Modeling the Action Potential in
a Squid Giant Axon
And how this relates to the beating
of your heart
Outline
1.
2.
3.
4.
The story of an action potential
Digression: Heartbeats and action potentials
Ion Channels
Three stages:
A. Polarization (and resting state)
B. Depolarization
C. Hyperpolarization
5. The equations for neurons
6. Back to action potentials in cardiac tissue
2. Digression: Heartbeats and action potentials
Relating ECGs to APs and Contractions
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
Action Potentials in Different Regions
of the Heart
Bachmann’s
Bundle
Gilmour, “Electrophysiology of the Heart”
2. Digression: Heartbeats and action potentials
The shape of the curve
Gilmour, “Electrophysiology of the Heart”
3. Ion channels
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)
4. Three stages
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
4A. Polarization
Polarized
4B. Depolarization
Depolarization
Gilmour, “Electrophysiology of the Heart”
4C. Repolarization
Repolarization
Gilmour, “Electrophysiology of the Heart”
5. The equations
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
I(t)
C – capacitance
E – equilibrium potential
g – conductance
I(t) – current applied at time t
gK
Equivalent
Circuit Model
gNa
gL
CM
K+
EK
ENa
Ermentrout, Mathematical Foundations of Neuroscience
EL
scitable.com
5. The equations for neurons
Hodgkin-Huxley Equations
m gate – sodium activation
h gate – sodium inactivation
n gate – potassium
Ermentrout, Mathematical Foundations of Neuroscience
5. The equations for neurons
Impact of diffusion
• Add in a term representing neighboring
areas/cells:
where D is the diffusion constant.
6. Back to action potentials in the heart
Action Potentials in Different Regions
of the Heart
Bachmann’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 Equations
Tusscher et al, “A Model for Human Ventricular Tissue,” 2005
6. Back to action potentials in the heart
4V Minimal Model
u is the cell membrane potential
v represents a fast channel gate
s and w represent slow channel gates
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