Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 1 / 34

Outline 1 Motivation 2 Complements 3 Stochastic Differential Equations 4 Link between SDE and PDE 5 Approximation of Solutions 6 Noisy Integrate and Fire Models 7 Complement: Point Poisson Processes E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 2 / 34

Motivation First models of individual neurons Simple Integrate-and-Fire (SIF) Model Leaky Integrate-and-Fire (LIF) Model E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 4 / 34

A first model of single neuron The integrate-and-fire neuronal model was introduced by Lapicque in 1907. The membrane equation where I L (V ): the leak current I syn(v, t): the synaptic current I ext(t): the external current C: membrane capacitance dv (t) C = I L (V ) + I syn (V, t) + I ext (t), dt A spike response is generated whenever the membrane potential reaches a fixed threshold V th. After the spike, V is reset to a fixed value V reset. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 5 / 34

Two simple examples Simple integrate-and-fire model The simplest form: the IF neuron has no leak current i.e. I L (V ) = 0. Leaky integrate-and-fire model I L (V ) = g L (V (t) E L ), where g L : the leak conductance E L : resting potential E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 6 / 34

Synaptic current First simple model of synaptic transmission An instantaneous rise or fall of the synaptic current at arrival of a presynaptic spike The PSC is described by a delta function with amplitude of efficiency J The total synaptic current stemming from N sym synaptic input channels takes the form N syn I syn (V, t) = I syn (t) = τ m J δ(t ti k ), where τ m = C. g L Asymptotic behaviour i=1 Assume the neuron receives a high barrage of Poissonian distributed and uncorrelated synaptic inputs Assume the amplitude J is small, i.e. k J << V th E L E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 7 / 34

Asymptotic behaviour Current I syn (t)dt µdt + σ τ m dw t, where µ = τ m JN syn ν syn σ 2 = τ m J 2 N syn ν syn where ν syn is the mean activation rate of each synapse. A continuous time limit equation τ m dv (t) = f (V (t))dt + I ext (t)dt + µdt + σ τ m dw t. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 8 / 34

Simple examples f (V ) = 0, I ext = 0 τ m dv (t) = µdt + σ τ m dw t. The potential V evolves as a Brownian Motion with constant drift. f (V ) = V, I ext = 0 τ m dv (t) = (µ V (t))dt + σ τ m dw t. The potential V evolves as an Ornstein Uhlenbeck process. Spiking times Recall that the considered neuron emits a spike at each time τ its potential hits threshold V th From a mathematical viewpoint, the spiking times are the first hitting time of constant threshold by a stochastic process τ = inf {t > 0, V (t) V th } E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 9 / 34

Levy s Characterization of Brownian Motion Theorem A stochastic process X = (X t ) t 0 is a standard Brownian Motion if and only if it is a continuous local martingale with [X] t = t. Theorem (Multi-dimensional Version) Let X = (X 1 t,, X n t ) t 0 be continuous local martingales such that [X i, X j ] t = tδ i,j. Then X is a standard n-dimensional Brownian Motion. Remark Condition in previous theorems are obviously characterization of Brownian motions. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 11 / 34

Itô Isometry Proposition Let H be an adapted process such that Then, ( T E 0 T 0 E ( H 2 s ) ds <. ) 2 T H s dw S = E ( H 2 ) s ds 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 12 / 34

Change of time Theorem (Dubins-Schwartz) Let M be a continuous locale Martingale such that M 0 = 0 a.s., and [M] =. We set τ s = inf{t 0, [M] t s}, then B s = M τs is an F τs -Brownian Motion with M t = B [M]t Remark The Dubins-Schwarz theorem, which shows that continuous martingales with unbounded (as time goes to infinity) quadratic variation ARE Brownian Motion, up to a (stochastic) time change. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 13 / 34

Stochastic Differential Equations dx s = b(s, X s )ds + σ(s, X s )dw s, where b and σ are predictable. Solution t t X t = X 0 + b(s, X s )ds + σ(s, X s )dw s, 0 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 15 / 34

Strong Solutions A strong solution of the SDE on the given probability space (Ω, F, F, P) and with respect to the fixed F Brownian motion (W t ) t 0 and initial condition ξ is a process (X t ) t 0 with continuous sample paths and with the following properties X is adapted to the filtration F P(X 0 = ξ) = 1 for every t 0 holds almost surely. ( t ) P b(s, X s ) + σ 2 (s, X s )ds < = 1 0 t t X t = X 0 + b(s, X s )ds + σ(s, X s )dw s 0 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 16 / 34

Weak Solutions A weak solution to SDE is a triple (X, W ), (ω, F, P),F where (Ω, F, P) is a probability space, and F is a filtration of sub-σ-fields of F satisfying the usual conditions. X is a continuous, F-adapted stochastic process W is an F-Brownian motion for every t 0 holds almost surely. ( t ) P b(s, X s ) + σ 2 (s, X s )ds < = 1 0 t t X t = X 0 + b(s, X s )ds + σ(s, X s )dw s 0 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 17 / 34

Strong Existence and Uniqueness Theorem Let T > 0 and b(.,.) : [0, T ] R n R n and σ(.,.) : [0, T ] R n R n m be measurable functions satisfying b(t, x) + σ(t, x) C(1 + x ); x R n, t [0, T ] for some constant C and such that b(t, x) b(t, y) + σ(t, x) σ(t, y) D x y ; x, y R n, t [0, T ]. Then the stochastic differential equation dx t = b(t, X t )dt + σ(t, X t )db t, X 0 = X 0 has a unique solution X, continuous in time, such that X is adapted to the filtration generated by X 0 and the Brownian Motion and [ ] T E X t 2 dt <. 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 18 / 34

Main Ideas of the Proof Uniqueness is obtained thanks to Gronwall Lemma. Existence: Picard iteration scheme. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 19 / 34

Infinitesimal Generator Associated to a Feller Process Let X be a Feller process; a function f in C 0 is said to belong to the domain D A of the infinitesimal generator of X if the limit Af (x) = lim t 0 E x (f (X t )) f (x) t exists in C 0. The operator A : D A C 0 is called the infinitesimal generator of the process X. Example (Diffusion Processes) The infinitesimal generator associated to the solution of a Stochastic Differential Equation dx t = b(x t )dt + σ(x t )dw t writes with Af (x) = i b i (x) x i f (x) + 1 2 a ij (x) = 2 a ij (x) f (x), x i x j i r σ ik (x)σ kj (x). k=1 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 20 / 34 j

Kolmogorov s backward equation Theorem Let f C 2 0 (Rn ). Define then u(t,.) D A for each t and u(t, x) = E x [f (X t )] u t = Au, t > 0, x Rn u(0, x) = f (x), x R n. where the right hand side is to be interpreted as A applied to the function x u(t, x). E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 22 / 34

Feynman-Kac formula Theorem Let f C 2 0 (Rn ) and q C(R n ). Assume that q is lower bounded. Put ( t ) ] v(t, x) = E [exp x q(x s )ds f (X t ). 0 v = Av qv, t > 0, x Rn t v(0, x) = f (x), x R n. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 23 / 34

Fokker Planck Equation Theorem Let X be an Itô diffusion in R n, solution of the stochastic differential equation dx t = b(x t )dt + σ(x t )dw t. Assume that P x (X t dy) = Γ(t, x, y)dy, for all x R n, t > 0. Assume that y Γ(t, x, y) is smooth for each t and x. Then, Γ satisfies the Kolmogorov forward equation (also known as Fokker Planck eq.) d dt Γ(t, x, y) = A Γ(t, x, y), where A is the adjoint of the operator A A φ(y) = i y i (b i (y)φ(y)) + 1 2 i j 2 y i y j [a ij (y)φ(y)]. Here a = σσ t, that is a ij (x) = r k=1 σ ik(x)σ kj (x). E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 24 / 34

Backward Kolmogorov Equation Theorem Let X be a solution of the Stochastic Differential Equation dx t = b(x t )dt + σ(x t )dw t. Denote Γ(t, x, y) = P(X t = y X 0 = x). Then, Γ Γ (t, x, y) = b(x) t x (t, x, y) + 1 2 a(x) 2 Γ (t, x, y) x 2 = AΓ(t, x, y), where the infinitesimal operator A acts here on the variable x and a = σσ t. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 25 / 34

Approximation of Solutions Euler scheme X δ 0 = X 0 X δ (k+1)δ = X δ kδ + b( X δ kδ)δ + σ( X δ kδ)(w (k+1)δ W kδ ) Theorem The numerical scheme is strongly convergent lim E ( X T X T δ ) = 0. δ 0 The numerical scheme is weakly convergent lim Eg(XT ) Eg( X T δ ) = 0. δ 0 E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 27 / 34

Rate of convergence Theorem Assume b and σ are C 4 functions with bounded derivatives, then E ( X T X T δ ) C T δ 1/2. Eg(X T ) Eg( X T δ ) C T δ. Remark: Romberg extrapolation Assume, we have an expansion of the error Eg(X T ) Eg( X T δ ) = CT 1 δ + CT 2 δ 2 + O(δ 3 ). Then a well chosen combination of X δ and X δ/2 gives an order 2 scheme. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 28 / 34

Simple Integrate-and-Fire Model Special case I ext = 0 dv t = σdw t. Exercise: Law of the first hitting time Compute the law of ( P (τ a t) = P sup 0 s t τ a := inf {t > 0, W t a}. ) W s a ( ) ( ) = P sup 0 s t W s a, W t a ( + P sup 0 s t W s a, W t < a ) = P (W t a) + P sup W s a, W t W τa < 0 0 s t ( ) = P (W t a) + P sup 0 s t W s a, W t W τa > 0 = 2P (W t a) E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 30 / 34

Ornstein Uhlenbeck Process dv t = λ(v V t )dt + σdw t V 0 = v Solve explicitly the equation Give the law of V t (the conditional law given V 0 ) Make explicit the associated stationary measure. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 31 / 34

Point Poisson Process (P.P.P.) Let us consider a set D (e.g. [0, T ] [0, K]). A realisation of a P.P.P. on D with intensity I(t, x) is a set of points of D. For all subset F of D, we denote by N F the number of points of the P.P.P. which are in F. Characterization of a Point Poisson Process for all F D, N F is a random variable (with value in N) with Poisson law of parameter I(t, x)dtdx, F for all subset F and G with empty intersection, N F and N G are independent random variables. A particular case is: the intensity I is equal to 1. For all subset F of D, the number N F of points of the P.P.P. in F is a Poisson random variable with parameter equal to the volume of F. E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 33 / 34

Point Poisson Process Recall that a Poisson random variable X of parameter λ has the following law: P(X = k) = exp( λ) λk k!. In particular, P(X = 0) = exp( λ). Let us consider a process (X t ; 0 t T ) and a non negative function φ, bounded from above by K. In order to simulate an event with probability exp( T 0 φ(x s)ds), we can use Point Poisson Processes. Indeed, the probability that the P.P.P. on [0, T ] [0, K] have no point below the curve t φ(x t ) is precisely exp( T 0 φ(x s)ds). E. Tanré (INRIA - Team Tosca) Mathematical Methods for Neurosciences November 26th, 2014 34 / 34