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