Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Inter Spike Intervals probability distribution and
Double Integral Processes
Olivier Faugeras
NeuroMathComp project team - INRIA/ENS Paris
Workshop on Stochastic Models in Neuroscience
18-22 January 2010
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
LIF models of neurons
I
Membrane potential:
τm dV (t) = − (V (t) − Vrest ) + Ie (t) dt + dIs (t)
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
LIF models of neurons
I
Membrane potential:
τm dV (t) = − (V (t) − Vrest ) + Ie (t) dt + dIs (t)
I
Synaptic currents:
τs dIs (t) = −Is (t)dt + σdWt
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Reaching the threshold
Integrate the linear SDE:
V (t) = Vrest (1 − e
t
−τ
m
)+
1
τm
Rt
0
e
s−t
τm Ie (s) ds+
t
Is (0) − τt
σ − τt
−
s − e τm ) +
e m
τm (e
1 − τs
τm τs
with α =
Olivier Faugeras
ISI and DIP
1
τm
−
t
Z
0
s
eα
Z
s
s0
e τs
dWs 0
ds
0
1
τs .
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Reaching the threshold
I
A spike is emitted when V (t) reaches the threshold θ(t)
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Reaching the threshold
I
A spike is emitted when V (t) reaches the threshold θ(t)
I
Same as first hitting time of
Z
Z t
s
Xt =
eα
0
Olivier Faugeras
ISI and DIP
s
s0
e τs
dWs 0
ds
0
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Reaching the threshold
I
A spike is emitted when V (t) reaches the threshold θ(t)
I
Same as first hitting time of
Z
Z t
s
Xt =
eα
0
I
s
s0
e τs
dWs 0
ds
0
to the deterministic boundary a(t)
t R t s−t
σ − τt
−
e m a(t) = θ(t)− Vrest 1−e τm + τ1m 0 e τm Ie (s) ds+
τm τs
t
!
t
Is (0)
−τ
−τ
e s −e m
1 − ττms
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Stopping times
Definition
A positive real random variable is called a stopping time with
respect to the filtration Ft provided that {τ ≤ t} ∈ Ft for all
t ≥ 0.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Stopping times and diffusion equations
I
SDE:
dX (t) = b(X , t)dt + B(X , t)dWt
X (0) = X0
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Stopping times and diffusion equations
I
SDE:
dX (t) = b(X , t)dt + B(X , t)dWt
X (0) = X0
I
Let E be a non-empty open or closed set of
Rn , then
{τ = inf | X (t) ∈ E }
t≥0
is a stopping time.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Stopping times and diffusion equations
I
SDE:
dX (t) = b(X , t)dt + B(X , t)dWt
X (0) = X0
I
Let E be a non-empty open or closed set of
Rn , then
{τ = inf | X (t) ∈ E }
t≥0
is a stopping time.
I
Connection between SDEs and PDEs through the
Feynman-Kac formulae.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
LIF models
Stopping times and diffusion equations
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
A neural network
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
For each neuron i, define X (i) (t) ≥ 0 to be the remaining time
until the next emission of a spike by neuron i if it does not
receive any spike meanwhile.
I
This process has a very simple dynamics:
dX (i) (t)
= −1
dt
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
At time t, the next spike will occur in neuron
i = Arg Minj∈{1...N} X (j) (t) at time t + X (i) (t).
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
At time t, the next spike will occur in neuron
i = Arg Minj∈{1...N} X (j) (t) at time t + X (i) (t).
I
At spike time, the membrane potential of the neuron that just
spiked is reset.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
At time t, the next spike will occur in neuron
i = Arg Minj∈{1...N} X (j) (t) at time t + X (i) (t).
I
At spike time, the membrane potential of the neuron that just
spiked is reset.
I
The countdown value is also reset to a value Yi corresponding
to the next spike time of this neuron if nothing occurs
meanwhile.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
At time t, the next spike will occur in neuron
i = Arg Minj∈{1...N} X (j) (t) at time t + X (i) (t).
I
At spike time, the membrane potential of the neuron that just
spiked is reset.
I
The countdown value is also reset to a value Yi corresponding
to the next spike time of this neuron if nothing occurs
meanwhile.
I
This value is a random variable, the reset random variable.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
I
At time t, the next spike will occur in neuron
i = Arg Minj∈{1...N} X (j) (t) at time t + X (i) (t).
I
At spike time, the membrane potential of the neuron that just
spiked is reset.
I
The countdown value is also reset to a value Yi corresponding
to the next spike time of this neuron if nothing occurs
meanwhile.
I
This value is a random variable, the reset random variable.
I
Depending upon the neurone model, its law is that of the first
hitting time of a Brownian, an IWP or a DIP to a
deterministic boundary.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
The interaction random variable ηij between neurons i and j is the
modification of the time to the next spike of neuron j caused by its
receiving a spike from neuron i.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Countdown process and reset variable
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Markov description of the network
For a large variety of IF and LIF models, the state of the network
can be described by a Markov chain (or process) (Touboul,
Faugeras, in preparation), e.g. (X (t), Is (t), t).
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Neural networks
Neural networks and queuing theory
I
A lot can probably be gained in the study of neural networks
by looking at the work in queuing theory.
I
The countdown process is called an hourglass model
(introduced by Marie Cottrell 1992).
I
Later studied in (Turova 1996, Asmussen and Turova 1998,
Cottrell and Turova 2000, Turova 2000).
I
In order to apply this modeling we need to define in each case
the reset and the interaction random variables.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Double Integrated Process
Definition (DIP)
Let f ∈ L2R(R) and g ∈ L1 (R). Let Mt be the martingale defined
t
by Mt := 0 f (s)dWs .
The double integral process (DIP) associated to the functions f
and g is defined for all t ≥ 0 by:
Z s
Z t
Z t
Xt =
g (s)Ms ds =
g (s)
f (u)dWu ds
0
Olivier Faugeras
ISI and DIP
0
0
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Double Integrated Process
The LIF model:
Z
Xt =
ISI and DIP
s
Z
s
u
e τs dWu
eα
0
Olivier Faugeras
t
ds
0
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Double Integrated Process
Proposition
The two-dimensional process (Xt , Mt ) is a Gaussian Markov
process.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Introduction
A special case, the IWP
Definition (IWP)
The Integrated Wiener Process is a special case of the DIP where
the functions f and g are identically equal to 1 :
Z t
Xt =
Ws ds Ms = Ws
0
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Introduction
A special case, the IWP
Its transition measure reads:
h
i
= pt (u, v ; x, y )du dv =
P Xt+s ∈ du, Wt+s ∈ dv Xs = x, Ws = y def
√
h 6
i
3
6
2
2
2
(v
−y
)
du dv
exp
−
(u−x
−ty
)
+
(u−x
−ty
)(v
−y
)−
πt 2
t3
t2
t
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Introduction
Describing the problem
15
a(t)
10
5
0
Xt
Wt
−5
−10
Olivier Faugeras
ISI and DIP
0
1
2
3
4
5
6
7
8
9
10
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Constant boundary
First hitting time to a constant boundary
I
Consider Ut = (Xt + x + ty , Wt + y ) where Xt is the standard
IWP
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Constant boundary
First hitting time to a constant boundary
I
Consider Ut = (Xt + x + ty , Wt + y ) where Xt is the standard
IWP
I
Denote by
τa = inf t > 0 ; Xt + x + ty = a
the first passage time at a of the first component of the
two-dimensional Markov process Ut .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Constant boundary
A bit of history
McKean (1963) computes the joint law of (τa , Wτa ) for x = a:
h
i
= P(a, y ) (τa ∈ dt; |Wτa | ∈ dz)
P τa ∈ dt ; |Wτa | ∈ dz U0 = (a, y ) def
3z
2
2
= √ e −(2/t)(y −|y |z+z )
π 2t 2
Z
4|y |z/t
e
0
−3θ/2
dθ
√
πθ
!
1[0,+∞) (z)dzdt
def
= ma (t, y , z)
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Constant boundary
A bit of history
Goldman (1971) computes the distribution of the random variable
τa in the case where x < a and y ≤ 0:
h
i
hr 3 3(a − x) 2
3
−y e −3(a−x−ty ) /(2t )
P τa ∈ dt U0 = (x, y ) = dt
3
8πt
t
Z +∞
Z tZ ∞ h
i
i
+
zdz
P τ0 ∈ ds ; |Wτ0 | ∈ dµU0 = (0, z) qt−s (x, y ; a, z)
0
0
0
where qt (x, y ; u, v ) = pt (x, y ; u, v ) − pt (x, y ; u, −v )
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Constant boundary
A bit of history
Lachal (1991) extends these results and gives the joint distribution
of the pair (τa , Wτa ) in all cases:
h
P(x,y ) [τa ∈ dt ; Wτa ∈ dz] = |z| pt (x, y ; a, z)−
Z tZ
0
0
+∞
m0 (s, −|z|, µ)pt−s (x, y ; a, −εµ) dµ ds
i
1A (z)dzdt
where A = [0, ∞) if x < a, A = (−∞, 0] if x > a, ε = sign(a − x)
and m0 (s, −|z|, µ) is given by McKean’s formula. We denote this
a (t, z).
density by lx,y
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
Lachal (1996) extends these results to the case of a cubic
boundary. Idea of the proof
I
Under a certain probability, the process Wt + β2 t 2 + αt + x is
a Wiener process (Girsanov theorem)
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
Lachal (1996) extends these results to the case of a cubic
boundary. Idea of the proof
I
Under a certain probability, the process Wt + β2 t 2 + αt + x is
a Wiener process (Girsanov theorem)
I
Under this probability, the process Xt + β6 t 3 + α2 t 2 + tx + y
has the law of an IWP.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
Lachal (1996) extends these results to the case of a cubic
boundary. Idea of the proof
I
Under a certain probability, the process Wt + β2 t 2 + αt + x is
a Wiener process (Girsanov theorem)
I
Under this probability, the process Xt + β6 t 3 + α2 t 2 + tx + y
has the law of an IWP.
I
The knowledge of the pdf of the first hitting time of the IWP
to a constant yields that of the hitting time of the IWP to a
cubic.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
Let τC be the first hitting time of the standard IWP to the cubic
curve C of equation
C (t − s) = a + b(t − s) +
Olivier Faugeras
ISI and DIP
β
α
(t − s)2 + (t − s)3 . t ≥ s
2
6
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
Theorem
Under the reference probability P, the law of the random variable
(τC , WτC ) satisfies the equation:
Ps,(x,y ) (τC ∈ dt, WτC ∈ dz) = d −α,−β (s, x, y −b; t, a, z−b−α(t−s)−
β
β
(t−s)2 )×Ps,(x,y −b) (τa ∈ dt, Wτa −b−α(τa −s)− (τa −s)2 ∈ dz)
2
2
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
The case of a cubic boundary
I
The function d α,β is given by the application of Girsanov’s
theorem.
I
The probability in the righthand side is that given by Lachal in
1991.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
Why a cubic?
I
In the proof the IWP comes from the stochastic integration of
the function α + βt with respect to the Brownian density.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
Why a cubic?
I
In the proof the IWP comes from the stochastic integration of
the function α + βt with respect to the Brownian density.
I
Had we chosen a polynomial of degree greater than 1, the
integration by parts would have produced higher-order
integrals of the Brownian motion.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
Why a cubic?
I
In the proof the IWP comes from the stochastic integration of
the function α + βt with respect to the Brownian density.
I
Had we chosen a polynomial of degree greater than 1, the
integration by parts would have produced higher-order
integrals of the Brownian motion.
I
This method does not generalize to polynomial boundaries of
degree larger than three.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Cubic boundary
Why a cubic?
I
In the proof the IWP comes from the stochastic integration of
the function α + βt with respect to the Brownian density.
I
Had we chosen a polynomial of degree greater than 1, the
integration by parts would have produced higher-order
integrals of the Brownian motion.
I
This method does not generalize to polynomial boundaries of
degree larger than three.
I
For general boundaries we perform a piecewise-cubic
approximation.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principle of the method
Compute the probability that the first hitting time of the IWP to a
continuous piecewise function is greater than t ∈ [tp , tp+1 [.
14
12
10
8
6
4
2
0
−2
Olivier Faugeras
ISI and DIP
tp
0
1
2
3
4
5
6
7
8
t
tp+1
9
10
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
I
Let C (t) be a continuous piecewise cubic function defined on
the interval [0, T ].
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
I
Let C (t) be a continuous piecewise cubic function defined on
the interval [0, T ].
I
Let (Ut )t≥0 = (Xt , Wt )t≥0 and τCs = inf {t > s | Xt = C (t)}
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
I
Let C (t) be a continuous piecewise cubic function defined on
the interval [0, T ].
I
Let (Ut )t≥0 = (Xt , Wt )t≥0 and τCs = inf {t > s | Xt = C (t)}
I
Fix t ∈ [0, T [, let p be the index of the bin t belongs to:
t ∈ [tp , tp+1 [
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
I
Let C (t) be a continuous piecewise cubic function defined on
the interval [0, T ].
I
Let (Ut )t≥0 = (Xt , Wt )t≥0 and τCs = inf {t > s | Xt = C (t)}
I
Fix t ∈ [0, T [, let p be the index of the bin t belongs to:
t ∈ [tp , tp+1 [
I
We use the strong Markov property of Ut to express τC0
recursively as an integral of a product of p + 1 terms
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
The event {Ut1 = u1 , τC0 ≥ t1 , U0 } is in F Ut1
Therefore
P τC0 ≥ t Ut1 = u1 , τC0 ≥ t1 , U0 = P τCt1 ≥ t Ut1 = u1
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
P τC0 ≥ t U0 =
Z
(2)
P τC0 ≥ t Ut1 = u1 , τC0 ≥ t1 , U0 P Ut1 ∈ du1 , τC0 ≥ t1 U0 =
Z
(2)
P τCt1 ≥ t Ut1 = u1 P Ut1 ∈ du1 , τC0 ≥ t1 U0
The first term in the integral is similar to the lefthand side of the
equation: proceed recursively
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
P
τC0
Z
≥ t U0 =
(4)
P
τCt2
≥ t Ut2 = u2
× P Ut2 ∈ du2 , τCt1 ≥ t2 |Ut1 = u1
× P Ut1 ∈ du1 , τC0 ≥ t1 U0
..
.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
P τC0 ≥ t U0 =
Z
(2p)
P τCtp ≥ t Utp = up
t
× P Utp ∈ dup , τCp−1 ≥ tp |Utp−1 = up−1
t
× P Utp−1 ∈ dup−1 , τCp−2 ≥ tp−1 |Utp−2 = up−2
× ...
× P Ut1 ∈ du1 , τC0 ≥ t1 U0
Olivier Faugeras
ISI and DIP
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Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
‘
n
o
t
t
{Utk ∈ duk , τCk−1 ≥ tk } = {Utk ∈ duk } \ Utk ∈ duk , τCk−1 < tk ,
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
P
t
Utk ∈ duk , τCk−1 ≥ tk |Utk−1 = uk−1
t
= P Utk ∈ duk |Utk−1 = uk−1 − P Utk ∈ duk , τCk−1 ≤ tk |Utk−1 = uk−1
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
= P Utk ∈ duk |Utk−1 = uk−1 −
Z tk P Utk ∈ duk , τCtk−1 ∈ ds|Utk−1 = uk−1
tk−1
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
= P Utk ∈ duk |Utk−1 = uk−1
Z tk Z
−
P Utk ∈ duk |τCtk−1 = s, Ws = y , Utk−1 = uk−1
t
R
k−1
t
× P τCk−1 ∈ ds, Ws ∈ dy Utk−1 = uk−1
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
= ptk −tk−1 (uk ; uk−1 )
Z tk Z
t
−
ptk −s (uk ; C (s), y )P τCk−1 ∈ ds, Ws ∈ dy Utk−1 = uk−1 duk
tk−1
Olivier Faugeras
ISI and DIP
R
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Continuous piecewise cubic function
Principles of the proof
Theorem
The law of the first hitting time of the IWP to a continuous
piecewise cubic boundary is given by the formula:
P τC0 ≥ t U0 =
Z
(2p)
P τCtp ≥ t|Utp = up
p
Y
ptk −tk−1 (uk ; uk−1 )
k=1
Z
−
tk
tk−1
Z
R
ptk −s (uk ; C (s), y )P
t
τCk−1
!
∈ ds, Ws ∈ dy Utk−1
duk
t
Note that P τCk−1 ∈ ds, Ws ∈ dy Utk−1 has been derived
previously.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principle of the method
15
10
Approximated
Boundary
5
Original
Boundary
0
IWP
−5
Olivier Faugeras
ISI and DIP
0
1
2
3
4
5
6
7
8
9
10
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principles of the proof
Let C : R 7→ R be a continuously differentiable function
Let also T > 0 and
0 = t0 < t1 < . . . < tn = T
be a partition, noted π, of the interval [0, T ].
Denote by δ π the mesh step defined as:
δ π = max{ti+1 − ti , i = 0 . . . n − 1}
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principles of the proof
Let Cπ be a cubic spline approximation of C
It is a C 2 interpolation of C which is an approximation of order
four, i.e.
kC − Cπ k∞,T = sup |C (t) − Cπ (t)| ≤ K (C )δ(π)4 ,
t∈[0,T ]
K (C ) is a function of C only.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principles of the proof
Theorem
The first hitting time of the IWP to the curve Cπ before T
converges in law to the first hitting time of the IWP to the curve
C before T .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principles of the proof
I
If C is C 2 the convergence is of the same order as the
approximation of C by the cubic function Cπ .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
General boundary
Principles of the proof
I
I
If C is C 2 the convergence is of the same order as the
approximation of C by the cubic function Cπ .
Let P(T , g ) = P Xt ≥ g (t) for some t ∈ [0, T ] . There
exists a constant K̃ (C , T ) such that:
|P(T , C ) − P(T , Cπ )| ≤ K̃ (C , T ) kC − Cπ k∞,T
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
Lemma
Let (Xt )t≥0 be a DIP. Assume that f (s) 6= 0 for all s ≥ 0. The
study of the hitting times of the DIP X is equivalent to the study
of the simpler process:
Z t
X̃t =
h(s)Ws ds,
0
where h is obtained from f and g .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
1. Mt =
Olivier Faugeras
ISI and DIP
Rt
0
f (s)dWs
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
1. Mt =
Rt
0
f (s)dWs
2. There exists a Brownian motion (Wt )t such that almost surely
(Dubins-Schwarz)
Mt = WhMit
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
1. Mt =
Rt
0
f (s)dWs
2. There exists a Brownian motion (Wt )t such that almost surely
(Dubins-Schwarz)
Mt = WhMit
Rt 2
3. Let Φ(t) = hMit = 0 f (s)ds
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
1. Mt =
Rt
0
f (s)dWs
2. There exists a Brownian motion (Wt )t such that almost surely
(Dubins-Schwarz)
Mt = WhMit
Rt 2
3. Let Φ(t) = hMit = 0 f (s)ds
4. Φ is one to one.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Simplified DIP
Simplified DIP
1. Mt =
Rt
0
f (s)dWs
2. There exists a Brownian motion (Wt )t such that almost surely
(Dubins-Schwarz)
Mt = WhMit
Rt 2
3. Let Φ(t) = hMit = 0 f (s)ds
4. Φ is one to one.
5.
Z
t
Xt =
g (s)Ms ds
0
L
Z
=
t
g (s)WΦ(s) ds
0
Z
=
0
Olivier Faugeras
ISI and DIP
Φ−1 (t)
g (Φ−1 (s))
Ws ds ≡
Φ0 Φ−1 (s)
Z
t
h(s)Ws ds
0
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principles of the method
Approximation principle
15
10
Original
Boundary
Control points
Cubic spline boundary
approximation
5
DIP
0
Approximated
DIP
−5
−10
Olivier Faugeras
ISI and DIP
0
1
2
3
4
5
6
7
8
9
10
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principles of the method
I
Let π be a partition of the interval [0, T ] with n intervals:
0 = t0 < t1 < t2 < . . . < tn = T
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principles of the method
I
Let π be a partition of the interval [0, T ] with n intervals:
0 = t0 < t1 < t2 < . . . < tn = T
I
Denote by hπ the piecewise constant approximation of h
defined by:
n−1
X
π
h (t) =
h(ti )1[ti ,ti+1 ) (t),
i=0
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principles of the method
I
Let π be a partition of the interval [0, T ] with n intervals:
0 = t0 < t1 < t2 < . . . < tn = T
I
Denote by hπ the piecewise constant approximation of h
defined by:
n−1
X
π
h (t) =
h(ti )1[ti ,ti+1 ) (t),
i=0
I
Denote by
Xπ
the associated DIP:
Z t
hπ (s)Ws ds.
Xtπ =
0
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principles of the method
I
Let π be a partition of the interval [0, T ] with n intervals:
0 = t0 < t1 < t2 < . . . < tn = T
I
Denote by hπ the piecewise constant approximation of h
defined by:
n−1
X
π
h (t) =
h(ti )1[ti ,ti+1 ) (t),
i=0
I
Denote by
Xπ
the associated DIP:
Z t
hπ (s)Ws ds.
Xtπ =
0
I
C π is the piecewise cubic approximation of the boundary.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
Proposition
The process Xtπ converges almost surely to the process Xt .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
Theorem
The first hitting time τ π of the process X π to the curve C π
converges in law to the first hitting time τC of the process X to
the curve C .
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
Theorem
Let h be a Lipschitz continuous real function, T > 0 and π a
partition of the interval [0, T ]
0 = t0 < t1 < . . . < tn = T
Let C be a continuously differentiable function. The first hitting
time τ π of the approximated process X π to a cubic spline
approximation of C on the partition π, denoted by C π , satisfies the
equation on the next slide
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
n
(2n) Y
(
x − x
k
k−1
, yk −yk−1 ; 0, 0 −
h(tk−1 )
k=1
Z tk Z
x − C π (s)
k
ptk −s
, yk − y ; 0, 0
h(tk−1 )
tk −1 R
)
P(τ π ≥ T |U0 ) =
Z
ptk −tk−1
Ps,(0,y ) (τ(C −xk−1 )/h(tk−1 ) ∈ ds, Ws ∈ dy ) dxk dyk
where P(τC ∈ ds, Ws ∈ dy ) is given by Lachal’s or McKean’s
density.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Principle of the method
Principle of the method
R2n when
I
The expressions we found involve an integral on
there are n + 1 points in the mesh.
I
Another approximation is done besides the previous ones.
I
We express the integral as an expectation and use a
Monte-Carlo algorithm to compute it.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Implementation
Implementation
Corollary
I
Let h be a Lipschitz continuous real function, (Xt , Wt )t≥0 be
a standard IWP-Brownian motion pair, T > 0 and π a
partition of the interval [0, T ].
I
Let C be a continuously differentiable function. The first
hitting time τ π of the approximated process X π to a cubic
spline approximation C π of C on the partition π can be
computed as the expectation:
i
h
P τ π ≥ t U0 = E θph,π (t, Xt1 , Wt1 , . . . , Xt , Wt )U0
I
The function θph,π is defined for t ∈ [tp−1 , tp [ on the next slide.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Implementation
Implementation
θph,π (x1 , y1 . . . , x, y ) :=
p−1
Y
(
ptk −s
π
xk −C (s)
h(tk−1 ) , yk
− z; 0, 0
xk −xk−1
h(tk−1 ) , yk
− yk−1 ; 0, 0
ptk −tk−1 (xk , yk , xk−1 , yk−1 )
×
ptk −tk−1 (xk , yk , xk−1 , yk−1 )
k=1
ptk −tk−1
Ps,(0,y ) (τ(C −x
pt−tp−1
s
x−xp−1
h(tp−1 ) , y
Z
tk
−
tk−1
Z
R
)
k−1 )/h(tk−1 )
− yp−1 ; 0, 0
∈ ds, Ws ∈ dz)
pt−tp−1 (x, y , xp−1 , yp−1 )
!
Z t Z pt−s x−C π (s) , y − z; 0, 0
h(tp−1 )
Ps,(0,z) (τ(C −xp−1 )/h(tp−1 ) ∈ ds, Ws ∈ dz)
−
tp−1 R pt−tp−1 (x, y , xp−1 , yp−1 )
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
Probability density function of the first hitting time of the IWP to the
cubic curve: t 7→ 1 − 2t − 2t 2 − t 3 with the intial condition
X0 = 0, W0 = 0. The total mass is 1 in this case.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Results
Results
Probability density function of the first hitting time of the IWP to the
cubic curve: t 7→ 1 − 12 t + t 3 with the intial condition X0 = 0, W0 = 0.
The total mass is ≈ 0.2578 in this case.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
Method of approximation of the probability distribution of the
first hitting time of a Double Integral Process (DIP) to a
curved boudary. This is the first result for this problem.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
1) We obtain a closed-form expression of the probability
distribution of the first hitting time of the Integrated Wiener
Process (IWP) to a continuous piecewise cubic boundary.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
1) We obtain a closed-form expression of the probability
distribution of the first hitting time of the Integrated Wiener
Process (IWP) to a continuous piecewise cubic boundary.
I
2) By approximating a general smooth boundary with a
piecewise cubic function we compute an approximation of the
probability distribution of the first hitting time of the IWP to
any smooth curved boundary, and prove convergence.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
3) By approximating the DIP with a piecewise IWP we
compute an approximation of the probability distribution of
the first hitting time of the DIP to any smooth curved
boundary, and prove convergence.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
3) By approximating the DIP with a piecewise IWP we
compute an approximation of the probability distribution of
the first hitting time of the DIP to any smooth curved
boundary, and prove convergence.
I
We sketch a numerical procedure based on Monte-Carlo
simulation to compute the probability distribution efficiently.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Conclusion
I
3) By approximating the DIP with a piecewise IWP we
compute an approximation of the probability distribution of
the first hitting time of the DIP to any smooth curved
boundary, and prove convergence.
I
We sketch a numerical procedure based on Monte-Carlo
simulation to compute the probability distribution efficiently.
I
These results have potential applications in many fields of
physics and biology.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris
Motivations
DIP
IWP
IWP hits boundary
DIP hits boundary
Numerics
Conclusion
Acknowledgements
I
From Touboul and Faugeras, Advances in Applied Probability,
2008.
I
Supported by European grant FACETS and ERC grant NerVi.
Olivier Faugeras
ISI and DIP
NeuroMathComp project team - INRIA/ENS Paris