One Rule to Grow Them All: A General Theory of Neuronal Branching and Its Practical Application

Abstract

Understanding the principles governing axonal and dendritic branching is essential for unravelling the functionality of single neurons and the way in which they connect. Nevertheless, no formalism has yet been described which can capture the general features of neuronal branching. Here we propose such a formalism, which is derived from the expression of dendritic arborizations as locally optimized graphs. Inspired by Ramón y Cajal's laws of conservation of cytoplasm and conduction time in neural circuitry, we show that this graphical representation can be used to optimize these variables. This approach allows us to generate synthetic branching geometries which replicate morphological features of any tested neuron. The essential structure of a neuronal tree is thereby captured by the density profile of its spanning field and by a single parameter, a balancing factor weighing the costs for material and conduction time. This balancing factor determines a neuron's electrotonic compartmentalization. Additions to this rule, when required in the construction process, can be directly attributed to developmental processes or a neuron's computational role within its neural circuit. The simulations presented here are implemented in an open-source software package, the “TREES toolbox,” which provides a general set of tools for analyzing, manipulating, and generating dendritic structure, including a tool to create synthetic members of any particular cell group and an approach for a model-based supervised automatic morphological reconstruction from fluorescent image stacks. These approaches provide new insights into the constraints governing dendritic architectures. They also provide a novel framework for modelling and analyzing neuronal branching structures and for constructing realistic synthetic neural networks.

On the functions, mechanisms, and malfunctions of intracortical contextual modulation

“…Ca 2+ indicates the apical integration site initiating calcium spikes that amplify response. Source: Layer 5 cell created with the TREES toolbox (Cuntz et al, 2010), courtesy of Hermann Cuntz.…”

A scaling law derived from optimal dendritic wiring

“…To interpret neural connectivity from morphological data, it is important to understand the relationship between dendrite shape and synaptic input distribution (1)(2)(3)(4). As early as the end of the 19th century (5), it was suggested that dendrites optimize connectivity in terms of cable length and conduction time costs, and a number of recent studies have supported the idea that optimal wiring explains dendritic branching patterns using simulations (6)(7)(8) or by reasoning from first principles (1, 2, 9, 10). However, although dendrite length is the most common measure for molecular studies of dendritic growth (11), its relationship to dendritic branching and the number of synaptic contacts has not been elucidated.…”

Optimal experimental design for sampling voltage on dendritic trees in the low-SNR regime

“…More generally, a heuristic approximation to the greedy algorithm which uses only the first iteration of the variance reductions performs as well as the full greedy approach, due to the local effects of each observation. This heuristic produces an order of magnitude speed increase, making the proposed methods potentially tractable in experimental settings in which dendritic reconstructions may be obtained in living preparations (Losavio et al, 2008;Cuntz et al, 2010).…”

“…We acknowledge that on-line reconstruction of dendritic trees remains a challenging task; however, significant research into automated dendrite reconstruction methods is underway. For example, the powerful TREES MATLAB toolbox was recently introduced for exactly this type of task (Cuntz et al, 2010). Similarly, (Losavio et al, 2008) demonstrate excellent good qualitative and quantitative performance using their ORION software for fully automatic morphological reconstruction.…”