SLIDE 1 Hodgkin-Huxley Model of Action Potentials
Differential Equations Math 210
SLIDE 2 Neuron
Dendrites Collect electrical signals Cell body Contains nucleus and
Axon Passes electrical signals
another cell or to an effector cell
SLIDE 3
Electrochemical Equilibrium
SLIDE 4
Action Potential
Axon membrane
potential difference V = Vin - Vout
When the axon is
excited, V spikes because sodium Na+ and potassium K+ ions flow through the membrane
SLIDE 5
Modeling the dynamics of an action potential
Alan Lloyd Hodgkin and Andrew
Huxley
Proposed model in 1952 Explains ionic mechanisms
underlying the initiation and propagation of action potential in the squid giant axon
Received the 1963 Nobel Prize in
Physiology or Medicine
SLIDE 6
Circuit model for axon membrane
Conductors or resistors represent the ion channels. Capacitors represent the ability of the membrane to store charge.
q(t) = the charge carried by particles in circuit at time t I(t) = the current (rate of flow of charge in the circuit) = dq/dt V(t) = the voltage difference in the electrical potential at time t R = resistance (property of a material that impedes flow of charge particles) g(V) = conductance = 1/R C = capacitance ( property of an element that physically separates charge) C R V
SLIDE 7
Physical relationships in a circuit
Ohm’s law: the voltage drop across a resistor is proportional
to the current through the resistor; R (or 1/g) is the factor or proportionality
Faraday’s law: the voltage drop across a capacitor is
proportional to the electric charge; 1/C is the factor of proportionality
SLIDE 8 Elements in parallel
For elements in parallel, the total current is equal to the sum of currents in
each branch; the voltage across each branch is then the same. Differentiate Faraday’s Law ( ) leads to
SLIDE 9
Hodgkin-Huxley Model
gL is constant gNa and gK are voltage-dependent
SLIDE 10
Ion channel gates
Membrane Ion channel “n” gates
SLIDE 11 Voltage dependency of gate position
n
(proportion in the
n - 1
(proportion in the
αn βn αn , βn are transition rate constants (voltage-dependent) αn = the # of times per second that a gate which is in the shut state opens βn = the # of times per second that a gate which is in the open state shuts Fraction of gates opening per second = αn(1 – n) Fraction of gates shutting per second = βnn The rate at which n changes: Equilibrium:
What is the behavior of n?
SLIDE 12 Gating variable
Solve initial value problem by separation of variables:
If αn or βn is large → time constant is short → n approaches n∞ rapidly If αn or βn is small → time constant is long → n approaches n∞ slowly
time constant
SLIDE 13 Gating Variables
K+ channel is controlled by 4 n activation gates: Na+ channel is controlled by 3 m activation gates and 1 h inactivation gate: Activation gate: open probability increases with depolarization Inactivation gate: open probability decreases with depolarization
dn dt = 1 τ n n∞ − n
( ) ⇒
dm dt = 1 τ m m∞ − m
( )
dh dt = 1 τ h h∞ − h
( ) gK = n4 gK
maximum K+ conductance
SLIDE 14
Steady state values
SLIDE 15
Time constants
SLIDE 16
Voltage step scenario
Given the voltage step above:
Sketch n as a function of time. What does n4 look like? Sketch m and h on the same graph as functions of time. What
does m3h look like?
SLIDE 17 How does the Hodgkin-Huxley model predict action potentials?
Depolarization fast in m gNa Na+ inflow
Positive Feedback
(results in upstroke of V) Depolarization Slow in n gK
Negative Feedback
(this and leak current repolarizes) K+ outflow Repolarization
