MDL Neuron Modeling
The Minimum Description Length (MDL) principle was developed by Rissanen in 1978 and it has its root in information theory [1]. It was generated from Shannon’s classic statistical communication theory [2] and Solomonoff’s inductive inference theory [3]. The original idea is to minimize the codes in terms of binary digits (bits) of a signal to be sent over a communication channel and be decoded on the other side of the channel.
The MDL principle can provide a complete estimation of the parameter model, including not only the number of its components and the parameters themselves. With the MDL model of backbones at its optimal value, the whole dendritic model can be accomplished by comprising the spine model based on the MDL pinciple. The spine models are secondary structure of the whole dendritic model since they are attached to the backbones. Therefore, a minimum description tree (MDT) will be created.
If combining the cost function of two description length together, the description length of the minimum description tree includes the two parts: backbone model description and spine model description, which constitute the two levels of dendrite model.
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Image Preprocessing Methods
Upon the acquisition of deconvoloved confocal microscopy 3D images, the basic 3D image preprocessing methods, such as thresholding, morphological image processing, connected component removal and flood filling [4], are used for noise reduction, image enhancement and regions of interest (ROI) selection. Since the skeletonization method is based on 3D intensity images and corresponding gradient vector field, it is necessary to derive smooth vector field without the affection of noise, while still keeping the original intensity ridges.
Image Preprocessing
- Thresholding
- Morphological image processing
- Connected component removal
- Flood filling
- Anisotropic diffusion
Anisotropic Diffusion
The algorithm used to extract the skeletons of the tubular objects is based on a computed gradient vector field with certain kernel. The original 3D intensity map contains noise corrupted part and it introduces irregular components to the gradient vector field. Therefore, it is necessary to smooth the 3D intensity map before deriving the vector field while it must preserve the ridges of neuronal structure at the same time.
Anisotropic diffusion can smooth out some unwanted small objects attached to the main part of the tubular objects, while it can overcome the edge shifting problem in isotropic linear diffusion. It is an iterative method that can produce smooth results while keeping useful edge structures [5].
Grayscale Skeletonization of Dendrites and Spines
The gradient vector field is computed from the smoothed and regularized 3D images after the anisotropic diffusion. The skeletonization method is derived from the vector field where critical points and high curvature points are located as seed points. The central lines of neurons and spines are extracted by path-line formation along intensity ridges with all the necessary seed points.
Gradient Vector Field and Critical Points
- Sobel kernel to compute vector field
- Jacobian matrix of the vectors
- Eigenvalues of Jacobian
- Critical points
- Attracting points
- Repelling points
- Saddle points
- High curvature points
Creation of Skeletons
The two kinds of seed points, saddles and high curvature points, constitute the starting points of flow integration. The iterative algorithm of path-line formation in vector fields move a step forward in the direction of the vector at each current position until the step size falls below a predetermined threshold. One can use a standard flow integration method to follow the path-lines [6]. Common integration schemes use either Euler schemes, Runge-Kutta second order (RK-2), or Runge-Kutta fourth order(RK-4).
Graph Structure and Morphology
In order to create connection relationships of skeleton points and make the dendrite and spine analyses easy to accomplish, a graph structure of the skeleton points needs to be constructed. The later MDL approaches are based on this graph model. Directly, the graph structure of dendritic backbones is obtained with a graph morphology method.
Minimum Spanning Tree with Intensity Weighted Edges
The skeleton points of dendrites and spines can be transformed into a graph structure that provides an easy way to manage the tree structure of neuron and it is straightforward to traverse the whole dendritic tree and measure the necessary data about it [7]. For instance, the graph can retrieve all the branch points immediately.
The vertices of graph are the whole 3D points of skeletons from streamline generation. Within curtain range of each point, graph edges are created and edge weights are computed based on point Euclidean distance and 3D image intensities.
Graph Morphology Methods
Graph erosion operation can change the morphology of the graph and remove unwanted trivial leaves of the tree and keep the major tree structure. After a sequence of erosion operations on the tree, original backbone structure will show up. However it is a little shorter than the true value, a dilation operation will bring it back to its original length.
MDL Method on Dendrites Structures
The structures of dendritic backbones can be approximately and concisely fitted by polynomial splines prior to any surface reconstruction if necessary. Currently, there are two commonly used ways, the piecewise polynomial form and B-spline form. With MDL principle, the best fitted B-spline functions can be achieved to be optimized backbone model. The MDL method can automatically and inherently prevent from overfitting and it can overcome the difficulty in estimation of both the parameters and the structure of a model that includes the number of parameters [8].
MDL on B-spline backbone models
The spline fitting was applied to real datasets. The original extracted dendritic backbone is not smooth at some places. They can be represented with piecewise B-Spline functions well. The order and complication of the B-spline model that is optimal can be determined by the MDL principle. On the one hand, simple spline functions don't fit the data well; on the other hand, high order spline functions are too complicated and have some unnecessary ripples.
MDL Method on Spines Structures
The data are described in terms of extracted features from the fluorescence Images. The features are assumed to be best representation of spines from both geometry and intensity properties of spines. The features under consideration are mean intensity of the branch, length of the branch, and mean vesselness of the branch, etc. The features are components of multi-dimensional vector space.
MDL on Dendritic Spine Structure
The descriptive language is based on feature descriptions and model description [9]. When the features are determined for each spine, the spine can be established. Assume the observed spine features are the sum of certain feature values and noise with some variances. These two components add up to the given feature observations exactly.
Spine Models with Prior Knowledge
Consider two factors in the prior distribution of spine models:
- The relationship of neighborhood of spines on a dendrite;
- The relationship of a spine with its dendritic backbone.
When prior knowledge comes into the model, more bits will be needed to describe additional information of the spines. Among other prior knowledge of dendritic structure, the first one is the spine local restriction. Assuming the neighboring two spines are not intending to be too close to each other; otherwise it can be a false detection. Under the other prior knowledge of spine positions on dendrites, the spines are protruding from backbones only and the protruding positions are not far from the backbone. Otherwise, it is not likely to be a spine. it can be called offbranch problem to restrict the spines not to be far away from the backbone lines.
References
- [1] J. Rissanen, Modeling by shortest data description, Automatica, vol. 14, no. 5, pp. 465-471, 1978.
- [2] C.E., Shannon, W. Weaver, The Mathematical Theory of Communication, The University of Illinois Press, 1949.
- [3] R.J. Solomonoff, A Formal Theory of Inductive Inference: Part I and II, Information and Control, 7, 1964, Part I: pp. 1-22, Part II: pp. 224-254.
- [4] D.F. Rogers, Procedural Elements for Computer Graphics, McGraw-Hill, Boston, MA, 1998.
- [5] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, no. 7 pp. 629–639, July 1990.
- [6] W. Schroeder, K. Martin and B. Lorensen, The Visualization Toolkit an Object Oriented Approach to 3D Graphics, Printice-Hall, Inc., 1998.
- [7] Cohen, A. R., Roysam, B., and Turner, J. N., Automated Tracing and Volume Measurements of Neurons from 3-D Confocal Fluorescence Microscopy Data, Journal of Microscopy, Vol. 173, Pt. 2, February 1994.
- [8] P. Grunwald, J. Myung, and M. Pitt, Advances in Minimum Description Length: Theory and Applications, MIT Press, 2004.
- [9] Y. G. Leclerc, Constructing simple stable descriptions for image partitioning, Int. J. Comput. Vision, vol. 3, no. 1, pp. 73-102, 1989.
