Workshop on Mathematical Physiology
June 14 - 25, 1999
University of British Columbia
MODELS (XPP and XTC input files):
Mark Pernarowski's polynomial model: pernarowski.ode
R.E. Plant's model (type II): plant81.ode
John Rinzel's modified FitzHugh-Nagumo model (type III): fhn-rinzel.ode
Arthur Sherman's generic burster (Type I): bmb.ode
Arthur Sherman's models for a 1D string of coupled bursters: manyburst.ode (coupled ODE model, for use with XPP) and qbc.xtc (PDE model, for use with XTC)
Introduction to Models of Bursting Electrical Activity in Pancreatic Beta Cells
I. Atwater, C.M. Dawson, A. Scott, G. Eddlestone, and E. Rojas, The nature of oscillatory behaviour in electrical activity from pancreatic beta-cell, in Biochemistry and Biophysics of the Pancreatic Beta-cell, Hormone and Metabolic Research Supplement Series 10, W.J. Malaisse and I.B. Taljedal, eds., Georg Thiem Verlag, Stuttgart, 1980, pp. 100-107. First descriptive model.
T.R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophys. J., 42 (1983), pp. 181-190. First mathematical model.
T.R. Chay and J. Keizer, Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell, Biophys. J., 48 (1985), pp. 815-827. Simplification of the first mathematical model.
J. Rinzel, Bursting oscillations in an excitable membrane model,
in Ordinary and Partial Differential Equations, Lecture Notes in Mathematics
1151, B.D. Sleeman and R.J. Jarvis, eds., Springer, New York, 1985, pp.
304-316. Fast-slow analysis.
Classification of Bursters
R. Bertram, M.J. Butte, T. Kiemel, and A. Sherman, Topological and phenomenological classification of bursting oscillations, Bull. Math. Biol., 57 (1995), pp. 413-439. Classification via two-parameter bifurcation diagrams.
D.L. Cook, D. Porte, and W.E. Crill, Voltage dependence of rhythmic plateau potentials of pancreatic islet cells, Am. J. Physiol., 240 (1981), E290-E296. Resetting experiments.
G. de Vries, Multiple bifurcations in a polynomial model of bursting electrical activity, J. Nonlinear Sci., 8 (1998), pp. 281-316. Bifurcation map, showing relationship between different types of bursters.
E. Izhikevich, Neural excitability, spiking, and bursting, preprint. Mammoth classification (120 types of bursters).
M. Pernarowski, Fast subsystem bifurcations in a slowly varying Lienard system exhibiting bursting, SIAM J. Appl. Math., 54 (1994), pp. 814-832. Polynomial model for bursting.
J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biomathematics 71, E. Teramoto and M. Yamaguti, eds., Springer-Verlag, New York, 1987, pp. 267-281. Type I, II, III.
J. Rinzel and Y.S. Lee, On different mechanisms for membrane potential bursting, in Nonlinear Oscillations in Biology and Chemistry, Lecture Notes in Biomathematics 66, H.G. Othmer, ed., Springer-Verlag, New York, 1986, pp. 19-33. Square-wave burster (type I) compared to parabolic burster (type II).
J. Rinzel and Y.S. Lee, Dissection of a model for neuronal parabolic
bursting, J. Math. Biology, 25 (1987), pp. 653-675. Fast-slow
analysis of a parabolic burster.
Coupling of Beta-Cells: Effects of Noise and Heterogeneity
G. de Vries, H.-R. Zhu, and A. Sherman, Diffusively coupled bursters: Effects of cell heterogeneity, Bull. Math. Biol., 60 (1998), pp. 1167-1199. Normal form analysis of a coupled pair of (non)identical bursters.
G. de Vries and A. Sherman, Channel sharing in pancreatic beta-cells revisited: Enhancement of emergent bursting by noise, J. Theor. Biol., 207 (2000), pp. 513-530. Investigation of bursting as an emergent phenomenon.
G. de Vries and A. Sherman, From spikers to bursters via coupling: Help from heterogeneity, Bull. Math. Biol., 63 (2001), pp. 371-391. Continuation of the study of bursting as an emergent phenomenon.
A. Sherman, Anti-phase, asymmetric and aperiodic oscillationsin excitable cells - I. Coupled bursters, Bull. Math. Biol., 56 (1994), pp. 811-835. Fast-slow analysis of a coupled pair of identical bursters.
A. Sherman and J. Rinzel, Model for synchronization of pancreatic beta-cells by gap junctions, Biophys. J., 59 (1991), pp. 547-559. Channel-sharing hypothesis - using cluster model, with finite coupling strength.
A. Sherman and J. Rinzel, Rhythmogenic effects of weak electrotonic coupling in neuronal models, Proc. Natl. Acad. Sci. USA, 89 (1992), pp. 2471-2474. Numerical study of a coupled pair of bursters.
A. Sherman, J. Rinzel and J. Keizer, Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing, Biophys. J., 54 (1988), pp. 411-425. Channel-sharing hypothesis - using multicell model, with infinite coupling strength.
P. Smolen, J. Rinzel and A. Sherman, Why pancreatic islets burst
but single beta-cells do not: the heterogeneity hypothesis, Biophys.
J., 64 (1993), pp. 1668-1680. Heterogeneity
hypothesis - using multicell model.
A. Sherman, Theoretical aspects of synchronized bursting in beta-cells, in Pacemaker Activity and Intercellular Communication, J.D. Huizinga, ed., CRC Press, Boca Raton, FL, 1995, pp. 323-337.
A. Sherman, Contributions of modeling to understanding stimulus-secretion
coupling in pancreatic beta-cells, Am. J. Phys., 271 (1996), pp. E362-E372.