1. Modeling the Action Potential in a Squid Giant Axon
And how this relates to the beating of your heart
Goal: Get an intuitive feel for what’s going on when action potentials are generated and diffuse through cells. I will try to avoid derivations, but can’t avoid complicated equations. We will focus on what the equations mean and how they act, not on how they are derived.
Note: the goal of the equations is to describe the phenomena, not to reflect an eternal truth (like the Pythagorean Theorem). That is, these equations are models, not theorems.

2. Outline The story of an action potential
Digression: Heartbeats and action potentials
Ion Channels
Three stages:
Polarization (and resting state)
Depolarization
Hyperpolarization
The equations for neurons
Back to action potentials in cardiac tissue
The action potentials we’re talking about, in parts 3 and 4, are neuron action potentials – the original HH model. But first, it’s nice to know where this is going, so a little about action potentials in the heart.

3. Relating ECGs to APs and Contractions
2. Digression: Heartbeats and action potentials
Relating ECGs to APs and Contractions
Gilmour, “Electrophysiology of the Heart”

4. Action Potentials in Different Regions of the Heart
2. Digression: Heartbeats and action potentials
Action Potentials in Different Regions of the Heart
Bachmann’s
Bundle
The heartbeat begins in the sinoatrial (SA) node, spreads into the right myocardial tissue and then through Bachmann’s bundle to the left atrium. The wavefronts converge on the AV node and on exiting it enter a special conduction system, the His-Purkinje system, which distributes activation quickly and widely to ventricular tissue.
Action potentials look a little different in the SA node, the atrium, the AV node, the Purkinje system, and the ventricle.
Gilmour, “Electrophysiology of the Heart”

5. The shape of the curve 2. Digression: Heartbeats and action potentials
Gilmour, “Electrophysiology of the Heart”

6. 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

7. HHSim and Resting Potentials
3. Ion channels
HHSim and Resting Potentials
Simulates electrical properties of a neuron
Guide
Software (on workshop laptops, use windows)

8. Three Stages Polarization (and resting state) Depolarization
Sodium-potassium pump and another view
Equilibrium potential determined by permeability to K+
Depolarization
Positive charge opens Na+ channels
Another view
Discussion
Repolarization
Na+ channels are deactivated
Important actors:
The sodium potassium pump
Sodium – the fast inward sodium current
Potassium – the slow outward potassium current

9. Polarized 4A. Polarization
The sodium pump forces sodium outside and potassium inside, against the diffusion gradient.
A certain number of potassium channels are open, meaning that the equilibrium potential across the membrane is determined by the potassium equilibrium potential.
Now, should we do the first problem in the exercises?

10. Depolarization 4B. Depolarization
When the transmembrane potential is reduced from the resting membrane potential to about -65 mV, the activation (m) gate opens almost instantaneously. The inactivation (h) gate is voltage sensitive also, and begins to close at the same time, but takes about 1-2 ms. So while the inactivation gate is still open, the Na+ can rush in.
Gilmour, “Electrophysiology of the Heart”

11. Repolarization 4C. Repolarization
While the h gate is closed, there is no possibility of activation (this is the
Gilmour, “Electrophysiology of the Heart”

12. 4. Five stages!!!
Another view
WH Freeman

13. How can we model this? As an electrical circuit
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

14. Equivalent Circuit Model I(t) CM EK ENa EL gL gK gNa 5. The equations
C – capacitance
E – equilibrium potential
g – conductance
I(t) – current applied at time t K+
Equivalent circuit model
Ermentrout, Mathematical Foundations of Neuroscience
scitable.com

15. Hodgkin-Huxley Equations
5. The equations for neurons
Hodgkin-Huxley Equations
If V is lower than equilibrium, dV/dt will be positive and the term will increase; if it’s higher, it will decrease.
The g terms represent the permeability of the membrane to the ions
dV/dt is inversely proportional to capacitance, which is the ratio of the change in the electric charge in a system to the change in its potential, so the rhs should describe the change in the charge at a given potential
Phi is a temperature factor
Current is the time derivative of charge
m gate – sodium activation
n gate – potassium
h gate – sodium inactivation
Ermentrout, Mathematical Foundations of Neuroscience

16. 5. The equations for neurons
Impact of diffusion
Add in a term representing neighboring areas/cells:
where D is the diffusion constant.

17. Action Potentials in Different Regions of the Heart
6. Back to action potentials in the heart
Action Potentials in Different Regions of the Heart
Bachmann’s
Bundle
The heartbeat begins in the sinoatrial (SA) node, spreads into the right myocardial tissue and then through Bachmann’s bundle to the left atrium. The wavefronts converge on the AV node and on exiting it enter a special conduction system, the His-Purkinje system, which distributes activation quickly and widely to ventricular tissue.
Gilmour, “Electrophysiology of the Heart”

18. 6. Back to action potentials in the heart
Muscle Contraction
Transmission of action potential by the neuromuscular junction
Action potential and muscle contraction

19. TNNP Equations 6. Back to action potentials in the heart
Tusscher et al, “A Model for Human Ventricular Tissue,” 2005

20. 4V Minimal Model u is the cell membrane potential
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.

21. 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