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A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block.

Yu N, Canavier CC - J Math Neurosci (2015)

Bottom Line: The two slow variables contribute as follows.A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking.The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau.

View Article: PubMed Central - PubMed

Affiliation: Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, New Orleans, LA 70112 USA ; Department of Mathematics and Computer Science, Lawrence Technological University, 21000 West 10 Mile Road, Southfield, MI 48075 USA.

ABSTRACT
Midbrain dopamine neurons exhibit a novel type of bursting that we call "inverted square wave bursting" when exposed to Ca(2+)-activated small conductance (SK) K(+) channel blockers in vitro. This type of bursting has three phases: hyperpolarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca(2+) channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.

No MeSH data available.


Related in: MedlinePlus

Fast-slow bifurcation diagrams for oscillatory plateau potentials. a Bifurcation analysis of the full model from Fig. 2b with . Solid and dotted lines represent the stable and unstable fixed points on the bifurcation diagram, respectively. Dots indicate bifurcation points, with SN denoting saddle-node bifurcation. Double and single arrows indicate the direction of fast and slow changes in voltage, respectively, on the closed curve (thin black lines) that represents the limit cycle trajectory. b Same as a except the bifurcation diagram is equivalent to the voltage cline in the reduced two-variable system, and the cline for the  pool is shown (dashed curve)
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Fig3: Fast-slow bifurcation diagrams for oscillatory plateau potentials. a Bifurcation analysis of the full model from Fig. 2b with . Solid and dotted lines represent the stable and unstable fixed points on the bifurcation diagram, respectively. Dots indicate bifurcation points, with SN denoting saddle-node bifurcation. Double and single arrows indicate the direction of fast and slow changes in voltage, respectively, on the closed curve (thin black lines) that represents the limit cycle trajectory. b Same as a except the bifurcation diagram is equivalent to the voltage cline in the reduced two-variable system, and the cline for the pool is shown (dashed curve)

Mentions: A bifurcation analysis was performed on the single-compartment model from Fig. 2b with , , and set to zero. The remaining state variables were . The trajectory (thin black closed curve with arrows) corresponding to Fig. 2b was replotted in Fig. 3a in the plane of membrane potential (v) and the fraction of ERG channels () that are not closed, but are in either the open or inactivated state. The choice of these coordinate axes was motivated by a previous modeling study [19] that suggested that the pool of ERG potassium channels was the appropriate slow variable for a fast/slow analysis. As noted in Sect. 2, the transition rates between the closed and open states ( and ) are much slower than the transition rates between open and inactivated states ( and ), so the inactivated channels remain approximately at steady state with respect to the open fraction as their sum () changes slowly. Fig. 3


A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block.

Yu N, Canavier CC - J Math Neurosci (2015)

Fast-slow bifurcation diagrams for oscillatory plateau potentials. a Bifurcation analysis of the full model from Fig. 2b with . Solid and dotted lines represent the stable and unstable fixed points on the bifurcation diagram, respectively. Dots indicate bifurcation points, with SN denoting saddle-node bifurcation. Double and single arrows indicate the direction of fast and slow changes in voltage, respectively, on the closed curve (thin black lines) that represents the limit cycle trajectory. b Same as a except the bifurcation diagram is equivalent to the voltage cline in the reduced two-variable system, and the cline for the  pool is shown (dashed curve)
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4385104&req=5

Fig3: Fast-slow bifurcation diagrams for oscillatory plateau potentials. a Bifurcation analysis of the full model from Fig. 2b with . Solid and dotted lines represent the stable and unstable fixed points on the bifurcation diagram, respectively. Dots indicate bifurcation points, with SN denoting saddle-node bifurcation. Double and single arrows indicate the direction of fast and slow changes in voltage, respectively, on the closed curve (thin black lines) that represents the limit cycle trajectory. b Same as a except the bifurcation diagram is equivalent to the voltage cline in the reduced two-variable system, and the cline for the pool is shown (dashed curve)
Mentions: A bifurcation analysis was performed on the single-compartment model from Fig. 2b with , , and set to zero. The remaining state variables were . The trajectory (thin black closed curve with arrows) corresponding to Fig. 2b was replotted in Fig. 3a in the plane of membrane potential (v) and the fraction of ERG channels () that are not closed, but are in either the open or inactivated state. The choice of these coordinate axes was motivated by a previous modeling study [19] that suggested that the pool of ERG potassium channels was the appropriate slow variable for a fast/slow analysis. As noted in Sect. 2, the transition rates between the closed and open states ( and ) are much slower than the transition rates between open and inactivated states ( and ), so the inactivated channels remain approximately at steady state with respect to the open fraction as their sum () changes slowly. Fig. 3

Bottom Line: The two slow variables contribute as follows.A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking.The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau.

View Article: PubMed Central - PubMed

Affiliation: Department of Cell Biology and Anatomy, Louisiana State University School of Medicine, New Orleans, LA 70112 USA ; Department of Mathematics and Computer Science, Lawrence Technological University, 21000 West 10 Mile Road, Southfield, MI 48075 USA.

ABSTRACT
Midbrain dopamine neurons exhibit a novel type of bursting that we call "inverted square wave bursting" when exposed to Ca(2+)-activated small conductance (SK) K(+) channel blockers in vitro. This type of bursting has three phases: hyperpolarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca(2+) channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K(+) current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.

No MeSH data available.


Related in: MedlinePlus