Coordinated learning of grid cell and place cell spatial and temporal properties: Multiple scales, attention and oscillations

Philosophical Transactions of the Royal Society B: Biological Sciences

Volumn 369, Issue 1635, 2014, Pages

  Grossberg, Stephen ^a   Pilly, Praveen K ^a  

^a Boston University   (United States)

Author keywords

Adaptive timing; Attention; Grid cells; Place cells; Self organizing map; Spatial navigation

Indexed keywords

HIERARCHICAL SYSTEM; LEARNING; NUMERICAL MODEL; OSCILLATION; SPATIOTEMPORAL ANALYSIS; TRAJECTORY;

ACTION POTENTIAL; ADAPTIVE TIMING; ANIMAL; ARTICLE; ATTENTION; BIOLOGICAL MODEL; BIOLOGICAL RHYTHM; COMPUTER SIMULATION; CYTOLOGY; DEPTH PERCEPTION; ENTORHINAL CORTEX; GRID CELLS; HIPPOCAMPUS; LEARNING; PHYSIOLOGY; PLACE CELLS; PYRAMIDAL NERVE CELL; RAT; SELF-ORGANIZING MAP; SPATIAL NAVIGATION;

ADAPTIVE TIMING; ATTENTION; GRID CELLS; PLACE CELLS; SELF-ORGANIZING MAP; SPATIAL NAVIGATION;

ACTION POTENTIALS; ANIMALS; BIOLOGICAL CLOCKS; COMPUTER SIMULATION; ENTORHINAL CORTEX; HIPPOCAMPUS; LEARNING; MODELS, NEUROLOGICAL; PYRAMIDAL CELLS; RATS; SPACE PERCEPTION;

EID: 84906921434     PISSN: 09628436     EISSN: 14712970     Source Type: Journal    
DOI: 10.1098/rstb.2012.0524     Document Type: Article

References (113)

1. Morris RGM, Garrud P, Rawlins JNP, O'Keefe J. 1982 Place navigation impaired in rats with hippocampal lesions. Nature 297, 681-683. (doi:10.1038/ 297681a0)
2. Davis S, Butcher SP, Morris RG. 1992 The NMDA receptor antagonist D-2-amino-5-phosphonopentanoate (D-AP5) impairs spatial learning and LTP in vivo at intracerebral concentrations comparable to those that block LTP in vivo. J. Neurosci. 12, 21-34.
3. Parron C, Save E. 2004 Evidence for entorhinal and parietal cortices involvement in path integration in the rat. Exp. Brain Res. 159, 349-359. (doi:10. 1007/s00221-004-1960-8)
4. O'Keefe J, Dostrovsky J. 1971 The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res. 34, 171-175. (doi:10.1016/0006-8993(71)90358-1)
5. Fenton AA, Kao H-Y, Neymotin SA, Olypher A, Vayntrub Y, Lytton WW, Ludvig N. 2008 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. J. Neurosci. 28, 11 250-11 262. (doi:10.1523/JNEUROSCI.2862-08.2008)
6. Henriksen EJ, Colgin LL, Barnes CA, Witter MP, Moser MB, Moser EI. 2010 Spatial representation along the proximodistal axis of CA1. Neuron 68, 127-137. (doi:10.1016/j.neuron.2010.08.042)
7. Park EH, Dvorak D, Fenton AA. 2011 Ensemble place codes in hippocampus: CA1, CA3, and dentate gyrus place cells have multiple place fields in large environments. PLoS ONE 6, e22349. (doi:10.1371/ journal.pone.0022349)
8. Etienne AS, Maurer R, Seguinot V. 1996 Path integration in mammals and its integration with visual landmarks. J. Exp. Biol. 199, 201-209.
9. Gothard KM, Skaggs WE, McNaughton BL. 1996 Dynamics of mismatch correction in the hippocampal ensemble code for space: interaction between path integration and environmental cues. J. Neurosci. 16, 8027-8040.
10. Chen C, King JA, Burgess N, O'Keefe J. 2013 How vision and movement combine in the hippocampal place code. Proc. Natl Acad. Sci. USA 110, 378-383. (doi:10.1073/pnas.1215834110)
11. Hafting T, Fyhn M, Molden S, Moser MB, Moser EI. 2005 Microstructure of a spatial map in the entorhinal cortex. Nature. 436, 801-806. (doi:10. 1038/nature03721)
12. 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. (doi:10.1038/nrn1932)
13. 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. (doi:10. 1126/science.1125572)
14. Brun VH, Solstad T, Kjelstrup KB, Fyhn M, Witter MP, Moser EI, Moser MB. 2008 Progressive increase in grid scale from dorsal to ventral medial entorhinal cortex. Hippocampus 18, 1200-1212. (doi:10.1002/hipo.20504)
15. Stensola H, Stensola T, Solstad T, Frøland K, Moser M-B, Moser EI. 2012 The entorhinal grid map is discretized. Nature 492, 72-78. (doi:10.1038/ nature11649)
16. O'Keefe J, Burgess N. 2005 Dual phase and rate coding in hippocampal place cells: theoretical significance and relationship to entorhinal grid cells. Hippocampus 15, 853-866. (doi:10.1002/ hipo.20115)
17. Gorchetchnikov A, Grossberg S. 2007 Space, time, and learning in the hippocampus: how fine spatial and temporal scales are expanded into population codes for behavioral control? Neural Netw. 20, 182-193. (doi:10.1016/j.neunet.2006. 11.007)
18. Rolls ET, Stringer SM, Elliot T. 2006 Entorhinal cortex grid cells can map to hippocampal place cells by competitive learning. Network 17, 447-465. (doi:10.1080/09548980601064846)
19. Molter C, Yamaguchi Y. 2008 Entorhinal theta phase precession sculpts dentate gyrus place fields. Hippocampus 18, 919-930. (doi:10.1002/ hipo.20450)
20. Savelli F, Knierim JJ. 2010 Hebbian analysis of the transformation of medial entorhinal grid-cell inputs to hippocampal place fields. J. Neurophysiol. 103, 3167-3183. (doi:10.1152/jn.00932.2009)
21. Grossberg S, Pilly PK. 2012 How entorhinal grid cells may learn multiple spatial scales from a dorsoventral gradient of cell response rates in a selforganizing map. PLoS Comput. Biol. 8, 31002648. (doi:10.1371/journal.pcbi.1002648.)
22. Mhatre H, Gorchetchnikov A, Grossberg S. 2012 Grid cell hexagonal patterns formed by fast selforganized learning within entorhinal cortex. Hippocampus 22, 320-334. (doi:10.1002/ hipo.20901)
23. Pilly PK, Grossberg S. 2012 How do spatial learning and memory occur in the brain? Coordinated learning of entorhinal grid cells and hippocampal place cells. J. Cogn. Neurosci. 24, 1031-1054. (doi:10.1162/jocn_a_00200)
24. Pilly PK, Grossberg S. 2013 How reduction of theta rhythm by medium septum inactivation may disrupt periodic spatial responses of entorhinal grid cells by reduced cholinergic transmission. Front. Neural Circuits 7, 173. (doi:10.3389/fncir. 2013.00173)
25. Pilly PK, Grossberg S. 2013 Spiking neurons in a hierarchical self-organizing map model can learn to develop spatial and temporal properties of entorhinal grid cells and hippocampal place cells. PLoS ONE 8, e0060599. (doi:10.1371/journal.pone. 0060599)
26. Grossberg S. 1976 Adaptive pattern classification and universal recoding. I. parallel development and coding of neural feature detectors. Biol. Cybern. 23, 121-134. (doi:10.1007/BF00344744)
27. Grossberg S. 1978 A theory of human memory: selforganization and performance of sensory-motor codes, maps, and plans. In Progress in theoretical biology, vol. 5 (eds R Rosen, F Snell), pp. 233-374. New York, NY: Academic Press.
28. von der Malsburg C. 1973 Self-organization of orientation sensitive cells in the striate cortex. Kybernetik 14, 85-100. (doi:10.1007/ BF00288907)
29. Royer S, Pare D. 2003 Conservation of total synaptic weight through balanced synaptic depression and potentiation. Nature 422, 518-522. (doi:10.1038/ nature01530)
30. 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. (doi:10.1126/ science.1188210)
31. Wills TJ, Cacucci F, Burgess N, O'Keefe J. 2010 Development of the hippocampal cognitive map in preweanling rats. Science 328, 1573-1576. (doi:10. 1126/science.1188224)
32. Olson S, Grossberg S. 1998 A neural network model for the development of simple and complex cell receptive fields within cortical maps of orientation and ocular dominance. Neural Netw. 11, 189-208. (doi:10.1016/S0893-6080(98)00003-3)
33. Grossberg G, Williamson JR. 2001 A neural model of how horizontal and interlaminar connections of visual cortex develop into adult circuits that carry out perceptual groupings and learning. Cereb. Cortex 11, 37-58. (doi:10.1093/cercor/11.1.37)
34. Grossberg S, Seitz A. 2003 Laminar development of receptive fields, maps, and columns in visual cortex: the coordinating role of the subplate. Cereb. Cortex 13, 852-863. (doi:10.1093/cercor/13.8.852)
35. Burgess N, Barry C, O'Keefe J. 2007 An oscillatory interference model of grid cell firing. Hippocampus 17, 801-812. (doi:10.1002/hipo.20327)
36. Hasselmo M, Giocomo L, Zilli E. 2007 Grid cell firing may arise from interference of theta frequency membrane potential oscillations in single neurons. Hippocampus 17, 1252-1271. (doi:10.1002/ hipo.20374)
37. Hafting T, Fyhn M, Bonnevie T, Moser MB, Moser EI. 2008 Hippocampus independent phase precession in entorhinal grid cells. Nature 453, 1248-1252. (doi:10.1038/nature06957)
38. Couey JJ et al. 2013 Recurrent inhibitory circuitry as a mechanism for grid formation. Nat. Neurosci. 16, 318-324. (doi:10.1038/nn.3310)
39. Pastoll H, Solanka L, van Rossum MC, Nolan MF. 2013 Feedback inhibition enables theta-nested gamma oscillations and grid firing fields. Neuron 77, 141-154. (doi:10.1016/j.neuron.2012.11.032)
40. Cao Y, Grossberg S. 2012 Stereopsis and 3D surface perception by spiking neurons in laminar cortical circuits: a method of converting neural rate models into spiking models. Neural Netw. 26, 75-98. (doi:10.1016/j.neunet.2011.10.010)
41. Giocomo LM, Zilli E, Fransen E, Hasselmo ME. 2007 Temporal frequency of subthreshold oscillations scales with entorhinal grid cell field spacing. Science 315, 1719-1722. (doi:10.1126/science.1139207)
42. 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. (doi:10. 1016/j.neuron.2008.10.044)
43. Giocomo LM, Hasselmo ME. 2008b Time constants of h current in layer II stellate cells differ along the dorsal to ventral axis of medial entorhinal cortex. J. Neurosci. 28, 9414-9425. (doi:10.1523/jneurosci. 3196-08.2008)
44. Yoshida M, Giocomo LM, Boardman I, Hasselmo ME. 2011 Frequency of subthreshold oscillations at different membrane potential voltages in neurons at different anatomical positions on the dorsoventral axis in the rat medial entorhinal cortex. J. Neurosci. 31, 12 683-12 694. (doi:10.1523/JNEUROSCI.1654-11.2011)
45. Giocomo LM, Hussaini SA, Zheng F, Kandel ER, Moser MB, Moser EI. 2011 Grid cells use HCN1 channels for spatial scaling. Cell 147, 1159-1170. (doi:10.1016/j.cell.2011.08.051)
46. Boehlen A, Heinemann U, Erchova I. 2010 The range of intrinsic frequencies represented by medial entorhinal cortex stellate cells extends with age. J. Neurosci. 30, 4585-4589. (doi:10.1523/jneurosci. 4939-09.2010)
47. 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. (doi:10.1002/hipo.20939)
48. Grossberg S, Schmajuk NA. 1989 Neural dynamics of adaptive timing and temporal discrimination during associative learning. Neural Netw. 2, 79-102. (doi:10.1016/0893-6080(89)90026-9)
49. Grossberg S, Merrill JWL. 1992 A neural network model of adaptively timed reinforcement learning and hippocampal dynamics. Cogn. Brain Res. 1, 3-38. (doi:10.1016/0926-6410(92)90003-A)
50. Grossberg S, Merrill JWL. 1996 The hippocampus and cerebellum in adaptively timed learning, recognition, and movement. J. Cogn. Neurosci. 8, 257-277. (doi:10.1162/jocn.1996.8.3.257)
51. Smith MC. 1968 CS-US interval and US intensity in classical conditioning of the rabbit's nictitating membrane response. J. Comp. Physiol. Psychol. 3, 679-687. (doi:10.1037/h0026550)
52. Roberts S. 1981 Isolation of an internal clock. J. Exp. Psychol. Anim. Behav. Process. 7, 242-268. (doi:10. 1037/0097-7403.7.3.242)
53. Roberts WA, Cheng K, Cohen JS. 1989 Timing light and tone signals in pigeons. J. Exp. Psychol. Anim. Behav. Process. 15, 23-25. (doi:10.1037/0097-7403.15.1.23)
54. Gibson J. 1991 The origins of scalar timing. Learn. Motiv. 22, 3-38. (doi:10.1016/0023-9690(91)90015-Z)
55. MacDonald CJ, Lepage KQ, Eden UT, Eichenbaum H. 2011 Hippocampal 'time cells' bridge the gap in memory for discontiguous events. Neuron 71, 737-749. (doi:10.1016/j.neuron.2011.07.012)
56. Fiala JC, Grossberg S, Bullock D. 1996 Metabotropic glutamate receptor activation in cerebellar Purkinje cells as substrate for adaptive timing of the classically conditioned eye blink response. J. Neurosci. 16, 3760-3774.
57. Brown J, Bullock D, Grossberg S. 1999 How the basal ganglia use parallel excitatory and inhibitory learning pathways to selectively respond to unexpected rewarding cues. J. Neurosci. 19, 10 502-10 511.
58. Hargreaves EL, Rao G, Lee I, Knierim JJ. 2005 Major dissociation between medial and lateral entorhinal input to dorsal hippocampus. Science 308, 1792-1794. (doi:10.1126/science.1110449)
59. Aminoff E, Gronau N, Bar M. 2007 The parahippocampal cortex mediates spatial and nonspatial associations. Cereb. Cortex 17, 1493-1503. (doi:10.1093/cercor/bhl078)
60. Kerr KM, Agster KL, Furtak SC, Burwell RD. 2007 Functional neuroanatomy of the parahippocampal region: the lateral and medial entorhinal areas. Hippocampus 17, 697-708. (doi:10.1002/ hipo.20315)
61. Eichenbaum H, Lipton PA. 2008 Towards a functional organization of the medial temporal lobe memory system: role of the parahippocampal and medial entorhinal cortical areas. Hippocampus 18, 1314-1324. (doi:10.1002/hipo.20500)
62. van Strien NM, Cappaert NLM, Witter MP. 2009 The anatomy of memory: an interactive overview of the parahippocampal-hippocampal network. Nat. Rev. Neurosci. 10, 272-282. (doi:10.1038/nrn2614)
63. Tulving E. 1972 Episodic and semantic memory. In Organization of memory (eds E Tulving, W Donaldson), pp. 381-402. New York, NY: Academic Press.
64. Tulving E, Thomson DC. 1973 Encoding specificity and retrieval processes in episodic memory. Psychol. Rev. 80, 352-373. (doi:10. 1037/h0020071)
65. Ranck Jr JB. 1984 Head-direction cells in the deep cell layers of dorsal presubiculum in freely moving rats. In Proc. Annual Conference of the Society for Neuroscience, Anaheim, CA. 10, 599.
66. Taube JS, Muller RU, Ranck Jr JB. 1990 Headdirection cells recorded from the postsubiculum in freely moving rats. I. Description and quantitative analysis. J. Neurosci. 10, 420-435.
67. Blair H, Sharp P. 1995 Anticipatory head direction signals in anterior thalamus: evidence for a thalamocortical circuit that integrates angular head motion to compute head direction. J. Neurosci. 15, 6260-6270.
68. Skaggs WE, Knierim J, Kudrimoti HS, McNaughton BL. 1995 A model of the neural basis of the rat's sense of direction. Adv. Neural Inf. Process. Syst. 7, 173-180.
69. Blair H, Sharp P. 1996 Visual and vestibular influences on head direction cells in the anterior thalamus of the rat. Behav. Neurosci. 10, 643-660. (doi:10.1037/0735-7044.110.4.643)
70. Redish AD, Elga AN, Touretzky DS. 1996 A coupled attractor model of the rodent head direction system. Netw. Comput. Neural Syst. 7, 671-685. (doi:10. 1088/0954-898X/7/4/004)
71. Goodridge JP, Touretzky DS. 2000 Modeling attractor deformation in the rodent head-direction system. J. Neurophysiol. 83, 3402-3410.
72. Boucheny C, Brunel N, Arleo A. 2005 A continuous attractor network model without recurrent excitation: maintenance and integration in the head direction cell system. J. Comput. Neurosci. 18, 205-227. (doi:10.1007/s10827-005-6559-y)
73. Song P, Wang XJ. 2005 Angular path integration by moving 'hill of activity': a spiking neuron model without recurrent excitation of the head-direction system. J. Neurosci. 25, 1002-1014. (doi:10.1523/ JNEUROSCI.4172-04.2005)
74. Fortenberry B, Gorchetchnikov A, Grossberg S. 2012 Learned integration of visual, vestibular, and motor cues in multiple brain regions computes head direction during visually-guided navigation. Hippocampus 22, 2219-2237. (doi:10.1002/ hipo.22040)
75. Krupic J, Burgess N, O'Keefe J. 2012 Neural representations of location composed of spatially periodic bands. Science 337, 853-857. (doi:10. 1126/science.1222403)
76. Caballero-Bleda M, Witter MP. 1993 Regional and laminar organization of projections from the presubiculum and parasubiculum to the entorhinal cortex: an anterograde tracing study in the rat. J. Comp. Neurol. 328, 115-129. (doi:10.1002/cne. 903280109)
77. Caballero-Bleda M, Witter MP. 1994 Projections from the presubiculum and the parasubiculum to morphologically characterized entorhinal-hippocampal projection neurons in the rat. Exp. Brain Res. 101, 93-108. (doi:10.1007/BF00243220)
78. Blair HT, Gupta KJ, Zhang K. 2008 Conversion of a phase-to a rate-coded position signal by a threestage model of theta cells, grid cells, and place cells. Hippocampus 18, 1239-1255. (doi:10.1002/ hipo.20509)
79. Thompson LT, Best PJ. 1990 Long-term stability of the place-field activity of single units recorded from the dorsal hippocampus of freely behaving rats. Brain Res. 509, 299-308. (doi:10.1016/0006-8993(90)90555-P)
80. Wilson MA, McNaughton BL. 1993 Dynamics of the hippocampal ensemble code for space. Science 261, 1055-1058. (doi:10.1126/science.8351520)
81. Muller RA. 1996 A quarter of a century of place cells. Neuron 17, 813-822. (doi:10.1016/S0896-6273(00)80214-7)
82. Frank LM, Stanley GB, Brown EN. 2004 Hippocampal plasticity across multiple days of exposure to novel environments. J. Neurosci. 24, 7681-7689. (doi:10. 1523/JNEUROSCI.1958-04.2004)
83. Grossberg S. 1980 How does a brain build a cognitive code? Psychol. Rev. 87, 1-51. (doi:10.1037/0033-295X.87.1.1)
84. Grossberg S. 2012 Adaptive resonance theory: how a brain learns to consciously attend, learn, and recognize a changing world. Neural Netw. 37, 1-47. (doi:10.1016/j.neunet.2012.09.017)
85. Carpenter GA, Grossberg S. 1987 A massively parallel architecture for a self-organizing neural pattern recognition machine. Comput. Vision Graph. 37, 54-115. (doi:10.1016/S0734-189X(87)80014-2)
86. Raizada R, Grossberg S. 2003 Towards a theory of the laminar architecture of cerebral cortex: computational clues from the visual system. Cereb. Cortex 13, 100-113. (doi:10.1093/cercor/13.1.100)
87. Desimone R. 1998 Visual attention mediated by biased competition in extrastriate visual cortex. Proc. R. Soc. Lond. B 353, 1245-1255. (doi:10. 1098/rstb.1998.0280)
88. Kentros CG, Agniotri NT, Streater S, Hawkins RD, Kandel ER. 2004 Increased attention to spatial context increases both place field stability and spatial memory. Neuron 42, 283-295. (doi:10. 1016/S0896-6273(04)00192-8)
89. Kentros C, Hargreaves E, Hawkins RD, Kandel ER, Shapiro M, Muller RV. 1998 Abolition of long-term stability of new hippocampal place cell maps by NMDA receptor blockade. Science 280, 2121-2126. (doi:10.1126/science.280.5372.2121)
90. Morris RGM, Frey U. 1997 Hippocampal synaptic plasticity: role in spatial learning or the automatic recording of attended experience? Proc. R. Soc. Lond. B 352, 1489-1503. (doi:10.1098/rstb.1997.0136)
91. Bonnevie T, Dunn B, Fyhn M, Hafting T, Derdikman D, Kubie JL, Roudi Y, Moser EI, Moser M-B. 2013 Grid cells require excitatory drive from the hippocampus. Nat. Neurosci. 16, 309-319. (doi:10.1038/nn.3311)
92. Grossberg S, Versace M. 2008 Spikes, synchrony, and attentive learning by laminar thalamocortical circuits. Brain Res. 1218, 278-312. (doi:10.1016/j. brainres.2008.04.024)
93. Berke JD, Hetrick V, Breck J, Green RW. 2008 Transient 23-to 30-Hz oscillations in mouse hippocampus during exploration of novel environments. Hippocampus 18, 519-529. (doi:10.1002/hipo.20435)
94. Grossberg S. 2009 Beta oscillations and hippocampal place cell learning during exploration of novel environments. Hippocampus 19, 881-885. (doi:10.1002/hipo.20602)
95. Buffalo EA, Fries P, Landman R, Buschman TJ, Desimone R. 2011 Laminar differences in gamma and alpha coherence in the ventral stream. Proc. Natl Acad. Sci. USA 108, 11 262-11 267. (doi:10. 1073/pnas.1011284108)
96. Buschman TJ, Miller EK. 2009 Serial, covert shifts of attention during visual search are reflected by the frontal eye fields and correlated with population oscillations. Neuron 63, 386-396. (doi:10.1016/j. neuron.2009.06.020)
97. Vertes RP, Kocsis B. 1997 Brainstem-diencephaloseptohippocampal systems controlling the theta rhythm of the hippocampus. Neuroscience 81, 893-926. (doi:10.1016/S0306-4522(97)00239-X)
98. Tóth K, Borhegyi Z, Freund TF. 1993 Postsynaptic targets of GABAergic hippocampal in the medial septum-diagonal band of broca complex. J. Neurosci. 13, 3712-3724.
99. Wang XJ. 2002 Pacemaker neurons for the theta rhythm and their synchronization in the septohippocampal reciprocal loop. J. Neurophysiol. 87, 889-900.
100. 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. (doi:10. 1126/science.1201652)
101. 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. (doi:10.1126/science. 1201685)
102. Burak Y, Fiete IR. 2006 Do we understand the emergent dynamics of grid cell activity? J. Neurosci. 26, 9352-9354. (doi:10.1523/JNEUROSCI.2857-06.2006)
103. Fuhs MC, Touretzky DS. 2006 A spin glass model of path integration in rat medial entorhinal cortex. J. Neurosci. 26, 4266-4276. (doi:10.1523/ JNEUROSCI.4353-05.2006)
104. 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. (doi:10.1142/ S0129065707001093)
105. Burgess N. 2008 Grid cells and theta as oscillatory interference. Theory and predictions. Hippocampus 18, 1157-1174. (doi:10.1002/hipo.20518)
106. Zilli EA. 2012 Models of grid cell spatial firing published 2005-2011. Front. Neural Circuits 6, 16. (doi:10.3389/fncir.2012.00016)
107. Kropff E, Treves A. 2008 The emergence of grid cells: intelligent design or just adaptation? Hippocampus 18, 1256-1269. (doi:10.1002/ hipo.20520)
108. Fyhn M, Hafting T, Treves A, Moser MB, Moser EI. 2007 Hippocampal remapping and grid realignment in entorhinal cortex. Nature 446, 190-194. (doi:10. 1038/nature05601)
109. Yartsev MM, Witter MP, Ulanovsky N. 2011 Grid cells without theta oscillations in the entorhinal cortex of bats. Nature 479, 103-107. (doi:10.1038/ nature10583)
110. Domnisoru C, Kinkhabwala AA, Tank DW. 2013 Membrane potential dynamics of grid cells. Nature 495, 199-204. (doi:10.1038/ nature11973)
111. Schmidt-Heiber C, Hausser M. 2013 Cellular mechanisms of spatial navigation in the medial entorhinal cortex. Nat. Neurosci. 16, 325-331. (doi:10.1038/nn.3340)
112. Muzzio IA, Levita L, Kulkami J, Monaco J, Kentros C, Stead M, Abbott LF, Kandel ER. 2009 Attention enhances the retrieval and stability of visuospatial and olfactory representations in the dorsal hippocampus. PLoS Biol. 7, e1000140. (doi:10.1371/ journal.pbio.1000140)
113. Gaudiano P, Grossberg S. 1991 Vector associative maps: unsupervised real-time error-based learning and control of movement trajectories. Neural Netw. 4, 147-183. (doi:10.1016/0893-6080(91)90002-M)