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 potentialThe story of an action potential 2.Digression: Heartbeats and action potentials 3.Ion Channels 4.Three stages: A.Polarization (and resting state) B.Depolarization C.Hyperpolarization 5.The equations for neurons 6.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 Heart Bachmanns 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 Sodium-potassium pump – Equilibrium potential determined by permeability to K+ Depolarization – Positive charge opens Na+ channels Repolarization – Na+ channels are deactivated 4. Three stages

Polarized 4A. Polarization

Depolarization 4B. Depolarization Gilmour, Electrophysiology of the Heart

Repolarization 4C. Repolarization Gilmour, Electrophysiology of the Heart

How can we model this? As an electrical circuit – Capacitance (the membranes 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) CMCM EKEK E Na ELEL gLgL gKgK g Na C – capacitance E – equilibrium potential g – conductance I(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 activation h 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 Heart Bachmanns Bundle Gilmour, Electrophysiology of the Heart 6. Back to action potentials in the heart

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

TNNP Equations 6. 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 potential v represents a fast channel gate s 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