Transcript Lecture 6: Stochastic models of channels, synapses

Lecture 7: Stochastic models of
channels, synapses
References:
Dayan & Abbott, Sects 5.7, 5.8
Gerstner & Kistler, Sect 2.4
C Koch, Biophysics of Computation Chs 4,8 (13)
A Destexhe, Z Mainen & T J Sejnowski, Ch 1 in
Methods in Neuronal Modeling, 2nd ed, C Koch and I
Segev, eds (MIT Press)
Stochastic models of channels
Single channels are stochastic, described by kinetic equations for
probabilities of being in different states
Stochastic models of channels
Single channels are stochastic, described by kinetic equations for
probabilities of being in different states
Example: the HH K channel:
HH K channel
Kinetic equations:
dp1
  n p2  4 n p1
dt
dp2
 4 n p1  2  n p3  (  n  3 n ) p2
dt
dp3
 3 n p2  3 n p4  (2  n  2 n ) p3
dt
dp4
 2 n p3  4 n p5  (3 n   n ) p4
dt
dp5
  n p4  4  n p5
dt
HH K channel
Kinetic equations:
dp1
  n p2  4 n p1
dt
dp2
 4 n p1  2  n p3  (  n  3 n ) p2
dt
dp3
 3 n p2  3 n p4  (2  n  2 n ) p3
dt
dp4
 2 n p3  4 n p5  (3 n   n ) p4
dt
dp5
  n p4  4  n p5
dt
Open probability:
n = p5
HH Na Channel
HH Na Channel
HH Na Channel
But in this picture, inactivation
only when activation gate is
open:
Na channel: Patlak model
Na channel: Patlak model
V-independent k1, k2, k3
Fits fast data a bit better than stochastic HH model
Synapses
Synapses
Conductances gated by presynaptic activity:
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
P  Ps Prel
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
P  Ps Prel
~ deterministic (many channels)
on postsynaptic side,
stochastic on presynaptic side
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
P  Ps Prel
~ deterministic (many channels)
on postsynaptic side,
stochastic on presynaptic side
Receptors: ionotropic and metabotropic
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
P  Ps Prel
~ deterministic (many channels)
on postsynaptic side,
stochastic on presynaptic side
Receptors: ionotropic and metabotropic
Synapses
Conductances gated by presynaptic activity:
I syn (t )  g syn (t )(V (t )  Vrev )
gs  gs P
P  Ps Prel
~ deterministic (many channels)
on postsynaptic side,
stochastic on presynaptic side
Receptors: ionotropic and metabotropic
Transmitters and Receptors
Main transmitters:
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
AMPA (Na, K)
NMDA (Na, K, Ca)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
AMPA (Na, K)
NMDA (Na, K, Ca)
GABA receptors
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
AMPA (Na, K)
NMDA (Na, K, Ca)
GABA receptors
GABAA (ionotropic, Cl)
GABAB (metabotropic, K)
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
AMPA (Na, K)
NMDA (Na, K, Ca)
GABA receptors
GABAA (ionotropic, Cl)
GABAB (metabotropic, K)
Ach receptors:
Transmitters and Receptors
Main transmitters:
glutamate (excitatory)
GABA (g-aminobutyric acid, inhibitory)
ACh (neuromuscular junction)
Noradrenaline (modulatory)
Receptor types (named after pharmacological agonists):
Glutamate receptors (both ionotropic) :
AMPA (Na, K)
NMDA (Na, K, Ca)
GABA receptors
GABAA (ionotropic, Cl)
GABAB (metabotropic, K)
Ach receptors:
nicotinic (ionotropic)
muscarinic (metabotropic)
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Transmitter: s constant for a short time, s >> s
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Transmitter: s constant for a short time, s >> s
Ps (t )  1  ( Ps (0)  1) exp(  st ),
0t T
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Transmitter: s constant for a short time, s >> s
Ps (t )  1  ( Ps (0)  1) exp(  st ),
0t T
Then =0, decay:
Ps (t )  Ps (T ) exp(  s (t  T )),
t T
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Transmitter: s constant for a short time, s >> s
Ps (t )  1  ( Ps (0)  1) exp(  st ),
0t T
Then =0, decay:
Ps (t )  Ps (T ) exp(  s (t  T )),
t T
Postsynaptic conductance
(AMPA receptor)
Kinetic equation:
dPs
  s (1  Ps )   s Ps
dt
Transmitter: s constant for a short time, s >> s
Ps (t )  1  ( Ps (0)  1) exp(  st ),
0t T
Then =0, decay:
Ps (t )  Ps (T ) exp(  s (t  T )),
t T
s = 0.93/ms
s = 0.19/ms
Other receptors
excitatory
inhibitory
Other receptors
excitatory
inhibitory
commonly fit by
Ps 
1
[exp( t /  1 )  exp( t /  2 )]
1  2
1   2
Other receptors
excitatory
inhibitory
commonly fit by
Ps 
1
[exp( t /  1 )  exp( t /  2 )]
1  2
limit  1   2 : Ps 
t
2
exp( t /  )
1   2
Other receptors
excitatory
inhibitory
commonly fit by
Ps 
1
[exp( t /  1 )  exp( t /  2 )]
1  2
limit  1   2 : Ps 
t
2
exp( t /  )
1   2
“-function”
NMDA receptors
Conductance is voltage-dependent
(raising voltage knocks out Mg ions that block channel at low V)
NMDA receptors
Conductance is voltage-dependent
(raising voltage knocks out Mg ions that block channel at low V)
I NMDA  g NMDA Ps (V  VNMDA )
NMDA receptors
Conductance is voltage-dependent
(raising voltage knocks out Mg ions that block channel at low V)
I NMDA  g NMDA Ps (V  VNMDA )
g NMDA 
0
g NMDA
[Mg 2 ]
1
exp( V / 16.13 mV)
3.57 mM
NMDA receptors
Conductance is voltage-dependent
(raising voltage knocks out Mg ions that block channel at low V)
I NMDA  g NMDA Ps (V  VNMDA )
g NMDA 
0
g NMDA
[Mg 2 ]
1
exp( V / 16.13 mV)
3.57 mM
NMDA receptors
Conductance is voltage-dependent
(raising voltage knocks out Mg ions that block channel at low V)
I NMDA  g NMDA Ps (V  VNMDA )
g NMDA 
Opening requires both preand postsynaptic depolarization:
Coincidence detector
(important for learning)
0
g NMDA
[Mg 2 ]
1
exp( V / 16.13 mV)
3.57 mM
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
dr
  r [T ](1  r )   r r
dt
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
dr
  r [T ](1  r )   r r
dt
Active receptor activates second messenger:
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
dr
  r [T ](1  r )   r r
dt
Active receptor activates second messenger:
ds
 kr   s s
dt
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
dr
  r [T ](1  r )   r r
dt
Active receptor activates second messenger:
ds
 kr   s s
dt
Cooperative binding of second messenger to K channel opens it
for current:
GABAB receptor kinetics
Simplest model for a metabotropic receptor:
Transmitter binding activates receptor:
dr
  r [T ](1  r )   r r
dt
Active receptor activates second messenger:
ds
 kr   s s
dt
Cooperative binding of second messenger to K channel opens it
for current:
I GABAB  g GABAB
s4
(V  VK )
s4  K
Presynaptic kinetics:
depression and facilitation
Presynaptic kinetics:
depression and facilitation
depression
(exc->exc synapses)
Presynaptic kinetics:
depression and facilitation
depression
(exc->exc synapses)
facilitation
(exc->inh synapses)
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
dPrel 1  Prel

 U (t  t s ) Prel
dt
r
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
dPrel 1  Prel

 U (t  t s ) Prel
dt
r
For presynaptic rate r(t),
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
dPrel 1  Prel

 U (t  t s ) Prel
dt
r
For presynaptic rate r(t),
dPrel 1  Prel

 Ur(t ) Prel
dt
r
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
dPrel 1  Prel

 U (t  t s ) Prel
dt
r
For presynaptic rate r(t),
dPrel 1  Prel

 Ur(t ) Prel
dt
r
For stationary rate, stationary solution is
Prel0 
1
1  Ur r
Synaptic depression
Dynamics of Prel controlled by depletion of synaptic vesicles:
dPrel 1  Prel

 U (t  t s ) Prel
dt
r
For presynaptic rate r(t),
dPrel 1  Prel

 Ur(t ) Prel
dt
r
For stationary rate, stationary solution is
Prel0 
1
1  Ur r
Response to change in
presynaptic rate
expand:
Prel  Prel0  Prel
Response to change in
presynaptic rate
expand:
Prel  Prel0  Prel
1

d
Prel    Urfinal Prel
dt
 r

Prel

 r Prel0, final
Response to change in
presynaptic rate
expand:
Prel  Prel0  Prel
1

d
Prel    Urfinal Prel
dt
 r

Prel

 r Prel0, final
Response to change in
presynaptic rate
expand:
Prel  Prel0  Prel
1

d
Prel    Urfinal Prel
dt
 r

Prel

 r Prel0, final
Responds to change in input, not much to absolute level
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Dynamics of x: depression (vesicle depletion)
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Dynamics of x: depression (vesicle depletion)
Dynamics of y: facilitation (need Ca influx to make release possible)
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Dynamics of x: depression (vesicle depletion)
Dynamics of y: facilitation (need Ca influx to make release possible)
dy y0  y

 f (1  y) (t  t s )
dt
f
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Dynamics of x: depression (vesicle depletion)
Dynamics of y: facilitation (need Ca influx to make release possible)
dy y0  y

 f (1  y) (t  t s )
dt
f
For stationary rate:
y 
y0  fr f
1  fr f
Synaptic facilitation
Prel = P(vesicle) P(release|vesicle)
x
y
Dynamics of x: depression (vesicle depletion)
Dynamics of y: facilitation (need Ca influx to make release possible)
dy y0  y

 f (1  y) (t  t s )
dt
f
For stationary rate:
y 
y0  fr f
1  fr f
Combined model
(Markram-Tsodyks)
Combined model
(Markram-Tsodyks)
Facilitation as before:
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Depression is proportional to Prob(release|vesicle) after spike:
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Depression is proportional to Prob(release|vesicle) after spike:
dx 1  x

 [ y  f (1  y )] (t  t s ) x
dt
r
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Depression is proportional to Prob(release|vesicle) after spike:
dx 1  x

 [ y  f (1  y )] (t  t s ) x
dt
r
With presynaptic rate r(t):
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Depression is proportional to Prob(release|vesicle) after spike:
dx 1  x

 [ y  f (1  y )] (t  t s ) x
dt
r
With presynaptic rate r(t):
dy y0  y

 f (1  y)r (t )
dt
f
Combined model
(Markram-Tsodyks)
Facilitation as before:
dy y0  y

 f (1  y) (t  t s )
dt
f
Depression is proportional to Prob(release|vesicle) after spike:
dx 1  x

 [ y  f (1  y )] (t  t s ) x
dt
r
With presynaptic rate r(t):
dy y0  y

 f (1  y)r (t )
dt
f
dx 1  x

 [ y  f (1  y )]r (t ) x
dt
r