Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Grid cells generate an analog error-correcting code for singularly precise neural computation

Nature Neuroscience volume 14, pages 1330–1337 (2011)Cite this article

Subjects

Abstract

Entorhinal grid cells in mammals fire as a function of animal location, with spatially periodic response patterns. This nonlocal periodic representation of location, a local variable, is unlike other neural codes. There is no theoretical explanation for why such a code should exist. We examined how accurately the grid code with noisy neurons allows an ideal observer to estimate location and found this code to be a previously unknown type of population code with unprecedented robustness to noise. In particular, the representational accuracy attained by grid cells over the coding range was in a qualitatively different class from what is possible with observed sensory and motor population codes. We found that a simple neural network can effectively correct the grid code. To the best of our knowledge, these results are the first demonstration that the brain contains, and may exploit, powerful error-correcting codes for analog variables.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Coding for location: GPC and CPCs.
Figure 2: Error correction and the structure of the GPC location-phase map.
Figure 3: Scaling of dmin and error correction.
Figure 4: Simple neural network can perform ongoing error correction for accurate path integration.

Similar content being viewed by others

References

  1. Softky, W.R. & Koch, C. The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. J. Neurosci. 13, 334–350 (1993).

    Article  CAS  Google Scholar 

  2. Stevens, C.F. & Wang, Y. Changes in reliability of synaptic function as a mechanism for plasticity. Nature 371, 704–707 (1994).

    Article  CAS  Google Scholar 

  3. Seung, H.S. Learning in spiking neural networks by reinforcement of stochastic synaptic transmission. Neuron 40, 1063–1073 (2003).

    Article  CAS  Google Scholar 

  4. Kao, M.H., Doupe, A.J. & Brainard, M.S. Contributions of an avian basal ganglia-forebrain circuit to real-time modulation of song. Nature 433, 638–643 (2005).

    Article  CAS  Google Scholar 

  5. Fiete, I.R. & Seung, H.S. Gradient learning in spiking neural networks by dynamic perturbation of conductances. Phys. Rev. Lett. 97, 048104 (2006).

    Article  Google Scholar 

  6. Hubel, D. Eye, Brain and Vision (Scientific American Library, 1988).

  7. Taube, J.S., Muller, R.U. & Ranck, J.B. Jr. Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J. Neurosci. 10, 420–435 (1990).

    Article  CAS  Google Scholar 

  8. O'Keefe, J. & Dostrovsky, J. The hippocampus as a spatial map: preliminary evidence from unit activity in the freely-moving rat. Brain Res. 34, 171–175 (1971).

    Article  CAS  Google Scholar 

  9. Wilson, M.A. & McNaughton, B.L. Dynamics of the hippocampal ensemble code for space. Science 261, 1055–1058 (1993).

    Article  CAS  Google Scholar 

  10. Schreiner, C.E., Read, H.L. & Sutter, M.L. Modular organization of frequency integration in primary auditory cortex. Annu. Rev. Neurosci. 23, 501–529 (2000).

    Article  CAS  Google Scholar 

  11. Seung, H.S. & Sompolinsky, H. Simple models for reading neuronal population codes. Proc. Natl. Acad. Sci. USA 90, 10749–10753 (1993).

    Article  CAS  Google Scholar 

  12. Abbott, L.F. & Dayan, P. The effect of correlated variability on the accuracy of a population code. Neural Comput. 11, 91–101 (1999).

    Article  CAS  Google Scholar 

  13. Sompolinsky, H., Yoon, H., Kang, K. & Shamir, M. Population coding in neuronal systems with correlated noise. Phys. Rev. E 64, 051904 (2001).

    Article  CAS  Google Scholar 

  14. Latham, P.E., Deneve, S. & Pouget, A. Optimal computation with attractor networks. J. Physiol. (Paris) 97, 683–694 (2003).

    Article  Google Scholar 

  15. Brunel, N. & Nadal, J.-P. Mutualinformation, fisher information and population coding. Neural Comput. 10, 1731–1757 (1998).

    Article  CAS  Google Scholar 

  16. Zhang, K. & Sejnowski, T. Neuronal tuning: to sharpen or broaden? Neural Comput. 11, 75–84 (1999).

    Article  CAS  Google Scholar 

  17. Bethge, M., Rotermund, D. & Pawelzik, K. Optimal short-term population coding: when fisher information fails. Neural Comput. 14, 2317–2351 (2002).

    Article  CAS  Google Scholar 

  18. MacKay, D. Information Theory, Inference and Learning Algorithms (Cambridge University Press, 2004).

  19. Shannon, C. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 623–656 (1948).

    Article  Google Scholar 

  20. Goblick, T. Theoretical limitations on the transmission of data from analog sources. IEEE Trans. Inf. Theory 11, 558–567 (1965).

    Article  Google Scholar 

  21. Hafting, T., Fyhn, M., Molden, S., Moser, M.-B. & Moser, E. Microstructure of a spatial map in the entorhinal cortex. Nature 436, 801–806 (2005).

    Article  CAS  Google Scholar 

  22. Fuhs, M.C. & Touretzky, D.S. A spin glass model of path integration in rat medial entorhinal cortex. J. Neurosci. 26, 4266–4276 (2006).

    Article  CAS  Google Scholar 

  23. Burgess, N., Barry, C. & O'Keefe, J. An oscillatory interference model of grid cell firing. Hippocampus 17, 801–812 (2007).

    Article  Google Scholar 

  24. Burak, Y. & Fiete, I.R. Accurate path integration in continuous attractor network models of grid cells. PLOS Comput. Biol. 5, e1000291 (2009).

    Article  Google Scholar 

  25. Fiete, I.R., Burak, Y. & Brookings, T. What grid cells convey about rat location. J. Neurosci. 28, 6856–6871 (2008).

    Article  Google Scholar 

  26. Deneve, S., Latham, P.E. & Pouget, A. Reading population codes: a neural implementation of ideal observers. Nat. Neurosci. 2, 740–745 (1999).

    Article  CAS  Google Scholar 

  27. Zhang, K. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. J. Neurosci. 16, 2112–2126 (1996).

    Article  CAS  Google Scholar 

  28. Tsodyks, M. Attractor neural network models of spatial maps in hippocampus. Hippocampus 9, 481–489 (1999).

    Article  CAS  Google Scholar 

  29. Xie, X., Hahnloser, R.H.R. & Seung, H.S. Double-ring network model of the head-direction system. Phys. Rev. E 66, 041902 (2002).

    Article  Google Scholar 

  30. Brun, V.H. et al. Impaired spatial representation in CA1 after lesion of direct input from entorhinal cortex. Neuron 57, 290–302 (2008).

    Article  CAS  Google Scholar 

  31. Paradiso, M.A. A theory for the use of visual orientation information which exploits the columnar structure of striate cortex. Biol. Cybern. 58, 35–49 (1988).

    Article  CAS  Google Scholar 

  32. Knierim, J.J., Kudrimoti, H.S. & McNaughton, B.L. Interactions between idiothetic cues and external landmarks in the control of place cells and head direction cells. J. Neurophysiol. 80, 425–446 (1998).

    Article  CAS  Google Scholar 

  33. Lever, C., Burton, S., Jeewajee, A., O'Keefe, J. & Burgess, N. Boundary vector cells in the subiculum of the hippocampal formation. J. Neurosci. 29, 9771–9777 (2009).

    Article  CAS  Google Scholar 

  34. Jin, D.Z. & Seung, H.S. Fast computation with spikes in a recurrent neural network. Phys. Rev. E. 65, 051922 (2002).

    Article  Google Scholar 

  35. Shamir, M. The temporal winner-take-all readout. PLOS Comput. Biol. 5, e1000286 (2009).

    Article  Google Scholar 

  36. Ma, W.J., Beck, J.M., Latham, P.E. & Pouget, A. Bayesian inference with probabilistic population codes. Nat. Neurosci. 9, 1432–1438 (2006).

    Article  CAS  Google Scholar 

  37. Hanzo, L., Liew, T. & Yeap, B. Turbo Coding, Turbo Equalisation And Space-Time Coding (Wiley-IEEE Press, 2002).

  38. Sudan, M. Ideal error-correcting codes: unifying algebraic and number-theoretic algorithms. in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (eds. Boztas, S. & Shparlinski, I.E.) 36–45 (Springer Berlin/Heidelberg, 2001).

  39. Amaral, D.G. & Witter, M.P. The three-dimensional organization of the hippocampal formation: a review of anatomical data. Neuroscience 31, 571–591 (1989).

    Article  CAS  Google Scholar 

  40. van Strien, N.M., Cappaert, N.L.M. & Witter, M.P. The anatomy of memory: an interactive overview of the parahippocampal-hippocampal network. Nat. Rev. Neurosci. 10, 272–282 (2009).

    Article  CAS  Google Scholar 

  41. McNaughton, B.L., Barnes, C.A. & O'Keefe, J. The contributions of position, direction, and velocity to single unit activity in the hippocampus of freely-moving rats. Exp. Brain Res. 52, 41–49 (1983).

    Article  CAS  Google Scholar 

  42. Remondes, M. & Schuman, E.M. Molecular mechanisms contributing to long-lasting synaptic plasticity at the temporoammonic-CA1 synapse. Learn. Mem. 10, 247–252 (2003).

    Article  Google Scholar 

  43. Takahashi, H. & Magee, J.C. Pathway interactions and synaptic plasticity in the dendritic tuft regions of CA1 pyramidal neurons. Neuron 62, 102–111 (2009).

    Article  CAS  Google Scholar 

  44. Carr, M.F., Jadhav, S.P. & Frank, L.M. Hippocampal replay in the awake state: a potential substrate for memory consolidation and retrieval. Nat. Neurosci. 14, 147–153 (2011).

    Article  CAS  Google Scholar 

  45. Burgess, N. Grid cells and theta as oscillatory interference: theory and predictions. Hippocampus 18, 1157–1174 (2008).

    Article  Google Scholar 

  46. Hafting, T., Fyhn, M., Bonnevie, T., Moser, M. & Moser, E. Hippocampus-independent phase precession in entorhinal grid cells. Nature 453, 1248–1252 (2008).

    Article  CAS  Google Scholar 

  47. Ahmed, O.J. & Mehta, M.R. The hippocampal rate code: anatomy, physiology and theory. Trends Neurosci. 32, 329–338 (2009).

    Article  CAS  Google Scholar 

  48. Jarsky, T., Roxin, A., Kath, W.L. & Spruston, N. Conditional dendritic spike propagation following distal synaptic activation of hippocampal ca1 pyramidal neurons. Nat. Neurosci. 8, 1667–1676 (2005).

    Article  CAS  Google Scholar 

  49. Ang, C.W., Carlson, G.C. & Coulter, D.A. Hippocampal CA1 circuitry dynamically gates direct cortical inputs preferentially at theta frequencies. J. Neurosci. 25, 9567–9580 (2005).

    Article  CAS  Google Scholar 

  50. Ristic, B., Arulampalam, S. & Gordon, N. Beyond the Kalman Filter: Particle Filters for Tracking Applications (Artech House Radar Library, 2004).

Download references

Acknowledgements

We are grateful to Y. Burak, S. Seung, A. Kepecs, P. Latham, A. Pouget and P. Dayan for thought-provoking questions and comments, S.H. Lee for helpful conversations, T. Lam for advice on a volume computation, D. Johnston, A. Huk, N. Priebe, A. Tan and members of our laboratory for comments on the manuscript, and A. Preston for useful pointers to the literature. I.F. is a Sloan Foundation Fellow, a Searle Scholar and a McKnight Scholar, and receives funding from the Office of Naval Research through the Multidisciplinary University Research Initiative.

Author information

Authors and Affiliations

  1. Center for Learning and Memory, University of Texas at Austin, Austin, Texas, USA

    Sameet Sreenivasan & Ila Fiete

  2. Department of Computer Science, Rensselaer Polytechnic Institute, Troy, New York, USA

    Sameet Sreenivasan

Authors
  1. Sameet Sreenivasan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ila Fiete
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

I.F. conceived the model. S.S. performed simulations and analyzed the data. I.F. and S.S. performed analytical calculations and wrote the manuscript.

Corresponding author

Correspondence to Ila Fiete.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Text and Figures

Supplementary Results (PDF 909 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sreenivasan, S., Fiete, I. Grid cells generate an analog error-correcting code for singularly precise neural computation. Nat Neurosci 14, 1330–1337 (2011). https://doi.org/10.1038/nn.2901

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nn.2901

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing