Slow/fast dynamics and periodic behaviors for mean-field excitable - PowerPoint PPT Presentation

Slow/fast dynamics and periodic behaviors for mean-field excitable systems Christophe Poquet Universit´ e Lyon 1 April 4 th , 2019 Workshop : Mean-field approaches to the dynamics of neuronal networks, EITN, Paris In collaboration with E. Lu¸ con (Universit´ e Paris Descartes) 1/18

Noisy excitable systems in interaction An excitable system : • possesses a stable rest position . • threshold phenomenon : after a sufficiently large perturbation, follows a complex trajectory before coming back to the rest state. 2/18

Noisy excitable systems in interaction An excitable system : • possesses a stable rest position . • threshold phenomenon : after a sufficiently large perturbation, follows a complex trajectory before coming back to the rest state. General observation : A large population of noisy excitable systems in mean field interaction may possess a synchronized periodic behavior . 2/18

Noisy excitable systems in interaction An excitable system : • possesses a stable rest position . • threshold phenomenon : after a sufficiently large perturbation, follows a complex trajectory before coming back to the rest state. General observation : A large population of noisy excitable systems in mean field interaction may possess a synchronized periodic behavior . Aim Rigorous proof of periodic behavior for noisy neurons in mean field interaction ? 2/18

Active rotators [Shinimoto, Kuramoto, 1986] Consider a population of N oscillators in S = R / (2 π Z ) with dynamics � N dϕ i,t = − δV ′ ( ϕ i,t ) dt − K sin( ϕ i,t − ϕ j,t ) dt + dB i,t . N j =1 3/18

Active rotators [Shinimoto, Kuramoto, 1986] Consider a population of N oscillators in S = R / (2 π Z ) with dynamics � N dϕ i,t = − δV ′ ( ϕ i,t ) dt − K sin( ϕ i,t − ϕ j,t ) dt + dB i,t . N j =1 Example of potential : V ( θ ) = θ − a cos( θ ) , V ′ ( θ ) = 1 + a sin( θ ) . a<1 a>1 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 3/18

Active rotators [Shinimoto, Kuramoto, 1986] Consider a population of N oscillators in S = R / (2 π Z ) with dynamics � N dϕ i,t = − δV ′ ( ϕ i,t ) dt − K sin( ϕ i,t − ϕ j,t ) dt + dB i,t . N j =1 Example of potential : V ( θ ) = θ − a cos( θ ) , V ′ ( θ ) = 1 + a sin( θ ) . a<1 a>1 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 � N 1 On any time interval [0 , T ] , the empirical measure µ N,t = i =1 δ ϕ i,t converges N weakly to the solution of � � � ∂ t µ t = 1 2 ∂ 2 + δ∂ θ ( µ t V ′ ) . sin( θ − ψ ) dµ t ( ψ ) θ µ t + K∂ θ µ t S 3/18

Active rotators [Shinimoto, Kuramoto, 1986] Consider a population of N oscillators in S = R / (2 π Z ) with dynamics � N dϕ i,t = − δV ′ ( ϕ i,t ) dt − K sin( ϕ i,t − ϕ j,t ) dt + dB i,t . N j =1 Example of potential : V ( θ ) = θ − a cos( θ ) , V ′ ( θ ) = 1 + a sin( θ ) . a<1 a>1 4 4 2 2 0 0 −2 −2 −4 −4 −6 −6 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 � N 1 On any time interval [0 , T ] , the empirical measure µ N,t = i =1 δ ϕ i,t converges N weakly to the solution of � � � ∂ t µ t = 1 2 ∂ 2 + δ∂ θ ( µ t V ′ ) . sin( θ − ψ ) dµ t ( ψ ) θ µ t + K∂ θ µ t S For accurate choices of parameters ( a may be larger than one) and δ small enough, this non-linear Fokker Planck PDE admits a limit cycle . [Giacomin, Pakdaman, Pellegrin and P., 2012] 3/18

Simulation : N = 4000 , K = 2 , a = 1 . 1 , δ = 0 . 5 4/18

Active rotators For δ = 0 we have � � � ∂ t µ t ( θ ) = 1 2 ∂ 2 θ p t ( θ ) + K∂ θ µ t sin( θ − ψ ) dµ t ( ψ ) , S and if K > 1 , the model admits moreover a stable curve of synchronized stationary solutions M 0 = { q ψ ( · ) : ψ ∈ S} , where q ψ ( · ) = q 0 ( · − ψ ) . 0.20 0.15 0.10 0.05 1 2 3 4 5 6 5/18

Active rotators For δ = 0 we have � � � ∂ t µ t ( θ ) = 1 2 ∂ 2 θ p t ( θ ) + K∂ θ µ t sin( θ − ψ ) dµ t ( ψ ) , S and if K > 1 , the model admits moreover a stable curve of synchronized stationary solutions M 0 = { q ψ ( · ) : ψ ∈ S} , where q ψ ( · ) = q 0 ( · − ψ ) . 0.20 0.15 0.10 0.05 1 2 3 4 5 6 For δ small the model admits an invariant curve M δ = { q δ ψ : ψ ∈ S} , perturbation of M 0 , with phase dynamics � � a ψ δ ˙ sin( ψ δ t ≈ δ 1 + t ) . a K 5/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 6/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 where     k 1 0 σ 1 0     ... ... • δ � 0 , K =  > 0 , σ =  > 0 ,   0 k d 0 σ d 6/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 where     k 1 0 σ 1 0     ... ... • δ � 0 , K =  > 0 , σ =  > 0 ,   0 k d 0 σ d • ( B i ) i =1 ...N family of standard independent Brownian motions, 6/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 where     k 1 0 σ 1 0     ... ... • δ � 0 , K =  > 0 , σ =  > 0 ,   0 k d 0 σ d • ( B i ) i =1 ...N family of standard independent Brownian motions, • F smooth, and • ( F ( x ) − F ( y )) · ( x − y ) � C | x − y | 2 , • F ( x ) · Kσ − 2 x � C 1 {| x | � r } − c | x | 2 . • | F ( x ) | � Ce ε | x | 2 , | ∂ xk F ( x ) | � Ce ε | x | 2 , | ∂xk F ( x ) | • lim | x |→ 0 F ( x ) · Kσ − 2 = 0 , | ∂ xk,xl F ( x ) | � Ce ε | x | 2 , 6/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 where     k 1 0 σ 1 0     ... ... • δ � 0 , K =  > 0 , σ =  > 0 ,   0 k d 0 σ d • ( B i ) i =1 ...N family of standard independent Brownian motions, • F smooth, and • ( F ( x ) − F ( y )) · ( x − y ) � C | x − y | 2 , • F ( x ) · Kσ − 2 x � C 1 {| x | � r } − c | x | 2 . • | F ( x ) | � Ce ε | x | 2 , | ∂ xk F ( x ) | � Ce ε | x | 2 , | ∂xk F ( x ) | • lim | x |→ 0 F ( x ) · Kσ − 2 = 0 , | ∂ xk,xl F ( x ) | � Ce ε | x | 2 , [Baladron, Fasoli, Faugeras, Touboul, 2012], [Lu¸ con, Stannat, 2014], [Bossy, Faugeras, Talay, 2015], [Mehri, Scheutzow, Stannat, � N 1 Zangeneh, 2018]] : On any time interval [0 , T ] , the empirical measure µ N,t = i =1 δ X i,t N converges weakly to the solution of � � � ∂ t µ t = ∇ · ( σ 2 ∇ µ t ) + ∇ · µ t K ( x − − δ ∇ · ( µ t F ) . R d zdµ t ( z ) 6/18

The model Consider a population of N interacting units in R d with dynamics   N � √  X i,t − 1  dt + dX i,t = δF ( X i,t ) dt − K X j,t 2 σdB i,t , N j =1 where     k 1 0 σ 1 0     ... ... • δ � 0 , K =  > 0 , σ =  > 0 ,   0 k d 0 σ d • ( B i ) i =1 ...N family of standard independent Brownian motions, • F smooth, and • ( F ( x ) − F ( y )) · ( x − y ) � C | x − y | 2 , • F ( x ) · Kσ − 2 x � C 1 {| x | � r } − c | x | 2 . • | F ( x ) | � Ce ε | x | 2 , | ∂ xk F ( x ) | � Ce ε | x | 2 , | ∂xk F ( x ) | • lim | x |→ 0 F ( x ) · Kσ − 2 = 0 , | ∂ xk,xl F ( x ) | � Ce ε | x | 2 , [Baladron, Fasoli, Faugeras, Touboul, 2012], [Lu¸ con, Stannat, 2014], [Bossy, Faugeras, Talay, 2015], [Mehri, Scheutzow, Stannat, � N 1 Zangeneh, 2018]] : On any time interval [0 , T ] , the empirical measure µ N,t = i =1 δ X i,t N converges weakly to the solution of � � � ∂ t µ t = ∇ · ( σ 2 ∇ µ t ) + ∇ · µ t K ( x − − δ ∇ · ( µ t F ) . R d zdµ t ( z ) µ t is the distribution of √ dX t = δF ( X t ) dt − K ( X t − E [ X t ]) dt + 2 σdB t . 6/18

A toy example � � x 2 − a Consider F ( x, y ) = with a ∈ R , b > 0 . − by p a p a 0 0 a > 0 a < 0 7/18

FitzHugh Nagumo Consider � � v − v 3 3 − w F ( v, w ) = , 1 c ( v + a − bw ) where a ∈ R , b, c > 0 . 8/18