過去的文獻中我們發現不同的神經模型皆展現了鎖模的現象，這暗示了神經 元中隱含了相角動力學的機制，然而現今的機器學習領域中卻忽略了此項重要 的特性。本論文將以保留了離散化相角動力學特性的Kinouchi-Copelli 模型為切 入點研究其在正方形晶格網絡中的動力學行為，並發現它展現了神經編碼中重 要的線性-非線性-卜瓦松(LNP)模型的特性。另一方面，再經過引入抑制型突觸 概念的延拓後，Kinouchi-Copelli 神經元網絡重現了在視網膜神經節細胞中墨西 哥帽型的接受域，這些結果都說明了引入相角動力學機制的神經元網絡，在研 究神經資訊處理的前潛力。

In previous literature, it found that different neuron models demonstrate the mode-locking phenomenon, this indicates there exists the phase dynamics behind neural activities. However, modern machine learning models only consider the rate model and drop this important property. In this thesis, we start from the Kinouchi-Copelli neuron located on the square lattice network and study its dynamic behaviors, which we observe that it presents the property of the LinearNonlinear-Poisson(LNP) model in neural encoding. On the other hand, after including the concept of inhibitory synapse, the Kinouchi-Copelli neuronal network produces the Mexican-hat shape receptive field which can be found in retina ganglion cells. Those results demonstrate the phase dynamics neuronal networks have the potential to study the information processing in neuroscience.

[1] H. H. Lin and M. P. A. Fisher, “Mode locking in quantum-hall-effect point contacts,” Phys. Rev. B, vol. 54, pp. 10593–10603, Oct 1996.
[2] G. Field and E. Chichilnisky, “Information processing in the primate retina: Circuitry and coding,” Annual Review of Neuroscience, vol. 30, no. 1, pp. 1– 30, 2007.
[3] T. Hosoya, S. A. Baccus, and M. Meister, “Dynamic predictive coding by the retina,” Nature, vol. 436, no. 7047, pp. 71–77, 2005.
[4] A. L. HODGKIN and A. F. HUXLEY, “A quantitative description of membrane current and its application to conduction and excitation in nerve,” The Journal of physiology, vol. 117, pp. 500–544, 08 1952.
[5] E. Izhikevich, “Simple model of spiking neurons,” IEEE Transactions on Neural Networks, vol. 14, no. 6, pp. 1569–1572, 2003.
[6] R. FitzHugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophysical Journal, vol. 1, no. 6, pp. 445–466, 1961.
[7] G. B. Ermentrout and N. Kopell, “Parabolic bursting in an excitable system coupled with a slow oscillation,” SIAM Journal on Applied Mathematics, vol. 46, no. 2, pp. 233–253, 1986.
[8] H. Wang, Y. Sun, Y. Li, and Y. Chen, “Influence of autapse on mode-locking structure of a hodgkin–huxley neuron under sinusoidal stimulus,” Journal of Theoretical Biology, vol. 358, no. 1, pp. 25–30, 2014.
[9] K. Yoshino, T. Nomura, K. Pakdaman, and S. Sato, “Synthetic analysis of periodically stimulated excitable and oscillatory membrane models,” Phys. Rev. E, vol. 59, pp. 956–969, Jan 1999.
[10] S. Coombes and P. C. Bressloff, “Mode locking and arnold tongues in integrate-and-fire neural oscillators,” Phys. Rev. E, vol. 60, pp. 2086–2096, Aug 1999.
[11] A. B. Givental, B. A. Khesin, J. E. Marsden, A. N. Varchenko, V. A. Vassiliev, O. Y. Viro, and V. M. Zakalyukin, eds., Small denominators. I. Mapping of the circumference onto itself, pp. 152–223. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009.
[12] C.-Y. Lin, P.-H. Chen, H.-H. Lin, and W.-M. Huang, “U(1) dynamics in neuronal activities,” 2021.
[13] A. L. Hodgkin, “The local electric changes associated with repetitive action in a non-medullated axon,” The Journal of physiology, vol. 107, pp. 165–181, 03 1948.
[14] O. Kinouchi and M. Copelli, “Optimal dynamical range of excitable networks at criticality,” Nature Physics, vol. 2, no. 5, pp. 348–351, 2006.
[15] M. Copelli and O. Kinouchi, “Intensity coding in two-dimensional excitable neural networks,” Physica A: Statistical Mechanics and its Applications, vol. 349, no. 3, pp. 431–442, 2005.
[16] R. S. Williamson, M. Sahani, and J. W. Pillow, “The equivalence of information-theoretic and likelihood-based methods for neural dimensionality reduction,” PLOS Computational Biology, vol. 11, pp. 1–31, 04 2015.
[17] W. Gerstner, W. M. Kistler, R. Naud, and L. Paninski, Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition. Cambridge University Press, 2014.
[18] P. W. Anderson, “A poor man's derivation of scaling laws for the kondo problem,” Journal of Physics C: Solid State Physics, vol. 3, pp. 2436–2441, dec 1970.
[19] J. E. Dowling and J. L. Dowling, Vision: How It Works and What Can Go Wrong. The MIT Press, 03 2016.
[20] C. G. Galizia and P.-M. Lledo, Neurosciences - From Molecule to Behavior: a university textbook. 01 2013.
[21] D. Cai, G. C. Deangelis, and R. D. Freeman, “Spatiotemporal receptive field organization in the lateral geniculate nucleus of cats and kittens,” Journal of Neurophysiology, vol. 78, no. 2, pp. 1045–1061, 1997.
[22] M. W. Cho and M. Choi, “A model for the receptive field of retinal ganglion cells,” Neural Networks, vol. 49, pp. 51–58, 2014.
[23] H. Wei, L. Wang, S. Wang, Y. Jiang, and J. Li, “A signal-processing neural model based on biological retina,” Electronics, vol. 9, no. 1, 2020.
[24] A. Farokhniaee and E. W. Large, “Mode-locking behavior of izhikevich neurons under periodic external forcing,” Phys. Rev. E, vol. 95, p. 062414, Jun 2017.
[25] C.-Y. Lin, “U(1) neuronal network,” 2020.
[26] B. S. Gutkin, G. B. Ermentrout, and A. D. Reyes, “Phase-response curves give the responses of neurons to transient inputs,” Journal of Neurophysiology, vol. 94, no. 2, pp. 1623–1635, 2005.
[27] C. Haldeman and J. M. Beggs, “Critical branching captures activity in living neural networks and maximizes the number of metastable states,” Phys. Rev. Lett., vol. 94, p. 058101, Feb 2005.
[28] M. S. A. Ferraz, H. L. C. Melo-Silva, and A. H. Kihara, “Optimizing information processing in neuronal networks beyond critical states,” PLOS ONE, vol. 12, pp. 1–12, 09 2017.
[29] D. R. Chialvo, “Emergent complex neural dynamics,” Nature Physics, vol. 6, no. 10, pp. 744–750, 2010.
[30] W. L. Shew, W. P. Clawson, J. Pobst, Y. Karimipanah, N. C. Wright, and R. Wessel, “Adaptation to sensory input tunes visual cortex to criticality,” Nature Physics, vol. 11, no. 8, pp. 659–663, 2015.
[31] R. Chaudhuri, K. Knoblauch, M.-A. Gariel, H. Kennedy, and X.-J. Wang, “A large-scale circuit mechanism for hierarchical dynamical processing in the primate cortex,” Neuron, vol. 88, no. 2, pp. 419–431, 2015.
[32] N. Sukenik, O. Vinogradov, E. Weinreb, M. Segal, A. Levina, and E. Moses, “Neuronal circuits overcome imbalance in excitation and inhibition by adjust- ing connection numbers,” Proceedings of the National Academy of Sciences, vol. 118, no. 12, p. e2018459118, 2021.
[33] J. H. van Hateren and D. L. Ruderman, “Independent component analysis of natural image sequences yields spatio-temporal filters similar to simple cells in primary visual cortex,” Proceedings of the Royal Society of London. Series B: Biological Sciences, vol. 265, no. 1412, pp. 2315–2320, 1998.
[34] D. Vishwanath and E. Blaser, “Retinal blur and the perception of egocentric distance,” Journal of Vision, vol. 10, pp. 26–26, 08 2010.