Modeling grid fields instead of modeling grid cells

An effective model at the macroscopic level and its relationship with the underlying microscopic neural system

Abstract

A neuron’s firing correlates are defined as the features of the external world to which its activity is correlated. In many parts of the brain, neurons have quite simple such firing correlates. A striking example are grid cells in the rodent medial entorhinal cortex: their activity correlates with the animal’s position in space, defining ‘grid fields’ arranged with a remarkable periodicity. Here, we show that the organization and evolution of grid fields relate very simply to physical space. To do so, we use an effective model and consider grid fields as point objects (particles) moving around in space under the influence of forces. We reproduce several observations on the geometry of grid patterns. This particle-like behavior is particularly salient in a recent experiment in which two separate grid patterns merge. We discuss pattern formation in the light of known results from physics of two-dimensional colloidal systems. Notably, we study the limitations of the widely used ‘gridness score’ and show how physics of 2d systems could be a source of inspiration, both for data analysis and computational modeling. Finally, we draw the relationship between our ‘macroscopic’ model for grid fields and existing ‘microscopic’ models of grid cell activity and discuss how a description at the level of grid fields allows to put constraints on the underlying grid cell network.

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Notes

  1. 1.

    Grid patterns with non homogeneous orientation have been shown by Stensola et al. (2015) but, in the absence of a local measure, could not be quantified.

  2. 2.

    Some slight differences, either of spacing or orientation, have been reported, but we neglect them for the present discussion.

  3. 3.

    A question not answered by this experiment is whether this position is fixed with respect to the walls or with respect to something else, e.g., distal cues.

  4. 4.

    Near the walls some grid distortion could come into play as we have shown, but let us assume here that the box is big enough so that we can neglect such edge effects.

  5. 5.

    So this model differs from the place cell case only by its boundary conditions (Spalla et al. 2019)

References

  1. Alekseĭ, A.A., Gorkov, L.P., Dzyaloshinski, I.E. (2012). Methods of quantum field theory in statistical physics. Courier Corporation.

  2. Barry, C., Hayman, R., Burgess, N., Jeffery, K.J. (2007). Experience-dependent rescaling of entorhinal grids. Nature Neuroscience, 10(6), 682.

    Article  CAS  PubMed  Google Scholar 

  3. Berezinskii, V. (1971). Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group i. classical systems. Soviet Physics - JETP, 32(3), 493–500.

    Google Scholar 

  4. Blair, H.T., Welday, A.C., Zhang, K. (2007). Scale-invariant memory representations emerge from moire interference between grid fields that produce theta oscillations: a computational model. Journal of Neuroscience, 27 (12), 3211–3229.

    Article  CAS  PubMed  Google Scholar 

  5. Boccara, C., Stella, F., Nardin, M., O’neill, J, Csicsvari, J. (2016). Goal remapping in grid cells. In: fENS Conference.

  6. Bonnevie, T., Dunn, B., Fyhn, M., Hafting, T., Derdikman, D., Kubie, J.L., Roudi, Y., Moser, E.I., Moser, M.B. (2013). Grid cells require excitatory drive from the hippocampus. Nature Neuroscience, 16(3), 309–317.

    Article  CAS  PubMed  Google Scholar 

  7. Burak, Y., & Fiete, I.R. (2009). Accurate path integration in continuous attractor network models of grid cells. PLoS Computational Biology, 5(2), e1000291.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Burgess, N., Barry, C., O’keefe, J. (2007). An oscillatory interference model of grid cell firing. Hippocampus, 17(9), 801–812.

    Article  PubMed  PubMed Central  Google Scholar 

  9. Carpenter, F., Manson, D., Jeffery, K., Burgess, N., Barry, C. (2015). Grid cells form a global representation of connected environments. Current Biology, 25(9), 1176–1182.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  10. Couey, J.J., Witoelar, A., Zhang, S.J., Zheng, K., Ye, J., Dunn, B., Czajkowski, R., Moser, M.B., Moser, E.I., Roudi, Y., et al. (2013). Recurrent inhibitory circuitry as a mechanism for grid formation. Nature Neuroscience, 16(3), 318–324.

    Article  CAS  PubMed  Google Scholar 

  11. Derdikman, D., Whitlock, J.R., Tsao, A., Fyhn, M., Hafting, T., Moser, M.B., Moser, E.I. (2009). Fragmentation of grid cell maps in a multicompartment environment. Nature neuroscience, 12(10), 1325–1332.

    Article  CAS  PubMed  Google Scholar 

  12. Dunn, B., Wennberg, D., Huang, Z., Roudi, Y. (2017). Grid cells show field-to-field variability and this explains the aperiodic response of inhibitory interneurons. arXiv:170104893.

  13. Fuhs, M.C., & Touretzky, D.S. (2006). A spin glass model of path integration in rat medial entorhinal cortex. Journal of Neuroscience, 26(16), 4266–4276.

    Article  CAS  PubMed  Google Scholar 

  14. Fyhn, M., & et al. (2004). Spatial representation in the entorhinal cortex. Science, 305(5688), 1258.

    Article  CAS  PubMed  Google Scholar 

  15. Giocomo, L.M., Zilli, E.A., Fransén, E., Hasselmo, M.E. (2007). Temporal frequency of subthreshold oscillations scales with entorhinal grid cell field spacing. Science, 315(5819), 1719–1722.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Guanella, A., Kiper, D., Verschure, P. (2007). A model of grid cells based on a twisted torus topology. International journal of neural systems, 17(04), 231–240.

    Article  PubMed  Google Scholar 

  17. Hafting, T., Fyhn, M., Bonnevie, T., Moser, M.B., Moser, E.I. (2008). Hippocampus-independent phase precession in entorhinal grid cells. Nature, 453(7199), 1248.

    Article  CAS  PubMed  Google Scholar 

  18. Hafting, T., & et al. (2005). Microstructure of a spatial map in the entorhinal cortex. Nature, 436(7052), 801–806.

    Article  CAS  PubMed  Google Scholar 

  19. Hägglund, M. (2017). Distortions and development of local spatial features of the grid, spring Hippocampal Research Conference.

  20. Halperin, B., & Nelson, D.R. (1978). Theory of two-dimensional melting. Physical Review Letters, 41(2), 121.

    Article  CAS  Google Scholar 

  21. Hardcastle, K., Ganguli, S., Giocomo, L.M. (2015). Environmental boundaries as an error correction mechanism for grid cells. Neuron, 86(3), 827–839.

    Article  CAS  PubMed  Google Scholar 

  22. Hasselmo, M.E., Giocomo, L.M., Zilli, E.A. (2007). Grid cell firing may arise from interference of theta frequency membrane potential oscillations in single neurons. Hippocampus, 17(12), 1252–1271.

    Article  PubMed  PubMed Central  Google Scholar 

  23. Kosterlitz, J.M., & Thouless, D.J. (1973). Ordering, metastability and phase transitions in two-dimensional systems. Journal of Physics C:, Solid State Physics, 6(7), 1181.

    Article  CAS  Google Scholar 

  24. Kropff, E., & Treves, A. (2008). The emergence of grid cells: Intelligent design or just adaptation? Hippocampus, 18(12), 1256–1269.

    Article  PubMed  Google Scholar 

  25. Krupic, J., Bauza, M., Burton, S., Lever, C., O’Keefe, J. (2014). How environment geometry affects grid cell symmetry and what we can learn from it. Philosophical Transactions of the Royal Society B, 369(1635), 20130188.

    Article  Google Scholar 

  26. Krupic, J., Bauza, M., Burton, S., Barry, C., O’Keefe, J. (2015). Grid cell symmetry is shaped by environmental geometry. Nature, 518(7538), 232–235.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  27. Langston, R.F., Ainge, J.A., Couey, J.J., Canto, C.B., Bjerknes, T.L., Witter, M.P., Moser, E.I., Moser, M.B. (2010). Development of the spatial representation system in the rat. Science, 328(5985), 1576–1580.

    Article  CAS  PubMed  Google Scholar 

  28. Manoharan, V.N. (2015). Colloidal matter: Packing, geometry, and entropy. Science, 349(6251), 1253751.

    Article  CAS  Google Scholar 

  29. McNaughton, B.L., Battaglia, F.P., Jensen, O., Moser, E.I., Moser, M.B. (2006). Path integration and the neural basis of the’cognitive map’. Nature reviews Neuroscience, 7(8), 663.

    Article  CAS  PubMed  Google Scholar 

  30. Merrill, D.A., Chiba, A.A., Tuszynski, M.H. (2001). Conservation of neuronal number and size in the entorhinal cortex of behaviorally characterized aged rats. Journal of Comparative Neurology, 438(4), 445–456.

    Article  CAS  PubMed  Google Scholar 

  31. Monasson, R., & Rosay, S. (2014). Crosstalk and transitions between multiple spatial maps in an attractor neural network model of the hippocampus: Collective motion of the activity. Physical Review E, Statistical, Nonlinear, and Soft Matter Physics, 89(3-1), 032803–032803.

    Article  CAS  PubMed  Google Scholar 

  32. Ocko, S.A., Hardcastle, K., Giocomo, L.M., Ganguli, S. (2018). Emergent elasticity in the neural code for space. Proceedings of the National Academy of Sciences, 115(50), E11798–E11806.

    Article  CAS  Google Scholar 

  33. Pastoll, H., Solanka, L., van Rossum, M.C., Nolan, M.F. (2013). Feedback inhibition enables theta-nested gamma oscillations and grid firing fields. Neuron, 77(1), 141–154.

    Article  CAS  PubMed  Google Scholar 

  34. Peierls, R. (1935). Quelques propriétés typiques des corps solides. Ann IH Poincare, 5, 177–222.

    Google Scholar 

  35. Sargolini, F., Fyhn, M., Hafting, T., McNaughton, B.L., Witter, M.P., Moser, M.B., Moser, E.I. (2006). Conjunctive representation of position, direction, and velocity in entorhinal cortex. Science, 312(5774), 758–762.

    Article  CAS  PubMed  Google Scholar 

  36. Si, B., & Treves, A. (2013). A model for the differentiation between grid and conjunctive units in medial entorhinal cortex. Hippocampus, 23(12), 1410–1424.

    Article  PubMed  Google Scholar 

  37. Si, B., Kropff, E., Treves, A. (2012). Grid alignment in entorhinal cortex. Biological cybernetics, pp 1–24.

  38. Solstad, T., Boccara, C.N., Kropff, E., Moser, M.B., Moser, E.I. (2008). Representation of geometric borders in the entorhinal cortex. Science, 322(5909), 1865–1868.

    Article  CAS  PubMed  Google Scholar 

  39. Spalla, D., Dubreuil, A., Rosay, S., Monasson, R., Treves, A. (2019). Can grid cell ensembles represent multiple spaces? submitted.

  40. Sprekeler, H. (2008). Slowness learning: Mathematical approaches and synaptic mechanisms. PhD thesis, Berlin, Humboldt-Univ.

  41. Stella, F., & Treves, A. (2015). The self-organization of grid cells in 3d. eLife, 4, e05913.

    Article  PubMed Central  Google Scholar 

  42. Stella, F., Si, B., Kropff, E., Treves, A. (2013). Grid cells on the ball. Journal of Statistical Mechanics:, Theory and Experiment, 2013(03), P03013.

    Article  Google Scholar 

  43. Stensola, H., Stensola, T., Solstad, T., Frøland, K., Moser, M.B., Moser, E.I. (2012). The entorhinal grid map is discretized. Nature, 492(7427), 72–78.

    Article  CAS  PubMed  Google Scholar 

  44. Stensola, T., Stensola, H., Moser, M.B., Moser, E.I. (2015). Shearing-induced asymmetry in entorhinal grid cells. Nature, 518(7538), 207–212.

    Article  CAS  PubMed  Google Scholar 

  45. Urdapilleta, E., Troiani, F., Stella, F., Treves, A. (2015). Can rodents conceive hyperbolic spaces? Journal of the Royal Society Interface, 12(107), 20141214.

    Article  PubMed Central  Google Scholar 

  46. Urdapilleta, E., Si, B., Treves, A. (2017). S elforganization of modular activity of grid cells. Hippocampus, 27(11), 1204–1213.

    Article  PubMed  PubMed Central  Google Scholar 

  47. Weber, S.N. (2018). https://www.gitlabtubittu-berlinde/simonweber/gridscore.

  48. Weber, S.N., & Sprekeler, H. (2018). Learning place cells, grid cells and invariances with excitatory and inhibitory plasticity. eLife, 7, e34560.

    Article  PubMed  PubMed Central  Google Scholar 

  49. Weber, S.N., & Sprekeler, H. (2019). A local measure of symmetry and orientation for individual spikes of grid cells. PLoS Computational Biology, 15(2), e1006804.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  50. Wernle, T., Waaga, T., Mørreaunet, M., Treves, A., Moser, M.B., Moser, E.I. (2018). Integration of grid maps in merged environments. Nature Neuroscience, 21(1), 92.

    Article  CAS  PubMed  Google Scholar 

  51. Widloski, J., & Fiete, I.R. (2014). A model of grid cell development through spatial exploration and spike time-dependent plasticity. Neuron, 83(2), 481–495.

    Article  CAS  PubMed  Google Scholar 

  52. Yartsev, M.M., Witter, M.P., Ulanovsky, N. (2011). Grid cells without theta oscillations in the entorhinal cortex of bats. Nature, 479(7371), 103.

    Article  CAS  PubMed  Google Scholar 

  53. Yoon, K., Buice, M.A., Barry, C., Hayman, R., Burgess, N., Fiete, I.R. (2013). Specific evidence of low-dimensional continuous attractor dynamics in grid cells. Nature Neuroscience, 16(8), 1077.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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Acknowledgements

We are grateful to Alessandro Treves, Rémi Monasson, Giuseppe D’Adamo, Thomas Gueudré and Henning Sprekeler for their remarks on the model and its relationship with physics. We thank Tanja Wernle for extensive discussion on the merging experiment; Bailu Si and Eugenio Urdapilleta for their code of the adaptation model. Many thanks also to John Nicholls. The idea of grid alignment via recurrent inhibitory connections was developed in discussions between Henning Sprekeler and S.W.. S.R. would like to thank the GRIDMAP project for financial support and the Abdus Salam International Centre for Theoretical Physics in Trieste for hospitality in the conclusive phase of this work. S.W. was funded by the German Federal Ministry for Education and Research, FKZ 01GQ1201.

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Correspondence to Sophie Rosay.

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Rosay, S., Weber, S. & Mulas, M. Modeling grid fields instead of modeling grid cells. J Comput Neurosci 47, 43–60 (2019). https://doi.org/10.1007/s10827-019-00722-8

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Keywords

  • Hippocampus
  • Grid cells
  • Effective model
  • Physics of 2d systems