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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

Calibration of dopamine neuron model. a The equivalent circuit for the conductance-based model with nonlinear conductance in parallel with the membrane capacitance . The maximal conductance and reversal potential of each current are indicated by  and , respectively. The arrows indicate time and voltage-dependent conductances. b Calibration of model K+ currents. b1 The parameters of the description of the A-type K+ current were adjusted to fit published voltage clamp data from nucleated membrane patches from SNc dopamine neurons (representative current traces from Fig. 11A3 of [28]). The conductance used for these simulations (120 μS/cm2) was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 100 ms steps from a holding potential of −100 mV to 50 mV in increments of 10 mV. b2 The parameters of the description of the ERG-type K+ current were adjusted to fit published voltage clamp data (Fig. 1A of [32]) from human channels heterologously expressed in Xenopus oocytes. The conductance used for these simulations was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 600 ms steps from a holding potential of −80 mV to −100 to 50 mV in increments of 10 mV. Tail currents were measured at −70 mV. c The model neuron exhibits slow pacemaker firing at 3.6 Hz under control conditions. d With  set to zero and  set to 35 pA, the model exhibits a Ca2+-dependent sinusoidal slow oscillatory potential (SOP)
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Fig1: Calibration of dopamine neuron model. a The equivalent circuit for the conductance-based model with nonlinear conductance in parallel with the membrane capacitance . The maximal conductance and reversal potential of each current are indicated by and , respectively. The arrows indicate time and voltage-dependent conductances. b Calibration of model K+ currents. b1 The parameters of the description of the A-type K+ current were adjusted to fit published voltage clamp data from nucleated membrane patches from SNc dopamine neurons (representative current traces from Fig. 11A3 of [28]). The conductance used for these simulations (120 μS/cm2) was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 100 ms steps from a holding potential of −100 mV to 50 mV in increments of 10 mV. b2 The parameters of the description of the ERG-type K+ current were adjusted to fit published voltage clamp data (Fig. 1A of [32]) from human channels heterologously expressed in Xenopus oocytes. The conductance used for these simulations was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 600 ms steps from a holding potential of −80 mV to −100 to 50 mV in increments of 10 mV. Tail currents were measured at −70 mV. c The model neuron exhibits slow pacemaker firing at 3.6 Hz under control conditions. d With set to zero and set to 35 pA, the model exhibits a Ca2+-dependent sinusoidal slow oscillatory potential (SOP)

Mentions: The model (Fig. 1a) consists of a fast spiking sodium current () [25, 26], an L-type calcium current () [27], a delayed rectifier () [28], a transient outward potassium current () [28], an ether-a-go-go-related potassium current () [29], a calcium-activated small conductance SK potassium current (), a nonspecific hyperpolarization-activated cation current (), and a leak current () that is comprised of a nonspecific () and calcium ion specific component (). A small applied stimulus current was required for one simulation, and converted from pA to intensive units using the diameter and of the cylindrical somatic compartment. The conductances for these currents () are in parallel with the membrane capacitance . The differential equations for transmembrane potential are as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\mathrm{m}} \frac{dv}{dt} =- I_{\mathrm{Na}} - I_{\mathrm{Ca},\mathrm{L}} - I_{\mathrm{K},\mathrm{DR}} - I_{\mathrm{K},\mathrm{A}} - I_{\mathrm{K},\mathrm {ERG}} - I_{\mathrm{K},\mathrm{SK}} - I_{\mathrm{H}} - I_{\mathrm{Leak}} + 0.1 I_{\mathrm{stim}} /\pi\, dL. $$\end{document}Cmdvdt=−INa−ICa,L−IK,DR−IK,A−IK,ERG−IK,SK−IH−ILeak+0.1Istim/πdL.Fig. 1


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)

Calibration of dopamine neuron model. a The equivalent circuit for the conductance-based model with nonlinear conductance in parallel with the membrane capacitance . The maximal conductance and reversal potential of each current are indicated by  and , respectively. The arrows indicate time and voltage-dependent conductances. b Calibration of model K+ currents. b1 The parameters of the description of the A-type K+ current were adjusted to fit published voltage clamp data from nucleated membrane patches from SNc dopamine neurons (representative current traces from Fig. 11A3 of [28]). The conductance used for these simulations (120 μS/cm2) was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 100 ms steps from a holding potential of −100 mV to 50 mV in increments of 10 mV. b2 The parameters of the description of the ERG-type K+ current were adjusted to fit published voltage clamp data (Fig. 1A of [32]) from human channels heterologously expressed in Xenopus oocytes. The conductance used for these simulations was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 600 ms steps from a holding potential of −80 mV to −100 to 50 mV in increments of 10 mV. Tail currents were measured at −70 mV. c The model neuron exhibits slow pacemaker firing at 3.6 Hz under control conditions. d With  set to zero and  set to 35 pA, the model exhibits a Ca2+-dependent sinusoidal slow oscillatory potential (SOP)
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4385104&req=5

Fig1: Calibration of dopamine neuron model. a The equivalent circuit for the conductance-based model with nonlinear conductance in parallel with the membrane capacitance . The maximal conductance and reversal potential of each current are indicated by and , respectively. The arrows indicate time and voltage-dependent conductances. b Calibration of model K+ currents. b1 The parameters of the description of the A-type K+ current were adjusted to fit published voltage clamp data from nucleated membrane patches from SNc dopamine neurons (representative current traces from Fig. 11A3 of [28]). The conductance used for these simulations (120 μS/cm2) was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 100 ms steps from a holding potential of −100 mV to 50 mV in increments of 10 mV. b2 The parameters of the description of the ERG-type K+ current were adjusted to fit published voltage clamp data (Fig. 1A of [32]) from human channels heterologously expressed in Xenopus oocytes. The conductance used for these simulations was chosen to match the amplitude of the currents from the voltage clamp data, obtained with 600 ms steps from a holding potential of −80 mV to −100 to 50 mV in increments of 10 mV. Tail currents were measured at −70 mV. c The model neuron exhibits slow pacemaker firing at 3.6 Hz under control conditions. d With set to zero and set to 35 pA, the model exhibits a Ca2+-dependent sinusoidal slow oscillatory potential (SOP)
Mentions: The model (Fig. 1a) consists of a fast spiking sodium current () [25, 26], an L-type calcium current () [27], a delayed rectifier () [28], a transient outward potassium current () [28], an ether-a-go-go-related potassium current () [29], a calcium-activated small conductance SK potassium current (), a nonspecific hyperpolarization-activated cation current (), and a leak current () that is comprised of a nonspecific () and calcium ion specific component (). A small applied stimulus current was required for one simulation, and converted from pA to intensive units using the diameter and of the cylindrical somatic compartment. The conductances for these currents () are in parallel with the membrane capacitance . The differential equations for transmembrane potential are as follows:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{\mathrm{m}} \frac{dv}{dt} =- I_{\mathrm{Na}} - I_{\mathrm{Ca},\mathrm{L}} - I_{\mathrm{K},\mathrm{DR}} - I_{\mathrm{K},\mathrm{A}} - I_{\mathrm{K},\mathrm {ERG}} - I_{\mathrm{K},\mathrm{SK}} - I_{\mathrm{H}} - I_{\mathrm{Leak}} + 0.1 I_{\mathrm{stim}} /\pi\, dL. $$\end{document}Cmdvdt=−INa−ICa,L−IK,DR−IK,A−IK,ERG−IK,SK−IH−ILeak+0.1Istim/πdL.Fig. 1

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