Lecture 4: more ion channels and their functions Na + channels: persistent K + channels: A current, slowly inactivating current, Ca-dependent K currents I C, I AHP Ca 2+ channels: low-threshold I T and high- threshold I L, non-ohmic currents Refs: Dayan and Abbott, Ch 6; Gerstner and Kistler, Sect.2.3, T F Weiss. Cellular Biophysics (MIT Press) Ch 7.

General formalism: ohmic channels General equation

General formalism: ohmic channels General equation Currents have form

General formalism: ohmic channels General equation Currents have form m : activating variables h : inactivating variables

General formalism: ohmic channels General equation Currents have form m : activating variables h : inactivating variables HH Na channel:

Persistent (noninactivating) Na channel

No h !

Persistent (noninactivating) Na channel No h !

Persistent (noninactivating) Na channel No h ! Increases neuronal excitability

K channels: “A currents” (same form as HH Na channel)

K channels: “A currents” (same form as HH Na channel) fast slow-inactivating current

K channels: “A currents” (same form as HH Na channel) fast slow-inactivating current 2 kinds of each

Effect of A currents  h ~ ms

Effect of A currents  h ~ ms Opposite direction from Na current: hyperpolarizes membrane

Effect of A currents  h ~ ms Opposite direction from Na current: hyperpolarizes membrane Slows spike initiation: have to wait for I A to inactivate:

Effect of A currents  h ~ ms Opposite direction from Na current: hyperpolarizes membrane Slows spike initiation: have to wait for I A to inactivate:

Type I and Type II neurons Type I: arbitrarily slow rate possible (fx with A current) Type II: minimum firing rate >0 (fx Standard HH)

Ca 2+ -dependent K conductances (1): I C

(persistent)

Ca 2+ -dependent K conductances (1): I C (persistent)

Ca 2+ -dependent K conductances (1): I C (persistent)

Ca 2+ -dependent K conductances (1): I C (persistent) Activation is [Ca 2+ ] -dependent

Ca 2+ -dependent K conductances (1): I C [Ca 2+ ] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0  mol/l (persistent) Activation is [Ca 2+ ] -dependent

Ca 2+ -dependent K conductances (1): I C [Ca 2+ ] = 0.1, 0,2, 0.5, 1.0, 2.0, 5.0  mol/l Contributes to repolarization after spikes (persistent) Activation is [Ca 2+ ] -dependent

Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger  Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m  slows down spiking Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ -dependent K conductances (2): I AHP Ca 2+ enters (through other channels) during action potentials Each spike  bigger , bigger m  slows down spiking Slow, no voltage dependence! After-hyperpolarization current

Ca 2+ currents (1): low-threshold I T

(ohmic approximation here, but see later)

Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later)

Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”)

Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”) Consequences: (1) “Post-inhibitory rebound”; fires “Ca spike” on release from hyperpolarization

Ca 2+ currents (1): low-threshold I T (ohmic approximation here, but see later) Closed at rest because h nearly 0 (channel is “inactivated”) unlike HH Na channel, which is closed because m nearly 0 (channel is “not activated”) Consequences: (1) “Post-inhibitory rebound”; fires “Ca spike” on release from hyperpolarization (2) Ca spikes can lead to Na spikes

Ca 2+ currents (2): high-threshold I L in ohmic approximation

Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation

Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential

Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential This activates Ca-dependent K current

Ca 2+ currents (2): high-threshold I L Persistent: in ohmic approximation Lets in some Ca 2+ with each action potential This activates Ca-dependent K current Ca 2+ dynamics:

Non-ohmic Ca currents Current through membrane:

Non-ohmic Ca currents Current through membrane: Diffusive part:  = ion density

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant  = ion density

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential Total current:

Non-ohmic Ca currents Current through membrane: Diffusive part: diffusion constant Drift in field:  = ion density v = velocity  = mobility, F = force z = valence, e = proton charge, V = electrostatic potential Total current: Nernst-Planck equation

Can also be written

Nernst-Planck equation Can also be written using Einstein relation

Nernst-Planck equation Can also be written using Einstein relation or

Nernst-Planck equation Can also be written using Einstein relation or where

Nernst-Planck equation Can also be written using Einstein relation or where is the electrochemical potential

Steady state: J = const Nernst-Planck equation:

Steady state: J = const Nernst-Planck equation: Use integrating factor

Steady state: J = const Nernst-Planck equation: Use integrating factor 

Steady state: J = const Nernst-Planck equation: Use integrating factor  Integrate from x 0 to x 1 :

Steady state: J = const Nernst-Planck equation: Use integrating factor  Integrate from x 0 to x 1 :

Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness

Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d

Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d

Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d Result:

Goldman-Hodgkin-Katz equation: assume constant field in membrane V = membrane potential, d = membrane thickness can integrate denominator x 1 = 0, x 2 = d Result: vanishes at reversal potential, by definition

Ohmic limit Using i.e.,

Ohmic limit Using i.e., 

Ohmic limit Using i.e.,  Now expand in V-V r :

Ohmic limit Using i.e.,  Now expand in V-V r :