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

• What can be modeled on population level?

• Which details are important?

• What kinds of population models do exist?

• Which one to choose?

Experiment. Thalamic neuron responses on 3 trials of visual stimulation by movie.

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

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

[R.M.Bruno, B.Sakmann // Science 312:1622-1627, 2006] Population PSTH of thalamic neurons’ responses to a 2-Hz sinusoidal deflection of their respective principal whiskers (n = 40 cells).

Commonly information is coded by firing rate

Whole-cell (WC) recording of a layer 2/3 neuron of the C2 cortical barrel column was performed simultaneously with measurement of VSD fluorescence under conventional optics in a urethane anesthetized mouse.

spontaneous activity

evoked activity

Commonly populations are localized in cortical space

F. Chavane, D. Sharon, D. Jancke, O.Marre, Y. Frégnac and A.

Grinvald // Frontiers in Systems Neuroscience, v.5, article 4, 1-26,

2011.

Local interactions in visual cortex

Voltage-sensitive Dye Optical Imaging

[W.Tsau, L.Guan, J.-Y.Wu, 1999]

• Evoked responses

• Oscillations

•Traveling waves

Pure population events observed in experiments:

• What can be modeled on population level?

• Which details are important?

• What kinds of population models do exist?

• Which one to choose?

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 for

are

from [L.Graham, 1999];

IAHP is from [N.Kopell et al., 2000]

Model of a pyramidal neuron

)()()()( 0 ttuVVtsIIIIIIIdtdV

C AHPLHMADRNa

HMADRNa IIIII ,,,,

)()(

,)(

)(

UyUy

dtdy

UxUx

dtdx

y

x

))(()()( ......... VtVtytxgI qp

Color noise model (Ornstein-Uhlenbeck process):

)(2 tdtd

MODEL

EXPЕRIМЕNТ

• ionic channel kinetics• input signal is 2-d

)()()()( 0 tIVVtgtu electrodeS SS S S tgts )()(

C

Vd

Vd

Vd

Vd

Vs

Vs

Is

Is

g=Id/(Vd-Vrev)

B

2-comp. neuron with synaptic currents at somas

dd

ms

drest

dd

m

s

Sd

restm

It

IG

VVVVdt

dV

GI

VVVVdtdV

31

))(2()(

)()(

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:m= 33 ms, = 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

TV

VGRXV

sX

)(TIRXV

SLX

02

2

VXV

TV

;0)0,( TV

Solution:

• neuron is spatially distributed

[A.V.Chizhov // Biophysics 2004]

Excitatory synaptic current:

Inhibitory synaptic current:

Non-dimensional synaptic conductances:

where

- rise and decay time constants - presynaptic firing rate

)()(2

2

tmdt

dm

dt

md preSS

SdS

rS

SdS

rS

)()()( NMDANMDANMDANMDA VVVftgi

)()( AMPAAMPAAMPA

NMDAAMPAE

VVtgi

iii

))062.0exp(57.3/1/(1)( VMgVf NMDA

)()( GABAGABAI VVtgi

dS

rS ,

s 1

)(tpreS

Pyramidal neurons

Interneurons

)()( tmgtg SSS Synaptic conductance:

NMDAGABAAMPAS ,,

NMDAGABAAMPAS ,,

• synaptic channel kinetics

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

Эксперимент. Зрительная кора. Карта ориентационной избирательности активности нейронов.

Модель “Pinwheels” карты ориентационной избирательности входных сигналов.

• spatial structure of connections

1 mm

• What can be modeled on population level?

• Which details are important?

• What kinds of population models do exist?

• Which one to choose?

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

(4000)

Стимулирующий ток

Direct Monte-Carlo simulation of individual neurons:

Firing-rate:

Probability Density Approach (PDA):

Nttn

tt

спайкиVVVV

tVVgItV

C

act

resetT

ILL

)(1)(

т , если

)()(

0

*)0,()(

))/,()((1

*)),((

)(*

*

:

dtHttv

dtdUUBUAttUH

VUgItU

tU

C

Htt

модельRD

m

LL

))((~)(

),(

))((

))(()(

tUft

VUgIdt

dUCor

tIfdtd

or

tIft

LL

Types of population models

(4000)

Assumption. Neurons are de-synchronized.

f

I

“f-I-curve”

Стимулирующий ток

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]X

SXFdt

Xd

)(

),( tX

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

XW

XXF

Xt

�

)(

S

W�

),,,( nhmVX

F

where is the common deterministic part,

is the noisy term.S

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]

N

ttn

tt

VVVV

ttIVVtsgdt

dVC

act

resetT

IrestL

)(1)(

then, if

)()()))(((

Leaky Integrate-and-Fire (LIF) neuron with noise (stochastic LIF)

Kolmogorov-Fokker-Planck eq. for ρ(t,V) of LIF-neurons

)(2)(

)(2

22

resetmV

Lrestm VV

VtsgtI

VVVt

TVVm

V

Vt

2

)(

VTVreset

ρ

Hz

0

))(/(

,/

tsg

gC

LIV

Lm

V

Stimulation current

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

0

*)0,()(

)()))(((*

*)),((*

spikelast thesince time theis*

dtHttv

tIVUtsgt

U

t

UC

ttUHtt

t

LL

Refractory Density model

[Chizhov et al. // Neurocomputing 2006]

Stimulation current

where according to

Spike Response Model (SRM):

Htt

0

**),()()0,( dttttt

*

0

*** ''',,t

dttIttktttU

)*),,(( TVttUHH

[W.Gerstner, W.Kistler, 2002]

Similar approach: Refractory Density model for SRM-neurons

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

iAHPLHMADR IIIIIIIt

U

t

UC

*

)(

)(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

t* is the time since the last spike;

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

0

*)( dtHtv

[Chizhov, Graham, Turbin // Neurocomputing, 2006][Chizhov, Graham // Phys. Rev. E, 2007][Chizhov, Graham // Phys. Rev. E, 2008]

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

NaI

Htt

iAHPLHMADR IIIIIIIt

U

t

UC

*

)(

)(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

Boundary conditions:

-- firing rate)()0,(0

tdtHt

).),((),(:

;),()0,(

;),()0,(

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

)0,(

***

*

*

dtttdUUttUt

Iforwttwtw

Iforxttxtx

IIIforytyxtx

UtU

TTTT

AHPresetT

MresetT

HADRresetreset

reset

),(),,(),,(),,( **** ttyyttxxttUUtt t* is the time since the last spike

AHPHMADR IIIIIfor

VUyxgI

,,,,

)( .........

**

*

tttdt

dt

tdt

d

)(1)exp(2

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

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

),()(1)(

2

m

*****

TerfT

TFUU

TTFdtdT

UB

ttggttgttgttgttgC

dtdUUBUAUH

T

AHPLHMADRm

m

-- Hazard function

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

[Chizhov, Graham // PRE 2007,2008]

[Chizhov et al. // Neurocomputing 2006]

2. Refractory density approach (t* - parameterization)

Application

3. Hazard function

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

0~

2~))((

~ 2

VVtU

Vtm

Single LIF neuron - Langevin equation

spikethenUVif T

)()( ttUVdt

dVm

0)( t)'()'()( 2 tttt m

Fokker-Planck equation

0),(~ TUt

0),(~ t

22)(exp1

),0(~

UVV

))()((ˆ)( tUtVtu

))((ˆ)( tUUtT T

0~

2

1~~

uu

utm

0))(,(~ tTt

0),(~ t

2exp1

),0(~ uu

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

Approximation:

)(

~

21

/)(~)(tTum

m utHtH

TUVm VtH

~

2)(

2

Self-similar solution (T=const)

shapeutpamplitudet ),(,)(

),()(),(~ utptut

)(),(~)(

tTduuttwhere

ptHup

puut

pm

)(~21

Tuup

tHwhere

21

)(~0),( Ttp

0),( tp

Assumption.

2exp1

),0( uup

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

ptAu

ppu

u

)(2

1

Tuu

ptAwhere

2

1)(

0),( Ttp

0),( tp

)(TAdt

dm

),(~= tHdtd

m

HA ~

Equivalent formulation:

Frozen Gaussian distribution (dT/dt = ∞)

)(),(~)(

tTduuttwhere

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

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

otherwise

tTtuifuut

,0

)()(,exp1

),(~2

)(~

2 TFdt

dT

dt

dT

dT

dB m

m

or

)erf(1

)exp(2)(

~ 2

T

TTFwhere

[x]+ for x>0 and zero otherwise

Bdt

dm

U(t) UT

)(

~

2

1ˆ

tTum uH

0~

2

1~~

uu

utm

Approximation of hazard function in arbitrary case

tt )(

Weak stimulus Strong stimulus

0))(,(~ tTt0),(~ t

2exp1

),0(~ uu

Approximation:

))((ˆ)( tUUtTгде T

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

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

)(2

)(~),(),(

UUtUgtT

tqUVtUguгде

tqdtdq

tTutqudtdu

tU

Ttot

tot

m

)(2 tdtd

0)( t)'()'()( tttt

0~1~~

~2

2

qq

qqu

ut m

Fokker-Planck eq.

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

TqTutqut

qutqut

ququk

kk

kk

qut 2)1(2

1exp

21

),,0(~ 22

Hazard-function in the case of colored noise

)(),( ttVIdtdV

C tot

Without noise: TUU

With noise:

or

or

shapeutpamplitudet ),(,)(),,()(),,(~ qutptqut 1),,(

)(

tTduqutpdqwhere

ptHqp

qpq

kpquut

pm )(~)( 2

2

dqqTtpTqtUHwhereT

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

. ),,(~=)(

~

dudqquttT

)/,(),( tUtUk m, ),~,(~ )~(

1))((~

~dqqTtTqtUH

T

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

),,(),,(

tTqtTutpqutp

qutpqutp

),(~= tHdtd

m

Self-similar solution (T=const)

Assumption. U(t) (or T(t)) is constant or slow. Then the shape of , which is , is invariable. ~ ),,( 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 H by A is green, by B is blue,by A+B is red, exact solution is black.

Hazard function in arbitrary case

tt )(

K=1:

K=8:

Weak stimulus

Weak stimulus

Strong stimulus

Strong stimulus

Simulations with CBRD-model

Non-adaptive neurons

(4000)

Single population: comparison of CBRD with Monte-Carlo

Single population: current-step stimulation. Color noise. Adaptive neurons.

LIF

Adaptive conductance-based neuron

with IM

Single population: oscillatory input

Single population: comparison of CBRD with analytical solution for stochastic LIF in steady-state

Firing rate depends on the noise time constant.

1'

00 )/(

)(exp=

uduadu

uauHu ma

m

)(/= LT

La VUgIa

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

(*)

Single population: color noise, comparison with “adiabatic approach”

Single cell level

Populations

Htt

iAHPLHMADR IIIIIIIt

U

t

UC

*

)(

)(

,)(

)(

*

*

U

yUy

t

y

t

y

U

xUx

t

x

t

x

y

x

)()0,(0

tdtFt

AHPHMADR IIIIIfor

VUyxgI

,,,,

)( .........

t* is the time since the last spike

CBRD

Large-scale simulations(NMM & FR-models

for EEG & MRI)

From CBRD to Firing-Rate model

IVUggdt

dUC LSL ))((

)(;)(

exp1

-)(

(steady) ;)(1)(

/

),()()(

2

2

m

1/)(

/)(

2

suddenUV

dt

dUUB

duuerfeUA

gC

dtdUUBUAt

T

UV

UV

um

Lm

T

reset

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

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

)()/1,/1(

)1()(

0101

2

201 tv

K

nnn

dt

dn

dt

nd

MMMMMM

Not-adaptive neurons Adaptive neurons

• What can be modeled on population level?

• Which details are important?

• What kinds of population models do exist?

• Which one to choose?

Monte-Carlo simulations:

conventionalFiring-Rate model:

CBRD:

Nttn

tt

спайкиVVVV

tVVggItV

C

КарлоМонтеМетод

act

resetT

ILSL

)(1)(

т , если

)())((

:

0

*)0,()(

))/,()((1

*)),((

))((*

*

:

dtHttv

dtdUUBUAttUH

VUggItU

tU

C

Htt

модельRD

m

LSL

)/,()()(

))((

:FR

dtdUUBUAt

VUggIdt

dUC

модель

LSL

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

Precision: 4 2 3 5Precision for non-stationary problems:

5 2 4 5Precision for adaptive neurons :

5 1 3 4Computational efficiency:

2 5 5 4Mathematical analyzability:

1 5 4 4

)/,()()(

)()()(

2)(

)()())((

:

2

22

dtdUUBUAt

tvgtgdt

tgddt

tgd

IIVVggItV

C

модельFR

SSS

SS

S

AHPMLSL

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

Simulations with FR-model

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

Approximation of synaptic current

)( spikesss

ss ttgg

dt

dg

)()( sss VVtgi

Presynaptic spike

Postsynaptic current

Excitatory synaptic current:

Inhibitory synaptic current:

Non-dimensional synaptic conductances:

where

- rise and decay time constants - presynaptic firing rate

)()(2

2

tmdt

dm

dt

md preSS

SdS

rS

SdS

rS

)()()( NMDANMDANMDANMDA VVVftgi

)()( AMPAAMPAAMPA

NMDAAMPAE

VVtgi

iii

))062.0exp(57.3/1/(1)( VMgVf NMDA

)()( GABAGABAI VVtgi

dS

rS ,

s 1

)(tpreS

Pyramidal neurons

Interneurons

)()()( tmtwgtg SSSS Synaptic conductance:

Approximation of synaptic current

NMDAGABAAMPAS ,,

NMDAGABAAMPAS ,,

There is no plasticity in the model reproducing the experimental monosynaptic IPSCs evoked by extracellular pulse trains.

Fig 1. IPSC-kinetics in the experiment and model. The maximum amplitudes of IPSC and IPSP in the model, shown at the right, are the same as registered in the experiment, 1.2nA and 14mV.

Fig 2. Paired-pulse modulation of IPSCs in the experiment and model.

Fig 3. Frequency-dependent IPSC modulation with repetitive stimulation in the experiment and model.

M.Vreugdenhil, J.G.R.Jefferys, M.R.Celio, B.Schwaller. Parvalbumin-Deficiency Facilitates Repetitive IPSCs and Gamma Oscillations in the Hippocampus. J Neurophysiol 89: 1414-1422, 2003.

Synaptic integration

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.

FS

νext

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) (adapted by permission from Macmillan Publishers 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 from Macmillan 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

),()()( titiVUgdt

dUC IELL

),()()( UBUAt

;)(1)(

-12/)(

2/)(

2

VT

Vreset

UV

UV

um duuerfeUA

2

2

2)(

exp2

1)(

V

T

V

UVdt

dUUB

))(()()( III VtVtgti

)()( 212

2

21 tggdt

dgdt

gdII

IIIIII

Simple model of interacting cortical interneurons, evoked by thalamus

Синаптические токи и проводимости:

Мембранный потенциал:

Популяционная частота спайков:

Firing-rate model of adaptive neuron population: «interictal» activity

EI

)(),( MAHP II

)(SI

)/,()()(

)()()(

2)(

))((

)()()()(

:

2

22

dtdUUBUAt

tvgtgdt

tgd

dt

tgd

VVtgI

IVVgIIIt

VC

модельFR

SSS

SS

S

SSS

SLLMAHP

Simulations with CBRD-model

with IM and IAHP

),)(()()(

),()()(

SSS

Sexti

VtUtgtI

tItItI

)(2

mS/cm 1

mV, 5

ms, 1

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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

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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 decreasing distribution 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

),,( yxt),,( yxt

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

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PSPs and PSCs evoked by extracellular stimulation and registered

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

[S.Karnup, A.Stelzer 1999] 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 muA stimulation 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.Corradetti 2001] 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.

B

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 disinhibited neocortical 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.

Conclusion

• «Gross» processes can be described by only population approach. When the dynamics of individual neurons is important, it should be modeled on a background of population activity.

• Any population model must correctly reproduce unsteady regimes.

• For a population of LIF-neurons one can choose the Fokker-Planck-based model.

• For conductance-based neurons the CBRD-model is recommended.

• As an approximate and simple model, a modified firing-rate (FR) model can be used (with «non-stationary term»).

Chizhov, Graham // Phys. Rev. E 2007 Chizhov, Graham // Phys. Rev. E 2008Chizhov et al. // Physics Letters A 2007Chizhov et al. // Neurocomputing 2006 Rodrigues et al. // Biol Cybern. 2010

Buchin, Chizhov // Opt.Memory 2010 Чижов // Мат. биол. и биоинф. 2010Бучин, Чижов // Биофизика 2010 Чижов // Биофизика 2002 Чижов // Биофизика 2004

Чижов // Нейрокомпьютеры 2004 Чижов, Грэм //Известия РАЕН 2004 Чижов и др. // Биофизика 2009 Чижов // Вестник СПбГУ 2009 Смирнова, Чижов // Биофизика 2011

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