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Condensing operators and periodic solutions of infinite delay impulsive evolution equations
The mixed-mode oscillations in Av-Ron-Parnas-Segel model
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 |
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.
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. Google Scholar |
[2] |
E. Av-Ron, H. Parnas and L. A. Segel, A basic biophysical model for bursting neurons, Biological Cybernetics, 69 (1993), 87-95. Google Scholar |
[3] |
E. Av-Ron, Modeling a small neuronal network: the lobster cardiac ganglion, Journal of Biological Systems, 3 (1995), 1087-1090. Google Scholar |
[4] | E. Av-Ron, H. Parnas and L. A. Segel, Modeling the Bursting Interneurons of the Lobster Cardiac Ganglion, Springer-Verlag, New York, 1995. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
H. G. Rotstein, Mixed-mode oscillations in single neurons, Encyclopedia of Computational Neuroscience, 2 (2014), 1-9. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
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. Google Scholar |
[2] |
E. Av-Ron, H. Parnas and L. A. Segel, A basic biophysical model for bursting neurons, Biological Cybernetics, 69 (1993), 87-95. Google Scholar |
[3] |
E. Av-Ron, Modeling a small neuronal network: the lobster cardiac ganglion, Journal of Biological Systems, 3 (1995), 1087-1090. Google Scholar |
[4] | E. Av-Ron, H. Parnas and L. A. Segel, Modeling the Bursting Interneurons of the Lobster Cardiac Ganglion, Springer-Verlag, New York, 1995. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[21] |
H. G. Rotstein, Mixed-mode oscillations in single neurons, Encyclopedia of Computational Neuroscience, 2 (2014), 1-9. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |













Parameter | Value | Parameter | Value |
0 | |||
-50 mV | |||
55 mV | 124 mV | ||
-31 mV | 0.065 | ||
-46 mV | 0.055 | ||
-20 mV | 0.2 |
Parameter | Value | Parameter | Value |
0 | |||
-50 mV | |||
55 mV | 124 mV | ||
-31 mV | 0.065 | ||
-46 mV | 0.055 | ||
-20 mV | 0.2 |
1 a) | 1 b) | 1 c) | 1 d) | 1 e) | 1 f) | 1 g) | 1 h) | |
120 | 120 | 120 | 120 | 250 | 250 | 250 | 250 | |
8 | 8 | 8 | 8 | 8 | 8 | 6 | 8 | |
0.25 | 1 | 1 | 8 | 1 | 8 | 8 | 8 | |
0.5 | 0.5 | 2 | 6 | 0.6 | 0.6 | 0.6 | 0.6 | |
-72 | -72 | -72 | -72 | -72 | -60 | -72 | -72 | |
1 | 10 | 10 | 40 | 40 | 40 | 40 | 40 | |
0.08 | 0.08 | 0.08 | 2 | 2 | 2 | 2 | 2 | |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
1 a) | 1 b) | 1 c) | 1 d) | 1 e) | 1 f) | 1 g) | 1 h) | |
120 | 120 | 120 | 120 | 250 | 250 | 250 | 250 | |
8 | 8 | 8 | 8 | 8 | 8 | 6 | 8 | |
0.25 | 1 | 1 | 8 | 1 | 8 | 8 | 8 | |
0.5 | 0.5 | 2 | 6 | 0.6 | 0.6 | 0.6 | 0.6 | |
-72 | -72 | -72 | -72 | -72 | -60 | -72 | -72 | |
1 | 10 | 10 | 40 | 40 | 40 | 40 | 40 | |
0.08 | 0.08 | 0.08 | 2 | 2 | 2 | 2 | 2 | |
0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
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