Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual...
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Neuron-Computer Interface in Dynamic-Clamp Experiments. Models of Neuronal Populations and Visual Cortex.
A.V.Chizhov
A.F.Ioffe Physical-Technical Institute of RAS, St.-Petersburg, Russia
1. Two-compartment neuron model
• Spiking activity as function of current
and conductance in-vivo, in-vitro и in-
silico
• “Firing-clamp” algorithm of estimation of
synaptic conductances
• Model of statistical ensemble of
Hodgkin-Huxley-like neurons - CBRD
• Model of primary visual cortex.
Mappings of models of a hypercolumn.
ModelExperiment
Models of single neurons and Dynamic-Clamp
- Leaky integrate-and-fire model
- 2-compartmental passive neuron model
- Hodgkin-Huxley neuron model
- Control parameters of neuron
- Dynamic-clamp
• Artificial synaptic current
• Artificial voltage-dependent current
• Synaptic conductance estimation
resetT
restL
VVVV
tIVVtsgdt
dVC
=>
+−+−=
then, if
)()))(((
Leaky Integrate-and-Fire neuron (LIF)
E X P E R I M E N T
LIF - M O D E L
)()()()( tIVVtgtI electrodeS
restSS +−= ∑∑=S
S tgts )()(L
m gC=τ
V is the membrane potential; I is the input (synaptic) current;s is the input (synaptic) conductance; C is the membrane capacity; gL is the membrane conductance; Vrest is the rest potential; VT is the threshold potential; Vreset is the reset potential.
Steady-state firing rate dependence on current and conductance
pA
Hz
0 100 200 3000
50
100
150
200LIF, no noiseLIF + noiseCBRD
s=
0
s=
2gL
−++−++
+=
resetLL
TLL
L
VsgIV
VsgIVC
sg
)/(
)/(ln
ν
LIF, no noise
LIF with noise
C
Vd
Vd
Vd
Vd
Vs
Vs
Id
Is
g
B
2-compartmental neuron with somatically registered PSC and PSP
+∂
∂−−+−−−=
−−+−−=
dd
ms
drest
dd
m
s
Sd
restm
It
IG
VVVVdt
dV
GI
VVVVdtdV
31
))(2()(
)()(
τρ
ρτ
ρτ
Figure Transient activation of somatic and delayed activation of dendritic inhibitory conductances in experiment (solid lines) and in the model (small circles). A, Experimental configuration.B, Responses to alveus stimulation without (left) and with (right) somatic V-clamp. C, In a different cell, responses to dynamic current injection in the dendrite; conductance time course (g) in green, 5-nS peak amplitude, Vrev=-85 mV.
A
[F.Pouille, M.Scanziani //Nature, 2004]
X=0 X=L
Vd
V0
Parameters of the model:τm= 33 ms, ρ = 3.5, Gs= 6 nS in B and 2.4 nS in C
Two boundary problems:• current-clamp to register PSP: • voltage-clamp to register PSC:
;0
∂∂+=
∂∂
= TV
VGRXV
sX
)(TIRXV
SLX
=∂∂
=
02
2
=+∂∂−
∂∂
VXV
TV
;0)0,( =TV
Solution:[A.V.Chizhov // Biophysics 2004]
PSC and PSP: single-compartmental neuron model
G
IV
dt
dV =+τ
Parameters found to fit PSC and PSP:
ms14=τ
How does the model fit to simultaneously recorded PSC and PSP?
Parameters found by fitting:
ms4.21=τnSG 27.1=
GIIIVdtdV stim
NMDAAMPA /)( .++=+τ
mSeg AMPA465.2144.0 −⋅=
mSegNMDA465.24.13 −⋅=
PSC and PSP: Model of concentrated soma and cylindrical dendrite (“model S-D”) [W.Rall, 1959]
Two boundary problems:
A) current-clamp to register PSP:
B) voltage-clamp to register PSC,
i.e.0
1
=∂∂−=
Xc X
V
RI
02
2
=+∂∂−
∂∂
VX
V
T
V
τ/tT =λ/xX =
0=X at the end of dendrite, : LX =
∂∂+=
∂∂
= T
VVGR
X
Vs
X 0
)(TIRX
V
LX
=∂∂
=
X=0 X=L
0)0,( =TV )(TIRX
V
LX
=∂∂
=
X=0 X=L
Parameters: 1=L
VRI p 1.0=10=α
3.0=sGR
.
)1exp()( TTITI p αα −=
irR λ=
at soma, :
At dendrite:
Subtracting (2), obtain:
Eqs. (1),(2) and (3) are equivalent to
PSC and PSP: 2-compartmental model
LXTVLXTVXTV L )()1()(),( 0 +−=
B) Voltage-clamp mode
Assume the potential V(X) to be linear, i.e.
VL
X=0 X=L
V=0
Model S-D
cL
L IIdT
dVV
RL+=
+
2
1
As
current through synapse is (1)
RLVXVRI Lc −=∂∂⋅−= 1
cc I
dT
dII
2
3
2
1 −−=
A) Current-clamp mode
X=0 X=L
VL
V0
RLG
VVV
dT
dV
s
L )( 00
0 −=+ RLVVI Lc )( 0−−=(2) because
IRLVVRLG
VdT
dVL
sL
L ⋅+−
+−=+ 2)(
12 0
+−=
++
++ c
c
ss
d
s
d Idt
dI
GV
G
G
dt
dV
G
G
dt
Vd3
12324 0
02
02
2 τττ
where RLGd /1= is the dendrite conductance
Model S-D
cLL II
dT
VVdVV
RL+=
+++ 2/)(
2
1 00
At soma:
(3)
At dendrite:
PSC and PSP: Fitting experimental PSP and PSC from [Karnup and Stelzer, 1999]
Parameters found by fitting, given fixed : ms 20=τ
for 2-compartmental model: nS 1.3=sG
nS .73=dG
nS 1.3=sG
nS 15=dG
for 1-compartmental model: nS 17=G nS 31=G
EPSC and EPSP IPSC and IPSP
( )[ ]LrG id λ/1=
Conclusion. Solution of voltage- and current-clamp boundary problems by 2-compartmental model describes well the PSP-on-PSC dependence.
V – somatic potential;Vd – dendritic potential;Is – registered on soma current through synapses located
near soma;Id – registered on soma current through synapses located
on dendrites;τm – membrane time constant; ρ – ratio of dendritic to somatic conductances;Gs – specific somatic conductance.
+
∂∂−−+−−−=
−−+−−=
dd
ms
drest
dd
m
s
s
drest
m
It
I
GVVVV
dt
dV
G
IVVVV
dt
dV
31
))(2()(
)()(
τρ
ρτ
ρτ
C
Vd
Vd
Vd
Vd
Vs
Vs
Id
IsFigure Transient activation of somatic and delayed activation of dendritic inhibitory conductances in experiment (solid lines) and in the model (small circles). A, Experimental configuration.B, Responses to alveus stimulation without (left) and with (right) somatic V-clamp. C, In a different cell, responses to dynamic current injection in the dendrite; conductance time course (g) in green, 5-nS peak amplitude, Vrev=-85 mV.
Parameters of the model:τm= 33 ms, ρ = 3.5, Gs= 6 nS in B and 2.4 nS in C
A
B[F.Pouille, M.Scanziani (2004) Nature, v.429(6993):717-23]
PSC and PSP: Fitting experimental PSP and PSC from [Pouille and Scanziani, 2004]
h
[Покровский, 1978]
φ≈0
rV(x) V(x+Δx)
im
jm
C
Внутри
Снаружи
V
gK
gNa
VNa
Vrest
VK
SLS IVVg +−− )(
Hodgkin-Huxley model
Approximations of ionic channels:
Parameters:
Set of experimental data for Hodgkin-Huxley approximations
Approximations for
are taken from [L.Graham, 1999]; IAHP
is from [N.Kopell et al., 2000]
SAHPLHMADRNa IIIIIIIIdt
dVC −−−−−−−−=
HMADRNa IIIII ,,,,
)()(
,)(
)(
UyUy
dtdy
UxUx
dtdx
y
x
τ
τ−=
−=
∞
∞
))(()()( ......... VtVtytxgI qp −=
Color noise model for synaptic current IS is the Ornstein-Uhlenbeck process:
)(2)(0 tItIdt
dISS
S σξττ +−=
Model with noise
E X P Е R I М Е N Т
Model of pyramidal neuron
Control parameters of neuron
)()()(),( tIVVtstVI restS +−−=
)()()()( tIVVtgtI electrodeS
restSS +−= ∑∑=
SS tgts )()(
Property: Neuron is controlled by two parameters[Pokrovskiy, 1978]
)(
)(
V
hVh
dt
dh
hτ−= ∞
)(
)(
V
mVm
dt
dm
mτ−= ∞
)(
)(
V
nVn
dt
dn
nτ−= ∞
2
2
x
Vk
∂∂+
[Hodgkin, Huxley, 1952]
Voltage-gated channels kinetics:
),())(())()(,(
))()(,(),()(
4
3
tVIVtVgVtVtVng
VtVtVhtVmgdt
tdVC
SLLKK
NaNa
+−−−−
−−−=
)())(()(),( tIVtVtgtVI electrodeS
SSS +−−= ∑
EXPERIMENT
)())(()(),( tIVtVtgtVIdtdV
C elS
SSchannelsionic +−−−= ∑
),())(()(),( 0 tuVtVtstVIdtdV
C channelsionic +−−−=
∑=S
S tgts )()(
,
)()()()( 0 tIVVtgtu elS
SS +−= ∑
The case of many voltage-independent synapses
“Current clamp”,V(t) is registered
“Voltage clamp”,I(t) is registered
Whole-cell patch-clamp:Current- and Voltage-Clamp modes
const
Warning! The input in current clamp corresponds to negative synaptic conductance!
Current-clamp is here!
• For artificial passive leaky channel s=const
• For artificial synaptic channel s(t) reflects the synaptic kinetics
• For voltage-gated channel s(V(t),t) is described by ODEs
Conductance clamp (Dynamic clamp):V(t) is registered,I(V,t) = s (V,t) (V(t)-Vus) + u is injected.
Whole-cell patch-clamp:Dynamic-Clamp mode
30 μs
Acquisition card
“Current clamp”Conductance clamp (Dynamic clamp):I(V(t))=s (V(t)-VDC)+u is injected
Dynamic clamp for synaptic current
[Sharp AA, O'Neil MB, Abbott LF, Marder E. Dynamic clamp: computer-generated conductances in real neurons. // J.Neurophysiol. 1993, 69(3):992-5]
)()( GABAGABA VVtgI −=( ) nSgsseegtg GABA
ttGABAGABA 8,15,5,)( max
21//max 21 ===+= −− ττττ
Dynamic clamp for spontaneous
potassium channels
Control
artificial K-channels∑ −=
=+++
iittg
gdtdg
dtgd
)(
)(
12max
212
2
21
δττ
ττττ
msms 200,5 21 == ττ
mVVK 70=
))(( KVVtgI −=
nSg 1max =
u, µA/cm2
s,m
S/c
m2
0 1 2 3
0.01
0.02
0.03
0.04
0.05
0.061101009080706050403020100
(1.7; 0.024)
(2.7; 0.06)
Hz
u, µA/cm2
s,m
S/c
m2
0 1 2 3
0.01
0.02
0.03
0.04
0.05
0.06
1101009080706050403020100
Hz
Hz(2.7; 0.06)
(1.7; 0.024)
Experiment: pyramidal cell of visual cortex in vivo
Model [Graham, 1999] of CA1 pyramidal neuron
u, mkA/cm2
s,m
S/c
m2
0 2 4 6 8 10
0.1
0.2
0.3
0.4
0.5
0.6 80
60
40
20
0
Hz
Dynamic clamp to study firing properties of
neuron
0 500 1000
-80
-60
-40
-20
0
20
V, m
V
t, ms0 500 1000
-80
-60
-40
-20
0
20
V,
mV
t, ms
Experiment
Model
u=7.7 mkA/cm2
S=0.4 mS/cm2
u=1.7 mkA/cm2
S=0.024 mS/cm2
u=2.7 mkA/cm2
S=0.06 mS/cm2
u=4 mkA/cm2
S=0.15 mS/cm2
Bottom point Top point
Divisive effect of shunting inhibition is due to spike threshold sensitivity to slow inactivation of sodium channels
∑ −∆+−=i
spikei
TTTT
ttVVV
dt
dV)(0 δ
τ
inhex GGRate∂∂
∂2
Total Response (all spikes during 500ms-step)
Only 1st spikes Only 1st interspike intervals
Hippocampal Pyramidal Neuron In Vitro
Dynamic clamp for voltage-gated current: compensation of INaP
[Vervaeke K, Hu H., Graham L.J., Storm J.F. Contrasting effects of the persistent Na+ current on neuronal excitability and spike timing, Neuron, v49, 2006]
Effect of “negative conductance” by INaP
'
and 0 where ,
then,0 where ,)(
)(
))((
a
VbbVaabVaI
VVVV
baVVgLet
VVVgI
NaP
NaP
NaP
NaPNaPNaP
−
∆=′>∆=′′−′−≈>∆∆−≈−
+≈−=
plays a role of negative conductance
Dynamic clamp for electric couplings
between real and modeled neurons
Medium electric conductance
High electric conductance
constg
VVgI
=−= )( modexp
“Threshold-Clamp”
Dynamic clamp for synaptic conductance estimations in-vivo
1s
20 mV
10 nS
5 nS
V
σ±V
IA GGABA :
EGAMPA :
Эксперимент [Lyle Graham et al.]: Внутриклеточные измерения patch-clamp в зрительной коре кошки in vivo. Стимул – движущаяся полоска.
Preferred direction Null direction
«Firing-Clamp» - method of synaptic
conductance estimation
Idea: a patched neuron is forced to spike with a constant rate; gE, gI, are estimated from values of subthreshold voltage and spike amplitude.
Threshold voltage, VT Peak voltage, V P
1 ms
τ(V)
MODEL
Measuring system is a neuron:
I/GL (mV)
G/G
L
0 20 40 60 800
1
2
3
4
5
6
4032241680
V peak (mV)
I/GL (mV)
G/G
L
20 30 40 50 60 70 800
1
2
3
4
5
6 -36-37-38-39-40-41-42-43
VT (mV)Firing-Clamp EXPERIMENT
Calibration:Firing-Clamp
Cel
l 16
_28
_28
Ce
ll 1
6_29
_40
Cel
l 16
_33_
14
VT Vpeak
I (pA)
G(n
S)
-200 0 2000
1
2
3
4
5
6
7
8
-20-40-60-80-100
VT=-0.02*I*G+0.209*I-1.46*G-51.9 (mV)
I (pA)
G(n
S)
-200 0 2000
1
2
3
4
5
6
7
8
110805020
-10
Vpeak=-0.024*I*G+0.22*I-6.74*G+64.3 (mV)
EXPERIMENT
Measurements:Firing-Clamp
Ce
ll 1
6_27
_50
Cel
l 16
_27
_5
I (pA)
G(n
S)
-200 0 2000
1
2
3
4
5
6
7
8
-20-40-60-80-100
VT=-0.02*I*G+0.209*I-1.46*G-51.9 (mV)
I (pA)
G(n
S)
-200 0 2000
1
2
3
4
5
6
7
8
110805020
-10
Vpeak=-0.024*I*G+0.22*I-6.74*G+64.3 (mV)
VT Vpeak
EXPERIMENT
Dynamic Clamp
• is necessary for measuring firing characteristics of neuron
• helps to create artificial ionic intrinsic or synaptic channels
• is necessary for estimation of input synaptic conductances in-vivo
Conclusions