1. ANUP UMRANIKAR A Phantom Bursting Mechanism for Episodic Bursting Richard Bertram, Joseph Rhoads, Wendy Cimbora

2. 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

3. 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

4. Mathematical Model Two slow variables interact with the fast subsystem Planar fast subsystem given by Expressions for ionic current are given by

5. 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

6. Fast Bursting

7. Fast Bursting – My Results

8. 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.

9. Slow Bursting

10. Slow Bursting – My Results

11. Episodic Bursting

12. Episodic Bursting – My Results

13. 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 model Also, 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

14. 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