anton v. chizhov ioffe physico-technical institute of ras, st.-petersburg lab. of neurophysics and...
TRANSCRIPT
Anton V. Chizhov
Ioffe Physico-Technical Institute of RAS,St.-Petersburg
Lab. of Neurophysics and Physiology, Université Rene Descartes, Paris
Single cell level
Populations
Large-scale simulations(NMM & FR-models
for EEG & MRI)
Rough 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
Refractory Density model for a population of
conductance-based neurons
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
Experimental evidences of population firing rate coding
[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.
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
What happens in the populations?
Membraneequations
Eq. for spatial connections
Conductance-based neuron
uVVstVIS )(),( 0
)()()()( 0 tIVVtgtu electrodeS
SS S
S tgts )()(
)(
)(
V
hVh
dt
dh
h
)(
)(
V
mVm
dt
dm
m
)(
)(
V
nVn
dt
dn
n
2
2
x
Vk
[Hodgkin, Huxley, 1952]
EXPERIMENT
Voltage-gated channels kinetics:
),())(())()(,(
))()(,(),()(
4
3
tVIVtVgVtVtVng
VtVtVhtVmgdt
tdVC
SLLKK
NaNa
M O D E L)())(()(),( tIVtVtgtVI electrodeS
SSS
Property: Neuron is controlled by two signals [Pokrovskiy, 1978]
C
Vd
Vd
Vd
Vd
Vs
Vs
Is
Is
g=Id/(Vd-Vrev)
B
2-compartmental neuron with somatically registered synaptic currents
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:[A.V.Chizhov // Biophysics 2004]
1 mm
Model. Response of 4 pinwheels to horizontal and then vertical oriented stimuli.
Experimental map of orientationally selective columns.
Population models
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 of population models:– Input – synaptic current (+conductance)– Infinite number of neurons– Output – population firing rate
• Types of population models:
1. Direct Monte-Carlo simulation of individual neurons
2. Firing-Rate models. Assumption. Neurons are de-synchronized.
3. Probability Density Approach (PDA)
)(I
N
tttn
tt act
Nt
);(1limlim)(
0
)(Uor
(4000)
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]
• 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
ttRIVdtdV
m
resetT
m
)(2
)(2
22
resetm VVV
RIVVt
Htt
0
**),()()0,( dttttt
*
0
*** ''',,t
dttIttktttU
)*),,(( TVttUHH
[W.Gerstner, W.Kistler, 2002]
TVVVt
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
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;
[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
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
2. Refractory density approach (t* - parameterization)
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
tdtFt
)(1)exp(2
)(~,),(~2-)(
)),(),(),(),(),(/(
),()(1)(
2
m
*****
TerfT
TFUU
TTFdtdT
UB
ttggttgttgttgttgC
dtdUUBUAUH
T
AHPLHMADRm
m
).),((),(:
;),()0,(
;),()0,(
;,,)0,(,)0,(
)0,(
***
*
*
dtttdUUttUt
Iforwttwtw
Iforxttxtx
IIIforytyxtx
UtU
TTTT
AHPresetT
MresetT
HADRresetreset
reset
),(),,(),,(),,( **** ttyyttxxttUUtt
-- 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
CBRD: 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
3. 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.
Verification of CBRD
Comparison with Monte-Carlo simulations
Non-adaptive neurons
(4000)
with IM
Oscillatory input
Cравнение c аналитическим стационарным решением
[Johannesma 1968]
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
)0,()()(
2)(
2
22 tgtg
dt
tdg
dt
tgdSS
SS
SS
[S.Karnup, A.Stelzer 2001]
Experiment
Recurrent network of pyramidal cells, including all-to-all connectivity by excitatory synapses
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
2
S
d
S
7mS/cm
-80mV,V
1ms,
1ms,
3ms,
Sg
CBRD: again Hazard-function
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
3. 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
Results: comparison with individual neuron simulation in case of step input
LIF
Adaptive conductance-based neuron
Results: stationary solution
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]
(*)
Application of CBRD: coupled 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
),()(
)()()( NMDANMDANMDANMDANMDA VVVftmgi
)()( AMPAAMPAAMPAAMPA
NMDAAMPAE
VVtmgi
iii
))062.0exp(57.3/1/(1)( VMgVf NMDA
jg
jm
jV
)()( GABAGABAGABAI VVtmgi
1))2exp(1(2)(S jj dj
rj ,
s 1
)(tj
Pyramidal neurons
Interneurons
Approximations of synaptic currents
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
),,,(
),,,(YXyxd
eYXyxW
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.
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
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
Эксперимент
Модель
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:
Visual cortex
Model. Response of 4 pinwheels to horizontal and then vertical oriented stimuli.
Experimental map of orientationally selective columns.
1 mm
Preferred direction Null direction
V
GE
GI
Refractory Density Approach (RDA)-based ring model for
HH-neurons with synaptic kinetics
RDA-based ring model for
LIF-neurons with synaptic kinetics
Kolmogorov-Fokker-Planck (KFP)-based ring model for
LIF-neurons with synaptic kinetics
2-D distributedRefractory Density Approach (RDA)-based model
for Hodgkin-Huxley (HH)-neurons with synaptic kinetics
KFP-based ring model for
LIF-neurons with instantaneous synaptic currents
Firing-Rate (FR)-based ring model with
instantaneous synaptic currents
=
Hierarchy of visual cortex models
2-d RD model
FR-shunt ring model
KFP-shunt ring model
RD-cos ring modelRD-exp ring model
Paremeters:FR ring model
Visual illusion (tilt after-effect)
Explanation:
Model
Conclusion
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)
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:
3 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):
Colleagues:
Lyle Graham
Adrien Schramm
Anatoliy Buchin
Elena Smirnova
Andrey Turbin