Abstract
Understanding brain computation requires assembling a complete catalog of its architectural components. Although the brain is organized into several anatomical and functional regions, it is ultimately the neurons in every region that are responsible for cognition and behavior. Thus, classifying neuron types throughout the brain and quantifying the population sizes of distinct classes in different regions is a key subject of research in the neuroscience community. The total number of neurons in the brain has been estimated for multiple species, but the definition and population size of each neuron type are still open questions even in common model organisms: the so called “cell census” problem. We propose a methodology that uses operations research principles to estimate the number of neurons in each type based on available information on their distinguishing properties. Thus, assuming a set of neuron type definitions, we provide a solution to the issue of assessing their relative proportions. Specifically, we present a three-step approach that includes literature search, equation generation, and numerical optimization. Solving computationally the set of equations generated by literature mining yields best estimates or most likely ranges for the number of neurons in each type. While this strategy can be applied towards any neural system, we illustrate its usage on the rodent hippocampus.



Similar content being viewed by others
References
Abernethy, J., & Hazan, E. (2015). Faster convex optimization: simulated annealing with an efficient universal barrier. http://proceedings.mlr.press/v48/abernethy16.pdf. Accessed 02 October 2019.
Armañanzas, R., & Ascoli, G. A. (2015). Towards the automatic classification of neurons. Trends in Neuroscience,38(5), 307–318.
Armstrong, C., Szabadics, J., Tamas, G., & Soltesz, I. (2011). Neurogliaform cells in the molecular layer of the dentate gyrus as feed-forward gamma-aminobutyric acidergic modulators of entorhinal-hippocampal interplay. The Journal of Comparative Neurology,519(8), 1476–1491.
Ascoli, G. A., & Wheeler, D. W. (2016). In search of a periodic table of the neurons: Axonal-dendritic circuitry as the organizing principle—Patterns of axons and dendrites within distinct anatomical parcels provide the blueprint for circuit-based neuronal classification. BioEssays,38(10), 969–976.
Attili, S. M., Silva, M. F. M., Nguyen, T., & Ascoli, G. A. (2019). Cell numbers, distribution, shape, and regional variation throughout the murine hippocampal formation from the adult brain Allen Reference Atlas. Brain Structure and Function,224, 2883–2897.
Audet, C., & Dennis, J. E., Jr. (2003). Analysis of generalized pattern searches. SIAM Journal on Optimization,13(3), 889–903.
Bartheld, C. S. V. (2001). Comparison of 2-D and 3-D counting: The need for calibration and common sense. Trends in Neurosciences,24(9), 504–506. https://doi.org/10.1016/s0166-2236(00)01960-3.
Bayer, S., Yackel, J., & Puri, P. (1982). Neurons in the rat dentate gyrus granular layer substantially increase during juvenile and adult life. Science,216, 890–892.
Bezaire, M. J., Raikov, I., Burk, K., Vyas, D., & Soltesz, I. (2016). Interneuronal mechanisms of hippocampal theta oscillations in a full-scale model of the rodent CA1 circuit. ELife. https://doi.org/10.7554/eLife.18566.001.
Bezaire, M. J., & Soltesz, I. (2013). Quantitative assessment of CA1 local circuits: Knowledge base for interneuron-pyramidal cell connectivity. Hippocampus,23(9), 751–785. https://doi.org/10.1002/hipo.22141.
Bhanu, B., & Peng, J. (2000). Adaptive integrated image segmentation and object recognition. IEEE Trans Syst Man Cybern Part C (Appl Rev),30, 427–441.
Bota, M., & Swanson, L. W. (2007). The neuron classification problem. Brain Research Reviews,56(1), 79–88.
Buckmaster, P. S., & Jongen-Relo, A. L. (1999). Highly specific neuron loss preserves lateral inhibitory circuits in the dentate gyrus of kainate-induced epileptic rats. Journal of Neuroscience,19(21), 9519–9529.
Byrd, R. H., Gilbert, J. C., & Nocedal, J. (2000). A trust region method based on interior point techniques for nonlinear programming. Mathematical Programming,89(1), 149–185.
Calhoun, M. E., Kurth, D., & Phinney, A. L. (1998). Hippocampal neuron and synaptophysin-positive bouton number in aging C57BL/6 mice. Neurobiology of Aging,19, 599–606.
Ceranik, K., Bender, R., Geiger, J. R., Monyer, H., Jonas, P., Frotscher, M., et al. (1997). A novel type of GABAergic interneuron connecting the input and the output regions of the hippocampus. Journal of Neuroscience,17(14), 5380–5394.
Coleman, T. F., & Li, Y. A. (1996). Reflective newton method for minimizing a quadratic function subject to bounds on some of the variables. SIAM Journal on Optimization,6(4), 1040–1058.
Conn, A. R., Gould, N. I. M., & Toint, P. (1997). A globally convergent augmented lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Mathematics of Computation,66(217), 261–288.
Ecker, J. R., Geschwind, D. H., Kriegstein, A. R., Ngai, J., Osten, P., Polioudakis, D., et al. (2017). The BRAIN initiative cell census consortium: Lessons learned toward generating a comprehensive brain cell atlas. Neuron,96, 542–557.
Erö, C., Gewaltig, C., Keller, M., & Markram, D. (2018). A cell atlas for the mouse brain. Frontiers in Neuroinformatics. https://doi.org/10.3389/fninf.2018.00084.
Fitting, S., Booze, R. M., Hasselrot, U., & Mactutus, C. F. (2009). Dose-dependent longterm effects of Tat in the rat hippocampal formation: A design-based stereological study. Hippocampus. https://doi.org/10.1002/hipo.20648.
Gill, P. E., Murray, W., & Wright, M. H. (1981). Practical optimization. Cambridge: Academic Press.
Grady, M. S., Charleston, J. S., Maris, D., Witgen, B. M., & Lifshitz, J. (2003). Neuronal and glial cell number in the hippocampus after experimental traumatic brain injury: Analysis by stereological estimation. Journal of Neurotrauma,20(10), 929–941.
Hamilton, D. J., Shepherd, G. M., Martone, M. E., & Ascoli, G. A. (2012). An ontological approach to describing neurons and their relationships. Frontiers in Neuroinformatics. https://doi.org/10.3389/fninf.2012.00015.
Hamilton, D. J., White, C. M., Rees, C. L., Wheeler, D. W., & Ascoli, G. A. (2017). Molecular fingerprinting of principal neurons in the rodent hippocampus: A neuroinformatics approach. Journal of Pharmaceutical and Biomedical Analysis,144(10), 269–278.
Han, Z. S. (1994). Electrophysiological and morphological differentiation of chandelier and basket cells in the rat hippocampal formation: A study combining intracellular recording and intracellular staining with biocytin. Neuroscience Research,19(1), 101–110.
Herculano-Houzel, S. (2009). The human brain in numbers: A linearly scaled-up primate brain. Frontiers in Human Neuroscience. https://doi.org/10.3389/neuro.09.031.2009.
Herculano-Houzel, S., Bartheld, C. S. V., Miller, D. J., & Kaas, J. H. (2015). How to count cells: The advantages and disadvantages of the isotropic fractionator compared with stereology. Cell and Tissue Research,360(1), 29–42.
Herculano-Houzel, S., Mota, B., & Lent, R. (2006). Cellular scaling rules for rodent brains. PNAS,103, 12138–12143.
Herculano-Houzel, S., Ribeiro, P., Campos, L., Valotta da Silva, A., Torres, L. B., Catania, K. C., et al. (2011). Updated neuronal scaling rules for the brains of Glires (rodents/lagomorphs). Brain, Behavior and Evolution,78, 302–314.
Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics,12(1), 55–67.
Hosseini-Sharifabad, M., & Nyengaard, J. R. (2007). Design-based estimation of neuronal number and individual neuronal volume in the rat hippocampus. Journal of Neuroscience Methods,162, 206–214.
Husmann, K., Lange, A., & Spiegel, E. (2017). The R package optimization: Flexible global optimization with simulated-annealing. Researchgate.net. Accessed 01 August 2019.
Insausti, A. M., Megas, M., & Crespo, D. (1998). Hippocampal volume and neuronal number in Ts65Dn mice: A murine model of down syndrome. Neuroscience Letters,253, 175–178.
Januszewski, M., Kornfeld, J., Li, P. H., Pope, A., Blakely, T., Lindsey, L., et al. (2018). High-precision automated reconstruction of neurons with flood-filling networks. Nature Methods,15(8), 605–610.
Kaae, S. S., Chen, F., Wegener, G., Madsen, T. M., & Nyengaard, J. R. (2012). Quantitative hippocampal structural changes following electroconvulsive seizure treatment in a rat model of depression. Synapse (New York, N. Y.),66, 667–676.
Kalai, A. T., & Vempala, S. (2006). Simulated annealing for convex optimization. Mathematics of Operations Research,31(2), 253–266. https://doi.org/10.1287/moor.1060.0194.
Keller, D., Meystre, J., Veettil, R. V., Burri, O., Guiet, R., Schurmann, F., et al. (2019). A derived positional mapping of inhibitory subtypes in the somatosensory cortex. Frontiers in Neuroanatomy. https://doi.org/10.3389/fnana.2019.00078.
Laarhoven, P. J. M. V., & Aarts, E. H. L. (1987). Simulated annealing. In Simulated annealing: Theory and applications (pp. 77–98). Dordrecht: Springer.
Lawson, C. L., & Hanson, R. J. (1995). Solving least squares problems. Classics in applied mathematics. Philadelphia: SIAM.
Lister, J. P., Tonkiss, J., Blatt, G. J., Kemper, T. L., Debassio, W. A., Galler, J. R., et al. (2006). Asymmetry of neuron numbers in the hippocampal formation of prenatally malnourished and normally nourished rats: A stereological investigation. Hippocampus,16, 946–958.
Lubke, J., Frotscher, M., & Spruston, N. (1998). Specialized electrophysiological properties of anatomically identified neurons in the hilar region of the rat fascia dentata. Journal of Neurophysiology,79(3), 1518–1534.
Moradi, K., & Ascoli, G. A. (2019). A comprehensive knowledge base of synaptic electrophysiology in the rodent hippocampal formation. Hippocampus. https://doi.org/10.1101/632760.
More, J. J. (1978). The Levenberg–Marquardt algorithm: Implementation and theory. Lecture Notes in Mathematics Numerical Analysis,45, 105–116. https://doi.org/10.1007/bfb0067700.
Morgan, R. J., Santhakumar, V., & Soltesz, I. (2007). Modeling the dentate gyrus. Progress in Brain Research,163, 639–658.
Mott, D. D., Turner, D. A., Okazaki, M. M., & Lewis, D. V. (1997). Interneurons of the dentate-hilus border of the rat dentate gyrus: Morphological and electrophysiological heterogeneity. Journal of Neuroscience,17(11), 3990–4005.
Mulders, W., West, M., & Slomianka, L. (1998). Neuron numbers in the presubiculum, parasubiculum, and entorhinal area of the rat. Journal of Comparative Neurology,385, 83–94.
Mullen, M. (2015). The Stark–Parker algorithm for bounded-variable least squares. https://cran.rproject.org/web/packages/bvls/bvls.pdf. Accessed 01 August 2019.
Mullen, M., & van Stokkum, H. M. (2015). The Lawson–Hanson algorithm for non-negative least squares (NNLS). https://cran.r-project.org/web/packages/nnls/nnls.pdf. Accessed 01 August 2019.
Murakami, T. C., Mano, T., Saikawa, S., Horiguchi, S. A., Shigeta, D., et al. (2018). A three-dimensional single-cell-resolution whole-brain atlas using CUBIC-X expansion microscopy and tissue clearing. Nature Neuroscience,21, 625–637.
Peng, H., Hawrylycz, M., Roskams, J., Hill, S., Spruston, N., Meijering, E., et al. (2015). BigNeuron: Large-scale 3D neuron reconstruction from optical microscopy images. Neuron,87(2), 252–256.
Peng, H., Roysam, B., & Ascoli, G. A. (2013). Automated image computing reshapes computational neuroscience. BMC Bioinformatics,14, 293.
Ramsden, M., Berchtold, N. C., Kesslak, J. P., Cotman, C. W., & Pike, C. J. (2003). Exercise increases the vulnerability of rat hippocampal neurons to kainate lesion. Brain Research,971, 239–244.
Rapp, P. R., & Gallagher, M. (1996). Preserved neuron number in the hippocampus of aged rats with spatial learning deficits. Proceedings of the National Academy of Sciences of the United States of America,93(18), 9926–9930.
Rasmussen, T., Schliemann, T., Sorensen, J. C., Zimmer, J., & West, M. J. (1996). Memory impaired aged rats: No loss of principal hippocampal and subicular neurons. Neurobiology of Aging,17, 143–147.
Russ, J. C., & Deho, R. T. (2001). Practical stereology. New York: Kluwer Academic.
Shepherd, M., G., Marenco, Luis, Hines, L., M., et al. (2019, February 7). Neuron names: A gene- and property-based name format, with special reference to cortical neurons. Frontiers, https://www.frontiersin.org/articles/10.3389/fnana.2019.00025/full. Accessed 24 October 2019.
Sousa, N., Madeira, M. D., & Paula-Barbosa, M. M. (1998). Effects of corticosterone treatment and rehabilitation on the hippocampal formation of neonatal and adult rats. An unbiased stereological study. Brain Research,794, 199–210.
Stark, P. B., & Parker, R. L. (1993). Bounded-variable least-squares: An algorithm and applications. http://digitalassets.lib.berkeley.edu/sdtr/ucb/text/394.pdf. Accessed 01 August 2019.
Tasic, B., Yao, Z., Smith, K. A., Graybuck, L., Nguyen, T., Bertagolli, D., et al. (2018). Shared and distinct transcriptomic cell types across neocortical areas. Nature,563(7729), 72–78.
Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society: Series B,58(1), 267–288.
Wang, L., Gordon, M. D., & Zhu, J. (2006). Regularized least absolute deviations regression and an effcient algorithm for parameter tuning. IEEE. https://ieeexplore.ieee.org/abstract/document/4053094. Accessed 01 August 2019.
West, M. J., Slomianka, L., & Gundersen, H. J. (1991). Unbiased stereological estimation of the total number of neurons in the subdivisions of the rat hippocampus using the optical fractionator. The Anatomical Record,231(4), 482–497.
Wheeler, D. W., et al. (2015). Hippocampome.org: A knowledge base of neuron types in the rodent hippocampus. Elife,4, 09960.
White, C. M., Rees, C. L., Wheeler, D. W., Hamilton, D. J., & Ascoli, G. A. (2019). Molecular expression profiles of morphologically defined hippocampal neuron types: Empirical evidence and relational inferences. Hippocampus. https://doi.org/10.1002/hipo.23165.
Williams, P. A., Larimer, P., Gao, Y., & Strowbridge, B. W. (2007). Semilunar granule cells: Glutamatergic neurons in the rat dentate gyrus with axon collaterals in the inner molecular layer. Journal of Neuroscience,27(50), 13756–13761.
Woodson, W., Nitecka, L., & Ben-Ari, Y. (1989). Organization of the GABAergic system in the rat hippocampal formation: A quantitative immunocytochemical study. The Journal of Comparative Neurology,280(2), 254–271.
Xiang, Y., Gubian, S., Suomela, B., & Hoeng, J. (2013). Generalized simulated annealing for global optimization: The GenSA package. The R Journal,5, 13.
Acknowledgements
This project is supported in parts by Grants R01NS39600 and U01MH114829. The authors are grateful to Drs. Diek Wheeler, Keivan Moradi, and Padmanabhan Seshaiyer for their help and many useful discussions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Attili, S.M., Mackesey, S.T. & Ascoli, G.A. Operations research methods for estimating the population size of neuron types. Ann Oper Res 289, 33–50 (2020). https://doi.org/10.1007/s10479-020-03542-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03542-7

