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.

Contents 1 Image Preprocessing Methods 1.1 Image Preprocessing 1.2 Anisotropic Diffusion 2 Grayscale Skeletonization of Dendrites and Spines 2.1 Gradient Vector Field and Critical Points 2.2 Creation of Skeletons 3 Graph Structure and Morphology 3.1 Minimum Spanning Tree with Intensity Weighted Edges 3.2 Graph Morphology Methods 4 MDL Method on Dendrites Structures 4.1 MDL on Dendritic Backbone Structures 4.2 MDL on B-spline backbone models 5 MDL Method on Spines Structures 5.1 MDL on Dendritic Spine Structure 5.2 Spine Models with Prior Knowledge 6 References

Image Preprocessing Methods

Image Preprocessing

• Thresholding
• Morphological image processing
• Connected component removal
• Flood filling
• Anisotropic diffusion

Anisotropic Diffusion

Grayscale Skeletonization of Dendrites and Spines

Gradient Vector Field and Critical Points

Creation of Skeletons

Graph Structure and Morphology

Minimum Spanning Tree with Intensity Weighted Edges

Graph Morphology Methods

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 [4].

MDL on Dendritic Backbone Structures

MDL on B-spline backbone models

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

Spine Models with Prior Knowledge

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] P. Grunwald, J. Myung, and M. Pitt, Advances in Minimum Description Length: Theory and Applications, MIT Press, 2004.