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Conceived and designed the experiments: HC FF AB MH. Performed the experiments: HC FF. Analyzed the data: HC FF. Wrote the paper: HC FF AB MH.

The authors have declared that no competing interests exist.

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.

More than a century has passed since Ramón y Cajal presented a set of fundamental biological laws of neuronal branching. He described how the shape of the core elements of the neural circuitry – axons and dendrites – are constrained by physical parameters such as space, cytoplasmic volume, and conduction time. The existence of these laws enabled him to organize his histological observations, to formulate the neuron doctrine, and to infer directionality in signal flow in the nervous system. We show that Cajal's principles can be used computationally to generate synthetic neural circuits. These principles rigorously constrain the shape of real neuronal structures, providing direct validation of his theories. At the same time, this strategy provides us with a powerful set of tools for generating synthetic neurons, as well as a model-based approach for automated reconstructions of neuronal trees from confocal image stacks.

Neuronal circuits are composed of a large variety of branched structures – axons and dendrites – forming a highly entangled web, reminiscent of a stochastic fractal

In parallel, the development of compartmental modelling techniques based on the theories of Wilfrid Rall have highlighted the importance of a neuron's precise branching morphology for its electrophysiological properties

These two lines of investigation raise the question as to whether computation plays the determining role in shaping the morphological appearance of neuronal branching structures. Alternatively, the simple laws of material and conduction time preservation of Ramón y Cajal could have more influence.

Using computational techniques it has become possible to construct synthetic neuronal structures

Here we explore the algorithm's general applicability and its potential to describe any type of dendritic branching. If the algorithm is sufficient to accurately mimic the essential structure of neuronal circuitry we can resolve the relative importance of computation and wiring constraints in shaping neuronal morphology. We can then claim that Ramón y Cajal's laws are sufficient for shaping neuronal morphology. Specific computation will then only play a subordinate role in determining a neuron's branching pattern. We show here that while Cajal's laws do represent a strict constraint on neuronal branching, a neuronal morphology has a certain freedom to operate within these constraints. Firstly, by adjusting the balance between the two wiring costs, a dendrite can efficiently set its electrotonic compartmentalization, a quantity attributable to computation. Secondly, the density profile of the spanning field in which a dendrite grows determines its shape dramatically. Thirdly, a few weaker constraints such as the suppression of multifurcations, the addition of spatial jitter or the sequential growth of sub-regions of a dendrite are helpful for reproducing the dendritic branching patterns of particular preparations. These additional constraints might shed light on further functional, computational, developmental or network determinants for certain dendritic structures, and more of these will follow when applying our method to many more preparations. Moreover, the simple principles presented in this study can be used to efficiently edit, visualize, and analyze neuronal trees. Finally, these approaches allow one to generate highly realistic synthetic branched structures, and to automatically reconstruct neuronal branching from microscopy image stacks.

Before generating complex neuronal morphologies, a simple formalism is required to compare and assess natural and synthetic neuronal trees. We derive such a formalism from graph theory: a neuronal tree is a graph which connects a set of labelled nodes via directed edges away from a root labelled “1” (

(A) The tree consists of cylinders or frusta (red) connecting each two nodes along the directed edges (away from the root node, arrows). Branch points and termination points represent the topology (topological points). A branch is a set of continuation points between two topological points. The labelling of the nodes is unique following three principles: hierarchical sorting, continuous labelling preserving sub-tree consistency and topological sorting (see text). (B) Rearrangement of node locations on a sample tree. Examples of equidistant node redistribution resulting in 10 or 20 µm resampling and a 20 µm resampling including length conservation (see text and “resampling” section of

When node labelling and distribution are attributed in a distinct manner, a unique representation of the tree is provided. By rearranging the node locations of a sample tree (

In order to incorporate Cajal's hypotheses about wiring optimization in our theoretical description of a neuronal tree, we implemented optimization procedures known from graph theory. This approach was previously shown to be successful for generating synthetic dendritic structures of fly interneurons

(A) The growth described by an extended minimum spanning tree algorithm (see text). Unconnected carrier points (red) are connected one by one to the nodes of a tree (black). Red dashed lines indicate three sample Euclidean distances to the nodes of the tree for sample point

In the following, we will apply this method of creating optimized graphs to reproduce morphologies in various neuronal preparations. The main effort will be to obtain an adequate set of carrier points for the application of the algorithm; this will prove to depend strongly on the density profile of the spanning field in the respective geometries. When additional constraints will be required in generating neurons in specific brain areas, this will provide clues pointing to actual computational or functional features of neuronal morphology.

Whereas our previous work was limited to insect dendrites

(A) Reconstruction of a starburst amacrine cell in the inner plexiform layer of the rabbit retina (data from

We next studied dendrites of hippocampal granule cells, which fill a three-dimensional volume rather than a plane (template data from

(A) Reconstructions of four sample hippocampal granule cells (data from

The two cases described in

(A) After rotating rat somatosensory cortex layer 2/3, 4 and 5 pyramidal cells to overlap, the limits of their individual regions were extracted: black shaded boxes show the mean limits in XY for the apical region; the black empty boxes delineate one standard deviation away from the mean. Corresponding red boxes duplicate this procedure for the basal dendrites. Cells are then scaled region-by-region to the mean limits of each region. Overlay colours describe local density (colormap see

Using this approach, dendritic morphologies of different pyramidal cells (

(A), (B) and (C) show sample synthetically generated model cells of layer 2/3, layer 4 and layer 5 cortical pyramidal cells respectively, all grown using the general strategy described in

Distributions of branching parameters to compare the synthetic pyramidal cells with their real counterparts are shown in

While branching statistics of starburst amacrine cells and hippocampal granule cells were moderately homogeneous, pyramidal cells exhibited stronger variations. Balancing factors leading to reasonable branching statistics ranged from 0.4 to 0.7. In the following we compare branching parameter distributions as in

The local circuitry ultimately determines the context in which neuronal trees grow. There are global boundaries given by the neural tissue such as layers, topography or physical borders. However, competition for inputs between neighbouring neurons also seems to play a major role. Competition is easily implementable in the greedy growth algorithm introduced here because of the iterative nature of the algorithm. This can then be considered as a greedy extension of the growth algorithm and should be applicable in the network context. When grown under competitive conditions in which trees connect to a carrier point one after the other, the immediate consequence is spatial tiling. This can be seen in 2D for example when trees were grown from starting points on a spatial grid in a homogeneous substrate of random carrier points (

(A) Nine synthetic neuronal trees grown competitively on a sample square substrate of homogeneously distributed random carrier points: the competitive greedy growth results automatically in tiling of the available space. (B) Three out of 16 neuronal trees grown competitively on random carrier points distributed on a ring: this simulates well the sharp borders of Purkinje cells in the cerebellum. Whether Purkinje cell dendrites actually tile in sagittal planes of the cerebellum remains to be determined. (C) Hippocampal granule cells from

As mentioned above, the network context also plays a major role in governing neuronal spanning fields and their density profiles. Arranging the hippocampal granule cells developed in

Finally, as shown previously

(A) Example of an additional application of the algorithm: automated model-based tree reconstruction from image stacks. Maximum intensity projection of tiled image stacks containing a sample sub-tree of a fluorescently labelled neuronal tree. Blue overlay in top panel corresponds to the output of a non-linear thresholding. The resulting binary matrix is then reduced to single points in space (green dots) via a skeletonization procedure. After a distance graph is obtained which describes the probability of a connection between these points due to the image data the points are used as carrier points for the growth algorithm to obtain the corresponding tree using the distance graph as an additional cost factor. After unlikely branches are removed the underlying tree structure is captured (green tree structure in the lower panel, see text for more detail; note absence of scale bar since this a sample image stack). (B) Maximum intensity projections of tiled 2-photon fluorescent image stacks acquired at 820 nm from primary visual cortex of a p13 JAX transgenic mouse (strain #007677,

We present a new framework for understanding dendritic branching in neurons based on the use of graph theory. Our results demonstrate that the laws of conservation of cytoplasm and conduction time formulated by Ramón y Cajal from simple observation represent a fundamental constraint to dendritic branching. Generation of synthetic tree structures using a simple algorithm derived from these constraints creates highly realistic neuronal branching structures across a wide range of neuronal types. Nevertheless, there remains flexibility within these natural constraints. Notably, the weighting of both components, the balancing factor – which determines a neuron's electrotonic compartmentalization – is an adjustable parameter and can differ from one cell type to the next. Within these constraints, neuronal processes can grow and adapt depending on their specific functionality. Most strikingly, the spanning field and density profile are key determinants of nerve cell individuality and depend on the sharp physical boundaries of the tissue and on the network input-output topography. Finally, additions to this rule are required in certain cases such as the suppression of multifurcations and the addition of spatial jitter. These might relate to the developmental bio-mechanistic growth process or specific computational features and we are confident we will find many more in further preparations.

The spanning field in which a dendrite grows plays a major role in defining the computational and functional features of axons and dendrites. This is reflected in its importance in the process of accurately reproducing single cell morphologies. Furthermore, to replicate dendrite regions such as the apical tuft of layer 5 pyramidal neurons or the primary dendrites of Purkinje cells, a timed growth process was required in which subparts of a region were grown in a second step. This could indicate a functional constraint governing neuronal outgrowth in these cells. However, two limitations of the greedy algorithm must be considered. Firstly, the growth process does not guarantee a global optimum since it is based on an algorithm which optimizes at the local level, adding carrier points one by one. Secondly, it does not involve volumetric considerations. Both cable diameter when optimizing the amount of material used and space packing issues in conjunction with axons and dendrites of other neurons as well as with glia cells are known to play a role in determining wiring properties in the brain

We show that spatial tiling as observed in many dendritic structures (

Based on the formalisms of optimized graphs, we have derived several new ways of representing dendritic structure and function. First, we show that graph resampling and labelling order lead to an objective representation for electrotonic compartmentalization. Simplified models which still faithfully represent the compartmentalization behaviour can be obtained with such a process. Second, taking advantage of their scalability, we derived generalized spanning fields and their density profile descriptions. These representations may be useful for comparing branching structures of different neuronal cell types. We show here how these can be useful for generating synthetic neuronal tree clones. Finally, as mentioned previously

In summary, we find that a simple growth algorithm which optimizes total cable length and the path length from any point to the root in an iterative fashion can generate synthetic dendritic trees that are indistinguishable from their real counterparts for a wide variety of neurons. This represents a direct validation of the fundamental constraints on neuronal circuit organization described originally by Cajal. Furthermore, this approach provides a new framework for understanding dendritic tiling, which is a direct consequence of using this algorithm. The availability of these tools in a comprehensive software package (the TREES toolbox; see

We have developed a software toolbox, the TREES toolbox (deposited at

The labelling of the nodes of a tree should be unique in order to, for example, compare the graphs of two trees topologically or electrotonically. In order to obtain such a unique labelling, nodes were first sorted according to their topological depth, chosen here to be the sum of the path length values of all children. Each node was then inserted in that order into a one-dimensional string, one-by-one directly behind its direct parent node. Subsequently, the resulting string of labels was mapped back onto the nodes of the tree. We refer to the Supplementary Information for more details on this subject (

The direct comparison of two trees along strict criteria also requires a unique distribution of node locations on the graph. We redistributed nodes on a tree structure with equal inter-nodal distances, a process we termed resampling. Each terminal branch was first lengthened by half the sampling distance. Then, starting at the root, extra nodes were positioned at integer multiples of the sampling distance along the path of the tree. All other nodes were then removed while maintaining the connectivity. During this process all segments become shorter or remain the same length; this is because a wriggly path is simplified by a straight line (which is by definition always shorter). In order to preserve the total branching length and the electrotonic properties all segments were then elongated to the given sampling length (

Rearranging the metrics of a tree after sorting the labels leads to generation of its unique electrotonic equivalent tree. To obtain the equivalent tree, its metrics were rearranged according to a circular dendrogram where the angle towards which a segment is directed within a circle around the root corresponds to the value of the label of its target node.

To obtain the electrotonic signature, the conductance matrix describing the axial conductances along the edges of the graph (following the adjacency matrix structure) and the membrane leak conductances (on the diagonal of the matrix) was simply inverted. The result is a potential matrix (in mV) corresponding to the resulting steady-state potential in one node when 1 nA current was injected in another node, i.e. the current transfer from any node to another. The passive axial and membrane conductances were 100 Ωcm and 2000 Ωcm^{2} for the sample insect dendrite sub-tree in ^{2} for the layer 5 pyramidal cells in

The branching growth was implemented as a greedy algorithm

Two-photon microscopy 3D image stacks containing neurons filled with a fluorescent dye or expressing a fluorescent protein were subject to local brightness level thresholding. After 3D skeletonization and sparsening of the resulting carrier points, these were submitted to the same greedy algorithm (started at a user defined dendrite root location) as used for obtaining synthetic dendrites. In the case of multiple entangled neurons as in

The TREES toolbox documentation.

(6.97 MB PDF)

We thank Karl Farrow, Yihwa Kim, Philipp Rautenberg, Martin O'Reilly and Sarah Rieubland for testing parts of the TREES toolbox software package; Kate Buchanan and Jesper Sjöström for providing the filled neurons in