The Annals of Applied Statistics

Statistical shape analysis of simplified neuronal trees

Adam Duncan, Eric Klassen, and Anuj Srivastava

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Neuron morphology plays a central role in characterizing cognitive health and functionality of brain structures. The problem of quantifying neuron shapes and capturing statistical variability of shapes is difficult because neurons differ both in geometry and in topology. This paper develops a mathematical representation of neuronal trees, restricting to the trees that consist of: (1) a main branch viewed as a parameterized curve in $\mathbb{R}^{3}$, and (2) some number of secondary branches—also parameterized curves in $\mathbb{R}^{3}$—which emanate from the main branch at arbitrary points. It imposes a metric on the representation space, in order to compare neuronal shapes, and to obtain optimal deformations (geodesics) across arbitrary trees. The key idea is to impose certain equivalence relations that allow trees with different geometries and topologies to be compared efficiently. The combinatorial problem of matching side branches across trees is reduced to a linear assignment with well-known efficient solutions. This framework is then applied to comparing, clustering, and classifying neurons using fully automated algorithms. The framework is illustrated on three datasets of neuron reconstructions, specifically showing geodesics paths and cross-validated classification between experimental groups.

Article information

Ann. Appl. Stat., Volume 12, Number 3 (2018), 1385-1421.

Received: August 2016
Revised: October 2017
First available in Project Euclid: 11 September 2018

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Neuron morphology elastic shape analysis tree registration neuron deformation tree geodesics


Duncan, Adam; Klassen, Eric; Srivastava, Anuj. Statistical shape analysis of simplified neuronal trees. Ann. Appl. Stat. 12 (2018), no. 3, 1385--1421. doi:10.1214/17-AOAS1107.

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  • Andersson-Engels, S., af Klinteberg, C., Svanberg, K. and Svanberg, S. (1997). In vivo fluorescence imaging for tissue diagnostics. Phys. Med. Biol. 42 815–824.
  • Ascoli, G. A., Donohue, D. E. and Halavi, M. (2007). NeuroMorpho.Org: A central resource for neuronal morphologies. J. Neurosci. 27 9247–9251.
  • Aydin, B., Pataki, G., Wang, H., Bullitt, E. and Marron, J. S. (2009). A principal component analysis for trees. Ann. Appl. Stat. 3 1597–1615.
  • Aydin, B., Pataki, G., Wang, H., Ladha, A. and Bullitt, E. (2011). Visualizing the structure of large trees. Electron. J. Stat. 5 405–420.
  • Bassell, G. J. and Warren, S. T. (2008). Fragile X syndrome: Loss of local mRNA regulation alters synaptic development and function. Neuron 60 201–214.
  • Billera, L. J., Holmes, S. P. and Vogtmann, K. (2001). Geometry of the space of phylogenetic trees. Adv. in Appl. Math. 27 733–767.
  • Bruveris, M. (2015). Optimal reparametrizations in the square-root velocity framework. SIAM Journal on Mathematical Analysis 48 4335–4354.
  • Chan-Palay, V. and Asan, E. (1989). Alterations in catecholamine neurons of the locus coeruleus in senile dementia of the Alzheimer type and in Parkinson’s disease with and without dementia and depression. The Journal of Comparative Neurology 287 373–392.
  • Chen, J.-R., Wang, B.-N., Tseng, G.-F., Wang, Y.-J., Huang, Y.-S. and Wang, T.-J. (2014). Morphological changes of cortical pyramidal neurons in hepatic encephalopathy. BMC Neuroscience 15 15.
  • Coleman, P. D. and Flood, D. G. (1987). Neuron numbers and dendritic extent in normal aging and Alzheimer’s disease. Neurobiol. Aging 8 521–545.
  • Cuntz, H., Forstner, F., Haag, J. and Borst, A. (2008). The morphological identity of insect dendrites. PLoS Comput. Biol. 4 e1000251.
  • Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis. Wiley, Chichester.
  • Duncan, A., Srivastava, A., Descombes, X. and Klassen, E. (2015). Geometric analysis of axonal tree structures. In DIFF-CV: Differential Geometric Techniques in Computer Vision.
  • Engle, E. C. (2008). Human genetic disorders of axon guidance. Neuron 60 201–214.
  • Feragen, A. (2012). Complexity of computing distances between geometric trees. In Structural, Syntactic, and Statistical Pattern Recognition: Joint IAPR International Workshop, SSPR&SPR 2012, Hiroshima, Japan, November 79, 2012. Proceedings 89–97. Springer, Berlin.
  • Feragen, A., Lauze, F. and Hauberg, S. (2015). Geodesic exponential kernels: When curvature and linearity conflict. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR).
  • Feragen, A., Hauberg, S., Nielsen, M. and Lauze, F. (2011). Means in spaces of tree-like shapes. In Computer Vision (ICCV), 2011 IEEE International Conference on 736–746.
  • Feragen, A., Owen, M., Petersen, J., Wille, M. M., Thomsen, L. H., Dirksen, A. and de Bruijne, M. (2013a). Tree-space statistics and approximations for large-scale analysis of anatomical trees. In Information Processing in Medical Imaging: 23rd International Conference, IPMI 2013, Asilomar, CA, USA, June 28–July 3, 2013. Proceedings 74–85. Springer, Berlin.
  • Feragen, A., Lo, P., de Bruijne, M., Nielsen, M. and Lauze, F. (2013b). Toward a theory of statistical tree-shape analysis. IEEE Trans. Pattern Anal. Mach. Intell. 35 2008–2021.
  • Feragen, A., Petersen, J., Owen, M., Lo, P., Thomsen, L. H., Wille, M. M., Dirksen, A. and de Bruijne, M. (2015). Geodesic atlas-based labeling of anatomical trees: Application and evaluation on airways extracted from CT. IEEE Trans. Med. Imag. 34 1212–1226.
  • Gibson, D. A. and Ma, L. (2011). Developmental regulation of axon branching in the vertebrate nervous system. Development 138 183–195.
  • Halavi, M., Hamilton, K. A., Parekh, R. and Ascoli, G. A. (2012). Digital reconstructions of neuronal morphology: Three decades of research trends. Frontiers in Neuroscience 6.
  • Heuman, H. and Wittum, G. (2009). The tree-edit-distance, a measure for quantifying neuronal morphology. Neuroinformatics 7 179–190.
  • Hirokawa, N., Niwa, S. and Tanaka, Y. (2010). Molecular motors in neurons: Transport mechanisms and roles in brain function, development, and disease. Neuron 68 610–638.
  • Jonker, R. and Volgenant, A. (1987). A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing 38 325–340.
  • Joshi, S. H., Klassen, E., Srivastava, A. and Jermyn, I. (2007). A novel representation for Riemannian analysis of elastic curves in Rn. In 2007 IEEE Conference on Computer Vision and Pattern Recognition 1–7.
  • Kabaso, D., Coskren, P., Henry, B., Hof, P. and Wearne, S. (2009). The electrotonic structure of pyramidal neurons contributing to prefrontal cortical circuits in macaque monkeys is significantly altered in aging. Cerebral Cortex 19 2248–2268.
  • Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. Wiley, Chichester.
  • Kurtek, S., Srivastava, A., Klassen, E. and Ding, Z. (2012). Statistical modeling of curves using shapes and related features. J. Amer. Statist. Assoc. 107 1152–1165.
  • Lahiri, S., Robinson, D. and Klassen, E. (2015). Precise matching of PL curves in $\mathbb{R}^{N}$ in the square root velocity framework. Geom. Imaging Comput. 2 133–186.
  • Ledderose, J., Sencion, L., Salgado, H., Arias-Carrion, O. and Trevino, M. (2014). A software tool for the analysis of neuronal morphology data. Int. Archive Medicine 7 1–9.
  • Liu, W., Srivastava, A. and Klassen, E. (2008). Joint shape and texture analysis of objects boundaries in images using a Riemannian approach. In Asilomar Conference on Signals, Systems, and Computers.
  • Liu, W., Srivastava, A. and Zhang, J. (2011). A mathematical framework for protein structure comparison. PLoS Comput. Biol. 7 e1001075, 10.
  • Medioni, C., Ramialison, M., Ephrussi, A. and Besse, F. (2014). Imp promotes axonal remodeling by regulating profilin mRNA during brain development. Curr. Biol. 24 793–800.
  • Mio, W. and Srivastava, A. (2004). Elastic-string models for representation and analysis of planar shapes. In Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004. 2 II–10–II–15.
  • Ntziachristos, V. (2006). Fluorescence molecular imaging. Annu Rev Biomed Eng 8 1–33.
  • Schoenberg, I. J. (1938). Metric spaces and positive definite functions. Trans. Amer. Math. Soc. 44 522–536.
  • Selkow, S. M. (1977). The tree-to-tree editing problem. Inform. Process. Lett. 6 184–186.
  • Srivastava, A. and Klassen, E. P. (2016). Functional and Shape Data Analysis. Springer, New York.
  • Srivastava, A., Klassen, E., Joshi, S. H. and Jermyn, I. H. (2011). Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33 1415–1428.
  • Suo, L., Lu, H., Ying, G., Capecchi, M. R. and Wu, Q. (2012). Protocadherin clusters and cell adhesion kinase regulate dendrite complexity through Rho GTPase. Journal of Molecular Cell Biology 4 362.
  • Tai, K. C. (1979). The tree-to-tree correction problem. J. Assoc. Comput. Mach. 26 422–433.
  • Wang, H. and Marron, J. S. (2007). Object oriented data analysis: Sets of trees. Ann. Statist. 35 1849–1873.
  • West, M. J., Coleman, P. D., Flood, D. G. and Troncoso, J. C. (1994). Differences in the pattern of hippocampal neuronal loss in normal ageing and Alzheimer’s disease. Lancet 344 769–772.
  • Whitehouse, P. J., Price, D. L., Struble, R. G., Clark, A. W., Coyle, J. T. and Delon, M. R. (1982). Alzheimer’s disease and senile dementia: Loss of neurons in the basal forebrain. Science 215 1237–1239.
  • Wu, C. H. et al. (2012). Mutations in the profilin 1 gene cause familial amyotrophic lateral sclerosis. Nature 488 499–503.
  • Zhang, K. (1996). A constrained edit distance between unordered labeled trees. Algorithmica 15 205–222.