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?

Commonly information
is coded by firing rate

[R.M.Bruno, B.Sakmann // Science 312:1622-1627, 2006]
Population PSTH of thalamic neurons’
responses to a 2-Hz sinusoidal deflection of their
[E.Aksay, R.Baker, H.S.Seung, D.W.Tank
Activity of a
J.Neurophysiol. 84:1035-1049, 2000]
respective principal whiskers (n = 40 cells).
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.

Commonly populations are
localized in cortical space

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.

Pure population events
observed in experiments:

• Evoked responses

• Oscillations

•Traveling waves

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

Synaptic
conductance
kinetics
GABA-IPSC AMPA-EPSC
AMPA-EPSC GABA-IPSC
Membrane
GABA-IPSP AMPA-EPSP equations
AMPA-EPSP GABA-IPSP
PSP
PSP
Threshold criterium
Spike
Spike

Population model
Firing rate
Firing rate

Eq. for spatial
connections

• ionic channel kinetics
• input signal is 2-d Model of a pyramidal neuron
dV
C = − I Na − I DR − I A − I M − I H − I L − I AHP − s(t ) (V − V0 ) + u(t ) + η (t )
dt
u(t ) = ∑S g S (t ) (VS − V0 ) + I electrode (t )
I ... = g... x p (t ) y q (t ) (V (t ) − V... ) s(t ) = ∑S g S (t )
dx x∞ (U ) − x
= ,
dt τ x (U )
dy y∞ (U ) − y
=
dt τ y (U )
Approximations for
I Na , I DR , I A , I M , I H
are
from [L.Graham, 1999]; EXPЕRIМЕNТ
IAHP is from [N.Kopell et al., 2000]

MODEL
Color noise model
(Ornstein-Uhlenbeck
process):

dη
τ = −η + 2τ σξ (t )
dt

• neuron is spatially distributed A

2-comp. neuron with synaptic currents at somas
15
0 B
PSC, exp.
PSP, exp. Vd
-50 PSP, model 2 10
PSP, model 1

PSP, mV
PSC, pA

Vd
-100
Vs Is
5
-150

-200 0
0 5 10
t, ms
15 C
Two boundary problems: ∂V  ∂V  g=Id/(Vd-Vrev) Vd
A) current-clamp to register PSP: ∂X = R Gs  V + ;
X =0  ∂T 
B) voltage-clamp to register PSC: V (T ,0) = 0; Vd
∂V
Vd Is
= R I S (T ) Vs
∂X X =L
V0 [F.Pouille,
∂V ∂ V 2 M.Scanziani
− +V = 0 //Nature, 2004]
∂T ∂X 2
X=0 X=L Figure Transient activation of somatic and delayed
Solution: activation of dendritic inhibitory conductances in
dV I [A.V.Chizhov //
τm = −(V − V rest ) + ρ (Vd − V ) − S Biophysics 2004]
experiment (solid lines) and in the model (small circles).
dt Gs A, Experimental configuration.
dV 1  ∂I d  B, Responses to alveus stimulation without (left) and with
τ m d = −(Vd − V rest ) − (2 + ρ )(Vd − V ) − τ m
 + 3I d 
 (right) somatic V-clamp.
dt ρ Gs  ∂t  C, In a different cell, responses to dynamic current injection
Parameters of the model: in the dendrite; conductance time course (g) in green, 5-nS
τm= 33 ms, ρ = 3.5, Gs= 6 nS in B and 2.4 nS in C peak amplitude, Vrev=-85 mV.

• spatial structure of connections

1 mm

Эксперимент. Зрительная кора. Карта
ориентационной избирательности
активности нейронов.

Модель “Pinwheels” карты
ориентационной
избирательности входных
сигналов.
Модель. Ответ зрительной коры на полосу горизонтальной,
а затем вертикальной ориентации.

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 (4000)

1 nact (t; t + ∆t )
ν (t ) = lim lim
∆t →0 N →∞ ∆t N

Direct Monte-Carlo simulation
of individual neurons: Types of population models
∂V
C = I − g L (V − VL ) + σ I ξ (t )
∂t
если V > V T , т V = Vreset и спайк
1 nact ( t + ∆t )
ν (t ) =
∆t N
Firing-rate:
ν (t ) = f ( I (t )) f “f-I-curve”
dν
or τ = −ν + f ( I (t ))
dt
dU
or C = I − g L (U − VL ),
dt I
ν (t ) = ~(U (t ))
f
Assumption. Neurons are de-synchronized.
Probability Density Approach (PDA):
(4000)
RD модель :
∂ρ ∂ρ
+ = − ρH
∂t ∂t *
 ∂U ∂U 
C +  = I − g L (U − VL )
 ∂t ∂t * 
1
H (U ( t , t*)) = ( A(U ) + B(U , dU / dt ))
τm
∞
v (t ) = ρ (t ,0) = ∫ ρ H dt *
+0

Idea of Probability Density Approach (PDA)
Single neuron equation (e.g. H-H model)
r
dX
= F(X ) + S
r r r
dt
r
where F is the common deterministic part,
r
S is the noisy term.
X = (V , m, h, n )
r
For classical H-H:

ρ ( X , t)
r
Eq. for neural density

∂ρ
∂t
=−
∂
∂X
r r
( ∂
r ⋅ F(X ) ρ + r
∂X
)  t ∂ρ 
⋅ W r 
 ∂X  [B.Knight 1972]
 
t r
where the matrix W represents the influence of noise S

Problem! The equation is multi-dimensional.
Particular cases are [A.Omurtag et al. 2000]
X ≡V - membrane potential [D.Nykamp, D.Tranchina 2000]
[N.Brunel, V.Hakim 1999], …

X ≡ t* - time passed since the last spike [J.Eggert, JL.Hemmen 2001]
[А.Чижов, А.Турбин 2003]

X ≡τ - time till the next spike [A.Turbin 2003]

ρ
Simplest 1-d PDAs
• Kolmogorov-Fokker-Planck eq. for ρ(t,V)
Hz
Leaky Integrate-and-Fire (LIF) neuron:

dV
τm = −V + RI (t ) + η (t ),
dt
if V > V T then V = Vreset 0 Vreset VT

< η (t ) >= 0, < η (t ) η (t ' ) >= τ m σ 2 δ (t − t ' ) ν

∂ρ ∂ σ 2 ∂2ρ
τm = [(V − RI ) ρ ] + + ν ⋅ δ (V − Vreset )
∂t ∂V 2 ∂V 2 Hz
σ 2 ∂ρ Problem! Voltage can not
ν (t ) = uniquely characterize
2 ∂V V =V T neuron’s state. 0 t
• Refractory density ρ(t,t*) for SRM - neurons
∂ρ ∂ρ ∞
+ ∗ = −ρ H ρ (t ,0) ≡ ν (t ) = ∫ ρ (t , t * ) dt *
∂t ∂t 0

H = H (U (t , t*), V T )
Spike Response Model (SRM):

U (t , t * ) = η (t * ) + ∫ k (t * , t ') I (t ' ) dt '
t*
[W.Gerstner, W.Kistler, 2002]
0

1-D Refractory Density
Approach for conductance-
based neurons (CBRD)
[A.V.Chizhov, L.J.Graham // Phys. Rev. E 2007, 2008]

1. Threshold single-neuron model

2. Refractory density approach (t* -
parameterization)

3. Hazard-function t* is the time since the last spike;
H ≈ A+ B ∂ρ ∂ρ
+ = −ρ H
∂t ∂t ∗
 ∂U ∂U 
C + *  = − I DR − I A − I M − I H − I L − I AHP − I i
H(U) = ‘frozen stationary’ + ‘self-similar’  ∂t ∂t 
solutions of Kolmogorov-Fokker-Planck eq. ∂x ∂x x (U ) − x
+ * = ∞ , ∞
∂t ∂t τ x (U )
for I&F neuron with white or color v (t ) = ρ H dt *
∫
noise-current ∂y ∂y y ∞ (U ) − y
+ = +0
∂t ∂t * τ y (U )

1. Threshold neuron model
Full single neuron model dV
C = − I Na − I DR − I A − I M − I H − I L − I AHP − I i
dt
Approximations for I Na , I DR , I A , I M , I H are taken from [L.Graham, 1999]; IAHP is from [N.Kopell et al., 2000]

Threshold model dU
C = − I Na − I DR − I A − I M − I H − I L − I AHP − I i
dt
dx x∞ (U ) − x dy y∞ (U ) − y
= , =
dt τ x (U ) dt τ y (U )
if U > V T then U = U reset = −40 mV
for I DR : x = x reset = 0.262, y = y reset = 0.473;
for IA : x = x reset = 0.743, y = y reset = 0.691;
for IH : y = y reset = 0.002;
for IM : x = x + ∆ x reset , ∆ x reset = 0.18 (1 − x );
for I AHP : w = w + ∆ w reset , ∆ w reset = 0.018 (1 − w).

Models of neuronal populations

2. Refractory density approach (t* - parameterization)
t* is the time since the last spike;
ρ = ρ (t , t * ), U = U (t , t * ), x = x (t , t * ), y = y (t , t * )

∂ρ ∂ρ d • ∂ • dt * ∂ • ∂ • ∂ •
+ ∗ = −ρ H = + = +
∂t dt ∂t * ∂t ∂t *
∂t ∂t
dt

 ∂U ∂U 
C + *  = − I DR − I A − I M − I H − I L − I AHP − I i
 ∂t ∂t 
∂x ∂x x∞ (U ) − x
+ *= ,
∂t ∂t τ x (U ) I ... = g ... x y (U − V... )
for I DR , I A , I M , I H , I AHP
∂y ∂y y∞ (U ) − y
+ =
∂t ∂t * τ y (U )
H (U ) = 1 τ m ( A(U ) + B (U , dU dt ) ) -- Hazard function
τ m = C /( g DR (t , t * ) + g A (t , t * ) + g M (t , t * ) + g H (t, t * ) + g L + g AHP (t , t * ))
Boundary conditions:
A(U ) = exp(6.1 ⋅ 10−3 − 1.12 T − 0.257 T 2 − 0.072 T 3 − 0.0117 T 4 ). ∞

B (U ) = -τ m 2
dT ~
F (T ), T=
U −U
T
, ~
F (T ) =
2 exp( −T ) 2
ρ (t ,0) = ∫ ρ F dt ∗ ≡ ν (t ) -- firing rate
dt σ π 1 + erf (T ) +0

U (t ,0) = U reset
x (t ,0) = x reset , y (t ,0) = y reset for I DR , I A , I H ;
Application x (t ,0) = x (t , t *T ) + ∆ x reset for I M ;
w(t ,0) = w(t , t *T ) + ∆ w reset for I AHP ;
t *T : U (t , t *T ) = U T ( dU (t , t *T ) dt ).

3. Hazard-function in the case of white noise-current
(First-time passage problem)
A – solution in case of steady stimulation (self-similar);
Approximation: H ≈ A+ B
B – solution in case of abrupt excitation

Single LIF neuron - Langevin equation
dV < η (t ) > = 0
τm = −V + U (t ) + η (t ) < η (t )η (t ' ) >= σ 2 τ m δ (t − t ' )
dt
if V < UT then spike

Fokker-Planck equation
~ ∂
∂ρ  ~
σ 2 ∂ρ 
τm ~−
+ (U (t ) − V ) ρ 2 ∂V  = 0 ρ (t,U T ) = 0
~
∂t ∂V   ρ (t ,−∞) = 0
~
σ 2 ∂ρ~
exp (− (V − U ) 2 σ 2 )
H (t ) ≡ − ~ 1
ρ (0,V ) =
2τ m ∂V V =U T πσ

u (t ) = (V (t ) − U (t )) σ
ˆ
T (t ) = (U T − U (t )) σ
ˆ
~ ~ ρ (t , T (t )) = 0
~
∂ρ ∂  ~ − 1 ∂ρ  = 0
τm + −uρ ρ (t ,−∞) = 0
~
∂t ∂u  2 ∂u 

exp(− u 2 )
~ 1
ρ (0, u ) =
~ 1 ∂ρ ~ π
H (t ) ≡ H (t ) / τ m = −
2τ m ∂u u =T ( t )

Self-similar solution (T=const)
Equivalent formulation:
~
ρ ( t , u ) = ρ (t ) p (t , u )
T (t )
where ρ (t ) = ∫ ~
ρ (t , u ) du
−∞

ρ (t ) − amplitude , p(t , u ) − shape
p (t , T ) = 0
∂p ∂  1 ∂p  ~ ~ 1 ∂p
τm + − u p − = H (t ) ⋅ p where H (t ) = − p(t ,−∞) = 0
∂t ∂u  2 ∂u 
 2 ∂u u = T exp(− u 2 )
1
p(0, u ) =
dρ ~ π
τm = − ρ H (t ),
dt
~
Assumption. U(t)=const (or T(t)=const). Notation: A ≡ H
~
Then the shape of ρ , which is p(t , u) , is invariable.
∂  1 ∂p  1 ∂p p (t , T ) = 0
 −u p−  = A(t ) ⋅ p where A(t ) = −
∂u  2 ∂u  2 ∂u u =T p(t ,−∞) = 0

dρ
τm = − ρ A(T )
dt

Frozen Gaussian distribution (dT/dt = ∞)

Assumption. 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).

dρ
τm = −ρ B
T (t )
where ρ (t ) = ∫ ~
ρ (t , u ) du
dt −∞

τ m dρ τ dρ  dT 
or B=− =− m
ρ dt ρ dT  dt  +
 
U(t) UT

~(t , u ) =  π exp(− u ), if
 1
For the simplicity, we consider the case of 2
u ( t ) < T (t )
arbitrary but monotonically increasing T(t) and ρ
the Gaussian distribution 
0, otherwise


τ m dρ  dT   dT  ~
B=− = −τ m 2   F (T )
ρ dT  dt  +
   dt  +

~ 2 exp( −T 2 )
where F (T ) =
π 1 + erf(T )
[x]+ for x>0 and zero otherwise

Approximation of hazard function in arbitrary case
~
∂ρ ∂  ~ ρ (t , T (t )) = 0
~ где T (t ) = (U T − U (t )) σ
τm + ~ − 1 ∂ρ  = 0
−uρ
ˆ
∂t ∂u  2 ∂u  ρ (t ,−∞) = 0
~
 
exp(− u 2 )
~ 1
1 ∂ρ ~ ρ (0, u ) =
H=
ˆ π
2τ m ∂u u =T ( t )
A – solution in case of steady stimulation (self-similar);
Approximation: H ≈ A+ B
B – solution in case of abrupt excitation
Weak stimulus Strong stimulus

Approximation of H by A is green, by B is blue, by A+B is red, exact solution is black. ν (t ) = ∂ρ ∂t

3. Hazard-function in the case of colored noise
dU Langevin equation
Without noise: C = − I tot (U , t ) U < UT
dt
dV du ~
With noise: C = − I tot (V , t ) + η (t ) V < UT τ m (U , t ) = −u + q(t ), u < T (t )
dt dt
dη dq
τ = −η + 2τ σ ξ (t ) or τ = − q + 2τ ξ (t )
dt dt
< ξ (t ) > = 0 где u = gtot (U , t )(V − U ) / σ , q = η (t ) / σ ,
< ξ (t ) ξ (t ' ) >= τ δ (t − t ' ) ~
T (t ) = g (U , t )(U T − U ) / σ
tot

Fokker-Planck eq.
∂ρ ∂  − u + q ~  ∂  q ~  1 ∂ ρ
~ ~ 2
ρ (t, u = ∞, q) = ρ (t , u, q = −∞) =
~ ~
+  τ ρ  + − ρ  − =0
∂t ∂u  m  ∂q  τ  τ ∂q
2 ~ ~ ~ ~
= ρ (t, u, q = +∞) = ρ (t , u = T , q ≤ T ) = 0

~ 1 ∞
~ ~ ~ ρ (t = 0, u, q) =
~ 1+ k
2π k
1+ k
exp [ 
− (1 + k )u 2 − q 2 + 2 qu  ]
H (U (t )) ≡
ρT
~
∫ ( q − T ) ρ (t, T , q) dq, k (U , t ) ≡ τ m (U , t )/τ
 2k 

or ~
T ∞
∞
ρ ( t , u, q ) = ρ ( t ) p ( t , u, q )
T (t )
~ where ∫−∞ dq ∫− ∞ p (t , u, q ) du = 1 ρ (t ) = ∫ ∫ ρ (t, u, q) dq du.
~
ρ (t ) − amplitude , p(t , u ) − shape −∞−∞

dρ ~ ∞
τm = − ρ H (t ), where ~ ~ ~
H (U (t )) ≡ ∫ ( q − T ) p (t , T , q ) dq
dt ~
T

∂p ∂ ∂ ∂2 p  ~ p(t , u = ∞, q) = p(t , u, q = −∞) =
τm = (u − q) p + k  qp + 2  + H (t ) p
∂t ∂u  ∂q ∂q  ~ ~
= p(t , u, q = +∞) = p(t , u = T (t ), q ≤ T (t )) = 0

Self-similar solution (T=const)
Assumption. U(t) (or T(t)) is constant or slow.
~
Then the shape of ρ , which is p(t , u, q), is invariable.

∂ ∂ ∂2 p  p(t, u = ∞, q) = p(t, u, q = −∞) =
(u − q) p + k  qp + 2  + A p = 0 ~ ~
∂u  ∂q ∂q  = p(t , u, q = +∞) = p(t, u = T , q ≤ T ) = 0
∞
~ ~
A = ∫ ( q − T ) p (t , T , q ) dq
where ~ 1+ k
T =T
~
T 2

q

u

A ∞ (T) = exp(0.0061 - 1.12 T - 0.257 T 2 - 0.072 T 3 - 0.0117 T 4 )

Hazard function in arbitrary case H ≈ A+ B
K=1: Weak stimulus Strong stimulus

K=8: Weak stimulus Strong stimulus

Approximation of H
by A is green,
by B is blue,
by A+B is red,
exact solution is black.
ν (t ) = ∂ρ ∂t

CBRD Single cell level

t* is the time since the last spike
∂ρ ∂ρ
+ ∗ = −ρ H
∂t ∂t
 ∂U ∂U 
C +  = − I DR − I A − I M − I H − I L − I AHP − I i
 ∂t ∂t * 
∂x ∂x x∞ (U ) − x Populations
+ *= , I ... = g ... x y (U − V... )
∂t ∂t τ x (U )
for I DR , I A , I M , I H , I AHP
∂y ∂y y∞ (U ) − y
+ *=
∂t ∂t τ y (U )
∞
ρ (t ,0) = ∫ ρ F dt ∗ ≡ ν (t )
+0

Large-scale simulations
(NMM & FR-models
for EEG & MRI)

Simulations. Current-step stimulation.
Comparison with Monte-Carlo.
Non-adaptive neurons

(4000)

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

Adaptive conductance-based neuron

Simulations. Oscillatory input.

with IM

Simulations. Constant current stimulation.
Comparison with analytical solution.

[Johannesma 1968]

Simulations. Constant current stimulation.
Color noise.
Comparison with analytical solution.
−1 (*)
 a  u ' τ m H (u )  
ν = τ m ∫0 exp − ∫0 du  /(a − u ′) du′
  a−u  
a = I a /g L (U T − VL )

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

Firing rate depends on the noise
time constant.

Interconnected populations

Synaptic current
kinetics

GABA-IPSC AMPA-EPSC
AMPA-EPSC GABA-IPSC
Membrane
GABA-IPSP AMPA-EPSP equations
AMPA-EPSP GABA-IPSP
PSP
PSP
Threshold criterium
Spike
Spike

Population model
Firing rate
Firing rate

Pyramidal neurons
Approximations of synaptic currents 200 AMPA-PSC 40 NMDA-PSC
(with PTX, APV) (with PTX, CNQX)
150 20
Vh=-40 mV
Vh=-80 mV experiment
model 0

PSC, pA
100

PSC, pA
experiment
model
Excitatory synaptic current: -20
50
iE = i AMPA + i NMDA Vh=-40 mV
-40

i AMPA = g AMPA m AMPA (t ) (V − V AMPA ) 0
Vh=+20 mV -60
Vh=+20 mV

iNMDA = g NMDAm NMDA (t ) f NMDA (V ) (V − VNMDA ) -50
-80
0 10 20 30 40 50 0 25 50 75 100
t, ms t, ms
gj - maximum specific conductance, 0
0
mj - non-dimensional conductance GABA-PSC
(with CNQX, D-AP5)
Vj - reversal potential -100
f NMDA (V ) = 1 /(1 + Mg / 3.57 exp( −0.062V )) -50 Vh=-64 mV fast GABA-A -IPSC

PSC, pA
(with CNQX, D,L-APV)

PSC, pA
-200
Vh=-60 mV
Inhibitory synaptic current:
-300
i I = g GABAmGABA (t ) (V − VGABA ) -100
experiment
model experiment
Non-dimensional synaptic conductances: -400 model

d 2m j dm j
ττ + (τ rj + τ d ) + m j = S (ν j ),
r d 0 10 20 30 40 50 -500
0 10 20 30 40 50
j j j t, ms t, ms
dt 2 dt
Interneurons
j = AMPA , GABA , NMDA 500 AMPA-PSC
(with PTX, APV) 150 NMDA-PSC
where S ( ν j ) = 2 ( 1 + exp( −2τ ν j ) ) − 1 τ = 1 µs 400
(with PTX, CNQX)
τ r , τ d - rise and decay time constants
j j experiment
100
Vh=-40 mV
300 Vh=-80 mV
ν j (t ) - firing rate on j-type axonal terminals model
PSC, pA

50

PSC, pA
200 experiment
0 model

100 -50
Vh=-40 mV
0 -100 Vh=+20 mV
Vh=+20 mV
-100 -150
0 10 20 30 40 50 0 25 50 75 100
t, ms t, ms

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

I i (t ) = I ext (t ) + I S (t ), Model
with IM and IAHP
I S (t ) = g S (t ) (U (t ) − VS ),
2
2 d g S (t ) dg (t )
τS + 2τ S S + g S (t ) = g S τ ρ (t ,0)
dt 2 dt
Experiment I = 150 pA
ext

τ S = 5.4 ms,
τ = 1 ms,
VS = 5 mV,
g S = 1 mS/cm 2
σ V = 2 mV ( at rest )

[S.Karnup, A.Stelzer 2001]

Simulations. Gamma rhythm. Recurrent network of interneurons,
including all-to-all connectivity by inhibitory synapses
τ S = 3ms,
d 2 g S (t ) dg (t )
I i (t ) = I ext (t ) + I S (t ), τ2
S + 2τ S S + g S (t ) = g S τ ν (t − τ d ) τ d = 1ms,
dt 2 dt τ = 1ms,
I S (t ) = g S (t ) (U (t ) − VS ), for density approach ν (t ) = ρ (t , t * = 0) VS = -80mV,
g S = 7mS/cm 2

Model Experiments Oscillations
Control (“Kainate”) +“Bicuculline”
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,
Spikes in single neurons spike trains and monosynaptic
responses .

Conductances The model reproduces the following
characteristics of gamma-oscillations :
frequency of population spikes
a single pyramidal cell does not fire
Power Spectrum of Extracellular Potentials every cycle
every interneuron fires every cycle
bic
con amplitude of EPSC is less than that
of IPSC
blockage of GABA-A receptors
[Khazipov, Holmes, 2003] reduces the frequency
Kainate-induced oscillations [A.Fisahn et al., 1998]
in CA3. Cholinergically induced
oscillations in CA3
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
[N.Hajos, J.Palhalmi, E.O.Mann, B.Nemeth, homogeneous in space along the
Spike timing of pyramidal and inhibitory cells. O.Paulsen, and T.F.Freund. J.Neuroscience, cortical surface (data not shown)
24(41):9127–9137, 2004]

Spatial connections

ϕij (t , x, y ) = ∫ ∫ ν i (t − d ( x, y , X , Y ) / c, X , Y ) W ( x, y , X , Y ) dX dY ,
d ( x, y , X , Y ) = ( x − X ) 2 + ( y − Y ) 2 Experiment:

φ ( t , x , y ) - firing rate on presynaptic terminals;
ν ( t , x , y ) - firing rate on somas.
Assumption: distances from soma to synapses have exponentially decreasing
distribution p(x) [B.Hellwig 2000].

d ( x , y , X ,Y )
−
W ( x, y , X , Y ) = e λ

∂ 2φ ∂φ 2 ∂ φ
2
∂ 2φ   2 ∂
+ 2γ + γ φ − c  2 + 2  =  γ + γ  ν (t , x, y )
2
 ∂x
∂t 2 ∂t  ∂y  
 ∂t 
[V.Jirsa, G.Haken 1996]
where γ = c/λ; c – the average velocity of spike [P.Nunez 1995]
propagation along the cortex surface by axons; [J.Wright, P.Robinson 1995]

λ – characteristic axon length. [D.Contreras, R.Llinas 2001]

Model Experiments Evoked responses
A B

The model reproduces postsynaptic currents
and postsynaptic potentials registered on
somas of pyramidal cells, namely:
monosynaptic EPSCs and EPSPs
[S.Karnup, A.Stelzer
1999] Effects of GABA-A disynaptic IPSC/Ps followed be EPSC/Ps
receptor blockade on
orthodromic potentials in CA1 polysynaptic EPSC/Ps
pyramidal cells. Superimposed
C
responses in a pyramidal cell
reduction of delays in polysynaptic EPSCs
soma before and after decay of excitation after II component of
application of picrotoxin (PTX,
100 muM). Control and PTX poly-EPSCs in presence of GABA-A receptor
recordings were obtained at V block.
rest (-64 mV; 150 muA
stimulation intensities; 1 mm The model predicts that the evoked responses
distance between stratum [B.Mlinar, are essentially non-homogeneous in space:
radiatum stimulation site and A.M.Pugliese,
perpendicular line through
stratum pyramidale recording R.Corradetti
site). The recordings were 2001] Components of
carried out in ‘minislices’ in complex synaptic
which the CA3 region was cut responses evoked in CA1
off by dissection. pyramidal neurones in the
presence of GABAA
receptor block.

PSPs and PSCs evoked by
extracellular stimulation and registered
at 3.5cm away, w/ and w/o kainate.
[V.Crepel, R.Khazipov,
Y.Ben-Ari, 1997]
In normal concentrations of Mg and in the
absence of CNQX, block of GABA-A Spatial profiles of membrane potential and
receptors induced a late synaptic response. firing rate in pyramids.

Model Experiments Waves
In the case of reduced GABA-reversal
potential VGABA= -50mV and stimulation Waves with unchanging chape and
by extracellular electrode we obtain a velocity are observed in cortical tissue
traveling wave of stable amplitude and in disinhibiting or overexciting
velocity 0.15 m/s. The velocity is much conditions. The waves are produced
less than the axon propagation velocity by complex interaction of pyramidal
(1m/s) and is A determined mostly by cells and interneurons. That is
synaptic interactions. confirmed by much lower speed of the
wave propagation comparing with the
140 voltage, pyramids
axon propagation velocity which is the
voltage, interneurons
120 firing rate, pyramids
firing rate, interneurons
coefficient in the wave-like equation.
100
80
-40 Analysis of wave solutions and more

mV
Hz

60 B detailed comparison with experiments
40
20 -60
are expected in future.
0
0 25 50 75 100
ms
[Leinekugel et al. 1998]. Spontaneous GDPs
propagate synchronously in both hippocampi from septal to
120 voltage, pyramids
temporal poles. Multiple extracellular field recordings from the CA3
100
voltage, interneurons
firing rate, pyramids region of the intact bilateral septohippocampal complex.
firing rate, interneurons
0.15m/s -40 Simultaneous extracellular field recordings at the four recording
80
sites indicated in the scheme. Corresponding electrophysiological
mV
Hz

60
traces (1–4) showing propagation of a GDP at a large time scale.
40
-60
20

0
10 20 30 40
mm

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

From CBRD to Firing-Rate model

Macro- and meso-scale
macro-scale meso-scale micro-scale

external granular
layer

external pyramidal
layer

internal granular
layer

internal pyramidal
layer

AP generation zone synapses

[S.Kiebel]
[C.Friston]

Not-adaptive neurons Firing-rate model Adaptive neurons
dU
C = −( g L + g S )(U − VL ) − I C
dU
= −( g L + g S )(U − VL ) − g M n 2 (t )(U − VM ) − g AHP w(t )(U − V AHP ) + I
dt dt
Hazard-function: Hazard-function:
ν (t ) = A (U ) + B(U , dU dt ) -- firing rate ν (t ) = A (U ) + B(U , dU dt ) -- firing rate
τ m = C / gL τ m = C / gL
−1 −1
   
T T
(V −U ) / σ (V −U ) / σ

A(U ) = τ m π ∫ e (1 + erf (u))du  ; A(U ) = τ m π ∫ e (1 + erf (u))du  ;
u2 u2
(steady) (steady)
   
 (V reset −U ) / σ   (V reset −U ) / σ 
1  dU   (V T − U )2  1  dU   (V T − U )2 
B(U ) = -τ m exp − ; ( sudden) B(U ) = -τ m exp − ; ( sudden)
π σ  dt  +
  
 σ2 
 π σ  dt  +
  
 σ2 

d 2w dw χ (1 − w)
τ 1 τ AHP
0
+ (τ 1 + τ AHP )
0
− w∞ + w = v(t )
K (1 / τ 1 ,1 / τ AHP )
AHP 2 AHP 0
dt dt AHP

d 2n dn ξ (1 − n)
τ1 τM
0
+ (τ 1 + τ M ) − n∞ + n =
0
v(t )
K (1 / τ 1 ,1 / τ M )
M 2 M 0
dt dt M

Oscillating input Oscillating input

[Chizhov, Rodrigues, Terry // Phys.Lett.A, 2007 ] [Чижов, Бучин // Нейроинформатика-2009 ]

Синаптические токи и проводимости:
Simple model of interacting
iE (t ) = g E (t ) (V (t ) − VE ) i I (t ) = g I (t ) (V (t ) − VI )
d 2 gE dg d 2gI dg cortical interneurons,
τ 1Eτ 2E + (τ 1E + τ 2E ) E + g E = τ g E ν ext (t ) ττ I I
+ (τ 1I + τ 2I ) I + g I = τ g I ν (t )
dt 2
dt
1 2
dt 2
dt evoked by thalamus
Мембранный потенциал:

dU
C = − g L (U − VL ) + i E (t ) + i I (t ),
dt Experiment
Популяционная частота спайков:

ν (t ) = A (U ) + B (U ),
-1
 (VT −U ) / σ V 2

A(U ) = τ m π ∫ e (1 + erf (u ) ) du  ;
u2
 
 (Vreset −U ) / σ V 2 
1  dU   (V T − U ) 2 
B(U ) = × exp −
 

2π σ V  dt  +
   2σ V2

Model

gE
νext
FS
ν gI

Рис. 12. Схема активности
популяции FS (fast spiking)
нейронов, возбуждаемых
внешним стимулом νext(t),
Рис. 13. Постсинаптический
приходящим из таламуса.
(моносинаптический) ток в FS-
Обозначения: ν(t) –
нейроне при слабой
популяционная частота спайков
таламической стимуляции
FS нейронов, gE(t), gI(t) –
током 30 µA и потенциале Рис. 14. Ответы FS-нейронов на таламическую стимуляцию
проводимости возбуждающих и
фиксации -88 mV в током 120 µA в эксперименте (слева) (adapted by permission from
тормозящих синапсов.
эксперименте (вверху) (adapted Macmillan Publishers Ltd: (Cruikshank et al., 2007), © 2007) и в
by permission from Macmillan модели (справа). A, B - постсинаптические токи при
Publishers Ltd: (Cruikshank et al., потенциале фиксации -88, -62, и -35 mV; C, D - синаптические
2007), copyright 2007) и в модели проводимости; E, F – постсинаптические потенциалы U и
(внизу). модельная популяционная частота ν.

Частотная модель популяции
адаптивных нейронов:
«интериктальная» активность
I AHP (ν ), I M (ν )
I
E
I S (ν )

FR модель :
∂V
C = I − I AHP (ν ) − I M (ν ) − g L (V − VL ) − I S (ν )
∂t
I S = g S (t )(V − VS )
d 2 g S (t ) d g S (t )
τS 2
2
+ 2τ S + g S (t ) = g Sτv(t )
dt dt
ν (t ) = A(U ) + B(U , dU / dt )

Monte-Carlo conventional
simulations: Firing-Rate modified Firing-Rate CBRD:
model: model (non-
Метод Монте − Карло :
∂V FR модель : stationary and RD модель :
C = I − ( g L + g S )(V − VL ) + σ I ξ ( t )
∂t adaptive): ∂ρ ∂ρ
dU + = − ρH
если V > V T , т V = Vreset и спайк C = I − ( g L + g S )(U − VL ) ∂t ∂t *
dt
1 nact ( t + ∆ t ) ν (t ) = A(U ) + B (U , dU / dt )
FR модель :  ∂U ∂U 
ν (t ) = C +  = I − ( g L + g S )(U − VL )
∆t ∂V  ∂t ∂t * 
N C = I − ( g L + g S )(V − VL ) − I M (ν ) − I AHP (ν )
∂t 1
2 H (U (t , t*)) = ( A(U ) + B(U , dU / dt ))
τS 2 d g S (t )
+ 2τ S
d g S (t )
+ g S (t ) = g Sτv (t ) τm
dt 2 dt ∞
ν (t ) = A(U ) + B (U , dU / dt ) v (t ) = ρ (t ,0) = ∫ ρ H dt *
+0

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

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