Quantifying neuronal spiking patterns
using continuous-space Markov chains
Nils Berglund
MAPMO, Université d’Orléans
CNRS, UMR 7349 & Fédération Denis Poisson
www.univ-orleans.fr/mapmo/membres/berglund
nils.berglund@math.cnrs.fr
Collaborators: Barbara Gentz (Bielefeld)
Christian Kuehn (Vienna), Damien Landon (Dijon)
ANR project MANDy, Mathematical Analysis of Neuronal Dynamics
Rhein-Main Kolloquium Stochastik
Gutenberg-Universität Mainz, February 1, 2013
Neurons and action potentials
Action potential [Dickson 00]
. Neurons communicate via patterns of spikes
in action potentials
1
Neurons and action potentials
Action potential [Dickson 00]
. Neurons communicate via patterns of spikes
in action potentials
. Question: effect of noise on interspike interval statistics?
. Poisson hypothesis: Exponential distribution
⇒ Markov property
1-a
ODE models for evolution of membrane potential
. Integrate-and-fire models
Conduction-based models
Hodgkin–Huxley model (1952)
C v̇ =
X
Ii(v)
α
β
Ii(v) = giϕi i ψi i (v − Ei)
ion channels i
ϕi, ψi: Gating variables, satisfy linear ODEs
Morris–Lecar model (1982)
C v̇= −gCam∗(v)(v − vCa) − gKw(v − vK) − gL(v − vL)
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
g
τ ẇ= α − βv − γw
2
ODE models for evolution of membrane potential
. Integrate-and-fire models
. Conduction-based models
n
o Hodgkin–Huxley model (1952)
C v̇ =
X
Ii(v)
α
β
Ii(v) = giϕi i ψi i (v − Ei)
ion channels i
ϕi, ψi:
Gating variables, satisfy linear ODEs
Morris–Lecar model (1982)
C v̇= −gCam∗(v)(v − vCa) − gKw(v − vK) − gL(v − vL)
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
g
τ ẇ= α − βv − γw
2-a
ODE models for evolution of membrane potential
. Integrate-and-fire models
. Conduction-based models
n
o Hodgkin–Huxley model (1952)
C v̇ =
X
Ii(v)
α
β
Ii(v) = giϕi i ψi i (v − Ei)
ion channels i
ϕi, ψi:
Gating variables, satisfy linear ODEs
n
o Morris–Lecar model (1982)
C v̇= −gCam∗(v)(v − vCa) − gKw(v − vK) − gL(v − vL)
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
g
τ ẇ= α − βv − γw
2-b
ODE models for evolution of membrane potential
. Integrate-and-fire models
. Conduction-based models
n
o Hodgkin–Huxley model (1952)
C v̇ =
X
Ii(v)
α
β
Ii(v) = giϕi i ψi i (v − Ei)
ion channels i
ϕi, ψi:
Gating variables, satisfy linear ODEs
n
o Morris–Lecar model (1982)
C v̇= −gCam∗(v)(v − vCa) − gKw(v − vK) − gL(v − vL)
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) =
n
o Fitzhugh–Nagumo model (1962)
C v̇= v − v 3 + w
g
τ ẇ= α − βv − γw
2-c
Random Poincaré map
Fold
Stable slow
manifold
Unstable slow
manifold
Stable slow
manifold
Folded node
Fold
3
Random Poincaré map
Fold
Unstable slow
manifold
Σ
Stable slow
manifold
Folded node
Fold
Successive intersections with Σ define Markov chain.
3-a
Random Poincaré map
In appropriate coordinates
dϕt = f (ϕt, xt) dt + σF (ϕt, xt) dWt
ϕ∈R
dxt = g(ϕt, xt) dt + σG(ϕt, xt) dWt
x∈E⊂Σ
. all functions periodic in ϕ (say period 1)
. f > c > 0 and σ small ⇒ ϕt likely to increase
. process may be killed when x leaves E
4
Random Poincaré map
In appropriate coordinates
dϕt = f (ϕt, xt) dt + σF (ϕt, xt) dWt
ϕ∈R
dxt = g(ϕt, xt) dt + σG(ϕt, xt) dWt
x∈E⊂Σ
. all functions periodic in ϕ (say period 1)
. f > c > 0 and σ small ⇒ ϕt likely to increase
. process may be killed when x leaves E
x
E
X1
X2
X0
ϕ
1
2
X0, X1, . . . form (substochastic) Markov chain
4-a
Random Poincaré map and harmonic measure
x
E
X1
X0
−M
ϕ
1
. τ : first-exit time of zt = (ϕt, xt) from D = (−M, 1) × E
. µz (A) = P z {zτ ∈ A}: harmonic measure (wrt generator L)
. [Ben Arous, Kusuoka, Stroock ’84]: under hypoellipticity cond,
µz admits (smooth) density h(z, y) wrt arclength on ∂D
5
Random Poincaré map and harmonic measure
x
E
X1
X0
−M
ϕ
1
. τ : first-exit time of zt = (ϕt, xt) from D = (−M, 1) × E
. µz (A) = P z {zτ ∈ A}: harmonic measure (wrt generator L)
. [Ben Arous, Kusuoka, Stroock ’84]: under hypoellipticity cond,
µz admits (smooth) density h(z, y) wrt arclength on ∂D
. Remark: Lz h(z, y) = 0 (kernel is harmonic)
. For B ⊂ E Borel set
P X0 {X1 ∈ B} = K(X0, B) :=
Z
B
K(X0, dy)
where K(x, dy) = h((0, x), y) dy =: k(x, y) dy
5-a
Fredholm theory
Consider integral operator K acting
. on L∞ via f 7→ (Kf )(x) =
Z
. on L1 via m 7→ (mK)(A) =
E
Z
k(x, y)f (y) dy = E x[f (X1)]
E
m(x)k(x, y) dx = P µ{X1 ∈ A}
6
Fredholm theory
Consider integral operator K acting
. on L∞ via f 7→ (Kf )(x) =
Z
. on L1 via m 7→ (mK)(A) =
E
Z
k(x, y)f (y) dy = E x[f (X1)]
E
m(x)k(x, y) dx = P µ{X1 ∈ A}
[Fredholm 1903]:
. If k ∈ L2, then K has eigenvalues λn of finite multiplicity
. Eigenfcts Khn = λnhn, h∗nK = λnh∗n form complete ON basis
[Perron, Frobenius, Jentzsch 1912, Krein–Rutman ’50, Birkhoff ’57]:
. Principal eigenvalue λ0 is real, simple, |λn| < λ0 ∀n > 1, h0 > 0
Spectral decomp: k(x, y) = λ0h0(x)h∗0(y) + λ1h1(x)h∗1(y) + . . .
⇒ P x{Xn ∈ dy|Xn ∈ E} = π0(dy) + O((|λ1|/λ0)n)
R
∗
where π0 = h0/ E h∗0 is quasistationary distribution (QSD)
6-a
Fredholm theory
Consider integral operator K acting
. on L∞ via f 7→ (Kf )(x) =
Z
. on L1 via m 7→ (mK)(A) =
E
Z
k(x, y)f (y) dy = E x[f (X1)]
E
m(x)k(x, y) dx = P µ{X1 ∈ A}
[Fredholm 1903]:
. If k ∈ L2, then K has eigenvalues λn of finite multiplicity
. Eigenfcts Khn = λnhn, h∗nK = λnh∗n form complete ON basis
[Perron, Frobenius, Jentzsch 1912, Krein–Rutman ’50, Birkhoff ’57]:
. Principal eigenvalue λ0 is real, simple, |λn| < λ0 ∀n > 1, h0 > 0
Spectral decomp: k(x, y) = λ0h0(x)h∗0(y) + λ1h1(x)h∗1(y) + . . .
⇒ P x{Xn ∈ dy|Xn ∈ E} = π0(dy) + O((|λ1|/λ0)n)
R
∗
where π0 = h0/ E h∗0 is quasistationary distribution (QSD)
6-b
Fredholm theory
Consider integral operator K acting
. on L∞ via f 7→ (Kf )(x) =
Z
. on L1 via m 7→ (mK)(A) =
E
Z
k(x, y)f (y) dy = E x[f (X1)]
E
m(x)k(x, y) dx = P µ{X1 ∈ A}
[Fredholm 1903]:
. If k ∈ L2, then K has eigenvalues λn of finite multiplicity
. Eigenfcts Khn = λnhn, h∗nK = λnh∗n form complete ON basis
[Perron, Frobenius, Jentzsch 1912, Krein–Rutman ’50, Birkhoff ’57]:
. Principal eigenvalue λ0 is real, simple, |λn| < λ0 ∀n > 1, h0 > 0
∗
n
∗
Spectral decomp: kn(x, y) = λn
0 h0 (x)h0 (y) + λ1 h1 (x)h1 (y) + . . .
⇒ P x{Xn ∈ dy|Xn ∈ E} = π0(dy) + O((|λ1|/λ0)n)
R
∗
where π0 = h0/ E h∗0 is quasistationary distribution (QSD)
6-c
Fredholm theory
Consider integral operator K acting
. on L∞ via f 7→ (Kf )(x) =
Z
. on L1 via m 7→ (mK)(A) =
E
Z
k(x, y)f (y) dy = E x[f (X1)]
E
m(x)k(x, y) dx = P µ{X1 ∈ A}
[Fredholm 1903]:
. If k ∈ L2, then K has eigenvalues λn of finite multiplicity
. Eigenfcts Khn = λnhn, h∗nK = λnh∗n form complete ON basis
[Perron, Frobenius, Jentzsch 1912, Krein–Rutman ’50, Birkhoff ’57]:
. Principal eigenvalue λ0 is real, simple, |λn| < λ0 ∀n > 1, h0 > 0
∗
n
∗
Spectral decomp: kn(x, y) = λn
0 h0 (x)h0 (y) + λ1 h1 (x)h1 (y) + . . .
⇒ P x{Xn ∈ dy|Xn ∈ E} = π0(dx) + O((|λ1|/λ0)n)
R
∗
where π0 = h0/ E h∗0 is quasistationary distribution (QSD)
[Yaglom ’47, Bartlett ’57, Vere-Jones ’62, . . . ]
6-d
How to estimate the principal eigenvalue
. “Trivial” bounds: ∀A ⊂ E with Lebesgue(A) > 0,
inf K(x, A) 6 λ0 6 sup K(x, E)
x∈A
x∈E
7
How to estimate the principal eigenvalue
. “Trivial” bounds: ∀A ⊂ E with Lebesgue(A) > 0,
inf K(x, A) 6 λ0 6 sup K(x, E)
x∈A
x∈E
Z
h0 (y)
dy 6 K(x∗ , E)
∗
h0 (x )
E
Z
Z
Z
λ0
h∗0 (y) dy =
h∗0 (x)K(x, A) dx > inf K(x, A)
h∗0 (y) dy
Proof: x∗ = argmax h0 ⇒ λ0 =
A
E
k(x∗ , y)
x∈A
A
7-a
How to estimate the principal eigenvalue
. “Trivial” bounds: ∀A ⊂ E with Lebesgue(A) > 0,
inf K(x, A) 6 λ0 6 sup K(x, E)
x∈A
x∈E
Z
h0 (y)
dy 6 K(x∗ , E)
∗
h0 (x )
E
Z
Z
Z
λ0
h∗0 (y) dy =
h∗0 (x)K(x, A) dx > inf K(x, A)
h∗0 (y) dy
Proof: x∗ = argmax h0 ⇒ λ0 =
A
E
k(x∗ , y)
x∈A
A
. Donsker–Varadhan-type bound:
1
where τ∆ = inf{n > 0 : Xn 6∈ E}
λ0 6 1 −
supx∈E E x[τ∆]
7-b
How to estimate the principal eigenvalue
. “Trivial” bounds: ∀A ⊂ E with Lebesgue(A) > 0,
inf K(x, A) 6 λ0 6 sup K(x, E)
x∈A
x∈E
Z
h0 (y)
dy 6 K(x∗ , E)
∗
h0 (x )
E
Z
Z
Z
λ0
h∗0 (y) dy =
h∗0 (x)K(x, A) dx > inf K(x, A)
h∗0 (y) dy
Proof: x∗ = argmax h0 ⇒ λ0 =
A
E
k(x∗ , y)
x∈A
A
. Donsker–Varadhan-type bound:
1
where τ∆ = inf{n > 0 : Xn 6∈ E}
λ0 6 1 −
supx∈E E x[τ∆]
. Bounds using Laplace transforms (see below)
7-c
Application: Stochastic FitzHugh–Nagumo equations
dxt =
1
σ1
(1)
[xt − x3
+
y
]
dt
+
dW
√
t
t
t
ε
ε
(2)
dyt = [a − xt] dt + σ2 dWt
. x ∝ membrane potential of neuron
. y ∝ proportion of open ion channels (recovery variable)
(1)
(2)
. Wt , Wt : independent
Wiener processes
q
. 0 < σ1, σ2 1, σ =
σ12 + σ22
8
Application: Stochastic FitzHugh–Nagumo equations
dxt =
1
σ1
(1)
[xt − x3
+
y
]
dt
+
dW
√
t
t
t
ε
ε
(2)
dyt = [a − xt] dt + σ2 dWt
. x ∝ membrane potential of neuron
. y ∝ proportion of open ion channels (recovery variable)
(1)
(2)
. Wt , Wt : independent
Wiener processes
q
. 0 < σ1, σ2 1, σ =
σ12 + σ22
ε = 0.1
2
δ = 3a2−1 = 0.02
σ1 = σ2 = 0.03
8-a
Application: Stochastic FitzHugh–Nagumo equations
dxt =
1
σ1
(1)
[xt − x3
+
y
]
dt
+
dW
√
t
t
t
ε
ε
(2)
dyt = [a − xt] dt + σ2 dWt
. x ∝ membrane potential of neuron
. y ∝ proportion of open ion channels (recovery variable)
(1)
(2)
. Wt , Wt : independent
Wiener processes
q
. 0 < σ1, σ2 1, σ =
σ12 + σ22
σ
ε3/4
Different regimes
σ = δ 3/2
σ = (δε)1/2
σ = δε1/4
[Muratov & Vanden Eijnden ’08]
ε1/2
δ
8-b
Small-amplitude oscillations (SAOs)
Definition of random number of SAOs N :
nullcline y = x3 − x
D
separatrix
P
F , parametrised by R ∈ [0, 1]
9
Small-amplitude oscillations (SAOs)
Definition of random number of SAOs N :
nullcline y = x3 − x
D
separatrix
P
F , parametrised by R ∈ [0, 1]
(R0, R1, . . . , RN −1) substochastic Markov chain with kernel
K(R0, A) = P R0 {Rτ ∈ A}
R ∈ F , A ⊂ F , τ = first-hitting time of F (after turning around P )
N = number of turns around P until leaving D
9-a
Main result 1
Theorem 1: [B & Landon, 2012]
If σ1, σ2 > 0, then λ0 < 1 and N is asymptotically geometric:
lim P µ0 {N = n + 1|N > n} = 1 − λ0
n→∞
10
Main result 1
Theorem 1: [B & Landon, 2012]
If σ1, σ2 > 0, then λ0 < 1 and N is asymptotically geometric:
lim P µ0 {N = n + 1|N > n} = 1 − λ0
n→∞
Proof:
. λ0 6 K(x∗ , E) < 1 by ellipticity (k bounded below)
R
µ
µ
0
0
. P {N > n} = P {Xn ∈ E} = E µ0 (dx)K n (x, E)
R
P µ0 {N > n} = P µ0 {Xn ∈ E} = E µ0 (dx)λn0 [1 + O((|λ1 |/λ0 )n )]
P µ0 {N > n} = P µ0 {Xn ∈ E} = λn0 [1 + O((|λ1 |/λ0 )n )]
R R
. P µ0 {N = n + 1} = E E µ0 (dx)K n (x, dy)[1 − K(y, E)]
P µ0 {N = n + 1} = λn0 (1 − λ0 )[1 + O((|λ1 |/λ0 )n )]
. Existence of spectral gap follows from positivity condition
Remark: If µ0 = π0 then N has geometric distribution and
1
P π0 {N = 1} = 1 − λ0 = π
E 0 [N ]
10-a
Main result 1
Theorem 1: [B & Landon, 2012]
If σ1, σ2 > 0, then λ0 < 1 and N is asymptotically geometric:
lim P µ0 {N = n + 1|N > n} = 1 − λ0
n→∞
Proof:
. λ0 6 K(x∗ , E) < 1 by ellipticity (k bounded below)
R
µ
µ
0
0
. P {N > n} = P {Xn ∈ E} = E µ0 (dx)K n (x, E)
R
P µ0 {N > n} = P µ0 {Xn ∈ E} = E µ0 (dx)λn0 [1 + O((|λ1 |/λ0 )n )]
P µ0 {N > n} = P µ0 {Xn ∈ E} = λn0 [1 + O((|λ1 |/λ0 )n )]
R R
. P µ0 {N = n + 1} = E E µ0 (dx)K n (x, dy)[1 − K(y, E)]
P µ0 {N = n + 1} = λn0 (1 − λ0 )[1 + O((|λ1 |/λ0 )n )]
. Existence of spectral gap follows from positivity condition
Remark: If µ0 = π0 then N has geometric distribution and
1
P π0 {N = 1} = 1 − λ0 = π
E 0 [N ]
10-b
Histograms of distribution of SAO number N (1000 spikes)
σ = ε = 10−4 , δ = 1.2 · 10−3 , . . . , 10−4
140
160
120
140
120
100
100
80
80
60
60
40
40
20
0
20
0
500
1000
1500
2000
2500
3000
3500
4000
600
0
0
50
100
150
200
250
300
1000
900
500
800
700
400
600
300
500
400
200
300
200
100
100
0
0
10
20
30
40
50
60
70
0
0
1
2
3
4
5
6
7
8
11
Main result 2
Theorem 2: [B & Landon 2012]
√
Assume ε and δ/ ε sufficiently small
√
2
1/4
2
There exists κ > 0 s.t. for σ 6 (ε
δ) / log( ε/δ)
. Principal eigenvalue:
(ε1/4δ)2
1 − λ0 6 exp −κ
σ2
. Expected number of SAOs:
1/4 δ)2
(ε
µ
E 0 [N ] > C(µ0) exp κ
σ2
where C(µ0) = probability of starting on F above separatrix
12
Main result 2
Theorem 2: [B & Landon 2012]
√
Assume ε and δ/ ε sufficiently small
√
2
1/4
2
There exists κ > 0 s.t. for σ 6 (ε
δ) / log( ε/δ)
. Principal eigenvalue:
(ε1/4δ)2
1 − λ0 6 exp −κ
σ2
. Expected number of SAOs:
1/4 δ)2
(ε
µ
E 0 [N ] > C(µ0) exp κ
σ2
where C(µ0) = probability of starting on F above separatrix
Proof:
. Construct a set A ⊂ E that the process is unlikely to leave
. Use the fact that for δ = 0, the deterministic system admits a first integral
. Apply the trivial bound
12-a
Transition from weak to strong noise
Linear approximation near separatrix:
dzt0 =
⇒
δ − σ12/ε
ε1/2
+ tzt0
σ1
σ2
(1)
(2)
dt − 3/4 t dWt + 3/4 dWt
ε
ε
ε1/4 (δ−σ12 /ε)
1/4
q
P{N = 1} ' Φ −π
σ12 +σ22
2
e−y /2
√
dy
Φ(x) =
−∞
2π
Z x
13
Transition from weak to strong noise
Linear approximation near separatrix:
dzt0 =
⇒
δ − σ12/ε
ε1/2
+ tzt0
σ1
σ2
(1)
(2)
dt − 3/4 t dWt + 3/4 dWt
ε
ε
ε1/4 (δ−σ12 /ε)
1/4
q
P{N = 1} ' Φ −π
σ12 +σ22
2
e−y /2
√
dy
Φ(x) =
−∞
2π
Z x
1
0.9
0.8
series
1/E(N)
P(N=1)
phi
∗: P{no SAO}
+: 1/E[N ]
◦: 1 − λ0
curve: x 7→ Φ(π 1/4x)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−1.5
−1
−0.5
0
0.5
1
1.5
2
−µ/σ
13-a
Back to the general case
[Joint work with Barbara Gentz, Christian Kuehn, in progress]
. Consider system of dim> 3 with several stable periodic orbits
. Noise can cause transitions between these orbits
. E.g. [Höpfner, Löcherbach & Thieullen ’12] show that a HH system with
noise will track any det solution with positive probability for some time
14
Back to the general case
[Joint work with Barbara Gentz, Christian Kuehn, in progress]
. Consider system of dim> 3 with several stable periodic orbits
. Noise can cause transitions between these orbits
. E.g. [Höpfner, Löcherbach & Thieullen ’12] show that a HH system with
noise will track any det solution with positive probability for some time
Pictures courtesy of K. Endler, master thesis, directed by R. Höpfner & M. Birkner
14-a
Back to the general case
[Joint work with Barbara Gentz, Christian Kuehn, in progress]
. Consider system of dim> 3 with several stable periodic orbits
. Noise can cause transitions between these orbits
. E.g. [Höpfner, Löcherbach & Thieullen ’12] show that a HH system with
noise will track any det solution with positive probability for some time
Pictures courtesy of K. Endler, master thesis, directed by R. Höpfner & M. Birkner
. Can we quantify transitions between deterministic patterns?
. Does the dynamics resemble some kind of Markov process
jumping between patterns?
. Spectral-theoretic approach inspired from reversible case
14-b
Laplace transforms
Given A ⊂ E, B ⊂ E ∪ {∆}, A ∩ B = ∅, x ∈ E and u ∈ C , define
τA = inf{n > 1 : Xn ∈ A}
σA = inf{n > 0 : Xn ∈ A}
x [euτA 1
Gu
(x)
=
E
{τA <τB } ]
A,B
u (x) = E x [euσA 1
HA,B
{σA <σB } ]
15
Laplace transforms
Given A ⊂ E, B ⊂ E ∪ {∆}, A ∩ B = ∅, x ∈ E and u ∈ C , define
τA = inf{n > 1 : Xn ∈ A}
σA = inf{n > 0 : Xn ∈ A}
x [euτA 1
Gu
(x)
=
E
{τA <τB } ]
A,B
u (x) = E x [euσA 1
HA,B
{σA <σB } ]
h
i−1
u
c
u
. GA,B (x) is analytic for |e | < supx∈(A∪B)c K(x, (A ∪ B) )
u
c, H u
u
. Gu
=
H
in
(A
∪
B)
=
1
in
A
and
H
A,B
A,B
A,B
A,B = 0 in B
. Feynman–Kac-type relation
u
KHA,B
= e−u Gu
A,B
15-a
Laplace transforms
Given A ⊂ E, B ⊂ E ∪ {∆}, A ∩ B = ∅, x ∈ E and u ∈ C , define
τA = inf{n > 1 : Xn ∈ A}
σA = inf{n > 0 : Xn ∈ A}
x [euτA 1
Gu
(x)
=
E
{τA <τB } ]
A,B
u (x) = E x [euσA 1
HA,B
{σA <σB } ]
h
i−1
u
c
u
. GA,B (x) is analytic for |e | < supx∈(A∪B)c K(x, (A ∪ B) )
u
c, H u
u
. Gu
=
H
in
(A
∪
B)
=
1
in
A
and
H
A,B
A,B
A,B
A,B = 0 in B
. Feynman–Kac-type relation
u
KHA,B
= e−u Gu
A,B
Proof:
u
(KHA,B
)(x)
h
i
X1 uσA
= E E e 1{σA <σB }
h
h
i
i
x
X1 uσA
x
X1 uσA
= E 1{X1 ∈A} E e 1{σA <σB } + E 1{X1 ∈Ac } E e 1{σA <σB }
x
x u(τA −1)
1{1<τA <τB }
= E 1{1=τA <τB } + E e
= E x eu(τA −1) 1{τA <τB } = e−u GuA,B (x)
x
⇒ if Gu
A,B varies little in A ∪ B, it is close to an eigenfunction
15-b
Heuristics
(inspired by [Bovier, Eckhoff, Gayrard, Klein ’04])
. Stable periodic orbits in x1, . . . , xN
S
. Bi small ball around xi, B = N
i=1 Bi
. Eigenvalue equation
(Kh)(x) = e−u h(x)
. Assume h(x) ' hi in Bi
B3
B1
B2
16
Heuristics
(inspired by [Bovier, Eckhoff, Gayrard, Klein ’04])
. Stable periodic orbits in x1, . . . , xN
S
. Bi small ball around xi, B = N
i=1 Bi
. Eigenvalue equation
(Kh)(x) = e−u h(x)
. Assume h(x) ' hi in Bi
Ansatz: h(x) =
N
X
j=1
B3
B1
B2
u
hj HB\B
(x) +r(x)
j ,Bj
16-a
Heuristics
(inspired by [Bovier, Eckhoff, Gayrard, Klein ’04])
. Stable periodic orbits in x1, . . . , xN
S
. Bi small ball around xi, B = N
i=1 Bi
. Eigenvalue equation
(Kh)(x) = e−u h(x)
. Assume h(x) ' hi in Bi
Ansatz: h(x) =
N
X
j=1
B3
B1
B2
u
hj HB\B
(x) +r(x)
j ,Bj
. x ∈ Bi: h(x) = hi +r(x)
. x ∈ B c: eigenvalue equation is satisfied (by Feynman–Kac)
. x = xi: eigenvalue equation yields by Feynman–Kac
hi =
N
X
j=1
hj Mij (u)
xi [euτB 1
Mij (u) = Gu
(x
)
=
E
{τB =τBj } ]
B\Bj ,Bj i
⇒ condition det(M − 1l) = 0 ⇒ N eigenvalues exp close to 1
If P{τB > 1} 1 then Mij (u) ' eu P xi {τB = τBj } =: eu Pij and P h ' e−u h
16-b
Control of the error term
The error term satisfies the boundary value problem
(Kr)(x) = e−u r(x)
r(x) = h(x) − hi
x ∈ Bc
x ∈ Bi
17
Control of the error term
The error term satisfies the boundary value problem
(Kr)(x) = e−u r(x)
r(x) = h(x) − hi
x ∈ Bc
x ∈ Bi
Lemma: For u s.t. Gu
B,E c exists, the unique solution of
(Kψ)(x) = e−u ψ(x)
x ∈ Bc
ψ(x) = θ(x)
x∈B
is given by ψ(x) = E x[euτB θ(XτB )] .
17-a
Control of the error term
The error term satisfies the boundary value problem
(Kr)(x) = e−u r(x)
r(x) = h(x) − hi
x ∈ Bc
x ∈ Bi
Lemma: For u s.t. Gu
B,E c exists, the unique solution of
(Kψ)(x) = e−u ψ(x)
x ∈ Bc
ψ(x) = θ(x)
x∈B
is given by ψ(x) = E x[euτB θ(XτB )] .
Proof:
. Show that T f (x) = E x [eu θ(X1 )1{X1 ∈B} ] + E x [eu f (X1 )1{X1 ∈B c } ]
is a contraction on L∞ (B c )
. Set ψ0 (x) = 0, ψn+1 (x) = T ψn (x) ∀n > 0
. Show by induction that ψn (x) = E x [euτB θ(XτB )1{τB 6n} ]
. ψ(x) = limn→∞ ψn (x) is fixed point of T ⇒ satisfies the bndry value problem
17-b
Control of the error term
The error term satisfies the boundary value problem
(Kr)(x) = e−u r(x)
r(x) = h(x) − hi
x ∈ Bc
x ∈ Bi
Lemma: For u s.t. Gu
B,E c exists, the unique solution of
(Kψ)(x) = e−u ψ(x)
x ∈ Bc
ψ(x) = θ(x)
x∈B
is given by ψ(x) = E x[euτB θ(XτB )] .
P
x
uτ
B
⇒ r(x) = E [e
θ(XτB )] where θ(x) = j [h(x) − hj ]1{x∈Bj }
To show that h(x) − hj is small in Bj : use Harnack inequalities
17-c
Conclusions
. Reduction to an N -state process in the sense that
P x{Xn ∈ Bi} =
N
X
∗ (B ) + O(|λ
n)
λn
h
(x)h
|
j
i
N
+1
j
j
j=1
. Residence times are approx exponential (provided system can
relax to QSD)
. Generically, eigenvalues λj are determined by “metastable
hierarchy” of periodic orbits
18
Conclusions
. Reduction to an N -state process in the sense that
N
X
P x{Xn ∈ Bi} =
∗ (B ) + O(|λ
n)
λn
h
(x)h
|
j
i
N
+1
j
j
j=1
. Residence times are approx exponential (provided system can
relax to QSD)
. Generically, eigenvalues λj are determined by “metastable
hierarchy” of periodic orbits
Open questions/outlook
. How to determine efficiently the Mij or Pij = P xi {τB = τBj }?
Large deviations – but not easy to implement and not very
precise
. Chaotic orbits?
18-a
Further reading
N.B. and Barbara Gentz, Noise-induced phenomena in slow-fast dynamical systems, A
sample-paths approach, Springer, Probability and
its Applications (2006)
N.B. and Barbara Gentz, 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., Stochastic dynamical systems in neuroscience, Oberwolfach Reports
8:2290–2293 (2011)
N.B., Barbara Gentz and Christian Kuehn, Hunting French Ducks in a Noisy
Environment, J. Differential Equations 252:4786–4841 (2012). arXiv:1011.3193
N.B. and Damien Landon, Mixed-mode oscillations and interspike interval
statistics in the stochastic FitzHugh–Nagumo model, Nonlinearity 25:2303–
2335 (2012). arXiv:1105.1278
N.B. and Barbara Gentz, On the noise-induced passage through an unstable
periodic orbit II: General case, preprint arXiv:1208.2557
www.univ-orleans.fr/mapmo/membres/berglund
19
Gérard Ben Arous, Shigeo Kusuoka, and Daniel W. Stroock, The Poisson
kernel for certain degenerate elliptic operators, J. Funct. Anal. 56:171–209
(1984).
Garrett Birkhoff, Extensions of Jentzsch’s theorem, Trans. Amer. Math.
Soc. 85:219–227 (1957).
Ivar Fredholm, Sur une classe d’équations fonctionnelles, Acta Math., 27:365–
390 (1903).
Robert Jentzsch, Über Integralgleichungen mit positivem Kern, J. f. d. reine
und angew. Math., 141:235–244 (1912).
Reinhard Höpfner, Eva Löcherbach, Michèle Thieullen, Transition densities
for stochastic Hodgkin-Huxley models, preprint arXiv:1207.0195 (2012).
Cyrill B. Muratov and Eric Vanden-Eijnden, Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle, Chaos
18:015111 (2008).
Esa Nummelin, General irreducible Markov chains and nonnegative operators,
Cambridge University Press, Cambridge, 1984.
20