BME 6938
Neurodynamics
Instructor: Dr Sachin S Talathi
Cable Equation: Transient Solution
~
dV (X,T) ¶ 2V (X,T)
+ V (X,T) = i ext (X,T)
2
dT
¶X
Green’s Function G(X,T) ~for infinite cable: solution of above equation for:
i ext = (a /2)d (X).d (T)
With initial condition: V(X,0) = (a /2)d (X) and Boundary condition:
G(X) = C.Q(T)e
æ
X2ö
-çT +
4T ÷ø
è
C =a
p
t
General Solution to Cable Equation:
t
V (X) =
¥
~
ò dT' ò dX'G(X - X';T - T') i
-¥
-¥
Hint: Use the formula:
ext
(X - X';T - T')
Graphical representation
Two points worth noticing:
1.The potential is described as a
Gaussian function centered at the site
of current injection that broadens and
shrinks in amplitude with time
2. Membrane potential measured from
further location reaches its maximum
value at later time
Ralls Model-Equivalent cylinder
Ralls Assumptions:
1. All membrane properties are the same
2. All terminal branches end with same boundry condition
3. All terminal branches end at same electrotonic distance
from soma
4. At every branch point 3/2 rule is obeyed
5. Any dendritic input must be delivered proportionally to
all branches at a given electrotonic distance
Practical Situation:
Choose L=average eletrotonic length of all
dendritic branches and
Also read the classic papers by Rall (posted on the web: Rall_1959, Rall_1962, Rall_1973)
Synaptic Integration
Model
for
current
injection into neuron
through synapse-alpha
function
Imp Points to Note:
1. Distal synaptic input produces measurable signals in the soma
2. Obvious fact, the measured EPSP at soma due to stimulation at distal input is smaller as
compared to closer to soma
3. Rise time to peaks is progressively delayed for inputs at increasing distance from soma
4. Decay times of all the inputs is the same, ie. Potential cannot decay slower than membrane time
constant
We know now that dendrites are not passive, what is the current view on dendritic integration.
Read the 2 review articles (Magee_2000, London_2005) posted on the website
Nonlinear Membrane:
dV( x,t)
d ¶ 2V( x,t)
CM
+ Iion (V( x,t),t) =
+ i ext ( x,t)
2
dt
4RA ¶x
I M (V( x,t),t) =
å
I iion
i
Ions: Na+, K+, Ca2+, Cl-
Re-visiting Goldman Equation: The idea
of time dependent conductance
zF
V=
Vm
RT
V Na +
zF
=
VNa +
RT
Na+ reversal potential
(Nernst Equilibrium Potential)
Membrane voltage and time dependent ion channel conductance
First order approximation
The 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
I have posted the original paper by HH where they proposed to above model and the experiments they conducted to develop the
now famous HH model for neuron membrane dynamics (HH_1952). I urge all of you to read this classic work; that led them to
receive Nobel Prize in Medicine (The only Noble Prize so far that the field of computational neuroscience have produced)
Kinetics of Gate Transition



Let P represent the fraction of gates within the ion
channel that are open at any given instant in time
1-P then represents the fraction that are closes at that
instant in time
If
and
are the rate constants we have
Open
P
Closed
1-P
Steady State:

Steady state implies
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 describe channel
dynamics of cell membrane HH had to determine the
following 3 things



Macro characteristics of channel type
The number and type of gates on each channel
The dependence of transition rate constants
on membrane voltage V
and
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
The 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
Current flows assumes Ohms Law
Goal: Find
and
Example: Time and Voltage dependence
of Potassium channel conductance
GK (V,t)
GNa (V,t)
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
CM
dV(x,t)
+ Iion (V(x,t),t) = i ext (x,t)
dt
Voltage Clamp: Eliminate Capacitive
Current
http://www.sinauer.com/neuroscience4e/animations3.1.html
Example of Voltage Clamp Recording
Clamped Voltage=20 mV
The sum of parts
Note the different time scale
Series of Voltage Clamp Expt
Selective blocking with pharmacological
agents
TTX: Tetrodotoxin; selectively blocks sodium channels
TEA: Tetraethylamonium; selectively blocks potassium channels
H-H experiments to test Ohms law
Foundation of Cellular Neurophysiology, Johnston and Wu
HH measurement of Na and K
conductance
Maximum
conductance
Gating variables
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:
Determining
and

Determine
and
Use the following relationship

Do empirical curve fitting to obtain

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