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W Rall

Publications and source records attributed to W Rall.

At least 19 recordsLinked to original sources
PubMed1998

Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations.

Important advances in experimental methods have made it possible to measure the electrical events in dendrites directly and to record optically from dendritic spines. These new techniques allow us to focus on the input region of the neuron and highlight the excitable properties of the dendritic membrane. Interestingly, some of the recent experimental findings were anticipated by earlier theoretical research, for example, the observation that some spines possess excitable channels that might generate local all-or-none events. Computer models were used previously to explore the conditions for initiating an action potential at the dendritic tree, in particular, at the spine head, and for active propagation between excitable spines and excitable dendritic arbors. The consequences for synaptic amplification, for the extent of active spread in the tree and for non-linear discriminations between different patterns of synaptic inputs were also considered. Here we review the biophysical insights gained from the theory and demonstrate how these elucidate the recent experimental results.

Animals↗
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↗
PubMed1991

Studies on cryopreservation of Cryptosporidium parvum.

Neonatal BALB/c mice received oocysts or sporozoites of Cryptosporidium parvum pretreated by a variety of cryopreservation protocols. Histologic sections of infected and control mice were examined to determine if pretreated organisms established infection in the intestine. Sporozoites were inoculated rectally, oocysts orally. Freshly excysted sporozoites were frozen in Hanks' balanced salt solution (HBSS) containing dimethylsulfoxide (DMSO) or glycerol at concentrations of 5%, 10%, or 15% at cooling rates of -1 C and -10 C per min. Other sporozoites were frozen to -70 C in the absence of cryoprotectant without controlled reduction of temperature, others placed in HBSS with 10% DMSO but not subjected to freezing, whereas others were placed in vitrification media containing 5.5 M propylene glycol, 6.5 M glycerol, or 8 M ethylene glycol for 1 min before resuspension in fresh HBSS and inoculation into mice. Intact oocysts were frozen without controlled reduction of temperature directly to -70 C in HBSS containing no cryoprotectant or in HBSS that contained 10% DMSO. Others were cooled at -0.3 C per min from 4 C to -70 C in HBSS with 5% or 10% DMSO. Still others were cooled at a rate of -1 C per min until reaching -40 C and then cooled at -10 C per min until reaching -70 C in HBSS with 7.5% DMSO. Oocysts and sporozoites not exposed to cryoprotectants were inoculated into mice orally and rectally, respectively, for control purposes. Only unfrozen oocysts and sporozoites not exposed to cryoprotectant, and some of the unfrozen oocysts and sporozoites exposed to 10% DMSO, successfully established infections in mice.

Animals↗
PubMed1988

Computational study of an excitable dendritic spine.

1. A compartmental model was employed to investigate the electrical behavior of a dendritic spine having excitable membrane at the spine head. Here we used the Hodgkin and Huxley equations to generate excitable membrane properties; in some cases the kinetics were modified to get a longer duration action potential. Passive membrane was assumed for both the spine stem and the dendritic shaft. Synaptic input was modeled as a transient conductance increase (alpha-function) that lies in series with a battery (that corresponds to an excitatory or inhibitory synaptic equilibrium potential). 2. Threshold conditions for an action potential at the spine head membrane were found to be sensitive to the membrane properties at the spine head and to the conductance loading provided by the spine stem and the dendritic tree. Increasing either the number or the open times of the excitable channels had the effect of lowering spike threshold voltage. Increasing the spine stem resistance (RSS) or increasing the input resistance at the spinal base (RSB) also lowered the spike threshold voltage. Because a preexisting dendritic depolarization reduced the spine stem current, this lowered the spike threshold voltage, and this threshold was also shown to be sensitive to the distribution of membrane potential along the dendrite. 3. For each set of spine and dendritic parameters, there was an optimal range of RSS values for which the excitable properties at the spine head membrane resulted in maximal amplification of the dendritic excitatory postsynaptic potential (EPSP), when compared with that produced by a corresponding passive spine. This optimal range depended (with nonlinear sensitivity) on the properties of the voltage-gated channels at the spine head membrane. The maximal amplification found (for each of several sets of parameters) ranged from two to thirteen times. 4. Near this optimal range of RSS values, there was maximal (nonlinear) sensitivity of the dendritic EPSP amplitude to small changes in RSS. A minor decrease resulted in a subthreshold response at the spine head, and this resulted in a large decrease in the EPSP amplitude at the spine base. Increasing the value of RSS above this optimal range decreased the amount of spine stem current flowing to the spine base (by Ohm's law); this decreased the EPSP amplitude at the spine base. The demonstration of this optimum agrees with earlier expectations and results. 5. Excitable dendritic spines can be seen to provide an anatomical arrangement that economizes both excitable and synaptic channels. A small number of these channels (located in spine head membrane) can produce a large dendritic depolarization.(ABSTRACT TRUNCATED AT 400 WORDS)

Action Potentials↗
PubMed1985

Synaptic amplification by active membrane in dendritic spines.

The suspected functional role of dendritic spines as loci of neuronal plasticity (possibly memory and learning) is greatly enriched when active membrane properties are assumed at the spine head. Computations with reasonable electrical and structural parameter values (corresponding to an optimal range for spine stem resistance) show that an active spine head membrane can provide very significant synaptic amplification and also strongly non-linear properties that could modulate the integration of input from many afferent sources.

Cell Membrane↗
PubMed1985

Signal enhancement in distal cortical dendrites by means of interactions between active dendritic spines.

Pyramidal neurons in the cerebral cortex characteristically give rise to an apical dendrite, whose distal dendritic branches in layer I are covered with spines. These spines are known to be sites of synaptic connections, but the physiological properties of the spines and the functional significance of their responses are still largely unknown. The main function attributed thus far to these synaptic responses, situated at a great distance from the neuronal cell body, is slow background modulation of impulse output in the axon. In pursuing computer simulation analysis of electrical properties of dendrites, we have obtained results suggesting interactions between distal dendritic spines. If the heads of dendritic spines have excitable membrane properties, the spread of current from one or several spines could bring adjacent spines to their thresholds for impulse generation. This could give rise to a sequence of spine head action potentials, representing a saltatory propagation, from one or more excitable spine heads to nearby excitable spine heads, in the distal dendritic branches. Both the amplification due to several spine action potentials and the possibility of propagation into more proximal branches would increase the efficacy of distal synaptic inputs. Because of nonlinear dependence upon several modifiable parameters (such as spine stem resistance and membrane excitability) and upon the spatio-temporal pattern of synaptic input, such contingent synaptic enhancement would be particularly relevant to cortical functions underlying information processing and to plasticity underlying learning and memory.

Dendrites↗
PubMed1983

Factors that control amplitude of EPSPs in dendritic neurons.

We have used a computer-based mathematical model of alpha-motoneurons and of group Ia synaptic input to them, based on anatomical and electrophysiological data from the cat spinal cord, in order to examine the effects of variations in neuron size and input resistance and of conductance magnitude and duration on the generation of excitatory postsynaptic potentials (EPSPs). The first set of calculations were designed to test the possible role of nonlinear EPSP summation in producing a differential distribution of posttetanic potentiation of group Ia EPSPs, described in the preceding paper (25; see also Refs. 26, 27). The results suggest that the negative correlations observed between the degree of posttetanic potentiation of Ia EPSPs and initial (pretetanic) EPSP amplitude as well as with the input resistance of the postsynaptic motoneurons can be explained in part by the inherent non-linearity between conductance change and the resultant potential change at chemical synapses. In a second set of calculations, we used the same model system to evaluate the effects produced by variations in neuronal membrane area, input resistance, and specific membrane resistivity, as well as of the density of excitatory synaptic input on the peak amplitude of EPSPs. With parameters constrained to match the properties of alpha-motoneurons and group Ia synaptic input, EPSP amplitudes were most sensitive to changes in synaptic density and were much less sensitive to alterations in neuron input resistance and specific membrane resistivity when synaptic density was constant.

Animals↗
PubMed1974

Changes of action potential shape and velocity for changing core conductor geometry.

The theoretical changes in shape and velocity of an action potential were computed in regions of changing core conductor geometry. Step decrease and step increase of diameter, branch points, and gradual taper or flare of diameter were studied. Results showed increase of both velocity and peak height as the action potential approaches a point of step decrease. A step increase causes decrease of both velocity and peak height with approach; propagation may either fail, succeed with brief delay, or, with longer delay, succeed in both forward and reverse directions. With branching, both the shape and the dimensionless velocity, tautheta/lambda, remain unchanged when the d(3/2) values are matched. Without such matching, the changes of shape and dimensionless velocity of an action potential correspond to those found for step decrease or step increase of diameter. For regions of flare or taper, it was found (for a specific previously defined class) that velocity changed in proportion with the changing length constant. A simple formula was found to predict how this proportionality constant depends upon the amount of flare or taper.

Action Potentials↗
PubMed1974

Transient response in a dendritic neuron model for current injected at one branch.

Mathematical expressions are obtained for the response function corresponding to an instantaneous pulse of current injected to a single dendritic branch in a branched dendritic neuron model. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. The response function when used in a convolution formula enables one to compute the voltage transient at any specified point in the dendritic tree for an arbitrary current injection at a given input location. A particular numerical example, for a brief current injection at a branch terminal, illustrates the attenuation and delay characteristics of the depolarization peak as it spreads throughout the neuron model. In contrast to the severe attenuation of voltage transients from branch input sites to the soma, the fraction of total input charge actually delivered to the soma and other trees is calculated to be about one-half. This fraction is independent of the input time course. Other numerical examples, which compare a branch terminal input site with a soma input site, demonstrate that, for a given transient current injection, the peak depolarization is not proportional to the input resistance at the injection site and, for a given synaptic conductance transient, the effective synaptic driving potential can be significantly reduced, resulting in less synaptic current flow and charge, for a branch input site. Also, for the synaptic case, the two inputs are compared on the basis of the excitatory post-synaptic potential (EPSP) seen at the soma and the total charge delivered to the soma.

Animals↗
PubMed1973

Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model.

Mathematical solutions and numerical illustrations are presented for the steady-state distribution of membrane potential in an extensively branched neuron model, when steady electric current is injected into only one dendritic branch. Explicit expressions are obtained for input resistance at the branch input site and for voltage attenuation from the input site to the soma; expressions for AC steady-state input impedance and attenuation are also presented. The theoretical model assumes passive membrane properties and the equivalent cylinder constraint on branch diameters. Numerical examples illustrate how branch input resistance and steady attenuation depend upon the following: the number of dendritic trees, the orders of dendritic branching, the electrotonic length of the dendritic trees, the location of the dendritic input site, and the input resistance at the soma. The application to cat spinal motoneurons, and to other neuron types, is discussed. The effect of a large dendritic input resistance upon the amount of local membrane depolarization at the synaptic site, and upon the amount of depolarization reaching the soma, is illustrated and discussed; simple proportionality with input resistance does not hold, in general. Also, branch input resistance is shown to exceed the input resistance at the soma by an amount that is always less than the sum of core resistances along the path from the input site to the soma.

Dendrites↗
PubMed1969

Time constants and electrotonic length of membrane cylinders and neurons.

A theoretical basis is provided for the estimation of the electrotonic length of a membrane cylinder, or the effective electrotonic length of a whole neuron, from electrophysiological experiments. It depends upon the several time constants present in passive decay of membrane potential from an initially nonuniform distribution over the length. In addition to the well known passive membrane time constant, tau(m) = R(m)C(m), observed in the decay of a uniform membrane potential, there exist many smaller time constants that govern rapid equalization of membrane potential over the length. These time constants are present also in the transient response to a current step applied across the membrane at one location, such as the neuron soma. Similar time constants are derived when a lumped soma is coupled to one or more cylinders representing one or more dendritic trees. Different time constants are derived when a voltage clamp is applied at one location; the effects of both leaky and short-circuited termination are also derived. All of these time constants are demonstrated as consequences of mathematical boundary value problems. These results not only provide a basis for estimating electrotonic length, L = [unk]/lambda, but also provide a new basis for estimating the steady-state ratio, rho, of cylinder input conductance to soma membrane conductance.

Membrane Potentials↗
PubMed1969

Distributions of potential in cylindrical coordinates and time constants for a membrane cylinder.

A mathematical problem relating to membrane cylinders is stated and solved; its implications are illustrated and discussed. The problem concerns the volume distribution, in cylindrical coordinates, of the electric potential inside and outside a membrane cylinder of finite length (with sealed ends), during passive decay of an initially nonuniform membrane potential. The time constants for equalization with respect to the angle, theta, are shown to be typically about ten thousand times smaller than the time constant, tau(m) = R(m)C(m), for uniform passive membrane potential decay. The time constants for equalization with respect to length are shown to agree with those from one-dimensional cable theory; typically, they are smaller than tau(m) by a factor between 2 and 10. The relation of the membrane current density, I(m)(theta, x, t), to the values (at the outer membrane surface) of the extracellular potential phi(e)(r, theta, x, t) and of partial differential(2)phi(e)/ partial differentialx(2), is examined and it is shown that these quantities are not proportional to each other, in general; however, under certain specified conditions, all three of these quantities are proportional with each other and with phi(i)(r, theta, x, t) and partial differential(2)phi(i)/ partial differentialx(2) (at the inner membrane surface). The relation of these results to those of one-dimensional cable theory is discussed.

Membrane Potentials↗