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

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

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.
Citation: Hinke M. Osinga, Arthur Sherman, Krasimira Tsaneva-Atanasova. Cross-currents between biology and mathematics: The codimension of pseudo-plateau bursting. Discrete & Continuous Dynamical Systems - A, 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. 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.

[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. 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. 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. 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.

[7]

M. Golubitsky, K. Josić and T. J. Kaper, An unfolding theory approach to bursting in fast-slow systems,, in, (2001), 277.

[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.

[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. 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. 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.

[12]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'', Applied Mathematical Sciences, 126 (1997).

[13]

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

[14]

J. Keener and J. Sneyd, "Mathematical Physiology,", 2nd edition, 8 (2009).

[15]

A. I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations,, Nonlinearity, 11 (1998), 1505. 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. 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. doi: 10.1137/S003613999223449X.

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model,, in, 1151 (1985), 304.

[19]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems,, in, (1987), 1578.

[20]

J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations,, in, (1998), 251.

[21]

J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting,, J. Math. Biol., 25 (1987), 653. 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).

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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.

[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. 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. 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. 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.

[7]

M. Golubitsky, K. Josić and T. J. Kaper, An unfolding theory approach to bursting in fast-slow systems,, in, (2001), 277.

[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.

[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. 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. 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.

[12]

F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'', Applied Mathematical Sciences, 126 (1997).

[13]

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

[14]

J. Keener and J. Sneyd, "Mathematical Physiology,", 2nd edition, 8 (2009).

[15]

A. I. Khibnik, B. Krauskopf and C. Rousseau, Global study of a family of cubic Liénard equations,, Nonlinearity, 11 (1998), 1505. 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. 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. doi: 10.1137/S003613999223449X.

[18]

J. Rinzel, Bursting oscillations in an excitable membrane model,, in, 1151 (1985), 304.

[19]

J. Rinzel, A formal classification of bursting mechanisms in excitable systems,, in, (1987), 1578.

[20]

J. Rinzel and B. Ermentrout, Analysis of neural excitability and oscillations,, in, (1998), 251.

[21]

J. Rinzel and Y. S. Lee, Dissection of a model for neuronal parabolic bursting,, J. Math. Biol., 25 (1987), 653. 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).

[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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/s003329900053.

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