1. Simulation of an action potential using the Hodgkin-Huxley Model in Python
Nathan Law
Medical Biophysics 3970
Western University
03/24/13
Supervised by
Dr. Andrea Soddu
Medical Physicist
The Department of Physics and Astronomy
COMA Science Group

2. Background Department of Medical Biophysics
A neuron is an excitable cell.
Electrical potentials (imbalance of charge) exist across all cells of the body. How?
Source: Neil Fraser:
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3. Department of Medical Biophysics
2) First note that the lipid bilayer is impermeable to ions. Therefore, we need specific transporters/channels to pass ions through. These channels can be in an open or a closed state.
-The Na+/K+ pumps maintains a high concentration of potassium ions inside the cell with respect to the extracellular environment (It pumps 3Na+ ions out and 2K+ ions in).
3) There is now a diffusion force, that causes K+ ions to leak out of the cell through K+ leak channels DOWN IT’S CONCENTRATION GRADIENT.
The more K+ ions that flow out of the cell, the more negatively charged the intracellular space becomes.
4) An electrical force is produced by the attraction of positive K+ ions back in to the intracellular space.
At electrochemical equilibrium: there exists an electrical potential  the electrical gradient is balanced by the diffusion gradient.
The resting membrane potential in most neurons is usually -69mV.
Source: Breedlove, et al., Biological Psychology, Fourth Edition, Sinauer Associates © 2008 Sinauer Associates and Sumanas, Inc.
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4. The Action Potential (AP)
rapid reversal of the resting membrane potential (RMP)  depolarization
permeability of membrane to ions changes with membrane potential (MP)
We can make use of the electrical potential carried by a neuron. We can use it to transmit a signal via propagation of a depolarization down the axon. This is called the action potential.
-A depolarization can be induced by some mechanical, chemical or electrical stimuli. For example, the baroreceptors in the carotid sinus are sensitive to pressure.
-there exists a minimum threshold for depolarization before an action potential fires
- Once an action potential fires  the permeability of the cell for certain ions changes
The figure shows the change in membrane potential with respect to time. The graph shows that the maximum change in membrane potential is about 100mV. This means that a neuron with a RMP of -69mV will peak at approximatly-40mV.
Source [1]: Neil Fraser:
Source [2]: Hodgkin-Huxley (1952): articles/PMC /pdf/jphysiol pdf
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5. *SHOW THIS SLIDE so that students don’t get confused that there are in fact two gates (inactivation & activation) gates in the voltage gated sodium channel
- Note that the activities of the activation and the inactivation gates are given by the gating variables m and h respectively.
Source: Candace Thompson:

6. Motivation Why model an action potential? estimate parameters
determine/prove correlation between variables
test new and hypothetical situations
make quantitative and qualitative predictions
Why use mathematical models?
Estimate parameters – duration of each phase of the action potential based on how we know ion channels open and close with respect to membrane voltage
determine/prove a correlation between variables – the Hodgkin-Huxley model showed that ionic conductance was responsible for the nerve action potential, and therefore, changes in membrane potential
testing new and hypothetical situations
Help us make quantitative and qualitative predictions
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7. Objectives
1) Develop a simulation for an action potential using Python based on the Hodgkin-Huxley model.
2) Compare and contrast the Hodgkin-Huxley model with empirical data based on the giant squid axon.
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8. Hypothesis
Hodgkin-Huxley model will closely replicate the empirical data based on a single action potential from the giant squid
variation of parameters will result in a better fitting AP spike
*this slide did not serve much purpose during the presentation and was therefore not shown
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9. Circuit Diagram Interpretation
Membrane potential in an axon can be modeled after the circuit diagram above.
An electrical current can travel from the extracellular space (outside) to the intracellular compartment (inside) through the capacitance, Cm, (lipid bilayer) or any one of the resistances, RNA, RK, or Rl, connected in parallel to the circuit (ion channels).
inject current I in to the cell from outside cell
Hodgkin-Huxley (1952): articles/PMC /pdf/jphysiol pdf
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10. Derivation of Equations
Ohm’s Law
We can solve for the current strictly due to open channels from one particular ion by the following:
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11. The Hodgkin-Huxley Model
A mathematical model that describes action potential initiation and propagation based on the giant squid axon
model is based on four first-order ordinary differential equations
(1)
(2)
The Hodgkin Huxley is an empirical model. Therefore, it is a model that is based on real observed/measured data (The data is from the giant squid axon). This data will come in to play when we solve the differential equations.
n controls potassium channel opening
m controls sodium channels opening
h controls sodium channels closing
n, m and h are dimensionless
CM is the membrane capacitance per unit area, gKbar, gNabar, and glbar are sodium and potassium conductances
Vk, VNa and Vl are the reversal potentials (a.k.a Nernst Potential: the potential of the cell membrane at which there is no net movement of a particular ion between the intracellular and extracellular space)
V is simply the membrane potential
The term on the right of equation (1) is called the capacity current. It’s the sum of the Na+, K+ and leak ion currents per cm^2 of membrane.
(3)
(4)
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12. Methods
differential equations from: A Quantitative Description of Membrane Current And It’s Application To Conduction And Excitation In Nerve: A. Hodgkin & A. Huxley (1952)
all parameters based on paper
Python (programming language)
Spyder (scientific Python development environment)
Python is a general purpose, high-level programming language
Spyder:
allows smooth transition from MATLAB to Python programming language - Spyder is an interactive scientific development environment
UNLIKE MATLAB - all integers, strings, arrays etc. in Python are called objects
You have to write code to indicate that an array of numbers you created is actually a vector. Otherwise, they will concatenate (become joined end to end, like a list).
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13. Methods Code development: Define Variables based on empirical data
Define a time scale and an array of membrane potentials
Input equations based on Hodgkin-Huxley model
Plot

14. Methods Define variables: Department of Medical Biophysics Variable
Value
vrest
0 #mV EK
-12 #mV
ENa
115 #mV El
#mV
gKbar
36 #mS/cm^2
gNabar
120 #mS/cm^2
glbar
0.3 #mS/cm^2 cm
1 #uF/cm^2 ts
100ms dt
0.025 v
(-100,250) #mV
‘E’ denotes Nernst Potential
- These values were derived from a series of voltage clamp experiments.
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15. Results Department of Medical Biophysics
This graph describes the behaviour of each ion channel with respect to membrane voltage. You have to think of each gate like a binary switch. 1 means that the gate is completely open. 0 means that the gate is closed. In order for the cell to allow Na+ to pass through the membrane, both the sodium inactivating gate and activating gate must be on (greater than 0)
note that gating variables depend on both time and voltage
For increasing voltage, both the value of n(0)  potassium channel and m(0)  sodium channel opening increases, whereas h(0)  sodium closing decreases.
Therefore, if there is some stimulus that causes the membrane to depolarize, the conductance of both sodium channels and potassium channels increases
The cell membrane becomes more and more positive, due to the opening of these channels which allow positive ions to flow in. If this positive feedback is large enough, then an action potential is initiated.
At high membrane potential, the activity of the sodium gate is inactivated due to the value h?
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16. Results
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17. Discussion Qualitative Analysis
there is a hump during the depolarization phase in the calculated model
the peak in the calculated model is sharper
during repolarization, the calculated model is not smooth
slope of the repolarization phase may be too steep
The model that I produced on Python is not a perfect representation of an action potential. It is simply based off of parameters from observed data and solutions to differential equations the closely replicate the behaviour of each ion channel.
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18. Discussion
Above: Original tracing of membrane action potential recorded at 9.1°C (Empirical Model)
Above: Change in Membrane Potential with Respect to Time (Hodgkin Huxley Model)

19. Conclusion
equations closely replicate the behaviour of a measured action potential
good approximation of electrical characteristics of excitable cells
not perfect

20. Future Research & Implications
Simplification of the model
Model groups of neurons, a bundle of axons, such as in a nerve

21. Acknowledgements
Dr. Andrea Soddu PhD

22. References
Breedlove, et al., Biological Psychology, Fourth Edition, Sinauer Associates © 2008 Sinauer Associates and Sumanas, Inc.
Thompson, C. (2013). Action Potentials. Medicine 2015 at Howard University College of Medicine. Retrieved March 8th, /picture jpg
Fraser, N. (1998). The Biological Neuron. Schematic of Biological Neuron. Retrieved March 5th,
Hodgkin, A. L.; Huxley, A. F. (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". The Journal of physiology 117 (4): 500– 544.PMC   PMID 

23. Questions? Comments?