Modelling Electrical Activity in Physiological Systems, 2012

Tutorial: Converting Between Plateau and Pseudo-Plateau Bursting

Richard Bertram

Department of Mathematicsand

Programs in Neuroscience and Molecular Biophysics Florida State University, Tallahassee, FL.

Modelling Electrical Activity in Physiological Systems, 2012

Coworkers and Collaborators

Joël Tabak (FSU)

Funding: NSF-DMS0917664 and NIH-DK043200

Wondimu Teka (FSU) Krasimira Tsaneva-Atanasova(Univ. Bristol)

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

These are Associated with Different Fast-Slow Bifurcation Structures

Fast-slow analysis of plateau or square-wave bursting

These are Associated with Different Fast-Slow Bifurcation Structures

Fast-slow analysis of pseudo-plateau or pituitary bursting

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

The Chay-Keizer Model

This well-studied model was developed to describeplateau bursting in pancreatic β-cells, but it hasalso been used as a template for this type of burstingin 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.

The Chay-Keizer ModelV=voltage (mV) t= time (msec) n= fraction of open delayed rectifying K+ channels

ICa = Ca2+ current

IK = delayed rectifying K+ current

IK(Ca) = Ca2 +-activated K+ current IK(ATP) = ATP-sensitive K+ current

The Chay-Keizer Model: Ca2+ Dynamics

c = free calcium concentration in the cytosol

c activates the K(Ca) channels;

Plateau Bursting with Standard Parameter Values

c is the slow variable, turningspiking on and off as it varies

The bursting can be analyzedby examining the subsystem of fast variables (V and n) withc treated as a parameter

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 fcyt .

2. Modify parameter values that change the upper part of the fast subsystem bifurcation structure. This requires changing

appropriate fast subsystem parameters.

Make the Delayed Rectifier Activate at a Higher Voltage

Increasing vn shifts the n curve to the right.

vn

Red = old curve Blue = new curve

Bifurcation Structure for Pseudo-Plateau Bursting Achieved by Increasing vn

vn increased from -20 mV to -12 mV, and c speededup by increasing fcyt from 0.00025 to 0.0135.

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.

Two-Parameter Bifurcation Structure: vn vs. c

Two-Parameter Bifurcation Structure: vn vs. c

Two-Parameter Bifurcation Structure: vn vs. c

Other Approaches

3. Decrease the delayed rectifier channel conductance

4. Increase Ca2+ channel conductance

In all four approaches, making the cell more excitable converts the plateau bursting to pseudo-plateau bursting.

2. Shift the Ca2+ activation curve leftward

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 asslow variables, then in the singular limita folded node singularity is created.

Thank You!