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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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Review about external stimulation of cells
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).

  • Putting a negative potential on the outside therefore draws (+) charges away from the membrane. (-) charges on the inside disperse and EM becomes more positive (depolarized).

  • The opposite process occurs under the anode--hyperpolarization.


Ap recorded at 3 positions
AP Recorded At 3 Positions

How would you calculate AP velocity?



Propagation of active responses
Propagation of Active Responses

Essentially, this shows electrotonic propagation between Na+ gates and regeneration at each gate.

The diagram above overemphasizes the distances and decay!


Understanding propagation

  • Propagation can occur no faster than the time it takes to depolarize the membrane to threshold.

  • Additionally, there is the time needed for gate allosteric changes.

Understanding Propagation


Propagation and the cable properties of the membrane

Propagation and the "Cable Properties" of the Membrane

Space Constant-- a measure of decay over distance.

2. Time Constant-- a measure of depolarization time


Space constant has to do with the distance over which a passive response propagates

Recall:

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.


Definition of the space constant
Definition of the Space Constant

= 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.


Determinants of the space constant
Determinants of the Space Constant

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:


What determines the rate that e m can change in one section of a membrane
What determines the rate that Em can change in one section of a membrane?

And now take the derivative with respect to time to get the rate of change of the membrane potential:

  • Thus, the rate that EM changes (the membrane polarizes or depolarizes) is:

    • directly proportional to the membrane current and

    • inversely proportional to the capacitance.


The time constant
The Time Constant

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


Calculation of the time constant
Calculation of the Time Constant

Recall:

t = R*C

Without getting into why, the measure of resistance over some distance is the geometric mean of membrane and length resistance:

Therefore:


The meaning of the time constant
The Meaning of the Time Constant

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


The effect of cell geometry on ap conduction velocity
The Effect of Cell Geometry On AP Conduction Velocity

membrane SA = 2 * r * π* L

i.e.,

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

i.e.,

Doubling the radius increases the x-sectional area by 4!


Cell geometry doubling radius effect on r m and r i
Cell Geometry, Doubling Radius Effect on Rm and Ri

(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.


Cell geometry effect on c m
Cell Geometry -- Effect on Cm

Since:

if the radius doubles, Cmdoubles.


Overall effect of doubling radius on t
Overall Effect of Doubling Radius on t

  • If the radius of the cell doubles:

    • Membrane area doubles and so does Cm.

    • Membrane area doubles so Rm decreases by half.

    • Internal volume quaduples and Riis cut to 0.25.


Myelin

Myelin

On axons of vertebrates -- but certainly not on all axons!


Capacitance in series
Capacitance in series

  • Myelin is essentially a bunch of capacitors in series.

    • Typically, about 50 capacitors (25 "turns", two lipid bilayers per turn)

  • Putting capacitors in series is like increasing the thickness of the dielectric. Recall that this decreases the capacitance (essentially there is less attraction of opposites between the two opposing conductors).

  • So, for series capacitance:

    When comparing cells of the same size with and without myelin:

    Cm = 1/50 Cm


    Effect of typical myelination on the time constant
    Effect of typical myelination on the time constant

    Myelination adds Rm in series without changing Ri and Ro.

    Change in time constant



    Change in space constant with changes in geometry e g dendrites and soma
    Change in Space Constant with Changes in Geometry (e.g., dendrites and soma)

    Double radius, 0.25Ri, Ro is same, Rm cut in half

    What does this mean? Why focus on dendrites and soma?