B1. Nervous System Physiology II: Passive Membrane Properties

Membrane Dynamics

• Neurons are excitable cells.
• The plasma membrane is a phospholipid bilayer.
  • Semipermeable: ions must cross the membrane via ion channels.
• Ion channels: membrane proteins with selective permeability for particular ions.
  • Leak channels: pores, remain open.
  • Voltage-gated ion channels: open/close in response to changes in membrane potential.
  • Ligand-gated ion channels: open in response to chemical signals binding.

Resting Membrane Potential (Vm)

• Membrane potential (Vm): difference in electrical charge between inside and outside of the cell.
  • Units: volts, mV.
  • Voltage measured as relative difference between inside and outside.
• Electrophysiology: method to measure membrane potential.
  • A recording electrode is inserted into a neuron.
  • Voltage is compared to a reference or ground electrode outside the neuron.
  • Stimulating electrodes can inject current.
• Ions that determine neuron membrane potential: $[K^+]$, $[Na^+]$ and $[Cl^-]$.
• Resting membrane potential (RMP, Vrest): Vm when a cell is at rest, not firing an action potential.
  • Steady-state: no net movement of charge.
  • RMP in most neurons = -60 to -70 mV.
• Na+-K+ ATPase maintains Vrest by building up $[Na^+]$ in ECF, $[K^+]$ in ICF.
  • 3 $Na^+$ out, 2 $K^+$ in.
  • Antiport: carrier protein that moves substances in opposite directions.
  • Active transport: $Na^+-K^+-ATPase$ hydrolyzes ATP to move ions against the concentration gradient.
• Two factors determine Vm:
  1. Electrochemical Gradient: uneven distribution of ions across the cell membrane
     • Electrical driving force: attraction and repulsion between charged particles.
     • Chemical driving force: diffusion, drives ions from a region of high concentration to low concentration due to entropy (2nd Law of Thermodynamics).
  2. Differences in membrane permeability: ability for ions to pass through the membrane
     • Membrane proteins have selective permeability: only certain ions can pass through most ion channels.
     • Permeability is determined by the number of ion channels.
• At rest, the membrane is more permeable to $K^+$, because # of $K^+$ leak channels > # of $Na^+$ leak channels.

Calculating $E_{ion}$

• Equilibrium potential ( $E_{ion}$ ): membrane potential that exactly opposes the concentration gradient.
  • Electrical and chemical forces are equal and opposite.
  • No net movement of ions.
• Electrochemical potential ($\mu_{ion}$): the sum of the electrical and chemical potentials
  • $\mu<em>{ion} = RT \ln \frac{[ion]</em>{inside}}{[ion]<em>{outside}} + zF (E</em>{inside} – E_{outside})$
  • z = ion’s charge ($K^+$ = 1, $Na^+$ = 1, $Ca^{++}$ = 2, $Cl^-$ = -1)
  • E = voltage
  • At equilibrium: $\mu_{ion} = 0$
  • $E<em>{ion} = \frac{RT}{zF} \ln \frac{[ion]</em>{outside}}{[ion]_{inside}}$ Nernst Equation (calculates the equilibrium potential)
    1. Is a voltage across the membrane
    2. There is no net flow of the ion across the membrane
• Nernst Equation: calculates equilibrium potential for a membrane permeable to one ion
  • R = gas constant
  • T = temperature in Kelvin
  • z = ion charge
  • F = Faraday constant
  • $[ion]_{out}$ = concentration of ion in ECF
  • $[ion]_{in}$ = concentration of ion in ICF
• At 37°C (body temperature):
  • $E<em>{ion} = \frac{61}{z} \log \frac{[ion]</em>{outside}}{[ion]_{inside}}$
  • Examples:
    • $E_{Cl} = -70$ mV
    • $[K^+]<em>{inside} = 145$ mM, $[K^+]</em>{outside} = 5$ mM, $E_K = -90$ mV
    • $[Na^+]<em>{inside} = 12$ mM, $[Na^+]</em>{outside} = 145$ mM, $E_{Na} = +66.5$ mV
• $F<em>{ion} = V</em>m - E_{ion}$
  • Driving forces on an ion are dependent on the difference between $V<em>m$ and $E</em>{ion}$
  • Example Values:
    • If $V<em>m$=-90 mV, $F</em>{K+} = V<em>m - E</em>{K+} = -90 mV - (-90 mV) = 0$ mV, so no net forces drive $K^+$
    • If $V<em>m$=+50 mV, $F</em>{K+} = V<em>m - E</em>{K+} = +50 mV - (-90 mV) = 140$ mV, so forces drive $K^+$ out of the cell
    • If $V<em>m$=-150 mV, $F</em>{K+} = V<em>m - E</em>{K+} = -150 mV - (-90 mV) = -60$ mV, so forces drive $K^+$ into the cell

Calculating Vm

• Vm for a membrane permeable to 1 ion is $E_{ion}$
  • $E_{K+} = -90$ mV
  • $E_{Na+} = +60$ mV
• Goldman-Hodgkin-Katz (GHK) Equation: calculates membrane potential resulting from the contribution of $E_{ions}$ of all ions, as a function of permeability
  • Permeability ~ relative ion contribution to membrane potential. Usually expressed as a ratio, e.g. $P<em>{K+}$ 1: $P</em>{Na+}$ .05 : $P_{Cl-}$ .45
• Vm is determined by combined equilibrium potentials and permeabilities of each ion
  • P = relative permeability of the membrane to the ion
  • $[ion]_{out}$ = concentration of ion in ECF
  • $[ion]_{in}$ = concentration of ion in ICF
• Number of leak ion channels determines the permeability of the membrane to specific ions in a neuron at rest
  • $P<em>{K+}: P</em>{Na+} = 20:1$
  • $E_{K+} = -90$ mV
  • $E_{Na+} = +65$ mV
• GHK Equation:
  • $V<em>m = 61 \log \frac{P</em>K [K^+]<em>{out} + P</em>{Na} [Na^+]<em>{out} + P</em>{Cl} [Cl^-]<em>{in}}{P</em>K [K^+]<em>{in} + P</em>{Na} [Na^+]<em>{in} + P</em>{Cl} [Cl^-]_{out}}$
• If the membrane is permeable to only one ion, GHK "becomes" Nernst
  *At $V<em>{rest}$ conditions *Cations going out of the cell = cations going into the cell ($I</em>K = -I_{Na}$)
  *Neuron is in steady-state (not in equilibrium for $Na^+$ or $K^+$
• Na+/K+ ATPase pumps
  • Set up and maintain the ion gradients
  • Due to asymmetry, add a small amount of (-) to Vm
  • Pumps are too slow to generate rapid repolarization

Electrical Current

• Electrical Current: ($I$), flow of electrical charge carried by an ion
  1. Measured in amperes (amps)
  2. Ion movement produces electrical signals
  3. $I$ is dependent on $F_{ion}$ and permeability Na +
• Current is dependent on $F_{ion}$
  • $F<em>{ion} = V</em>m - E_{ion}$
  • At $E_{ion}$, net $I = 0$
• $I$ is dependent on conductance/resistance
  • Conductance: (G) ease with which ions flow across the membrane
    • Units: Siemens
    • Conductance determined by # open ion channels
    • Stimuli alter permeability → ions flow with electrochemical forces
  • Resistance: (R) difficulty with which ions flow across the membrane
    • Units: ohms
    • Resistance is determined by # closed ion channels
    • Stimuli alter permeability → ions flow with electrochemical forces
    • $G = \frac{1}{R}$
• Resistance: (R) force that opposes flow
  • Inverse of conductance ($\frac{1}{G}$)
  • Units: ohms ($\Omega$)
  • In neurons:
    • Membrane resistance ($R_m$)
    • Cytoplasm resistance ($R_i$)
• Ohm’s Law: states that current flow is directly proportional to the electrical potential difference between 2 points and conductance
  • $V = IR$
  • $I = GV$
  • $V = potential difference in volts$
  • $I = current in amperes$
  • $G = conductance$
  • $R = resistance in ohms$
• Ionic Current ($I_{ion}$) number of ions (amount of charge) crossing the membrane
  1. Force ($F_{ion}$)
     • $F<em>{ion} = (V</em>m – E_{ion})$
       • $V<em>m$ far from $E</em>{ion}$: large force to move the ion
       • $V<em>m = E</em>{ion}$: no force to move the ion, no net movement, the system is in equilibrium for that ion
  2. Conductance ($G_{ion}$) similar to permeability
  3. Current Equation
     • $I<em>{ion} = G</em>{ion} (V<em>m - E</em>{ion})$
     • $I_{ion}$ requires:
       • a driving force ($F_{ion}$)
       • a pathway ($G_{ion}$)

Changes to Vm

1. Depolarization: increase in Vm
   • The membrane becomes more permeable to $Na^+$
   • Inward $I_{Na+}$ with electrochemical gradient
   • Vm above RMP
2. Repolarization and Hyperpolarization: decrease in Vm
   • The membrane becomes more permeable to $K^+$
   • Outward $I<em>{K+}$ with electrochemical gradient or inward $I</em>{Cl-}$
   • Hyperpolarize: Vm below RMP