Skip to main content
SpringerLink
Log in

Computational Models of Grid Cell Firing

  • Chapter
  • First Online:
Hippocampal Microcircuits

Part of the book series: Springer Series in Computational Neuroscience ((NEUROSCI))

Abstract

Grid cells in the medial entorhinal cortex (mEC) fire whenever the animal enters a regular triangular array of locations that cover its environment. Since their discovery, several models that can account for these remarkably regular spatial firing patterns have been proposed. These generally fall into one of three classes, generating grid cell firing patterns either by oscillatory interference, through continuous attractor dynamics, or as a result of spatially modulated input from a place cell population. Neural network simulations have been used to explore the implications and predictions made by each class of model, while subsequent experimental data have allowed their architecture to be refined. Here, we describe implementations of two classes of grid cell model – oscillatory interference and continuous attractor dynamics – alongside a hybrid model that incorporates the principal features of each. These models are intended to be both parsimonious and make testable predictions. We discuss the strengths and weaknesses of each model and the predictions they make for future experimental manipulations of the grid cell network in vivo.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Alonso A, Llinas RR (1989) Subthreshold Na+-dependent theta-like rhythmicity in stellate cells of entorhinal cortex layer II. Nature 342:175–177

    Article  CAS  PubMed  Google Scholar 

  • Andrzejak RG, Bicanski A (2007) Forming place cells through feedforward input from grid cells – a computational model. Soc Neurosci Abst 753:1

    Google Scholar 

  • Barry C, Lever C, Hayman R, Hartley T, Burton S, O’Keefe J, Jeffery K, Burgess N (2006) The boundary vector cell model of place cell firing and spatial memory. Rev Neurosci 17:71–97

    Article  PubMed  PubMed Central  Google Scholar 

  • Barry C, Hayman R, Burgess N, Jeffery K (2007) Experience-dependent rescaling of entorhinal grids. Nat Neurosci 10:682–684

    Article  CAS  PubMed  Google Scholar 

  • Barry C, Ginsberg LL, O’Keefe J, Burgess N (2012a) Grid cell firing patterns signal environmental novelty by expansion. PNAS 109:17687–17692

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Barry C, Bush D, O’Keefe J, Burgess N (2012b) Models of grid cells and theta oscillations. Nature 488:E1

    Article  CAS  PubMed  Google Scholar 

  • Bittner KC, Grienberger C, Vaidya SP, Milstein AD, Macklin JJ, Suh J, Tonegawa S, Magee JC (2015) Conjunctive input processing drives feature selectivity in hippocampal CA1 neurons. Nat Neurosci 18:1133–1142

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Blair HT, Gupta K, Zhang K (2008) Conversion of a phase- to a rate-coded position signal by a three stage model of theta cells, place cells, and grid cells. Hippocampus 18:1239–1255

    Article  PubMed  PubMed Central  Google Scholar 

  • Blair HT, Wu D, Cong J (2014) Synchronization coding by ring attractors: a theoretical framework for oscillatory neurocomputing. Philos Trans R Soc Lond B 369:20120526

    Article  Google Scholar 

  • Boccara CN, Sargolini F, Thoresen VH, Solstad T, Witter MP, Moser EI, Moser MB (2010) Grid cells in pre- and parasubiculum. Nat Neurosci 13:987–994

    Article  CAS  PubMed  Google Scholar 

  • Bonnevie T, Dunn B, Fyhn M, Hafting T, Derdikmann D, Kubie JL, Roudi Y, Moser EI, Moser M-B (2013) Grid cells require excitatory drive from the hippocampus. Nat Neurosci 16:309–317

    Article  CAS  PubMed  Google Scholar 

  • Brandon MP, Bogaard AR, Libby CP, Connerney MA, Gupta K, Hasselmo ME (2011) Reduction of theta rhythm dissociates grid cell spatial periodicity from directional tuning. Science 332:595–599

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Buetfering C, Allen K, Monyer H (2014) Parvalbumin interneurons provide grid cell–driven recurrent inhibition in the medial entorhinal cortex. Nat Neurosci 17:710–718

    Article  CAS  PubMed  Google Scholar 

  • Burak Y, Fiete IR (2009) Accurate path integration in continuous attractor network models of grid cells. PLoS Comput Biol 5:e1000291

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Burgess N, Barry C, Jeffery KJ, O’Keefe J (2005) A grid and place cell model of path integration utilizing phase precession versus theta. Computational cognitive neuroscience conference poster; Washington, DC: http://f1000.com/posters/browse/summary/225

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

    Article  PubMed  PubMed Central  Google Scholar 

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

    Article  PubMed  PubMed Central  Google Scholar 

  • Burgess CP, Burgess N (2014) Controlling phase noise in oscillatory interference models of grid cell firing. J Neurosci 34:6224–6232

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Bush D, Burgess N (2014) A hybrid oscillatory interference/continuous attractor network model of grid cell firing. J Neurosci 34:5065–5079

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Bush D, Barry C, Manson D, Burgess N (2015) Using grid cells for navigation. Neuron 87:507–520

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Cei A, Girardeau G, Drieu C, El Kanbi K, Zugaro M (2014) Reversed theta sequences of hippocampal cell assemblies during backward travel. Nat Neurosci 17:719–724

    Article  CAS  PubMed  Google Scholar 

  • Chen G, Manson D, Cacucci F, Wills TJ (2016) Absence of visual input results in the disruption of grid cell firing in the mouse. Curr Biol 26:2335

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Climer JR, Newman EL, Hasselmo ME (2013) Phase coding by grid cells in unconstrained environments: two-dimensional phase precession. Eur J Neurosci 38:2526–2541

    Article  PubMed  PubMed Central  Google Scholar 

  • Conklin J, Eliasmith C (2005) An attractor network model of path integration in the rat. J Comput Neurosci 18:183–203

    Article  PubMed  Google Scholar 

  • Couey JJ, Witoelar A, Zhang S-J, Zheng K, Ye J, Dunn B, Czajkowski R, Moser M-B, Moser EI, Roudi Y, Witter MP (2013) Recurrent inhibitory circuitry as a mechanism for grid formation. Nat Neurosci 16:318–324

    Article  CAS  PubMed  Google Scholar 

  • Derdikman D, Whitlock JR, Tsao A, Fyhn M, Hafting T, Moser M-B, Moser EI (2009) Fragmentation of grid cell maps in multicompartment environment. Nat Neurosci 12:1325–1332

    Article  CAS  PubMed  Google Scholar 

  • Dhillon A, Jones R (2000) Laminar differences in recurrent excitatory transmission in the rat entorhinal cortex in vitro. Neuroscience 99:413–422

    Article  CAS  PubMed  Google Scholar 

  • Doeller CF, Barry C, Burgess N (2010) Evidence for grid cells in a human memory network. Nature 463:657–661

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Dordek Y, Soudry D, Meir R, Derdikman D (2016) Extracting grid cell characteristics from place cell inputs using non-negative principal component analysis. elife 5:e10094

    Article  PubMed  PubMed Central  Google Scholar 

  • Domnisoru C, Kinkhabwala AA, Tank DW (2013) Membrane potential dynamics of grid cells. Nature 495:199–204

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Eliav T, Geva-Sagiv M, Finkelstein A, Yartsev M, Rubin A, Las L, Ulanovksy N (2015) Synchronicity without rhythmicity in the hippocampal formation of behaving bats. Soc Neurosci Abstr 632:01

    Google Scholar 

  • Ekstrom AD, Caplan JB, Shattuck K, Fried I, Kahana MJ (2005) Human hippocampal theta activity during virtual navigation. Hippocampus 15:881–889

    Article  PubMed  Google Scholar 

  • Evans T, Bicanski A, Bush D, Burgess N (2016) How environment and self-motion combine in neural representations of space. J Physiol 594:6535

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fiete IR, Burak Y, Brookings T (2008) What grid cells convey about rat location. J Neurosci 28:6858–6871

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fuchs EC, Neitz A, Pinna R, Melzer S, Caputi A, Monyer H (2016) Local and distant input controlling excitation in layer II of the medial entorhinal cortex. Neuron 89:194–208

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fuhs MC, Touretzky DS (2006) A spin glass model of path integration in rat medial entorhinal cortex. J Neurosci 26:4266–4276

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Fyhn M, Hafting T, Treves A, Moser M-B, Moser EI (2007) Hippocampal remapping and grid realignment in entorhinal cortex. Nature 446:190–194

    Article  CAS  PubMed  Google Scholar 

  • Fyhn M, Hafting T, Witter MP, Moser EI, Moser MB (2008) Grid cells in mice. Hippocampus 18:1230–1238

    Article  PubMed  Google Scholar 

  • Garden DLF, Dodson PD, O’Donnell C, White MD, Nolan MF (2008) Tuning of synaptic integration in the medial entorhinal cortex to the Organization of Grid Cell Firing Fields. Neuron 60:875–889

    Article  CAS  PubMed  Google Scholar 

  • Gatome CW, Slomianka L, Lipp HP, Amrein I (2010) Number estimates of neuronal phenotypes in layer II of the medial entorhinal cortex of rat and mouse. Neuroscience 170:156–165

    Article  CAS  PubMed  Google Scholar 

  • Giocomo LM, Zilli EA, Fransén E, Hasselmo ME (2007) Temporal frequency of subthreshold oscillations scales with entorhinal grid cell field spacing. Science 315:1719–1722

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Giocomo LM, Moser M-B, Moser EI (2011) Computational models of grid cells. Neuron 71:589–603

    Article  CAS  PubMed  Google Scholar 

  • Guanella A, Kiper D, Verschure P (2007) A model of grid cells based on a twisted torus topology. Int J Neural Syst 17:231–240

    Article  PubMed  Google Scholar 

  • Hafting T, Fyhn M, Molden S, Moser MB, Moser EI (2005) Microstructure of a spatial map in the entorhinal cortex. Nature 436:801–806

    Article  CAS  PubMed  Google Scholar 

  • Hafting T, Fyhn M, Bonnevie T, Moser MB, Moser EI (2008) Hippocampus-independent phase precession in entorhinal grid cells. Nature 453:1248–1252

    Article  CAS  PubMed  Google Scholar 

  • Hardcastle K, Ganguli S, Giocomo LM (2015) Environmental boundaries as an error correction mechanism for grid cells. Neuron 86:1–13

    Article  CAS  Google Scholar 

  • Hasselmo ME (2008) Grid cell mechanisms and function: contributions of entorhinal persistent spiking and phase resetting. Hippocampus 18:1116–1126

    Google Scholar 

  • Heys JG, Rangarajan KV, Dombeck DA (2014) The functional micro-organization of grid cells revealed by cellular-resolution imaging. Neuron 84:1079–1090

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Horiuchi TK, Moss CF (2015) Grid cells in 3-D: reconciling data and models. Hippocampus 25:1489–1500

    Article  PubMed  Google Scholar 

  • Jacobs J, Weidemann CT, Miller JF, Solway A, Burke JF, Wei X, Suthana N, Sperling MR, Sharan AD, Fried I, Kahana MJ (2013) Direct recordings of grid-like neuronal activity in human spatial navigation. Nat Neurosci 16:1188–1190

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Jeewajee A, Barry C, O'Keefe J, Burgess N (2008) Grid cells and theta as oscillatory interference: electrophysiological data from freely-moving rats. Hippocampus 18:1175–1185

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Jeewajee A, Barry C, Douchamps V, Manson D, Lever C, Burgess N (2014) Theta phase precession of grid and place cell firing in open environments. Philos Trans R Soc B 369:20120532

    Article  CAS  Google Scholar 

  • Killian NJ, Jutras MJ, Buffalo EA (2012) A map of visual space in the primate entorhinal cortex. Nature 491:761–764

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Kitamura T, Pignatelli M, Suh J, Kohara K, Yoshiki A, Abe K, Tonegawa S (2014) Island cells control temporal association memory. Science 343:896–901

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Koenig J, Linder AN, Leutgeb JK, Leutgeb S (2011) The spatial periodicity of grid cells is not sustained during reduced theta oscillations. Science 332:592–595

    Article  CAS  PubMed  Google Scholar 

  • Kropff E, Treves A (2008) The emergence of grid cells: intelligent design or just adaptation? Hippocampus 18:1256–1269

    Article  PubMed  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Langston RF, Ainge JA, Couey JJ, Canto CB, Bjerknes TL, Witter MP, Moser EI, Moser MB (2010) Development of the spatial representation system in the rat. Science 328:1576–1580

    Article  CAS  PubMed  Google Scholar 

  • Lee D, Lin BJ, Lee AK (2012) Hippocampal place fields emerge upon single-cell manipulation of excitability during behavior. Science 337:849–853

    Article  CAS  PubMed  Google Scholar 

  • Lengyel M, Szatmáry Z, Érdi P (2003) Dynamically detuned oscillations account for the coupled rate and temporal code of place cell firing. Hippocampus 13:700–714

    Article  PubMed  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • McFarland WL, Teitelbaum H, Hedges EK (1975) Relationship between hippocampal theta activity and running speed in the rat. J Comp Physiol Psychol 88:324–328

    Article  CAS  PubMed  Google Scholar 

  • McNaughton BL, Battaglia FP, Jensen O, Moser EI, Moser MB (2006) Path integration and the neural basis of the cognitive map. Nat Rev Neurosci 7:663–678

    Article  CAS  PubMed  Google Scholar 

  • Mhatre H, Gorchetchnikov A, Grossberg S (2012) Grid cell hexagonal patterns formed by fast self-organized learning within entorhinal cortex. Hippocampus 22:320–334

    Article  PubMed  Google Scholar 

  • Naumann RK, Ray S, Prokop S, Las L, Heppner FL, Brecht M (2015) Conserved size and periodicity of pyramidal patches in layer 2 of medial/caudal entorhinal cortex. J Comp Neurol 524:783–806

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Navratilova Z, Giocomo LM, Fellous JM, Hasselmo ME, McNaughton BL (2012) Phase precession and variable spatial scaling in a periodic attractor map model of medial entorhinal grid cells with realistic after-spike dynamics. Hippocampus 22:772–789

    Article  PubMed  Google Scholar 

  • O'Keefe J, Nadel L (1978) The Hippocampus as a cognitive map. Oxford University Press, Oxford

    Google Scholar 

  • O'Keefe J, Recce ML (1993) Phase relationship between hippocampal place units and the EEG theta rhythm. Hippocampus 3:317–330

    Article  CAS  PubMed  Google Scholar 

  • Ólafsdóttir HF, Carpenter F, Barry C (2016) Coordinated grid and place cell replay during rest. Nat Neurosci 19:792–794

    Article  PubMed  CAS  Google Scholar 

  • Orchard J (2015) Oscillator-interference models of path integration do not require theta oscillations. Neural Comput 27:548–560

    Article  PubMed  Google Scholar 

  • Pastoll H, Ramsden H, Nolan MF (2012) Intrinsic electrophysiological properties of entorhinal cortex stellate cells and their contribution to grid firing fields. Front Neural Circuits 6:1–21

    Article  Google Scholar 

  • Pastoll H, Solanka L, van Rossum MCW, Nolan MF (2013) Feedback inhibition enables theta-nested gamma oscillations and grid firing fields. Neuron 77:141–154

    Article  CAS  PubMed  Google Scholar 

  • Pérez-Escobar JA, Kornienko O, Latuske P, Kohler L, Allen K (2016) Visual landmarks sharpen grid cell metric and confer context specificity to neurons of the medial entorhinal cortex. elife 5: e16937

    Google Scholar 

  • Raudies F, Brandon MP, Chapman GW, Hasselmo ME (2015) Head direction is coded more strongly than movement direction in a population of entorhinal neurons. Brain Res 1621:355–367

    Article  CAS  PubMed  Google Scholar 

  • Ray S, Naumann R, Burgalossi A, Tang Q, Schmidt H, Brecht M (2014) Grid-layout and theta-modulation of layer 2 pyramidal neurons in medial entorhinal cortex. Science 343:891–896

    Article  CAS  PubMed  Google Scholar 

  • Reifenstein ET, Kempter R, Schreiber S, Stemmler MB, Herz AV (2012) Grid cells in rat entorhinal cortex encode physical space with independent firing fields and phase precession at the single-trial level. PNAS 109:6301–6306

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Reifenstein E, Stemmler M, Herz AVM, Kempter R, Schreiber S (2014) Movement dependence and layer specificity of entorhinal phase precession in two-dimensional environments. PLoS One 9:e100638

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  • Remme MW, Lengyel M, Gutkin BS (2010) Democracy-independence trade-off in oscillating dendrites and its implications for grid cells. Neuron 66:429–437

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Rivas J, Gaztelu JM, García-Austt E (1996) Changes in hippocampal cell discharge patterns and theta rhythm spectral properties as a function of walking velocity in the Guinea pig. Exp Brain Res 108:113–118

    Article  CAS  PubMed  Google Scholar 

  • Rolls ET, Stringer SM, Elliot T (2006) Entorhinal cortex grid cells can map to hippocampal place cells by competitive learning. Netw Comput Neural Syst 447:447–465

    Article  Google Scholar 

  • Samsonovich A, McNaughton BL (1997) Path integration and cognitive mapping in a continuous attractor neural network model. J Neurosci 17:5900–5920

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Sargolini F, Fyhn M, Hafting T, McNaughton BL, Witter MP, Moser MB, Moser EI (2006) Conjunctive representation of position, direction, and velocity in entorhinal cortex. Science 312:758–762

    Article  CAS  PubMed  Google Scholar 

  • Savelli F, Yoganarasimha D, Knierim JJ (2008) Influence of boundary removal on the spatial representations of the medial entorhinal cortex. Hippocampus 18:1270–1282

    Article  PubMed  PubMed Central  Google Scholar 

  • Schmidt-Hieber C, Häusser M (2013) Cellular mechanisms of spatial navigation in the medial entorhinal cortex. Nat Neurosci 16:325–331

    Article  CAS  PubMed  Google Scholar 

  • Schmidt-Hieber C, Häusser M (2014) How to build a grid cell. Philos Trans R Soc B 369:20120520

    Article  Google Scholar 

  • Schmidt-Hieber C, Toleikyte G, Aitchison L, Roth A, Clark BA, Branco T, Häusser M (2017) Active dendritic integration as a mechanism for robust and precise grid cell firing. Nat Neurosci 20:1114–1121

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Schmidt-Hieber C, Nolan MF (2017) Synaptic integrative mechanisms for spatial cognition. Nat Neurosci 26:1483–1492

    Article  CAS  Google Scholar 

  • Sheffield ME, Dombeck DA (2015) Calcium transient prevalence across the dendritic arbour predicts place field properties. Nature 517:200–204

    Article  CAS  PubMed  Google Scholar 

  • Shipston-Sharman O, Solanka L, Nolan MF (2016) Continuous attractor network models of grid cell firing based on excitatory–inhibitory interactions. J Physiol 594:6547.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Solanka L, van Rossum MCW, Nolan MF (2015) Noise promotes independent control of gamma oscillations and grid firing within a recurrent attractor network. elife 4:e06444

    Article  PubMed Central  Google Scholar 

  • Solstad T, Boccara CN, Kropff E, Moser MB, Moser EI (2008) Representation of geometric borders in the entorhinal cortex. Science 322:1865–1868

    Article  CAS  PubMed  Google Scholar 

  • Stemmler M, Mathis A, Herz AVM (2015) Connecting multiple spatial scales to decode the population activity of grid cells. Sci Adv 1:e1500816

    Article  PubMed  PubMed Central  Google Scholar 

  • Stensola H, Stensola T, Solstad T, Frøland K, Moser MB, Moser EI (2012) The entorhinal grid map is discretized. Nature 492:72–78

    Article  CAS  PubMed  Google Scholar 

  • Stensola T, Stensola H, Moser M-B, Moser EI (2015) Shearing-induced asymmetry in entorhinal grid cells. Nature 518:207–212

    Article  CAS  PubMed  Google Scholar 

  • Sun C, Kitamura T, Yamamoto J, Martin J, Pignatelli M, Kitch LJ, Schnitzer MJ, Tonegawa S (2015) Distinct speed dependence of entorhinal island and ocean cells, including respective grid cells. PNAS 112:9466–9471

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Tocker G, Barak O, Derdikman D (2015) Grid cells correlation structure suggests organized feedforward projections into superficial layers of the medial entorhinal cortex. Hippocampus 25:1599–1613

    Article  PubMed  Google Scholar 

  • Vanderwolf CH (1969) Hippocampal electrical activity and voluntary movement in the rat. EEG Clin Neurophysiol 26:407–418

    Article  CAS  Google Scholar 

  • Watrous AJ, Lee DJ, Izadi A, Gurkoff GG, Shahlaie K, Ekstrom AD (2013) A comparative study of human and rat hippocampal low-frequency oscillations during spatial navigation. Hippocampus 23:656–661

    Article  PubMed  PubMed Central  Google Scholar 

  • Welday AC, Shlifer IG, Bloom ML, Zhang K, Blair HT (2011) Cosine directional tuning of theta cell burst frequencies: evidence for spatial coding by oscillatory interference. J Neurosci 31:16157–16176

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Welinder PE, Burak Y, Fiete IR (2008) Grid cells: the position code, neural network models of activity, and the problem of learning. Hippocampus 18:1283–1300

    Article  PubMed  Google Scholar 

  • Wells CE, Amos DP, Jeewajee A, Douchamps V, Rodgers RJ, O’Keefe J, Burgess N, Lever C (2013) Novelty and anxiolytic drugs dissociate two components of hippocampal theta in behaving rats. J Neurosci 33:8650–8667

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Widlowski J, Fiete IR (2014) A model of grid cell development through spatial exploration and spike time-dependent plasticity. Neuron 83:481–495

    Article  CAS  Google Scholar 

  • Wills TJ, Cacucci F, Burgess N, O’Keefe J (2010) Development of the hippocampal cognitive map in pre-weanling rats. Science 328:1573–1576

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Winter SS, Mehlman ML, Clark BJ, Taube JS (2015) Passive transport disrupts grid signals in the Parahippocampal cortex. Curr Biol 25:2493–2502

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Yartsev MM, Witter MP, Ulanovsky N (2011) Grid cells without theta oscillations in the entorhinal cortex of bats. Nature 479:103–107

    Article  CAS  PubMed  Google Scholar 

  • Yoon K, Buice MA, Barry C, Hayman R, Burgess N, Fiete IR (2013) Specific evidence of low-dimensional continuous attractor dynamics in grid cells. Nat Neurosci 16:1077–1084

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Yoon K, Lewallen S, Kinkhabwala AA, Tank DW, Fiete IR (2016) Grid cell responses in 1D environments assessed as slices through a 2D lattice. Neuron 89:1086–1099

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  • Zilli EA (2012) Models of grid cell spatial firing published 2005–2011. Front Neural Circuits 6:16

    Article  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Andrej Bicanski, Neil Burgess, Talfan Evans, Robin Hayman, Matt Nolan and Freyja Ólafsdóttir for helpful comments and discussion during the preparation of this chapter. This work was supported by a grant from the ERC (StG 678790 NEWRON to C.S.-H.).

Author information

Authors and Affiliations

  1. UCL Institute of Cognitive Neuroscience, London, UK

    Daniel Bush

  2. UCL Institute of Neurology, London, UK

    Daniel Bush

  3. Institut Pasteur, Paris, France

    Christoph Schmidt-Hieber

Authors
  1. Daniel Bush
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Schmidt-Hieber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Daniel Bush or Christoph Schmidt-Hieber .

Editor information

Editors and Affiliations

  1. School of Computer Science, University of Lincoln, Lincoln, UK

    Vassilis Cutsuridis

  2. School of Natural Sciences, Department of Computing Science and Mathematics, University of Stirling, Stirling, UK

    Bruce P. Graham

  3. Centre for Discovery Brain Sciences, University of Edinburgh, Edinburgh, UK

    Stuart Cobb

  4. Institute for Integrative Neuroanatomy, Charité – Universitätsmedizin Berlin, Berlin, Germany

    Imre Vida

Resources

Resources

A list of freely available code for various grid cell simulations is given below.

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bush, D., Schmidt-Hieber, C. (2018). Computational Models of Grid Cell Firing. In: Cutsuridis, V., Graham, B., Cobb, S., Vida, I. (eds) Hippocampal Microcircuits. Springer Series in Computational Neuroscience. Springer, Cham. https://doi.org/10.1007/978-3-319-99103-0_16

Download citation

Publish with us

Policies and ethics

Search

Navigation