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!