# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 570-608. doi: 10.1137/S1111111102419130. [2] M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, in "Bifurcation Theory and Spatio-Temporal Pattern Formation," Fields Institute Communications, 49, Amer. Math. Soc., Providence, RI, (2006), 39-63. [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), 1131-1162. 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), 015107, 8 pp. [5] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765. 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, to appear. [7] E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284. [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: http://cmvl.cs.concordia.ca/. [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), 015115, 14 pp. [10] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. [11] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Springer, 1983. [12] J. Guckenheimer, M. Wechselberger and L.-S. Young, Chaotic attractors of relaxation oscillations, Nonlinearity, 19 (2006), 701-720. doi: 10.1088/0951-7715/19/3/009. [13] J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes, Chaos, 18 (2008), 015108, 9 pp. [14] J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355-1377. 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), 92-128. doi: 10.1137/100801950. [16] R. Haiduc, Horseshoes in the forced van der Pol system, Nonlinearity, 22 (2009), 213-237. doi: 10.1088/0951-7715/22/1/011. [17] C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems'' (ed. R. Johnson), Lecture Notes in Mathematics, Springer, New York, (1995), 44-120. [18] M. Krupa and M. Wechselberger, Local analysis near a folded saddle-node singularity, Journal of Differential Equations, 248 (2010), 2841-2888. [19] C. Kuehn, On decomposing mixed-mode oscillations and their return maps, Chaos, 21 (2011) 033107, 15 pp. [20] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Springer-Verlag, New York, 2004. [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), 319-339. 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), 38-43. 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), 505-519. 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), 1301-1314. 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), 152-159. doi: 10.1016/j.tem.2005.03.003. [26] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453. [27] P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, Journal of Differential Equations, 200 (2004), 69-104. [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), 211-222. 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), 1292-1311. 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), 1418-1450. 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), 436-451. 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), 131-144. 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), 1133-1146. 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), 443-458. doi: 10.1007/s10827-010-0226-7. [35] M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal of Applied Dynamical Systems, 4 (2005), 101-139. doi: 10.1137/030601995. [36] M. Wechselberger and W. Weckesser, Bifurcations of mixed-mode oscillations in a stellate cell model, Physica D, 238 (2009), 1598-1614. 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), 829-850. doi: 10.3934/dcdss.2009.2.829. [38] M. Wechselberger, À propos de canards (Apropos canards), Transactions of the American Mathematical Society, 364 (2012), 3289-3309. 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), 2852-2870. 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), 570-608. doi: 10.1137/S1111111102419130. [2] M. Brøns, M. Krupa and M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon, in "Bifurcation Theory and Spatio-Temporal Pattern Formation," Fields Institute Communications, 49, Amer. Math. Soc., Providence, RI, (2006), 39-63. [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), 1131-1162. 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), 015107, 8 pp. [5] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765. 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, to appear. [7] E. J. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), 265-284. [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: http://cmvl.cs.concordia.ca/. [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), 015115, 14 pp. [10] N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. [11] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,'' Springer, 1983. [12] J. Guckenheimer, M. Wechselberger and L.-S. Young, Chaotic attractors of relaxation oscillations, Nonlinearity, 19 (2006), 701-720. doi: 10.1088/0951-7715/19/3/009. [13] J. Guckenheimer, Return maps of folded nodes and folded saddle-nodes, Chaos, 18 (2008), 015108, 9 pp. [14] J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables, SIAM Journal of Applied Dynamical Systems, 7 (2008), 1355-1377. 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), 92-128. doi: 10.1137/100801950. [16] R. Haiduc, Horseshoes in the forced van der Pol system, Nonlinearity, 22 (2009), 213-237. doi: 10.1088/0951-7715/22/1/011. [17] C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems'' (ed. R. Johnson), Lecture Notes in Mathematics, Springer, New York, (1995), 44-120. [18] M. Krupa and M. Wechselberger, Local analysis near a folded saddle-node singularity, Journal of Differential Equations, 248 (2010), 2841-2888. [19] C. Kuehn, On decomposing mixed-mode oscillations and their return maps, Chaos, 21 (2011) 033107, 15 pp. [20] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," 3rd edition, Springer-Verlag, New York, 2004. [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), 319-339. 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), 38-43. 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), 505-519. 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), 1301-1314. 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), 152-159. doi: 10.1016/j.tem.2005.03.003. [26] P. Szmolyan and M. Wechselberger, Canards in $\mathbb{R}^3$, Journal of Differential Equations, 177 (2001), 419-453. [27] P. Szmolyan and M. Wechselberger, Relaxation oscillations in $\mathbb{R}^3$, Journal of Differential Equations, 200 (2004), 69-104. [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), 211-222. 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), 1292-1311. 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), 1418-1450. 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), 436-451. 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), 131-144. 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), 1133-1146. 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), 443-458. doi: 10.1007/s10827-010-0226-7. [35] M. Wechselberger, Existence and bifurcation of canards in $\mathbb{R}^3$ in the case of a folded node, SIAM Journal of Applied Dynamical Systems, 4 (2005), 101-139. doi: 10.1137/030601995. [36] M. Wechselberger and W. Weckesser, Bifurcations of mixed-mode oscillations in a stellate cell model, Physica D, 238 (2009), 1598-1614. 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), 829-850. doi: 10.3934/dcdss.2009.2.829. [38] M. Wechselberger, À propos de canards (Apropos canards), Transactions of the American Mathematical Society, 364 (2012), 3289-3309. 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), 2852-2870. doi: 10.1016/S0006-3495(03)70014-9.
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