Signal Propagation

The negative electrode (cathode) is the stimulator. Review: About external stimulation of cells: 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 E M becomes more positive (depolarized). The opposite process occurs under the anode--hyperpolarization.

AP Recorded At 3 Positions How would you calculate AP velocity?

Propagation of Graded (non-active) Responses

Essentially, this shows electrotonic propagation between Na + gates and regeneration at each gate. The diagram above overemphasizes the distances and decay! Propagation of Active Responses

Understanding Propagation 1.Propagation can occur no faster than the time it takes to depolarize the membrane to threshold. 2.Additionally, there is the time needed for gate allosteric changes.

Propagation and the "Cable Properties" of the Membrane 1.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. λ is 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. Recall:

= 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: E x = E o * e -(x/λ) = E o * e -1 ≈ 0.37 * E o Thus, the space constant is the distance over which the potential decays to ≈ 37% of its original value. Definition 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: Determinants of the Space Constant

Thus, the rate that E M changes (the membrane polarizes or depolarizes) is: directly proportional to the membrane current and inversely proportional to the capacitance. What determines the rate that E m 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:

i m is related to resistance (for a given E) and C m is determined by membrane characteristics. 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 The Time Constant

Recall:  = R*C Without getting into why, the measure of resistance over some distance is the geometric mean of membrane and length resistance: Calculation of the Time Constant Therefore:

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  = RC: The Meaning of the Time Constant Thus,  is the time required for 63% change in E m. How could RC =  -- don’t they have different units? Obviously they do -- its an equation! But let's see: R has units of (J*s)/coulombs 2 and C has units of coulombs 2 / J Therefore R*C = J*s / coulombs 2 *C has units of coulombs 2 / J = s

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 = r 2 *π i.e., Doubling the radius increases the x-sectional area by 4 !

Cell Geometry, Doubling Radius Effect on R m and R i (more R in parallel) So, if the radius doubles, A doubles, G m doubles and R m is halved. (a lot more G in parallel) thus if the radius doubles, x-sectional area quadruples, G i increases by 4-fold and therefore R i decreases to 1/4.

Cell Geometry -- Effect on C m Since: if the radius doubles, C m doubles.

If the radius of the cell doubles: Membrane area doubles and so does C m. Membrane area doubles so R m decreases by half. Internal volume quadruples and R i is cut to Overall Effect of Doubling Radius on 

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

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). When comparing cells of the same size with and without myelin: C m = 1/50 C m So, for series capacitance:

Effect of typical myelination on the time constant Myelination adds R m in series without changing R i and R o. Change in time constant

Myelination and Change in the Space Constant? Clearly an increase!

Change in Space Constant with Changes in Geometry (e.g., dendrites and soma) Double radius, 0.25R i, R o is same, R m cut in half What does this mean? Why focus on dendrites and soma?