Wilfrid Rall’s research while affiliated with The National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health and other places

The advent of microelectrode recordings from neurons with branching dendritic trees presented a new problem for neurophysiologists: how to interpret the data obtained from these non-isopotential structures. An important step was the development of mathematical models of dendrites. This chapter provides an historical perspective of the development of such models, paying attention to the progression from passive, linear models to the more complex case of neurons with extensive branching and nonlinear properties.

The sections in this article are:

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.Trends Neurosci. (1998) 21, 453–460

This chapter focuses on different electrotonic models of neuronal dendrites and single neuron computation to reduce the number degrees of freedom. The areas that help in reducing the degrees of freedom include the importance for modeling studies of having good estimates of the electrotonic structure of a cell, the dynamic range of computational possibilities available to a neuron, by considering its possible resting states, assuming that a real neuron ever can be considered to be at rest, and variables that may be important for producing modification in dendritic spines. Dendritic models concerned with computation must make assumptions about the morphological and electrotonic structure of the neuronal dendrites. The dynamic range of computational possibilities for a neuron is immense.

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=