Where Are We Going With All This?

  • Chapter
  • First Online:
Mathematical Tools for Neuroscience

Part of the book series: Lecture Notes in Morphogenesis ((LECTMORPH))

  • 563 Accesses

Abstract

What are the implications of using geometric techniques for understanding the brain? After assessing the current limitations, the relationship with complementary techniques are outlined and it is argued that neuroscience is in need of a mathematics more suited to the behaviour of neurons as opposed to mathematics suited to our thought processes.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Block, B.A.: Evolutionary novelties: how fish have built a heater out of muscle. Am. Zool. 31, 726–742 (1991)

    Article  Google Scholar 

  2. Brainard, M.S., Doupe, A.J.: Translating birdsong: songbirds as a model for basic and applied medical research. Annu. Rev. Neurosci. 36, 726–742 (2013). https://doi.org/10.1146/annurev-neuro-060909-152826

    Article  Google Scholar 

  3. Bruce, J.W.: Seeing-the mathematical viewpoint. Math. Intell. 6, 18–25 (1984)

    Article  MathSciNet  Google Scholar 

  4. Butler, A.B., Hodos, W.: Comparative Vertebrate Anatomy. John Wiley & Sons, Inc., Hoboken, New Jersey (2005)

    Google Scholar 

  5. Churchland, M., Cunningham, J., Kaufman, M., Foster, J., Nuyujukian, P., Ryu, S., Shenoy, K.: Neural population dynamics during reaching. Nature 487, 1–56 (2012)

    Article  Google Scholar 

  6. Doya, K., Boyle, M., Selverston, A.: Map** between neural and physical activities of the lobster gastric mill. Adv. Neural Inf. Process. Syst. 5, 913–920 (1993)

    Google Scholar 

  7. Han, Q., Hong, J.X.: Isometric embedding of Riemannian manifolds in Euclidean spaces. In: Mathematical Surveys and Monographs, vol. 130. American Mathematical Society, Providence (2006). https://doi.org/10.1090/surv/130

  8. Helmholtz, H.: Treatise on Physiological Optics. Translated by J.P.C. Southall. Reprinted by Dover, New York (1910)

    Google Scholar 

  9. Marzban, C., Yurtsever, U.: Baby morse theory in data analysis. In: Proceedings of the 2011 Workshop on Knowledge Discovery, Modelling and Simulation, pp. 15–21. San Diego, Association for Computing Machinery, New York (2011). https://doi.org/10.1145/2023568.2023576

  10. Miller, J.: Understanding and misunderstanding extraocular muscle pulleys. J. Vis. 7, 1–15 (2007). https://doi.org/10.1167/7.11.10

    Article  Google Scholar 

  11. Montgomery, R.: Gauge theory of the falling cat. In: Enos, M.J. (ed.) Fields Inst. Commun. Dynamics and Control of Mechanical Systems: The Falling Cat and Related Problems, pp. 195–281. Fields Institute Communications (1993). https://doi.org/10.1090/fic/001

  12. Nassim, C.: Lessons from the Lobster. MIT Press, Cambridge (2018)

    Book  Google Scholar 

  13. Noakes, L., Mees, A.: Dynamical signatures. Physica D 58, 243–250 (1992). https://doi.org/10.1016/0167-2789(92)90112-Z

    Article  MathSciNet  Google Scholar 

  14. Rodieck, R.: The First Steps in Seeing. Sinauer Associates, Inc., Sunderland (1998)

    Google Scholar 

  15. Sanguinetti, G., Citti, G., Sarti, A.: A model of natural image edge co-occurrence in the rototranslation group. J. Vis. 10, 1–16 (2010). https://doi.org/10.1167/10.14.37

    Article  Google Scholar 

  16. Sillar, K., Picton, L., Heitler, W.: The Neuroethology of Predation and Escape. John Wiley & Sons, Ltd., Chichester (2016)

    Book  Google Scholar 

  17. Sussillo, D., Barak, O.: Opening the black box: low-dimensional dynamics in high-dimensional recurrent neural networks. Neural Comput. 25, 626–649 (2013)

    Article  MathSciNet  Google Scholar 

  18. Wang, J., Narain, D., Hosseini, E., Jazayeri, M.: Flexible timing by temporal scaling of cortical responses. Nat. Neurosci. 21, 102–110 (2018). https://doi.org/10.1038/s41593-017-0028-6

    Article  Google Scholar 

  19. Wilson, H.R.: Spikes, Decision and Actions. Oxford University Press, Oxford (1999)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Clement, R.A. (2022). Where Are We Going With All This?. In: Mathematical Tools for Neuroscience. Lecture Notes in Morphogenesis. Springer, Cham. https://doi.org/10.1007/978-3-030-98495-3_8

Download citation

Publish with us

Policies and ethics

Navigation