1. Tutorial: Converting Between Plateau and Pseudo-Plateau Bursting Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University, Tallahassee, FL. Modelling Electrical Activity in Physiological Systems, 2012

2. Coworkers and Collaborators Joël Tabak (FSU) Funding: NSF-DMS0917664 and NIH-DK043200 Wondimu Teka (FSU) Krasimira Tsaneva-Atanasova (Univ. Bristol)

3. Two Classes of Bursting Oscillations Guinea pig trigeminal motoneuron (Del Negro et al., J. Neurophysiol., 81(4): 1478, 1999) Plateau bursting S. S. Stojilkovic, Biol. Res., 39(3): 403, 2006 Pseudo-plateau bursting

4. These are Associated with Different Fast- Slow Bifurcation Structures Fast-slow analysis of plateau or square-wave bursting

5. These are Associated with Different Fast- Slow Bifurcation Structures Fast-slow analysis of pseudo-plateau or pituitary bursting

6. How Can Neuron-Like Plateau Bursting be Converted to Pituitary-Like Pseudo- Plateau Bursting? Published in Teka et al., Bull. Math. Biol., 73:1292, 2011

7. The Chay-Keizer Model This well-studied model was developed to describe plateau bursting in pancreatic β-cells, but it has also been used as a template for this type of bursting in other cells, such as neurons. T. R. Chay and J. Keizer, Biophys. J., 42:181, 1983 We use a variation of this that includes a K(ATP) current and that has lower dimensionality.

8. The Chay-Keizer Model V=voltage (mV) t= time (msec) n= fraction of open delayed rectifying K + channels I Ca = Ca 2+ current I K = delayed rectifying K + current I K(Ca) = Ca 2 + -activated K + current I K(ATP) = ATP-sensitive K + current

9. The Chay-Keizer Model: Ca 2+ Dynamics c = free calcium concentration in the cytosol c activates the K(Ca) channels;

10. Plateau Bursting with Standard Parameter Values c is the slow variable, turning spiking on and off as it varies The bursting can be analyzed by examining the subsystem of fast variables (V and n) with c treated as a parameter

11. Moving From Plateau to Pseudo-Plateau 1. Make the slow variable, c, much faster. This results in short burst duration and the burst trajectory moves rapidly along the fast subsystem bifurcation structure. To get this, just increase f cyt. 2.Modify parameter values that change the upper part of the fast subsystem bifurcation structure. This requires changing appropriate fast subsystem parameters.

12. Make the Delayed Rectifier Activate at a Higher Voltage Increasing v n shifts the n  curve to the right. vnvn Red = old curve Blue = new curve

13. Bifurcation Structure for Pseudo-Plateau Bursting Achieved by Increasing v n v n increased from -20 mV to -12 mV, and c speeded up by increasing f cyt from 0.00025 to 0.0135.

14. Bursting Types Depend on the Order of Bifurcations c-values at the bifurcation points:  plateau bursting: supHB < LSN < HM < USN  Transtion bursting: LSN < subHB < HM < USN  Pseudo-plateau bursting: LSN < HM < subHB < USN By using a two-parameter bifurcation diagram, we can determine the parameter regions for these bursting patterns.

15. Two-Parameter Bifurcation Structure: v n vs. c

18. Other Approaches 3. Decrease the delayed rectifier channel conductance 4. Increase Ca 2+ channel conductance In all four approaches, making the cell more excitable converts the plateau bursting to pseudo-plateau bursting. 2. Shift the Ca 2+ activation curve leftward

19. Why Does it Work? Teka et al., J. Math. Neurosci., 1:12, 2011 If one treats V as the sole fast variable and n and c as slow variables, then in the singular limit a folded node singularity is created.

20. Thank You!