Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


What can topology tell us about the neural code?
HTML articles powered by AMS MathViewer

by Carina Curto PDF
Bull. Amer. Math. Soc. 54 (2017), 63-78 Request permission


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.
  • Physiology or Medicine 1981—Press Release, 2014, Nobel Media AB.
  • 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, DOI 10.1214/15-AOAS886
  • 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.
  • 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.
  • N. Burgess, The 2014 Nobel Prize in Physiology or Medicine: A Spatial Model for Cognitive Neuroscience, Neuron 84 (2014), no. 6, 1120–1125.
  • 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, DOI 10.1162/NECO_{a}_{0}0538
  • 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.
  • J. Cruz, C. Giusti, V. Itskov, and W. Kronholm, On open and closed convex codes, arXiv:1609.03502v1 [math.CO], 2016.
  • 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.
  • 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.
  • Carina Curto and Vladimir Itskov, Cell groups reveal structure of stimulus space, PLoS Comput. Biol. 4 (2008), no. 10, e1000205, 13. MR 2457124, DOI 10.1371/journal.pcbi.1000205
  • 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, DOI 10.1007/s11538-013-9860-3
  • Y. Dabaghian, V. L. Brandt, and L. M. Frank, Reconceiving the hippocampal map as a topological template, Elife 3 (2014), e03476.
  • 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.
  • E. Colin de Verdiere, G. Ginot, and X. Goaoc, Multinerves and Helly Numbers of Acyclic Families, Symposium on Computational Geometry - SoCG ’12 (2012).
  • 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, DOI 10.4310/HHA.2014.v16.n1.a14
  • Robert Ghrist, Barcodes: the persistent topology of data, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 1, 61–75. MR 2358377, DOI 10.1090/S0273-0979-07-01191-3
  • Chad Giusti and Vladimir Itskov, A no-go theorem for one-layer feedforward networks, Neural Comput. 26 (2014), no. 11, 2527–2540. MR 3243436, DOI 10.1162/NECO_{a}_{0}0657
  • 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, DOI 10.1007/s10827-016-0608-6
  • 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, DOI 10.1073/pnas.1506407112
  • 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.
  • V. Itskov, Personal communication, 2015.
  • 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.
  • 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.
  • 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.
  • C. Lienkaemper, A. Shiu, and Z. Woodstock, Obstructions to convexity in neural codes, Available online at
  • Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
  • 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.
  • J. O’Keefe and L. Nadel, The hippocampus as a cognitive map, Clarendon Press Oxford, 1978.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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).
  • B. Stolz, Computational topology in neuroscience, Master’s thesis, University of Oxford, 2014.
Similar Articles
  • Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 54-XX, 92-XX
  • Retrieve articles in all journals with MSC (2010): 54-XX, 92-XX
Additional Information
  • Carina Curto
  • Affiliation: Department of Mathematics, The Pennsylvania State University
  • MR Author ID: 663228
  • Email:
  • 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.
  • © Copyright 2016 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 54 (2017), 63-78
  • MSC (2010): Primary 54-XX, 92-XX
  • DOI:
  • MathSciNet review: 3584098