Download The Physics of the Brain

Synaptic transmission
• Presynaptic release of neurotransmitter
• Quantal analysis
• Postsynaptic receptors
• Single channel transmission
• Models of AMPA and NMDA receptors
• Analysis of two state models
• Realistic models
Synaptic transmission:
CNS synapse
PNS synapse
Neuromuscular junction
Much of what we
know comes from the
more accessible large
synapses of the
neuromuscular
junction.
This synapse never
shows failures.
Different sizes and shapes
I. Presynaptic release
II. Postsynaptic, channel
openings.
I. Presynaptic release: The Quantal Hypothesis
I. Presynaptic release
A single spontaneous
release event – mini.
Mini amplitudes, recorded
postsynaptically are
variable.
Assumption: minis result
from a release of a single
‘quanta’.
The variability can come
from recording noise or
from variability in
quantal size.
Quanta = vesicle
A single mini
Induced release is
multi-quantal
Statistics of the quantal hypothesis:
•N available vesicles
•Pr- prob. Of release
Binomial statistics:
P( K | N )  ( Pr ) (1  Pr )
K
N K
 N 
K
Binomial statistics: Examples
N

P( K | N )  ( Pr ) (1  Pr )  K 
 
K
K
•N available vesicles
•Pr- prob. Of release
mean:
 K  Pr  N
variance:
  N  Pr (1  Pr )
2
Note – in real data, the
variance is larger
Example of cortical quantal release
Yoshimura Y, Kimura F, Tsumoto T, 1999
Short term synaptic dynamics:
depression
facilitation
Synaptic depression:
• Nr- vesicles available for release.
• Pr- probability of release.
• Upon a release event NrPr of the vesicles are
moved to another pool, not immediately
available (Nu).
• Used vesicles are recycled back to available
pool, with a time constant τu
dN r
  Pr N r (t  ti )  N u /  u
dt
N u  N r  NT
Nr
Nu
1/τu
Therefore:
dN r
  Pr N r (t  ti )  ( NT  N r ) /  u
dt
And for many AP’s:
dN r
  Pr N r   (t  ti )  ( NT  N r ) /  u
i
dt
Show examples of short term depression.
How might facilitation work?
II. Postsynaptic, channel openings.
There are two major types of excitatory glutamate
receptors in the CNS:
•AMPA receptors
And
• NMDA receptors
Openings, look like:
but actually
Openings, look like:
How do we model this?




Nr 




  [Glu]




  [Glu]

dN s
  s (Glu )  ( N r  N s )   s N s
dt
How do we model this?
A simple option:
dPs
  s (Glu )  (1  Ps )   s Ps
dt
Assume for simplicity that:
 s (Glu)  k  [Glu ]
 s  constent
Furthermore, that glutamate is briefly at a high
value Gmax and then goes back to zero.
dPs
  s (Glu )  (1  Ps )   s Ps
dt
Assume for simplicity that:
 s (Glu)  k  [Glu ]
 s  constent
Examine two extreme cases:
1) Rising phase, kGmax>>βs:
dPs
 k [Glu ]  (1  Ps )
dt
Ps (t )  (k [Glu ]  Ps (0))(1  exp( t  k [Glu]))  Ps (0)
Rising phase, time constant= 1/(k[Glu])
Ps (t )  (kGmax  Ps (0))(1  exp( t  k[Glu]))  Ps (0)
τrise
Where the time constant,
τrise = 1/(k[Glu])
2) Falling phase, [Glu]=0:
dPs
   s Ps
dt
Ps (t )  Ps (max)  exp(   s t )
rising phase
combined
Simple algebraic form of synaptic conductance:
Ps  Pmax B(exp( t / 1 )  exp( t /  2 ))
Where B is a normalization constant, and τ1 > τ2 is
the fall time.
Or the even simpler ‘alpha’ function:
Ps 
Pmax t
s
which peaks at t= τs
exp( t /  s )
Variability of synaptic conductance
through N receptors
(do on board)
A more realistic model of an AMPA receptor
Closed
Bound 1
K1[Glu]
K-1
Open
Bound 2
K2[Glu]
K3
K-2
K-3
Kd
K-d
Desensitized 1
Markov model as in Lester and Jahr, (1992), Franks et. al.
(2003).
NMDA receptors are also voltage dependent:
Jahr and Stevens; 90


[ Mg ]
GNMDA  1 
exp( V / 16.13) 
 3.57mM

2
1
Can this also be done with a dynamical equation?
Why is the use this algebraic form justified?
NMDA model is both ligand and voltage dependent
Homework 4.
a.Implement a 2 state, stochastic, receptor
  [Glu]

Assume α=1, β=0.1, and glue is 1 between times 1 and 2.
Run this stochastic model many times from time 0 to 30, show
the average probability of being in an open state
(proportional to current).
b. Implement using an ODE a model to calculate the average
current, compare to a. and to analytical curve
c. Implement using an ODE the following 5-state
receptor:
Closed
Bound 1
K1[Glu]
K2[Glu]
K-1
K-2
Bound 2
K3
Open
K-3
Kd
K-d
Desensitized 1
K1=13;
K2=13;
K3=2.7;
Kd=0.9
[mM/msec];
[mM/msec];
[1/msec];
[1/msec];
K-1=5.9*(10^(-3)); [1/ms]
K-2=86; [1/msec]
K-3=0.2; [ 1/msec]
K-d=0.9
Assume there are two pulses of [Glu]= ?, for a duration of 0.2 ms
each, 10 ms apart.
Show the resulting currents
Summary