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