August  2012, 32(8): 2879-2912. doi: 10.3934/dcds.2012.32.2879

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

Received  June 2011 Published  March 2012

Mixed mode oscillations (MMOs) are complex oscillatory waveforms that naturally occur in physiologically relevant dynamical processes. MMOs were studied in a model of electrical bursting in a pituitary lactotroph [34] where geometric singular perturbation theory and bifurcation analysis were combined to demonstrate that the MMOs arise from canard dynamics. In this work, we extend the analysis done in [34] and consider bifurcations of canard solutions under variations of key parameters. To do this, a global return map induced by the flow of the equations is constructed and a qualitative analysis given. The canard solutions act as separatrices in the return maps, organising the dynamics along the Poincaré section. We examine the bifurcations of the return maps and demonstrate that the map formulation allows for an explanation of the different MMO patterns observed in the lactotroph model.
Citation: Theodore Vo, Richard Bertram, Martin Wechselberger. Bifurcations of canard-induced mixed mode oscillations in a pituitary Lactotroph model. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2879-2912. doi: 10.3934/dcds.2012.32.2879
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.  Google Scholar

[2]

M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon,, in, 49 (2006), 39.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[10]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, Journal of Differential Equations, 31 (1979), 53.   Google Scholar

[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.  Google Scholar

[13]

J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes,, Chaos, 18 (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Haiduc, Horseshoes in the forced van der Pol system,, Nonlinearity, 22 (2009), 213.  doi: 10.1088/0951-7715/22/1/011.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, Journal of Differential Equations, 177 (2001), 419.   Google Scholar

[27]

P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbbR^3$,, Journal of Differential Equations, 200 (2004), 69.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon,, in, 49 (2006), 39.   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[10]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, Journal of Differential Equations, 31 (1979), 53.   Google Scholar

[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.  Google Scholar

[13]

J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes,, Chaos, 18 (2008).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

R. Haiduc, Horseshoes in the forced van der Pol system,, Nonlinearity, 22 (2009), 213.  doi: 10.1088/0951-7715/22/1/011.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. Szmolyan and M. Wechselberger, Canards in $\mathbbR^3$,, Journal of Differential Equations, 177 (2001), 419.   Google Scholar

[27]

P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbbR^3$,, Journal of Differential Equations, 200 (2004), 69.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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