A Fokker-Planck Model in for Two
Interacting Populations of Neurons
J.A. Carrillo1, S. Cordier2, S.Mancini2
1
2
ICREA, Universitat Autònoma de Barcelona
MAPMO, Fédération Denis Poisson, Université d’Orléans
Stochastic Models in Neuroscience
CIRM, 18 January 2010
Outline
• The Wilson-Cowan and Deco-Martı̀ model
• The stochastic ODE system and its Fokker-Planck equation
• Stationnary solution (equilibrium)
• Time dependent problem and Generalized relative entropy
• Numerical simulations
• Slow-fast behavior
Wilson-Cowan model
. Two neurons populations/cells
. Neglect spatial interactions
. Time dynamics
E(t) = excitatory cells prop. activ/unit time at time t
I(t) = inhibitory cells prop. activ/unit time at time t
Model : The value of E(t + τ ) and I(t + τ ) is proportional to number of cells
which are sensitive (not refractory) and which also receive at least threshold,
θ, excitation at time t.
[WC] H.R. Wilson, J.D. Cowan, Biophysical Journal (1972)
Wilson-Cowan, the model
. Excit./Inhibit. sensitives prop. of cells (r refractory time):
1−
Z t
t−r
E(t0)dt0 ,
1−
Z t
t−r
I(t0)dt0
. Sub-pop. prop. receiving at least θ excitation/time, as a fonction of the
mean excitation x(t) => response fonction (sigmoide):
S(x) =
Z x
θ
D(θ)dθ ,
D(θ) = thresholds distribution
. α(t) decreasing rate of stimuli effect, ci coeff.connexion, Ie(t), Ii(t) stimuli;
E(t + τ ) = 1 −
I(t + τ ) = 1 −
Z t
t−r
Z t
t−r
E(t0)dt0 Se
I(t0)dt0 Si
Z t
−∞
Z t
| −∞
α(t − t0)[c1E(t0) − c2I(t0) + Ie(t0)]dt0
α(t − t0)[c3E(t0) − c4I(t0) + Ii(t0)]dt0
{z
x(t)
}
Wilson-Cowan, simplified
Time averaging ⇒:
Z t
t−r
E(t0)dt0 → rE(t) ,
Z t
−∞
α(t − t0)E(t0)dt0 → kE(t)
First order expansion in τ = 0 ⇒
dE
τ
= −E + (1 − rE)Se(kc1E − kc2I + kIe(t))
dt
dI
0
τ
= −I + (1 − rI)Si(k0c3E − k0c4I + k0Ii(t))
dt
. Hysteresis multiple loops and limit cycles.
. Results are independent of the choice of the sigmoide.
The Deco-Martı̀ model
w+ excitation coeff. between neurons of the same population
w− excitation coeff. between neurons of different populations
wI inhibition coeff. between all neuronal populations
Synaptic force between population i and j:

w − w
I
+
wij =
w− − w
I
i=j
i 6= j
⇒
w11 = w22, w12 = w21
. Unbiased : λ2 = λ1
. Biased : λ2 = λ1 + 0.1
[DM] G.Deco, D.Marti, Biological Cybernetics (2007)
The associated stochastic system
. ν1 = ν1(t), ν2 = ν2(t) firing rates of the 2 sub-populations:
. ξ = ξ(t) white noise of amplitude β (brownian motion with variance β 2/2).

P
τ ν˙ = −ν + φ λ +
1
1
j=1,2 w1j νj + ξ
1
P
τ ν˙ = −ν + φ λ +
2
2
2
j=1,2 w2j νj + ξ
where φ(x) is the sigmoide (response fct.), strictly monotone and bounded:
φ(x) =
νc
,
1 + exp(−α(x/νc − 1))
α, νc ∈ R
The Fokker-Planck equation
Let dxt = f (xt )dt + g(xt )dBt a ods syst. and L =
P
fi (x)∂xi + (1/2)
P
dij ∂xi j the generator
of T , then u(x, t) = Tt ϕ(x) satisfies ∂t u = Lu .
def
. f (t, ν1, ν2) probability distribution, t ≥ 0 and ν = (ν1, ν2) ∈ Ω = R2
+
.
β2
∂tf + ∇ · (F f ) −
∆f = 0
2
F f = (−ν + Φ(Λ + W · ν)) f
!
2
β
Ff −
∇f · n = 0
2
. flux incoming F · n ≤ 0 (H1)
R
. normalization Ω f dν = 1 (H2)
(F P )
No explicit equilibrium solution
Remark : The flux F is not the gradient of a potential A : ∂ν2 F1 6= ∂ν1 F2
. No explicit solution of the stationnary problem associated to (F P ).
. If one proves the existence, uniqueness and positivity of a solution f∞ of
the stationnary problem, then it is possible to split F in the sum of a gradient
and non-gradient terms: let A = − log f∞ (ie. f∞ = e−A),

∇ · F e−A −
and so:
β2
2
!
∇(e−A) = 0,
hence

!


2


β
−A

∇· F +
∇A e 
=0
2


|
{z
}
G
β2
F = − ∇A + G
2
Remark : G is s.t. ∇·(Ge−A) = 0, but we do not have an explicit form for G
[ACJ] A. Arnold, E. Carlen, Q. Ju, Communication in Stochastic Analysis (2008)
Stationnary problem
We first consider the stationnary problem associated to (F P )
Af = −
β2
2
∆f + ∇ · (F f ) = 0 ,
Ff −
β2
2
!
∇f
·n=0
(S)
Theorem: If (H1) and (H2) hold,
then there exists an unique positive solution f∞(ν) to (S).
Proof: Based on the Krein-Rutman theorem :
• T : L2 (Ω) → L2 (Ω), s.t. ∀g ∈ L2 (Ω), T g = f , with f the unique solution of :
Af + ρf = g
β2
∇f ) · n = 0
(F f −
2
in Ω,
on ∂Ω
2,2
• T : H 2 → H 2 is a compact operator, and T : K → K str. pos., with K = W+
(Ω).
• KR th. =⇒ r(T ) > 0 and ∃ g > 0 s.t. T g = r(T )g. We have,
Af + ρf = λf,
Af = (λ − ρ)f
1
f = r(T )g > 0, λ =
and
r(T )
Z
⇒ (λ − ρ)
f dx = 0
⇒ ρ=λ
Ω
⇒ Af = 0.
Time dependent problem
We consider here the parabolic problem:
∂tf + Af = 0 ,
Ff −
β2
2
!
∇f
·n=0
(P )
with the initial condition: f0(·) ∈ L2(Ω)
Theorem: (P ) admits an unique solution f (t, ν1, ν2).
Proof: Let a(t, f, g) be the bi-linear form associated to A:
Z
Z
β2
a(t, f, g) =
∇f · ∇g dν −
f F · ∇g dν , ∀ f, g ∈ H 1 (Ω) ,
Ω 2
Ω
•
a(t, f, g) is continuous,
•
a(t, f, g) + ρ < f, g > is coercive for ρ ∈ R large enough.
Remark : The maximum principle does not apply !
(F has negative divergence ∇ · F ≤ 0)
(a)
Generalized relative entropy
Theorem: Let f1, f2 > 0 solutions to (P ), and g > 0 a solution of the
dual problem:


β2
∂tg = −F · ∇g − 2 ∆g,
 ∂g = 0
on ∂Ω
∂n
in Ω × [0, T ],
Then, ∀ H convex
d
f2
gf1H(
dt | Ω
f
{z 1
Z
Hg (f2 |f1 )
Proof:
2
f2 f2 00
dν = −
gf1H ( ) ∇( ) dν ≤ 0 ,
2
f
f
Ω
1
1
}
{z
}
|
Dg (f2 |f1 )
Z
β2
(GRE)
β2
f1
β2
2
∂
H]
=
−∇
·
gf
H]
+
∇
·
g
∇
H
−
gf1 H 00 |∇ (f2 /f1 )|2
[gf
[F
1
1
∂t
2
2
g
|
{z
}
=0 integr. by part
Lemme: The solution f to (P ) satisfies:
Proof: Integration of d/dt
R
Ω
fg
Z
Ω
f (t, ν) dν =
Z
Ω
[MMP] P. Michel, S.Mischler, B.Perthame, J.Math. Pures Appl. (2005).
f0(ν) dν.
Corollaries
Corollary 1: If f0(ν) > 0, then f (t, ν) > 0 ∀t.
Proof: We choose in (GRE): g = 1, f1 = f∞ , f2 solution of (P ) with f2 (0, ν) > 0,
and H(f2 /f1 ) = ε(f2 /f1 )− . Then:
Z
d
f∞ (f2 /f1 )− dν ≤ 0.
dt Ω
|
{z
}
h(t)
h(0) = 0, h(t) is decreasing and h(t) is positive ⇒ h(t) = 0 ∀t
Corollary 2: If (H1) holds and f is a solution to (P ) with initial condition f0, then
lim
Z
t→∞ Ω
|f (t, ν) − f∞(ν)|2 dν = 0
Proof: We choose in (GRE): g = 1, f1 = f∞ , f2 = f and H(s) = s2 /2;
and applying the Aubin-Lions theorem
Numerical approximation - finite difference method
Let f k (i, j) = f (k∆t, ni , nj ) with ni = (i + 21 )∆N1 , i = 0...N1 − 1 and nj = (j + 12 )∆N2 , j =
0...N2 − 1. Then, the Fokker-Planck equation is discretised by :
f k+1 (i, j) = f k (i, j)
k
k
+ ∆t F (i + 1/2, j) − F (i − 1/2, j) /∆N1
+ ∆t Gk (i, j + 1/2) − Gk (i, j − 1/2) /∆N2 ,
where F k (i + 12 , j), Gk (i, j + 12 ) are the flux at the interfaces :
F k (i + 1/2, j) =
−ni+1/2 + Φ(λ + w11 ni+1/2 + w12 nj ) f k (i + 1/2, j)
β2
−
f k (i + 1, j) − f k (i, j) ,
2∆N1
Gk (i, j + 1/2) = −nj+1/2 + Φ(λ + w21 ni + w22 nj+1/2 ) f k (i, j + 1/2)
β2
f k (i, j + 1) − f k (i, j) .
−
2∆N2
and we choose linear interpolation for f at the interfaces:
f k (i + 1, j) + f k (i, j)
f k (i, j + 1) + f k (i, j)
k
f (i + 1/2, j) =
,
f (i, j + 1/2) =
.
2
2
Remark: Adaptatif ∆t (gain factor 100) ⇒ for i, j s.t. f k (i, j) 6= 0 and F k (i, j) 6= 0:
k
f k (i, j)
∆t = min
i,j 2|F k (i, j)|
Computed quantities
Marginales of f (t, ν1, ν2) with respect to ν2 and ν1 :
N1(t, ν1) =
Z ν
M
0
f (t, ν1, ν2)dν2 ,
N2(t, ν2) =
Z ν
M
0
f (t, ν1, ν2)dν1.
First order moments:
µi(t) =
Z Z
Ω
νif (ν1, ν2, t)dν1dν2,
i = 1, 2
Second order moments:
γij (t) =
Z Z
Ω
νiνj f (ν1, ν2, t)dν1dν2,
i, j = 1, 2.
Probabilities ρi(t) to belong to the domains Ωi, i = 1, 2, 3 :
ρi(t) =
Z Z
Ωi
f (ν1, ν2, t)dν1dν2.
We have N1 = N2 = 200 discretisation points, and stop computations when
we get a 10−10 difference between to successive iterations.
β = 0.1, α = 4, νc = 20, λ1 = 15, r = 0.3, w+ = 2.35, wI = 1.9, τ = 10−2.
Stationnary solutions
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Contour levels for the distribution f (ν1 , ν2 ) at equilibrium.
Left: unbiased, symmetry, concentrations of ”mass” in S1 = (1.3, 5.9) and S3 = (5.9, 1.3).
Right: biased, no symmetry, concentration of ”mass” in S1 = (1.1, 6.6) and S3 = (5.5, 1.6)
Convergence to steady state
1
0
−1
−2
−3
−4
−5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Convergence towards the steady state, in the unbiased case, in logarithmic scale.
Linear regression done on the last quarter of the curve has a slope of -0.08 with a
standard deviation of 0.004.
After a small transition period, the convergence of the solution towards its
stationary state has an exponential behavior.
Marginales time evolution
40
50
35
40
30
25
30
20
20
15
10
10
5
0
0
0
1
2
initial
0.56 sec
1.15 sec
3
4
5
6
7
3.75 sec
8
9
10
0
1
2
initial
0.56 sec
1.15 sec
3
4
5
6
3.4 sec
7
8
9
10
Initial condition: Gaussian centered in S = (3, 3), near the instable point S2 .
Left : unbiased, evolution towards a symmetric double picked distribution in S1 and S3 .
Right: biased, evolution towards a double picked distribution, a bigger one in S1 and a
smaller one in S3 .
Probabilities in subdomains Ωi
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.5
rho1
rho2
rho3
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0
0.5
rho1
rho2
rho3
1.0
1.5
2.0
2.5
Domains : Ω1 = [0, 2] × [5, 10], Ω2 = [2, 5] × [2, 5], Ω3 = [5, 10] × [0, 2]
initial condition: Gaussian centered near the unstable point S2 , dans Ω2 .
Left: unbiased, time evolution of probablities ρi , ρ1 = ρ3 .
Right: biased, ρ3 almost equal to zero.
3.0
3.5
Escaping time
. 1 dim: Kramers law E(T ) = exp(H/β 2 ) , H= potential height, β = noise
. dim > 1, no general formulation
Let f (0, ν1 , ν2 ) a Gaussian centered in S1 and β = 0.2, ..., 1.
Let T be the exit time=time needed to move half of the mass from Ω1 to Ω3 : ρ1 (T ) < 2ρ3 (T ).
We remark that, in our problem, T has an exponential behavior :
β
T
0.2
12.91
0.3
3.33
0.4
1.69
0.5
1.12
0.6
0.80
0.7
0.59
0.8
0.49
0.9
0.37
3.0
2.5
2.0
1.5
1.0
0.5
0.0
−0.5
−1.0
−1.5
−1.8
−1.6
−1.4
−1.2
−1.0
−0.8
−0.6
−0.4
−0.2
−0.0
Escaping time T as fonction of β in log scaling.
Computing times become rather long....
1
0.30
Slow-fast structure of the dynamics
Blue lines are numerical approximations of the solution of
the o.d.e

P
ν˙ = −ν + φ λ +
1
1
j=1,2 w1j νj P
ν˙ = −ν + φ λ +
2
2
j=1,2 w2j νj
they highlight the slow manifold to which belong the
equilibrium points of the
system
Realization of a trajectory for
the ODEs, starting in (5, 5)
⇒ fast towards the manifold,
slow on the manifold
[BG] N.Berglund, B.Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems.
A Sample-Paths Approach. Springer, Probability and its Applications (2005)
One-dimensional reduction
The change of variables x = ν1 + ν2, y = ν1 − ν2, in the unbiased case, leads
to the following system:

εẋ = h(x, y) + ξ
ẏ = g(x, y) + ξ
,
where
ε=
∂y g
|S2
∂x h
Let x∗(y) be the solution of h(x, y) = 0, then we can write the problem in
a unidimensional equation defined on the slow manifold:
ẏ = g(x∗(y), y) + ξ
Again, we associate an one-dimensional Fokker-Planck equation. Now, the
stationnary solution is given explicitely by:
!
exp
−G
,
2
β
G = ∂y g(x∗(y), y)
In this case, we can say exact things on the escaping time too.
Comparison : 2D vs. simplified model
Plot of the y-marginal of the 2D equilibrium state and the stationary solution of the reduced
slow fast system in the unbiased case and β = 0.3
Comparison : simplified model vs. moment system
Left : ν1 -marginal obtained from the Deco and Martı́ moment system, for various value of
β in [0.1, 0.6] in the biased case.
Right : y-marginal obtained for the reduced system in similar conditions.
Conclusions - Perspectives
• We propose a kinetic model for the time evolution of two interacting
populations (of neurons), derived from a system of Wilson-Cowan type.
This model describes, statistically, both equilibrium and evolution.
• We present theoretical results about existence, uniqueness, positivity and
trend to equilibrium for the F-P equation, which as a non-potential flux.
• Study (in progress) of the simplified 1D model using the slow-fast reduction, in agreement with [Deco et Martı̀] and 2D model for escaping
time, trend toward equilibrium