BME 6938 Neurodynamics 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 properties satisfying Ohms law Nonlinear membrane Ions: Na +,K +,Ca 2+,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 in The cable equation

Voltage Clamp: Eliminate capacitive current

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