Neurophysics
- part 1 -
Adrian Negrean
adrian.negrean@cncr.vu.nl
Contents
1. Aim of this class
2. A first order approximation of neuronal biophysics
1. Introduction
2. Electro-chemical properties of neurons
3. Ion channels and the Action Potential
4. The Hodgkin-Huxley model
Introduction
(4)
(3)
1) Cell body
2) Axon
(5)
(1)
3) Apical dendrite
4) Basal dendrite
5) Synapses
(2)
histological staining of a single neuron in a rat brain (photo
credits: Cristiaan de Kock, CNCR, VU, Amsterdam)
Let’s watch a brief video clip for a quick check of what you
(might) know about neurons:
http://www.youtube.com/watch?v=DF04XPBj5uc&feature=related
• Neurons connect to each other via synapses
• Fast communication (a few milliseconds) occurs through Action Potentials
• Action Potentials release neurotransmitters at the synapse
• Neurotransmitters bind to “special gates” on the other side of the synapse and let
in a “flood of charged particles” which start up a new electrical signal in the
receiving neuron
• A bunch of neurons forming intricate connections allows us to think imaginatively
Q: How is this possible ? physically speaking ?!
• There is a great diversity of neurons, and
people are still struggling to classify them
Electro-chemical properties of neurons
• Membrane capacitance
“ parallel plate
capacitor “
C
 0 A
d
• Ion-channels:
1) Leak-channels
2) Voltage-gated ion-channels
3) Ligand-gated ion-channels
4) Metabotropic ion-channels
Structure of a voltage-gated potassium
selective ion-channel from the Kv1.2 gene
• Electrical circuit of a simple cell having:
1) Capacitance C
2) Leak ion-channels R
3) Membrane potential Vm
…keeping in mind that there
are ions instead of electrons
• Ions and electrochemical potentials:
A) A Cl- semi-permeable membrane separates a salty solution from
water
B) As time passes, Cl- diffuse, creating a potential difference
A
B
E=0
Na+ClNa+ClNa+Cl-
E=ECl
water
Cl- permeable
membrane
Na+ClNa+
Na+
ClCl-
• Nernst’s equation for equilibrium potentials:
Eion
 [ion ]out
RT

 ln
zF
 [ion ]in



where z is the valence of the ion, R is the gas constant (8.315 J K-1mol-1), T is the
temperature (K), F is Faraday’s constant (96.485 C mol-1) and [ion]o and [ion]i are the
concentrations of the ion inside and outside of the cell respectively
The membrane potential of a simple cell can be calculated using this
formula if the membrane is permeable only to one ion type.
• Goldman-Hodgkin-Katz equation for equilibrium potentials:
(generalization of Nernst’s equation for membranes permeable to multiple
ions simultaneously)
  
  
 
 
  
   
 pK K  o  p Na Na  o  pCl Cl  i
RT
Vm 
 ln



F
p
K

p
Na

p
Cl
i
i
o
Na
Cl
 K
for example, the membrane relative permeability coefficients for the Squid
Giant Axon are pK : pNa : pCl = 1.00 : 0.04 : 0.45 when the axon is at rest.
…so if you know the ion concentrations
inside and outside the cell at rest
…and the relative permeability ratios
…then using the GHK equation you can
calculate the resting membrane potential of
the cell
but the simplest thing to do in
practice is to just measure it
Intracellular and extracellular concentrations of
different ions (millimoles) given in parentheses
for a typical mammalian neuron and their Nernst
equilibrium potentials (mV).
• Ion-channels:
- membrane-bound
proteins
- conduct ions across
the membrane
- are selective for
certain ions
- they open/close in
response to a wide
range of stimuli:
a) electrical
b) mechanical
c) chemical
d) thermal
e) optical
f) intracellular
• Ion-channel gating is a stochastic process
example: an Ohmic leak channel
C: Gramicidin A peptide has been added to a
phospholipid bilayer membrane to form transmembrane channels that allow passage of ions.
A: The formation of functional Gramicidin A
channels can be seen as random step-increases
in current when a potential difference is applied
to the membrane. B: The size of the current
steps is related to the applied potential through
Ohm’s law.
• when describing Ohmic single channel leak currents, the reversal potential
has to be also taken into account:
single channel
current
membrane potential
I L  g L E  Erev 
single channel
conductance
reversal potential for
the ions involved
• for the great majority of ion-channels, the single channel conductance is
not constant but depends on the membrane potential
voltage-gated ion-channels
Q: So what are voltage-gated ion-channels good for ?
A: Action Potentials, among other many interesting examples
watch video:
The nerve impulse
An Action Potential propagates down the axon, and causes the release of a
neurotransmitter at the synaptic cleft
watch video:
Action Potential propagation
Finally the released neurotransmitter binds to ion-channels in the postsynaptic neuron, and depolarizes the cell.
watch video:
Synaptic transmission
Let’s have a closer look at what the ion-channels do during an Action Potential:
watch video:
Ion-Channels involved in the Action Potential
main points:
1) Voltage-gated ion-channels have gates
2) These gates open/close with different
speeds
3) The opening / closing of ion-channels is
actually a change in the coformational state of
the protein
• Single-channel kinetics involved in AP production
K+ channels
- the opening of the channel requires 4 independent subunits to
change conformation
PK  n
probability for
channel to be open
k
k=4
probability of a subunit to
change conformation
- the subunit open probability n is related to the membrane potential through
a first order kinetic scheme
closed / open subunit conformation
voltage-dependent
transition rates
- in practice the voltage-dependent transition rates are fitted to measured
data, still in this case, thermodynamic arguments give a good result
here’s the thermodynamic argument:
1) the subunit contains a charged domain q that couples to the
transmembrane electric field E=V/d
2) a movement of the subunit means that a fraction Bα from the charge
q moved within the electric field E doing a work of q BαV
3) Boltzmann statistics says that the probability to make a transition to
a state of higher energy, qBαV is proportional to:
temperature
exp(qBV / k BT )
Boltzmann’s constant
energy separating the two states
thus
n (V )  A exp(qBV / kBT )
(in Chemistry this kind of equation is known as
Arrhenius’s law describing reaction rates)
- a similar equation can be written also for the reverse transition rate
Na+ channels
k=3
PNa  m k h
- in addition to three activation subunits m they also have an inactivation
subunit h that closes the channel after a while, even if the activation subunits
are open
- same thermodynamic arguments as for the K+ channels
The Hodgkin-Huxley model of AP’s
- describes the phenomenon of Action Potential generation in neurons
- the model was applied to the Squid Giant Axon and later generalized to
other neurons
(1)
(2)
(3)
(4)
time
(5)
using Kirchhoff’s laws:
(the dot is a time derivative)
I Na
IK
IL



 
 

CV  I inj  g K n4 (V  EK )  g Na m3h(V  ENa )  g L (V  EL )
capacitor current
- now we have to add the single-channel gating kinetics for both Na+ and K+
as described before:
n  n (V )(1  n)  n (V )n
 n (V )  0.01
10  V
 10  V 
exp
 1
10


  m (V )(1  m)  m (V )m
m
 n (V )  0.125exp
h  h (V )(1  h)  h (V )h
 m (V )  0.1
- and specify also the channel
conductances and reversal
potentials
 V 

 80 
25  V
 25  V 
exp
 1
 10 
 V 

 18 
 m (V )  4 exp
 V 

20


 h (V )  0.07exp
 h (V ) 
1
 30  V 
exp
 1
10


and the result is: