing capacitance-based monitoring of microbial growth.

Microbiological impedance devices are routinely used by food and manufacturing industries, and public health agencies to measure microbiological growth. Factors contributing to increases and decreases in capacitance at the culture medium-electrode interface are poorly understood. To objectively evaluate the effects of temperature, cell density and medium conductivity on capacitance, admittance values from an impedance device were standardized; capacitance was converted to susceptance to allow unit comparisons with conductance. Although increases in temperature increased susceptance, a linear relationship could not be established between the change of susceptance with temperature and conductance of the medium. Cell density by itself had no measureable effect on susceptance or conductance, indicating that cells did not impede the movement of ions in the medium or around the electrode. In a low conductivity medium, increases in conductance by the addition of ions resulted in a concomitant increase of susceptance values. However, in a high conductivity medium, increases in conductance resulted in little or no increase of susceptance values because ions saturated the electrode surface. Susceptance increased when Escherichia coli, Pseudomonas aeruginosa, Alcaligenes faecalis and Staphylococcus aureus were grown in high conductivity media because protons produced by metabolically active bacteria balance more charge on the electrode than other ions. Increases in susceptance due to bacterial growth and metabolism in low conductivity media were attributed to both increases in protons and ionic metabolites. These results indicate that capacitance may provide a better measure of microbial growth and metabolism than conductance.     

A nonlinear biphasic model of flow-controlled infusion in brain: Fluid transport and tissue deformation analyses

A biphasic nonlinear mathematical model is proposed for the concomitant fluid transport and tissue deformation that occurs during constant flow rate infusions into brain tissue. The model takes into account material and geometrical nonlinearities, a hydraulic conductivity dependent on strain, and nonlinear boundary conditions at the infusion cavity. The biphasic equations were implemented in a custom written code assuming spherical symmetry and using an updated Lagrangian finite element algorithm. Results of the model showed that both, geometric and material nonlinearities play an important role in the physics of infusions, yielding important differences from infinitesimal analyses. Geometrical nonlinearities were mainly due to the significant enlargement of the infusion cavity, while variations of the parameters that describe the degree of nonlinearity of the stress–strain curve yielded significant differences in all distributions. For example, a parameter set showing stiffening under tension yielded maximum values of radial displacement and porosity not localized at the infusion cavity. On the other hand, a parameter set showing softening under tension yielded a slight decrease in the fluid velocity for a three-fold increase in the flow rate, which can be explained by the substantial increase of the infusion cavity, not considered in linear analyses. This study strongly suggests that significant enlargement of the infusion cavity is a real phenomenon during infusions that may produce collateral damage to brain tissue. Our results indicate that more experimental tests have to be undertaken in order to determine material nonlinearities of brain tissue over a range of strains. With better understanding of these nonlinear effects, clinicians may be able to develop protocols that can minimize the damage to surrounding tissue.     

Measurement of the membrane potential in small cells using patch clamp methods

The resting membrane potential, E(m), of mammalian cells is a fundamental physiological parameter. Even small changes in E(m) can modulate excitability, contractility and rates of cell migration. At present accurate, reproducible measurements of E(m) and determination of its ionic basis remain significant challenges when patch clamp methods are applied to small cells. In this study, a mathematical model has been developed which incorporates many of the main biophysical principles which govern recordings of the resting potential of 'small cells'. Such a prototypical cell (approx. capacitance, 6 pF; input resistance 5 GΩ) is representative of neonatal cardiac myocytes, and other cells in the cardiovascular system (endothelium, fibroblasts) and small cells in other tissues, e.g. bone (osteoclasts) articular joints (chondrocytes) and the pancreas (β cells). Two common experimental conditions have been examined: (1) when the background K(+) conductance is linear; and (2) when this K(+) conductance is highly nonlinear and shows pronounced inward rectification. In the case of a linear K(+) conductance, the presence of a "leakage" current through the seal resistance between the cell membrane and the patch pipette always depolarizes E(m). Our calculations confirm that accurate characterization of E(m) is possible when the seal resistance is at least 5 times larger than the input resistance of the targeted cell. Measurement of E(m) under conditions in which the main background current includes a markedly nonlinear K(+) conductance (due to inward rectification) yields complex and somewhat counter-intuitive findings. In fact, there are at least two possible stable values of resting membrane potential for a cell when the nonlinear, inwardly rectifying K(+) conductance interacts with the seal current. This type of bistable behavior has been reported in a variety of small mammalian cells, including those from the heart, endothelium, smooth muscle and bone. Our theoretical treatment of these two common experimental situations provides useful mechanistic insights, and suggests practical methods by which these significant limitations, and their impact, can be minimized.     

Structural dynamics and resonance in plants with nonlinear stiffness

Although most biomaterials are characterized by strong stiffness nonlinearities, the majority of studies of plant biomechanics and structural dynamics focus on the linear elastic range of their behavior. In this paper, the effects of hardening (elastic modulus increases with strain) and softening (elastic modulus decreases with strain) nonlinearities on the structural dynamics of plant stems are investigated. A number of recent studies suggest that trees, crops, and other plants often uproot or snap when they are forced by gusting winds or waves at their natural frequency. This can be attributed to the fact that the deflections of the plant, and hence mechanical stresses along the stem and root system, are greatest during resonance. To better understand the effect of nonlinear stiffness on the resonant behavior of plants, plant stems have been modeled here as forced Duffing oscillators with softening or hardening nonlinearities. The results of this study suggest that the resonant behavior of plants with nonlinear stiffness is substantially different from that predicted by linear models of plant structural dynamics. Parameter values were considered over a range relevant to most plants. The maximum amplitudes of deflection of the plant stem were calculated numerically for forcing frequencies ranging from zero to twice the natural frequency. For hardening nonlinearities, the resonant behavior was 'pushed' to higher frequencies, and the maximum deflection amplitudes were lower than for the linear case. For softening nonlinearities, the resonant behavior was pushed to lower frequencies, and the maximum deflection amplitudes were higher than for the linear case. These nonlinearities could be beneficial or detrimental to the stability of the plant, depending on the environment. Damping had the effect of drastically decreasing deflection amplitudes and reducing the effect of the nonlinearities.     

Retinal and cortical nonlinearities combine to produce masking in V1 responses to plaids

The visual response of a cell in the primary visual cortex (V1) to a drifting grating stimulus at the cell’s preferred orientation decreases when a second, perpendicular, grating is superimposed. This effect is called masking. To understand the nonlinear masking effect, we model the response of Macaque V1 simple cells in layer 4Cα to input from magnocellular Lateral Geniculate Nucleus (LGN) cells. The cortical model network is a coarse-grained reduction of an integrate-and-fire network with excitation from LGN input and inhibition from other cortical neurons. The input is modeled as a sum of LGN cell responses. Each LGN cell is modeled as the convolution of a spatio-temporal filter with the visual stimulus, normalized by a retinal contrast gain control, and followed by rectification representing the LGN spike threshold. In our model, the experimentally observed masking arises at the level of LGN input to the cortex. The cortical network effectively induces a dynamic threshold that forces the test grating to have high contrast before it can overcome the masking provided by the perpendicular grating. The subcortical nonlinearities and the cortical network together account for the masking effect. Melinda Koelling is formerly from Center for Neural Science and Courant Institute, New York University.     

Chemical networks with inflows and outflows: A positive linear differential inclusions approach

Certain mass‐action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state‐dependent linear time‐varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to a ubiquitous biochemical reaction network with inflows and outflows, known as the futile cycle. We also provide a characterization of exponential stability of general homogeneous switched systems which is not only of interest in itself, but also plays a role in the analysis of the futile cycle. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009     

Experimental analysis of neuronal dynamics in cultured cortical networks and transitions between different patterns of activity

A general method for developing data-based, stochastic nonlinear models of neurons by means of extended functional series expansions was applied to neural activities of pigeon auditory nerve fibers responding to Gaussian white noise stimuli. To determine Wiener series representations of the investigated neurons the fast orthogonal search algorithm was used. The results suggest that nonlinearities are only instantaneous and that the signal transduction of the investigated sensory system can be described by cascades of dynamic linear and static nonlinear devices. However, only slight improvements result from the nonlinear terms. Considerable improvements are, nevertheless, possible by generalizing the ordinary Wiener series, so that prior neural activity can be taken into account. These extended series were used to develop stochastic models of spiking neurons. The models are able to generate realistic interspike interval distributions and rate-intensity functions. Finally, it will be shown that the irregularity in real and modeled action potential trains has advantages concerning the decoding of neural responses. Received: 16 April 1996 / Accepted in revised form: 23 October 1996     

Modeling environment for numerical simulation of applied electric fields on biological cells

The application of electric pulses in cells increases membrane permeability. This phenomenon is called electroporation. Current electroporation models do not explain all experimental findings: part of this problem is due to the limitations of numerical methods. The Equivalent Circuit Method (ECM) was developed in an attempt to solve electromagnetic problems in inhomogeneous and anisotropic media. ECM is based on modeling of the electrical transport properties of the medium by lumped circuit elements as capacitance, conductance, and current sources, representing the displacement, drift, and diffusion current, respectively. The purpose of the present study was to implement a 2-D cell Model Development Environment (MDE) of ionic transport process, local anisotropy around cell membranes, biological interfaces, and the dispersive behaviour of tissues. We present simulations of a single cell, skeletal muscle, and polygonal cell arrangement. Simulation of polygonal form indicates that the potential distribution depends on the geometrical form of cell. The results demonstrate the importance of the potential distributions in biological cells to provide strong evidences for the understanding of electroporation.     

Temporal processing and adaptation in the songbird auditory forebrain

Songbird auditory neurons must encode the dynamics of natural sounds at many volumes. We investigated how neural coding depends on the distribution of stimulus intensities. Using reverse-correlation, we modeled responses to amplitude-modulated sounds as the output of a linear filter and a nonlinear gain function, then asked how filters and nonlinearities depend on the stimulus mean and variance. Filter shape depended strongly on mean amplitude (volume): at low mean, most neurons integrated sound over many milliseconds, while at high mean, neurons responded more to local changes in amplitude. Increasing the variance (contrast) of amplitude modulations had less effect on filter shape but decreased the gain of firing in most cells. Both filter and gain changes occurred rapidly after a change in statistics, suggesting that they represent nonlinearities in processing. These changes may permit neurons to signal effectively over a wider dynamic range and are reminiscent of findings in other sensory systems.     

Anisotropic hydraulic permeability under finite deformation

The structural organization of biological tissues and cells often produces anisotropic transport properties. These tissues may also undergo large deformations under normal function, potentially inducing further anisotropy. A general framework for formulating constitutive relations for anisotropic transport properties under finite deformation is lacking in the literature. This study presents an approach based on representation theorems for symmetric tensor-valued functions and provides conditions to enforce positive semidefiniteness of the permeability or diffusivity tensor. Formulations are presented, which describe materials that are orthotropic, transversely isotropic, or isotropic in the reference state, and where large strains induce greater anisotropy. Strain-induced anisotropy of the permeability of a solid-fluid mixture is illustrated for finite torsion of a cylinder subjected to axial permeation. It is shown that, in general, torsion can produce a helical flow pattern, rather than the rectilinear pattern observed when adopting a more specialized, unconditionally isotropic spatial permeability tensor commonly used in biomechanics. The general formulation presented in this study can produce both affine and nonaffine reorientations of the preferred directions of material symmetry with strain, depending on the choice of material functions. This study addresses a need in the biomechanics literature by providing guidelines and formulations for anisotropic strain-dependent transport properties in porous-deformable media undergoing large deformations.     

Nonlinear quantum theory of stimulated Cherenkov radiation of transverse electromagnetic waves from a low-density relativistic electron beam in a dielectric medium

A nonlinear quantum theory of stimulated Cherenkov radiation of transverse electromagnetic waves from a low-density relativistic electron beam in an isotropic dielectric medium is presented. A quantum model based on the Klein-Gordon equation is used. The growth rates of beam instabilities caused by the effect of stimulated Cherenkov radiation have been determined in the linear approximation. Mechanisms of the nonlinear saturation of relativistic quantum Cherenkov beam instabilities have been analyzed and the corresponding analytical solutions have been obtained.     

Vestibular integrator neurons have quadratic functions due to voltage dependent conductances

The nonlinear properties of the dendrites of the prepositus hypoglossi nucleus (PHN) neurons are essential for the operation of the vestibular neural integrator that converts a head velocity signal to one that controls eye position. A novel system of frequency probing, namely quadratic sinusoidal analysis (QSA), was used to decode the intrinsic nonlinear behavior of these neurons under voltage clamp conditions. Voltage clamp currents were measured at harmonic and interactive frequencies using specific nonoverlapping stimulation frequencies. Eigenanalysis of the QSA matrix reduces it to a remarkably compact processing unit, composed of just one or two dominant components (eigenvalues). The QSA matrix of rat PHN neurons provides signatures of the voltage dependent conductances for their particular dendritic and somatic distributions. An important part of the nonlinear response is due to the persistent sodium conductance (g_NaP), which is likely to be essential for sustained effects needed for a neural integrator. It was found that responses in the range of 10 mV peak to peak could be well described by quadratic nonlinearities suggesting that effects of higher degree nonlinearities would add only marginal improvement. Therefore, the quadratic response is likely to sufficiently capture most of the nonlinear behavior of neuronal systems except for extremely large synaptic inputs. Thus, neurons have two distinct linear and quadratic functions, which shows that piecewise linear?+?quadratic analysis is much more complete than just piecewise linear analysis; in addition quadratic analysis can be done at a single holding potential. Furthermore, the nonlinear neuronal responses contain more frequencies over a wider frequency band than the input signal. As a consequence, they convert limited amplitude and bandwidth input signals to wider bandwidth and more complex output responses. Finally, simulations at subthreshold membrane potentials with realistic PHN neuron models suggest that the quadratic functions are fundamentally dominated by active dendritic structures and persistent sodium conductances.     

Gating characteristics of a steeply voltage-dependent gap junction channel in rat Schwann cells

The gating properties of macroscopic and microscopic gap junctional currents were compared by applying the dual whole cell patch clamp technique to pairs of neonatal rat Schwann cells. In response to transjunctional voltage pulses (Vj), macroscopic gap junctional currents decayed exponentially with time constants ranging from < 1 to < 10 s before reaching steady-state levels. The relationship between normalized steady-state junctional conductance (Gss) and (Vj) was well described by a Boltzmann relationship with e-fold decay per 10.4 mV, representing an equivalent gating charge of 2.4. At Vj > 60 mV, Gss was virtually zero, a property that is unique among the gap junctions characterized to date. Determination of opening and closing rate constants for this process indicated that the voltage dependence of macroscopic conductance was governed predominantly by the closing rate constant. In 78% of the experiments, a single population of unitary junctional currents was detected corresponding to an unitary channel conductance of approximately 40 pS. The presence of only a limited number of junctional channels with identical unitary conductances made it possible to analyze their kinetics at the single channel level. Gating at the single channel level was further studied using a stochastic model to determine the open probability (Po) of individual channels in a multiple channel preparation. Po decreased with increasing Vj following a Boltzmann relationship similar to that describing the macroscopic Gss voltage dependence. These results indicate that, for Vj of a single polarity, the gating of the 40 pS gap junction channels expressed by Schwann cells can be described by a first order kinetic model of channel transitions between open and closed states.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Signaling network triggers and membrane physical properties control the actin cytoskeleton-driven isotropic phase of cell spreading

Cell spreading is regulated by signaling from the integrin receptors that activate intracellular signaling pathways to control actin filament regulatory proteins. We developed a hybrid model of whole-cell spreading in which we modeled the integrin signaling network as ordinary differential equations in multiple compartments, and cell spreading as a three-dimensional stochastic model. The computed activity of the signaling network, represented as time-dependent activity levels of the actin filament regulatory proteins, is used to drive the filament dynamics. We analyzed the hybrid model to understand the role of signaling during the isotropic phase of fibroblasts spreading on fibronectin-coated surfaces. Simulations showed that the isotropic phase of spreading depends on integrin signaling to initiate spreading but not to maintain the spreading dynamics. Simulations predicted that signal flow in the absence of Cdc42 or WASP would reduce the spreading rate but would not affect the shape evolution of the spreading cell. These predictions were verified experimentally. Computational analyses showed that the rate of spreading and the evolution of cell shape are largely controlled by the membrane surface load and membrane bending rigidity, and changing information flow through the integrin signaling network has little effect. Overall, the plasma membrane acts as a damper such that only ∼5% of the actin dynamics capability is needed for isotropic spreading. Thus, the biophysical properties of the plasma membrane can condense varying levels of signaling network activities into a single cohesive macroscopic cellular behavior.     

Influence of conductance changes on patch clamp capacitance measurements using a lock-in amplifier and limitations of the phase tracking technique.

We characterized the influence of conductance changes on whole-cell patch clamp capacitance measurements with a lock-in amplifier and the limitations of the phase-tracking method by numerical computer simulations, error formulas, and experimental tests. At correct phase setting, the artifacts in the capacitance measurement due to activation of linear conductances are small. The cross talk into the capacitance trace is well approximately by the second-order term in the Taylor expansion of the admittance. In the case of nonlinear current-voltage relationships, the measured conductance corresponds to the slope conductance in the range of the sine wave amplitude, and the cross talk into the capacitance trace corresponds to the second-order effect of the slope conductance. The finite gating kinetics of voltage-dependent channels generate phase-shifted currents. These lead to major artifacts in the capacitance measurements when the angular frequency of the sine wave is close to the kinetic rate constant of the channel. However, when the channel kinetics are sufficiently slow, or sufficiently fast, the cross talk is still close to the second-order effect of the measured conductance. The effects of activation of voltage-dependent currents on the capacitance measurements may be estimated, provided a detailed characterization of the kinetics and voltage dependence is available. A phase error of the lock-in amplifier of a few degrees leads to significant projections. The phase-tracking method can be used to keep the phase aligned only during periods of low membrane conductance. However, nonideal properties of the equivalent circuit, in particular the fast capacitance between the pipette and the bath solutions, may lead to large phase errors when the phase-tracking method is used, depending on the electrical properties of the cell. In this article we provide practical values, setting the range where possible artifacts are below defined limits. For proper evaluation of capacitance measurements, the capacitance and conductance traces should always be displayed together.     

Model Studies of the Role of Mechano-sensitive Currents in the Generation of Cardiac Arrhythmias

Mechano-electrical feedback is studied by incorporating linear, instantaneously activating mechano-sensitive conductances into single cardiac cell models, as well as one- and two-dimensional cardiac network models. The models qualitatively reproduce effects of maintained mechanical stretch on experimentally measured action potential characteristics such as amplitude, maximum diastolic potential, peak upstroke velocity, and conduction velocity. Models are also used to simulate stretch-induced depolarizations, action potentials, and arrhythmias produced by pulsatile volume changes in left ventricle of dog. The mechano-sensitive conductance threshold for a stretch-induced action potential is closely related to the magnitude of the time-independent K⁺current,I_K1, which offsets inward mechano-sensitive current. Activation of mechano-sensitive conductances in small, spatially localized region of cells can evoke graded depolarizations, propagating ectopic beats, and if timed appropriately, spiral reentrant waves. Mechano-sensitive conductance changes required to evoke these responses are well within the physiologically plausible range. Results therefore indicate that many mechano-electrical feedback effects can be modeled using linear, instantaneously activating mechano-sensitive conductances. As an example of how stretch can occur in real human hearts, magnetic resonance images with saturation tagging are used to reconstruct the three-dimensional left ventricular wall motion. In patients with infarcts or recent ischemic events, “paradoxical deformation” is observed in that regions of myocardium are stretched rather than contracted during systole. In contrast, normal hearts contract uniformly with no stretch during systole. Paradoxical deformations in ischemic hearts may therefore present one possible substrate for the mechanically induced arrhythmias modeled above.