Hermann Cuntz

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

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 Cajals 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 neurons electrotonic compartmentalization. Additions to this rule, when required in the construction process, can be directly attributed to developmental processes or a neurons 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.

Optimization principles of dendritic structure.

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. We show that dendritic branching topology can be well described by minimizing the path length from the neurons 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.

The morphological identity of insect dendrites

Dendrite morphology, a neurons anatomical fingerprint, is a neuroscientists 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.

Robust coding of flow-field parameters by axo-axonal gap junctions between fly visual interneurons

Complex flight maneuvers require a sophisticated system to exploit the optic flow resulting from moving images of the environment projected onto the retina. In the flys visual course control center, the lobula plate, 10 so-called vertical system (VS) cells are thought to match, with their complex receptive fields, the optic flow resulting from rotation around different body axes. However, signals of single VS cells are unreliable indicators of such optic flow parameters in the context of their noisy, texture-dependent input from local motion measurements. Here we propose an alternative encoding scheme based on network simulations of biophysically realistic compartmental models of VS cells. The simulations incorporate recent data about the highly selective connectivity between VS cells consisting of an electrical axo-axonal coupling between adjacent cells and a reciprocal inhibition between the most distant cells. We find that this particular wiring performs a linear interpolation between the output signals of VS cells, leading to a robust representation of the axis of rotation even in the presence of textureless patches of the visual surround.

A scaling law derived from optimal dendritic wiring

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.

The TREES Toolbox-Probing the Basis of Axonal and Dendritic Branching

It has now been 100 years since Ramon y Cajal described the remarkable diversity of neuronal branching. Only recently, however, have a number of rigorous formalisms emerged providing an accurate quantitative description of axonal and dendritic morphologies. We have launched a freely distributed open-source software package, the TREES toolbox, written in Matlab (Mathworks, Natick, MA), in order to help to pool together the resources offered by a wide variety of novel approaches to studying dendritic and axonal branching that have recently become available. This package introduces a simple general description of neuronal morphology as a graph and provides the basic tools to edit, visualize and analyze neuronal trees in the basis of this description. We then implement our own approach, assuming that neuronal branching can largely be expressed by local optimization of total wiring and conduction distances. We provide the corresponding modular extendable tools to automatically reconstruct neuronal branching from microscopy image stacks and to generate synthetic branched structures. The package is complemented by an extensive user interface to facilitate the generation, visualization and editing of neuronal tree structures. The TREES toolbox is structured to make it easy for other groups to integrate their own code in order to implement their own specific applications. Accurate predictions of computation in single neurons are nowadays well known to require detailed morphological representations. Tools for compartmental modelling such as NEURON, Genesis and neuroConstruct have recently facilitated the modelling of small and large neural circuits involving detailed compartmental models of the neurons. Also, a new trend highlighting the importance of morphology for better understanding of network connectivity adds to the appeal of acquiring morphologies in their full level of detail. However, obtaining the morphologies of all neurons present in one network currently remains an insurmountable hurdle. On the other hand, a number of computational methods have

Computing the local field potential (LFP) from integrate-and-fire network models

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.

Neural image processing by dendritic networks

Convolution is one of the most common operations in image processing. Based on experimental findings on motion-sensitive visual interneurons of the fly, we show by realistic compartmental modeling that a dendritic network can implement this operation. In a first step, dendritic electrical coupling between two cells spatially blurs the original motion input. The blurred motion image is then passed onto a third cell via inhibitory dendritic synapses resulting in a sharpening of the signal. This enhancement of motion contrast may be the central element of figure–ground discrimination based on relative motion in the fly.

A New Approach for Determining Phase Response Curves Reveals that Purkinje Cells Can Act as Perfect Integrators

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

Eigenanalysis of a neural network for optic flow processing

Flies gain information about self-motion during free flight by processing images of the environment moving across their retina. The visual course control center in the brain of the blowfly contains, among others, a population of ten neurons, the so-called vertical system (VS) cells that are mainly sensitive to downward motion. VS cells are assumed to encode information about rotational optic flow induced by self-motion (Krapp and Hengstenberg 1996 Nature 384 463?6). Recent evidence supports a connectivity scheme between the VS cells where neurons with neighboring receptive fields are connected to each other by electrical synapses at the axonal terminals, whereas the boundary neurons in the network are reciprocally coupled via inhibitory synapses (Haag and Borst 2004 Nat. Neurosci.?7 628?34; Farrow et al 2005 J. Neurosci.?25 3985?93; Cuntz et al 2007 Proc. Natl Acad. Sci. USA). Here, we investigate the functional properties of the VS network and its connectivity scheme by reducing a biophysically realistic network to a simplified model, where each cell is represented by a dendritic and axonal compartment only. Eigenanalysis of this model reveals that the whole population of VS cells projects the synaptic input provided from local motion detectors on to its behaviorally relevant components. The two major eigenvectors consist of a horizontal and a slanted line representing the distribution of vertical motion components across the flys azimuth. They are, thus, ideally suited for reliably encoding translational and rotational whole-field optic flow induced by respective flight maneuvers. The dimensionality reduction compensates for the contrast and texture dependence of the local motion detectors of the correlation-type, which becomes particularly pronounced when confronted with natural images and their highly inhomogeneous contrast distribution.