Linear Cable Equation Research Articles

Linear and Nonlinear Behaviors of the Photoreceptor Coupled Network.

Photoreceptors are electrically coupled to one another, and the spatiotemporal properties of electrical synapses in a two-dimensional retinal network are still not well studied, because of the limitation of the single electrode or pair recording techniques which do not allow simultaneously measuring responses of multiple photoreceptors at various locations in the retina. A multiple electrode recording system is needed. In this study, we investigate the network properties of the two-dimensional rod coupled array of the salamander retina (both sexes were used) by using the newly available multiple patch electrode system that allows simultaneous recordings from up to eight cells and to determine the electrical connectivity among multiple rods. We found direct evidence that voltage signal spread in the rod-rod coupling network in the absence of I h (mediated by HCN channels) is passive and follows the linear cable equation. Under physiological conditions, I h shapes the network signal by progressively shortening the response time-to-peak of distant rods, compensating the time loss of signal traveling from distant rods to bipolar cell somas and facilitating synchronization of rod output signals. Under voltage-clamp conditions, current flow within the coupled rods follows Ohm's law, supporting the idea that nonlinear behaviors of the rod network are dependent on membrane voltage. Rod-rod coupling is largely symmetrical in the 2D array, and voltage-clamp blocking the next neighboring rod largely suppresses rod signal spread into the second neighboring rod, suggesting that indirect coupling pathways play a minor role in rod-rod coupling.

Numerical solution for the variable order linear cable equation with Bernstein polynomials

In this paper, Bernstein polynomials method is proposed for the numerical solution of a class of variable order fractional linear cable equation. In this paper, we adopted Bernstein polynomials basis defined on the interval [0,R] to solve the equations defined on the section Ω=[0,X]×[0,T]. The main characteristic behind this approach in this paper is that we derive two kinds of operational matrixes of Bernstein polynomials. With the operational matrixes, the initial equation is transformed into the products of several dependent matrixes which can also be viewed as the system of linear equations after dispersing the variable. By solving the linear system of algebraic equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

Stability analysis of TLP tethers

Compliant offshore TLPs are essentially meant for deep oil/gas exploration and are usually constructed on the seashore and then towed down to the particular location for anchorage. They are connected to the sea bed by means of pretension cables. The increased use of TLPs in deep waters and necessity of reduction of usually high value of pretension make the effect of variable tension in the tether dynamics more significant. This work presents the dynamic analysis of tethers and TLPs considering the linearly varying tension along the tether length. The modal analysis considers a linear cable equation for tether modeling subjected to tension which varies along its length. A Mathieu stability analysis is then performed for TLPs of different shapes and different water depth vis-à-vis of 527.8, 872, and 1200 m respectively to obtain the amplitudes of tether vibrations. The unstable modes of vibration are also verified. The resultant modal forms for the tether's dynamic model are then obtained in form of Bessel's function. From the numerical studies conducted it is seen that increased tether tension not only leads to a stable platform but also improves the stability due to increased hydrodynamic loading contributing to added mass. From the studies conducted it is also seen that the triangular configuration TLPs with increased initial pretension are more stable compared to four leg TLP in the first mode of vibration.

A spatial stochastic neuronal model with Ornstein-Uhlenbeck input current.

We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein-Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neuron's surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process.

A new computational method for cable theory problems

We discuss a new computational procedure for solving the linear cable equation on a tree of arbitrary geometry. The method is based on a simple set of diagrammatic rules implemented using an efficient computer algorithm. Unlike most other methods, this technique is particularly useful for determining the short-time behavior of the membrane potential. Examples are presented and the convergence and accuracy of the method are discussed.

Simple diagrammatic rules for solving dendritic cable problems

Abstract I present a set of diagrammatic rules for constructing a series solution of the linear cable equation on a dendritic tree of arbitrary geometry. The rules determine the Green's function for the time-dependent cable equation directly as a function of time and are particularly useful in the short-time limit. They can be implimented by computer algorithm and results from such a program are shown.

The path integral for dendritic trees

We construct the path integral for determining the potential on any dendritic tree described by a linear cable equation. This is done by generalizing Brownian motion from a line to a tree. We also construct the path integral for dendritic structures with spatially-varying and/or time-dependent membrane conductivities due, for example, to synaptic inputs. The path integral allows novel computational techniques to be applied to cable problems. Our analysis leads ultimately to an exact expression for the Green's function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules providing a fast and efficient method for solving complex cable problems.

Analysis of lumped and distributed elements models of cut muscle fibers in vaseline or sucrose gap preparations

A general method of finding the time course and the steady state distribution of potential in Vaseline or sucrose gap preparations is given by making use of the linear cable equation. The general solution has been found analytically in terms of its Laplace transform and then numerically inverted. Two particular experimental situations, namely the single gap and the double gap preparations, have been analyzed. The results have been compared with the solutions of the commonly used lumped elements models. While for the double gap no large errors are introduced by the lumped model, for the single gap there are significant differences. The dependence of the voltage distribution on various electrical and geometrical parameters has been examined. It is suggested that the proposed mathematical treatment might be used by experimenters as a reference to assess the validity of simplified lumped models.