Post on 12-Jan-2016
transcript
Lecture 3: linearizing the HH equations
HH system is 4-d, nonlinear. For some insight, linearize around a (subthreshold) resting state.
(Can vary resting voltage V0 by varying constant injected current I0.)
Ref: C Koch, Biophysics of Computation, Ch 10
Full Hodgkin-Huxley model
extNaNaKKLL IVVhmgVVngVVgdtdV
C )()()( 34
Full Hodgkin-Huxley model
extNaNaKKLL IVVhmgVVngVVgdtdV
C )()()( 34
hVhdtdh
VmVmdtdm
VnVndtdn
V hmn )()()()()()(
Full Hodgkin-Huxley model
extNaNaKKLL IVVhmgVVngVVgdtdV
C )()()( 34
hVhdtdh
VmVmdtdm
VnVndtdn
V hmn )()()()()()(
Full Hodgkin-Huxley model
extNaNaKKLL IVVhmgVVngVVgdtdV
C )()()( 34
hVhdtdh
VmVmdtdm
VnVndtdn
V hmn )()()()()()(
4 coupled nonlinear differential equations
Spikes, threshold, subthreshold dynamics
threshold propertyspike
Spikes, threshold, subthreshold dynamics
threshold propertyspike
sub- and suprathresholdregions
Linearizing the current equation:
00003
004
0 ))(()())(()( IVVVhVmgVVVngVVg NaNaKKLL
Equilibrium: V0, I0
Linearizing the current equation:
titiext VVtVIItI
e)(e)( 00
00003
004
0 ))(()())(()( IVVVhVmgVVVngVVg NaNaKKLL
Equilibrium: V0, I0
Small perturbations:
Linearizing the current equation:
titiext VVtVIItI
e)(e)( 00
00003
004
0 ))(()())(()( IVVVhVmgVVVngVVg NaNaKKLL
VVhVmgVnggVCiI NaKL )]()()([ 003
04
Equilibrium: V0, I0
Small perturbations:
Linearizing the current equation:
titiext VVtVIItI
e)(e)( 00
00003
004
0 ))(()())(()( IVVVhVmgVVVngVVg NaNaKKLL
VVhVmgVnggVCiI NaKL )]()()([ 003
04
VV
nVVVng KK ))((4 00
3
Equilibrium: V0, I0
Small perturbations:
Linearizing the current equation:
titiext VVtVIItI
e)(e)( 00
00003
004
0 ))(()())(()( IVVVhVmgVVVngVVg NaNaKKLL
VVhVmgVnggVCiI NaKL )]()()([ 003
04
VV
nVVVng KK ))((4 00
3
VV
hVVVmgV
V
mVVVhVmg NaNaNaNa ))(())(()(3 00
3000
2
Equilibrium: V0, I0
Small perturbations:
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(from with
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
from with
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
VVnndtd
Vn )(1)( 00
from with
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
VVnndtd
Vn )(1)( 00
titi ntnVtV
e)(,e)(
from with
Harmonic time dependence:
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
VVnndtd
Vn )(1)( 00
titi ntnVtV
e)(,e)(
VVnnVi n )(]1)([ 00
from with
Harmonic time dependence:
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
VVnndtd
Vn )(1)( 00
titi ntnVtV
e)(,e)(
1)()(
0
0
ViVn
Vn
n
VVnnVi n )(]1)([ 00
from with
Harmonic time dependence:
solution:
Linearized equations for gating variables)()( Vnn
dtdn
Vn VVVnVnn 00 ,)(
VdV
dnVnnVn
dt
ndV
dV
dV n
n
)()()( 000
VVnndtd
Vn )(1)( 00
titi ntnVtV
e)(,e)(
1)()(
0
0
ViVn
Vn
n
)()](/)(exp[)](/)([)( 000 tVVttVVntdtn nn
t
VVnnVi n )(]1)([ 00
from with
Harmonic time dependence:
solution:
or
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
For sigmoidal
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
For sigmoidal
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
1)()()](1)[(4
0
0004
ViVVVVnVng
n
KnK
For sigmoidal
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
1)()()](1)[(4
0
0004
ViVVVVnVng
n
KnK
)()](1[4)(ˆ1)(
ˆ000
4
0, KnKK
n
KactiveK VVVnVngg
ViVg
I
For sigmoidal
like a current
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
1)()()](1)[(4
0
0004
ViVVVVnVng
n
KnK
)()](1[4)(ˆ1)(
ˆ000
4
0, KnKK
n
KactiveK VVVnVngg
ViVg
I
VgIVi KactiveKn ˆ)1)(( ,0
For sigmoidal
like a current
i.e.
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
1)()()](1)[(4
0
0004
ViVVVVnVng
n
KnK
)()](1[4)(ˆ1)(
ˆ000
4
0, KnKK
n
KactiveK VVVnVngg
ViVg
I
VgIVi KactiveKn ˆ)1)(( ,0
VIgg
ViactiveK
KK
n
,0
ˆ
1
ˆ
)(
For sigmoidal
like a current
i.e. or
So back in current equation
VVi
VnVVVngV
V
nVVVng
nKKKK
1)(
)())((4))((4
0
000
300
3
)](exp[11
)(nn VV
Vn
)](1)[( VnVndV
dnn
1)()()](1)[(4
0
0004
ViVVVVnVng
n
KnK
)()](1[4)(ˆ1)(
ˆ000
4
0, KnKK
n
KactiveK VVVnVngg
ViVg
I
VgIVi KactiveKn ˆ)1)(( ,0
VIgg
ViactiveK
KK
n
,0
ˆ
1
ˆ
)(
)()](1)[(4
)(
ˆ
)(
0004
00
KnK
n
K
n
VVVnVng
V
g
VL
For sigmoidal
like a current
i.e. or
equation for an RLseries circuit with
Equivalent circuit component
Full linearized equation:
VVi
VhVV
Vi
VmVVVhVmg
Vi
VnVVVnggCiI
h
Nah
m
NamNa
n
KnKL
1)(
))(1)((
1)(
))(1)((31)()(
1)(
))(1)((41)(
0
00
0
0000
3
0
000
4
Full linearized equation:
VVi
VhVV
Vi
VmVVVhVmg
Vi
VnVVVnggCiI
h
Nah
m
NamNa
n
KnKL
1)(
))(1)((
1)(
))(1)((31)()(
1)(
))(1)((41)(
0
00
0
0000
3
0
000
4
VA )(
Full linearized equation:
VVi
VhVV
Vi
VmVVVhVmg
Vi
VnVVVnggCiI
h
Nah
m
NamNa
n
KnKL
1)(
))(1)((
1)(
))(1)((31)()(
1)(
))(1)((41)(
0
00
0
0000
3
0
000
4
VA )( A()= 1/R() = admittance
Full linearized equation:
VVi
VhVV
Vi
VmVVVhVmg
Vi
VnVVVnggCiI
h
Nah
m
NamNa
n
KnKL
1)(
))(1)((
1)(
))(1)((31)()(
1)(
))(1)((41)(
0
00
0
0000
3
0
000
4
VA )( A()= 1/R() = admittance
Equivalent circuit forNa terms:
Impedance() for HH squid neuron
|:)(| R
(=2f)
Impedance() for HH squid neuron
|:)(| R experiment:
(=2f)
Impedance() for HH squid neuron
|:)(| R experiment:
(=2f)Band-pass filtering (like underdampedharmonic oscillator)
Cortical pyramidal cell (model)
(log scale)
Damped oscillations
Responses to different current steps: