Optimal Population Codes for Space: Grid Cells Outperform Place Cells

Rodents use two distinct neuronal coordinate systems to estimate their position: place fields in the hippocampus and grid fields in the entorhinal cortex. Whereas place cells spike at only one particular spatial location, grid cells fire at multiple sites that correspond to the points of an imaginary hexagonal lattice. We study how to best construct place and grid codes, taking the probabilistic nature of neural spiking into account. Which spatial encoding properties of individual neurons confer the highest resolution when decoding the animal's position from the neuronal population response? A priori, estimating a spatial position from a grid code could be ambiguous, as regular periodic lattices possess translational symmetry. The solution to this problem requires lattices for grid cells with different spacings; the spatial resolution crucially depends on choosing the right ratios of these spacings across the population. We compute the expected error in estimating the position in both the asymptotic limit, using Fisher information, and for low spike counts, using maximum likelihood estimation. Achieving high spatial resolution and covering a large range of space in a grid code leads to a trade-off: the best grid code for spatial resolution is built of nested modules with different spatial periods, one inside the other, whereas maximizing the spatial range requires distinct spatial periods that are pairwisely incommensurate. Optimizing the spatial resolution predicts two grid cell properties that have been experimentally observed. First, short lattice spacings should outnumber long lattice spacings. Second, the grid code should be self-similar across different lattice spacings, so that the grid field always covers a fixed fraction of the lattice period. If these conditions are satisfied and the spatial “tuning curves” for each neuron span the same range of firing rates, then the resolution of the grid code easily exceeds that of the best possible place code with the same number of neurons.

[1]  A. Treves,et al.  Distinct Ensemble Codes in Hippocampal Areas CA3 and CA1 , 2004, Science.

[2]  E. L. Lehmann,et al.  Theory of point estimation , 1950 .

[3]  M. Stemmler,et al.  Multiscale codes in the nervous system: the problem of noise correlations and the ambiguity of periodic scales. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Roland Vollgraf,et al.  From grids to places , 2007, Journal of Computational Neuroscience.

[5]  M. Fyhn,et al.  Progressive increase in grid scale from dorsal to ventral medial entorhinal cortex , 2008, Hippocampus.

[6]  M. Hasselmo,et al.  Coupled Noisy Spiking Neurons as Velocity-Controlled Oscillators in a Model of Grid Cell Spatial Firing , 2010, The Journal of Neuroscience.

[7]  Charlotte N. Boccara,et al.  Grid cells in pre- and parasubiculum , 2010, Nature Neuroscience.

[8]  Christian F. Doeller,et al.  Evidence for grid cells in a human memory network , 2010, Nature.

[9]  W. Newsome,et al.  The Variable Discharge of Cortical Neurons: Implications for Connectivity, Computation, and Information Coding , 1998, The Journal of Neuroscience.

[10]  K. Jeffery,et al.  Experience-dependent rescaling of entorhinal grids , 2007, Nature Neuroscience.

[11]  G. Buzsáki Rhythms of the brain , 2006 .

[12]  William R. Softky,et al.  The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs , 1993, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[13]  May-Britt Moser,et al.  Place cells, spatial maps and the population code for memory , 2005, Current Opinion in Neurobiology.

[14]  Alexander Mathis,et al.  Resolution of nested neuronal representations can be exponential in the number of neurons. , 2012, Physical review letters.

[15]  M J West,et al.  Neuron numbers in the presubiculum, parasubiculum, and entorhinal area of the rat , 1997, The Journal of comparative neurology.

[16]  Herz Andreas,et al.  Single-run phase precession in entorhinal grid cells , 2010 .

[17]  Christian W. Eurich,et al.  Multidimensional Encoding Strategy of Spiking Neurons , 2000, Neural Computation.

[18]  J. O’Keefe,et al.  Grid cells and theta as oscillatory interference: Electrophysiological data from freely moving rats , 2008, Hippocampus.

[19]  J. O’Keefe,et al.  An oscillatory interference model of grid cell firing , 2007, Hippocampus.

[20]  G. Buzsáki,et al.  Gamma (40-100 Hz) oscillation in the hippocampus of the behaving rat , 1995, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[21]  Ila R Fiete,et al.  What Grid Cells Convey about Rat Location , 2008, The Journal of Neuroscience.

[22]  R. Muller,et al.  Place cell discharge is extremely variable during individual passes of the rat through the firing field. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[23]  Christian W. Eurich,et al.  Representational Accuracy of Stochastic Neural Populations , 2002, Neural Computation.

[24]  B L McNaughton,et al.  Dynamics of the hippocampal ensemble code for space. , 1993, Science.

[25]  Boris S. Gutkin,et al.  Democracy-Independence Trade-Off in Oscillating Dendrites and Its Implications for Grid Cells , 2010, Neuron.

[26]  A. Treves,et al.  Hippocampal remapping and grid realignment in entorhinal cortex , 2007, Nature.

[27]  Simon M Stringer,et al.  Entorhinal cortex grid cells can map to hippocampal place cells by competitive learning , 2006, Network.

[28]  Paul F M J Verschure,et al.  Prediction of the position of an animal based on populations of grid and place cells: a comparative simulation study. , 2007, Journal of integrative neuroscience.

[29]  Herz Andreas Movement Related Statistics of Grid Cell Firing , 2010 .

[30]  R. Zemel,et al.  Inference and computation with population codes. , 2003, Annual review of neuroscience.

[31]  Peter E. Latham,et al.  Narrow Versus Wide Tuning Curves: What's Best for a Population Code? , 1999, Neural Computation.

[32]  Yoram Burak,et al.  Triangular lattice neurons may implement an advanced numeral system to precisely encode rat position over large ranges , 2006 .

[33]  Terrence J. Sejnowski,et al.  Neuronal Tuning: To Sharpen or Broaden? , 1999, Neural Computation.

[34]  G. Einevoll,et al.  From grid cells to place cells: A mathematical model , 2006, Hippocampus.

[35]  T. Hafting,et al.  Finite Scale of Spatial Representation in the Hippocampus , 2008, Science.

[36]  H Sompolinsky,et al.  Simple models for reading neuronal population codes. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[37]  G. Buzsáki,et al.  Intrinsic Circuit Organization and Theta–Gamma Oscillation Dynamics in the Entorhinal Cortex of the Rat , 2010, The Journal of Neuroscience.

[38]  Emilio Salinas,et al.  Vector reconstruction from firing rates , 1994, Journal of Computational Neuroscience.

[39]  Yoram Burakyy,et al.  Accurate Path Integration in Continuous Attractor Network Models of Grid Cells , 2009 .

[40]  R. Passingham The hippocampus as a cognitive map J. O'Keefe & L. Nadel, Oxford University Press, Oxford (1978). 570 pp., £25.00 , 1979, Neuroscience.

[41]  Torkel Hafting,et al.  Conjunctive Representation of Position, Direction, and Velocity in Entorhinal Cortex , 2006, Science.

[42]  M. Paradiso,et al.  A theory for the use of visual orientation information which exploits the columnar structure of striate cortex , 2004, Biological Cybernetics.

[43]  M. Fyhn,et al.  Spatial Representation in the Entorhinal Cortex , 2004, Science.

[44]  Lisa M. Giocomo,et al.  Temporal Frequency of Subthreshold Oscillations Scales with Entorhinal Grid Cell Field Spacing , 2007, Science.

[45]  Neil Burgess,et al.  Optimal configurations of spatial scale for grid cell firing under noise and uncertainty , 2014, Philosophical Transactions of the Royal Society B: Biological Sciences.

[46]  K. Jeffery Self-localization and the entorhinal–hippocampal system , 2007, Current Opinion in Neurobiology.

[47]  Nicolas Brunel,et al.  Mutual Information, Fisher Information, and Population Coding , 1998, Neural Computation.

[48]  Sachin S. Deshmukh,et al.  Theta modulation in the medial and the lateral entorhinal cortices. , 2010, Journal of neurophysiology.

[49]  William W Lytton,et al.  Unmasking the CA1 Ensemble Place Code by Exposures to Small and Large Environments: More Place Cells and Multiple, Irregularly Arranged, and Expanded Place Fields in the Larger Space , 2008, The Journal of Neuroscience.

[50]  Alexander Mathis,et al.  How good is grid coding versus place coding for navigation using noisy, spiking neurons? , 2010, BMC Neuroscience.

[51]  Daniel L. Schacter,et al.  Spatial Representation in the Entorhinal Cortex , 2004 .

[52]  Kenneth D Harris,et al.  Theta-Mediated Dynamics of Spatial Information in Hippocampus , 2008, The Journal of Neuroscience.

[53]  Mark C. Fuhs,et al.  A Spin Glass Model of Path Integration in Rat Medial Entorhinal Cortex , 2006, The Journal of Neuroscience.

[54]  Frank Loren From grid cells to place cells: a generic and robust principle accounts for multiple spatial Maps , 2010 .

[55]  Bruce L. McNaughton,et al.  Path integration and the neural basis of the 'cognitive map' , 2006, Nature Reviews Neuroscience.

[56]  Yasser Roudi,et al.  Network mechanisms of grid cells , 2014, Philosophical Transactions of the Royal Society B: Biological Sciences.

[57]  J. O'Keefe,et al.  The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. , 1971, Brain research.

[58]  R. Muller,et al.  Attention-Like Modulation of Hippocampus Place Cell Discharge , 2010, The Journal of Neuroscience.

[59]  Kamran Diba,et al.  Temporal delays among place cells determine the frequency of population theta oscillations in the hippocampus , 2010, Proceedings of the National Academy of Sciences.

[60]  R. Muller,et al.  Head-direction cells recorded from the postsubiculum in freely moving rats. II. Effects of environmental manipulations , 1990, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[61]  Peter Dayan,et al.  Fast Population Coding , 2007, Neural Computation.

[62]  M. Nolan,et al.  Tuning of Synaptic Integration in the Medial Entorhinal Cortex to the Organization of Grid Cell Firing Fields , 2008, Neuron.

[63]  Edvard I. Moser,et al.  Grid Cells and Neural Coding in High-End Cortices , 2013, Neuron.

[64]  Alessandro Treves,et al.  A model for the differentiation between grid and conjunctive units in medial entorhinal cortex , 2013, Hippocampus.

[65]  Matthew A. Wilson,et al.  Neural Representation of Spatial Topology in the Rodent Hippocampus , 2013, Neural Computation.

[66]  Lisa M. Giocomo,et al.  Grid cell firing may arise from interference of theta frequency membrane potential oscillations in single neurons , 2007, Hippocampus.

[67]  Jadin C. Jackson,et al.  Network dynamics of hippocampal cell‐assemblies resemble multiple spatial maps within single tasks , 2007, Hippocampus.

[68]  Jason Cong,et al.  Oscillatory neurocomputing with ring attractors: a network architecture for mapping locations in space onto patterns of neural synchrony , 2014, Philosophical Transactions of the Royal Society B: Biological Sciences.

[69]  R U Muller,et al.  Head-direction cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis , 1990, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[70]  T. Hafting,et al.  Microstructure of a spatial map in the entorhinal cortex , 2005, Nature.

[71]  R. K. Simpson Nature Neuroscience , 2022 .

[72]  Jonathan R. Whitlock,et al.  Fragmentation of grid cell maps in a multicompartment environment , 2009, Nature Neuroscience.

[73]  J. O’Keefe Place units in the hippocampus of the freely moving rat , 1976, Experimental Neurology.

[74]  M. Moser,et al.  Pattern Separation in the Dentate Gyrus and CA3 of the Hippocampus , 2007, Science.

[75]  Herz Andreas Exponential Scaling of Nested Neuronal Representations , 2011 .

[76]  Alessandro Treves,et al.  The role of competitive learning in the generation of DG fields from EC inputs , 2009, Cognitive Neurodynamics.

[77]  F. Attneave Some informational aspects of visual perception. , 1954, Psychological review.

[78]  N. Schmajuk Cognitive maps , 1998 .

[79]  Yonina C. Eldar,et al.  Bayesian Filtering in Spiking Neural Networks: Noise, Adaptation, and Multisensory Integration , 2009, Neural Computation.

[80]  N. Burgess Grid cells and theta as oscillatory interference: Theory and predictions , 2008, Hippocampus.

[81]  W. Michael Brown,et al.  Optimal Neuronal Tuning for Finite Stimulus Spaces , 2004, Neural Computation.

[82]  Matthias Bethge,et al.  Optimal Short-Term Population Coding: When Fisher Information Fails , 2002, Neural Computation.

[83]  Alessandro Treves,et al.  The emergence of grid cells: Intelligent design or just adaptation? , 2008, Hippocampus.

[84]  T. Hafting,et al.  Hippocampus-independent phase precession in entorhinal grid cells , 2008, Nature.

[85]  Stephen Grossberg,et al.  Grid cell hexagonal patterns formed by fast self‐organized learning within entorhinal cortex , 2012, Hippocampus.

[86]  J. O’Keefe,et al.  Phase relationship between hippocampal place units and the EEG theta rhythm , 1993, Hippocampus.