Models of neuronal populations
description
Transcript of 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
• 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
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ms
drest
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m
s
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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):
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tt
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or
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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
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)(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
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VtsgtI
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V
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2
)(
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ρ
Hz
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tsg
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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
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U
t
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*
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*
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yUy
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t
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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
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iAHPLHMADR IIIIIIIdt
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Full single neuron model
Threshold model
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Boundary conditions:
-- firing rate)()0,(0
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).),((),(:
;),()0,(
;),()0,(
;,,)0,(,)0,(
)0,(
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,,,,
)( .........
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*
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2
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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
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
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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
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tItItI
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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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