BME 6938Neurodynamics
Instructor: Dr Sachin S. Talathi
Recap
• XPPAUTO introduction• Linear cable theory
– Cable equation– Boundary and Initial Conditions– Steady State Analysis– Transient Analysis
• Rall model-Equivalent cylinder
Nonlinear membrane
Linear cable propertiessatisfying Ohms law
Nonlinear membrane
Ions: Na+,K+,Ca2+,Cl-
In general a nonlinear function in voltage and time
Revisiting Goldman Eq.
Permeability of the membrane changes as function of voltage and time
Gate Model
• HH proposed the gate model to provide a quantitative framework for determining the time and membrane potential dependent properties of ion channel conductance
• The Assumptions in the Gate Model:– Membrane comprise of aqueous pores through which the ions
flow down their concentration gradient– These pores contain voltage sensitive gates that close and
open dependent on trans membrane potential– The transition from closed to open state and vice-versa follow
first order kinetics with rate constants: and
Kinetics of gate transition
• Let p represent the fraction of gates within the ion channel that are in open state at any given instant in time
• 1-p represents the remaining fraction of the gates that are in closed state
• If represents the transition rate for gate to go from closed to open state and represents the transition rate for gate to go from open to closed stat, we have
Open p
Closed 1-p
Steady state
The transient solution can then be obtained as:
OR
Multiple gates
• If a ion channel is comprised of multiple gates; then each and every gate must be open for the channel to conduct ion flow.
• The probability of gate opening then is given by:
• Gate Classification– Activation Gate: p(t,V) increases with membrane
depolarization– Inactivation Gate: p(t,V) decreases with membrane
depolarization
The unknowns
• In order to use the gate model to determine the ion channel dynamics, HH had to estimate the following 3 quantities– Macro characteristics of channel type – The number and type of gates on a given ion channel– The transition rate constants &
Macro characteristics include: Reversal potential, maximum conductance and ion specificy
The experiments
• Two important factors permitted HH analysis as they set about to design experiments to find the unknowns– Giant Squid Axon (Diameter approx 0.5 mm), allowed for
the use of crude electronics of 1950’s (Squid axon’s utility for of nerve properties is credited to J.Z Young (1936) )
– Development of feed back control device called the voltage clamp capable of holding the membrane potential to a desired value
Before we look into the experiments; lets have a look at
the model proposed by HH to describe the dynamics of squid axon cell membrane
HH model
• HH proposed the parallel conductance model wherein the membrane current is divided up into four separate contributions– Current carried by sodium ions– Current carried by potassium ions– Current carried by other ions (mainly chloride and
designated as leak currents)– The capacitive current
We have already seen this idea being utilized in GHK equations
The equivalent circuit
Goal: Find &
Results
The Experiments
Space clamp: Eliminate axial dependence of membrane voltage
• Stimulate along the entire length of the axon• Can be done using a pair
of electrodes as shown• Provides complete axial symmetry Result:
Eliminate the axial component inThe cable equation
Voltage Clamp: Eliminate capacitive current
http://www.sinauer.com/neuroscience4e/animations3.1.html
Example of Voltage Clamp Recording
Sum of parts
Series of Voltage clamp expts
Selectively blocking specific currents
H-H experiments to test Ohms law
HH measurement of Na and K conductance
Gating variables
Maximum conductance
Functional fitting to gate variable
• We see from last slide• Na comprise of activation and inactivation• K comprise of only activation term• HH fit the the time dependent components of the
conductance such that
Activation gate Inactivation gate
m,n and h are gate variables and follow first order kinetics of the gate model
Gate model for m,n and h
Activation: Inactivation:
• Determine and• Use the following relationship
• Do empirical curve fitting to obtain
Estimating gate model parameters
Profiles of fitted transition functions
Summary of HH experiments
• Determine the contributions to cell membrane current from constituent ionic components
• Determine whether Ohms law can be applied to determine conductances
• Determine time and voltage dependence of sodium and potassium conductances
• Use gate model to fit gating variables• Use equations from gate model to determine the
voltage dependent transition rates
The complete HH model
Success of HH model
• 150 years of animal electricity problem solved; in terms of a quantitative description of the process of generation of an action potential
• Correct form of experimentally observed action potential shape (on average 8 hours per 5 ms of the solution)
• Predicted the speed of action potential propagation correctly (we haven’t talked about this in the class)
Process of action potential generation