**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)

__REFERENCES:__

**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.

**Review Articles**

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.