Transcript Comparing the ODE and PDE Models of the Hodgkin

Comparing the ODE and PDE
Models of the Hodgkin-Huxley
Equation
Sarah Arvey, Haley Rosehill
Calculus 114
History of Hodgkin-Huxley Model
• Hodgkin and Huxley experimented on
squid giant axon and discovered how the
signal is produced within the neuron
• Model was published in Journal of
Physiology (1952)
• Hodgkin and Huxley awarded the 1963
Nobel Prize for model
Physical shape of a Neuron
•
•
•
•
•
Dendrites
Nucleus
Cell body
Myelin
Axon
– Variety of gates
• Synaptic Terminal
Brief Biology Background of a
Neuron
• A message is sent down the axon
• The axon membrane contains a variety of
gates.
• The gates slowly and continually open so
sodium and potassium ions can get
through the gates
• The rate at which the ions are pumped
across the membrane establishes the
“resting potential” (-70 mV)
Action Potential
Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif
Action Potential
Taken from C. George Boeree: www.ship.edu/~cgboeree
Ordinary
Differential
Equations
VS.
• Model phenomena that
evolve continuously in time
• Equations in which the
unknown element is a
function, rather than a
number
• Involves one independent
variable
Partial
Differential
Equations
• Involves two or more
independent variables
• Can track a function over
space and time
ODE of Hodgkin-Huxley
• Measures action potential at a given time
• Membrane potential
– Based on sodium, potassium and leakage
– Clamp method
Action Potential
Taken http://artsci-ccwin.concordia.ca/psychology/psyc358/Lectures/figures/act_pot1/s_ociloAP.gif
The Model
I = (m^3)(h) GNa (ENa - E ) + (n^4) GK (EK - E ) + GL (EL - E )
The parameter names in bold are fixed variables.
I : the total ionic current across the membrane
m : the probability that 1 of the 3 required activation particles has
contributed to the activation of the Na gate (m^3 : the probability that
all 3 activation particles have produced an open channel)
h : the probability that the 1 inactivation particle has not caused the
Na gate to close
G_Na : Maximum possible Sodium Conductance (about 120
mOhms^-1/cm2)
E : total membrane potential (about -60 mV)
E_Na : Na membrane potential (about 55 mV)
n : the probability that 1 of 4 activation particles has influenced the
state of the K gate.
G_K : Maximum possible Potassium Conductance (about 36
mOhms^-1/cm2)
E_K : K membrane potential (about -72 mV)
G_L : Maximum possible Leakage Conductance (about .3 mOhms^1/cm2)
E_L : Leakage membrane potential (about -50 mV)
M, H, and N are
variables. 3
variables? How is it
an ODE?
The Variable Functions
• Dm/dt= am(1-m)-bmm
• Dh/dt= ah(1-h)-bhh
• Dn/dt= an(1-n)-bnn
• All ODE’s thus Hodgkin and Huxley is a
system of ODE’s
PDE of Hodgkin-Huxley
• Analysis of a traveling pulse
• Measures the state of the action potential
over time and space
• Can be taken in respect to m, h, or n
The Actual Model
What is this?!?
• a= radius of axon
• p= resistance of the intracellular space
• The x variable is that of space
- just as single variable functions have
higher order derivative, so do multivariable functions
+/- of ODE
Positive Aspects
• Simple
• Gives total ionic
current at a specific
time
• Tracks excitability and
conductance of a
neuron
Negative Aspects
• Does not give
membrane potential
over space
– No true idea of action
potential activity
+/- of PDE
Positive Aspects
Negative Aspects
• More telling of the
• Confusing
action potential’s
activity
-space and time
• Tracks excitability and
conductance via wave
pulse
WE
LIKE
Which model is
THE
PDE!!!!
better?
References
http://www.math.niu.edu/~rusin/known-math/index/34XX.html
http://artsciccwin.concordia.ca/psychology/psyc358/Lecture
s/figures/act_pot1/s_ociloAP.gif
Segel, Lee A. “Biological Waves.” Mathematical Models in
Molecular and Cellular Biology. New York: Cambridge
University Press, 1980.
http://retina.anatomy.upenn.edu/~lance/modelmath/hogkin_
huxley.html
Muratov, C.B. “A Quantative Approximation Scheme for the
Traveling Wave Solutions in the Hogkin-Huxley Model.”
Biophysical Journal. Newark, New Jersey: University
Heights, 2000.
http://www.ship.edu/~cgboeree
http://tutorial.math.lamar.edu/AllBrowsers/2415/HighOrderP
artialDerivs.asp