Signal Propagation. Review: About external stimulation of cells:. The negative electrode ( cathode ) is the stimulator. At rest, the outside of the cell has a positive charge (there is an equal amount of negative charges across the inside of the membrane).
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The negative electrode (cathode) is the stimulator.
How would you calculate AP velocity?
Essentially, this shows electrotonic propagation between Na+ gates and regeneration at each gate.
The diagram above overemphasizes the distances and decay!
Space Constant-- a measure of decay over distance.
2. Time Constant-- a measure of depolarization time
This is a negative exponential decay. Mathematically:
x is the distance from some point of interest.
λis the decay (rate) constant, -- here the space constant. It is the distance to decay to some value (to be explained below)Space Constant -- has to do with the distance over which a passive response propagates.
= the distance over which a signal decays some amount.
This distance is defined by setting the variable distance x equal to the space constant (i.e., x = λ )and then solving the equation:
Ex = Eo * e -(x/λ) = Eo * e -1 ≈ 0.37 * Eo
Thus, the space constant is the distance over which the potential decays to ≈ 37% of its original value.
Resistances in and out of the cell and membrane resistance are the main determinants.
The space constant is proportional to the harmonic and geometric means of these resistances:
And now take the derivative with respect to time to get the rate of change of the membrane potential:
im is related to resistance (for a given E) and
Cmis determined by membrane characteristics.
We have seen these two factors previously in something called the time constant.
What is this constant precisely?
Defined: the amount of time it takes to charge or discharge the membrane capacitance by 63%
Importance -- obviously this is crucial to conducting a regenerating potential because voltage-gated Na+ channels can only open after the membrane has depolarized to above their threshold
t = R*C
Without getting into why, the measure of resistance over some distance is the geometric mean of membrane and length resistance:
If we look for an expression that tells us how long it takes for a given voltage change, we can start with:
Let us determine the voltage change we will get if t = RC:
Thus, t is the time required for 63% change in Em.
How could RC = t -- don’t they have different units?
Obviously they do -- its an equation! But let's see:
R has units of (J*s)/coulombs2 and C has units of coulombs2 / J
Therefore R*C = J*s / coulombs2 *C has units of coulombs2 / J = s
membrane SA = 2 * r * π* L
So doubling the radius doubles the membrane SA for a unit of length (Which we will assume to be very small, dL.)
X-sectional area = r2 *π
Doubling the radius increases the x-sectional area by 4!
(more R in parallel)
So, if the radius doubles, A doubles,Gm doubles and Rm is halved.
(a lot more G in parallel)
thus if the radius doubles,x-sectional area doubles, Gi increases by 4-fold and therefore Ri decreases to 1/4.
if the radius doubles, Cmdoubles.
On axons of vertebrates -- but certainly not on all axons!
So, for series capacitance:
When comparing cells of the same size with and without myelin:
Cm = 1/50 Cm
Myelination adds Rm in series without changing Ri and Ro.
Change in time constant
Clearly an increase!
Double radius, 0.25Ri, Ro is same, Rm cut in half
What does this mean? Why focus on dendrites and soma?