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What can topology tell us about the neural code?

Author: Carina Curto
Journal: Bull. Amer. Math. Soc. 54 (2017), 63-78
MSC (2010): Primary 54-XX, 92-XX
Published electronically: September 27, 2016
MathSciNet review: 3584098
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Abstract | References | Similar Articles | Additional Information

Abstract: Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze emerging data sets. Consistent with this trend, the last decade has seen an uptick in the use of topological ideas and methods in neuroscience. In this paper I will survey recent applications of topology in neuroscience, and explain why topology is an especially natural tool for understanding neural codes.

References [Enhancements On Off] (What's this?)

  • [1] Physiology or Medicine 1981--Press Release, 2014, Nobel Media AB.
  • [2] Paul Bendich, J. S. Marron, Ezra Miller, Alex Pieloch, and Sean Skwerer, Persistent homology analysis of brain artery trees, Ann. Appl. Stat. 10 (2016), no. 1, 198–218. MR 3480493,
  • [3] E. N. Brown, L. M. Frank, D. Tang, M. C. Quirk, and M. A. Wilson, A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells, J. Neurosci. 18 (1998), 7411-7425.
  • [4] J. Brown and T. Gedeon, Structure of the afferent terminals in terminal ganglion of a cricket and persistent homology, PLoS ONE 7 (2012), no. 5.
  • [5] N. Burgess, The 2014 Nobel Prize in Physiology or Medicine: A Spatial Model for Cognitive Neuroscience, Neuron 84 (2014), no. 6, 1120-1125.
  • [6] Zhe Chen, Stephen N. Gomperts, Jun Yamamoto, and Matthew A. Wilson, Neural representation of spatial topology in the rodent hippocampus, Neural Comput. 26 (2014), no. 1, 1–39. MR 3155578,
  • [7] H. Choi, Y. K. Kim, H. Kang, H. Lee, H.-J. Im, E. Edmund Kim, J.-K. Chung, D. S. Lee, et al., Abnormal metabolic connectivity in the pilocarpine-induced epilepsy rat model: a multiscale network analysis based on persistent homology, NeuroImage 99 (2014), 226-236.
  • [8] J. Cruz, C. Giusti, V. Itskov, and W. Kronholm, On open and closed convex codes, arXiv:1609.03502v1 [math.CO], 2016.
  • [9] C. Curto, E. Gross, J. Jeffries, K. Morrison, M. Omar, Z. Rosen, A. Shiu, and N. Youngs, What makes a neural code convex?, Available online at, 2016.
  • [10] C. Curto, E. Gross, J. Jeffries, K. Morrison, Z. Rosen, A. Shiu, and N. Youngs, Algebraic signatures of convex and non-convex codes, In preparation, 2016.
  • [11] Carina Curto and Vladimir Itskov, Cell groups reveal structure of stimulus space, PLoS Comput. Biol. 4 (2008), no. 10, e1000205, 13. MR 2457124,
  • [12] Carina Curto, Vladimir Itskov, Alan Veliz-Cuba, and Nora Youngs, The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes, Bull. Math. Biol. 75 (2013), no. 9, 1571–1611. MR 3105524,
  • [13] Y. Dabaghian, V. L. Brandt, and L. M. Frank, Reconceiving the hippocampal map as a topological template, Elife 3 (2014), e03476.
  • [14] Y. Dabaghian, F. Mémoli, L. Frank, and G. Carlsson, A topological paradigm for hippocampal spatial map formation using persistent homology, PLoS Comp. Bio. 8 (2012), no. 8, e1002581.
  • [15] E. Colin de Verdiere, G. Ginot, and X. Goaoc, Multinerves and Helly Numbers of Acyclic Families, Symposium on Computational Geometry - SoCG '12 (2012).
  • [16] Steven P. Ellis and Arno Klein, Describing high-order statistical dependence using “concurrence topology,” with application to functional MRI brain data, Homology Homotopy Appl. 16 (2014), no. 1, 245–264. MR 3211745,
  • [17] Robert Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 61–75. MR 2358377,
  • [18] Chad Giusti and Vladimir Itskov, A no-go theorem for one-layer feedforward networks, Neural Comput. 26 (2014), no. 11, 2527–2540. MR 3243436,
  • [19] Chad Giusti, Robert Ghrist, and Danielle S. Bassett, Two’s company, three (or more) is a simplex, J. Comput. Neurosci. 41 (2016), no. 1, 1–14. MR 3517602,
  • [20] Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov, Clique topology reveals intrinsic geometric structure in neural correlations, Proc. Natl. Acad. Sci. USA 112 (2015), no. 44, 13455–13460. MR 3429279,
  • [21] D. H. Hubel and T. N. Wiesel, Receptive fields of single neurons in the cat's striate cortex, J. Physiol. 148 (1959), no. 3, 574-591.
  • [22] V. Itskov, Personal communication, 2015.
  • [23] A. Khalid, B. S. Kim, M. K. Chung, J. C. Ye, and D. Jeon, Tracing the evolution of multi-scale functional networks in a mouse model of depression using persistent brain network homology, Neuroimage 101 (2014), 351-363.
  • [24] E. Kim, H. Kang, H. Lee, H.-J. Lee, M.-W. Suh, J.-J. Song, S.-H. Oh, and D. S. Lee, Morphological brain network assessed using graph theory and network filtration in deaf adults, Hear. Res. 315 (2014), 88-98.
  • [25] H. Lee, M. K. Chung, H. Kang, B.-N. Kim, and D. S. Lee, Discriminative persistent homology of brain networks, Biomedical Imaging: From Nano to Macro, 2011 IEEE International Symposium on, IEEE, 2011, pp. 841-844.
  • [26] C. Lienkaemper, A. Shiu, and Z. Woodstock, Obstructions to convexity in neural codes, Available online at
  • [27] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
  • [28] J. O'Keefe and J. Dostrovsky, The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat, Brain Res. 34 (1971), no. 1, 171-175.
  • [29] J. O'Keefe and L. Nadel, The hippocampus as a cognitive map, Clarendon Press Oxford, 1978.
  • [30] G. Petri, P. Expert, F. Turkheimer, R. Carhart-Harris, D. Nutt, P. J. Hellyer, and F. Vaccarino, Homological scaffolds of brain functional networks, J. Roy. Soc. Int. 11 (2014), no. 101, 20140873.
  • [31] V. Pirino, E. Riccomagno, S. Martinoia, and P. Massobrio, A topological study of repetitive co-activation networks in in vitro cortical assemblies., Phys. Bio. 12 (2014), no. 1, 016007-016007.
  • [32] J. Rinzel, Discussion: Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and update, Bull. Math. Biol. 52 (1990), no. 1/2, 5-23.
  • [33] G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson, and D. L. Ringach, Topological analysis of population activity in visual cortex, J. Vis. 8 (2008), no. 8, 11.
  • [34] G. Spreemann, B. Dunn, M. B. Botnan, and N. A. Baas, Using persistent homology to reveal hidden information in neural data, arXiv:1510.06629 [q-bio.NC] (2015).
  • [35] B. Stolz, Computational topology in neuroscience, Master's thesis, University of Oxford, 2014.

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Additional Information

Carina Curto
Affiliation: Department of Mathematics, The Pennsylvania State University

Received by editor(s): April 26, 2016
Published electronically: September 27, 2016
Additional Notes: This is a slightly expanded write-up of my talk for the Current Events Bulletin, held at the 2016 Joint Mathematics Meetings in Seattle, Washington.
Article copyright: © Copyright 2016 American Mathematical Society