Skip to main content
SpringerLink
Log in

Towards a Mathematical Model of the Brain

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

This article presents an idealized mathematical model of the cerebral cortex, focusing on the dynamical interaction of neurons. The author proposes a network architecture more consistent with neuroanatomy than in previous studies, borrows ideas from nonequilibrium statistical mechanics and calls attention to the fact that the brain is a large and complex dynamical system. The ideas proposed are illustrated with a realistic model of the visual cortex.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Baddeley, R., Abbott, L.F., Booth, M.C.A., Sengpiel, F., Freeman, T., Wakeman, E.A., Rolls, E.T.: Responses of neurons in primary and inferior temporal visual cortices to natural scenes. Proc. R Soc. Lond. B 264, 1775–1783 (1997)

    ADS  Google Scholar 

  2. Beaulieu, C., Kisvarday, Z., Somogyi, P., Cynader, M., Cowey, A.: Quantitative distribution of GABA-immunopositive and -immunonegative neurons and synapses in the monkey striate cortex (area 17). Cereb. Cortex 2, 295–309 (1992)

    Google Scholar 

  3. Binzegger, T., Douglas, R., Martin, K.: Topology and dynamics of the canonical circuit of cat V1. Neural Netw. 22, 1071–78 (2009)

    Google Scholar 

  4. Börgers, C., Kopell, N.: Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity. Neural Comput. 15(3), 509–538 (2003)

    MATH  Google Scholar 

  5. Bressloff, P.: Waves in Neural Media: From single neurons to neural fields. Lecture Notes in Math. Modeling in the Life Sciences, Springer (2014)

  6. Brunel, N., Hakim, V.: Fast global oscillations in networks of integrate-and-fire neurons with low firing rates. Neural Comput. 11(7), 1621–1671 (1999)

    Google Scholar 

  7. Brunel, N.: Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J. Comput. Neurosci. 3, 183–208 (2000)

    MATH  Google Scholar 

  8. Brunel, N., Wang, X.J.: What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. J. Neurophysiol. 90(1), 415–430 (2003)

    Google Scholar 

  9. Buice, M.A., Cowan, J.: Statistical mechanics of the neocortex. Prog. Biophys. Mol. Biol. 99(2–3), 53–86 (2009)

    Google Scholar 

  10. Burns, S.P., Xing, D., Shapley, R.M.: Gamma-band activity in the local field potential of V1 cortex: a “clock or filtered noise? J. Neurosci. 31, 9658–64 (2011)

    Google Scholar 

  11. Cai, D., Tao, L., Rangan, A.V., McLaughlin, D.W., et al.: Kinetic theory for neuronal network dynamics. Commun. Math. Sci. 4(1), 97–127 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Cardin, J.A.: Snapshots of the brain in action: local circuit operations through the lens of oscillations. J. Neurosci. 36, 10496–10504 (2006)

    Google Scholar 

  13. Chariker, L., Young, L.-S.: Emergent spike patterns in neuronal populations. J. Comput. Neurosci. 38(1), 203–220 (2015)

    MATH  Google Scholar 

  14. Chariker, L., Shapley, R., Young, L.-S.: Orientation selectivity from very sparse LGN inputs in a comprehensive model of macaque V1 cortex. J. Neurosci. 36, 12368–12384 (2016)

    Google Scholar 

  15. Chariker, L., Shapley, R., Young, L.-S.: Rhythm and synchrony in a cortical network model. J. Neurosci. 38(40), 8621–8634 (2018)

    Google Scholar 

  16. Chariker, L., Shapley, R., Young, L.-S.: Contrast response in a comprehensive network model of V1. Under Review

  17. De Groot, S., Mazur, P.: Nonequilibrium Thermodynamics. North Holland, Amsterdam (1962)

    MATH  Google Scholar 

  18. Eckmann, J.P., Young, L.-S.: Nonequilibrium energy profiles for a class of 1D models. Commun. Math. Phys. 262(1), 237–267 (2006)

    ADS  MATH  Google Scholar 

  19. Gerstein, G.L., Mandelbrot, B.: Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41 (1964)

    Google Scholar 

  20. Gray, C.M., Singer, W.: Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. USA 86, 1698–1702 (1989)

    ADS  Google Scholar 

  21. Heeger, D.: Theory of cortical function. Proc. Natl. Acad. Sci. USA 114, 1773–1782 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Henrie, J.A., Shapley, R.: Lfp power spectra in v1 cortex: the graded effect of stimulus contrast. J. Neurophysiol. 94(1), 479–490 (2005)

    Google Scholar 

  23. Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117(4), 500–44 (1952)

    Google Scholar 

  24. Holmgren, C., Harkany, T., Svennenfors, B., Zilberter, Y.: Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. J. Physiol. 551, 139–53 (2003)

    Google Scholar 

  25. Hubel, D., Wiesel, T.: Receptive fields, binocular interaction and functional architecture in the cats visual cortex. J. Physiol. 160, 106–154 (1962)

    Google Scholar 

  26. Joglekar, M., Mejias, J., Yang, G.R., Wang, X.-J.: Inter-areal balanced amplification enhances signal propagation in a large-scale circuit model of the primate cortex. Neuron 98(1), 222–234.e8 (2018)

    Google Scholar 

  27. Koch, C.: Biophysics of Computation. Oxford Univ Press, Oxford (1999)

    Google Scholar 

  28. Li, Y., Chariker, L., Young, L.-S.: How well do reduced models capture the dynamics in models of interacting neurons? J. Math. Biol. 78, 83 (2018). https://doi.org/10.1007/s00285-018-1268-0

    MathSciNet  MATH  Google Scholar 

  29. McLaughlin, D., Shapley, R., Shelley, M., Wielaard, D.J.: A Neuronal Network Model of sharpening and dynamics of orientation tuning in an input layer of macaque primary visual cortex. Proc. Natl. Acad. Sci. USA 97, 8087–8092 (2000)

    ADS  Google Scholar 

  30. Ostojic, S.: Interspike interval distributions of spiking neurons driven by fluctuating inputs. J. Neurophysiol. 106, 361–373 (2011)

    Google Scholar 

  31. Oswald, A.M., Reyes, A.: Maturation of intrinsic and synaptic properties of layer 2/3 pyramidal neurons in mouse auditory cortex. J. Neurophysiol. 99, 2998–3008 (2008)

    Google Scholar 

  32. Rangan, A.V., Young, L.-S.: Emergent dynamics in a model of visual cortex. J. Comput. Neurosci. 35, 155–167 (2013)

    MathSciNet  Google Scholar 

  33. Rangan, A.V., Young, L.-S.: Dynamics of spiking neurons: between homogeneity and synchrony. J. Comput. Neurosci. 34(3), 433–460 (2013)

    MathSciNet  MATH  Google Scholar 

  34. Tso, D.Y., Frostig, R.D., Lieke, E.E., Grinvald, A.: Functional organization of primate visual cortex revealed by high resolution optimal imaging. Science 249(4967), 417–420 (1990)

    ADS  Google Scholar 

  35. Ungerleider, L.G., Desimone, R.: Cortical connections of visual area MT in the macaque. J. Comput. Neurol. 248(2), 190–222 (1986)

    Google Scholar 

  36. Ungerleider, L.G., Galkin, T.W., Desimone, R., Gattass, R.: Cortical connections of area V4 in the macaque. Cereb. Cortex 3, 477–99 (2008)

    Google Scholar 

  37. van Vreeswijk, C., Sompolinsky, H.: Chaotic balanced state in a model of cortical circuits. Neural Comput. 10(6), 1321–1371 (1998)

    Google Scholar 

  38. Wilson, H., Cowan, J.D.: Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12(1), 1–24 (1972)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Lai-Sang Young

  2. School of Mathematics and School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Lai-Sang Young

Authors
  1. Lai-Sang Young
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lai-Sang Young.

Additional information

Communicated by Ivan Corwin.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is partially supported by NSF Grants 1734854 and 1901009.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Young, LS. Towards a Mathematical Model of the Brain. J Stat Phys 180, 612–629 (2020). https://doi.org/10.1007/s10955-019-02483-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-019-02483-1

Keywords

Search

Navigation