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Neuron is a typical dynamic system, therefore, it is quite natural to study the firing behaviors of neurons by using the dynamical system theory. Two kinds of firing patterns, i.e., the periodic spiking and the periodic bursting, are the limit cycle oscillators from the point of view of nonlinear dynamics. The simplest way to describe the limit cycle is to use the phase of the oscillator. A complex state space model can be mapped into a one-dimensional phase model by phase transformation, which is helpful for obtaining the analytical solution of the oscillator system. The response characteristics of the oscillator system in the motion state of the limit cycle to the external stimuli can be characterized by the phase response curve. A phase response curve illustrates the transient change in the cycle period of an oscillation induced by a perturbation as a function of the phase at which it is received. Now it is widely believed that the phase response curve provides a new way to study the behavior of the neuron. Existing studies have shown that the phase response curve of the periodic spiking can be divided into two types, which are closely related to the bifurcation mechanism of neurons from rest to repetitive firing. However, there are few studies on the relationship between the phase response curve and the bifurcation type of the periodic bursting. Clearly, the first prerequisite to understand this relationship is to calculate the phase response curve of the periodic bursting. The existing algorithms for computing the phase response curve are often unsuccessful in the periodic bursting. In this paper, we present a method of calculating the phase response curve, namely the direct algorithm with square wave perturbation. The phase response curves of periodic spiking and periodic bursting can be obtained by making use of the direct algorithm, which is verified in the four neuron models of the Hodgkin-Huxley, FitzHugh-Nagumo, Morris-Lecar and Hindmarsh-Rose. This algorithm overcomes the limitations to other algorithms in the application. The calculation results show that the phase response curve of the periodic spiking is determined by the bifurcation type. We find a suprathreshold periodic oscillation starting from a Hopf bifurcation and terminating at a saddle homoclinic orbit bifurcation as a function of the applied current strength in the Morris-Lecar model, and its phase response curve belongs to Type II. A large amount of calculation indicates that the relative size of the phase response and its positive or negative value depend only on the time of imposing perturbation, and the phase response curve of periodic bursting is more complicated than that of periodic spiking.
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Keywords:
- phase response curve /
- spiking /
- bursting /
- bifurcation
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[17] Yang Z Q, Guan T T, Gan C B, Zhang J Y 2011 Acta Phys. Sin. 60 110202 (in Chinese) [杨卓琴, 管亭亭, 甘春标, 张矫瑛 2011 物理学报 60 110202]
[18] Longtin A 1993 J. Stat. Phys. 70 309
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[21] Yu Y, Wang W, Wang J, Liu F 2001 Phys. Rev. E 63 021907
[22] Liu F, Wang J, Wang W 1999 Phys. Rev. E 59 3453
[23] Ermentrout B 1996 Neural Comput. 8 979
[24] Gutkin B S, Ermentrout G B, Reyes A D 2005 J. Neurophysiol. 94 1623
[25] Hastings J W, Sweeney B M 1958 Biol. Bull. 115 440
[26] Johnson C H 1999 Chronobiol. Int. 16 711
[27] Ikeda N 1982 Biol. Cybern. 43 157
[28] Tsalikakis D G, Zhang H G, Fotiadis D I, Kremmydas G P, Michalis L K 2007 Comput. Biol. Med. 37 8
[29] Ermentrout G B, Kopell N 1991 J. Math. Biol. 29 195
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[31] Stiger T, Danzl P, Moehlis J, Netoff T I 2010 J. Med. Devices 4 027533
[32] Shi X, Zhang J D 2016 Chin. Phys. B 25 060502
[33] Schultheiss N W, Prinz A A, Butera R J 2012 Phase Response Curves in Neuroscience (New York: Springer) p3
[34] Ermentrout G B 2002 Simulating, Analyzing, and Animating Dynamical Systems: a Guide to XPPAUT for Researchers and Students (Philadelphia: SIAM) p226
[35] Govaerts W, Sautois B 2006 Neural Comput. 18 817
[36] Sherwood W E, Guckenheimer J 2010 SIAM J. Appl. Dyn. Syst. 9 659
[37] Novicenko V, Pyragas K 2011 Nonlinear Dynam. 67 517
[38] Ermentrout G B, Terman D H 2010 Mathematical Foundations of Neuroscience (New York: Springer) p51
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[1] Hodgkin A L, Huxley A F 1952 J. Physiol. 117 500
[2] FitzHugh R 1961 Biophys. J. 1 445
[3] Nagumo J, Arimoto S, Yoshizawa S 1962 Proc. IRE 50 2061
[4] Morris C, Lecar H 1981 Biophys. J. 35 193
[5] Hindmarsh J L, Rose R M 1984 Proc. R. Soc. London Ser. B 221 87
[6] Xu L F, Li C D, Chen L 2016 Acta Phys. Sin. 65 240701 (in Chinese) [徐泠风, 李传东, 陈玲 2016 物理学报 65 240701]
[7] Holden A V, Fan Y S 1992 Chaos Soliton. Fract. 2 221
[8] Holden A V, Fan Y S 1992 Chaos Soliton. Fract. 2 349
[9] Holden A V, Fan Y S 1992 Chaos Soliton. Fract. 2 583
[10] Fan Y S, Holden A V 1993 Chaos Soliton. Fract. 3 439
[11] Izhikevich E M 2000 Int. J. Bifurcat. Chaos 10 1171
[12] Gong P L, Xu J X 2001 Phys. Rev. E 63 031906
[13] Ding X L, Li Y Y 2016 Acta Phys. Sin. 65 210502 (in Chinese) [丁学利, 李玉叶 2016 物理学报 65 210502]
[14] Gu H G, Zhu Z, Jia B 2011 Acta Phys. Sin. 60 100505 (in Chinese) [古华光, 朱洲, 贾冰 2011 物理学报 60 100505]
[15] Jin Q T, Wang J, Wei X L, Deng B, Che Y Q 2011 Acta Phys. Sin. 60 098701 (in Chinese) [金淇涛, 王江, 魏熙乐, 邓斌, 车艳秋 2011 物理学报 60 098701]
[16] Wang H X, Wang Q Y, Lu Q S 2011 Chaos Soliton. Fract. 44 667
[17] Yang Z Q, Guan T T, Gan C B, Zhang J Y 2011 Acta Phys. Sin. 60 110202 (in Chinese) [杨卓琴, 管亭亭, 甘春标, 张矫瑛 2011 物理学报 60 110202]
[18] Longtin A 1993 J. Stat. Phys. 70 309
[19] Braun H A, Wissing H, Schfer K, Hirsch M C 1994 Nature 367 270
[20] Wiesenfeld K, Moss F 1995 Nature 373 33
[21] Yu Y, Wang W, Wang J, Liu F 2001 Phys. Rev. E 63 021907
[22] Liu F, Wang J, Wang W 1999 Phys. Rev. E 59 3453
[23] Ermentrout B 1996 Neural Comput. 8 979
[24] Gutkin B S, Ermentrout G B, Reyes A D 2005 J. Neurophysiol. 94 1623
[25] Hastings J W, Sweeney B M 1958 Biol. Bull. 115 440
[26] Johnson C H 1999 Chronobiol. Int. 16 711
[27] Ikeda N 1982 Biol. Cybern. 43 157
[28] Tsalikakis D G, Zhang H G, Fotiadis D I, Kremmydas G P, Michalis L K 2007 Comput. Biol. Med. 37 8
[29] Ermentrout G B, Kopell N 1991 J. Math. Biol. 29 195
[30] Ermentrout G B 1992 SIAM J. Appl. Math. 52 1665
[31] Stiger T, Danzl P, Moehlis J, Netoff T I 2010 J. Med. Devices 4 027533
[32] Shi X, Zhang J D 2016 Chin. Phys. B 25 060502
[33] Schultheiss N W, Prinz A A, Butera R J 2012 Phase Response Curves in Neuroscience (New York: Springer) p3
[34] Ermentrout G B 2002 Simulating, Analyzing, and Animating Dynamical Systems: a Guide to XPPAUT for Researchers and Students (Philadelphia: SIAM) p226
[35] Govaerts W, Sautois B 2006 Neural Comput. 18 817
[36] Sherwood W E, Guckenheimer J 2010 SIAM J. Appl. Dyn. Syst. 9 659
[37] Novicenko V, Pyragas K 2011 Nonlinear Dynam. 67 517
[38] Ermentrout G B, Terman D H 2010 Mathematical Foundations of Neuroscience (New York: Springer) p51
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