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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
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Coworkers and Collaborators
Joël Tabak (FSU)
Funding: NSF-DMS0917664 and NIH-DK043200
Wondimu Teka (FSU) Krasimira Tsaneva-Atanasova(Univ. Bristol)
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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
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These are Associated with Different Fast-Slow Bifurcation Structures
Fast-slow analysis of plateau or square-wave bursting
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These are Associated with Different Fast-Slow Bifurcation Structures
Fast-slow analysis of pseudo-plateau or pituitary bursting
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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
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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.
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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
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The Chay-Keizer Model: Ca2+ Dynamics
c = free calcium concentration in the cytosol
c activates the K(Ca) channels;
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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
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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.
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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
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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.
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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.
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Two-Parameter Bifurcation Structure: vn vs. c
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Two-Parameter Bifurcation Structure: vn vs. c
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Two-Parameter Bifurcation Structure: vn vs. c
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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
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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.
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Thank You!