Inter Spike Intervals probability distribution and Double Integral - - PowerPoint PPT Presentation

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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

LIF models of neurons

◮ Membrane potential:

τmdV (t) =

• − (V (t) − Vrest) + Ie(t)
• dt + dIs(t)

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

LIF models of neurons

◮ Membrane potential:

τmdV (t) =

• − (V (t) − Vrest) + Ie(t)
• dt + dIs(t)

◮ Synaptic currents:

τsdIs(t) = −Is(t)dt + σdWt

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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

τm ) + 1 τm

t

0 e s−t τm Ie(s) ds+

Is(0) 1 − τm

τs

(e− t

τs − e− t τm ) +

σ τmτs e− t

τm

t e

s α

s e

s′ τs dWs′

• ds

with α =

1 τm − 1 τs .

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Reaching the threshold

◮ A spike is emitted when V (t) reaches the threshold θ(t)

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Reaching the threshold

◮ A spike is emitted when V (t) reaches the threshold θ(t) ◮ Same as first hitting time of

Xt = t e

s α

s e

s′ τs dWs′

• ds

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Reaching the threshold

◮ A spike is emitted when V (t) reaches the threshold θ(t) ◮ Same as first hitting time of

Xt = t e

s α

s e

s′ τs dWs′

• ds

◮ to the deterministic boundary a(t)

σ τmτs e− t

τm a(t) = θ(t)−

• Vrest
• 1−e− t

τm

• + 1

τm

t

0 e s−t τm Ie(s) ds+

Is(0) 1 − τm

τs

• e− t

τs − e− t τm

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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.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Stopping times and diffusion equations

◮ SDE:

dX(t) = b(X, t)dt + B(X, t)dWt X(0) = X0 ❘

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Stopping times and diffusion equations

◮ SDE:

dX(t) = b(X, t)dt + B(X, t)dWt X(0) = X0

◮ Let E be a non-empty open or closed set of ❘n, then

{τ = inf

t≥0 | X(t) ∈ E}

is a stopping time.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Stopping times and diffusion equations

◮ SDE:

dX(t) = b(X, t)dt + B(X, t)dWt X(0) = X0

◮ Let E be a non-empty open or closed set of ❘n, then

{τ = inf

t≥0 | X(t) ∈ E}

is a stopping time.

◮ Connection between SDEs and PDEs through the

Feynman-Kac formulae.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion LIF models

Stopping times and diffusion equations

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

A neural network

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ 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.

◮ This process has a very simple dynamics:

dX (i)(t) dt = −1

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ At time t, the next spike will occur in neuron

i = Arg Minj∈{1...N} X (j)(t) at time t + X (i)(t).

◮ At spike time, the membrane potential of the neuron that just

spiked is reset.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ At time t, the next spike will occur in neuron

i = Arg Minj∈{1...N} X (j)(t) at time t + X (i)(t).

◮ At spike time, the membrane potential of the neuron that just

spiked is reset.

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ At time t, the next spike will occur in neuron

i = Arg Minj∈{1...N} X (j)(t) at time t + X (i)(t).

◮ At spike time, the membrane potential of the neuron that just

spiked is reset.

◮ The countdown value is also reset to a value Yi corresponding

to the next spike time of this neuron if nothing occurs meanwhile.

◮ This value is a random variable, the reset random variable.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

◮ At time t, the next spike will occur in neuron

i = Arg Minj∈{1...N} X (j)(t) at time t + X (i)(t).

◮ At spike time, the membrane potential of the neuron that just

spiked is reset.

◮ The countdown value is also reset to a value Yi corresponding

to the next spike time of this neuron if nothing occurs meanwhile.

◮ This value is a random variable, the reset random variable. ◮ 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.

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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.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Countdown process and reset variable

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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).

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Neural networks

Neural networks and queuing theory

◮ A lot can probably be gained in the study of neural networks

by looking at the work in queuing theory.

◮ The countdown process is called an hourglass model

(introduced by Marie Cottrell 1992).

◮ Later studied in (Turova 1996, Asmussen and Turova 1998,

Cottrell and Turova 2000, Turova 2000).

◮ In order to apply this modeling we need to define in each case

the reset and the interaction random variables.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Double Integrated Process

Definition (DIP)

Let f ∈ L2(❘) and g ∈ L1(❘). Let Mt be the martingale defined by Mt := t

0 f (s)dWs.

The double integral process (DIP) associated to the functions f and g is defined for all t ≥ 0 by: Xt = t g(s)Msds = t g(s) s f (u)dWu

• ds

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Double Integrated Process

The LIF model: Xt = t e

s α

s e

u τs dWu

• ds

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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.

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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 : Xt = t Ws ds Ms = Ws

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Introduction

A special case, the IWP

Its transition measure reads: P

• Xt+s ∈ du, Wt+s ∈ dv
• Xs = x, Ws = y

def = pt(u, v; x, y)du dv = √ 3 πt2 exp

• − 6

t3 (u−x−ty)2+ 6 t2 (u−x−ty)(v−y)−2 t (v−y)2 du dv

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Introduction

Describing the problem

1 2 3 4 5 6 7 8 9 10 −10 −5 5 10 15

Xt Wt a(t) Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Constant boundary

First hitting time to a constant boundary

◮ Consider Ut = (Xt + x + ty, Wt + y) where Xt is the standard

IWP

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Constant boundary

First hitting time to a constant boundary

◮ Consider Ut = (Xt + x + ty, Wt + y) where Xt is the standard

IWP

◮ 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.

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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: P

• τa ∈ dt ; |Wτa| ∈ dz
• U0 = (a, y)

def = P(a, y)(τa ∈ dt; |Wτa| ∈ dz) = 3z π √ 2t2 e−(2/t)(y2−|y|z+z2) 4|y|z/t e−3θ/2 dθ √ πθ

• ✶[0,+∞)(z)dzdt

def

= ma(t, y, z)

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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: P

• τa ∈ dt
• U0 = (x, y)
• = dt
• 3

8πt3 3(a − x) t −y

• e−3(a−x−ty)2/(2t3)

+ +∞ zdz t ∞ P

• τ0 ∈ ds ; |Wτ0| ∈ dµ
• U0 = (0, z)
• qt−s(x, y; a, z)
• where qt(x, y; u, v) = pt(x, y; u, v) − pt(x, y; u, −v)

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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

• f the pair (τa, Wτa) in all cases:

P(x,y) [τa ∈ dt ; Wτa ∈ dz] = |z|

• pt(x, y; a, z)−

t +∞ m0(s, −|z|, µ)pt−s(x, y; a, −εµ) dµ ds

• ✶A(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 density by la

x,y(t, z).

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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

◮ Under a certain probability, the process Wt + β 2 t2 + αt + x is

a Wiener process (Girsanov theorem)

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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

◮ Under a certain probability, the process Wt + β 2 t2 + αt + x is

a Wiener process (Girsanov theorem)

◮ Under this probability, the process Xt + β 6 t3 + α 2 t2 + tx + y

has the law of an IWP.

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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

◮ Under a certain probability, the process Wt + β 2 t2 + αt + x is

a Wiener process (Girsanov theorem)

◮ Under this probability, the process Xt + β 6 t3 + α 2 t2 + tx + y

has the law of an IWP.

◮ 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.

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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) + α 2 (t − s)2 + β 6 (t − s)3. t ≥ s

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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 2 (t−s)2)×Ps,(x,y−b)(τa ∈ dt, Wτa−b−α(τa−s)−β 2 (τa−s)2 ∈ dz)

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Cubic boundary

The case of a cubic boundary

◮ The function dα,β is given by the application of Girsanov’s

theorem.

◮ The probability in the righthand side is that given by Lachal in

1991.

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Cubic boundary

Why a cubic?

◮ In the proof the IWP comes from the stochastic integration of

the function α + βt with respect to the Brownian density.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Cubic boundary

Why a cubic?

◮ In the proof the IWP comes from the stochastic integration of

the function α + βt with respect to the Brownian density.

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Cubic boundary

Why a cubic?

◮ In the proof the IWP comes from the stochastic integration of

the function α + βt with respect to the Brownian density.

◮ Had we chosen a polynomial of degree greater than 1, the

integration by parts would have produced higher-order integrals of the Brownian motion.

◮ This method does not generalize to polynomial boundaries of

degree larger than three.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Cubic boundary

Why a cubic?

◮ In the proof the IWP comes from the stochastic integration of

the function α + βt with respect to the Brownian density.

◮ Had we chosen a polynomial of degree greater than 1, the

integration by parts would have produced higher-order integrals of the Brownian motion.

◮ This method does not generalize to polynomial boundaries of

degree larger than three.

◮ For general boundaries we perform a piecewise-cubic

approximation.

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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[.

1 2 3 4 5 6 7 8 9 10 −2 2 4 6 8 10 12 14

t

tp tp+1

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

◮ Let C(t) be a continuous piecewise cubic function defined on

the interval [0, T].

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

◮ Let C(t) be a continuous piecewise cubic function defined on

the interval [0, T].

◮ Let (Ut)t≥0 = (Xt, Wt)t≥0 and τ s C = inf {t > s | Xt = C(t)}

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

◮ Let C(t) be a continuous piecewise cubic function defined on

the interval [0, T].

◮ Let (Ut)t≥0 = (Xt, Wt)t≥0 and τ s C = inf {t > s | Xt = C(t)} ◮ Fix t ∈ [0, T[, let p be the index of the bin t belongs to:

t ∈ [tp, tp+1[

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

◮ Let C(t) be a continuous piecewise cubic function defined on

the interval [0, T].

◮ Let (Ut)t≥0 = (Xt, Wt)t≥0 and τ s C = inf {t > s | Xt = C(t)} ◮ Fix t ∈ [0, T[, let p be the index of the bin t belongs to:

t ∈ [tp, tp+1[

◮ We use the strong Markov property of Ut to express τ 0 C

recursively as an integral of a product of p + 1 terms

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

The event {Ut1 = u1, τ 0

C ≥ t1, U0} is in FUt1

Therefore P

• τ 0

C ≥ t

• Ut1 = u1, τ 0

C ≥ t1, U0

• = P
• τ t1

C ≥ t

• Ut1 = u1
• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

P

• τ 0

C ≥ t

• U0
• =

(2) P

• τ 0

C ≥ t

• Ut1 = u1, τ 0

C ≥ t1, U0

• P
• Ut1 ∈ du1, τ 0

C ≥ t1

• U0
• =

(2) P

• τ t1

C ≥ t

• Ut1 = u1
• P
• Ut1 ∈ du1, τ 0

C ≥ t1

• U0
• The first term in the integral is similar to the lefthand side of the

equation: proceed recursively

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

P

• τ 0

C ≥ t

• U0
• =

(4) P

• τ t2

C ≥ t

• Ut2 = u2
• × P
• Ut2 ∈ du2, τ t1

C ≥ t2|Ut1 = u1

• × P
• Ut1 ∈ du1, τ 0

C ≥ t1

• U0
• .

. .

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Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

P

• τ 0

C ≥ t

• U0
• =

(2p) P

• τ tp

C ≥ t

• Utp = up
• × P
• Utp ∈ dup, τ tp−1

C

≥ tp|Utp−1 = up−1

• × P
• Utp−1 ∈ dup−1, τ tp−2

C

≥ tp−1|Utp−2 = up−2

• × . . .

× P

• Ut1 ∈ du1, τ 0

C ≥ t1

• U0
• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

‘ {Utk ∈ duk, τ tk−1

C

≥ tk} = {Utk ∈ duk}\

• Utk ∈ duk, τ tk−1

C

< tk

• ,

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Continuous piecewise cubic function

Principles of the proof

P

• Utk ∈ duk, τ tk−1

C

≥ tk|Utk−1 = uk−1

• = P
• Utk ∈ duk|Utk−1 = uk−1
• − P
• Utk ∈ duk, τ tk−1

C

≤ tk|Utk−1 = uk−1

• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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
• −

tk

tk−1

P

• Utk ∈ duk, τ tk−1

C

∈ ds|Utk−1 = uk−1

• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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
• −

tk

tk−1

• ❘

P

• Utk ∈ duk|τ tk−1

C

= s, Ws = y, Utk−1 = uk−1

• × P
• τ tk−1

C

∈ ds, Ws ∈ dy

• Utk−1 = uk−1
• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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)

− tk

tk−1

• ❘

ptk−s(uk; C(s), y)P

• τ tk−1

C

∈ ds, Ws ∈ dy

• Utk−1 = uk−1

duk

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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

• τ 0

C ≥ t

• U0
• =

(2p) P

• τ tp

C ≥ t|Utp = up

• p
• k=1
• ptk−tk−1(uk; uk−1)

− tk

tk−1

• ❘

ptk−s(uk; C(s), y)P

• τ tk−1

C

∈ ds, Ws ∈ dy

• Utk−1

duk Note that P

• τ tk−1

C

∈ ds, Ws ∈ dy

• Utk−1
• has been derived

previously.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion General boundary

Principle of the method

1 2 3 4 5 6 7 8 9 10 −5 5 10 15

IWP Approximated Boundary Original Boundary Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion General boundary

Principles of the proof

Let C : ❘ → ❘ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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. C − Cπ∞,T = sup

t∈[0,T]

|C(t) − Cπ(t)| ≤ K(C)δ(π)4, K(C) is a function of C only.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion General boundary

Principles of the proof

◮ If C is C 2 the convergence is of the same order as the

approximation of C by the cubic function Cπ. P

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion General boundary

Principles of the proof

◮ 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 ˜ K(C, T) such that: |P(T, C) − P(T, Cπ)| ≤ ˜ K(C, T) C − Cπ∞,T

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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) = 0 for all s ≥ 0. The study of the hitting times of the DIP X is equivalent to the study

• f the simpler process:

˜ Xt = t h(s)Wsds, where h is obtained from f and g.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Simplified DIP

Simplified DIP

• 1. Mt =

t

0 f (s)dWs

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Simplified DIP

Simplified DIP

• 1. Mt =

t

0 f (s)dWs

• 2. There exists a Brownian motion (Wt)t such that almost surely

(Dubins-Schwarz) Mt = WMt

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Simplified DIP

Simplified DIP

• 1. Mt =

t

0 f (s)dWs

• 2. There exists a Brownian motion (Wt)t such that almost surely

(Dubins-Schwarz) Mt = WMt

• 3. Let Φ(t) = Mt =

t

0 f 2(s)ds

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Simplified DIP

Simplified DIP

• 1. Mt =

t

0 f (s)dWs

• 2. There exists a Brownian motion (Wt)t such that almost surely

(Dubins-Schwarz) Mt = WMt

• 3. Let Φ(t) = Mt =

t

0 f 2(s)ds

• 4. Φ is one to one.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Simplified DIP

Simplified DIP

• 1. Mt =

t

0 f (s)dWs

• 2. There exists a Brownian motion (Wt)t such that almost surely

(Dubins-Schwarz) Mt = WMt

• 3. Let Φ(t) = Mt =

t

0 f 2(s)ds

• 4. Φ is one to one.

5. Xt = t g(s)Ms ds

L

= t g(s)WΦ(s)ds = Φ−1(t) g(Φ−1(s)) Φ′ Φ−1(s) Ws ds ≡ t h(s)Wsds

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principles of the method

1 2 3 4 5 6 7 8 9 10 −10 −5 5 10 15 Approximation principle DIP Approximated DIP Original Boundary Cubic spline boundary approximation Control points

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principles of the method

◮ Let π be a partition of the interval [0, T] with n intervals:

0 = t0 < t1 < t2 < . . . < tn = T ✶

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principles of the method

◮ Let π be a partition of the interval [0, T] with n intervals:

0 = t0 < t1 < t2 < . . . < tn = T

◮ Denote by hπ the piecewise constant approximation of h

defined by: hπ(t) =

n−1

• i=0

h(ti)✶[ti,ti+1)(t),

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principles of the method

◮ Let π be a partition of the interval [0, T] with n intervals:

0 = t0 < t1 < t2 < . . . < tn = T

◮ Denote by hπ the piecewise constant approximation of h

defined by: hπ(t) =

n−1

• i=0

h(ti)✶[ti,ti+1)(t),

◮ Denote by X π the associated DIP:

X π

t =

t hπ(s)Ws ds.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principles of the method

◮ Let π be a partition of the interval [0, T] with n intervals:

0 = t0 < t1 < t2 < . . . < tn = T

◮ Denote by hπ the piecewise constant approximation of h

defined by: hπ(t) =

n−1

• i=0

h(ti)✶[ti,ti+1)(t),

◮ Denote by X π the associated DIP:

X π

t =

t hπ(s)Ws ds.

◮ C π is the piecewise cubic approximation of the boundary.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Results

Results

Proposition

The process X π

t converges almost surely to the process Xt.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Results

Results

P(τ π ≥ T|U0) = (2n)

n

• k=1
• ptk−tk−1

xk − xk−1 h(tk−1) , yk−yk−1; 0, 0

• −

tk

tk−1

• ❘

ptk−s xk − C π(s) h(tk−1) , yk − y; 0, 0

• Ps,(0,y)(τ(C−xk−1)/h(tk−1) ∈ ds, Ws ∈ dy)
• dxkdyk

where P(τC ∈ ds, Ws ∈ dy) is given by Lachal’s or McKean’s density.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Principle of the method

Principle of the method

◮ The expressions we found involve an integral on ❘2n when

there are n + 1 points in the mesh.

◮ Another approximation is done besides the previous ones. ◮ We express the integral as an expectation and use a

Monte-Carlo algorithm to compute it.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Implementation

Implementation

Corollary

◮ 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].

◮ 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: P

• τ π ≥ t
• U0
• = ❊
• θh,π

p (t, Xt1, Wt1, . . . , Xt, Wt)

• U0
• ◮ The function θh,π

p

is defined for t ∈ [tp−1, tp[ on the next slide.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion Implementation

Implementation

θh,π

p

(x1, y1 . . . , x, y) :=

p−1

• k=1

ptk−tk−1

• xk−xk−1

h(tk−1) , yk − yk−1; 0, 0

• ptk−tk−1(xk, yk, xk−1, yk−1)

− tk

tk−1

• ❘

ptk−s

• xk−C π(s)

h(tk−1) , yk − z; 0, 0

• ptk−tk−1(xk, yk, xk−1, yk−1) Ps,(0,ys)(τ(C−xk−1)/h(tk−1) ∈ ds, Ws ∈ dz)
• ×

pt−tp−1 x−xp−1

h(tp−1) , y − yp−1; 0, 0

• pt−tp−1(x, y, xp−1, yp−1)

− t

tp−1

• ❘

pt−s

• x−C π(s)

h(tp−1) , y − z; 0, 0

• pt−tp−1(x, y, xp−1, yp−1) Ps,(0,z)(τ(C−xp−1)/h(tp−1) ∈ ds, Ws ∈ dz)
• Olivier Faugeras

NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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 → 1 − 2t − 2t2 − t3 with the intial condition X0 = 0, W0 = 0. The total mass is 1 in this case.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

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 → 1 − 1

2t + t3 with the intial condition X0 = 0, W0 = 0.

The total mass is ≈ 0.2578 in this case.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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.

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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 NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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.

◮ We sketch a numerical procedure based on Monte-Carlo

simulation to compute the probability distribution efficiently.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Conclusion

◮ 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.

◮ We sketch a numerical procedure based on Monte-Carlo

simulation to compute the probability distribution efficiently.

◮ These results have potential applications in many fields of

physics and biology.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP

Motivations DIP IWP IWP hits boundary DIP hits boundary Numerics Conclusion

Acknowledgements

◮ From Touboul and Faugeras, Advances in Applied Probability,

2008.

◮ Supported by European grant FACETS and ERC grant NerVi.

Olivier Faugeras NeuroMathComp project team - INRIA/ENS Paris ISI and DIP