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Modeling the Action Potential in a Squid Giant Axon
And how this relates to the beating of your heart
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
Relating ECGs to APs and Contractions
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
Action Potentials in Different Regions of the HeartBachmann’s
Bundle
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
The shape of the curve
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
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
HHSim and Resting Potentials
• Simulates electrical properties of a neuron
• Guide
• Software (on workshop laptops, use windows)
3. Ion channels
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
Polarized4A. Polarization
Depolarization4B. Depolarization
Gilmour, “Electrophysiology of the Heart”
Repolarization4C. Repolarization
Gilmour, “Electrophysiology of the Heart”
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
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
Hodgkin-Huxley Equations
m gate – sodium activationh gate – sodium inactivation
n gate – potassium
5. The equations for neurons
Ermentrout, Mathematical Foundations of Neuroscience
Impact of diffusion
• Add in a term representing neighboring areas/cells:
where D is the diffusion constant.
5. The equations for neurons
Action Potentials in Different Regions of the HeartBachmann’s
Bundle
Gilmour, “Electrophysiology of the Heart”
6. Back to action potentials in the heart
Muscle Contraction
• Transmission of action potential by the neuromuscular junction
• Action potential and muscle contraction
6. Back to action potentials in the heart
TNNP Equations6. Back to action potentials in the heart
Tusscher et al, “A Model for Human Ventricular Tissue,” 2005
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.
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