Modeling the Action Potential in a Squid Giant Axon

1. Modeling the Action Potential in a Squid Giant Axon And how this relates to the beating of your heart

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

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

4. 2. Digression: Heartbeats and action potentials Action Potentials in Different Regions of the Heart Bachmann’s Bundle Gilmour, “Electrophysiology of the Heart”

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

6. 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

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

8. 4. Three stages Three Stages • Polarization (and resting state) • 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

9. 4A. Polarization Polarized

10. 4B. Depolarization Depolarization Gilmour, “Electrophysiology of the Heart”

11. 4C. Repolarization Repolarization Gilmour, “Electrophysiology of the Heart”

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

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

15. 5. The equations for neurons Hodgkin-Huxley Equations 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. 6. Back to action potentials in the heart Action Potentials in Different Regions of the Heart Bachmann’s Bundle 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. 6. Back to action potentials in the heart TNNP Equations Tusscher et al, “A Model for Human Ventricular Tissue,” 2005

20. 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