models of neuronal populations

Models of neuronal populations

Anton V. Chizhov

Ioffe Physico-Technical Institute of RAS,St.-Petersburg

Definitions:

Population is a great number of similar neurons receiving similar input

Population activity (=population firing rate) is the number of spikes per unit time per total number of neurons

Neurons

Neuronal populations

Large-scale simulations(NMM & FR-models

for EEG & MRI)

Overview

• Experimental evidences of population firing rate coding

• Conductance-based neuron model

• Probability Density Approach (PDA)

• Conductance-Based Refractory Density approach (CBRD)

- threshold neuron- t*-parameterization - Hazard-function for white noise- Hazard-function for colored noise

• Simulations of coupled populations

• Firing-Rate model

• Hierarchy of visual cortex models

• What can be modeled on population level?

• Which details are important?

• What kinds of population models do exist?

• Which one to choose?

[E.Aksay, R.Baker, H.S.Seung, D.W.Tank \\

J.Neurophysiol. 84:1035-1049, 2000] Activity of a position neuron during spontaneoussaccades and fixations in the dark. A: horizontal eye position (top 2 traces), extracellular recording (middle), and firingrate (bottom) of an area I position neuronduring a scanning pattern of horizontal eyemovements.

[R.M.Bruno, B.Sakmann // Science 312:1622-1627, 2006 ]

Population PSTH of thalamic neurons’responses to a 2-Hz sinusoidal deflection of theirrespective principal whiskers (n = 40 cells).

Commonly information is coded by firing rate

Whole-cell (WC) recording of a layer2/3 neuron of the C2 cortical barrelcolumn was performed simultaneouslywith measurement of VSD fluorescence under conventional opticsin a urethane anesthetized mouse.

Commonly populations are localized in cortical space

Voltage-sensitive Dye Optical Imaging[W.Tsau, L.Guan, J.-Y.Wu, 1999]

Pure population events observed in experiments:

• Evoked responses

• Oscillations

•Traveling waves

GABA-IPSC AMPA-EPSCAMPA-EPSC

AMPA-EPSP

AMPA-EPSP

GABA-IPSP

GABA-IPSC

GABA-IPSPPSP

PSP

Firing rateFiring rate

SpikeSpike

Threshold criterium

Population model

Synaptic conductance kinetics

Membraneequations

Eq. for spatial connections

Approximations forare

from [L.Graham, 1999]; IAHP is from [N.Kopell et al., 2000]

Model of a pyramidal neuron

)()()()( 0 ttuVVtsIIIIIIIdt

dVC AHPLHMADRNa η++−−−−−−−−−=

HMADRNa IIIII ,,,,

)()(

,)(

)(

U

yUy

dt

dy

UxUx

dtdx

y

x

τ

τ−=

−=

∞

∞

))(()()( ......... VtVtytxgI qp −=

Color noise model (Ornstein-Uhlenbeckprocess):

)(2 tdtd σξτηητ +−=

MODEL

EXPЕRIМЕNТ

• ionic channel kinetics• input signal is 2-d

)()()()( 0 tIVVtgtu electrodeS SS +−=∑∑=

S S tgts )()(

• synaptic channel kinetics

C

Vd

Vd

Vd

Vd

Vs

Vs

Is

Is

g=Id/(Vd-Vrev)

B

t, ms

PS

C,p

A

PS

P,m

V

0 5 10 15-200

-150

-100

-50

0

0

5

10

15

PSC, exp.PSP, exp.PSP, model 2PSP, model 1

2-comp. neuron with synaptic currents at somas

+∂

∂−−+−−−=

−−+−−=

dd

ms

drest

dd

m

s

Sd

restm

It

I

GVVVV

dt

dV

G

IVVVV

dt

dV

31

))(2()(

)()(

τρ

ρτ

ρτ

Figure Transient activation of somatic and delayedactivation 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 injectionin 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:A) current-clamp to register PSP: B) voltage-clamp to register PSC:

;0

∂∂+=

∂∂

= T

VVGR

X

Vs

X

)(TIRX

VS

LX

=∂∂

=

02

2

=+∂∂−

∂∂

VXV

TV

;0)0,( =TV

Solution:

• neuron is spatially distributed

[A.V.Chizhov // Biophysics 2004]

Модель. Ответ зрительной коры на полосу горизонтальной, а затем вертикальной ориентации.

Эксперимент. Зрительная кора. Картаориентационной избирательности

активности нейронов.

Модель “Pinwheels” картыориентационной

избирательности входных

сигналов.

• spatial structure of connections

1 mm

Population models

• Definition

A population is a set of similar neurons

receiving a common input and dispersed due to noise and intrinsic parameter distribution.

• Common assumptions:

– Input – synaptic current (+conductance)

– Infinite number of neurons

– Output – population firing rate

N

tttn

tt act

Nt

);(1limlim)(

0

∆+∆

=∞→→∆

ν

(4000)

Direct Monte-Carlo simulationof individual neurons:

Firing-rate:

Probability Density Approach (PDA):

N

ttn

tt

спайкиVVVV

tVVgIt

VC

act

resetT

ILL

)(1)(

т , если

)()(

∆+∆

=

=>

+−−=∂∂

ν

ξσ

∫∞

+

==

+=

−−=

∂∂+

∂∂

−=∂∂+

∂∂

0

*)0,()(

))/,()((1

*)),((

)(*

*

:

dtHttv

dtdUUBUAttUH

VUgItU

tU

C

Htt

модельRD

m

LL

ρρ

τ

ρρρ

))((~)(

),(

))((

))(()(

tUft

VUgIdt

dUCor

tIfdtd

or

tIft

LL

=

−−=

+−=

=

ν

νντ

ν

Types of population models

(4000)

Assumption. Neurons are de-synchronized.

f

I

“f-I-curve”

where the matrix represents the influence of noise

Problem! The equation is multi-dimensional.

Particular cases are

- membrane potential

- time passed since the last spike

- time till the next spike

Idea of Probability Density Approach (PDA)

For classical H-H:

[A.Turbin 2003]τ≡X

SXFdt

Xd rrr

r

+= )(

),( tXr

ρ

Single neuron equation (e.g. H-H model)

( )

∂∂⋅

∂∂+⋅

∂∂−=

∂∂

XW

XXF

Xtr

t

r

rr

r

ρρρ)(

Sr

Wt

),,,( nhmVX =r

Fr

where is the common deterministic part,is the noisy term.S

r

Eq. for neural density

*tX ≡

VX ≡

[B.Knight 1972]

[A.Omurtag et al. 2000][D.Nykamp, D.Tranchina 2000][N.Brunel, V.Hakim 1999], …

[J.Eggert, JL.Hemmen 2001][А.Чижов, А.Турбин 2003]

• Kolmogorov-Fokker-Planck eq. for ρ(t,V)Leaky Integrate-and-Fire (LIF) neuron:

• Refractory density ρ(t,t*) for SRM - neurons

Spike Response Model (SRM):

)'()'()(,0)(

),()(

2 ttttt

VVthenVVif

ttRIVdt

dV

m

resetT

m

−>=<>=<

=>

++−=

δστηηη

ητ

[ ] )(2

)(2

22

resetm VVV

RIVVt

−⋅+∂∂+−

∂∂=

∂∂ δνρσρρτ

Htt

ρρρ −=∂∂+

∂∂

∗ ∫∞

=≡0

** ),()()0,( dttttt ρνρ

( ) ( ) ( ) ( )∫+=*

0

*** ''',,t

dttIttktttU η

)*),,(( TVttUHH =

[W.Gerstner, W.Kistler, 2002]

TVVVt

=∂∂= ρσν

2)(

2

VTVreset

ρ

Hz

0

ν

Hz

0 t

Problem! Voltage can not uniquely characterize neuron’s state.

Simplest 1-d PDAs

1-D Refractory Density Approach for conductance-

based neurons (CBRD)

1. Threshold single-neuron model

2. Refractory density approach (t* -parameterization)

3. Hazard-function

Htt

ρρρ −=∂∂+

∂∂

∗

iAHPLHMADR IIIIIIIt

U

t

UC −−−−−−−=

∂∂+

∂∂

*

)(

)(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

τ

τ−=

∂∂+

∂∂

−=∂∂+

∂∂

∞

∞

t* is the time since the last spike;

[A.V.Chizhov, L.J.Graham // Phys. Rev. E 2007, 2008]

H(U) = ‘frozen stationary’ + ‘self-similar’solutions of Kolmogorov-Fokker-Planck eq. for I&F neuron with white or color noise-current

BAH +≈

∫∞

+

=0

*)( dtHtv ρ

Approximations for are taken from [L.Graham, 1999]; IAHP is from [N.Kopell et al., 2000]

1. Threshold neuron model

iAHPLHMADRNa IIIIIIIIdt

dVC −−−−−−−−=

HMADRNa IIIII ,,,,

iAHPLHMADR IIIIIIIdt

dUC −−−−−−−=

Full single neuron model

Threshold model

).1(018.0,:

);1(18.0,:

;002.0:

;691.0,743.0:

;473.0,262.0:

40

wwwwwIfor

xxxxxIfor

yyIfor

yyxxIfor

yyxxIfor

mVUUthenVUif

resetresetAHP

resetresetM

resetH

resetresetA

resetresetDR

resetT

−=∆∆+=

−=∆∆+=

==

====

====−==>

)(

)(,

)(

)(

U

yUy

dt

dy

U

xUx

dt

dx

yx ττ−=−= ∞∞

NaI−

2. Refractory density approach (t* - parameterization)

Htt

ρρρ −=∂∂+

∂∂

∗

iAHPLHMADR IIIIIIIt

U

t

UC −−−−−−−=

∂∂+

∂∂

*

)(

)(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

τ

τ−=

∂∂+

∂∂

−=∂∂+

∂∂

∞

∞

Boundary conditions:

-- firing rate)()0,(0

tdtFt νρρ ≡= ∫∞

+

∗

( )

)(1)exp(2

)(~,),(~2-)(

)),(),(),(),(),(/(

),()(1)(

2

m

*****

Terf

TTF

UUTTF

dt

dTUB

ttggttgttgttgttgC

dtdUUBUAUH

T

AHPLHMADRm

m

+−=−==

+++++=

+=

πστ

ττ

).),((),(:

;),()0,(

;),()0,(

;,,)0,(,)0,(

)0,(

***

*

*

dtttdUUttUt

Iforwttwtw

Iforxttxtx

IIIforytyxtx

UtU

TTTT

AHPresetT

MresetT

HADRresetreset

reset

=

∆+=

∆+=

===

),(),,(),,(),,( **** ttyyttxxttUUtt ==== ρρ

-- Hazard function

AHPHMADR IIIIIfor

VUyxgI

,,,,

)( ......... −=

**

*

tttdt

dt

tdt

d

∂•∂+

∂•∂=

∂•∂+

∂•∂=•

t* is the time since the last spike;

). 0.0117 0.072 0.257 1.1210(6.1exp=)( 4323 TTTTUA −−−−⋅ −

Application

A – solution in case of steady stimulation (self-similar);B – solution in case of abrupt excitation

0~

2~))((

~ 2

=

∂∂−−

∂∂+

∂∂

VVtU

Vtm

ρσρρτ

Single LIF neuron - Langevin equation

spikethenUVif T<

)()( ttUVdt

dVm ητ ++−=

0)( =>< tη)'()'()( 2 tttt m −>=< δτσηη

Fokker-Planck equation

0),(~ =TUtρ

0),(~ =−∞tρ

( )22)(exp1

),0(~ σσπ

ρ UVV −−=

σ))()((ˆ)( tUtVtu −=

σ))((ˆ)( tUUtT T −=

0~

2

1~~

=

∂∂−−

∂∂+

∂∂

uu

utm

ρρρτ0))(,(~ =tTtρ

0),(~ =−∞tρ

( )2exp1

),0(~ uu −=π

ρ

3. Hazard-function in the case of white noise-current(First-time passage problem)

BAH +≈Approximation:

)(

~

21

/)(~)(tTum

m utHtH

=∂∂−=≡ ρ

ττ

TUVm VtH

=∂∂−≡ ρ

τσ ~

2)(

2

Self-similar solution (T=const)

shapeutpamplitudet −− ),(,)(ρ

),()(),(~ utptut ρρ = ∫ ∞−=

)(),(~)(

tTduuttwhere ρρ

ptHup

puut

pm ⋅=

∂∂−−

∂∂+

∂∂

)(~21τ

Tuu

ptHwhere

=∂∂−=

21

)(~0),( =Ttp

0),( =−∞tp

Assumption.

( )2exp1

),0( uup −=π

U(t)=const (or T(t)=const). Notation:Then the shape of , which is , is invariable. ρ~ ),( utp

ptAu

ppu

u⋅=

∂∂−−

∂∂

)(2

1

Tuu

ptAwhere

=∂∂−=

2

1)(

0),( =Ttp

0),( =−∞tp

)(TAdt

dm ρρτ −=

),(~= tHdt

dm ρρτ −

HA ~≡

Equivalent formulation:

Frozen Gaussian distribution (dT/dt = ∞)

∫ ∞−=

)(),(~)(

tTduuttwhere ρρ

T(t) decreases fast.The initial Gaussian distribution remains almost unchanged except cutting at u=T.The hazard function in this case is H=B(T,dT/dt).

Assumption.

+

−=−=dt

dT

dT

d

dt

dB mm ρ

ρτρ

ρτ

For the simplicity, we consider the case of arbitrary but monotonically increasing T(t) and the Gaussian distribution

( )

<−=

otherwise

tTtuifuut

,0

)()(,exp1

),(~2

πρ

)(~

2 TFdt

dT

dt

dT

dT

dB m

m

++

−=

−= τρρ

τ

or

)erf(1)exp(2

)(~ 2

T

TTFwhere

+−=

π[x]+ for x>0 and zero otherwise

Bdt

dm ρρτ −=

U(t) UT

)(

~

2

1ˆ

tTum uH

=∂∂= ρ

τ

0~

2

1~~

=

∂∂−−

∂∂+

∂∂

uu

utmρρρτ

Approximation of hazard function in arbitrary case

BAH +≈

tt ∂∂= ρν )(

Weak stimulus Strong stimulus

0))(,(~ =tTtρ0),(~ =−∞tρ

( )2exp1

),0(~ uu −=π

ρ

Approximation:

σ))((ˆ)( tUUtTгдеT −=

A – solution in case of steady stimulation (self-similar);B – solution in case of abrupt excitation

Approximation of H by A is green, by B is blue, by A+B is red, exact solution is black.

Langevin equation

TUV <

),( tUIdt

dUC tot−=

σσησ

ξττ

τ

/))(,()(~,/)(,/))(,(

)(2

)(~),(),(

UUtUgtT

tqUVtUguгде

tqdtdq

tTutqudtdu

tU

Ttot

tot

m

−=

=−=

+−=

<+−=

)(2 tdtd ξστηητ +−=

0)( =>< tξ)'()'()( tttt −>=< δτξξ

0~1~~

~2

2

=∂∂−

−∂∂+

+−∂∂+

∂∂

qq

qqu

ut m

ρτ

ρτ

ρτ

ρFokker-Planck eq.

0)~,~,(~),,(~),,(~),,(~

=≤==+∞===−∞==∞=

TqTutqut

qutqut

ρρρρ

[ ]

+−+−++== ququkkk

k

kqut 2)1(

21

exp21

),,0(~ 22

πρ

3. Hazard-function in the case of colored noise

)(),( ttVIdtdV

C tot η+−=

Without noise: TUU <

With noise:

or

or

shapeutpamplitudet −− ),(,)(ρ),,()(),,(~ qutptqut ρρ = 1),,(

)( =∫∫ ∞−

∞

∞−

tTduqutpdqwhere

ptHqp

qpq

kpquut

pm )(~)( 2

2

+

∂∂+

∂∂+−

∂∂=

∂∂τ

dqqTtpTqtUHwhereT

),~,( )~())((~~

−≡ ∫∞

. ),,(~=)(

~

dudqquttT

ρρ ∫∫∞

∞−∞−

ττ )/,(),( tUtUk m≡, ),~,(~ )~(

1))((~

~dqqTtTqtUH

T

ρρ

−≡ ∫∞

0))(~),(~,(),,(

),,(),,(

=≤==+∞===−∞==∞=

tTqtTutpqutp

qutpqutp

),(~= tHdt

dm ρρτ −

Self-similar solution (T=const)

Assumption. U(t) (or T(t)) is constant or slow. Then the shape of , which is , is invariable. ρ~ ),,( qutp

0)( 2

2

=+

∂∂+

∂∂+−

∂∂

pAqp

qpq

kpquu 0)~,~,(),,(

),,(),,(

=≤==+∞===−∞==∞=

TqTutpqutp

qutpqutp

dqqTtpTqAwhereT

),~,( )~(~

−= ∫∞

u

q

)T0.0117 -T0.072 -T0.257 - T1.12-exp(0.0061 (T)A 432=∞

21~ k

TT+=

Approximation of Hby A is green, by B is blue,by A+B is red,exact solution is black.

Hazard function in arbitrary case

tt ∂∂= ρν )(

K=1:

K=8:

BAH +≈Weak stimulus

Weak stimulus

Strong stimulus

Strong stimulus

Single cell level

Populations

Htt

ρρρ −=∂∂+

∂∂

∗

iAHPLHMADR IIIIIIIt

U

t

UC −−−−−−−=

∂∂+

∂∂

*

)()(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

τ

τ−=

∂∂+

∂∂

−=∂∂+

∂∂

∞

∞

)()0,(0

tdtFt νρρ ≡= ∫∞

+

∗

AHPHMADR IIIIIfor

VUyxgI

,,,,

)( ......... −=

t* is the time since the last spike

CBRD

Large-scale simulations(NMM & FR-models

for EEG & MRI)

Simulations by CBRD-model

Simulations. Current-step stimulation. Comparison with Monte-Carlo.

Non-adaptive neurons

(4000)

Simulations. Current-step stimulation. Color noise. Adaptive neurons.

LIF

Adaptive conductance-based neuron

with IM

Simulations. Oscillatory input.

Simulations. Constant current stimulation.Comparison with analytical solution.

[Johannesma 1968]

Simulations. Constant current stimulation.Color noise. Comparison with analytical solution.

Firing rate depends on the noise time constant.

1'

00 )/(

)(exp=

−

′′−

−− ∫∫ uduadu

uauHu ma

m

ττν

)(/= LT

La VUgIa −

dots – Monte-Carlosolid – eq.(*)dash – adiabatic approach [Moreno-Bote, Parga 2004]

(*)

GABA-IPSC AMPA-EPSCAMPA-EPSC

AMPA-EPSP

AMPA-EPSP

GABA-IPSP

GABA-IPSC

GABA-IPSPPSP

PSP

Firing rateFiring rate

SpikeSpike

Membraneequations

Threshold criterium

Population model

Synaptic current kinetics

Interconnected populations

Excitatory synaptic current:

- maximum specific conductance, - non-dimensional conductance

- reversal potential

Inhibitory synaptic current:

Non-dimensional synaptic conductances:

where

- rise and decay time constants- firing rate on j-type axonal terminals

NMDAGABAAMPAj

jjjd

jrj

jdj

rj Sm

dt

dm

dt

md

,,

2

2

),()(

=

=+++ νττττ

)()()( NMDANMDANMDANMDANMDA VVVftmgi −=t, ms

PS

C,p

A

0 10 20 30 40 50-50

0

50

100

150

200

Vh=-80 mV

Vh=+20 mV

Vh=-40 mV

AMPA-PSC(with PTX, APV)

experimentmodel

t, ms

PS

C,p

A

0 25 50 75 100-80

-60

-40

-20

0

20

40

Vh=+20 mV

Vh=-40 mV

NMDA-PSC(with PTX, CNQX)

Vh=+20 mV

Vh=-40 mV


Vh=+20 mV

Vh=-40 mV


experimentmodel

)()( AMPAAMPAAMPAAMPA

NMDAAMPAE

VVtmgi

iii

−=+=

))062.0exp(57.3/1/(1)( VMgVf NMDA −+=

jg

jm

jV

)()( GABAGABAGABAI VVtmgi −=

1))2exp(1(2)(S jj −ντ−+=νdj

rj ττ ,

sµτ 1=

)(tjν

t, ms

PS

C,p

A

0 10 20 30 40 50-500

-400

-300

-200

-100

0

Vh=-60 mV

fast GABA-A -IPSC(with CNQX, D,L-APV)

experimentmodel

t, ms

PS

C,p

A

0 10 20 30 40 50

-100

0

100

200

300

400

500

Vh=-80 mV

Vh=+20 mV

Vh=-40 mV

AMPA-PSC(with PTX, APV)

experimentmodel

t, ms

PS

C,p

A

0 25 50 75 100

-150

-100

-50

0

50

100

150 NMDA-PSC(with PTX, CNQX)NMDA-PSC(with PTX, CNQX)

Vh=+20 mV

Vh=-40 mV


experimentmodel

Pyramidal neurons

Interneurons

Approximations of synaptic currents

t, ms

PS

C,p

A

0 10 20 30 40 50

-100

-50

0

Vh=-64 mV

GABA-PSC(with CNQX, D-AP5)

experimentmodel

with IM and IAHP

),)(()()(

),()()(

SSS

Sexti

VtUtgtI

tItItI

−=+=

)(2

mS/cm 1

mV, 5

ms, 1

ms, 5.4

150

2

S

restatmV

g

V

pAI

V

S

S

ext

==

====

σ

ττ

)0,()()(

2)(

2

22 tgtg

dt

tdg

dt

tgdSS

SS

SS ρτττ =++

[S.Karnup, A.Stelzer 2001]

Experiment

Simulations. Interictal activity. Recurrent network of pyramidal cells,including all-to-all connectivity by excitatory synapses.

Model

Simulations. Gamma rhythm. Recurrent network of interneurons ,including all-to-all connectivity by inhibitory synapses

),)(()()(

),()()(

SSS

Sexti

VtUtgtI

tItItI

−=+=

)0,()(

)()()(

2)(

*

2

22

==

−=++

tttapproachdensityfor

tgtgdt

tdg

dt

tgddSS

SS

SS

ρν

τντττ

2

S

d

S

7mS/cm

-80mV,V

1ms,

1ms,

3ms,

=

====

Sg

τττ

OscillationsModel ExperimentsControl (“Kainate”) +“Bicuculline”

Spikes in single neurons

Conductances

Power Spectrum of Extracellular Potentials

Spike timing of pyramidal and inhibitory cells.

[Khazipov, Holmes, 2003]Kainate-induced oscillations in CA3.

[A.Fisahn et al., 1998] Cholinergically induced oscillations in CA3

[N.Hajos, J.Palhalmi, E.O.Mann, B.Nemeth, O.Paulsen, and T.F.Freund. J.Neuroscience, 24(41):9127–9137, 2004]

conbic

All the simulations were done with a single set of parameters. All the parameters except synaptic maximum conductances have been obtained by fitting to experimental registration of elementary events such as patch-electrode current-induced traces, spike trains and monosynaptic responses .

The model reproduces the following characteristics of gamma-oscillations :

� frequency of population spikes

� a single pyramidal cell does not fire every cycle

� every interneuron fires every cycle

� amplitude of EPSC is less than that of IPSC

� blockage of GABA-A receptors reduces the frequency

� peak of pyramidal cell’s firing frequency corresponds to the descending phase of EPSC and the ascending phase of IPSC

� firing of interneurons follows the firing of pyramidal cells

� gamma-oscillations are homogeneous in space along the cortical surface (data not shown)

Spatial connections

22 )()(),,,( YyXxYXyxd −+−=

- firing rate on presynaptic terminals;- firing rate on somas.

Assumption: distances from soma to synapses have exponentially decreasingdistribution p(x) [B.Hellwig 2000].

[V.Jirsa, G.Haken 1996][P.Nunez 1995] [J.Wright, P.Robinson 1995]

),,(2 22

2

2

222

2

2

yxttyx

ctt

νγγφφφγφγφ

∂∂+=

∂∂+

∂∂−+

∂∂+

∂∂

),,( yxtφ),,( yxtν

where γ = c/λ; c – the average velocity of spike propagation along the cortex surface by axons; λ – characteristic axon length. [D.Contreras, R.Llinas 2001]

Experiment:

, ),,,(),,/),,,((=),,( dYdXYXyxWYXcYXyxdtyxt iij −∫∫ νϕ

λ),,,(

),,,(YXyxd

eYXyxW−

=

PSPs and PSCs evoked by extracellular stimulation and registered

at 3.5cm away, w/ and w/o kainate.

[S.Karnup, A.Stelzer1999] Effects of GABA-A receptor blockade on orthodromic potentials in CA1 pyramidal cells. Superimposed responses in a pyramidal cell soma before and after application of picrotoxin (PTX, 100 muM). Control and PTX recordings were obtained at V rest (-64 mV; 150 muAstimulation intensities; 1 mm distance between stratum radiatum stimulation site and perpendicular line through stratum pyramidale recording site). The recordings were carried out in ‘minislices’ in which the CA3 region was cut off by dissection.

[V.Crepel, R.Khazipov, Y.Ben-Ari, 1997] In normal concentrations of Mg and in the absence of CNQX, block of GABA-A receptors induced a late synaptic response.

BA

C

[B.Mlinar,A.M.Pugliese, R.Corradetti2001] Components of complex synaptic responses evoked in CA1 pyramidal neurones in the presence of GABAA receptor block.

The model reproduces postsynaptic currents and postsynaptic potentials registered on somas of pyramidal cells, namely:

� monosynaptic EPSCs and EPSPs

� disynaptic IPSC/Ps followed be EPSC/Ps

� polysynaptic EPSC/Ps

� reduction of delays in polysynaptic EPSCs

� decay of excitation after II component of poly-EPSCs in presence of GABA-A receptor block.

The model predicts that the evoked responses are essentially non-homogeneous in space:

Spatial profiles of membrane potential and firing rate in pyramids.

Evoked responsesModel Experiments

WavesIn the case of reduced GABA-reversal potential VGABA= -50mV and stimulation by extracellular electrode we obtain a traveling wave of stable amplitude and velocity 0.15 m/s. The velocity is much less than the axon propagation velocity (1m/s) and is determined mostly by synaptic interactions.

ms

Hz mV

0 25 50 75 1000

20

40

60

80

100

120

140 voltage, pyramidsvoltage, interneuronsfiring rate, pyramidsfiring rate, interneurons

-40

-60

B

mm

Hz mV

10 20 30 400

20

40

60

80

100

120 voltage, pyramidsvoltage, interneuronsfiring rate, pyramidsfiring rate, interneurons

0.15m/s -40

-60

Fig.5. Wave propagating from the site of extracellular stimulation at right border of the “slice”.A, Evoked responses of pyramidal cells and interneurons at the site of stimulation. B, Profiles of mean voltage and firing rate in pyramidal cells and interneurons at the time 200 ms after the stimulus.

A

[Leinekugel et al. 1998]. Spontaneous GDPs propagate synchronously in both hippocampi from septal to temporal poles. Multiple extracellular field recordings from the CA3 region of the intact bilateral septohippocampal complex. Simultaneous extracellular field recordings at the four recording sites indicated in the scheme. Corresponding electrophysiological traces (1–4) showing propagation of a GDP at a large time scale.

[D.Golomb, Y.Amitai, 1997]Propagation of discharges in disinhibitedneocortical slices.

Model Experiments

Waves with unchanging chape and velocity are observed in cortical tissue in disinhibiting or overexciting conditions. The waves are produced by complex interaction of pyramidal cells and interneurons. That is confirmed by much lower speed of the wave propagation comparing with the axon propagation velocity which is the coefficient in the wave-like equation.Analysis of wave solutions and more detailed comparison with experiments are expected in future.

From CBRD to Firing-Rate model

Macro- and meso-scale

internal granularlayer

internal pyramidallayer

external pyramidallayer

external granularlayer

AP generation zone synapses

macro-scale meso-scale micro-scale

[S.Kiebel][C.Friston]

IVUggdt

dUC LSL −−+−= ))((

( )

)(;)(

exp1

-)(

(steady) ;)(1)(

/

),()()(

2

2

m

1/)(

/)(

2

suddenUV

dt

dUUB

duuerfeUA

gC

dtdUUBUAt

T

UV

UV

um

Lm

T

reset

−−

=

+=

=+=

+

−−

−∫

σσπτ

πτ

τν

σ

σ

Hazard-function:-- firing rate

Oscillating input

Firing-rate model

[Chizhov, Rodrigues, Terry // Phys.Lett.A, 2007 ]

( )

)(;)(

exp1

-)(

(steady) ;)(1)(

/

),()()(

2

2

m

1/)(

/)(

2

suddenUV

dt

dUUB

duuerfeUA

gC

dtdUUBUAt

T

UV

UV

um

Lm

T

reset

−−

=

+=

=+=

+

−−

−∫

σσπτ

πτ

τν

σ

σ

Hazard-function:-- firing rate

Oscillating input

[Чижов, Бучин // Нейроинформатика-2009 ]

IVUtwgVUtngVUggdt

dUC AHPAHPMMLSL +−−−−−+−= ))(())(())(( 2

)()/1,/1(

)1()(

0101

2

201 tv

K

www

dt

dw

dt

wd

AHPAHPAHPAHPAHPAHP ττ

χττττ −=+−++ ∞

)()/1,/1(

)1()(

0101

2

201 tv

K

nnn

dt

dn

dt

nd

MMMMMM ττ

ξττττ −=+−++ ∞

Not-adaptive neurons Adaptive neurons

Рис. 12. Схема активности

популяции FS (fast spiking) нейронов, возбуждаемых

внешним стимулом νext(t), приходящим из таламуса. Обозначения: ν(t) –популяционная частота спайков

FS нейронов, gE(t), gI(t) –проводимости возбуждающих и

тормозящих синапсов.

FSν

ext

ν gI

gE

Experiment

Model

Рис. 13. Постсинаптический

(моносинаптический) ток в FS-нейроне при слабой

таламической стимуляции

током 30 µA и потенциале

фиксации -88 mV в

эксперименте (вверху) (adaptedby permission from MacmillanPublishers Ltd: (Cruikshank et al., 2007), copyright 2007) и в модели

(внизу).

Рис. 14. Ответы FS-нейронов на таламическую стимуляцию

током 120 µA в эксперименте (слева) (adapted by permission fromMacmillan Publishers Ltd: (Cruikshank et al., 2007), © 2007) и в

модели (справа). A, B - постсинаптические токи при

потенциале фиксации -88, -62, и -35 mV; C, D - синаптические

проводимости; E, F – постсинаптические потенциалы U и

модельная популяционная частота ν.

))(()()( EEE VtVtgti −=

)()( 212

2

21 tggdt

dgdt

gd extEE

EEEEEE ντττττ =+++

),()()( titiVUgdt

dUC IELL ++−−=

),()()( UBUAt +=ν

( ) ;)(1)(

-12/)(

2/)(

2

+= ∫

−

−

VT

Vreset

UV

UV

um duuerfeUA

σ

σπτ

−−×

=+

2

2

2)(

exp21

)(V

T

V

UV

dt

dUUB

σσπ

))(()()( III VtVtgti −=

)()( 212

2

21 tggdt

dg

dt

gdII

IIIIII ντττττ =+++

Simple model of interacting cortical interneurons, evoked by thalamus

Синаптические токи и проводимости:

Мембранный потенциал:

Популяционная частота спайков:

Частотная модель популяции

адаптивных нейронов: «интериктальная» активность

EI

)(),( νν MAHP II

)(νSI

)/,()()(

)()()(

2)(

))((

)()()()(

:

2

22

dtdUUBUAt

tvgtgdt

tgd

dt

tgd

VVtgI

IVVgIIIt

VC

модельFR

SSS

SS

S

SSS

SLLMAHP

+=

=++

−=

−−−−−=∂∂

ν

τττ

ννν

Monte-Carlo simulations:

conventionalFiring-Rate model:

CBRD:

N

ttn

tt

спайкиVVVV

tVVggIt

VC

КарлоМонтеМетод

act

resetT

ILSL

)(1)(

т , если

)())((

:

∆+∆

=

=>

+−+−=∂∂

−

ν

ξσ

∫∞

+

==

+=

−+−=

∂∂+

∂∂

−=∂∂+

∂∂

0

*)0,()(

))/,()((1

*)),((

))((*

*

:

dtHttv

dtdUUBUAttUH

VUggIt

U

t

UC

Htt

модельRD

m

LSL

ρρ

τ

ρρρ

)/,()()(

))((

:FR

dtdUUBUAt

VUggIdt

dUC

модель

LSL

+=

−+−=

ν

Mathematical complexity:104 ODEs 1 ODE a few ODEs 1-d PDEs

Precision:4 2 3 5

Precision for non-stationary problems:5 2 4 5

Precision for adaptive neurons :5 1 3 4

Computational efficiency:2 5 5 4

Mathematical analyzability:1 5 4 4

)/,()()(

)()()(

2)(

)()())((

:

2

22

dtdUUBUAt

tvgtgdt

tgd

dt

tgd

IIVVggItV

C

модельFR

SSS

SS

S

AHPMLSL

+=

=++

−−−+−=∂∂

ν

τττ

νν

modified Firing-Rate model (non-stationary and adaptive):

models of neuronal populations

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