American Institute of Mathematical Sciences

Advanced
August  2012, 32(8): 2853-2877. doi: 10.3934/dcds.2012.32.2853

Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting

Hinke M. Osinga 1, , Arthur Sherman 2, and  Krasimira Tsaneva-Atanasova 3,

1. 

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142
2. 

Laboratory of Biological Modeling, N.I.D.D.K. National Institutes of Health, 12A SOUTH DR MSC 5621, Bethesda, MD 20892-5621, United States
3. 

Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom

Received  May 2011 Revised  August 2011 Published  March 2012

A great deal of work has gone into classifying bursting oscillations, periodic alternations of spiking and quiescence modeled by fast-slow systems. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A rigorous classification approach is to characterize the bifurcations found in the neighborhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. Fold/homoclinic bursting, along with most other burst types of interest, has been shown to occur near a singularity of codimension three by examining bifurcations of a cubic Liénard system; hence, these types of bursting have at most codimension three. Modeling and biological considerations suggest that fold/homoclinic bursting should be found near fold/subHopf bursting, a more recently identified burst type whose codimension has not been determined yet. One would expect that fold/subHopf bursting has the same codimension as fold/homoclinic bursting, because models of these two burst types have very similar underlying bifurcation diagrams. However, no codimension-three singularity is known that supports fold/subHopf bursting, which indicates that it may have codimension four. We identify a three-dimensional slice in a partial unfolding of a doubly-degenerate Bodganov-Takens point, and show that this codimension-four singularity gives rise to almost all known types of bursting.
Keywords: bursting, Fast-slow systems, ordinary differential equations., bifurcation theory.
Mathematics Subject Classification: Primary: 70K70, 70K45; Secondary: 34C2.
Citation: Hinke M. Osinga, Arthur Sherman, Krasimira Tsaneva-Atanasova. Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2853-2877. doi: 10.3934/dcds.2012.32.2853
References:
[1]

W. B. Adams and J. A. Benson, The generation and modulation of endogenous rhythmicity in the Aplysia bursting pacemaker neurone R15, Prog. Biophys. Molec. Biol., 46 (1985), 1-49. doi: 10.1016/0079-6107(85)90011-2.

[2]

R. Bertram, M. J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439.

[3]

J. Best, A. Borisyuk, J. Rubin, D. Terman and M. Wechselberger, The dynamic range of bursting in a model respiratory pacemaker network, SIAM J. Appl. Dyn. Syst., 4 (2005), 1107-1139. doi: 10.1137/050625540.

[4]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic $\beta$ cell, Biophys. J., 42 (1983), 181-190. doi: 10.1016/S0006-3495(83)84384-7.

[5]

L. Duan, Q. Lu and Q. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomputing, 72 (2008), 341-351. doi: 10.1016/j.neucom.2008.01.019.

[6]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Springer Lect. Notes Math., 1480 (1991), 1489-1500.

[7]

M. Golubitsky, K. Josić and T. J. Kaper, An unfolding theory approach to bursting in fast-slow systems, in "Global Analysis of Dynamical Systems'' (eds. H. W. Broer, B. Krauskopf and G. Vegter), Institute of Physics Publishing, Bristol, (2001), 277-308.

[8]

F. van Goor, Y.-X. Li and S. S. Stojilkovic, Paradoxical role of large-conductance calcium-activated K$^+$ (BK) channels in controlling action potential-driven $Ca^{2+}$ entry in anterior pituitary cells, J. Neurosci., 16 (2001), 5902-5915.

[9]

F. van Goor, D. Zivadinovic, A. Martinez-Fuentes and S. Stojilkovic, Dependence of pituitary hormone secretion on the pattern of spontaneous voltage-gated calcium influx. Cell type-specific action potential secretion coupling, J. Biol. Chem., 276 (2001), 33840-33846. doi: 10.1074/jbc.M105386200.

[10]

J. Hindmarsh and M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[11]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952), 205-249.

[12]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997.

[13]

E. M. Izhikevich, Neural excitability, spiking and bursting, Intl. J. Bifurc. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[14]

J. Keener and J. Sneyd, "Mathematical Physiology," 2nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2009.

[15]

A. I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity, 11 (1998), 1505-1519. doi: 10.1088/0951-7715/11/6/005.

[16]

A. P. LeBeau, A. B. Rabson, A. E. McKinnon and J. Sneyd, Analysis of a reduced model of corticotroph action potentials, J. Theoretical Biol., 192 (1998), 319-339. doi: 10.1006/jtbi.1998.0656.

[17]

M. Pernarowski, Fast subsystem bifurcations in a slowly varying Liénard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), 814-832. doi: 10.1137/S003613999223449X.

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations'' (Dundee, 1984) (eds. B. D. Sleeman and R. D. Jarvis), Lect. Notes Math., 1151, Springer, Berlin, (1985), 304-316.

[19]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in "Proc. Intl. Cong. Math., Vol. 1, 2" (Berkeley, Calif., 1986) (ed. A. M. Gleason), American Mathematical Society, Providence, RI, (1987), 1578-1593.

[20]

J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modeling" (eds. C. Koch and I. Segev), The MIT Press, (1998), 251-291.

[21]

J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting, J. Math. Biol., 25 (1987), 653-675. doi: 10.1007/BF00275501.

[22]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E (3), 71 (2005), 056214, 9 pp.

[23]

J. V. Stern, H. M. Osinga, A. LeBeau and A. Sherman, Resetting behavior in a model of burting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus, Bull. Math. Biol., 70 (2008), 68-88. doi: 10.1007/s11538-007-9241-x.

[24]

J. Tabak, N. Toporikova, M. E. Freeman and R. Bertram, Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents, J. Comput. Neurosci., 22 (2007), 211-222. doi: 10.1007/s10827-006-0008-4.

[25]

W. Teka, K. Tsaneva-Atanasova, R. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition, Bull. Math. Biol., 73 (2011), 1292-1311. doi: 10.1007/s11538-010-9559-7.

[26]

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 Comput., 20 (2008), 436-451. doi: 10.1162/neco.2007.08-06-310.

[27]

K. Tsaneva-Atanasova, H. M. Osinga, T. Rieß and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theoretical Biol., 264 (2010), 1133-1146. doi: 10.1016/j.jtbi.2010.03.030.

[28]

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, J. Neurophysiology, 98 (2007), 131-144. doi: 10.1152/jn.00872.2006.

[29]

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

[30]

G. de Vries, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), 281-316. doi: 10.1007/s003329900053.

show all references

References:
[1]

W. B. Adams and J. A. Benson, The generation and modulation of endogenous rhythmicity in the Aplysia bursting pacemaker neurone R15, Prog. Biophys. Molec. Biol., 46 (1985), 1-49. doi: 10.1016/0079-6107(85)90011-2.

[2]

R. Bertram, M. J. Butte, T. Kiemel and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), 413-439.

[3]

J. Best, A. Borisyuk, J. Rubin, D. Terman and M. Wechselberger, The dynamic range of bursting in a model respiratory pacemaker network, SIAM J. Appl. Dyn. Syst., 4 (2005), 1107-1139. doi: 10.1137/050625540.

[4]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic $\beta$ cell, Biophys. J., 42 (1983), 181-190. doi: 10.1016/S0006-3495(83)84384-7.

[5]

L. Duan, Q. Lu and Q. Wang, Two-parameter bifurcation analysis of firing activities in the Chay neuronal model, Neurocomputing, 72 (2008), 341-351. doi: 10.1016/j.neucom.2008.01.019.

[6]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Springer Lect. Notes Math., 1480 (1991), 1489-1500.

[7]

M. Golubitsky, K. Josić and T. J. Kaper, An unfolding theory approach to bursting in fast-slow systems, in "Global Analysis of Dynamical Systems'' (eds. H. W. Broer, B. Krauskopf and G. Vegter), Institute of Physics Publishing, Bristol, (2001), 277-308.

[8]

F. van Goor, Y.-X. Li and S. S. Stojilkovic, Paradoxical role of large-conductance calcium-activated K$^+$ (BK) channels in controlling action potential-driven $Ca^{2+}$ entry in anterior pituitary cells, J. Neurosci., 16 (2001), 5902-5915.

[9]

F. van Goor, D. Zivadinovic, A. Martinez-Fuentes and S. Stojilkovic, Dependence of pituitary hormone secretion on the pattern of spontaneous voltage-gated calcium influx. Cell type-specific action potential secretion coupling, J. Biol. Chem., 276 (2001), 33840-33846. doi: 10.1074/jbc.M105386200.

[10]

J. Hindmarsh and M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. London B, 221 (1984), 87-102. doi: 10.1098/rspb.1984.0024.

[11]

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London), 117 (1952), 205-249.

[12]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997.

[13]

E. M. Izhikevich, Neural excitability, spiking and bursting, Intl. J. Bifurc. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840.

[14]

J. Keener and J. Sneyd, "Mathematical Physiology," 2nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2009.

[15]

A. I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations, Nonlinearity, 11 (1998), 1505-1519. doi: 10.1088/0951-7715/11/6/005.

[16]

A. P. LeBeau, A. B. Rabson, A. E. McKinnon and J. Sneyd, Analysis of a reduced model of corticotroph action potentials, J. Theoretical Biol., 192 (1998), 319-339. doi: 10.1006/jtbi.1998.0656.

[17]

M. Pernarowski, Fast subsystem bifurcations in a slowly varying Liénard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), 814-832. doi: 10.1137/S003613999223449X.

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model, in "Ordinary and Partial Differential Equations'' (Dundee, 1984) (eds. B. D. Sleeman and R. D. Jarvis), Lect. Notes Math., 1151, Springer, Berlin, (1985), 304-316.

[19]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in "Proc. Intl. Cong. Math., Vol. 1, 2" (Berkeley, Calif., 1986) (ed. A. M. Gleason), American Mathematical Society, Providence, RI, (1987), 1578-1593.

[20]

J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations, in "Methods in Neuronal Modeling" (eds. C. Koch and I. Segev), The MIT Press, (1998), 251-291.

[21]

J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting, J. Math. Biol., 25 (1987), 653-675. doi: 10.1007/BF00275501.

[22]

A. Shilnikov, R. L. Calabrese and G. Cymbalyuk, Mechanism of bistability: Tonic spiking and bursting in a neuron model, Phys. Rev. E (3), 71 (2005), 056214, 9 pp.

[23]

J. V. Stern, H. M. Osinga, A. LeBeau and A. Sherman, Resetting behavior in a model of burting in secretory pituitary cells: Distinguishing plateaus from pseudo-plateaus, Bull. Math. Biol., 70 (2008), 68-88. doi: 10.1007/s11538-007-9241-x.

[24]

J. Tabak, N. Toporikova, M. E. Freeman and R. Bertram, Low dose of dopamine may stimulate prolactin secretion by increasing fast potassium currents, J. Comput. Neurosci., 22 (2007), 211-222. doi: 10.1007/s10827-006-0008-4.

[25]

W. Teka, K. Tsaneva-Atanasova, R. Bertram and J. Tabak, From plateau to pseudo-plateau bursting: Making the transition, Bull. Math. Biol., 73 (2011), 1292-1311. doi: 10.1007/s11538-010-9559-7.

[26]

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 Comput., 20 (2008), 436-451. doi: 10.1162/neco.2007.08-06-310.

[27]

K. Tsaneva-Atanasova, H. M. Osinga, T. Rieß and A. Sherman, Full system bifurcation analysis of endocrine bursting models, J. Theoretical Biol., 264 (2010), 1133-1146. doi: 10.1016/j.jtbi.2010.03.030.

[28]

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, J. Neurophysiology, 98 (2007), 131-144. doi: 10.1152/jn.00872.2006.

[29]

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

[30]

G. de Vries, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), 281-316. doi: 10.1007/s003329900053.

[1]

Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[2]

Zhuoqin Yang, Tingting Guan. Bifurcation analysis of complex bursting induced by two different time-scale slow variables. Conference Publications, 2011, 2011 (Special) : 1440-1447. doi: 10.3934/proc.2011.2011.1440
[3]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281
[4]

Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[5]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
[6]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[7]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[8]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[9]

Zvi Artstein. Averaging of ordinary differential equations with slowly varying averages. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 353-365. doi: 10.3934/dcdsb.2010.14.353
[10]

Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779
[11]

Serge Nicaise. Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021057
[12]

Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365
[13]

Ilya Schurov. Duck farming on the two-torus: Multiple canard cycles in generic slow-fast systems. Conference Publications, 2011, 2011 (Special) : 1289-1298. doi: 10.3934/proc.2011.2011.1289
[14]

Anatoly Neishtadt, Carles Simó, Dmitry Treschev, Alexei Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 621-650. doi: 10.3934/dcdsb.2008.10.621
[15]

Stefano Maset. Conditioning and relative error propagation in linear autonomous ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2879-2909. doi: 10.3934/dcdsb.2018165
[16]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[17]

Aeeman Fatima, F. M. Mahomed, Chaudry Masood Khalique. Conditional symmetries of nonlinear third-order ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 655-666. doi: 10.3934/dcdss.2018040
[18]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control and Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
[19]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39
[20]

Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control and Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (171)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy