Equivalent Cable Research Articles

Study on the subsea umbilical cable equivalent device’s hydraulic transmission characteristics

ABSTRACT To develop an equivalent device of the umbilical cable, in which the hydraulic transmission characteristics are the same as the real umbilical cable. Simulation and experimental methods were used to study the hydraulic transmission characteristics. Theoretical relations of hydraulic resistance, hydraulic capacitance, and hydraulic inductance were derived by using the similarity principle. Simulation results showed that by comparing the valve operational time and the subsea accumulator pressure characteristics of the umbilical cable and the equivalent umbilical cable, the equivalent umbilical cable model can simulate the true umbilical valve operation characteristics. Finally, the pressure relief in the absence of an actuator or in the presence of a single actuator, was achieved by using an umbilical equivalent device. The results indicated that the umbilical equivalent device could simulate the true umbilical hydraulic transmission characteristics, making it useful for replacing the true umbilical cable in the subsea product integrity tests.

Tension determination for suspenders of arch bridge based on multiple vibration measurements concentrated at one end

Considering the specific arrangement of boundary constraints for the suspenders of arch bridge, this research extends a recently developed method to carry out the determination of suspender tension simply with multiple measurements concentrated at one end. The virtual hinged boundary of the equivalent cable is first decided by the optimal shifting parameter solved from best fitting the identified mode shape ratios with an unsymmetrical formulation. The effective vibration length for the suspender with a hinged connection at the higher end is taken as the distance between the virtual and actual hinged boundaries to directly estimate its tension. On the other hand, an iteration algorithm using a finite element model is developed to find the convergent tension for the suspender with a fixed constraint at the arch end. The feasibility and accuracy of the proposed method are extensively verified with demonstrative numerical examples, laboratory experiments, and practical applications in real bridges.

Simplified modeling of EM field coupling to complex cable bundles

Abstract. In this contribution, the procedure "Equivalent Cable Bundle Method" is used for the simplification of large cable bundles, and it is extended to the application on differential signal lines. The main focus is on the reduction of twisted-pair cables. Furthermore, the process presented here allows to take into account cables with wires that are situated quite close to each other. The procedure is based on a new approach to calculate the geometry of the simplified cable and uses the fact that the line parameters do not uniquely correspond to a certain geometry. For this reason, an optimization algorithm is applied.

Dendritic Signals Command Firing Dynamics in a Mathematical Model of Cerebellar Purkinje Cells

Dendrites of cerebellar Purkinje cells (PCs) respond to brief excitations from parallel fibers with lasting plateau depolarizations. It is unknown whether these plateaus are local events that boost the synaptic signals or they propagate to the soma and directly take part in setting the cell firing dynamics. To address this issue, we analyzed a likely mechanism underlying plateaus in three representations of a reconstructed PC with increasing complexity. Analysis in an infinite cable suggests that Ca plateaus triggered by direct excitatory inputs from parallel fibers and their mirror signals, valleys (putatively triggered by the local feed forward inhibitory network), cannot propagate. However, simulations of the model in electrotonic equivalent cables prove that Ca plateaus (resp. valleys) are conducted over the entire cell with velocities typical of passive events once they are triggered by threshold synaptic inputs that turn the membrane current inward (resp. outward) over the whole cell surface. Bifurcation analysis of the model in equivalent cables, and simulations in a fully reconstructed PC both indicate that dendritic Ca plateaus and valleys, respectively, command epochs of firing and silencing of PCs.

Determination of the location and magnitude of synaptic conductance changes in spinal motoneurons by impedance measurements.

The relation between impedance change and the location and magnitude of a tonic synaptic conductance was examined in compartmental motoneuron models based on previously published data. The dependency of motoneuron impedance on system time constant (tau), electrotonic length (L), and dendritic-to-somatic conductance ratio (rho) was examined, showing that the relation between impedance phase and rho differed markedly between models with uniform and nonuniform membrane resistivity. Dendritic synaptic conductances decreased impedance magnitude at low frequencies; at higher frequencies, impedance magnitude increased. The frequency at which the change in impedance magnitude reversed from a decrease to an increase-the reversal frequency, F(r)-was a good estimator of electrotonic synaptic location. A measure of the average normalized impedance change at frequencies less than F(r), cuDeltaZ, estimated relative synaptic conductance. F(r) and cuDeltaZ provided useful estimates of synaptic location and conductance in models with nonuniform (step, sigmoidal) and uniform membrane resistivity. F(r) also provided good estimates of spatial synaptic location on the equivalent cable in both step and sigmoidal models. Variability in relations between F(r), cuDeltaZ, and conductance location and magnitude between neurons was reduced by normalization with rho and tau. The effects on F(r) and cuDeltaZ of noise in experimental recordings, different synaptic distributions, and voltage-dependent conductances were also assessed. This study indicates that location and conductance of tonic dendritic conductances can be estimated from F(r), cuDeltaZ, and basic electrotonic motoneuron parameters with the exercise of suitable precautions.

Analytical and numerical construction of equivalent cables

The mathematical complexity experienced when applying cable theory to arbitrarily branched dendrites has lead to the development of a simple representation of any branched dendrite called the equivalent cable. The equivalent cable is an unbranched model of a dendrite and a one-to-one mapping of potentials and currents on the branched model to those on the unbranched model, and vice versa. The piecewise uniform cable, with a symmetrised tri-diagonal system matrix, is shown to represent the canonical form for an equivalent cable. Through a novel application of the Laplace transform it is demonstrated that an arbitrary branched model of a dendrite can be transformed to the canonical form of an equivalent cable. The characteristic properties of the equivalent cable are extracted from the matrix for the transformed branched model. The one-to-one mapping follows automatically from the construction of the equivalent cable. The equivalent cable is used to provide a new procedure for characterising the location of synaptic contacts on spinal interneurons.

Dendritic subunits determined by dendritic morphology.

A theoretical framework is presented in which arbitrarily branched dendritic structures with nonhomogeneous membrane properties and nonuniform geometry can be transformed into an equivalent unbranched structure (equivalent cable). Rall's equivalent cylinder is seen to be one part of the equivalent cable in the special case of dendrites satisfying the Rall criteria. The relation between the branched dendrite and its equivalent unbranched representation is uniquely defined by an invertible mapping that connects configurations of inputs on the branched structure with those on the unbranched structure, and conversely. This mapping provides a new definition of dendritic subunit and provides a mechanism for characterizing local and nonlocal signal processing within dendritic structures.

Comparison of alternative designs for reducing complex neurons to equivalent cables.

Reduction of the morphological complexity of actual neurons into accurate, computationally efficient surrogate models is an important problem in computational neuroscience. The present work explores the use of two morphoelectrotonic transformations, somatofugal voltage attenuation (AT cables) and signal propagation delay (DL cables), as bases for construction of electrotonically equivalent cable models of neurons. In theory, the AT and DL cables should provide more accurate lumping of membrane regions that have the same transmembrane potential than the familiar equivalent cables that are based only on somatofugal electrotonic distance (LM cables). In practice, AT and DL cables indeed provided more accurate simulations of the somatic transient responses produced by fully branched neuron models than LM cables. This was the case in the presence of a somatic shunt as well as when membrane resistivity was uniform.

Passive electrical cable properties and synaptic excitation of tiger salamander retinal ganglion cells.

The passive electrical properties of 17 ON-OFF retinal ganglion cells were derived from electrophysiological recordings. The parameters for each cells' equivalent model were obtained from the transient current responses to small step changes in clamp potential. Thirteen of the cells could be adequately approximated by a spherical soma connected to an equivalent dendritic cable. Estimates for the cell input conductance (GN), membrane time constant (tau m), the dendritic-to-soma conductance ratio (rho), and the normalized electrotonic length (L) were obtained (mean +/- standard deviation, n = 13): GN = 580 +/- 530 pS, tau m = 97 +/- 72 ms, rho = 2.8 +/- 2.8, and L = 0.34 +/- 0.13. Series resistance averaged 32 +/- 11 M omega. The mean of the derived soma diameters was 18 +/- 6 microns and the mean diameter and length of the equivalent cables were 1.4 +/- 0.6 and 470 +/- 90 microns, respectively. The average of the specific membrane conductances, 1.67 +/- 1.08 S/cm2, corresponded to a membrane resistivity of 60 k omega. cm2. Computer simulations of synaptic inputs were performed on a representative model, with an electrode at the soma and using the worst-case configuration, in which all synaptic inputs were confined to the tips of the dendrites. We draw three conclusions from the modeling: (1) Under voltage clamp, fast spontaneous EPSCs would be significantly attenuated and slowed while the time course of the slower, light-evoked non-NMDA and NMDA EPSCs would be minimally distorted by dendritic filtering. (2) Excitatory synaptic reversal potentials can be accurately determined under voltage clamp. (3) In the absence of GABAergic and glycinergic inhibition, the efficacy at the soma of excitatory conductance changes is essentially independent of their dendritic location. The specific membrane resistivity appears to represent a good compromise between having a small membrane time constant and minimal EPSP attenuation.

On trees as equivalent cables.

By means of a suitable transformation, any passive dendritic tree may be reduced to an equivalent, possibly non-uniform cable. Under certain conditions the equivalent cable has disjoint sections of which only one communicates with the soma. Inputs that map on to the disconnected sections cannot be seen by the soma. Ralls's equivalent cylinder and its generalizations emerge naturally as the simplest cases of this behaviour. Even where, as is more usual, decomposition does not occur exactly the equivalent cable together with the input mapping from the tree to the cable provides a readily visualisable and intuitively appealing description of quite subtle relationships on the tree. The structure of the equivalent cable is dominated by approximate geometric symmetries of the tree. These symmetries cause well-defined subspaces of the total space of synaptic inputs to arrive at the soma at different times, thus allowing them, in principle, to be reflected, for example in the temporal statistics of the neurons' spike output.

Voltage clamp calculations for myelinated and demyelinated axons.

Exact cable theory is used to calculate voltage distributions along fully myelinated axons and those with various patterns of demyelination. The model employed uses an R-C circuit for the soma, an equivalent cable for the dendrites, a myelinated axon with n internodes and a cable representing telodendria. For the case of a voltage clamp at the soma, a system of 2n + 1 equations must be solved to obtain the potential distribution and this is done for arbitrary n. An explicit calculation is performed for one internode whereas computer-generated solutions are obtained for several internodes. The relative importance of the position of a single demyelinated internode is determined. An approximate expression is given for the critical internodal length necessary for action potential generation.

Propagation of action potentials along complex axonal trees. Model and implementation

Axonal trees are typically morphologically and physiologically complicated structures. Because of this complexity, axonal trees show a large repertoire of behavior: from transmission lines with delay, to frequency filtering devices in both temporal and spatial domains. Detailed theoretical exploration of the electrical behavior of realistically complex axonal trees is notably lacking, mainly because of the absence of a simple modeling tool. AXONTREE is an attempt to provide such a simulator. It is written in C for the SUN workstation and implements both a detailed compartmental modeling of Hodgkin and Huxley-like kinetics, and a more abstract, event-driven, modeling approach. The computing module of AXONTREE is introduced together with its input/output features. These features allow graphical construction of arbitrary trees directly on the computer screen, and superimposition of the results on the simulated structure. Several numerical improvements that increase the computational efficiency by a factor of 5-10 are presented; most notable is a novel method of dynamic lumping of the modeled tree into simpler representations ("equivalent cables"). AXONTREE's performance is examined using a reconstructed terminal of an axon from a Y cell in cat visual cortex. It is demonstrated that realistically complicated axonal trees can be handled efficiently. The application of AXONTREE for the study of propagation delays along axonal trees is presented in the companion paper (Manor et al., 1991).

A generalized tapering equivalent cable model for dendritic neurons

A mathematical model has been developed which collapses a dendritic neuron of complex geometry into a single electrotonically tapering equivalent cable. The modified cable equation governing the transient distribution of subthreshold membrane potential in a branching tree is transformed, becoming amenable to analytic solution. This transformation results in a Riccati differential equation whose six solutions (expressed in terms of elementary functions) control the amount and degree of taper found in the equivalent cable model. To illustrate the theory, an analytic solution (in series form) of the modified cable equation is obtained for a voltage-clamp present at the soma of a quadratically tapering equivalent cable whose distal end is sealed.

Analysis of a Postsynaptic Scheme based on a Tapering Equivalent Cable Model

An analytic solution of a modified cable equation with reversal potentials is used to explore nonlinear synaptic effects in passive dendritic trees of arbitrary geometry. It is shown that shunting inhibition can be effective when located off the direct path between the excitation and the soma. It is also shown that a peripherally placed excitatory input juxtaposed with a shunting inhibitory input may produce a voltage-peak minimum at the soma in order to suppress the initiation of an action potential at the axon hillock. The applicability of this postsynaptic scheme as a basis for a directionally selective signal generation in the retina is discussed.

Model study of the spread of electrotonic potential in cardiac tissue.

This model study describes the electrotonic response of a cable model of cardiac tissue stimulated at one point. The stimulus is applied intracellularly in the form of a 2 ms pulse of current of near threshold amplitude. The attenuation of the electrotonic potential with distance and its mode of propagation along the cable are compared for equivalent passive, continuous and discontinuous cables. The three structures have the same basic physical and electrical characteristic and they differ either with respect to being active or passive or to the presence or absence of intercellular gap junctions. In the continuous cable a just subthreshold stimulus produces a local active response which propagates more slowly and is attenuated less rapidly with distance than in a passive cable. The spatial decrement of the local response in a discontinuous cable is faster than in a continuous cable of equal average resistivity. It is suggested that the larger time constant of the foot of the action potential observed in the longitudinal direction in cardiac muscle could be due in part to the electrotonic spread of the local response from the site of stimulation.

Electrotonic parameters of rat dentate granule cells measured using short current pulses and HRP staining.

The passive electrotonic parameters of nerve cells in the dentate gyrus of the rat were studied in vitro. Intracellular recordings from 30 granule cells and 3 pyramidal basket cells followed by intracellular injection of horseradish peroxidase (HRP), allowed calculations of input resistance (RN), membrane time constant (tau m), electrotonic length (L), ratio of dendritic to somatic conductance (rho), membrane specific capacitance and resistance (Rm, Cm), and specific axoplasmic resistance (Ri). The analysis of the voltage decays from long saturating (100 ms) and short (0.5 ms) current pulses showed that the short-pulse method gave better resolution for the measurement of the time constants and avoided some of the time-dependent nonlinearities but required larger currents than the long pulse. Morphological analysis of 49 branching points taken from the dendritic trees of granule cells showed that the branching power, n, is equal to 1.56 +/- 0.186 and was fairly constant throughout the tree. Given the fact that all dendrites have approximately the same length and number of branch points, the granule cell dendritic tree can be meaningfully collapsed into an equivalent cable. Moreover, electrophysiological data suggested that the cable had a "sealed" end or at least a high-impedance termination. Based on an equivalent cable model with a sealed end and a lumped soma impedance, a method was implemented to analyze the multiexponential decays from hyperpolarizing current pulses and to solve the equations of the model. This was done successfully in only 40% of the cells and yielded the following mean values for L = 1.13 and rho = 7.58. From the measurements of the soma surface area (S) and the equivalent cable diameter (D), the average specific membrane parameters were calculated: Rm = 2,726 alpha x cm2, Cm = 5.24 microF/cm2, Ri = 101 alpha x cm. The input resistance and time constant of the granule cells as measured from the short-pulse technique averaged to RN 58.57 M alpha and tau m = 16.21 ms. The failure of the model to fit 60% of the cells was interpreted to be due to the presence of a somatic shunt resulting from electrode injury, tonic synaptic activity, a lower somatic membrane specific resistance, or electronic coupling.(ABSTRACT TRUNCATED AT 400 WORDS)

Limits of usefulness of electrophysiological methods for estimating dendritic length in neurones

The present report deals with methodological problems concerning the electrophysiological analysis of multipolar neurones. In a nerve cell in which the dendrites may be considered as being equivalent to a single cable, the electrotonic length of the equivalent cable may be determined by analyzing the potential response to a step of current injected into the soma (Rall, 1969). In the present model study it is shown that this kind of analysis is unsuitable for discriminating between neurones with different dendritic lengths in excess of about 1.5-2.0 length constants.

On Optimal Analog Repeater Systems

In this paper, the optimal analog repeater which transmits maximum mutual information (MI-system) is discussed. The solution is obtained by conventional variational method and the functional structure of the resulting optimal is investigated. The consists of a whitening filter, an equivalent cable, and a system equalizer, the concepts of which are newly introduced and defined in the paper. For comparison, the repeater with minimum mean-square error (ME-system) is analyzed by the analogous method. It is shown that the whitening filter and the equivalent cable are invariant for both systems and that the only part which varies with the optimizing criterion is the system equalizer. Finally, these results are applied to the special case of single stage and the relation to Shannon's water pouring theorem is discussed.

Minimum weight design of elastic cables

This paper is concerned with the minimum weight design of an elastic cable of variable cross section with both ends positioned at the same level. The cable is under the action of a uniformly distributed load. With the deflection of the cable given and the stress in the cable restricted, our objective is to determine the optimum distribution of the cross section. Calculus of variations with equality and inequality constraints is employed to find the necessary conditions of optimality. Efficiency of the optimum design evaluated from the comparison of an optimum cable with an equivalent cable of uniform cross section is studied. The best sag-span ratio of the optimum cable is also determined.

Physical basis of recording fast versus slow cortical waves

It is not yet knownwhy, in the presence of Well-marked, slow (postsynaptic) waves at all points of the cortex, spike discharges are observed relatively rarely, and only in the immediate neighborhood of the cell body. This relationship between fast and slow waves in the extraeellular medium is still less clear if it is remembered that the true (intracellular) amplitude of the action potential is 6-10 times greater than the amplitude of postsynaptie potentials. To study this problem the mathematical model previously suggested (1), describing the distribution of extracellular potential around a pyramidal neuron of the cortex during the onset of an excitatory postsynaptie potential (EPSP} on its soma, was used. Lorente de N6 (2) showed that pyramidal neurons play a leading role in formation of the extracellular field in the cortex. Nonpyramidal neurons form afield of "latent type," which cannot be detected even at a very short distance from the soma. In the present calculations, the pyramidal neuron was replaced by an equivalent homogeneous cable, placed in a three-dimensiona l conductor. This substitution made it possible to assess the role of currents leaving the apical dendrite in formation of the extracellular field. Morphological data show that the basal dendrites are shorter and thinner than the apical {3}. Currents leaving the basal dendrites are therefore small and cannot significantly change the general pattern of distribution, which is mainly determined by currents leaving the apical dendrite. It has been shown that where an EPSP arises on the soma, replacement of the apical dendrite by a homogeneous cable can be justified as follows. Ramification of the apical dendrite begins at adistance of 2-3 k from the soma ( k being the length constant of the dendrite}, which introduces little if any change into the pattern of distribution of the currents. It was assumed that because of the closeness of the buildup time of the EPSP {v = 5 msec (4}) and the time constant of the membrane {Tin = 3.2-8.5 msec (5, 6)), the distribution of amplitude of the extracellular potential in this case differs little from its distribution during the formation of a steady potential on the soma membrane. In the case of a steady potential on the soma, the capacitance of the membrane is evidently unimportant. Conversely, the capacitance Cm of the cable per unit length is important for distribution of currents flowing in the cable with a rapidly changing potential at its input. In this paper the extracellular field of an equivalent cable, generated by an "excitation battery" E0 (t) connected to its end, is examined making allowance for the capacitance of the cable. From the results, the effect of capacitance on the extracellular field