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June  2017, 10(3): 487-504. doi: 10.3934/dcdss.2017024

The mixed-mode oscillations in Av-Ron-Parnas-Segel model

Bo Lu 1,, , Shenquan Liu 1,, , Xiaofang Jiang 1, , Jing Wang 1, and  Xiaohui Wang 2,

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China
2. 

School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author

Received  January 2016 Revised  December 2016 Published  February 2017

Fund Project: The first author is supported by NSF of China under 11172103 and 11572127.

Mixed-mode oscillations (MMOs) as complex firing patterns with both relaxation oscillations and sub-threshold oscillations have been found in many neural models such as the stellate neuron model, HH model, and so on. Based on the work, we discuss mixed-mode oscillations in the Av-Ron-Parnas-Segel model which can govern the behavior of the neuron in the lobster cardiac ganglion. By using the geometric singular perturbation theory we first explain why the MMOs exist in the reduced Av-Ron-Parnas-Segel model. Then the mixed-mode oscillatory phenomenon and aperiodic mixed-mode behaviors in the model have been analyzed numerically. Finally, we illustrate the influence of certain parameters on the model.

Keywords: Geometric singular perturbation theory, mixed-mode oscillations, InterSpike Intervals, Av-Ron-Parnas-Segel model, lobster cardiac ganglion.
Mathematics Subject Classification: Primary: 37N25, 37N30; Secondary: 92C20.
Citation: Bo Lu, Shenquan Liu, Xiaofang Jiang, Jing Wang, Xiaohui Wang. The mixed-mode oscillations in Av-Ron-Parnas-Segel model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 487-504. doi: 10.3934/dcdss.2017024
References:
[1]

E. Av-Ron, H. Parnas and L. A. Segel, A minimal biophysical model for an excitable and oscillatory neuron, Biological Cybernetics, 65 (1991), 487-500. 

[2]

E. Av-Ron, H. Parnas and L. A. Segel, A basic biophysical model for bursting neurons, Biological Cybernetics, 69 (1993), 87-95. 

[3]

E. Av-Ron, Modeling a small neuronal network: the lobster cardiac ganglion, Journal of Biological Systems, 3 (1995), 1087-1090. 

[4] E. Av-Ron, H. Parnas and L. A. Segel, Modeling the Bursting Interneurons of the Lobster Cardiac Ganglion, Springer-Verlag, New York, 1995. 
[5]

A. Berlind, Monoamine pharmacology of the lobster cardiac ganglion, Comparative Biochemistry and Physiology Part C Toxicology and Pharmacology, 128 (2001), 377-390.  doi: 10.1016/S1532-0456(00)00210-6.

[6]

M. Brøns and M. Krupa, Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63. 

[7]

M. Brøns, T. J. Kaper and H. G. Rotstein, Mixed mode oscillations due to the generalized canard phenomenon, Journal of Nonlinear Science, 18 (2008), 015101. 

[8]

T. H. Bullock and C. A. Terzuolo, Diverse forms of activity in the somata of spontaneous and integrating ganglion cells, Journal of Physiology, 138 (1957), 341-364. 

[9]

M. Desroches, J. Guckenheimer and B. Krauskopf, Mixed-mode oscillations with multiple time scales, SIAM Reviews, 54 (2012), 211-288.  doi: 10.1137/100791233.

[10]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[11]

J. Grasman, Relaxation oscillations, Pflügers Archiv European Journal of Physiology, 463 (2009), 561-569. 

[12]

K. R. Graziani, J. L. Hudson and R. A. Schmitz, The Belousov-Zhabotinskii reaction in a continuous flow reactor, The Chemical Engineering Journal, 12 (1976), 9-21.  doi: 10.1016/0300-9467(76)80013-5.

[13]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117) (1952), 500-544. 

[14] C. K. R. T. Jones, Geometric Singular Perturbation Theory in Dynamical Systems, Springer-Verlag, Berlin/New York, 1995.  doi: 10.1007/BFb0095239.
[15]

B. Lu, S.Q. Liu and X. L. Liu, Bifurcation and spike adding transition in Chay-Keizer model, International Journal of Bifurcation and Chaos, 26 (2016), 1650090, 13pp.  doi: 10.1142/S0218127416500905.

[16]

A. Milik, P. Szmolyan and H. Löffelmann, Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator, International Journal of Bifurcation and Chaos, 8 (1997), 505-519.  doi: 10.1142/S0218127498000322.

[17]

T. Otani and T. H. Bullock, Effects of presetting the membrane potential of the soma of spontaneous and integrating ganglion cells, Physiological Zoology, 32 (1959), 104-114.  doi: 10.1086/physzool.32.2.30155393.

[18]

V. Petrov, S. Scott and K. Showalter, Mixed-mode oscillations in chemical systems, Journal of Chemical Physics, 97 (1992), 6191-6198.  doi: 10.1063/1.463727.

[19]

R. E. Plant, The effects of calcium2+ on bursting neurons. A modeling study, Biophysical Journal, 21 (1978), 217-237.  doi: 10.1016/S0006-3495(78)85521-0.

[20]

J. Rinzel, Excitation dynamics: insights from simplified membrane models, Federation Proceedings, 44 (1985), 2944-2946. 

[21]

H. G. Rotstein, Mixed-mode oscillations in single neurons, Encyclopedia of Computational Neuroscience, 2 (2014), 1-9. 

[22]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), 5-32.  doi: 10.1007/s00422-007-0153-5.

[23]

J. Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales, Chaos, 18 (2008), 015105, 12pp..  doi: 10.1063/1.2789564.

[24]

K. Shimizu, Y. Saito and M. Sekikawa, Complex mixed-mode oscillations in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation, Physica D: Nonlinear Phenomena, 241 (2012), 1518-1526.  doi: 10.1016/j.physd.2012.05.014.

[25]

H. Susumu and B. T. Holmes, Intracellular potentials in pacemaker and integrative neurons of the lobster cardiac ganglion, Journal of Cellular Physiology, 50 (1957), 25-47. 

[26]

P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, Journal of Differential Equations, 200 (2004), 69-104.  doi: 10.1016/j.jde.2003.09.010.

[27]

T. Vo, R. Bertram and J. Tabak, Mixed-mode oscillations as a mechanism for pseudo-plateau bursting, Journal of Computational Neuroscience, 28 (2010), 443-458.  doi: 10.1007/s10827-010-0226-7.

[28]

M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal on Applied Dynamical Systems, 4 (2005), 101-139.  doi: 10.1137/030601995.

[29]

M. Wechselberger, Á propos de canards (Apropos canards), Transactions of the American Mathematical Society, 364 (2012), 3289-3309.  doi: 10.1090/S0002-9947-2012-05575-9.

[30]

M. Wechselberger, J. Mitry and J. Rinzel, Canard theory and excitability, Lecture Notes in Mathematics, 2101 (2013), 89-132.  doi: 10.1007/978-3-319-03080-7_3.

[31]

H. L. Wu and S. Q. Liu, Dynamical analysis of lobster model for cardiac ganglion, Journal of Dynamics and Control, 10 (2012), 168-170. 

show all references

References:
[1]

E. Av-Ron, H. Parnas and L. A. Segel, A minimal biophysical model for an excitable and oscillatory neuron, Biological Cybernetics, 65 (1991), 487-500. 

[2]

E. Av-Ron, H. Parnas and L. A. Segel, A basic biophysical model for bursting neurons, Biological Cybernetics, 69 (1993), 87-95. 

[3]

E. Av-Ron, Modeling a small neuronal network: the lobster cardiac ganglion, Journal of Biological Systems, 3 (1995), 1087-1090. 

[4] E. Av-Ron, H. Parnas and L. A. Segel, Modeling the Bursting Interneurons of the Lobster Cardiac Ganglion, Springer-Verlag, New York, 1995. 
[5]

A. Berlind, Monoamine pharmacology of the lobster cardiac ganglion, Comparative Biochemistry and Physiology Part C Toxicology and Pharmacology, 128 (2001), 377-390.  doi: 10.1016/S1532-0456(00)00210-6.

[6]

M. Brøns and M. Krupa, Mixed mode oscillations due to the generalized canard phenomenon, Fields Institute Communications, 49 (2006), 39-63. 

[7]

M. Brøns, T. J. Kaper and H. G. Rotstein, Mixed mode oscillations due to the generalized canard phenomenon, Journal of Nonlinear Science, 18 (2008), 015101. 

[8]

T. H. Bullock and C. A. Terzuolo, Diverse forms of activity in the somata of spontaneous and integrating ganglion cells, Journal of Physiology, 138 (1957), 341-364. 

[9]

M. Desroches, J. Guckenheimer and B. Krauskopf, Mixed-mode oscillations with multiple time scales, SIAM Reviews, 54 (2012), 211-288.  doi: 10.1137/100791233.

[10]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[11]

J. Grasman, Relaxation oscillations, Pflügers Archiv European Journal of Physiology, 463 (2009), 561-569. 

[12]

K. R. Graziani, J. L. Hudson and R. A. Schmitz, The Belousov-Zhabotinskii reaction in a continuous flow reactor, The Chemical Engineering Journal, 12 (1976), 9-21.  doi: 10.1016/0300-9467(76)80013-5.

[13]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Journal of Physiology, 117) (1952), 500-544. 

[14] C. K. R. T. Jones, Geometric Singular Perturbation Theory in Dynamical Systems, Springer-Verlag, Berlin/New York, 1995.  doi: 10.1007/BFb0095239.
[15]

B. Lu, S.Q. Liu and X. L. Liu, Bifurcation and spike adding transition in Chay-Keizer model, International Journal of Bifurcation and Chaos, 26 (2016), 1650090, 13pp.  doi: 10.1142/S0218127416500905.

[16]

A. Milik, P. Szmolyan and H. Löffelmann, Geometry of Mixed-Mode Oscillations in the 3-D Autocatalator, International Journal of Bifurcation and Chaos, 8 (1997), 505-519.  doi: 10.1142/S0218127498000322.

[17]

T. Otani and T. H. Bullock, Effects of presetting the membrane potential of the soma of spontaneous and integrating ganglion cells, Physiological Zoology, 32 (1959), 104-114.  doi: 10.1086/physzool.32.2.30155393.

[18]

V. Petrov, S. Scott and K. Showalter, Mixed-mode oscillations in chemical systems, Journal of Chemical Physics, 97 (1992), 6191-6198.  doi: 10.1063/1.463727.

[19]

R. E. Plant, The effects of calcium2+ on bursting neurons. A modeling study, Biophysical Journal, 21 (1978), 217-237.  doi: 10.1016/S0006-3495(78)85521-0.

[20]

J. Rinzel, Excitation dynamics: insights from simplified membrane models, Federation Proceedings, 44 (1985), 2944-2946. 

[21]

H. G. Rotstein, Mixed-mode oscillations in single neurons, Encyclopedia of Computational Neuroscience, 2 (2014), 1-9. 

[22]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), 5-32.  doi: 10.1007/s00422-007-0153-5.

[23]

J. Rubin and M. Wechselberger, The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales, Chaos, 18 (2008), 015105, 12pp..  doi: 10.1063/1.2789564.

[24]

K. Shimizu, Y. Saito and M. Sekikawa, Complex mixed-mode oscillations in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation, Physica D: Nonlinear Phenomena, 241 (2012), 1518-1526.  doi: 10.1016/j.physd.2012.05.014.

[25]

H. Susumu and B. T. Holmes, Intracellular potentials in pacemaker and integrative neurons of the lobster cardiac ganglion, Journal of Cellular Physiology, 50 (1957), 25-47. 

[26]

P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, Journal of Differential Equations, 200 (2004), 69-104.  doi: 10.1016/j.jde.2003.09.010.

[27]

T. Vo, R. Bertram and J. Tabak, Mixed-mode oscillations as a mechanism for pseudo-plateau bursting, Journal of Computational Neuroscience, 28 (2010), 443-458.  doi: 10.1007/s10827-010-0226-7.

[28]

M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal on Applied Dynamical Systems, 4 (2005), 101-139.  doi: 10.1137/030601995.

[29]

M. Wechselberger, Á propos de canards (Apropos canards), Transactions of the American Mathematical Society, 364 (2012), 3289-3309.  doi: 10.1090/S0002-9947-2012-05575-9.

[30]

M. Wechselberger, J. Mitry and J. Rinzel, Canard theory and excitability, Lecture Notes in Mathematics, 2101 (2013), 89-132.  doi: 10.1007/978-3-319-03080-7_3.

[31]

H. L. Wu and S. Q. Liu, Dynamical analysis of lobster model for cardiac ganglion, Journal of Dynamics and Control, 10 (2012), 168-170. 

Figure 1.  The time courses of the membrane potential at different parameters.
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Figure 2.  The critical manifold of the reduced ARPS model (5)-(7). a) The cubic-shaped surface critical manifold and trajectory of system (5)-(7) at $\tau_x=2.6$. b) Folded curves on the critical manifold.
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Figure 3.  Folded curves of the critical manifold projection. a) Folded curves projection on the $(x,~v)$ plane. b) Magnification of projection for $-0.002<x<0.005$.
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Figure 4.  Bifurcation diagram of ISIs with $\tau_x$. a) $1^1$ MMO pattern at $\tau_x=2.55$; b) $1^2$ MMO pattern at $\tau_x=2.65$; c)$1^3$ MMO pattern at $\tau_x=2.7$; d)$1^{17}$ MMO pattern at $\tau_x=2.805$.
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Figure 5.  The graph of staircase function about the firing number with increasing of $\tau_x$ . b) The variation of the largest Lyapunov exponent (LLE) with $\tau_x$.
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Figure 6.  Exotic MMOs are obtained from the ARPS model with the different values of $\tau_x$. a)-e) A large variety of MMOs are observed at $\tau_x=2.7, 2.4, 2.07, 2.0$ and $1.2$, respectively. f) The solution at $\tau_x=2.7$ projected to the $(V,w)$-plane. The subgraph is amplified image for $0.315<w<0.355$.
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Figure 7.  Firing number diagram with $\tau_x$.
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Figure 8.  MMOs with bursts under variation of $\tau_x$.
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Figure 9.  The variation of the sub-threshold oscillations under the parameter $V_K$. a)-c) MMO pattern firing at $\tau_x = 2.7, ~V_K=-71,-73,-74$ mV, respectively. d) The period doubling bifurcation diagram of $V_K$ vs ISIs at $\tau_x = 2.7$ ms.
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Figure 10.  The period doubling bifurcation diagram of $V_K$ vs ISIs at the firing pattern 1.
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Figure 11.  The period adding bifurcation diagram of $V_K$ vs ISIs at the firing pattern 8 as shown in Fig.1 h).
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Figure 12.  The period doubling bifurcation diagram of $g_K(Ca)$ vs ISIs at the firing pattern 1. a) $0.27<g_{K(Ca)}<0.37$. b) $0.292<g_{K(Ca)}<0.305$.
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Figure 13.  MMO pattern in the ARPS model. a) $2^4$ MMO pattern firing at $\tau_x = 2.4$ and $g_{K(Ca)}=0.38$. b) $1^4$ MMO pattern firing at $\tau_x = 10, \, g_{K(Ca)}=1,\, \lambda= 0.04$ and $g_K=11$.
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Table 1.  Parameters of ARPS model
ParameterValueParameterValue
$C_m$$1 ~{\rm \mu F\cdot cm^{-2}}$$I$0 $\mu$ A
$g_L$$0.3~ {\rm mS\cdot cm^{-2}}$$V_L$-50 mV
$V_{Na}$55 mV$V_{Ca}$124 mV
$V_{\frac{1}{2}}^{(m)}$-31 mV$a^{(m)}$0.065
$V_{\frac{1}{2}}^{(w)}$-46 mV$a^{(w)}$0.055
$V_{\frac{1}{2}}^{(x)}$-20 mV$a^{(x)}$0.2
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ParameterValueParameterValue
$C_m$$1 ~{\rm \mu F\cdot cm^{-2}}$$I$0 $\mu$ A
$g_L$$0.3~ {\rm mS\cdot cm^{-2}}$$V_L$-50 mV
$V_{Na}$55 mV$V_{Ca}$124 mV
$V_{\frac{1}{2}}^{(m)}$-31 mV$a^{(m)}$0.065
$V_{\frac{1}{2}}^{(w)}$-46 mV$a^{(w)}$0.055
$V_{\frac{1}{2}}^{(x)}$-20 mV$a^{(x)}$0.2
Table 2.  Parameter values of different firing patterns in the ARPS model
1 a)1 b)1 c)1 d)1 e)1 f)1 g)1 h)
$g_{Na}$120120120120250250250250
$g_{K}$88888868
$g_{K(Ca)}$0.251181888
$g_{Ca}$0.50.5260.60.60.60.6
$V_{K}$-72-72-72-72-72-60-72-72
$\tau_{x}$110104040404040
$\lambda$0.080.080.08222 2 2
$I$000000400
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1 a)1 b)1 c)1 d)1 e)1 f)1 g)1 h)
$g_{Na}$120120120120250250250250
$g_{K}$88888868
$g_{K(Ca)}$0.251181888
$g_{Ca}$0.50.5260.60.60.60.6
$V_{K}$-72-72-72-72-72-60-72-72
$\tau_{x}$110104040404040
$\lambda$0.080.080.08222 2 2
$I$000000400
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