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W R Holmes

Publications and source records attributed to W R Holmes.

15 recordsLinked to original sources
PubMed1992

Interpretation of time constant and electrotonic length estimates in multicylinder or branched neuronal structures.

1. We have investigated the theoretical and practical problems associated with the interpretation of time constants and the estimation of electrotonic length with equivalent cylinder formulas for neurons best represented as multiple cylinders or branched structures. Two analytic methods were used to compute the time constants and coefficients of passive voltage transients (and time constants of current transients under voltage clamp). One method, suitable for simple geometries, involves analytic solutions to boundary value problems. The other, suitable for neurons of any geometric complexity, is an algebraic approach based on compartmental models. Neither of these methods requires the simulation of transients. 2. We computed the time constants and coefficients of voltage transients for several hypothetical neurons and also for a spinal motoneuron whose morphology was characterized from serial reconstructions. These time constants and coefficients were used to generate voltage transients. Then exponential peeling, nonlinear regression, and transform methods were applied to these transients to test how well these procedures estimate the underlying time constants and coefficients. 3. For a serially reconstructed motoneuron with 732 compartments, we found that the theoretical and peeled tau 0 values were nearly equal, but the theoretical tau 1 was much larger than the peeled tau 1. The theoretical tau 1 could not be peeled because it was associated with a coefficient, C1, that had a very small value. In fact, there were 156 time constants between 1.0 and 6.0 ms, most of which had very small coefficients; none had a coefficient larger than 2% of the signal. The peeled value of tau 1 (called tau 1 peel) can be viewed as some sort of a weighted average of the time constants having the largest coefficients. 4. We studied simple hypothetical neurons to determine what interpretation could be applied to the multitude of theoretical time constants. We found that after tau 0, there was a group of time constants associated with eigenfunctions that were odd (or approximately odd) functions with respect to the soma. These time constants could be interpreted as "equalizing" time constants along particular paths between different pairs of dendritic terminals in the neuron. After this group of time constants, there was one that we call tau even because it was associated with an eigen-function that was approximately even with respect to the soma. This tau even could be interpreted as an equalizing time constant for charge equalization between proximal membrane (soma and proximal dendrites) and distal membrane (including all distal dendrites).4=

Animals↗
PubMed1992

Electrotonic length estimates in neurons with dendritic tapering or somatic shunt.

1. Compartmental models were used to compute the time constants and coefficients of voltage and current transients in hypothetical neurons having tapering dendrites or soma shunt and in a serially reconstructed motoneuron with soma shunt. These time constants and coefficients were used in equivalent cylinder formulas for the electrotonic length, L, of a cell to assess the magnitude of the errors that result when the equivalent cylinder formulas are applied to neurons with dendritic tapering or soma shunt. 2. Of all the formulas for a cylinder (with sealed ends), the most commonly used formula, which we call L tau 0/tau 1 (the formula uses the current-clamp time constants tau 0 and tau 1), was the most robust estimator of L in structures that tapered linearly. When the diameter at the end of the cylinder was no less than 20% of the initial diameter, L tau 0/tau 1 underestimated the actual L by at most 10%. 3. The equivalent cylinder formulas for a cylinder were applied to neurons modeled as a cylinder with a shunted soma at one end. The formula for L based solely on voltage-clamp time constants gave an exact estimate of L. However, the second voltage-clamp time constant cannot be reliably obtained experimentally for neurons studied thus far. Of the remaining formulas, L tau 0/tau 1 was again the most robust estimator of L. This formula overestimated L with the size of the overestimates depending on beta, rho beta = 1, and the actual L of the cylinder, where beta is the soma shunt factor, and rho beta = 1 is the dendritic-to-somatic conductance ratio when beta = 1 (no shunt). When the actual L was 0.5 and the soma shunt was large, this formula overestimated L by two- to threefold, but when the actual L was 1.5, the overestimate was only 10-15% regardless of the size of the shunt. 4. In neurons modeled as two cylinders with soma shunt, the L tau 0/tau 1 value computed with the actual tau 0 and tau 1 values overestimated the average L by two to six times when soma shunt was large. However, the L tau 0/tau 1 estimates computed with tau 0 and tau 1 values estimated with the exponential fitting program DISCRETE from voltage transients computed for these neuron models were never this large because of inherent problems in estimating closely spaced time constants from data.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗
PubMed1992

Estimating the electrotonic structure of neurons with compartmental models.

1. A procedure based on compartmental modeling called the "constrained inverse computation" was developed for estimating the electrotonic structure of neurons. With the constrained inverse computation, a set of N electrotonic parameters are estimated iteratively with use of a Newton-Raphson algorithm given values of N parameters that can be measured or estimated from experimental data. 2. The constrained inverse computation is illustrated by several applications to the basic example of a neuron represented as one cylinder coupled to a soma. The number of unknown parameters estimated was different (ranging from 2 to 6) when different sets of constraints were chosen. The unknowns were chosen from the following: dendritic membrane resistivity Rmd, soma membrane resistivity Rms, intracellular resistivity Ri, membrane capacity Cm, dendritic membrane area AD, soma membrane area As, electrotonic length L, and resistivity-free length, rfl (rfl = 2l/d1/2 where l and d are length and diameter of the cylinder). The values of the unknown parameters were estimated from the values of an equal number of known parameters, which were chosen from the following: the time constants and coefficients of a voltage transient tau 0, tau 1, ..., C0, C1, ..., voltage-clamp time constants tau vc1, tau vc2, ..., and input resistance RN. Note that initially, morphological data were treated as unknown, rather than known. 3. When complete morphology was not known, parameters from voltage and current transients, combined with the input resistance were not sufficient to completely specify the electrotonic structure of the neuron. For a neuron represented as a cylinder coupled to a soma, there were an infinite number of combinations of Rmd, Rms, Ri, Cm, AS, AD, and L that could be fitted to the same voltage and current transients and input resistance. 4. One reason for the nonuniqueness when complete morphology was not specified is that the Ri estimate is intrinsically bound to the morphology. Ri enters the inverse computation only in the calculation of the electrotonic length of a compartment. The electrotonic length of a compartment is l[4 Ri/(dRmd)]1/2, where l and d are the length and diameter of the compartment. Without complete morphology, the inverse computation cannot distinguish between a change in d or l and a change in Ri. Even when morphology is known, the accuracy of the Ri estimate obtained by any fitting procedure is affected by systematic errors in length and diameter measurements (i.e., tissue shrinkage); the Ri estimate is inversely proportional to the length measurement and proportional to the square root of the diameter measurement.(ABSTRACT TRUNCATED AT 400 WORDS)

Animals↗
PubMed1990

Is the function of dendritic spines to concentrate calcium?

Although dendritic spines are thought to play an important role in synaptic transmission and plasticity, their function remains unknown. Theoretical investigations of spine function have focused on the large electrical resistance provided by the narrow constriction of the spine neck. However, this narrow constriction is also thought to provide a large diffusional resistance. The importance of this diffusional resistance was investigated theoretically with models. When calcium currents were activated on dendritic spines, peak spine head Ca2+ concentration was an order of magnitude larger in 'long-thin' spines than in 'mushroom-shaped' or 'stubby' spines. The same currents activated on dendrites produced even smaller local Ca2+ concentration changes. Although the diffusional resistance of the spine neck was important for producing these differences in [Ca2+], the amplitude and duration of the Ca2+ current relative to the number of Ca2+ binding sites determined whether Ca2+ would be concentrated near synapses. Given the importance of Ca2+ for long-term potentiation, the ability of spines to concentrate Ca2+ may play a key role in processes leading to learning and memory storage.

Animals↗
PubMed1990

Insights into associative long-term potentiation from computational models of NMDA receptor-mediated calcium influx and intracellular calcium concentration changes.

1. Because induction of associative long-term potentiation (LTP) in the dentate gyrus is thought to depend on Ca2+ influx through channels controlled by N-methyl-D-aspartate (NMDA) receptors, quantitative modeling was performed of synaptically mediated Ca2+ influx as a function of synaptic coactivation. The goal was to determine whether Ca2+ influx through NMDA-receptor channels was, by itself, sufficient to explain associative LTP, including control experiments and the temporal requirements of LTP. 2. Ca2+ influx through NMDA-receptor channels was modeled at a synapse on a dendritic spine of a reconstructed hippocampal dentate granule cell when 1-115 synapses on spines at different dendritic locations were activated eight times at frequencies of 10-800 Hz. The resulting change in [Ca2+] in the spine head was estimated from the Ca2+ influx with the use of a model of a dendritic spine that included Ca2+ buffers, pumps, and diffusion. 3. To use a compelling model of synaptic activation, we developed quantitative descriptions of the NMDA and non-NMDA receptor-mediated conductances consistent with available experimental data. The experimental data reported for NMDA and non-NMDA receptor-channel properties and data from other non-LTP experiments that separated the NMDA and non-NMDA receptor-mediated components of synaptic events proved to be limiting for particular synaptic variables. Relative to the non-NMDA glutamate-type receptors, 1) the unbinding of transmitter from NMDA receptors had to be slow, 2) the transition from the bound NMDA receptor-transmitter complex to the open channel state had to be even slower, and 3) the average number of NMDA-receptor channels at a single activated synapse on a single spine head that were open and conducting at a given moment in time had to be very small (usually less than 1). 4. With the use of these quantitative synaptic conductance descriptions. Ca2+ influx through NMDA-receptor channels at a synapse was computed for a variety of conditions. For a constant number of pulses, Ca2+ influx was calculated as a function of input frequency and the number of coactivated synapses. When few synapses were coactivated, Ca2+ influx was small, even for high-frequency activation. However, with larger numbers of coactivated synapses, there was a steep increase in Ca2+ influx with increasing input frequency because of the voltage-dependent nature of the NMDA receptor-mediated conductance. Nevertheless, total Ca2+ influx was never increased more than fourfold by increasing input frequency or the number of coactivated synapses.(ABSTRACT TRUNCATED AT 400 WORDS)

Action Potentials↗
PubMed1989

Effects of uniform and non-uniform synaptic 'activation-distributions' on the cable properties of modeled cortical pyramidal neurons.

Knowledge of the resting potential and input resistance reveal little about the electrotonic structure of nerve cells since that structure is governed by the background distribution of activated conductances. The background distribution of activated conductances (or 'activation-distribution') is commonly assumed to be uniform, but there is much evidence to suggest that the 'activation-distribution' of cortical pyramidal cells in non-uniform. We investigated effects of uniform and non-uniform activation-distributions with simulations employing passive cable models of an HRP-injected cortical pyramidal neuron. The consequences of 5 different activation-distributions on the effectiveness of synaptic inputs and the electrophysiological properties of the neuron were compared. With non-uniform activation-distributions, (i) the resting membrane potential was non-uniform (with difference of 10-15 mV or more found between soma and distal dendrites), (ii) the electrotonic distances to distal synapses were smaller than with a uniform distribution, and (iii) a two-fold range of variation was seen in the effectiveness of distal synaptic inputs. Differences in time constants, tau 0 and tau 1, obtained from an analysis of transients and in electrotonic length, L, were also found with different activation-distributions. These differences were difficult to assess due to the inherent difficulty in estimating tau 1 (as demonstrated here) and the inappropriateness of the usual formula for L for these cells. Reducing afferent activity (as might happen in tissue slice) increased the effectiveness of distal inputs and reduced the differences in resting potential seen in the neuron. It is concluded that the effectiveness of synaptic inputs and the electrophysiological properties of a neuron can be quite different when the activation-distribution is non-uniform rather than uniform.

Animals↗
PubMed1989

The role of dendritic diameters in maximizing the effectiveness of synaptic inputs.

The role of dendritic diameters in maximizing the effectiveness of synaptic inputs was examined in a cell represented as a single cylinder and in a cortical pyramidal cell using mathematical models and computer stimulations. For current input into one end of a cylinder of fixed physical length, the maximum potential at the other end of the cylinder was obtained when the cylinder diameter was chosen so that the electrotonic length, L, of the cylinder was 2.98. For a steady-state or transient synaptic conductance change into the end of a cylinder, the maximum potential at the other end occurred for a value of L less than 2.98; how much less depended on the magnitude of the conductance change. In the model of the reconstructed cortical pyramidal cell, synaptic inputs at proximal, mid-dendritic, and distal locations were most effective for different, particular sets of dendritic diameters. For each synaptic input, there is a set (probably non-unique) of dendritic diameters which maximizes the effectiveness of that input. Paradoxically, a synaptic input at a given physical distance from the soma may produce a larger change in soma potential when it is at a longer electrotonic distance from the soma than at a shorter one. The dendritic diameters determine which inputs are operating at maximal effectiveness. Changes in Rm or Ri or changes in synaptic conductance magnitude or time course may shift the loci of inputs operating at maximal effectiveness. This would change the weighting of synaptic inputs and possibly affect neuronal function.

Cerebral Cortex↗
PubMed1986

A continuous cable method for determining the transient potential in passive dendritic trees of known geometry.

Models using cable equations are increasingly employed in neurophysiological analyses, but the amount of computer time and memory required for their implementation are prohibitively large for many purposes and many laboratories. A mathematical procedure for determining the transient voltage response to injected current or synaptic input in a passive dendritic tree of known geometry is presented that is simple to implement since it is based on one equation. It proved to be highly accurate when results were compared to those obtained analytically for dendritic trees satisfying equivalent cylinder constraints. In this method the passive cable equation is used to express the potential for each interbranch segment of the dendritic tree. After applying boundary conditions at branch points and terminations, a system of equations for the Laplace transform of the potential at the ends of the segments can be readily obtained by inspection of the dendritic tree. Except for the starting equation, all of the equations have a simple format that varies only with the number of branches meeting at a branch point. The system of equations is solved in the Laplace domain, and the result is numerically inverted back to the time domain for each specified time point (the method is independent of any time increment delta t). The potential at any selected location in the dendritic tree can be obtained using this method. Since only one equation is required for each interbranch segment, this procedure uses far fewer equations than comparable compartmental approaches. By using significantly less computer memory and time than other methods to attain similar accuracy, this method permits extensive analyses to be performed rapidly on small computers. One hopes that this will involve more investigators in modeling studies and will facilitate their motivation to undertake realistically complex dendritic models.

Dendrites↗