Most Probable Escape Path Method and its Application to Leaky Integrate And Fire Neurons

Transcription

1 Most Probable Escape Path Method and its Application to Leaky Integrate And Fire Neurons Simon Fugmann Humboldt University Berlin 13/02/06

2 Outline The Most Probable Escape Path (MPEP) Motivation General Equations The Leaky Integrate And Fire Neuron The Model Overdamped Brownian Particle in a Harmonic Potential Well Periodic Driving α-stimulus Summary, To Do Summary Open questions

3 Outline The Most Probable Escape Path (MPEP) Motivation General Equations The Leaky Integrate And Fire Neuron The Model Overdamped Brownian Particle in a Harmonic Potential Well Periodic Driving α-stimulus Summary, To Do Summary Open questions

4 Motivation Motivation Kramer s Escape Problem Metastable Potential Wall Small White Noise Noice induced escape flux: j e U T

5 Motivation Motivation Kramer s Escape Problem Metastable Potential Wall Small White Noise Noice induced escape flux: j e U T Dynamics q + Γ q + U = f (t) f (t) = 0 f (t) f (t ) = 2ΓT δ(t t ) T U

6 Motivation Potential Well The aim is to predict analytically the particle s movement to the top initially situated in the bottom well under influence of small white noise.

7 General Equations Path Integral Representation Of Transition Probability p tr (q f, t f, q i, t i ) = Df (t) P [f (t)] δ(q(t f ) q f ) q(t i )=q i with P [ξ(t)] representing the probability density of a given noise realization f (t)

8 General Equations Path Integral Representation Of Transition Probability p tr (q f, t f, q i, t i ) = Df (t) P [f (t)] δ(q(t f ) q f ) q(t i )=q i with P [ξ(t)] representing the probability density of a given noise realization f (t) for a given noise intensity D noise one has P [f (t)] = 1 Z exp( S [f ] /D noise)

9 General Equations Transformation from noise variables to dynamical variables for white noise one has S [f ] = 1 2 t 0 dτ f 2 (τ)

10 General Equations Transformation from noise variables to dynamical variables for white noise one has S [f ] = 1 2 t 0 resulting in dτ f 2 (τ) p tr = qf q i Dq(t) J f q (q) 1 Z exp( S [q] /D noise) J is the Jacobian of the transformation

11 General Equations Direct Transition On Optimal Path p tr = p exp( S min /D noise ) S min Minimum of the functional S among all trajectories p preexponential factor depending on noise relatively weakly t i = 0 and t f = t tr

12 General Equations Lagrangian and Action S = ttr 0 Action dt L (q, q, q, t)

13 General Equations Lagrangian and Action S = ttr 0 L = 1 4Γ Action dt L (q, q, q, t) Lagrangian ( q + Γ q + du ) 2 dq

14 General Equations Lagrangian and Action S = ttr 0 L = 1 4Γ Action dt L (q, q, q, t) Lagrangian ( q + Γ q + du ) 2 dq Activation Energy S min = min (S)

15 General Equations Necessary condition for the extrema of the functional S δs = 0

16 General Equations Necessary condition for the extrema of the functional S δs = 0 leading to the Euler-Poisson-Equation L q d dt ( ) L + d 2 q dt 2 ( ) L = 0 q

17 General Equations Necessary condition for the extrema of the functional S δs = 0 leading to the Euler-Poisson-Equation L q d dt ( ) L + d 2 q dt 2 d 4 q dt 4 + d 2 ( q dt 2 2 d 2 ) U dq 2 Γ2 + ( ) L = 0 q ( ) dq 2 d 3 U dt dq 3 + d 2 U du dq 2 dq = 0

18 General Equations d 4 q dt 4 + d 2 ( q dt 2 2 d 2 ) U dq 2 Γ2 + ( ) dq 2 d 3 U dt dq 3 + d 2 U du dq 2 dq = 0 4th-order differential equation with boundary conditions ( ) ( ) q(0) q(ttr ) = i and = j q(0) q(t tr )

19 General Equations Minimization of S over t tr δs δt tr = 0

20 General Equations Minimization of S over t tr δs δt tr = 0 Soskin: For the given S, carrying out an integration by parts twice and using the Euler-Poisson-Equation, one derives by introducing an quasi-energy Ẽ ( L Ẽ = L + q d dt ( )) L q + L q q q Ẽ = 0 along the solution, also been shown Ẽ 0

21 General Equations Time-reversal Process For an assumed known optimal solution q(τ) = q(t τ) with ( q(0) q(0) q + Γ q + U = 0 ) = j and ( q(ttr ) q(t tr ) ) = i

22 General Equations Time-reversal Process For an assumed known optimal solution q(τ) = q(t τ) with ( q(0) q(0) q + Γ q + U = 0 ) = j and ( q(ttr ) q(t tr ) ) = i One can show: ( Γ + ( Γ 2 (Γ ) 2) /2) = 2ΓẼ

23 General Equations Given that Ẽ = 0, that equation can be integrated explicitly, leading to d 2 q ( ) Γ(τ t) d q dτ 2 + Γ coth 2 dτ + du( q) = 0 d q q(τ) = q(t τ)

24 General Equations Given that Ẽ = 0, that equation can be integrated explicitly, leading to d 2 q ( ) Γ(τ t) d q dτ 2 + Γ coth 2 dτ + du( q) = 0 d q q(τ) = q(t τ) q(τ) τ=0 = q(t tr ) q(τ) τ=0 = q(t tr ) in case of wall boundary q(τ) τ=0 = q(t tr )! = 0

25 General Equations For further studies see e.g. Soskin et al., Phys. Rev. Lett 86, 1665 (2001) Soskin et al., Chaos 11, 595 (2001)

26 Outline The Most Probable Escape Path (MPEP) Motivation General Equations The Leaky Integrate And Fire Neuron The Model Overdamped Brownian Particle in a Harmonic Potential Well Periodic Driving α-stimulus Summary, To Do Summary Open questions

27 The Model RC Circuit I (t) = U R + C U with τ m U = U + RI, τm = RC U - membrane potential, τ m - membrane time constant

28 Overdamped Brownian Particle in a Harmonic Potential Well General Expression u = u + µ + 2Dξ(t)

29 Overdamped Brownian Particle in a Harmonic Potential Well General Expression u = u + µ + 2Dξ(t) with V = 1 2 (u µ)2 leading to u = dv du + 2Dξ(t) 3 parameters: u r reset value of the potential, u f threshold, µ external current

30 Overdamped Brownian Particle in a Harmonic Potential Well Two Different Regimes Deterministic Regime Excitable Regime

31 Overdamped Brownian Particle in a Harmonic Potential Well L = 1 2 ( u + dv ) 2 du

32 Overdamped Brownian Particle in a Harmonic Potential Well L = 1 2 ( u + dv ) 2 du δs = 0 0 = ü V V with Ṽ = 1 ( dv ) 2 2 du ü + dv e du = 0 Hamiltonian System E = u2 2 + Ṽ u = ± 2E + ( ) dv 2 du

33 Overdamped Brownian Particle in a Harmonic Potential Well L = 1 2 ( u + dv ) 2 du δs = 0 0 = ü V V with Ṽ = 1 ( dv ) 2 2 du ü + dv e du = 0 Hamiltonian System E = u2 2 + Ṽ u = ± 2E + ( ) dv 2 du t = uf u r du 2E + ( ) E(t) dv 2 du

34 Overdamped Brownian Particle in a Harmonic Potential Well S(t) = E + ( ) dv 2 du 2E + ( ) + dv du dv 2 du du

35 Overdamped Brownian Particle in a Harmonic Potential Well Explicite result S(t) = E + ( ) dv 2 du 2E + ( ) + dv du dv 2 du du S(t) = u2 f u2 r µ 4 2 (u f u r ) (u f µ) 2E + (u f µ) (u r µ) 2E + (u r µ) 2

36 Overdamped Brownian Particle in a Harmonic Potential Well u r = 0, u f = 1.2, µ = 0.8

37 Periodic Driving External Current with small periodic perturbation µ(t) = µ 0 + δ sin(ωt) δs = 0 ü u µ + µ = 0 with boundary conditions u(0) = u r and u(t tr ) = u f u(t) u(t) L(u, u, t) S(t tr )

38 Periodic Driving External Current with small periodic perturbation µ(t) = µ 0 + δ sin(ωt) δs = 0 ü u µ + µ = 0 with boundary conditions u(0) = u r and u(t tr ) = u f u(t) u(t) L(u, u, t) S(t tr ) S(t tr ) = c(t tr ) 2 (e 2ttr 1) c(t) = u r µ 0 + δω ω sinh(t) (u r e t u f + µ 0 (1 e t ) + δ ω (sin(ωt) ωcos(ωt) + ωet ))

39 Periodic Driving Example 1 u r = 0, u f = 1.2, µ 0 = 0.8, ω = 15, D = 0.1, δ = 0.1

40 Periodic Driving Example 2 u r = 0, u f = 1.0, µ 0 = 0.8, ω = 10, D = 0.05, δ = 0.1

41 α-stimulus Idea Neuron s reaction to random α-input of other neurons N neurons, coupling strength c j and random spike times t j per neuron fixed delay

42 α-stimulus N µ(t) = c j (t t j ) exp( (t t j ) + )Θ(t t j ) t j j=1

43 α-stimulus c(t) = µ(t) = N c j j=1 1 sinh(t) (u f u r e t t j (t t j ) exp( (t t j ) + )Θ(t t j ) L = 1 2 ( u + u µ(t))2 = 2 c(t tr ) 2 e 2t S(t tr ) = c(t tr ) 2 (e 2ttr 1) NX X c j j=1 2 (t t j ) 2 t j exp( (t t j )+ )Θ(t t j ))

44 α-stimulus c(t) = µ(t) = N c j j=1 1 sinh(t) (u f u r e t t j (t t j ) exp( (t t j ) + )Θ(t t j ) L = 1 2 ( u + u µ(t))2 = 2 c(t tr ) 2 e 2t S(t tr ) = c(t tr ) 2 (e 2ttr 1) NX X c j j=1 2 (t t j ) 2 t j exp( (t t j )+ )Θ(t t j )) to test the results creation of random t j -array (uniform distribution), both for analytical calculus and simulation

45 α-stimulus Example 1 u r = 0.2, u f = 1.0, = 2, D = 0.1, N = 8

46 α-stimulus Example 2 u r = 0.2, u f = 1.2, = 2, D = 0.1, N = 10

47 Outline The Most Probable Escape Path (MPEP) Motivation General Equations The Leaky Integrate And Fire Neuron The Model Overdamped Brownian Particle in a Harmonic Potential Well Periodic Driving α-stimulus Summary, To Do Summary Open questions

48 Summary by applicating of MPEP to some simple models analytical results can be derived results accord with simulations qualitatively quantitative differences

49 Open questions preexponential factor interacting neurons application to more complicated neuronal models

50 Open questions Thank You for Your Attention!