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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Block, B.A.: Evolutionary novelties: how fish have built a heater out of muscle. Am. Zool. 31, 726–742 (1991)
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
Bruce, J.W.: Seeing-the mathematical viewpoint. Math. Intell. 6, 18–25 (1984)
Butler, A.B., Hodos, W.: Comparative Vertebrate Anatomy. John Wiley & Sons, Inc., Hoboken, New Jersey (2005)
Churchland, M., Cunningham, J., Kaufman, M., Foster, J., Nuyujukian, P., Ryu, S., Shenoy, K.: Neural population dynamics during reaching. Nature 487, 1–56 (2012)
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)
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
Helmholtz, H.: Treatise on Physiological Optics. Translated by J.P.C. Southall. Reprinted by Dover, New York (1910)
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
Miller, J.: Understanding and misunderstanding extraocular muscle pulleys. J. Vis. 7, 1–15 (2007). https://doi.org/10.1167/7.11.10
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
Nassim, C.: Lessons from the Lobster. MIT Press, Cambridge (2018)
Noakes, L., Mees, A.: Dynamical signatures. Physica D 58, 243–250 (1992). https://doi.org/10.1016/0167-2789(92)90112-Z
Rodieck, R.: The First Steps in Seeing. Sinauer Associates, Inc., Sunderland (1998)
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
Sillar, K., Picton, L., Heitler, W.: The Neuroethology of Predation and Escape. John Wiley & Sons, Ltd., Chichester (2016)
Sussillo, D., Barak, O.: Opening the black box: low-dimensional dynamics in high-dimensional recurrent neural networks. Neural Comput. 25, 626–649 (2013)
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
Wilson, H.R.: Spikes, Decision and Actions. Oxford University Press, Oxford (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
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
DOI: https://doi.org/10.1007/978-3-030-98495-3_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-98494-6
Online ISBN: 978-3-030-98495-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)