The morphoelectrotonic transform: a graphical approach to dendritic function

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.