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### 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 properties

satisfying Ohms law

Nonlinear membrane

In general a nonlinear function in voltage and time

Ions: Na+,K+,Ca2+,Cl-

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

Steady state

The transient solution can then be obtained as:

OR

Open

p

Closed

1-p

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 &

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

http://www.sinauer.com/neuroscience4e/animations3.1.html

Functional fitting to gate variable

Inactivation gate

Activation gate

m,n and h are gate variables and follow first order kinetics of the gate model

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

Estimating gate model parameters

Determine and

Use the following relationship

Do empirical curve fitting to obtain

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

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)

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