在危急的情況下，能夠準確且快速地做出判斷是對於生物體的生存而言至關重
要的能力。過去的許多研究利用Random dots motion task 來研究準確度與速度
之間的取捨(trade-off)。其中，吸引子網路能夠成功的解釋實驗數據，且其所預
測的準確性與反應時間之平均值與實驗數據吻合。然而，新的實驗數據與吸引
子模型的其中一項假設不吻合。此實驗數據顯示決策模型中的抑制型神經應與
刺激型神經擁有同樣程度的選擇性，然而吸引子模型用的卻是沒有選擇性的抑
制型神經。因此，在此研究中我們修正了此一假設，並且比較修正前後的模型
之間的計算能力的差別。我們用了一個簡化版的模型來研究這兩個模型，並且
發現只有有選擇性的模型才能產生一個隨時變的位能井。時變性讓選擇性模型
得以在初期時小心衡量資訊，並在後期時快速做出決定。因此，選擇性模型得
以做出比無選擇性模型更快且更準確的決定。

The ability to decide swiftly and accurately in an urgent scenario is crucial for an organism’s survival. Past studies have used the random dots motion task to quantify the speed-accuracy trade-off that plagues most decision making tasks, and the attractor model is a model that has successfully captured the average accuracy and reaction time as seen in the experimental data. However, new experimental findings call into question the validity of the attractor model’s network configuration. In particular, it is shown that the inhibitory neurons in the posterior parietal cortex of mice are as selective to decision making results as the excitatory neurons, which is a direct contradiction to the attractor model’s assumption of unselective inhibition. In this study, we investigate a revised model of the classic attractor model, and analyze in detail the differences in computational ability that it brings. We proposed a reduced model for both the unselective and selective models, and showed that the selective model alone produces a time-varying energy landscape. This time dependence of the energy landscape allows the selective model to integrate information carefully in initial stages, then quickly converge to an attractor once the choice is clear. This results in the selective model having a more efficient speed-accuracy trade-off that is unreachable by unselective models.

Contents . . . i
List of Tables . . . iii
List of Figures . . . iv
1 Introduction . . . 1
2 Methods . . . 6
2.1 Neural Network Model . . . 6
2.1.1 Two-Variable Model . . . 6
2.1.2 Full Unselective/Selective Model . . . 9
2.2 Simulation Protocol . . . 12
2.3 Reduced Model . . . 13
2.3.1 Linear Relation of SG and S+ . . . 13
2.3.2 Quadratic Relation of SG− and S− . . . 15
2.3.3 Energy Landscape . . . 16
2.4 Statistical Analysis . . . 17
2.4.1 Speed-Accuracy Trade-off . . . 17
2.4.2 Non-Regret Choice . . . 18
3 Results . . . 19
3.1 Comparison of UM and SM without GABA Dynamics . . . 19
3.2 Non-trivial GABA Dynamics . . . 20
3.3 Reduction via Empirical Relations . . . 21
3.4 Relaxation Dynamics of SG+ and SG− . . . 23
3.5 Linear Reduced Models . . . 27
3.6 Quadratic Reduced Models . . . 27
4 Discussion . . . 34
Reference . . . 37

[1] Matjaž Jogan and Alan A. Stocker. A new two-alternative forced choice
method for the unbiased characterization of perceptual bias and discriminability.
Journal of Vision, 14(3):20–20, March 2014.
[2] Praveen K. Pilly and Aaron R. Seitz. What a difference a parameter makes:
A psychophysical comparison of random dot motion algorithms. Vision Research,
49(13):1599–1612, July 2009.
[3] Mark E. Mazurek, Jamie D. Roitman, Jochen Ditterich, and Michael N.
Shadlen. A Role for Neural Integrators in Perceptual Decision Making. Cerebral
Cortex, 13(11):1257–1269, November 2003.
[4] Alexander C. Huk and Michael N. Shadlen. Neural Activity in Macaque
Parietal Cortex Reflects Temporal Integration of Visual Motion Signals during
Perceptual Decision Making. J. Neurosci., 25(45):10420–10436, November
2005. Publisher: Society for Neuroscience Section: Behavioral/Systems/
Cognitive.
[5] Xiao-Jing Wang. Neural dynamics and circuit mechanisms of decisionmaking.
Current Opinion in Neurobiology, 22(6):1039–1046, December 2012.
[6] Xiao-Jing Wang. Decision Making in Recurrent Neuronal Circuits. Neuron,
60(2):215–234, October 2008.
[7] Rajesh P. N. Rao. Decision Making Under Uncertainty: A Neural Model
Based on Partially Observable Markov Decision Processes. Front. Comput.
Neurosci., 4, 2010. Publisher: Frontiers.
[8] Jeffrey M. Beck, Wei Ji Ma, Roozbeh Kiani, Tim Hanks, Anne K. Churchland,
Jamie Roitman, Michael N. Shadlen, Peter E. Latham, and Alexandre
Pouget. Probabilistic Population Codes for Bayesian Decision Making. Neuron,
60(6):1142–1152, December 2008.
[9] Christian K. Machens, Ranulfo Romo, and Carlos D. Brody. Flexible control
of mutual inhibition: a neural model of two-interval discrimination. Science,
307(5712):1121–1124, February 2005.
[10] Chung-Chuan Lo and Xiao-Jing Wang. Cortico-basal ganglia circuit mechanism
for a decision threshold in reaction time tasks. Nat Neurosci, 9(7):956–
963, July 2006.
[11] Rafal Bogacz, Eric Brown, Jeff Moehlis, Philip Holmes, and Jonathan D.
Cohen. The physics of optimal decision making: a formal analysis of models of
performance in two-alternative forced-choice tasks. Psychol Rev, 113(4):700–
765, October 2006.
[12] Drew Fudenberg, Whitney Newey, Philipp Strack, and Tomasz Strzalecki.
Testing the drift-diffusion model. PNAS, 117(52):33141–33148, December
2020.
[13] Xiao-Jing Wang. Probabilistic Decision Making by Slow Reverberation in
Cortical Circuits. Neuron, 36(5):955–968, December 2002.
[14] R. A. Heath. The Ornstein-Uhlenbeck model for decision time in cognitive
tasks: an example of control of nonlinear network dynamics. Psychol Res,
63(2):183–191, 2000.
[15] A. R. Pike. Stochastic Models of Choice Behaviour: Response Probabilities
and Latencies of Finite Markov Chain Systems1. British
Journal of Mathematical and Statistical Psychology, 19(1):15–32, 1966.
_eprint: https:// bpspsychub.onlinelibrary.wiley.com/ doi/ pdf/ 10.1111/
j.2044-8317.1966.tb00351.x.
[16] Kong-Fatt Wong, Alexander C. Huk, Michael N. Shadlen, and Xiao-Jing
Wang. Neural circuit dynamics underlying accumulation of time-varying evidence during perceptual decision making. Front. Comput. Neurosci., 1, 2007.
Publisher: Frontiers.
[17] Kong-Fatt Wong and Alexander C. Huk. Temporal dynamics underlying
perceptual decision making: Insights from the interplay between an attractor
model and parietal neurophysiology. Front. Neurosci., 2, 2008. Publisher:
Frontiers.
[18] Farzaneh Najafi, Gamaleldin F. Elsayed, Robin Cao, Eftychios Pnevmatikakis,
Peter E. Latham, John P. Cunningham, and Anne K. Churchland.
Excitatory and Inhibitory Subnetworks Are Equally Selective during
Decision-Making and Emerge Simultaneously during Learning. Neuron,
105(1):165–179.e8, January 2020.
[19] David M. Green and John A. Swets. Signal detection theory and psychophysics.
Signal detection theory and psychophysics. John Wiley, Oxford,
England, 1966. Pages: xi, 455.
[20] Kazuhiro Sohya, Katsuro Kameyama, Yuchio Yanagawa, Kunihiko Obata,
and Tadaharu Tsumoto. GABAergic neurons are less selective to stimulus
orientation than excitatory neurons in layer II/III of visual cortex, as revealed
by in vivo functional Ca2+ imaging in transgenic mice. J Neurosci,
27(8):2145–2149, February 2007.
[21] Cristopher M. Niell and Michael P. Stryker. Highly Selective Receptive Fields
in Mouse Visual Cortex. J. Neurosci., 28(30):7520–7536, July 2008. Publisher:
Society for Neuroscience Section: Articles.
[22] Aaron M. Kerlin, Mark L. Andermann, Vladimir K. Berezovskii, and R. Clay
Reid. Broadly tuned response properties of diverse inhibitory neuron subtypes
in mouse visual cortex. Neuron, 67(5):858–871, September 2010.
[23] Davi D. Bock, Wei-Chung Allen Lee, Aaron M. Kerlin, Mark L. Andermann,
Greg Hood, Arthur W. Wetzel, Sergey Yurgenson, Edward R. Soucy,
Hyon Suk Kim, and R. Clay Reid. Network anatomy and in vivo physiology
of visual cortical neurons. Nature, 471(7337):177–182, March 2011. Number:
7337 Publisher: Nature Publishing Group.
[24] Sonja B. Hofer, Ho Ko, Bruno Pichler, Joshua Vogelstein, Hana Ros, Hongkui
Zeng, Ed Lein, Nicholas A. Lesica, and Thomas D. Mrsic-Flogel. Differential
connectivity and response dynamics of excitatory and inhibitory neurons in
visual cortex. Nature Neuroscience, 14(8):1045–1052, August 2011. Number:
8 Publisher: Nature Publishing Group.
[25] Petr Znamenskiy, Mean-Hwan Kim, Dylan R. Muir, Maria Florencia Iacaruso,
Sonja B. Hofer, and Thomas D. Mrsic-Flogel. Functional selectivity and
specific connectivity of inhibitory neurons in primary visual cortex. bioRxiv,
page 294835, April 2018. Publisher: Cold Spring Harbor Laboratory Section:
New Results.
[26] Caroline A. Runyan, James Schummers, Audra Van Wart, Sandra J.
Kuhlman, Nathan R. Wilson, Z. Josh Huang, and Mriganka Sur. Response
features of parvalbumin-expressing interneurons suggest precise roles for subtypes
of inhibition in visual cortex. Neuron, 67(5):847–857, September 2010.
[27] Alexandra K. Moore and Michael Wehr. Parvalbumin-expressing inhibitory
interneurons in auditory cortex are well-tuned for frequency. J Neurosci,
33(34):13713–13723, August 2013.
[28] Greg Allen, Roderick Mccoll, Holly Barnard, Wendy Ringe, James Fleckenstein,
and Munro Cullum. Magnetic resonance imaging of cerebellar–prefrontal
and cerebellar–parietal functional connectivity. NeuroImage, 28:39–
48, November 2005.
[29] Lucas Pinto and Yang Dan. Cell-Type-Specific Activity in Prefrontal Cortex
during Goal-Directed Behavior. Neuron, 87(2):437–450, July 2015.
[30] Matthew Lovett-Barron, Patrick Kaifosh, Mazen A. Kheirbek, Nathan
Danielson, Jeffrey D. Zaremba, Thomas R. Reardon, Gergely F. Turi, René
Hen, Boris V. Zemelman, and Attila Losonczy. Dendritic inhibition in the
hippocampus supports fear learning. Science, 343(6173):857–863, February
2014.
[31] Valérie Ego-Stengel and Matthew A. Wilson. Spatial selectivity and theta
phase precession in CA1 interneurons. Hippocampus, 17(2):161–174, 2007.
[32] Kong-Fatt Wong and Xiao-Jing Wang. A Recurrent Network Mechanism
of Time Integration in Perceptual Decisions. J. Neurosci., 26(4):1314–1328,
January 2006.
[33] Fred Rieke, David Warland, Rob de Ruyter van Steveninck, and William
Bialek. Spikes: Exploring the Neural Code. Computational Neuroscience.
The Quarterly Review of Biology, 73(4):537–537, December 1998. Publisher:
The University of Chicago Press.
[34] Srdjan Ostojic and Nicolas Brunel. From spiking neuron models to linearnonlinear
models. PLoS Comput Biol, 7(1):e1001056, January 2011.
[35] M. Perouansky and Y. Yaari. Kinetic properties of NMDA receptor-mediated
synaptic currents in rat hippocampal pyramidal cells versus interneurones. J
Physiol, 465:223–244, June 1993.
[36] Norman H. Lam, Thiago Borduqui, Jaime Hallak, Antonio C. Roque, Alan
Anticevic, John H. Krystal, Xiao-Jing Wang, and John D. Murray. Effects
of Altered Excitation-Inhibition Balance on Decision Making in a Cortical
Circuit Model. bioRxiv, page 100347, January 2017. Publisher: Cold Spring
Harbor Laboratory Section: New Results.
[37] Cheng-Te Wang, Chung-Ting Lee, Xiao-Jing Wang, and Chung-Chuan
Lo. Top-Down Modulation on Perceptual Decision with Balanced Inhibition
through Feedforward and Feedback Inhibitory Neurons. PLOS ONE,
8(4):e62379, April 2013. Publisher: Public Library of Science.
[38] Chung-Chuan Lo. Dynamic tuning of perceptual decision making in a cortical
circuit model by balanced synaptic input. Frontiers, January 2010.
[39] Alex Roxin and Anders Ledberg. Neurobiological Models of Two-Choice Decision
Making Can Be Reduced to a One-Dimensional Nonlinear Diffusion
Equation. PLOS Computational Biology, 4(3):e1000046, March 2008. Publisher:
Public Library of Science.
[40] Jochen Ditterich. Evidence for time-variant decision making. Eur J Neurosci,
24(12):3628–3641, December 2006.
[41] Paul Cisek, Geneviève Aude Puskas, and Stephany El-Murr. Decisions
in Changing Conditions: The Urgency-Gating Model. J. Neurosci.,
29(37):11560–11571, September 2009. Publisher: Society for Neuroscience
Section: Articles.
[42] David Thura, Julie Beauregard-Racine, Charles-William Fradet, and Paul
Cisek. Decision making by urgency gating: theory and experimental support.
Journal of Neurophysiology, 108(11):2912–2930, September 2012. Publisher:
American Physiological Society.
[43] Mattia Rigotti, Omri Barak, Melissa R. Warden, Xiao-Jing Wang,
Nathaniel D. Daw, Earl K. Miller, and Stefano Fusi. The importance of mixed
selectivity in complex cognitive tasks. Nature, 497(7451):585–590, May 2013.
[44] Joseph K. Jun, Paul Miller, Adrián Hernández, Antonio Zainos, Luis Lemus,
Carlos D. Brody, and Ranulfo Romo. Heterogenous Population Coding of a
Short-Term Memory and Decision Task. J. Neurosci., 30(3):916–929, January
2010. Publisher: Society for Neuroscience Section: Articles.
[45] Mikio C. Aoi, Valerio Mante, and Jonathan W. Pillow. Prefrontal cortex
exhibits multidimensional dynamic encoding during decision-making. Nature
Neuroscience, 23(11):1410–1420, November 2020. Number: 11 Publisher:
Nature Publishing Group.