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Modeling the Action Potential in a Squid Giant Axon
And how this relates to the beating of your heart
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Outline
1. The story of an action potential2. Digression: Heartbeats and action potentials3. Ion Channels4. Three stages:
A. Polarization (and resting state)B. DepolarizationC. Hyperpolarization
5. The equations for neurons6. Back to action potentials in cardiac tissue
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Relating ECGs to APs and Contractions
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
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Action Potentials in Different Regions of the HeartBachmann’s
Bundle
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
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The shape of the curve
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
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Ion channels
• Permanent: always open
• Voltage-gated: the state is determined by the nearby membrane potential
• Ligand-gated: the state is determined by molecules bound to the gate
3. Ion channels
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HHSim and Resting Potentials
• Simulates electrical properties of a neuron
• Guide
• Software (on workshop laptops, use windows)
3. Ion channels
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Three Stages
• Polarization (and resting state)– Sodium-potassium pump– Equilibrium potential determined by permeability
to K+• Depolarization– Positive charge opens Na+ channels
• Repolarization– Na+ channels are deactivated
4. Three stages
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Polarized4A. Polarization
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Depolarization4B. Depolarization
Gilmour, “Electrophysiology of the Heart”
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Repolarization4C. Repolarization
Gilmour, “Electrophysiology of the Heart”
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How can we model this?• As an electrical circuit– Capacitance (the membrane’s ability to store a
charge)– Current (the ions flowing through the membrane)– Resistance to (conductance of) Na+, K+, and other
ions– Equilibrium potential for each type of ion
• With differential equations expressing the change in voltage with given values of the other variables
5. The equations
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K+
I(t)
CM
EK ENa EL
gLgK gNa
C – capacitanceE – equilibrium potential g – conductanceI(t) – current applied at time t
Equivalent Circuit Model
scitable.com
5. The equations
Ermentrout, Mathematical Foundations of Neuroscience
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Hodgkin-Huxley Equations
m gate – sodium activationh gate – sodium inactivation
n gate – potassium
5. The equations for neurons
Ermentrout, Mathematical Foundations of Neuroscience
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Impact of diffusion
• Add in a term representing neighboring areas/cells:
where D is the diffusion constant.
5. The equations for neurons
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Action Potentials in Different Regions of the HeartBachmann’s
Bundle
Gilmour, “Electrophysiology of the Heart”
6. Back to action potentials in the heart
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Muscle Contraction
• Transmission of action potential by the neuromuscular junction
• Action potential and muscle contraction
6. Back to action potentials in the heart
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TNNP Equations6. Back to action potentials in the heart
Tusscher et al, “A Model for Human Ventricular Tissue,” 2005
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4V Minimal Model
u is the cell membrane potentialv represents a fast channel gates and w represent slow channel gates
6. Back to action potentials in the heart
Grosu et al, “From Cardiac Cells to Genetic Regulatory Networks,” 2009.
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Summary
• Hodgkin-Huxley model: The sodium/potassium pump, sodium channels, and potassium channels
• TNNP: Many many channels
• 4V Minimal model: Summarizes channels into fast inward, slow inward, and slow outward