### 2013 Action Potential Modeling in PYTHON

```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
Background
Source: Neil Fraser: http://vv.carleton.ca/~neil/neural/neuron-a.html
Department of Medical Biophysics
Source: Breedlove, et al., Biological Psychology, Fourth Edition, Sinauer Associates
© 2008 Sinauer Associates and Sumanas, Inc.
Department of Medical Biophysics
The Action Potential (AP)
• rapid reversal of the resting membrane
potential (RMP)  depolarization
• permeability of membrane to ions changes
with membrane potential (MP)
Source [1]: Neil Fraser: http://vv.carleton.ca/~neil/neural/neuron-a.html
Source [2]: Hodgkin-Huxley (1952): http://www.ncbi.nlm.nih.gov/pmc/
articles/PMC1392413/pdf/jphysiol01442-0106.pdf
Department of Medical Biophysics
Source: Candace Thompson: http://www.studyblue.com/notes/note/n/actionpotentials/deck/1448117
Motivation
• Why model an action potential?
• estimate parameters
• determine/prove correlation between
variables
• test new and hypothetical situations
• make quantitative and qualitative
predictions
Department of Medical Biophysics
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.
Department of Medical Biophysics
Circuit Diagram Interpretation
Hodgkin-Huxley (1952): http://www.ncbi.nlm.nih.gov/pmc/
articles/PMC1392413/pdf/jphysiol01442-0106.pdf
Department of Medical Biophysics
Derivation of Equations
V  iR
Ohm’s Law
(V m  V y )  i y  R y
iy 
(V m  V y )
conductanc
e, g 
1
R
Ry
i y  g y (V m  V y )
Department of Medical Biophysics
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
I  CM
dV
 g K n (V  V K )  g Na m h (V  V Na )  g l (V  V l )
4
3
dt
dn
(1)
  n (1  n )   n n
(2)
  m (1  m )   m m
(3)
  h (1  h )   h h
(4)
dt
dm
dt
dh
dt
Department of Medical Biophysics
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)
Department of Medical Biophysics
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
Methods
• Define variables:
Variable
Value
vrest
0 #mV
EK
-12 #mV
ENa
115 #mV
El
10.613 #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
Department of Medical Biophysics
Results
Department of Medical Biophysics
Results
Department of Medical Biophysics
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
Department of Medical Biophysics
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)
Conclusion
• equations closely replicate the behaviour of
a measured action potential
• good approximation of electrical
characteristics of excitable cells
• not perfect
Future Research &
Implications
• Simplification of the model
• Model groups of neurons, a bundle of axons,
such as in a nerve
Acknowledgements
• Dr. Andrea Soddu PhD
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, 2013.
http://classconnection.s3.amazonaws.com/117/flashcards/693117/jpg
/picture1321109256503.jpg
• Fraser, N. (1998). The Biological Neuron. Schematic of Biological
Neuron. Retrieved March 5th, 2013.
http://vv.carleton.ca/~neil/neural/neuron-a.html
• 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 1392413. PMID 12991237