Edited by: Jaap van Pelt, Center for Neurogenomics and Cognitive Research, Netherlands
Reviewed by: Robert C. Cannon, Textensor Limited, UK; Eduardo Fernandez, Universidad Miguel Hernández de Elche, Spain
*Correspondence: Artur Luczak, Department of Neuroscience, Canadian Centre for Behavioural Neuroscience, University of Lethbridge, 4401 University Drive, Lethbridge, AB, Canada T1K 3M4. e-mail:
This is an open-access article subject to an exclusive license agreement between the authors and the Frontiers Research Foundation, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.
Neurons have complex branching systems which allow them to communicate with thousands of other neurons. Thus understanding neuronal geometry is clearly important for determining connectivity within the network and how this shapes neuronal function. One of the difficulties in uncovering relationships between neuronal shape and its function is the problem of quantifying complex neuronal geometry. Even by using multiple measures such as: dendritic length, distribution of segments, direction of branches, etc, a description of three dimensional neuronal embedding remains incomplete. To help alleviate this problem, here we propose a new measure, a shape diffusiveness index (SDI), to quantify spatial relations between branches at the local and global scale. It was shown that growth of neuronal trees can be modeled by using diffusion limited aggregation (DLA) process. By measuring “how easy” it is to reproduce the analyzed shape by using the DLA algorithm it can be measured how “diffusive” is that shape. Intuitively, “diffusiveness” measures how tree-like is a given shape. For example shapes like an oak tree will have high values of SDI. This measure is capturing an important feature of dendritic tree geometry, which is difficult to assess with other measures. This approach also presents a paradigm shift from well-defined deterministic measures to model-based measures, which estimate how well a model with specific properties can account for features of analyzed shape.
Information in the brain is processed by highly interconnected neuronal networks, where a typical cortical neuron receives signals from 1000 to 10000 neurons. To send and receive signals from such a large number of cells, neurons adopted elaborated branching structures. By tracing the extent and direction of dendrites and axons it was possible to start discovering the basic organizational principles of the brain (Ramon y Cajal,
Considering the significance of neuronal arborization, many researchers have attempted to describe multiple aspects of neuronal geometry in a quantitative manner (for review see Uylings and van Pelt,
To generate tree-like structures we used a diffusion limited aggregation (DLA) model which is especially suitable to reproduce branching structures. It was shown that this DLA-based model was able to reproduce the spatial embedding of multiple neuronal types: granule cells, Purkinje cells, pyramidal cells, and dendritic and axonal trees of interneurons (Luczak,
To reproduce shape using the DLA algorithm, the position of the first particle of the future aggregate must be defined. This particle called “seed” is positioned at the origin of the object to be reproduced (Figure
For computational efficiency, instead of one moving particle,
The example of typical DLA is shown in Figure
As explained before, a randomly moving particle after hitting the aggregate either will become a new particle of the aggregate, or will be deleted depending on whether or not the place of contact overlaps with the reproduced shape (Figure
The same algorithm can also be applied to reproduce 3D shapes. In that case, particles move randomly on a 3D grid instead of 2D. An example of DLA generated in 3D is in Figure
The distribution of number of hits as shown in Figure
Interestingly, the distribution of hits for DLA could be well approximated by log-normal distribution:
To express “diffusiveness” of an object, as measured with
Files with intracellularly labeled, reconstructed, and digitalized neurons were obtained from the Duke-Southampton on-line archive of neuronal morphology (
To illustrate values of SDI for different 2D objects we investigated a branching structure, square, and line (Figure
Reproducing line results in much lower values of SDI (SDI = 0.29 ± 0.04). Due to lack of screening from other branches, all parts of the line are hit by particles (Figure
For solid shapes like a square in Figure
For that the aggregate grows fast until it reaches the border of the square. Note that DLA cannot fill out completely solid shapes as growing branches prevent other particles from filling gaps. The distribution of hits for solid objects is composed of two parts (Figure
For example, studying an oak tree at scales of millimeters is appropriate to investigate shape of a leaf or bark patterns, but to study branching patterns of a tree, the scale of meters is more suitable. In the same way a given object may be shaped by diffusive-like processes only at a particular scale. Therefore it is important to investigate the shape diffusiveness at different spatial scales. To measure at what spatial resolution a given object is, the most similar to the DLA shape a SDI can be calculated for different sizes of particles creating aggregate. For computational convenience, instead of increasing size of moving particles in relation to the reproduced object, the size of the object can be decreased, while keeping particle size the same. An example of SDI for a DLA at different scales is shown in Figure
To evaluate diffusiveness of neuronal shape we analyzed dendritic trees of four different neuronal types: pyramidal cells, interneurons, Purkinje cells, and granule cells (Figure
The values of SDI across scales and neuronal types are summarized in Figure
Although measuring diffusiveness of a shape provides new and interesting information
Predicted type | Actual neuronal type | |||
---|---|---|---|---|
Interneuron | Purkinje | Granule | Pyramidal | |
Interneuron | 12 | 0 | 5 | 2 |
Purkinje | 0 | 3 | 0 | 0 |
Granule | 1 | 0 | 29 | 3 |
Pyramidal | 0 | 0 | 4 | 50 |
Neurons have elaborate 3D shapes, so while analyses in 2D are good for method development and an initial assessment, they may not give accurate reflection of 3D embedding. For that reason we repeated calculations of SDI for the same cells in 3D. Due to exponentially larger computational time and the computer memory demands required to generate DLA in 3D, neurons were analyzed only at five spatial scales (2, 4, 8, 16, and 32 μm). The results for all neuronal types are summarized in Figure
Predicted type | Actual neuronal type | |||
---|---|---|---|---|
Interneuron | Purkinje | Granule | Pyramidal | |
Interneuron | 11 | 0 | 6 | 0 |
Purkinje | 0 | 3 | 0 | 0 |
Granule | 2 | 0 | 30 | 9 |
Pyramidal | 0 | 0 | 2 | 46 |
In this paper we propose a Diffusion Limited Aggregate model as a “benchmark” for “diffusive” shape. Although DLA grows by connecting particles diffusing in space, DLA can also be seen as diffusing into space, where dendrites have the highest probability of growing in the direction of the largest local concentration of “trophic” particles. As a result, DLA forms complex tree-like shapes. Note that the seemingly simple concept of “tree-like shape” is in fact very difficult to quantitatively describe. For example, we would consider as a tree only a shape with a particular type of connectivity pattern, and with particular spatial distribution of segments, branching angles, relative lengths, orientation, etc. By using the DLA model to reproduce analyzed objects, we can quantify the tree-like resemblance of an object by simply measuring performance of the DLA algorithm. Thus this approach presents a conceptual change where the use a computational model allows to assess complex properties of an object, which otherwise would be very difficult to quantify with any other existing measures.
In general there is no single best measure to evaluate similarities between two objects or even graphs (Loncaric,
Performance of DLA algorithms could be measured in variety of ways, e.g.: how quickly it can cover shape; how completely it covers; how broad is the distribution of hits, etc. From all of the different measures we tried, the distance to the hit distribution of DLA provided the most reliable measure of similarity to DLA-like shapes. Because it is not convenient to use a distribution which is described with >50 numbers (probability for each hit values) we tried to fit the hit distribution of DLA with variety of known functions. We found a log-normal distribution to be well fitting all parts of the experimental distribution for 2D and 3D analyses. It is not obvious why a log-normal distribution, which describes the multiplicative product of many independent random variables, would be the best here, but the analytical investigation of the exact formula for the hit distribution of DLA and its relation to log-normal distribution is beyond the scope of this paper. Likely, part of the explanation is that the outermost parts of the aggregate have exponentially higher probability of being hit by randomly moving particles than the more inner parts of aggregate.
As explained in the Methods section, the hit distribution is calculated for range of hits 1–50. Nevertheless in some cases there are also parts of the reproduced shape which were never hit by diffusing particle (for example see area marked in black in Figures
Shape diffusiveness index is a complex measure which cannot be easily expressed with an equation. Instead SDI is defined based on hit distribution resulted from a probabilistic, iterative algorithm. SDI also cannot be directly related to a simple measure like mean segment lengths, because SDI depends on a non-trivial combination of a variety of parameters like relative spatial distribution of segments, connectivity pattern, branching angles, etc. Another difficulty with relating SDI to other measures is its non-monotonic dependence on spatial resolution, which allows the investigation of the spatial scale for branching processes. For example, for a granule cell, as shown in Figure
Calculating SDI at multiple scales proved to be a reliable measure to discriminate neuronal types (Tables
Shape diffusiveness index seems to be in good agreement with an intuitive assessment of the similarity between an analyzed shape and DLA. Nevertheless, what is really needed is a measure of similarity not to an artificial structure, but to a specific type of shape. For example, it would be beneficial to have a measure describing if a particular neuron looks like a normal and healthy Purkinje cell. For instance, such a measure would be of interest when screening for irregularities in dendritic morphology caused by disease, drug, and/or aging. To achieve this, an approach suggested by this study would be to generate not a generic DLA model, but instead a DLA-based model of a specific neuronal type, and this tailored model would be used to reproduce an analyzed shape (modifying density of particles in space will result in DLAs resembling different neuronal types). Importantly, the method proposed here is not restricted to DLA models only. Most likely, any other model of neuronal growth could be successfully used to quantify the similarity of a given shape to, e.g., pyramidal neuron. It could be implemented in an analogous fashion to the DLA model, where by evaluating “how easy” it is to reproduce shape of a given neuron by using model proposed by Ascoli and Krichmar (
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This work was supported by NSERC and AHFMR grants.