Optimization principles of dendritic structure

Abstract

Background: Dendrites are the most conspicuous feature of neurons. However, the principles determining their structure are poorly understood. By employing cable theory and, for the first time, graph theory, we describe dendritic anatomy solely on the basis of optimizing synaptic efficacy with minimal resources.

Results: We show that dendritic branching topology can be well described by minimizing the path length from the neuron's dendritic root to each of its synaptic inputs while constraining the total length of wiring. Tapering of diameter toward the dendrite tip--a feature of many neurons--optimizes charge transfer from all dendritic synapses to the dendritic root while housekeeping the amount of dendrite volume. As an example, we show how dendrites of fly neurons can be closely reconstructed based on these two principles alone.

Figure 1

Equalization of charge transfer in a model of a reconstructed HSS cell of the fly visual system. (A) Current transfer from all dendritic locations to the dendritic root. (B) Local input conductance (inverse of input resistance, 1/R_IN(x)). (C) Ratio of voltage at the dendritic root and the voltage generated at the dendrite locations, where the input current is applied. (D) Voltage ratios plotted against the inverse of the local input resistances follow a linear relationship expressing the proportionality suggested by equation (1). Colour scale in A-C ranges from 0 (blue), to maximal (red) current transfer (A), input conductance (B) and voltage ratio (C). Reference point for dendritic root is indicated by an arrow in A.

Figure 2

Diameter optimization for optimal current transfer in simplified dendritic cable models. (A) Optimal diameters for maximal charge transfer in four unbranched cable models, each composed of six segments of equal length (200, 300, 400 and 500 μm from bottom to top) and all attached to a cylindrical axon (20 μm in diameter, 2 mm long). Scale x : 1 mm y : 10 μm. Red: diameter tapering. (B) Diameter optimization in branched structures with six 300 μm-long segments each, sorted by error size (marked values) as defined in Equation (4). Part of the axon at the bottom of each tree is cut for presentation. (C) Dendrite diameter tapering for all models shown in B, when the path from root to terminal is normalized.

Figure 3

Optimal tapering follows quadratic decay. (A) Normalized optimal diameters (black dots) in cable pieces of different lengths divided into 10 segments each. In all cases a quadratic equation (red lines) could well describe the course of tapering. The fixed diameter of the first segment corresponding to the axon piece is not shown. (B) Changing the size of the leak (length of the first segment) did not alter the relative course of tapering.

Figure 4

Rules for optimizing dendritic branching. (A) Minimum spanning tree for randomly distributed points in the convex hull of dendritic territory of HSS neurons. Longest path is drawn in bold. (B) Extended minimum spanning tree, minimizing both the total path length from all synaptic locations to the root and the total wiring length. (C) Branching and termination points as putative sites for synaptic contacts for the HSS dendritic tree shown in Figure 1D, same algorithm as in B, using the putative synaptic sites from C. Dendritic root is marked with a circle. (E, F) Dendrograms representing the topology of the reconstructed HSS neuron and the artificially constructed dendritic tree shown in D, respectively.

Figure 5

Validation and quantification for optimizing parameters. (A) Current transfer to the root in artificially constructed dendritic tree shown in Figure 4D with tapering diameters. (B) Current transfer in reconstructed HSS dendritic tree assuming constant (2.3 μm) diameter for all dendritic branches (the maximal electrotonic distance in this case was similar to that of the HSS model with tapering dendritic diameter). (C) Current transfer in reconstructed dendrite, in which dendritic topology only minimized the total wiring from root to all points shown in Figure 4C but with tapering dendritic tree. (compare these graphs with Figure 1A, all having the same colour scale). In all cases the axon of the HSS model was appended to the artificially reconstructed dendrites. (D) Distributions of normalized current transfer in the different nodes of the dendrites of the real HSS model (black) and the three reconstructed dendrites (A – grey, B – red, C – blue).