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Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting
Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model
1. | School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia |
2. | Department of Mathematics and Programs in Neuroscience and Molecular Biophysics, Florida State University, Tallahassee, FL, United States |
References:
[1] |
K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser, The forced van der Pol equation. II. Canards in the reduced system,, SIAM Journal of Applied Dynamical Systems, 2 (2003), 570.
doi: 10.1137/S1111111102419130. |
[2] |
M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon,, in, 49 (2006), 39.
|
[3] |
M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node,, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1131.
doi: 10.1137/070708810. |
[4] |
M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled Fitzhugh-Nagumo system,, Chaos, 18 (2008).
|
[5] |
M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems,, Nonlinearity, 23 (2010), 739.
doi: 10.1088/0951-7715/23/3/017. |
[6] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales,, SIAM Review, (). Google Scholar |
[7] |
E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.
|
[8] |
E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, K. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, AUTO-07P: Continuation and bifurcation software for ordinary differential equations., Available from: \url{http://cmvl.cs.concordia.ca/}., (). Google Scholar |
[9] |
I. Erchova and D. J. McGonigle, Rhythms of the brain: An examination of mixed mode oscillation approaches to the analysis of neurophysiological data,, Chaos, 18 (2008).
|
[10] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, Journal of Differential Equations, 31 (1979), 53.
|
[11] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', Springer, (1983). Google Scholar |
[12] |
J. Guckenheimer, M. Wechselberger and L.-S. Young, Chaotic attractors of relaxation oscillations,, Nonlinearity, 19 (2006), 701.
doi: 10.1088/0951-7715/19/3/009. |
[13] |
J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes,, Chaos, 18 (2008).
|
[14] |
J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables,, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355.
doi: 10.1137/080718528. |
[15] |
J. Guckenheimer and C. Scheper, A geometric model for mixed-mode oscillations in a chemical system,, SIAM Journal of Applied Dynamical Systems, 10 (2011), 92.
doi: 10.1137/100801950. |
[16] |
R. Haiduc, Horseshoes in the forced van der Pol system,, Nonlinearity, 22 (2009), 213.
doi: 10.1088/0951-7715/22/1/011. |
[17] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44. Google Scholar |
[18] |
M. Krupa and M. Wechselberger, Local analysis near a folded saddle-node singularity,, Journal of Differential Equations, 248 (2010), 2841.
|
[19] |
C. Kuehn, On decomposing mixed-mode oscillations and their return maps,, Chaos, 21 (2011). Google Scholar |
[20] |
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 3rd edition, (2004). Google Scholar |
[21] |
A. P. LeBeau, A. B. Robson, A. E. McKinnon and J. Sneyd, Analysis of a reduced model of corticotroph action potentials,, Journal of Theoretical Biology, 192 (1998), 319.
doi: 10.1006/jtbi.1998.0656. |
[22] |
J. E. Lisman, Bursts as a unit of neural information: Making unreliable synapses reliable,, Trends in Neuroscience, 20 (1997), 38.
doi: 10.1016/S0166-2236(96)10070-9. |
[23] |
A. Milik, P. Szmolyan, H. Loeffelmann and E. Groeller, Geometry of mixed-mode oscillations in the 3-d autocatalator,, International Journal of Bifurcation and Chaos, 8 (1998), 505.
doi: 10.1142/S0218127498000322. |
[24] |
H. M. Osinga and K. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium,, Journal of Neuroendocrinology, 22 (2010), 1301.
doi: 10.1111/j.1365-2826.2010.02083.x. |
[25] |
S. S. Stojilkovic, H. Zemkova and F. Van Goor, Biophysical basis of pituitary cell type-specific $Ca^{2+}$ signaling-secretion coupling,, Trends in Endocrinology and Metabolism, 16 (2005), 152.
doi: 10.1016/j.tem.2005.03.003. |
[26] |
P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, Journal of Differential Equations, 177 (2001), 419.
|
[27] |
P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbbR^3$,, Journal of Differential Equations, 200 (2004), 69.
|
[28] |
J. Tabak, N. Toporikova, M. E. Freeman and R. Bertram, Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents,, Journal of Computational Neuroscience, 22 (2007), 211.
doi: 10.1007/s10827-006-0008-4. |
[29] |
W. Teka, K. Tsaneva-Atanasova, R. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition,, Bulletin of Mathematical Biology, 73 (2011), 1292.
doi: 10.1007/s11538-010-9559-7. |
[30] |
D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes,, SIAM Journal of Applied Mathematics, 51 (1991), 1418.
doi: 10.1137/0151071. |
[31] |
N. Toporikova, J. Tabak, M. E. Freeman and R. Bertram, A-type $K^+$ current can act as a trigger for bursting in the absence of a slow variable,, Neural Computation, 20 (2008), 436.
doi: 10.1162/neco.2007.08-06-310. |
[32] |
K. Tsaneva-Atanasova, A. Sherman, F. Van Goor and S. S. Stojilkovic, Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: Experiments and theory,, Journal of Neurophysiology, 98 (2007), 131.
doi: 10.1152/jn.00872.2006. |
[33] |
K. Tsaneva-Atanasova, H. M. Osinga, T. Rieb and A. Sherman, Full system bifurcation analysis of endocrine bursting models,, Journal of Theoretical Biology, 264 (2010), 1133.
doi: 10.1016/j.jtbi.2010.03.030. |
[34] |
T. Vo, R. Bertram, J. Tabak and M. Wechselberger, Mixed mode oscillations as a mechanism for pseudo-plateau bursting,, Journal of Computational Neuroscience, 28 (2010), 443.
doi: 10.1007/s10827-010-0226-7. |
[35] |
M. Wechselberger, Existence and bifurcation of canards in $\mathbbR^3$ in the case of a folded node,, SIAM Journal of Applied Dynamical Systems, 4 (2005), 101.
doi: 10.1137/030601995. |
[36] |
M. Wechselberger and W. Weckesser, Bifurcations of mixed-mode oscillations in a stellate cell model,, Physica D, 238 (2009), 1598.
doi: 10.1016/j.physd.2009.04.017. |
[37] |
M. Wechselberger and W. Weckesser, Homoclinic clusters and chaos associated with a folded node in a stellate cell model,, DCDS-S, 2 (2009), 829.
doi: 10.3934/dcdss.2009.2.829. |
[38] |
M. Wechselberger, À propos de canards (Apropos canards),, Transactions of the American Mathematical Society, 364 (2012), 3289.
doi: 10.1090/S0002-9947-2012-05575-9. |
[39] |
M. Zhang, P. Goforth, R. Bertram, A. Sherman and L. Satin, The $Ca^{2+}$ dynamics of isolated mouse $\beta$-cells and islets: Implications for mathematical models,, Biophysical Journal, 84 (2003), 2852.
doi: 10.1016/S0006-3495(03)70014-9. |
show all references
References:
[1] |
K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva and W. Weckesser, The forced van der Pol equation. II. Canards in the reduced system,, SIAM Journal of Applied Dynamical Systems, 2 (2003), 570.
doi: 10.1137/S1111111102419130. |
[2] |
M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon,, in, 49 (2006), 39.
|
[3] |
M. Desroches, B. Krauskopf and H. M. Osinga, The geometry of slow manifolds near a folded node,, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1131.
doi: 10.1137/070708810. |
[4] |
M. Desroches, B. Krauskopf and H. M. Osinga, Mixed-mode oscillations and slow manifolds in the self-coupled Fitzhugh-Nagumo system,, Chaos, 18 (2008).
|
[5] |
M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems,, Nonlinearity, 23 (2010), 739.
doi: 10.1088/0951-7715/23/3/017. |
[6] |
M. Desroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. M. Osinga and M. Wechselberger, Mixed-mode oscillations with multiple time scales,, SIAM Review, (). Google Scholar |
[7] |
E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems,, Congr. Numer., 30 (1981), 265.
|
[8] |
E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, K. E. Oldeman, R. C. Paffenroth, B. Sandstede, X. J. Wang and C. Zhang, AUTO-07P: Continuation and bifurcation software for ordinary differential equations., Available from: \url{http://cmvl.cs.concordia.ca/}., (). Google Scholar |
[9] |
I. Erchova and D. J. McGonigle, Rhythms of the brain: An examination of mixed mode oscillation approaches to the analysis of neurophysiological data,, Chaos, 18 (2008).
|
[10] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, Journal of Differential Equations, 31 (1979), 53.
|
[11] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'', Springer, (1983). Google Scholar |
[12] |
J. Guckenheimer, M. Wechselberger and L.-S. Young, Chaotic attractors of relaxation oscillations,, Nonlinearity, 19 (2006), 701.
doi: 10.1088/0951-7715/19/3/009. |
[13] |
J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes,, Chaos, 18 (2008).
|
[14] |
J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables,, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355.
doi: 10.1137/080718528. |
[15] |
J. Guckenheimer and C. Scheper, A geometric model for mixed-mode oscillations in a chemical system,, SIAM Journal of Applied Dynamical Systems, 10 (2011), 92.
doi: 10.1137/100801950. |
[16] |
R. Haiduc, Horseshoes in the forced van der Pol system,, Nonlinearity, 22 (2009), 213.
doi: 10.1088/0951-7715/22/1/011. |
[17] |
C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44. Google Scholar |
[18] |
M. Krupa and M. Wechselberger, Local analysis near a folded saddle-node singularity,, Journal of Differential Equations, 248 (2010), 2841.
|
[19] |
C. Kuehn, On decomposing mixed-mode oscillations and their return maps,, Chaos, 21 (2011). Google Scholar |
[20] |
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", 3rd edition, (2004). Google Scholar |
[21] |
A. P. LeBeau, A. B. Robson, A. E. McKinnon and J. Sneyd, Analysis of a reduced model of corticotroph action potentials,, Journal of Theoretical Biology, 192 (1998), 319.
doi: 10.1006/jtbi.1998.0656. |
[22] |
J. E. Lisman, Bursts as a unit of neural information: Making unreliable synapses reliable,, Trends in Neuroscience, 20 (1997), 38.
doi: 10.1016/S0166-2236(96)10070-9. |
[23] |
A. Milik, P. Szmolyan, H. Loeffelmann and E. Groeller, Geometry of mixed-mode oscillations in the 3-d autocatalator,, International Journal of Bifurcation and Chaos, 8 (1998), 505.
doi: 10.1142/S0218127498000322. |
[24] |
H. M. Osinga and K. Tsaneva-Atanasova, Dynamics of plateau bursting depending on the location of its equilibrium,, Journal of Neuroendocrinology, 22 (2010), 1301.
doi: 10.1111/j.1365-2826.2010.02083.x. |
[25] |
S. S. Stojilkovic, H. Zemkova and F. Van Goor, Biophysical basis of pituitary cell type-specific $Ca^{2+}$ signaling-secretion coupling,, Trends in Endocrinology and Metabolism, 16 (2005), 152.
doi: 10.1016/j.tem.2005.03.003. |
[26] |
P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, Journal of Differential Equations, 177 (2001), 419.
|
[27] |
P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbbR^3$,, Journal of Differential Equations, 200 (2004), 69.
|
[28] |
J. Tabak, N. Toporikova, M. E. Freeman and R. Bertram, Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents,, Journal of Computational Neuroscience, 22 (2007), 211.
doi: 10.1007/s10827-006-0008-4. |
[29] |
W. Teka, K. Tsaneva-Atanasova, R. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition,, Bulletin of Mathematical Biology, 73 (2011), 1292.
doi: 10.1007/s11538-010-9559-7. |
[30] |
D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes,, SIAM Journal of Applied Mathematics, 51 (1991), 1418.
doi: 10.1137/0151071. |
[31] |
N. Toporikova, J. Tabak, M. E. Freeman and R. Bertram, A-type $K^+$ current can act as a trigger for bursting in the absence of a slow variable,, Neural Computation, 20 (2008), 436.
doi: 10.1162/neco.2007.08-06-310. |
[32] |
K. Tsaneva-Atanasova, A. Sherman, F. Van Goor and S. S. Stojilkovic, Mechanism of spontaneous and receptor-controlled electrical activity in pituitary somatotrophs: Experiments and theory,, Journal of Neurophysiology, 98 (2007), 131.
doi: 10.1152/jn.00872.2006. |
[33] |
K. Tsaneva-Atanasova, H. M. Osinga, T. Rieb and A. Sherman, Full system bifurcation analysis of endocrine bursting models,, Journal of Theoretical Biology, 264 (2010), 1133.
doi: 10.1016/j.jtbi.2010.03.030. |
[34] |
T. Vo, R. Bertram, J. Tabak and M. Wechselberger, Mixed mode oscillations as a mechanism for pseudo-plateau bursting,, Journal of Computational Neuroscience, 28 (2010), 443.
doi: 10.1007/s10827-010-0226-7. |
[35] |
M. Wechselberger, Existence and bifurcation of canards in $\mathbbR^3$ in the case of a folded node,, SIAM Journal of Applied Dynamical Systems, 4 (2005), 101.
doi: 10.1137/030601995. |
[36] |
M. Wechselberger and W. Weckesser, Bifurcations of mixed-mode oscillations in a stellate cell model,, Physica D, 238 (2009), 1598.
doi: 10.1016/j.physd.2009.04.017. |
[37] |
M. Wechselberger and W. Weckesser, Homoclinic clusters and chaos associated with a folded node in a stellate cell model,, DCDS-S, 2 (2009), 829.
doi: 10.3934/dcdss.2009.2.829. |
[38] |
M. Wechselberger, À propos de canards (Apropos canards),, Transactions of the American Mathematical Society, 364 (2012), 3289.
doi: 10.1090/S0002-9947-2012-05575-9. |
[39] |
M. Zhang, P. Goforth, R. Bertram, A. Sherman and L. Satin, The $Ca^{2+}$ dynamics of isolated mouse $\beta$-cells and islets: Implications for mathematical models,, Biophysical Journal, 84 (2003), 2852.
doi: 10.1016/S0006-3495(03)70014-9. |
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