ANUP UMRANIKAR

A Phantom Bursting Mechanism for Episodic Bursting

Richard Bertram, Joseph Rhoads, Wendy Cimbora

Introduction

Bursting = Periods of electrical spiking followed by periods of rest

Bursting is observed in cells such as R15 neuron of aplysia Thalamic neurons Pyramidal neurons Trigeminal neurons Pancreatic beta-cells Pituitary gonadotrophs

Episodic (or Compound) Bursting

Complex form of bursting observed in beta-cells of islets of Langerhans in pancreas and GnRH of pituitary gland

Episodes of several bursts followed by long silent phases or ‘deserts’

Paper discusses episodic bursting using a minimal model

Depending on location in parameter space, model produces fast, slow and episodic bursting

Mathematical Model

Two slow variables interact with the fast subsystem

Planar fast subsystem given by

Expressions for ionic current are given by

Parameter Values

In paper, all simulations and bifurcations were calculated using XPPAUT software package; CVODE numerical method used to solve differential equations

I’ve used MATLAB for simulations; used ode15s to solve differential equations

Fast Bursting

Fast Bursting – My Results

Fast Bursting – Bifurcation Diagram

Fast/slow analysis of fast bursting (s2 = 0.49). The solid portion of the z-curve represents branches of stable steady states. Dashed curves represent unstable steady states. The two branches of filled circles represent the maximum and minimum values of periodic solutions. The green dot-dashed curve is the s1 nullcline. HB=supercritical Hopf bifurcation, HM=homoclinic bifurcation, LK=lower knee, UK=upper knee.

Slow Bursting

Slow Bursting – My Results

Episodic Bursting

Episodic Bursting – My Results

Conclusion

Model described in minimal, with two fast and two slow variables

Slow variables represent activation variables of hyperpolarizing K+ currents. However, similar behavior could be achieved by defining slow variables in other ways, such as inactivation variables of depolarizing currents or as a combination of activation and deactivation

Behaviors not restricted to specific details of this modelAlso, more complex neurons or endocrine cell models can

be achieved using this minimal model, as long as model possesses at least two slow variables with disparate time scales

References

Bertram, R., Rhoads, J., Cimbora, W., 2008. A phantom bursting mechanism for episodic bursting. Bull. Math. Biol. 70, 1979-1993

Bertram, R., Previte, J., Sherman, A., Kinard, T.A., Satin, L.S., 2000. The phantom burster model for pancreatic β-cells. Biophys. J. 79, 2880–2892