空間資訊在情節記憶中扮演很重要的角色，首先，在1970年代時，神經科
學家John O’Keefe在海馬迴組織中，發現了位置細胞，當動物在空間中特定的
位置時，位置細胞會發放動作電位，而在2000年左右，神經科學家Moser在解
剖學與測量技術的進步下，成功的在內嗅皮質層發現格狀細胞，格狀細胞在動
物自身位於空間重複排列的網格頂點會引發動作電位，這兩個基本的細胞，使
的動物能使用此資訊記住資訊或者應用在自我導航的功能上。
在此篇論文中，回顧整個空間資訊在海馬迴組織的研究發展與其模擬的結
果，並以其研究成果為基礎，發展出新的格狀細胞模型，使用更為真實的模擬
資料與模擬軟體，重現格狀細胞映射到位置細胞的腦部功能。
本篇論文所使用的映射學習演算法為競爭學習，競爭學習為非監督式學
習，主要用於分群上的演算法。此演算法所使用的資訊主要是區域的資訊，也
就是前級與後級的反應，這樣的原理符合赫布理論，相較於傳統的類神經演算
法，更接近生物可能使用的演算法。

Spatial information plays an important role in episodic memory. In the last decade,
anatomical observations and neurophysiological results give a great detail about grid
cells in the medical entorhinal cortex. In this thesis, we propose a new grid cell model
whose response field is much similar to real grid cell response field. In the simulation
of mapping medical entorhinal cortex grid cells to dentate place cells, we use statistical
properties which are extracted from real observation data. We show that this new grid
cell model with those properties can render almost all DG cells to become one field
place cells and to have a suitable peak field size using competitive learning algorithm.

1 Introduction 11
2 Memory 13
2.1 Basic Idea of Memory . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Hippocampus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Functions and Properties . . . . . . . . . . . . . . . . . . . . 14
2.3 Hippocampus Simulation . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Main purpose of simulation . . . . . . . . . . . . . . . . . . 16
2.3.2 Pair Associate Learning . . . . . . . . . . . . . . . . . . . . 16
2.4 Network Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Training Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.1 Initialize Process . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5.2 Run Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Learning Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.1 Delta Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6.2 The Generalized Delta Rule: Backpropagation . . . . . . . . 24
2.6.3 GeneRec . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6.4 Rate-based Hebbian Learning . . . . . . . . . . . . . . . . . 28
2.6.5 BCM Theory Model . . . . . . . . . . . . . . . . . . . . . . 29
2.6.6 The eXtended Contrastive Attractor Learning Algorithm . . . 30
2.6.7 Algorithm Used in Simulation . . . . . . . . . . . . . . . . . 31
2.7 The Drawbacks of O’Reilly Hippocampus Model . . . . . . . . . . . 32
2.7.1 Lack of Spatial Information in Input Pattern . . . . . . . . . . 32
2.7.2 Structure of Connection between Hippocampus and Entorhinal
Cortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Spatial Information Simulation 34
3.1 EC Grid Cells Map to DG Place Cells . . . . . . . . . . . . . . . . . 34
3.1.1 Neural Network Model . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 Input Layer Pattern Properties . . . . . . . . . . . . . . . . . 34
3.2 Competitive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Weight Update Algorithm . . . . . . . . . . . . . . . . . . . 36
3.3 Problems in Rolls Simulation . . . . . . . . . . . . . . . . . . . . . . 39
4 The Properties of Spatial Information in Hippocampus System 47
4.1 Grid Cell in the Entorhinal Cortex . . . . . . . . . . . . . . . . . . . 47
4.1.1 The Grid Cell Properties . . . . . . . . . . . . . . . . . . . . 48
4.1.2 The Field Size and Spacing . . . . . . . . . . . . . . . . . . . 49
4.1.3 The Orientation of Grid Cell . . . . . . . . . . . . . . . . . . 51
4.1.4 The Phase of Grid Cell . . . . . . . . . . . . . . . . . . . . . 51
4.2 Head Direction Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Simulation Detail and Result 58
5.1 Neuron model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1.1 Neuron Input . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.1.2 Neuron Activation Parameter . . . . . . . . . . . . . . . . . . 59
5.1.3 Neuron Activation Function . . . . . . . . . . . . . . . . . . 60
5.1.4 Neuron Input Detail . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2.1 Moser Model Equation . . . . . . . . . . . . . . . . . . . . . 65
5.2.2 My Model Equation (Product Model) . . . . . . . . . . . . . 66
5.3 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Analysis Input Pattern by Inner Product . . . . . . . . . . . . 67
5.3.2 Random Variable of EC Firing Rate . . . . . . . . . . . . . . 70
5.3.3 Cross Correlation Analysis Method . . . . . . . . . . . . . . 77
5.4 Simulation Detail and Result . . . . . . . . . . . . . . . . . . . . . . 84
5.4.1 Moser rat data . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.2 Average number of one peak . . . . . . . . . . . . . . . . . . 95
5.4.3 Normalize VS no Normalize . . . . . . . . . . . . . . . . . . 100
5.4.4 Inner product analysis . . . . . . . . . . . . . . . . . . . . . 106
5.4.5 Connection preference . . . . . . . . . . . . . . . . . . . . . 108
6 Conclusion 112
Bibliography 113

[1] J. O’Keefe, J. Dostrovsky, The hippocampus as a spatial map.
Preliminary evidence from unit activity in the freely-moving
rat, Brain Research, Volume 34, Issue 1, 1971, Pages 171-175,
ISSN 0006-8993, https://doi.org/10.1016/0006-8993(71)90358-1.
(http://www.sciencedirect.com/science/article/pii/0006899371903581)
[2] John O’Keefe, Place units in the hippocampus of the freely moving
rat, Experimental Neurology, Volume 51, Issue 1, 1976, Pages 78-
109, ISSN 0014-4886, https://doi.org/10.1016/0014-4886(76)90055-8.
(http://www.sciencedirect.com/science/article/pii/0014488676900558)
[3] Knierim, J. J. (2015). The hippocampus. Current Biology, 25(23), R1116-R1121.
https://doi.org/10.1016/j.cub.2015.10.049
[4] Barnes, J. M., & Underwood, B. J. (1959). ” Fate” of first-list associations in
transfer theory. Journal of experimental psychology, 58(2), 97.
[5] McCloskey, M., & Cohen, N. J. (1989). Catastrophic Interference in Connectionist
Networks: The Sequential Learning Problem. In G. H. Bower (Ed.), The
Psychology of Learning and Motivation, Vol. 24 (pp. 109–164). San Diego, CA:
Academic Press.
[6] https://github.com/emer/emergent
[7] Hasselmo ME, Bodelon C,Wyble BP (2002) A proposed function for hippocampal
theta rhythm: separate phases of encoding and retrieval enhance reversal of
prior learning. Neural Computation 14: 793–818.
[8] http://fourier.eng.hmc.edu/e161/lectures/nn/node13.html
[9] Rolls, E. T., Stringer, S. M., &Elliot, T. (2006). Entorhinal cortex
grid cells can map to hippocampal place cells by competitive
learning. Network: Computation in Neural Systems, 17(4), 447–465.
https://doi.org/10.1080/09548980601064846
[10] Commins, S. (2018). Spatial Navigation. In Behavioural Neuroscience
(pp. 172-184). Cambridge: Cambridge University Press.
doi:10.1017/9781316221655.015
[11] Grieves RM, Jeffery KJ. The representation of space in the brain. Behav Processes.
2017;135:113-131. doi:10.1016/j.beproc.2016.12.012
[12] Kerdels, J., &Peters, G. (2018). A Survey of Entorhinal Grid Cell Properties.
1–34. Retrieved from http://arxiv.org/abs/1810.07429
[13] Hafting, T., Fyhn, M., Molden, S., Moser, M. B., &Moser, E. I. (2005).
Microstructure of a spatial map in the entorhinal cortex. Nature, 436(7052),
801–806. https://doi.org/10.1038/nature03721
[14] Giocomo, L. M., Stensola, T., Bonnevie, T., VanCauter, T., Moser,
M. B., &Moser, E. I. (2014). Topography of head direction cells
in medial entorhinal cortex. Current Biology, 24(3), 252–262.
https://doi.org/10.1016/j.cub.2013.12.002
[15] Kobro-Flatmoen, Asgeir, and Menno P.Witter. ”Neuronal chemo-architecture of
the entorhinal cortex: A comparative review.” European Journal of Neuroscience
50.10 (2019): 3627-3662.
[16] 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.‘
[17] Brun, V. H., Solstad, T., Kjelstrup, K. B., Fyhn, M., Witter, M. P., Moser,
E. I., &Moser, M. B. (2008). Progressive increase in grid scale from dorsal
to ventral medial entorhinal cortex. Hippocampus, 18(12), 1200–1212.
https://doi.org/10.1002/hipo.20504
[18] Harland, B., Contreras, M., &Fellous, J.-M. (2018). A Role for the Longitudinal
Axis of the Hippocampus in Multiscale Representations of Large and Complex
Spatial Environments and Mnemonic Hierarchies. The Hippocampus - Plasticity
and Functions. https://doi.org/10.5772/intechopen.71165
[19] Witter, M. P., &Moser, E. I. (2006). Spatial representation and the architecture
of the entorhinal cortex. Trends in Neurosciences, 29(12), 671–678.
https://doi.org/10.1016/j.tins.2006.10.003
[20] Brandon, M. P., Koenig, J., &Leutgeb, S. (2014). Parallel and convergent processing
in grid cell, head-direction cell, boundary cell, and place cell networks.
Wiley Interdisciplinary Reviews: Cognitive Science, 5(2), 207–219.
https://doi.org/10.1002/wcs.1272
[21] Nilssen, E. S., Doan, T. P., Nigro, M. J., Ohara, S., &Witter, M. P. (2019). Neurons
and networks in the entorhinal cortex: A reappraisal of the lateral and medial
entorhinal subdivisions mediating parallel cortical pathways. Hippocampus,
29(12), 1238–1254. https://doi.org/10.1002/hipo.23145
[22] 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.
https://doi.org/10.1038/nature11649
[23] Jung, M.W., Wiener, S.I., and McNaughton, B.L. (1994). Comparison of spatial
firing characteristics of units in dorsal and ventral hippocampus of the rat. J.
Neurosci. 14, 7347–7356.
[24] Kjelstrup, K.B., Solstad, T., Brun, V.H., Hafting, T., Leutgeb, S., Witter, M.P.,
Moser, E.I., and Moser, M.B. (2008). Finite scale of spatial repre- sentation in
the hippocampus. Science 321, 140–143.
[25] O’Reilly, R. C., Munakata, Y., Frank, M. J., Hazy, T. E., and Contributors (2012).
Computational Cognitive Neuroscience. Wiki Book, 4th Edition (2020). URL:
https://github.com/CompCogNeuro/ed4
[26] Solstad, Trygve, Edvard I. Moser, and Gaute T. Einevoll. ”From grid cells to
place cells: a mathematical model.” Hippocampus 16.12 (2006): 1026-1031.
[27] Neher T, Azizi AH, Cheng S (2017) From grid cells to place
cells with realistic field sizes. PLOS ONE 12(7): e0181618.
https://doi.org/10.1371/journal.pone.0181618
[28] Si B, Treves A. The role of competitive learning in the generation of DG
fields from EC inputs. Cognitive Neurodynamics. 2009; 3(2):177187. Available
from: http://dx.doi.org/10.1007/s11571-009-9079-z. PMID: 19301148
[29] de Almeida L, Idiart M, Lisman JE. The inputoutput transformation
of the hippocampal granule cells: From grid cells to place
fields. The Journal of Neuroscience. 2009; 29(23):75047512. Available
from: http://www.jneurosci.org/content/29/23/7504.abstract.
https://doi.org/10.1523/JNEUROSCI.6048-08. 2009 PMID: 19515918
[30] de Almeida L, Idiart M, Lisman JE. The single place fields of CA3 cells:
A two-stage transformation from grid cells. Hippocampus. 2012 Feb;
22(2):200208. Available from: http://doi.wiley.com/10.1002/ hipo.20882
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=3690668&tool=
pmcentrez&rendertype=abstract. PMID: 20928834
[31] Azizi AH, Schieferstein N, Cheng S. The transformation from grid cells to
place cells is robust to noise in the grid pattern. Hippocampus. 2014 Aug;
24(8):9129. Available from: http://www.ncbi.nlm.nih. gov/pubmed/24866281.
https://doi.org/10.1002/hipo.22306 PMID: 24866281
[32] Neunuebel JP, Knierim JJ. Spatial firing correlates of physiologically distinct
cell types of the rat dentate gyrus. The Journal of neuroscience: the official
journal of the Society for Neuroscience. 2012 Mar; 32(11):384858. Available
from: http://www.jneurosci.org/content/32/11/3848.long. https://doi.org/10.
1523/JNEUROSCI.6038-11.2012
[33] Mizuseki K, Royer S, Diba K, BuzsaˆA ki G. Activity dynamics and behavioral
correlates of CA3 and CA1 hippocampal pyramidal neurons. Hippocampus.
2012; 22(8):16591680. Available from: http://dx.doi. org/10.1002/hipo.22002.
PMID: 22367959
[34] The open access dataset of Stensola, H.,
2012, ”The entorhinal grid map is discretized”
https://archive.norstore.no/pages/public/datasetDetail.jsf?id=10.11582/2018.00027
[35] The analysis function from Moser’s lab. https://bitbucket.org/cncntnu/bnt/src/master/