神经元模型对比分析

徐泠风; 李传东; 陈玲

doi:10.7498/aps.65.240701

摘要

近年来，生物神经元模型的建立与应用已经获得了越来越多的关注，逐渐成为神经科学的一个重要分支.神经元模型不仅在仿生学、存储器设计、逻辑运算、信号处理等方面有重大应用，对分析研究神经系统的动力学特性也具有重要意义.本文总结了自1907年第一个神经元模型建立以来的发展历程，归纳出17种最具代表性的数学模型，分为电导依赖模型和非电导依赖模型进行比较分析，重点展示包括最新神经芯片TrueNorth上的神经元在内的5种经典模型，分析其仿真特性，以及电路实现的需求，方便研究者根据具体需求选择和改进神经元模型.

关键词:

Abstract

In recent years,the modeling and application of biological neurons have gained more and more attention.By now, the research on neuron models has become one of the most important branches of neuroscience.Neuron models can be used in various areas,such as biomimetic applications,memory design,logical computing,and signal processing. Furthermore,it is significant to study the dynamic characteristics of neural system by using neuron models.In this paper,the historical development of neuron models is reviewed.The neuron models have experienced three development stages.In the pioneering stage,a group of scientists laid the experimental and theoretical foundation for later research. Then,the whole study started to blossom after the publication of Hodgkin-Huxley model.In the 1970s and 1980s,various models were proposed.One of the research focuses was the simulation of neural repetitive spiking.Since the 1990s, researchers have paid more attention to setting up models that are both physiologically meaningful and computationally effective.The model put forward by Izhikevich E M has been proved to solve the problem successfully.Recently,IBM presented a versatile spiking neuron model based on 1272 ASIC gates.The model,both theoretically understandable and physically implementable,has been used as a basic building block in IBM's neuro-chip TrueNorth.In the paper, seventeen neuron models worth studying are listed.To give a more explicit explanation,these models are classified as two groups,namely conductance-dependent and conductance-independent models.The former group's goal is to model the electrophysiology of neuronal membrane,while the latter group is only to seek for capturing the input-output behavior of a neuron by using simple mathematical abstractions.The complexity and features of each model are illustrated in a chart,while the prominent repetitive spiking curves of each model are also exhibited.Five of the models are further detailed,which are the Hodgkin-Huxley model,the Integrate-and-fire model,the Fitzhugh-Nagumo model,the Izhikevich model,and the most recent model used by IBM in its neuro-chip TrueNorth.Finally,three questions are put forward at the end of the paper,which are the most important problems that today's researchers must consider when setting up new neuron models.In conclusion,the feasibility of physical implementation remains to be a challenge to all researchers. Through the aforementioned work,the authors aim to provide a reference for the study that follows,helping researchers to compare those models in order to choose the properest one.

Keywords:

作者及机构信息

1.
西南大学电子信息工程学院, 重庆 400715;

2.
中国科学技术大学信息科学技术学院, 合肥 230027

通信作者: 李传东, licdswu@163.com

基金项目: 国家自然科学基金（批准号：61374078，61503307）、重庆市基础与前沿技术研究项目（批准号：cstc2015jcyjBX0052，cstc2016jcyjA0261）、中央高校基本科研业务费专项资金（批准号：XDJK2015C079）和博士后科学基金（批准号：2016M590854）资助的课题.

Authors and contacts

1.
College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China;

2.
School of Information Science and Technology, University of Science and Technology of China, Hefei 230027, China

Corresponding author: Li Chuan-Dong, licdswu@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61374078, 61503307), the Chongqing Research Program of Basic Research and Frontier Technology, China (Grant Nos. cstc2015jcyjBX0052, cstc2016jcyjA0261), the Fundamental Research Funds for the Central Universities, China (Grant No. XDJK2015C079), and the China Postdoctoral Science Foundation (Grant No. 2016M590854).

参考文献

[1]	Durstewitz D, Seamans J K, Sejnowski T J 2000 Nat. Neurosci. 3 1184
[2]	Cassidy A S, Merolla P, Arthur J V, Esser S K, Jackson B, Alvarez-Icaza R, Datta P, Sawada Jun, Wong T M, Feldman V, Amir A, Rubin D B, Akopyan F, McQuinn E, Risk W P, Modha D S 2013 The 2013 International Joint Conference on Neural Networks (IJCNN) Dallas, USA, August 4-9, 2013 p1
[3]	Smith G D, Cox C L, Sherman S M, Rinzel J 2000 J. Neurophysiol. 83 588
[4]	Fourcaud-Trocmé D, Hansel D, van Vreeswijk C, Brunel N 2003 J. Neurosci. 23 11628
[5]	Lapicque L 1907 J. Physiol. Pathol. Gen. 9 620
[6]	Abbott L F 1999 Brain Res. Bull. 50 303
[7]	Brunel N, van Rossum M C W 2007 Biol. Cybern. 97 337
[8]	McCulloch W S, Pitts W 1943 Bull. Math. Biol. 5 115
[9]	Bernstein J 1902 Pflgers Arch. 92 521
[10]	Hodgkin A L 1939 J. Physiol. 94 560
[11]	Hodgkin A L, Katz B 1949 J. Physiol. 108 37
[12]	Bonhoeffer K F 1948 J. Gen. Physiol. 32 69
[13]	Hodgkin A L 1948 J. Physiol. 107 165
[14]	Hebb D O 1949 The Organization of Behavior:A Neuropsychological Theory (1st Ed.) (London:Chapman & Hall) pp17-78
[15]	Hodgkin A L, Huxley A F, Katz B 1952 J. Physiol. 116 424
[16]	Eccles J C, Eccles R M, Lundberg A 1957 J. Physiol. 137 22
[17]	Rosenblatt F 1958 Psychol. Rev. 65 386
[18]	Fitzhugh R 1960 J. Gen. Physiol. 43 867
[19]	Nagumo J, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061
[20]	Fuortes M G F, Mantegazzini F 1962 J. Gen. Physiol. 45 1163
[21]	Stein R B 1965 Biophys. J. 5 173
[22]	Geisler C D, Goldberg J M 1966 Biophys. J. 6 53
[23]	Rall W 1967 J. Neurophysiol. 30 1138
[24]	Stein R B 1967 Proc. R. Soc. Lond. B:Biol. Sci. 167 64
[25]	Knight B W 1972 J. Gen. Physiol. 59 734
[26]	Kernell D, Sj·holm H 1973 Acta Physiol. Scand. 87 40
[27]	Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
[28]	Shapiro B I, Lenherr F K 1972 Biophys. J. 12 1145
[29]	Krinskii V I, Iu M K 1973 Biofizika 18 506
[30]	Krinskii V I, Iu M K 1973 Biofizika 18 878
[31]	Plant R E, Kim M 1976 Biophys. J. 16 227
[32]	Plant R E 1976 Computer Programs Biomed. 6 85
[33]	Rinzel J 1978 Studies in Mathematical Biology (1st Ed.) (Washington:Mathematical association of America) pp1-66
[34]	Fitzhugh R 1961 Biophys. J. 1 445
[35]	Connor J A, Stevens C F 1971 J. Physiol. 213 31
[36]	Connor J A, Walter D, Mckown R 1977 Biophys. J. 18 81
[37]	Morris C, Lecar H 1981 Biophys. J. 35 193
[38]	Rinzel J, Troy W C 1982 J. Chem. Phys. 76 1775
[39]	Chay T R 1985 Physica D 16 233
[40]	Chay T R, Keizer J 1983 Biophys. J. 42 181
[41]	Ermentrout G B, Kopell N 1986 SIAM J. Appl. Math. 46 233
[42]	Hindmarsh J L, Rose R M 1982 Nature 296 162
[43]	Hindmarsh J L, Rose R M 1984 Proc. R. Soc. Lond. B:Biol. Sci. 221 87
[44]	Rose R M, Hindmarsh J L 1985 Proc. R. Soc. Lond. B:Biol. Sci. 225 161
[45]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 267
[46]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 289
[47]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 313
[48]	Chay T R, Rinzel J 1985 Biophys. J. 47 357
[49]	Connor J A, Stevens C F 1971 J. Physiol. 213 1
[50]	Connor J A, Stevens C F 1971 J. Physiol. 213 21
[51]	Connor J 1975 J. Neurophysiol. 38 922
[52]	Jack J J B, Noble D, Tsien R W 1975 Electric Current Flow in Excitable Cells (1st Ed.) (Oxford:Clarendon Press) pp132-224
[53]	Connors B W, Gutnick M J, Prince D A 1982 J. Neurophysiol. 48 1302
[54]	Plant R E 1978 Biophys. J. 21 217
[55]	Rinzel J 1985 Ordinary and Partial Differential Equations (1st Ed.) (Berlin:Springer-Verlag) pp304-316
[56]	Rinzel J 1985 Fed. Proc. 44 2944
[57]	Mircea S, Jones E G, Llinás R R 1990 Thalamic Oscillations and Signaling (1st Ed.) (New York:John Wiley) pp1-43
[58]	Rinzel J 1987 Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (1st Ed.) (Berlin:Springer) pp267-281
[59]	Connors B W, Gutnick M J 1990 Trends Neurosci. 13 99
[60]	Rall W 1989 Methods in Neuronal Modeling:from Ions to Networks (2nd Ed.) (Cambridge:MIT Press) pp9-62
[61]	Ermentrout B 1996 Neural Comput. 8 979
[62]	Hoppensteadt F C, Izhikevich E M 1997 Weakly Connected Neural Networks (1st Ed.) (New York:Springer-Verlag) pp25-101
[63]	Rinzel J, Ermentrout G B 1989 Methods in Neuronal Modeling (1st Ed.) (Cambridge:MIT Press) pp135-169
[64]	Roth A, Häusser M 2001 J. Physiol. 535 445
[65]	Chay T R 1991 Biopolymers 31 1483
[66]	Stevens C F, Zador A M 1998 Proceedings of 5th Joint Symposium on Neural Computation San Diego, USA, May 16, 1998 p172
[67]	Wilson H R 1999 J. Theor. Biol. 200 375
[68]	Izhikevich E M 1999 IEEE Trans. Neural Netw. 10 499
[69]	Izhikevich E M 2001 SIAM Rev. 43 315
[70]	Izhikevich E M, Hoppensteadt F 2004 Int. J. Bifurcat. Chaos 14 3847
[71]	Abbott L F, van Vreeswijk C 1993 Phys. Rev. E 48 1483
[72]	Destexhe A, Rudolph M, Pare D 2003 Nat. Rev. Neurosci. 4 739
[73]	Avoli M, Hwa G G, Lacaille J C, Olivier A, Villemure J G 1994 Exp. Brain Res. 98 135
[74]	Hutcheon B, Miura R M, Puil E 1996 J. Neurophysiol. 76 683
[75]	Mainen Z F, Sejnowski T J 1996 Nature 382 363
[76]	Bower J M, Beeman D 1998 The Book of Genesis (1st Ed.) (New York:Springer) pp51-130
[77]	Destexhe A, Mainen Z F, Sejnowski T J 1994 J. Comput. Neurosci. 1 195
[78]	Hutcheon B, Yarom Y 2000 Trends Neurosci. 23 216
[79]	Latham P E, Richmond B J, Nelson P G, Nirenberg S 2000 J. Neurophysiol. 83 808
[80]	Pike F G, Goddard R S, Suckling J M, Ganter P, Kasthuri N, Paulsen O 2000 J. Physiol. 529 205
[81]	Hansel D, Mato G 2001 Phys. Rev. Lett. 86 4175
[82]	Izhikevich E M 2004 IEEE Trans. Neural Netw. 15 1063
[83]	Segev I, Fleshman J W, Burke R E 1989 Methods in Neuronal Modeling (1st Ed.) (Cambridge:MIT Press) pp63-96
[84]	Pinsky P F, Rinzel J 1994 J. Comput. Neurosci. 1 39
[85]	Wang X J, Buzsáki G 1996 J. Neurosci. 16 6402
[86]	Hutcheon B, Miura R M, Puil E 1996 J. Neurophysiol. 76 698
[87]	Manor Y, Rinzel J, Segev I, Yarom Y 1997 J. Neurophysiol. 77 2736
[88]	Koch C, Segev I 1998 Methods in Neuronal Modeling:From Ions to Networks (2nd Ed.) (Cambridge:MIT Press) pp93-136
[89]	Feng J, Brown D 2000 Bull. Math. Biol. 62 467
[90]	Feng J 2001 Neural Netw. 14 955
[91]	Kistler W M, Gerstner W, van Hemmen J L 1997 Neural Comput. 9 1015
[92]	Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171
[93]	Izhikevich E M 2001 Neural Netw. 14 883
[94]	Gerstner W, Kistler W M 2002 Spiking Neuron Models:Single Neurons, Populations, Plasticity (1st Ed.) (Cambridge:Cambridge University Press) pp31-146
[95]	Izhikevich E M 2003 IEEE Trans. Neural Netw. 14 1569
[96]	Jolivet R, Lewis T J, Gerstner W 2004 J. Neurophysiol. 92 959
[97]	Brette R, Gerstner W 2005 J. Neurophysiol. 94 3637
[98]	Mihalas S, Niebur E 2009 Neural Comput. 21 704
[99]	McCormick D A, Wang Z, Huguenard J 1993 Cereb. Cortex 3 387
[100]	Lumer E D 1998 Cereb. Cortex 8 553
[101]	Yang Z Q 2010 Acta Phys. Sin. 59 5319 (in Chinese)[杨卓琴2010 59 5319]
[102]	Liang X B, Liu X S, Liu A Z, Wang B L 2009 Acta Phys. Sin. 58 5065 (in Chinese)[梁晓冰, 刘希顺, 刘安芝, 王博亮2009 58 5065]
[103]	Wang H Q, Yu L C, Chen Y 2009 Acta Phys. Sin. 58 5070 (in Chinese)[王慧巧, 俞连春, 陈勇2009 58 5070]

施引文献

[1]	Durstewitz D, Seamans J K, Sejnowski T J 2000 Nat. Neurosci. 3 1184
[2]	Cassidy A S, Merolla P, Arthur J V, Esser S K, Jackson B, Alvarez-Icaza R, Datta P, Sawada Jun, Wong T M, Feldman V, Amir A, Rubin D B, Akopyan F, McQuinn E, Risk W P, Modha D S 2013 The 2013 International Joint Conference on Neural Networks (IJCNN) Dallas, USA, August 4-9, 2013 p1
[3]	Smith G D, Cox C L, Sherman S M, Rinzel J 2000 J. Neurophysiol. 83 588
[4]	Fourcaud-Trocmé D, Hansel D, van Vreeswijk C, Brunel N 2003 J. Neurosci. 23 11628
[5]	Lapicque L 1907 J. Physiol. Pathol. Gen. 9 620
[6]	Abbott L F 1999 Brain Res. Bull. 50 303
[7]	Brunel N, van Rossum M C W 2007 Biol. Cybern. 97 337
[8]	McCulloch W S, Pitts W 1943 Bull. Math. Biol. 5 115
[9]	Bernstein J 1902 Pflgers Arch. 92 521
[10]	Hodgkin A L 1939 J. Physiol. 94 560
[11]	Hodgkin A L, Katz B 1949 J. Physiol. 108 37
[12]	Bonhoeffer K F 1948 J. Gen. Physiol. 32 69
[13]	Hodgkin A L 1948 J. Physiol. 107 165
[14]	Hebb D O 1949 The Organization of Behavior:A Neuropsychological Theory (1st Ed.) (London:Chapman & Hall) pp17-78
[15]	Hodgkin A L, Huxley A F, Katz B 1952 J. Physiol. 116 424
[16]	Eccles J C, Eccles R M, Lundberg A 1957 J. Physiol. 137 22
[17]	Rosenblatt F 1958 Psychol. Rev. 65 386
[18]	Fitzhugh R 1960 J. Gen. Physiol. 43 867
[19]	Nagumo J, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061
[20]	Fuortes M G F, Mantegazzini F 1962 J. Gen. Physiol. 45 1163
[21]	Stein R B 1965 Biophys. J. 5 173
[22]	Geisler C D, Goldberg J M 1966 Biophys. J. 6 53
[23]	Rall W 1967 J. Neurophysiol. 30 1138
[24]	Stein R B 1967 Proc. R. Soc. Lond. B:Biol. Sci. 167 64
[25]	Knight B W 1972 J. Gen. Physiol. 59 734
[26]	Kernell D, Sj·holm H 1973 Acta Physiol. Scand. 87 40
[27]	Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
[28]	Shapiro B I, Lenherr F K 1972 Biophys. J. 12 1145
[29]	Krinskii V I, Iu M K 1973 Biofizika 18 506
[30]	Krinskii V I, Iu M K 1973 Biofizika 18 878
[31]	Plant R E, Kim M 1976 Biophys. J. 16 227
[32]	Plant R E 1976 Computer Programs Biomed. 6 85
[33]	Rinzel J 1978 Studies in Mathematical Biology (1st Ed.) (Washington:Mathematical association of America) pp1-66
[34]	Fitzhugh R 1961 Biophys. J. 1 445
[35]	Connor J A, Stevens C F 1971 J. Physiol. 213 31
[36]	Connor J A, Walter D, Mckown R 1977 Biophys. J. 18 81
[37]	Morris C, Lecar H 1981 Biophys. J. 35 193
[38]	Rinzel J, Troy W C 1982 J. Chem. Phys. 76 1775
[39]	Chay T R 1985 Physica D 16 233
[40]	Chay T R, Keizer J 1983 Biophys. J. 42 181
[41]	Ermentrout G B, Kopell N 1986 SIAM J. Appl. Math. 46 233
[42]	Hindmarsh J L, Rose R M 1982 Nature 296 162
[43]	Hindmarsh J L, Rose R M 1984 Proc. R. Soc. Lond. B:Biol. Sci. 221 87
[44]	Rose R M, Hindmarsh J L 1985 Proc. R. Soc. Lond. B:Biol. Sci. 225 161
[45]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 267
[46]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 289
[47]	Rose R M, Hindmarsh J L 1989 Proc. R. Soc. Lond. B:Biol. Sci. 237 313
[48]	Chay T R, Rinzel J 1985 Biophys. J. 47 357
[49]	Connor J A, Stevens C F 1971 J. Physiol. 213 1
[50]	Connor J A, Stevens C F 1971 J. Physiol. 213 21
[51]	Connor J 1975 J. Neurophysiol. 38 922
[52]	Jack J J B, Noble D, Tsien R W 1975 Electric Current Flow in Excitable Cells (1st Ed.) (Oxford:Clarendon Press) pp132-224
[53]	Connors B W, Gutnick M J, Prince D A 1982 J. Neurophysiol. 48 1302
[54]	Plant R E 1978 Biophys. J. 21 217
[55]	Rinzel J 1985 Ordinary and Partial Differential Equations (1st Ed.) (Berlin:Springer-Verlag) pp304-316
[56]	Rinzel J 1985 Fed. Proc. 44 2944
[57]	Mircea S, Jones E G, Llinás R R 1990 Thalamic Oscillations and Signaling (1st Ed.) (New York:John Wiley) pp1-43
[58]	Rinzel J 1987 Mathematical Topics in Population Biology, Morphogenesis and Neurosciences (1st Ed.) (Berlin:Springer) pp267-281
[59]	Connors B W, Gutnick M J 1990 Trends Neurosci. 13 99
[60]	Rall W 1989 Methods in Neuronal Modeling:from Ions to Networks (2nd Ed.) (Cambridge:MIT Press) pp9-62
[61]	Ermentrout B 1996 Neural Comput. 8 979
[62]	Hoppensteadt F C, Izhikevich E M 1997 Weakly Connected Neural Networks (1st Ed.) (New York:Springer-Verlag) pp25-101
[63]	Rinzel J, Ermentrout G B 1989 Methods in Neuronal Modeling (1st Ed.) (Cambridge:MIT Press) pp135-169
[64]	Roth A, Häusser M 2001 J. Physiol. 535 445
[65]	Chay T R 1991 Biopolymers 31 1483
[66]	Stevens C F, Zador A M 1998 Proceedings of 5th Joint Symposium on Neural Computation San Diego, USA, May 16, 1998 p172
[67]	Wilson H R 1999 J. Theor. Biol. 200 375
[68]	Izhikevich E M 1999 IEEE Trans. Neural Netw. 10 499
[69]	Izhikevich E M 2001 SIAM Rev. 43 315
[70]	Izhikevich E M, Hoppensteadt F 2004 Int. J. Bifurcat. Chaos 14 3847
[71]	Abbott L F, van Vreeswijk C 1993 Phys. Rev. E 48 1483
[72]	Destexhe A, Rudolph M, Pare D 2003 Nat. Rev. Neurosci. 4 739
[73]	Avoli M, Hwa G G, Lacaille J C, Olivier A, Villemure J G 1994 Exp. Brain Res. 98 135
[74]	Hutcheon B, Miura R M, Puil E 1996 J. Neurophysiol. 76 683
[75]	Mainen Z F, Sejnowski T J 1996 Nature 382 363
[76]	Bower J M, Beeman D 1998 The Book of Genesis (1st Ed.) (New York:Springer) pp51-130
[77]	Destexhe A, Mainen Z F, Sejnowski T J 1994 J. Comput. Neurosci. 1 195
[78]	Hutcheon B, Yarom Y 2000 Trends Neurosci. 23 216
[79]	Latham P E, Richmond B J, Nelson P G, Nirenberg S 2000 J. Neurophysiol. 83 808
[80]	Pike F G, Goddard R S, Suckling J M, Ganter P, Kasthuri N, Paulsen O 2000 J. Physiol. 529 205
[81]	Hansel D, Mato G 2001 Phys. Rev. Lett. 86 4175
[82]	Izhikevich E M 2004 IEEE Trans. Neural Netw. 15 1063
[83]	Segev I, Fleshman J W, Burke R E 1989 Methods in Neuronal Modeling (1st Ed.) (Cambridge:MIT Press) pp63-96
[84]	Pinsky P F, Rinzel J 1994 J. Comput. Neurosci. 1 39
[85]	Wang X J, Buzsáki G 1996 J. Neurosci. 16 6402
[86]	Hutcheon B, Miura R M, Puil E 1996 J. Neurophysiol. 76 698
[87]	Manor Y, Rinzel J, Segev I, Yarom Y 1997 J. Neurophysiol. 77 2736
[88]	Koch C, Segev I 1998 Methods in Neuronal Modeling:From Ions to Networks (2nd Ed.) (Cambridge:MIT Press) pp93-136
[89]	Feng J, Brown D 2000 Bull. Math. Biol. 62 467
[90]	Feng J 2001 Neural Netw. 14 955
[91]	Kistler W M, Gerstner W, van Hemmen J L 1997 Neural Comput. 9 1015
[92]	Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171
[93]	Izhikevich E M 2001 Neural Netw. 14 883
[94]	Gerstner W, Kistler W M 2002 Spiking Neuron Models:Single Neurons, Populations, Plasticity (1st Ed.) (Cambridge:Cambridge University Press) pp31-146
[95]	Izhikevich E M 2003 IEEE Trans. Neural Netw. 14 1569
[96]	Jolivet R, Lewis T J, Gerstner W 2004 J. Neurophysiol. 92 959
[97]	Brette R, Gerstner W 2005 J. Neurophysiol. 94 3637
[98]	Mihalas S, Niebur E 2009 Neural Comput. 21 704
[99]	McCormick D A, Wang Z, Huguenard J 1993 Cereb. Cortex 3 387
[100]	Lumer E D 1998 Cereb. Cortex 8 553
[101]	Yang Z Q 2010 Acta Phys. Sin. 59 5319 (in Chinese)[杨卓琴2010 59 5319]
[102]	Liang X B, Liu X S, Liu A Z, Wang B L 2009 Acta Phys. Sin. 58 5065 (in Chinese)[梁晓冰, 刘希顺, 刘安芝, 王博亮2009 58 5065]
[103]	Wang H Q, Yu L C, Chen Y 2009 Acta Phys. Sin. 58 5070 (in Chinese)[王慧巧, 俞连春, 陈勇2009 58 5070]

[1]	黄彬源, 贺志, 陈雨. 基于依赖强度Dicke模型的量子电池充电性能. , 2023, 72(18): 180301. doi: 10.7498/aps.72.20230578
[2]	梁艳美, 陆博, 古华光. 利用双慢变量的快慢变量分离分析新脑皮层神经元Wilson模型的复杂电活动. , 2022, 71(23): 230502. doi: 10.7498/aps.71.20221416
[3]	吴静, 潘春宇. 感性神经元模型及其动力学特性研究. , 2022, 71(4): 048701. doi: 10.7498/aps.71.20211626
[4]	吴静, 潘春宇. 感性神经元模型及其动力学特性研究. , 2021, (): . doi: 10.7498/aps.70.20211626
[5]	王月娥, 吴保卫, 汪锐. 切换系统的异步镇定: 相邻模型依赖平均驻留时间. , 2015, 64(5): 050201. doi: 10.7498/aps.64.050201
[6]	靳钊, 乔丽萍, 郭晨, 王江安, 刘策. 单轴应变Si(001)任意晶向电子电导有效质量模型. , 2013, 62(5): 058501. doi: 10.7498/aps.62.058501
[7]	金淇涛, 王江, 伊国胜, 李会艳, 邓斌, 魏熙乐, 车艳秋. 经颅磁刺激感应外电场作用下最小神经元模型放电起始动态机理分析. , 2012, 61(11): 118701. doi: 10.7498/aps.61.118701
[8]	王兴元, 任小丽, 张永雷. 参数未知神经元模型的全阶与降阶最优同步. , 2012, 61(6): 060508. doi: 10.7498/aps.61.060508
[9]	金淇涛, 王江, 魏熙乐, 邓斌, 车艳秋. 最小神经元模型放电起始动态控制与分析. , 2011, 60(9): 098701. doi: 10.7498/aps.60.098701
[10]	陈军, 李春光. 禁忌学习神经元模型的电路设计及其动力学研究. , 2011, 60(2): 020502. doi: 10.7498/aps.60.020502
[11]	陈军, 李春光. 具有自适应反馈突触的神经元模型中的混沌:电路设计. , 2011, 60(5): 050503. doi: 10.7498/aps.60.050503
[12]	赵丽霞, 张鹤鸣, 胡辉勇, 戴显英, 宣荣喜. 应变Si电子电导有效质量模型. , 2010, 59(9): 6545-6548. doi: 10.7498/aps.59.6545
[13]	刘勇, 谢勇. 分数阶FitzHugh-Nagumo模型神经元的动力学特性及其同步. , 2010, 59(3): 2147-2155. doi: 10.7498/aps.59.2147
[14]	汪茂胜, 黄万霞, 崔执凤. 二维映射神经元模型中的相干双共振. , 2010, 59(7): 4485-4489. doi: 10.7498/aps.59.4485
[15]	刘志宏, 周玉荣, 张安英, 庞小峰. 色关联噪声驱动下非线性神经元模型的相干共振. , 2010, 59(2): 699-704. doi: 10.7498/aps.59.699
[16]	汪茂胜. 二维映射神经元模型中频率依赖的随机共振. , 2009, 58(10): 6833-6837. doi: 10.7498/aps.58.6833
[17]	全荣辉, 韩建伟, 黄建国, 张振龙. 电介质材料辐射感应电导率的模型研究. , 2007, 56(11): 6642-6647. doi: 10.7498/aps.56.6642
[18]	宋杨, 赵同军, 刘金伟, 王向群, 展永. 高斯白噪声对神经元二维映射模型动力学的影响. , 2006, 55(8): 4020-4025. doi: 10.7498/aps.55.4020
[19]	常胜江, 刘, 张文伟, 申金媛, 翟宏琛, 张延. 适用于神经元状态非等概率分布的神经网络模型及其光学实现. , 1998, 47(7): 1101-1109. doi: 10.7498/aps.47.1101
[20]	万梅香, 王萍. VSC电导测量方法的物理模型. , 1986, 35(1): 82-88. doi: 10.7498/aps.35.82

计量

文章访问数: 10775
PDF下载量: 1334
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板