Dynamics of elapsed time inhomogeneous
neuron network model
Moon-Jin Kang a , Benoit Perthame b Delphine Salort c
a Department
b Sorbonne
c Sorbone
of Mathematics, Texas University, Austin
Universités, UPMC Univ Paris 06, UMR 7598, CNRS, INRIA,
Laboratoire Jacques-Louis Lions
Universités, UPMC Univ Paris 06, UMR 7238, CNRS, Laboratoire de
Biologie Computationnelle et Quantitative
Received *****; accepted after revision +++++
Presented by
Abstract
Models for neural networks have been proposed which describe the probability to
find a neuron for which time s has elapsed since the last discharge. These are written
under the form of nonlinear age-structured equation where the total network activity
modulates the firing rate. Here, we consider an inhomogeneous network of networks
with variability on the refractory period. We give conditions on the connectivity
leading to total desynchronization of the network. To cite this article: A. Name1,
A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
Résumé
Dynamique de réseaux de neurones inhomogènes structurés en âge. Pour
décrire l’activité de réseaux de neurones, des modèles qui représentent la probabilité
qu’un neurone ait passé le temps s depuis sa dernière décharge, ont été proposés.
Ce sont des équations structurées en âge, nonlinéaires où l’activité totale du réseau
contrôle le taux de décharge. Ici, nous considérons un réseau inhomogène de réseaux
prenant en compte la variabilité des périodes réfractaires. Nous donnons une condition sur la connectivité qui conduit à la désynchronisation totale. Pour citer cet
article : A. Name1, A. Name2, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
Email addresses: [email protected] (Moon-Jin Kang),
[email protected] (Benoit Perthame),
[email protected] (Delphine Salort).
Preprint submitted to the Académie des sciences
September 25, 2015
1
Introduction
A possible mathematical description of neural networks uses the time s past
since the last discharge and leads to write a nonlinear age-structured equation
for the probability of neurons in state s ≥ 0 at time t. This age-structured type
description of neural network have been studied by several authors [10], [11],
[12], [13] and the question on the link of this model with finite size models
have been studied in [2], [15]. Here we consider an infinite inhomogeneous
network parametrized by a real number σ, each network has a specific intrinsic
dynamic and refractory period. We assume that each neural network itself is
governed by an elapsed time model without delay [7], [8], [11], [12], [13] and
that all the networks are coupled via their mean activity. For simplicity we
also assume that the synaptic weights are all the same, of intensity J. These
assumptions lead to write infinitely many coupled neural networks on the
probability density n(s, σ, t) under the form




∂t n + ∂s n + pσ (s, X(t))n = 0,



R∞
N
(σ,
t)
:=
n(s
=
0,
σ,
t)
=
0 pσ (s, X(t))n(s, σ, t)ds,



R


 X(t) = J
N (σ, t)dσ.
(1)
R
We complete this equation with a Cauchy data which satisfies
n(s, σ, 0) = n0 (s, σ) ≥ 0,
Z∞
n0 ds = g(σ),
0
Z
gdσ = 1,
(2)
R
where g(σ) represents the probability density of neural networks parametrized
by σ. Notice that the intrinsic dynamic of the neurons is entirely determined
by the function pσ .
For simplicity, we assume that pσ is a piecewise constant function as follows :
pσ (s, x) =


 aσ ,
for s < s∗ (σ, x)

b
for s > s∗ (σ, x),
σ,
(3)
with aσ < bσ and 0 < a := minσ aσ , b := maxσ bσ < ∞ and s∗ a positive
function satisfying s∗ ∈ L∞ (R; Cb1 (R+ )), and we express that the network
activity increases the discharge rate with
∂x s∗ (σ, x) ≤ 0 for all σ, x.
(4)
2
Here, s∗ (σ, x) represents the length of the refractory period and includes randomness due to external noise. Therefore aσ is a small rate of discharge and
bσ a large value depending on the intrinsic dynamic of the neurons under
consideration. Finally, we assume that
0 ≤ n0 (s, σ) ≤ bσ g(σ) for all s > 0, σ ∈ R,
(5)
With these assumptions, it is standard to prove the existence of a unique
solution n ∈ C(R+ ; L1 (R)), which satisfies for all t > 0, s > 0, σ ∈ R,
Z∞
0 ≤ n(s, σ, t) ≤ bσ g(σ),
n(t, s, σ)ds = g(σ).
(6)
0
The aim of this note is to study the asymptotic behavior on n(t) and more
precisely, to give conditions on the connectivity parameter J for desynchronization. We begin by studying the possible steady states and conclude by
proving the exponential decay to the steady state for small connectivities J.
2
Steady states
For equation (1), we are going to prove that, for J small enough, there is a
unique stationary state n̄σ , with activity X ∗ . Since, it has to satisfy
∗
∂s n̄σ + pσ (s, X )n̄σ = 0,
+∞
Z
n̄σ (s)ds = g(σ),
0
we obtain the semi-explicit formula n̄σ (s) = n̄σ (0)e−
n̄σ (0) = g(σ)
Z∞
e−
Rs
0
pσ (τ,X ∗ )dτ
ds
−1
= g(σ)
0
Rs
0
pσ (τ X ∗ )dτ
with
−1
1
1
1
∗
∗
− ( − )e−aσ s (σ,X ) .
aσ
aσ b σ
To investigate the existence and uniqueness of X ∗ , we use the function
FJ (x) = J
Z
R
g(σ)
−1
1
1
1
∗
− ( − )e−aσ s (σ,x) dσ.
aσ
aσ b σ
Notice that the relation for X ∗ satisfies
∗
X =J
Z
n̄σ (0)dσ ⇐⇒ X ∗ = FJ (X ∗ ).
R
3
(7)
We have FJ (0) > 0 and FJ (+∞) ≤ J g(σ)bσ dσ < +∞; hence there exists at
least one fixed point and thus, we obtain the existence of X ∗ . For uniqueness,
we observe that, from (4), we can also compute
R
0<
FJ0 (x)
Z
=J
( abσσ − 1)e−aσ s
g(σ) R
1
aσ
−
( a1σ
−
∗ (σ,x)
∂x s∗ (σ, x)
1
)e−aσ s∗ (σ,x)
bσ
2 dσ.
From the inequality
1
1
1
1
∗
− ( − )e−aσ s (σ,x) ≥ ,
aσ
aσ b σ
bσ
we obtain that
FJ0 (x)
≤
Z∞
J
g(σ)∂x s∗ (σ, x)bσ (bσ − aσ )dσ ,
∞
−∞
and we conclude the uniqueness under the (simple) condition
Z∞
J
g(σ)∂x s∗ (σ, ·)bσ (bσ − aσ )dσ ∞
< 1.
(8)
−∞
Remark 1 Obviously, if J(b − a) is small enough, the assumption (8) is satisfied. It is also possible to give more general conditions for the uniqueness but
(8) turns out to be a good comprise for the study of desynchronization rate.
3
Large time behavior
As for the uniqueness of the steady state, the desynchronisation property can
also be obtained when the connectivity J is small enough since we have the
following.
Theorem 3.1 Assume (8) and that there exists β > 0 such that
βσ := aσ −
R
∞
R
∞
∞
2Jbσ 1 − J
∗
−∞ g(σ)bσ (bσ − aσ )∂x s (σ, ·)dσ ∗
−∞ bσ (bσ − aσ )g(σ)∂x s (σ, ·)dσ ∞
Then the following exponential decay estimate holds
Z∞ Z∞
−∞ 0
−βt
|n − n̄σ |(t)dsdσ ≤ e
Z∞ Z∞
−∞ 0
4
|n0 − n̄σ |dsdσ.
≥ β > 0.
We believe that, for the model under consideration, a size condition on the
connectivity J is necessary for desynchronization. Indeed, for the standard
elapsed time model, periodic solutions (or self-sustained activity) occur for
larger values of J, see [12]. This synchronization effect is desirable for several
models of networks and has been widely studied, see [1], [6], [3], [4], [5], [9],
[14].
Proof of Theorem 3.1. Let φ := n − n̄σ . Since φ satisfies
∂t |φ| + ∂s |φ| + pσ (s, X ∗ )|φ| ≤ |pσ (s, X ∗ ) − pσ (s, X(t))|n,
we may integrate w.r.t. s and σ and obtain
∞ ∞
Z∞
Z∞ Z∞
Z∞ Z∞
d Z Z
∗
|φ|dsdσ ≤
|φ(s = 0)|dσ+
|pσ (s, X )−pσ (s, X(t))|ndsdσ−
pσ (s, X ∗ )|φ|dsdσ.
dt
−∞ 0
−∞
−∞ 0
−∞ 0
Since
φ(s = 0) =
Z∞
∗
[pσ (s, X(t))n−pσ (s, X )n̄σ ]ds =
0
Z∞
∗
Z∞
(pσ (s, X(t))−pσ (s, X ))nds+ pσ (s, X ∗ )φds,
0
0
we have
d R∞ R∞
dt −∞ 0
|φ|dsdσ ≤ 2
R∞ R∞
−∞ 0
−
Since
d
dt
R∞
0
R∞ R∞
−∞ 0
R ∞ R ∞
∗
−∞ 0 pσ (s, X )φdsdσ
pσ (s, X ∗ )|φ|dsdσ.
φds = 0, for all σ, we get
R∞ R∞
−∞ 0
|pσ (s, X ∗ ) − pσ (s, X(t))|ndsdσ +
|φ|dsdσ ≤ 2
R∞ R∞
−∞ 0
|pσ (s, X ∗ ) − pσ (s, X(t))|ndsdσ
R ∞ R ∞
+ −∞
0 (pσ (s, X ∗ ) − aσ )φdsdσ
−
R∞ R∞
−∞ 0
(pσ (s, X ∗ ) − aσ )|φ|dsdσ −
R∞ R∞
−∞ 0
aσ |φ|dsdσ
and so
∞ ∞
Z∞ Z∞
Z∞ Z∞
d Z Z
∗
|φ|dsdσ ≤ 2
|pσ (s, X ) − pσ (s, X(t))|ndsdσ −
aσ |φ|dsdσ.(9)
dt
−∞ 0
−∞ 0
−∞ 0
|
{z
I
}
To control the term I, we use (3) to get, with χ the indicator function,
|pσ (s, X ∗ )−pσ (s, X(t))| ≤ (bσ −aσ )χs∈[min(s∗ (σ,X(t)),s∗ (σ,X ∗ )),max(s∗ (σ,X(t)),s∗ (σ,X ∗ ))] .
5
From this inequality, and using (6), we conclude that
I ≤2
≤2
R∞ R∞
−∞ 0
(bσ − aσ )nχs∈[min(s∗ (σ,X(t)),s∗ (σ,X ∗ )),max(s∗ (σ,X(t)),s∗ (σ,X ∗ ))] dsdσ
R∞
−∞ bσ g(σ)(bσ
R
≤ 2
∞
−∞
− aσ )|s∗ (σ, X(t)) − s∗ (σ, X ∗ )|dσ
g(σ)bσ (bσ − aσ )∂x s∗ (σ, ·)dσ |X(t) − X ∗ |.
∞
The last inequality follows from the sign condition on ∂x s∗ (σ, ·) in assumption
(4). Similarly, we can upper bound the activities difference as
|X(t) − X ∗ | ≤ J
R∞ R∞
pσ (X(t))|φ|dsdσ + J
≤J
R∞ R∞
bσ |φ|dsdσ + J −∞ 0
−∞ 0
R
∞
R∞ R∞
−∞ 0
−∞ bσ (bσ
|pσ (X(t)) − pσ (X ∗ )|n̄σ dsdσ
− aσ )g(σ)∂x s∗ (σ, ·)dσ |X(t) − X ∗ |.
∞
Thanks to the condition (8), we can conclude our estimate on I with
|X(t) − X | ≤
Z∞
J
∗
1 − J ∂x
R∞
−∞ bσ (bσ
− aσ )g(σ)s∗ (σ, ·)dσ bσ |φ|dsdσ.
∞ 0
Including the estimate on I in (9), and recalling the definition of βσ in Theorem
3.1, we find
∞ ∞
Z∞ Z∞
d Z Z
|φ|dsdσ ≤ −
βσ |φ|dsdσ
dt
−∞ 0
−∞ 0
which ends the proof of Theorem 3.1.
Acknowledgements
M.-J. Kang was supported by the Foundation Sciences Mathématiques de
Paris as a postdoctoral fellowship, and by an AMS-Simons Travel Grant. B.
Perthame and D. Salort are supported by the french ”ANR blanche” project
Kibord: ANR-13-BS01-0004.
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