1. Hodgkin-Huxley David Wallace Croft, M.Sc. Atzori Lab, U.T. Dallas
2011 Aug 19 Fri
Copyright 2011 David Wallace Croft.
This work is licensed under a Creative Commons Attribution 3.0 United States License.

2. Overview
The Hodgkin-Huxley equations provide a biologically realistic model of neural action potentials
The model is described using differential equations based on laws of physics and experimental measurements
A computer simulation is demonstrated which uses Euler's Method for numerical integration with a very small time step

3. The axon is giant, not the squid
Giant Squid Axon
The axon is giant, not the squid

4. Demonstration

5. Demonstration Injected current drives action potentials / spikes
Spiking ceases due to adaptation
Spiking resumes when injected current is toggled off and then back on
Intermittent current pulses are less likely to cause an action potential immediately following an action potential during the refractory period
The membrane voltage changes rapidly between time steps during action potentials

6. Equivalent Circuit
Arne Nordmann,

7. Math Notation and Symbols
Δ Delta: difference; a step-wise change
x' Prime: first derivative of x(t)

8. Physics Equations c = A • C membrane capacitance = area • capacitivity
g = A • G channel conductance = area • conductivity
i = g • ( v - E ) current = conductance • ( voltage - reversal potential )
Δq = Δt • (-i) Δ charge = simulation time step • negative current
Δv = Δq / c Δ membrane voltage = Δ charge / capacitance

9. Hodgkin-Huxley Equations 1
Voltage-dependent parameters
β ℎ = 1 𝑒 30−𝑉 β 𝑚 =4⋅ 𝑒 −𝑉 β 𝑛 =0.125⋅ 𝑒 −𝑉 80
α ℎ =0.07⋅ 𝑒 −𝑉 α 𝑚 = 0.1⋅ 25−𝑉 𝑒 25−𝑉 10 −1 α 𝑛 = 0.01⋅ 10−𝑉 𝑒 10−𝑉 10 −1

10. Hodgkin-Huxley Equations 2
τ ℎ = 1 α ℎ + β ℎ τ 𝑚 = 1 α 𝑚 + β 𝑚 τ 𝑛 = 1 α 𝑛 + β 𝑛
ℎ ∞ = α ℎ α ℎ + β ℎ 𝑚 ∞ = α 𝑚 α 𝑚 + β 𝑚 𝑛 ∞ = α 𝑛 α 𝑛 + β 𝑛
ℎ′= ℎ ∞ −ℎ τ ℎ 𝑚′= 𝑚 ∞ −𝑚 τ 𝑚 𝑛′= 𝑛 ∞ −𝑛 τ 𝑛
𝐶⋅𝑉′=− 𝑔 𝐾 ⋅ 𝑛 4 ⋅ 𝑉 𝑚 − 𝐸 𝐾 − 𝑔 𝑁𝑎 ⋅ 𝑚 3 ⋅ℎ⋅ 𝑉 𝑚 − 𝐸 𝑁𝑎 − 𝑔 𝐿 ⋅ 𝑉 𝑚 − 𝐸 𝐿 + 𝐼 𝑠𝑦𝑛𝑎𝑝𝑡𝑖𝑐

11. Potassium Channel

12. Simulation Parameters

13. Synaptic Currents

14. Hodgkin-Huxley Currents

15. References
Croft (2010) "HhNeuronImp.java",
Croft (2008) "Neuro",
Trappenberg (2010) "Section 2.3 The generation of action potentials: Hodgkin-Huxley equations", Fundamentals of Computational Neuroscience, 2nd Ed., Oxford University Press
Wikipedia, "Action potential"
Wikipedia, "Hodgkin-Huxley model"