Geometric singular perturbation theory
for stochastic differential equations
with applications to neuroscience
Nils Berglund
MAPMO, Université d’Orléans
CNRS, UMR 6628 et Fédération Denis Poisson
www.univ-orleans.fr/mapmo/membres/berglund
Collaborateurs:
Stéphane Cordier, Damien Landon, Simona Mancini, MAPMO, Orléans
Barbara Gentz, University of Bielefeld
Christian Kuehn, Max Planck Institute, Dresden
Projet ANR MANDy, Mathematical Analysis of Neuronal Dynamics
GdT Neuromathématiques et modèles de perception
IHP, Paris, 15 mars 2011
Plan
1. Deterministic
. Modeling neurons
. Slow–fast dynamical systems
. Excitability : Types I and II
2. Stochastic
. Mathematical tools
. Sample-path approach
. Application to excitable systems
Neuron : Excitable system
. Single neuron communicates by generating action potential
. Excitable: small change in parameters yields spike generation
1
ODE models for action potential generation
• Hodgkin–Huxley model (1952)
• Morris–Lecar model (1982)
C v̇= −gCam∗(v)(v − vCa) − gKw(v − vK) − gL(v − vL) + I(t)
τw (v)ẇ= −(w − w∗(v))
1+tanh((v−v1 )/v2 )
τ
,
τ
(v)
=
,
w
2
cosh((v−v3 )/v4 ))
1+tanh((v−v3 )/v4 )
w∗(v) =
2
m∗(v) =
• Fitzhugh–Nagumo model (1962)
C v̇= v − v 3 + w + I(t)
g
τ ẇ= α − βv − γw
For C/g τ : slow–fast systems of the form
εv̇= f (v, w)
ẇ= g(v, w)
2
Deterministic slow–fast systems
εẋ= f (x, y)
x : fast variable
ẏ= g(x, y)
y : slow variable
ε 1: Singular perturbation theory
Qualitative analysis: nullclines f = 0 and g = 0
x
f <0
g<0
f >0
g<0
f <0
g>0
f >0
g>0
y
3
Example: Van der Pol oscillator
1 x3
ẋ = y + x − 3
ẏ = −εx
t7→εt
⇐⇒
x00 +ε−1/2(x2 −1)x0 +x = 0
1 x3
εẋ = y + x − 3
ẏ = −x
4
x00 +ε−1/2(x2 −1)x0 +x = 0
Example: Van der Pol oscillator
1 x3
ẋ = y + x − 3
ẏ = −εx



y
t7→εt
⇐⇒
ẏ = 0
ẏ = −x



y
ε→0
3
x
ẋ = y + x − 1
3
1 x3
εẋ = y + x − 3
⇐⇒
/
ε→0
3)
y = −(x − 1
x
3
ẏ = −x
x
⇒ ẋ =
1 − x2
4-a
x00 +ε−1/2(x2 −1)x0 +x = 0
Example: Van der Pol oscillator
1 x3
ẋ = y + x − 3
t7→εt
⇐⇒
ẏ = −εx



y
1 x3
εẋ = y + x − 3
ẏ = −x



y
ε→0
3
x
ẋ = y + x − 1
3
⇐⇒
/
ε→0
3)
y = −(x − 1
x
3
ẏ = −x
ẏ = 0
x
⇒ ẋ =
2
1
−
x
x
x
y
y
4-b
x00 +ε−1/2(x2 −1)x0 +x = 0
Example: Van der Pol oscillator
1 x3
ẋ = y + x − 3
t7→εt
⇐⇒
ẏ = −εx
3
εẋ=y + x − 1
x
3
ẏ=−x
x
y
Relaxation oscillations
x
x
y
y
4-c
Quantitative results
Stable slow manifold: f = 0, ∂xf < 0
Tikhonov (1952) / Fenichel (1979):
Orbits converge to ε-neighbourhood of stable slow manifold
Dynamic bifurcations: f = 0, ∂xf = 0 ⇒ local analysis
x
x
ε2/3
x
ε1/2
ε1/3
y
ε1/2
y
y
Saddle–node
Transcritical
Pitchfork
f (x, y) = −x2 − y + . . .
f (x, y) = −x2 + y 2 + . . . f (x, y) = yx − x3 + . . .
5
Excitability of type I
. Stable equilibrium point at intersection of f = 0 and g = 0
. Close to a saddle–node-on-invariant-circle (SNIC) bifurcation
. At bifurcation, periodic solutions appear
. Period diverges at bifurcation point
. Example: Morris–Lecar model
6
Excitability of type II
. Stable equilibrium point at intersection of f = 0 and g = 0
. Close to a Hopf bifurcation
. At bifurcation, periodic solutions appear
. Period converges at bifurcation point
. Canard (french duck) phenomenon
. Example: Fitzhugh–Nagumo model
7
Adding noise
1
σ
f (xt, yt) dt + √ dWt
ε
ε
dyt= g(xt, yt) dt + σ 0 dWt0
dxt=
Wt, Wt0: Brownian motions (independent) ⇒ Ẇt, Ẇt0: white noises
Different mathematical methods :
. PDEs ⇒ evolution of probability density, exit from domain
. Large deviations ⇒ rare events, exit from domain
. Stochastic analysis ⇒ sample-path properties
. ...
8
Noise and partial differential equations
dxt = f (xt) dt + σ dWt
x ∈ Rn
2 ∆ϕ
Generator: Lϕ = f · ∇ϕ + 1
σ
2
2 ∆ϕ
σ
Adjoint: L∗ϕ = ∇ · (f ϕ) + 1
2
Kolmogorov forward or Fokker–Planck equation: ∂tµ = L∗µ
where µ(x, t) = probability density of xt
Exit problem:
Given D ⊂ R n, characterise
τD = inf{t > 0 : xt 6∈ D}
Fact: u(x) = E x[τD ] satisfies

Lu(x) = −1
u(x) = 0
x∈D
x ∈ ∂D
Similar boundary value problems give distribution of exit time
and exit location
9
Noise and partial differential equations
dxt = f (xt) dt + σ dWt
x ∈ Rn
2 ∆ϕ
Generator: Lϕ = f · ∇ϕ + 1
σ
2
2 ∆ϕ
σ
Adjoint: L∗ϕ = ∇ · (f ϕ) + 1
2
Kolmogorov forward or Fokker–Planck equation: ∂tµ = L∗µ
where µ(x, t) = probability density of xt
Exit problem:
Given D ⊂ R n, characterise
x τD
τD = inf{t > 0 : xt 6∈ D}
Fact: u(x) = E x[τD ] satisfies

Lu(x) = −1
u(x) = 0
D
x∈D
x ∈ ∂D
Similar boundary value problems give distribution of exit time
and exit location
9-a
Noise and large deviations
dxt = f (xt) dt + σ dWt
x ∈ Rn
Large deviation principle: Probability of sample path xt being
2
close to given curve ϕ : [0, T ] → R n behaves like e−I(ϕ)/σ
Rate function: (or action functional or cost functional)
1 T
I[0,T ](ϕ) =
kϕ̇t − f (ϕt)k2 dt
2 0
Z
10
Noise and large deviations
dxt = f (xt) dt + σ dWt
x ∈ Rn
Large deviation principle: Probability of sample path xt being
2
close to given curve ϕ : [0, T ] → R n behaves like e−I(ϕ)/σ
Rate function: (or action functional or cost functional)
1 T
I[0,T ](ϕ) =
kϕ̇t − f (ϕt)k2 dt
2 0
Z
Application to exit problem: (Wentzell, Freidlin 1969)
Assume D contains unique equilibrium point x?
. Cost to reach y ∈ ∂D: V (y) = inf inf{I[0,T ](ϕ) : ϕ0 = x?, ϕT = y}
T >0
. Gradient case: f (x) = −∇V (x) ⇒ V (y) = 2(V (y) − V (x?))
1
. Mean first-exit time: E[τD ] ∼ exp 2 inf V (y)
σ y∈∂D
10-a
Noise and stochastic analysis
x ∈ Rn
dxt = f (xt) dt + σ(x) dWt
Integral form for solution:
xt = x0 +
Z t
0
f (xs) ds +
Z t
0
σ(xs) dWs
where the second integral is the Itô integral
11
Noise and stochastic analysis
x ∈ Rn
dxt = f (xt) dt + σ(x) dWt
Integral form for solution:
xt = x0 +
Z t
0
f (xs) ds +
Z t
0
σ(xs) dWs
where the second integral is the Itô integral
Application to the exit problem:
The Itô integral is a martingale ⇒ its maximum can be
controlled in terms of variance at endpoint (Doob) :
Z
)
!2#
" Z
t
T
1
P sup σ(xs) dWs > δ 6 2 E
σ(xs) dWs
δ
0
0
t∈[0,T ]
(
Itô
isometry:
"
Z T
E
0
σ(xs) dWs
!2#
=
Z T
0
E[σ(xs)2] ds
11-a
Application to slow–fast systems
1
σ
dxt= f (xt, yt) dt + √ dWt
ε
ε
dyt= g(xt, yt) dt + σ 0 dWt0
Use different methods
. Near stable slow manifold (f = 0, ∂xf < 0)
. Near bifurcation points (f = 0, ∂xf = 0)
. Far from slow manifold (f 6= 0)
12
Near stable slow manifold
dxt =
1
σ
f (xt, t) dt + √ dWt
ε
ε
Slow–fast system with yt = t
If ∃ stable slow manif: f (x?(t), t) = 0,
If ∃ stable slow manif.: a?(t) = ∂xf (x?(t), t) 6 −a0
then ∃ adiabatic solution: x̄(t, ε) = x?(t) + O(ε) of εẋ = f (x, t)
13
Near stable slow manifold
dxt =
1
σ
f (xt, t) dt + √ dWt
ε
ε
Slow–fast system with yt = t
If ∃ stable slow manif: f (x?(t), t) = 0,
If ∃ stable slow manif.: a?(t) = ∂xf (x?(t), t) 6 −a0
then ∃ adiabatic solution: x̄(t, ε) = x?(t) + O(ε) of εẋ = f (x, t)
Observation: Let ā(t, ε) = ∂xf (x̄(t, ε), t) = a?(t) + O(ε)
Consider linearised equation at x̄(t, ε):
1
σ
dξt = ā(t, ε)ξt dt + √ dWt
ε
ε
ξt: gaussian process with variance σ 2v(t), s.t. εv̇ = 2ā(t, ε)v + 1
Asymptotically, v(t) ' v ?(t) = 1/2|ā(t, ε)|
q
B(h): strip of width ' h v ?(t, ε) around x̄(t, ε)
13-a
Near stable slow manifold
dxt =
1
σ
f (xt, t) dt + √ dWt
ε
ε
Theorem: [B. & Gentz, PTRF 2002]
n
o
2 /2σ 2
2
2
−κ
h
C(t, ε)e −
6 P leaving B(h) before time t 6 C(t, ε)e−κ+h /2σ
κ± = 1 ∓ O(h)
s
C(t, ε) =
h
Z
i
h
2 /σ 2
2 1 t
−h
ā(s, ε) ds 1 + error of order e
t/ε
πε 0
σ
x? (t)
xt
B(h)
x̄(t, ε)
13-b
e.g. f (x, y) = −y − x2
Saddle–node bifurcation
σ σc = ε1/2
σ σc = ε1/2
x
x
B(h)
y
y
Deterministic case σ = 0: Solutions stay at distance ε1/3 above
bifurcation point until time ε2/3 after bifurcation.
Theorem: [B. & Gentz, Nonlinearity 2002]
1. If σ σc:
until time
2. If σ σc:
transition
Paths likely to stay in B(h)
ε2/3 after bifurcation, maximal spreading σ/ε1/6.
Transition typically for t −σ 4/3
2
probability > 1 − e−cσ /ε|log σ|
14
Excitability of type I
Near bifurcation point:
σ δ 3/4
σ δ 3/4
1
σ
2
dxt= (yt − xt ) dt + √ dWt
ε
ε
dyt= (δ − yt) dt
Global behaviour:
15
Excitability of type I
Time series of −xt:
. σ δ 3/4: rare spikes, times between spikes ∼ exponentially
3/2 2
distributed, mean waiting time of order eδ /σ
⇒ Poisson point process
. σ δ 3/4: frequent spikes, more regularly spaced, waiting time
of order |log σ|
16
Excitability of type II
Near bifurcation point:
1
σ
(yt − x2
)
dt
+
dWt
√
t
ε
ε
dyt= (δ − xt) dt
dxt=
√
. δ > ε: equilibrium (δ, δ 2) is a node
Similar behaviour as before, crossover at σ ∼ δ 3/2
√
. δ < ε: equilibrium (δ, δ 2) is a focus. Two-dimensional problem
17
Noise-induced MMOs
[D. Landon, PhD thesis, in progress]
Conjectured bifurcation diagram [Muratov and Vanden Eijnden (2007)] :
σ
ε3/4
σ = δ 3/2
σ = (δε)1/2
σ = δε1/4
ε1/2
δ
18
Noise-induced MMOs
[D. Landon, PhD thesis, in progress]
Conjectured bifurcation diagram [Muratov and Vanden Eijnden (2007)] :
σ
ε3/4
σ = δ 3/2
σ = (δε)1/2
σ = δε1/4
ε1/2
δ
Work in progress :
. Prove bifurcation diagram is correct
. Characterize interspike time statistics and spike train statistics
. Characterize distribution of mixed-mode patterns
18-a
References
• N. B. & B. Gentz, Pathwise description of dynamic
pitchfork bifurcations with additive noise, Probab.
Theory Related Fields 122, 341–388 (2002)
•
, A sample-paths approach to noise-induced
synchronization: Stochastic resonance in a doublewell potential, Ann. Appl. Probab. 12, 1419-1470
(2002)
•
, The effect of additive noise on dynamical
hysteresis, Nonlinearity 15, 605–632 (2002)
•
, Noise-induced phenomena in slow-fast dynamical systems, A sample-paths approach, Springer,
Probability and its Applications (2006)
•
, Stochastic dynamic bifurcations and excitability, in C. Laing and G. Lord, (Eds.), Stochastic
methods in Neuroscience, p. 65-93, Oxford University Press (2009)
• N. B., B. Gentz & Christian Kuehn, Hunting French
Ducks in a Noisy Environment, hal-00535928, submitted (2010)
19