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Home , Dendrite, Electrotonic potential, Resting potential, Rm (Unix)

Passive electrical properties of the

Ricardo C. Araneda [email protected]/4-4539 Fall 2004

This lecture take notes from the following books The following books were used for the lecture:

“From neuron to brain” Nicholls et al., 3rd .

“ channels of excitable membranes” B. Hille, 3rd Ed.

“Fundamental ” Zigmond et al.

“Principles of neuroscience” Kandel et al., 4th Ed. Review of basic concepts Experiment 1

Cell membrane ◆Concentration Gradient NaCl NaCl 10 mM 100 mM in out Measure

◆Concentration Gradient NaCl NaCl 10 mM 100 mM in out = ?? Measure membrane potential

◆Concentration Gradient NaCl NaCl ◆ membrane is equally 10 mM 100 mM permeable in out to both Na and Cl

V= ?? Membrane is selectively permeable to

in out + Na + Na _ _ _ Cl Cl Cl Na + _ + Cl Concentration Gradient Na + Na _ Cl _ + Cl Na

Na + Na + _ _ _ Cl Cl Cl Na + Membrane is selectively permeable to Sodium

in out + Na + Na _ _ _ Cl Cl Cl Na + _ + Cl Concentration Gradient Na

Na + _ Cl _ + Cl Na

Na + Na + _ _ _ Cl Cl Cl Na + Membrane is selectively permeable to Sodium

in out + Na + Na _ _ _ Cl Cl Cl Na + _ + Cl Concentration Gradient Na

Cl Excess of Na+ - Excess of positive negative ions Potential difference _ + Cl Na

Na + Na + _ _ _ Cl Cl Cl Na + Membrane potential

◆Concentration Gradient across the membrane

◆Membrane is selectively permeable to ions

NaCl NaCl V = RT log [out] F [in] 10 mM 100 mM

in out VNa = + 58

VCl = - 58 mV Membranes as capacitors

Internal conducting External conducting solution solution (ions) (ions)

Thin insulating layer (membrane, 4nm) Membranes as Resistors

Internal conducting External conducting solution solution (ions) (ions)

Ion channels Voltage-gated, NT- gated etc. Electrical model of the

Ion channel Membrane RESISTOR CAPACITOR Capacitor

+ Q + _ A= Area _Q + + + _ _ + + _ _ _ + d = distance of plate separation Capacitance

C = Q/V / or Farads (F)

electrostatic C = eο*A/d eο permittivity A = C d = C Capacitor rubber membrane

Water pressure Release of pressure

Small Capacitance Capacitors in parallel add larger Capacitance Resistance

A = Area

l = length Ohm’s law

R = V/I Ohms (Ω) R = r*l/A r resistivity

l = R A = R

For the same current, a larger R produces larger V Resistors

R1 For ion channels is better to think in terms of conductance

R1 = 1/g1

As the # of Rs in parallel increases RT decreases! R1 1/RT = 1/R1 +1/R2

More (open) channels in the R2 membrane conductance gT = g1 +g2

R1 R2 RT = R1 + R2

Long, thin parts of a neuron have large resistance! Some useful equations

Current I = /second Amperes (A)

Ohm’s law V= IR

Capacitance C = Q/V Coulombs/ (F)

Voltage across capacitor V = Q/C

Changing the voltage ∆V = ∆Q/C in a capacitor

We change the charge Ic = ∆Q/∆t by passing current

The change in V depends ∆V = Ic*∆t/C on the duration of Ic Also remember...

Current likes to flow through the path with resistance

R = 100 Ω

I2 IT I1

R = 1 Ω And

IT = I1 + I2 Electrical model of the cell membrane

Capacitive Ionic membrane membrane current current Ic Ii

Membrane current Im = Ii + Ic Im Effects of passing current on circuits containing R and C

V changes instantaneously with I

V changes linearly in with I

V changes exponentially with a time constant = RC RC circuits

37% of Vo

(E and V are the same)

For a rising exponential -t/RC V = VO*(1-e ) Experiment 2 Passing current and recording the membrane potential from a paramecium current-passing electrode Negative current makes the membrane potential more negative recording electrode hyperpolarization

Positive current makes the membrane potential more positive “” Linear relationship between current and voltage

20 current-passing electrode

recording electrode 15 )

(mV 10 e g a t l o

V 5

0 0.05 0.1 0.15 0.2 Current (nA) “electrotonic potential” V= I*Rin

Input Resistance Rin = 100 MΩ Specific membrane resistance cross section of a cell To compare cell with different sizes The specific membrane resistance (resistance per area) 2 RM = Ω*cm 2 depends on the # of channels per cm

More channels RM smaller

For a spherical cell

2 Rin = RM /4πa a = radius

Rin determines how much the cell depolarizes in response to a steady current 2 Example same RM = 2000 Ω*cm

a a

Cell diameter is 5 µm

your numbers here... Cell diameter is 50 µm -4 a = 25 µm = 25*10 cm Rin = 637 MΩ 2 Rin = RM/4πa Rin is larger in a 2 -4 2 smaller cell Rin = 2000 Ω*cm /4π(25*10 cm)

Rin = 25 MΩ Specific membrane capacitance of biological membranes

2 CM = 1 µF/cm For a cell -80 mV how many ions is this?

CM = Q/V -6 Q = 10 C/V* 0.08 V -8 2 Q = 8*10 C/cm 5 Faraday constant ≈ 10 Coulombs/ 23 Avogadro’s number = 6.02*10 mole-1 11 2 Then this is 4.8*10 ions/cm Is this a lot??? Let’s assume the cell is 50 µm in diameter -4 a = 25 µm = 25*10 cm 2 50 surface of sphere A = 4πa -5 2 A = 7.85*10 cm

11 2 7 4.8*10 ions in 1 cm So total is 4*10 ions

-8 3 The volume of this cell is 6.55*10 cm

-6 Then this number of ions is ~ 10 M If KCl inside is 120 mM this means that only ~1/120,000 ions is in excess! Large cell Small cell large capacitance small capacitance

“CM” is the same (same membrane)

For a spherical cell, the input capacitance

2 Cin = CM*4πa a = radius

More charge (current) is required to change the voltage across a large cell In summary

2 Rin = RM /4πa 2 τ = Rin*CM Cin = CM*4πa The product of input Capacitance and Resistance (τ) determines the time it takes change the potential Notice that (τ) is not affected by “a” Real

How are signals affected by the passive properties of the membrane ? Experiment 3

Local potentials are graded

Almost no potential change is observed, why? Current in and

Current pulse

rm

Section of or of determined length (x). In this case 1 cm. ra

ra axial resistance (Ω/cm) rm membrane resistance (Ω*cm) Axial resistance increases with distance (x)

Total axial resistance

rX = ra*x (x = 4) ra1 ra2 ra3 ra4 (remember RT = R1 + R2)

Near the site of injection, the current flows

VO through rm (less resistance)

Then VO = Im * rm What is the value of V at increasing distances from the site of current injection?

voltage-recording current-passing electrode electrode -x/λ V = VO*e

λ is the length constant

λ =√ (rm/ra) (cm) 37% (1/e) Increasing m increases λ λ r Decreasing ra increases λ

i.e. V is closer to VO For 1 cm of cytoplasm (dendrites or axon)

3 r (Ω*cm) resistive property of 1 cm of cytoplasm (dendrites or axon)

2 ra = r/(pa ) rm = Rm/(2pa)

1 cm

λ =√ (rm/ra) λ =√R m*a If Rm and r are 2r constant

λ =√ K a The length constant is proportional to the square root of the radius of the process

For neurons is usually 0.1 to 1 mm Effect of length constant

For a dendrite or axon with the same diameter as the length constant increases the potential decreases less with distance. Effect of diameter

For a dendrite or axon with increasing diameters the length constant increases and the potential decreases less with distance. The passive properties of membranes and axon diameter affect the speed of conduction of action potentials

speed of conduction of action potentials is inversely related to ra*cm

speed of conduction is increased by increasing the diameter of the axon decreases ra

The giant axon of the squid 1 mm ! Myelination, the alternative to increasing the diameter of the axon.

Glial cell wrap around axons many times (20-160 times) this like adding 320 membranes (in series). This increases Rm and decreases Cm