# A scaling law derived from optimal dendritic wiring

^{a}Wolfson Institute for Biomedical Research and Department of Neuroscience, Physiology, and Pharmacology, University College London, London WC1E 6BT, United Kingdom;^{b}Institute of Clinical Neuroanatomy, Neuroscience Center, Goethe-University, D-60590 Frankfurt/Main, Germany; and^{c}Ernst Strüngmann Institute for Neuroscience in Cooperation with Max Planck Society, D-60528 Frankfurt/Main, Germany

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Edited by Eve Marder, Brandeis University, Waltham, MA, and approved May 18, 2012 (received for review January 9, 2012)

## Abstract

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.

One 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.

## Results

### Relating Total Dendritic Length to Optimal Wiring.

We assume that a dendritic tree of total length connects target points distributed over a volume (Fig. 1*A*). Each target point occupies an average volume . A tree that optimizes wiring will tend to connect points to their nearest neighbors, which are on average located at distances proportional to . We need at least such dendritic sections to make up the tree. The total length of these sections sums up to

This result shows that a 2/3 power law relationship between and (12) provides a lower bound for the total dendritic length, where is a proportionality constant. Approximating the volume around each target point by a sphere, then , 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 have been several proofs (12, 13) that in a minimum spanning tree in dimensions—the canonical model of a tree constructed to minimize wiring length— scales as a power of the number of target points or branch points. In summary, the wiring minimization hypothesis predicts a 2/3 power law between and *bp* and a 2/3 power law between and . By contrast, a process that randomly connects target points without optimizing wiring yields a power law with exponent 1. More interestingly, a canonical model of biological fractal trees previously introduced by West et al. (14) predicts a 4/3 power law between and or *bp* (for simple proof see *SI Text* and Fig. S2).

To study the scaling properties of neuronal dendritic trees, we took advantage of synthetic dendrites generated using an extension of the minimum spanning tree (MST) algorithm that we have previously shown can reproduce a wide range of dendrite morphologies (6⇓–8). In addition to minimizing the total length to connect a set of target points to a tree as discussed above, this procedure introduces a cost to minimize all path lengths from any target point toward the root along the tree. This additional cost is parameterized with the balancing factor *bf*, and we previously showed that *bf* values between 0.1 and 0.85 reproduce realistic dendritic morphologies (*Materials and Methods*). When target points were randomly distributed within a spherical volume and connected to a tree to minimize these costs, we found that Eq. 1 provides a tight lower bound for total dendritic length, particularly for low *bf* (Fig. 1*B*). With increasing *bf* the exponent in the power law increased from 0.66 to 0.72 for *bf* = 0.9, the maximal realistic balancing factor. The respective mean square errors between the curves from the model and our predicted equation increased from 1% to 5.2% as *bf* was increased.

### Relationship Between Number of Branch Points and Target Points.

The number of branch points grew proportionally to the number of target points (Fig. 1*C*) with proportionality constant *bp*/*n*. Such a proportional relationship is not surprising: It occurs in simple *n*-ary trees as well as in randomly generated trees (15). However, the particular value of the proportionality constant emerges as another direct consequence of optimal wiring as shown previously by Steele et al. (13), for the simplest case of *bf* = 0 (*SI Text* and Fig. S3)—although no analytical formula exists to compute it and it must be derived empirically. Surprisingly, we found that *bp*/*n* was independent of the geometrical arrangement of target point distributions. Specifically, the ratio did not change regardless of whether target points were distributed inhomogeneously, the root was displaced from the center of the sphere, or the physical boundaries of the sphere were replaced by those of a cube (Fig. 1*C*). However, the ratio depended in a linear manner on the model parameter *bf* (Fig. 1*C*, *Top Inset*), which we have previously shown to vary between different classes of dendrites (8). Importantly therefore, when two of the three quantities *bp*, , or *bf* are known, the third can therefore be inferred independently of tree conditions or metric scale. The 2/3 power is therefore equally present between *bp* and in the model (Fig. 1*D*), with powers ranging from 0.66 to 0.72 and a mean square error below 2% between the prediction and the model for all *bf*. To summarize, our algorithm that was shown previously (8) to reproduce many realistic neuronal morphologies, confirms the presence of the 2/3 power law between and and between and *bp* in synthetic trees.

We next tested whether the morphological power law relationship also has functional consequences. Using a model for scaling of dendritic diameters (6), we determined the passive electrotonic properties of the dendritic trees in our sample. The number of independent electrical compartments (Fig. 1*E*, fitted power: 0.67 for *bf* = 0.2) and total electrotonic length (Fig. S4) were also found to scale with the number of puncta governed by a 2/3 power law when *bf* > 0 (*Materials and Methods* and *SI Text*).

### General Applicability of the 2/3 Power Law.

To test for the presence of our predicted power law in real neuronal morphologies, we analyzed all available dendritic trees of all different cell types from the NeuroMorpho database (16). In these reconstructions, it is unclear what the target points of the dendrites are, but branch points can unambiguously be counted. We therefore plotted the wiring length of these cells against the number of branch points (Fig. 2). The power that best described the trends between dendrite length and number of branch points obtained for all cell types individually is very close to 2/3, 0.72 ± 0.10, and residual norms confirmed the goodness of fit (Fig. 2, *Right*). We replaced by *bp* in Eq. 1 and compensated for the decrease in points by using the *bp*/*n* ratio derived from the simulations in Fig. 1*C* to obtain our lower bound for dendrite length (Fig. 2, *Left*, black line). Although providing a lower bound, the resulting equation well described the data (with a root mean square error of 6.4%) and was very similar to the best fit (Fig. 2, *Left*, red line) with a power of 0.70 (*SI Text* and Fig. S5).

### Relation Between Dendritic Length and Number of Synapses in Real Neurons.

So far, we have discussed that neurons optimize wiring length to connect up target points without specific reference to what the target points are. The obvious candidates are the synapses the dendrite receives from its presynaptic targets. To experimentally assess the scaling of wiring with synapse number is challenging: During development, axons and dendrites grow in parallel and synapse locations are therefore not static and predefined, but move around and are a result of neurite outgrowth itself. To avoid this problem, we studied the space-filling growth process during neurogenesis in the adult animal in periglomerular (PG) neurons of the olfactory bulb, which integrates neurons into an existing circuit where axonal inputs and network architecture are presumably fixed. Reconstructions of these neurons show that the dendritic branching complexity, as expressed by the number of branch points *bp*, increases during maturation (17) (Fig. 3 *A* and *B*, red). However, the immature dendrite already covers the full target volume—on average 178,000 μm^{3}—at an early stage (Fig. 3 *A* and *B*, black; summary in Fig. 3*C*), approximately filling the spherical glomerulus with a 35-μm radius, and this volume does not change significantly during the increase in branching complexity. Synapse locations were labeled using GFP-tagged PSD95 markers (18) and reconstructed in conjunction with the entire dendritic trees. Because synapse numbers are known in this dataset, the volume is well defined, and the complexity of the dendrites increases during maturation, these data are ideal to test our predictions. The relationship between and for both real (18) and synthetic PG dendrites (Fig. 3*D*) is bounded by the 2/3 power law (with a root mean square error of 1.8%), confirming our prediction. Additionally, in the reconstructed PG neurons the number of branch points *bp* increased with the number of synaptic puncta with a constant ratio *bp*/*n*, which matched that predicted by our analysis of synthetic dendrites (Fig. 3*D*, *Inset*). This result indicates a quantitative match between synaptic puncta and target points of the minimum spanning tree algorithm. To test the qualitative use of synaptic puncta as target points, we further grew synthetic trees directly onto synapse locations obtained from the experiment. Using a *bf* = 0.2, chosen to match the number of branch points, most of the real tree structure was accounted for by the location of the synapses, indicating that they are connected optimally in the real dendrite with this choice for *bf* (Fig. 3*E*, black and red trees, respectively; *SI Text* and Fig. S6). We therefore conclude that the target points for synaptic contacts within the dendritic tree structure predicted by our algorithm correspond closely to the location of putative synapses observed in real PG neurons.

## Discussion

We have derived from first principles a simple equation that relates the most fundamental measures of dendrite branching: the total length, the number of branch points, the spanning volume, and the number of synapses. We show that this equation holds for all dendrite and axon reconstructions available in the neuromorpho.org database, including over 6,000 reconstructions from 140 datasets. This power law relationship shows that dendrites grow to fill a target space in an optimal manner and, similar to a minimum spanning tree, use the least amount of wiring to reach all synaptic contacts. For the example of newborn neurons in the adult olfactory bulb, we show that the power law also holds and describes how synapse locations define dendritic morphology.

Although caution must be applied to their interpretation (19), power laws often describe fundamental scaling properties in biology (14, 20) and in particular in the brain (21⇓⇓⇓⇓–26). Many of these studies build on the fundamental principle that cable length and brain volume follow an inverse cubic power (27). We have extended this work by demonstrating a 2/3 power relationship between cable length and synapse number and provide an exact lower bound for the cable length of a dendrite when its synapse number and its spanning volume are known. We show that this power law holds not only for dendritic trees generated using our minimum spanning tree algorithm (8), but also for a wide range of real neurons. Also, our calculations predict an equation with a 1/2 power law for planar dendrites within a given surface, and both data and model are consistent with this notion (*SI Text* and Fig. S7). These facts strengthen the validity of our algorithm as a method for construction of dendritic trees and suggest that it captures a fundamental principle of the growth of dendritic trees. This constructive algorithmic approach is in contrast to other analysis methods for determining space filling by neuronal processes (2–4, 28). The power law relationship thus provides an important additional constraint that must be fulfilled by other methods for generating dendritic trees. What is the underlying biological mechanism for growing optimal trees? We speculate that the molecular machinery used in dendritic pathfinding implements a developmental program computationally equivalent to a minimum spanning tree algorithm. For instance, the Dscam protein family in *Drosophila* (29, 30) prevents dendrites from one neuron from bundling and crossing over, features that would be incompatible with minimal wiring.

The 2/3 power law we have found for neuronal dendrites challenges the well-known model by West et al. (14) of allometric scaling based on optimal flow through the tree, which predicts a 4/3 power. This difference indicates that tree structures of dendrites are functionally distinct from distribution networks such as vasculatures for which that model has been shown to work (14) and that design constraints other than resource distribution may be important determinants of dendritic shape. At the very least, the assumption of invariant terminal units in the West et al. model (14) does not seem to hold for dendritic trees, which implies that the growth algorithm for vascular, bronchial, and botanical trees is distinct from that of neurons.

The structural constraints on dendritic morphology we have demonstrated are likely to have implications for the computations implemented by neural circuits. First, we have demonstrated that the power law is reflected in the passive integrative properties of neurons such as functional compartmentalization, meaning that it places constraints on information processing. Furthermore, an interesting consequence of minimizing wiring to target synapses is that synapses that are close together are more likely to be linked by the same stretch of dendrite and therefore involved in a local computation (31⇓–33), such that geometry is a key determinant of information processing. There is increasing evidence that such sophisticated local processing may be carried out within the dendritic tree (34, 35), with nonlinear interactions between synaptic inputs shaping the output of the neuron (36⇓⇓–39). We therefore predict it will be fruitful to study how the scaling laws of wiring and branching place constraints on the overall computational power of single neurons.

## Materials and Methods

### Wiring Algorithm.

Optimal wiring was implemented as previously described (6) to minimize both total cable length and the cost for path lengths from any target point along the tree toward the root (6). The second cost is weighted by the balancing factor *bf*, the only parameter required by the wiring algorithm: . These methods have been successfully used for a wide variety of dendrites (8, 39) and all algorithms are available in the TREES toolbox (8, 39).

### Morphological Modeling of Simplified Trees.

Various simplified 2D and 3D geometries were used in Fig. 1*C* and Figs. S2*D* and S7*C* (*Top Insets*):

(

*a*) A “circular” and “spherical” arrangement with a root in the center and target points homogeneously randomly distributed.(

*b*) Corresponding “inhomogeneous” arrangements, similar to the circular arrangement except that the target points were distributed in an inhomogeneous manner.(

*c*) The “off-center” arrangement, similar to the circular arrangement except that the root was displaced from the center.(

*d*) Corresponding “square” and “cubic” arrangements confined to boundaries with straight lines to form a square.

The target points were then connected for all cases in the same way as previously described. For the electrotonic analysis, a quadratic diameter taper according to ref. 6 (using the function “quaddiameter_tree” of the TREES toolbox) was mapped onto the dendrites, and sample specific membrane resistances of and axial resistances of were used.

### Anatomical Analysis of Other Cell Types.

All data from the NeuroMorpho database (www.neuromorpho.org; version as of October 7, 2011) (16) were used for Fig. 2. These data include 142 datasets and 6,577 reconstructions (6 were discarded because the files could not be read) of dendrites and axons. Of the 142 datasets, 74 datasets had enough reconstructions (at least 10) to fit the power law. Dendritic spanning volumes were estimated using the convex hull of the dendritic tree. To fit a power law to the data, the number of branch points and total length were first normalized by their respective volume. Then the log of the data was used in a linear regression. From simulation in Fig. 1*D* a slightly inhomogeneous power distribution above 2/3 was expected with the power increasing slightly with higher *bf*. The power in the fitted power law including all 6,577 reconstructions from NeuroMorpho was 0.70 (Fig. 2), which is consistent with this result.

### Anatomical Reconstructions of PG Neurons.

Reconstructions of mouse PG neurons were obtained from a published dataset (17, 18, 41). Briefly, in these studies, neurons were infected with a GFP or PSD95-GFP marker gene, the latter labeling putative synapse locations, by viral injection into newborn PG cells. These cells were left to migrate toward their specific glomerulus for 10 d and subsequently 3D image stacks were collected through a craniotomy in 12-h intervals, using a two-photon microscope. Reconstructions of tree morphologies and synapse locations were obtained from these image stacks, using Neurolucida (Mbf Bioscience). The morphologies were imported into our Matlab software package, the TREES toolbox (8, 40), and further analyzed.

## Acknowledgments

We thank Yoav Livneh and Adi Mizrahi for providing olfactory glomerulus reconstructions and for discussions that inspired this study. We also thank Alexander Borst, Peter Dayan, Mickey London, Charles Mathy, Sarah Rieubland, Arnd Roth, Christoph Schmidt-Hieber, Robert Sinclair, and Chuck Stevens for helpful discussions. This work was supported by grants from the Alexander von Humboldt Foundation, the Max Planck Society, the Gatsby Charitable Foundation, the European Research Council, and the Wellcome Trust.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: hermann.neuro{at}gmail.com.

Author contributions: H.C., A.M., and M.H. designed research; H.C. and A.M. performed research; H.C. and A.M. contributed new reagents/analytic tools; H.C. and A.M. analyzed data; and H.C., A.M., and M.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1200430109/-/DCSupplemental.

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