Electrotonic structure of dendrites plays a critical role in neuronal computation and plasticity. In this article we develop two novel measures of electrotonic structure that describe intraneuronal signaling in dendrites of arbitrary geometry. The log-attenuation Lij measures the efficacy, and the propagation delay Pij the speed, of signal transfer between any two points i and j. These measures are additive, in the sense that if j lies between i and k, the total distance Lik is just the sum of the partial distances: Lik = Lij + Ljk, and similarly Pik = Pij + Pjk. This property serves as the basis for the morphoelectrotonic transform (MET), a graphical mapping from morphological into electrotonic space. In a MET, either Pij or Lij replace anatomical distance as the fundamental unit and so provide direct functional measures of intraneuronal signaling. The analysis holds for arbitrary transient signals, even those generated by nonlinear conductance changes underlying both synaptic and action potentials. Depending on input location and the measure of interest, a single neuron admits many METs, each emphasizing different functional consequences of the dendritic electrotonic structure. Using a single layer 5 cortical pyramidal neuron, we illustrate a collection of METs that lead to a deeper understanding of the electrical behavior of its dendritic tree. We then compare this cortical cell to representative neurons from other brain regions (cortical layer 2/3 pyramidal, region CA1 hippocampal pyramidal, and cerebellar Purkinje). Finally, we apply the MET to electrical signaling in dendritic spines, and extend this analysis to calcium signaling within spines. Our results demonstrate that the MET provides a powerful tool for obtaining a rapid and intuitive grasp of the functional properties of dendritic trees.

SUMMARYAND CONCLUSIONS1. The spread of electrical signals in pyramidal neurons from the CA1 field of rat hippocampus was investigated through multicompartmental modeling based on three-dimensional morphometric reconstructions of four of these cells. These models were used to dissect the electrotonic architecture of these neurons, and to evaluate the equivalent cylinder approach that this laboratory and others have previously applied to them. Robustness of results was verified by the use of wide ranges of values of specific membrane resistance (R,) and cytoplasmic resistivity.2. The anatomy exhibited extreme departures from a key assumption of the equivalent cylinder model, the so-called "3/2 power law. ' ' 3. The compartmental models showed that the frequency distribution of steady-state electrotonic distances between the soma and the dendritic terminations was multimodal, with a large range and a sizeable coefficient of variation. This violated another central assumption of the equivalent cylinder model, namely, that all terminations are electrotonically equidistant from the soma. This finding, which was observed both for "centrifugal"(away from the soma) and ' 'centripetal" (toward the soma) spread of electrical signals, indicates that the concept of an equivalent electrotonic length for the whole dendritic tree is not appropriate for these neurons.4. The multiple peaks in the electrotonic distance distributions, whether for centrifugal or centripetal voltage transfer, were clearly related to the laminar organization of synaptic afferents in the CA1 region.5. The results in the three preceding paragraphs reveal how little of the electrotonic architecture of these neurons is captured by a simple equivalent cylinder model. The multicompartmental model is more appropriate for exploring synaptic signaling and transient events in CA1 pyramidal neurons.6. There was significant attenuation of synaptic potential, current, and charge as they spread from the dendritic tree to the soma. Charge suffered the least and voltage suffered the most attenuation. Attenuation depended weakly on R, and strongly on synaptic location. Delay of time to peak was more distorted for voltage than for current and was more affected by R,.7. Adequate space clamp is not possible for most of the synapses on these cells. Application of a somatic voltage clamp had no significant effect on voltage transients in the subsynaptic membrane.8. The possible existence of steep voltage gradients within the dendritic tree is consistent with the idea that there can be some degree of local processing and that different regions of the neuron may function semiautonomously. These spatial gradients are potentially relevant to synaptic plasticity in the hippocampus, and they also suggest caution in interpreting some neurophysiological results.