Lecture 2 Membrane potentials Ion channels Hodgkin-Huxley model References: Dayan and Abbott, Gerstner and Kistler,

Cell membranes

Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2

Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance

Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance Typical A = mm 2  C ~.1 – 1 nF

Cell membranes Lipid bilayer, 3-4 nm thick  capacitance c = C/A ~ 10 nF/mm 2 Ion channels  conductance Typical A = mm 2  C ~.1 – 1 nF Q=CV, Q= 10 9 ions  |V| ~ 65 mV

Membrane potential Fixed potential  concentration gradient

Membrane potential Fixed potential  concentration gradient Concentration difference  Potential difference Concentration difference maintained by ion pumps, which are transmembrane proteins

Nernst potential Concentration ratio for a specific ion (inside/outside):  = 1/k B T  ( q = proton charge, z = ionic charge in units of q )

Nernst potential Concentration ratio for a specific ion (inside/outside):  = 1/k B T  ( q = proton charge, z = ionic charge in units of q ) No flow at this potential difference Called Nernst potential or reversal potential for that ion

Reversal potentials Note: V T = k B T/q = (for chemists) RT/F ~ 25 mv Some reversal potentials: K: mV Na: +50 mV Cl: mV Ca: 150 mV Rest potential: ~ -65 mV ~2.5 V T

Effective circuit model for cell membrane

( C, g i, I ext all per unit area) (“point model”: ignores spatial structure)

Effective circuit model for cell membrane ( C, g i, I ext all per unit area) (“point model”: ignores spatial structure) g i can depend on V, Ca concentration, synaptic transmitter binding, …

Ohmic model One g i = g = const or

Ohmic model One g i = g = const or membrane time const

Ohmic model One g i = g = const or Start at rest: V= V 0 at t=0 membrane time const

Ohmic model One g i = g = const or Final state: Start at rest: V= V 0 at t=0 membrane time const

Ohmic model One g i = g = const or Final state: Start at rest: V= V 0 at t=0 Solution: membrane time const

channels are stochastic

Many channels: effective g = g open * (prob to be open) * N

Voltage-dependent channels

K channel Open probability: 4 independent, equivalent, conformational changes

K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation:

K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation: Rearrange:

K channel Open probability: 4 independent, equivalent, conformational changes Kinetic equation: Rearrange: relaxation time: asymptotic value

Thermal rates: u 1, u 2 : barriers

Thermal rates: u 1, u 2 : barriers Assume linear in V :

Thermal rates: u 1, u 2 : barriers Assume linear in V : 

Thermal rates: u 1, u 2 : barriers Assume linear in V :  Simple model: a n =b n, c 1 =c 2

Thermal rates: u 1, u 2 : barriers Assume linear in V :  Simple model: a n =b n, c 1 =c 2 Similarly,

Hodgkin-Huxley K channel

(solid: exponential model for both  and  Dashed: HH fit)

Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture)

Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits:

Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits:

Transient conductance: HH Na channel 4 independent conformational changes, 3 alike, 1 different (see picture) HH fits: m is fast (~.5 ms) h,n are slow (~5 ms)

Hodgkin-Huxley model

Parameters: g L = mS/mm 2 g K = 0.36 mS/mm 2 g Na = 1.2 ms/mm 2 V L = mV V K = -77 V Na = 50 mV

Spike generation Current flows in, raises V  m increases (h slower to react)  g Na increases  more Na current flows in  …  V rises rapidly toward V Na Then h starts to decrease  g Na shrinks  V falls, aided by n opening for K current Overshoot, recovery

Spike generation Current flows in, raises V  m increases (h slower to react)  g Na increases  more Na current flows in  …  V rises rapidly toward V Na Then h starts to decrease  g Na shrinks  V falls, aided by n opening for K current Overshoot, recovery Threshold effect

Spike generation (2)

Regular firing, rate vs I ext

Step increase in current

Noisy input current, refractoriness