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). 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 How would you calculate AP velocity?

Propagation of Graded (non-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.

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. λ is the decay (rate) constant, -- here the space constant. It is the distance to decay to some value (to be explained below)

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 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 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: Note that when many channels are opened and the membrane is far displaced from Eion for those channels, I is large as is dE/dt 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 im is related to resistance (for a given E) and Cm 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

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 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: How could RC = t -- don’t they have different units? R has units of (J*s)/coulombs2 and C has units of coulombs2 / J Therefore R*C has units of J*s / coulombs2 and C has units of coulombs2 / J = s 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 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 Rm and Ri (more R in parallel) So, if the radius doubles, A doubles, Gm doubles (Rm is halved). (a lot more G in parallel) thus if the radius doubles, x-sectional area quadruples, Gi increases by 4-fold (Ri decreases to ¼).

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

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 quadruples and Ri is cut to 0.25.

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

Capacitance in series Cm = 1/50 Cm So, for series capacitance: 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 Myelination adds Rm in series without changing Ri and Ro. Change in time constant

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

Change in Space Constant with Changes in Geometry (e. g 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?