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.

Leaky integrate-and-fire (LIF) network models are commonly used to study how the spiking dynamics of neural networks changes with stimuli, tasks or dynamic network states. However, neurophysiological studies in vivo often rather measure the mass activity of neuronal microcircuits with the local field potential (LFP). Given that LFPs are generated by spatially separated currents across the neuronal membrane, they cannot be computed directly from quantities defined in models of point-like LIF neurons. Here, we explore the best approximation for predicting the LFP based on standard output from point-neuron LIF networks. To search for this best “LFP proxy”, we compared LFP predictions from candidate proxies based on LIF network output (e.g, firing rates, membrane potentials, synaptic currents) with “ground-truth” LFP obtained when the LIF network synaptic input currents were injected into an analogous three-dimensional (3D) network model of multi-compartmental neurons with realistic morphology, spatial distributions of somata and synapses. We found that a specific fixed linear combination of the LIF synaptic currents provided an accurate LFP proxy, accounting for most of the variance of the LFP time course observed in the 3D network for all recording locations. This proxy performed well over a broad set of conditions, including substantial variations of the neuronal morphologies. Our results provide a simple formula for estimating the time course of the LFP from LIF network simulations in cases where a single pyramidal population dominates the LFP generation, and thereby facilitate quantitative comparison between computational models and experimental LFP recordings in vivo.

Correlated network activity plays a critical role in the development of many neural circuits. Purkinje cells are among the first neurons to populate the cerebellar cortex, where they sprout exuberant axon collaterals. Here we use multiple patch-clamp recordings targeted with two-photon microscopy to characterize monosynaptic connections between Purkinje cells of juvenile mice. We show that Purkinje cell axon collaterals project asymmetrically in the sagittal plane, directed away from the lobule apex. Based on our anatomical and physiological characterization of this connection, we construct a network model that robustly generates waves of activity traveling along chains of connected Purkinje cells. Consistent with the model, we observe traveling waves of activity in Purkinje cells in sagittal slices from young mice that require GABAA receptor-mediated transmission and intact Purkinje cell axon collaterals. These traveling waves are absent in adult animals, suggesting they play a developmental role in wiring the cerebellar cortical microcircuit.

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.

Dendrite morphology, a neuron's anatomical fingerprint, is a neuroscientist's asset in unveiling organizational principles in the brain. However, the genetic program encoding the morphological identity of a single dendrite remains a mystery. In order to obtain a formal understanding of dendritic branching, we studied distributions of morphological parameters in a group of four individually identifiable neurons of the fly visual system. We found that parameters relating to the branching topology were similar throughout all cells. Only parameters relating to the area covered by the dendrite were cell type specific. With these areas, artificial dendrites were grown based on optimization principles minimizing the amount of wiring and maximizing synaptic democracy. Although the same branching rule was used for all cells, this yielded dendritic structures virtually indistinguishable from their real counterparts. From these principles we derived a fully-automated model-based neuron reconstruction procedure validating the artificial branching rule. In conclusion, we suggest that the genetic program implementing neuronal branching could be constant in all cells whereas the one responsible for the dendrite spanning field should be cell specific.

The wide diversity of dendritic trees is one of the most striking features of neural circuits. Here we develop a general quantitative theory relating the total length of dendritic wiring to the number of branch points and synapses. We show that optimal wiring predicts a 2/3 power law between these measures. We demonstrate that the theory is consistent with data from a wide variety of neurons across many different species and helps define the computational compartments in dendritic trees. Our results imply fundamentally distinct design principles for dendritic arbors compared with vascular, bronchial, and botanical trees.computational neuroscience | branching | dendrite | morphology | minimum spanning tree O ne of the main roles of dendrites is to connect a neuron to its synaptic inputs. To interpret neural connectivity from morphological data, it is important to understand the relationship between dendrite shape and synaptic input distribution (1-4). 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 (6-8) 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. Understanding this relationship should provide crucial constraints for circuit structure and function. Here we directly test the hypothesis that neurons wire up a space in an optimal way by studying the consequences for dendrite length and branching complexity. We derive a simple equation that directly relates dendrite length with the number of branch points, dendrite spanning volume, and number of synapses. ResultsRelating Total Dendritic Length to Optimal Wiring. We assume that a dendritic tree of total length L connects n target points distributed over a volume V (Fig. 1A). Each target point occupies an average volume V =n. A tree that optimizes wiring will tend to connect points to their nearest neighbors, which are on average located at distances proportional to ðV =nÞ 1=3 . We need at least n such dendritic sections to make up the tree. The total length L of these sections sums up toThis result shows that a 2/3 power law relationship between L and n (12) provides a lower bound for the total dendritic length, where c is a proportionality constant. Approximating the volume around each target point by a sphere, then c ¼ ð3=4πÞ 1=3 , and each dendritic section corresponds to the radius of a sphere, giving(SI Text and Fig. S1). Importantly, assuming a constant ratio between the number of branch points, bp and the number of target points (which is addressed later), this assumption also results in a 2/3 power law between wiring length and the number of branch points. Supporting these intuitive derivations of power laws, there ...

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