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

Hinke M. Osinga; Arthur Sherman; Krasimira Tsaneva-Atanasova

doi:10.3934/dcds.2012.32.2853

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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

Abstract
Related Papers

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997. Google Scholar
[13]	E. M. Izhikevich, Neural excitability, spiking and bursting, Intl. J. Bifurc. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar
[14]	J. Keener and J. Sneyd, "Mathematical Physiology," 2^nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	G. de Vries, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), 281-316. doi: 10.1007/s003329900053. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[12]	F. C. Hoppensteadt and E. M. Izhikevich, "Weakly Connected Neural Networks,'' Applied Mathematical Sciences, 126, Springer-Verlag, New York, 1997. Google Scholar
[13]	E. M. Izhikevich, Neural excitability, spiking and bursting, Intl. J. Bifurc. Chaos Appl. Sci. Engrg., 10 (2000), 1171-1266. doi: 10.1142/S0218127400000840. Google Scholar
[14]	J. Keener and J. Sneyd, "Mathematical Physiology," 2^nd edition, Interdisciplinary Applied Mathematics, 8, Springer-Verlag, New York, 2009. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	G. de Vries, Multiple bifurcations in a polynomial model of bursting oscillations, J. Nonlinear Sci., 8 (1998), 281-316. doi: 10.1007/s003329900053. Google Scholar

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