Path Integral methods for solving stochastic problems. Carson C. Chow, NIH

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1 Path Integral methods for solving stochastic problems Carson C. Chow, NIH

2 Why? Often in neuroscience we run into stochastic ODEs of the form dx dt = f(x)+g(x)η(t) η(t) =0 η(t)η(t ) = δ(t t )

3 Examples Integrate-and-fire neuron with noise dv dt = I g Lv g NL (v)+ Dη(t) Set v v R when v = v T Want to know mean and variance, mean and variance of time to fire, etc.

4 Decision-theory model du dt = f(u)+g(t)+ση(t) u(0) = b Decision is made when u reaches ±θ

5 Population of neurons with noise t a = a + f ( w(x, y)a(y)dy ) + η(x, t) η(x, t) =0 η(x, t)η(x,t ) = δ(x x )δ(t t ) Again interested in mean, variance, correlations, any statistic

6 Solving equations Traditional methods include Ito and Stratonovich calculus or Fokker-Planck equation For most nonlinear stochastic equations, closed form solutions do not exist Need to be solved with perturbation theory, which is nontrivial Path integral and field theory approaches were designed for perturbation theory

7 Generating Function Generating function lets you calculate moments of a probability density P(x) by taking derivatives moments x n = x n P (x)dx generating function x n = 1 Z[0] Z[λ] = e λx = n λ n Z[λ] λ=0 e λx P (x)dx

8 Goals Derive a generating function for moments of a stochastic process Solve for moments perturbatively using diagrammatic methods.

9 Computing Generating functions e.g. Normal or Gaussian distribution P (x) e (x a) 2 2σ 2 Z[λ] = e (x a) 2 2σ 2 +λx dx Most important trick: complete the square

10 (x a)2 2σ 2 x c is given by + λx = A(x x c ) 2 + B ( ) d (x a)2 dx 2σ 2 + λx =0 A = 1 2σ 2 x c = λσ 2 + a B = x2 c 2σ 2 a2 2σ 2 = λ2 σ λa Z[λ] = e (x λσ 2 a) 2 2σ 2 +λa+ λ2 σ 2 2 dx = Z[0]e λa+ λ2 σ 2 2 Z[0] = 2πσ

11 x = d dλ eλa+ λ 2 σ 2 2 x 2 = d2 dλ 2 eλa+ λ 2 σ 2 2 = a λ=0 = a 2 + σ 2 λ=0 x 3 = d3 dλ 3 eλa+ λ 2 σ 2 2 x 4 = d4 dλ 4 eλa+ λ 2 σ 2 2 = a 3 +3aσ 2 λ=0 = a 4 +6a 2 σ 2 +3σ 4 λ=0

12 Cumulant generating function W [λ] = ln Z[λ] C n = dn dλ n W [λ] λ=0 For Gaussian W [λ] =λa (λ2 σ 2 ) + ln Z[0] C 1 = x = a C 2 = x 2 x 2 var(x) =σ 2 C n =0, n > 2

13 Generalize to multi-dimensions Z[λ i ]= dx i e 1 2 Pj,k x jk 1 jk x k+ P j λ jx j K 1 jk (K 1 ) jk i Assume K is symmetric and positive, thus it has full set of orthonormal eigenvectors K ij v α j = ω α v α i v α j v β j = δ αβ j j Expand x k = α c α v α k λ k = α d α v α k x j K jk x k = j c α ω β c β v α j v β k = α,β c α ω β c β δ αβ = α ω α c 2 α j,k α,β

14 Z[λ i ]= = = α i α dc α e P α ( 1 2 ω αc 2 α +d αc α ) dx i e 1 2 Pj,k x jk 1 jk x k+ P j λ jx j dc α e 1 2 ω αc 2 α +d αc α = Z[0] α e 1 2 ω 1 α d2 α Z[0] = (2π det K) n/2 = Z[0]e P jk 1 2 λ jk jk λ k j K 1 ij K jk = δ ik Cumulant generating function W [λ i ] = ln Z[λ i ]

15 Moments i x i = 1 Z[0] i Z[λ i ] λ i λi =0 Example x 2 3x 7 x 36 = 1 Z[0] 2 λ 2 3 λ 7 exp λ 36 ij 1 2 λ ik 1 ij λ j λi =0 = K 3,3 K 7,36 +2K 3,7 K 3,36

16 Wick s Theorem 2s 1 x i = 1 = Z[0] 2s 1 Z[λ i ] λ i all possible pairings λi =0 K i1,i 2 K i2s 1 i 2s x a x b x c x d = x a x b x c x d + x a x b x c x d + x a x b x c x d = K ab K cd + K ad K bc + K ac K bd make all pairings or contractions

17 Continuum limit Z[λ i ]= dx i e 1 2 Pj,k x jk 1 jk x k+ P j λ jx j i Let i t x i x(t) λ i λ(t) K ij K(s, t) dt i Z[λ(t)] = Dx(t)e 1 2 dx i Dx(t) i R x(s)k 1 (s,t)x(t)dsdt+ R λ(t)x(t)dt = Z[0]e R 1 2 λ(s)k(s,t)λ(t)dsdt

18 functional derivative λ i δ δλ(t) δλ(s) δλ(t) = δ(s t) Moments x(t 1 )x(t 2 ) = 1 Z[0] δ δλ(t 1 ) δ δλ(t 2 ) Z[λ(t)] = K(t 1,t 2 ) i x(t i ) = 1 Z[0] i δ δλ(t i ) Z[λ(t)] = all possible contractions K(t i1,t i2 ) K(t i2s 1,t ti2s )

19 Can generalize to higher dimensions x(t) ϕ( t) λ(t) J( t) t R d dt d d t Z[λ( t)] = Dϕe 1 2 R ϕ( s)k 1 ( s, t)ϕ( t)d d sd d t+ R λ( t)ϕ( t)d d t = Z[0]e R 1 2 λ( s)k( s, t)λ( t)d d sd d t

20 Field Theory Consider an arbitrary (infinite dimensional) probability distribution P [ϕ( t)] = e S[ϕ( t)] Z[J(ϕ( t)] = Dϕe S[φ]+J ϕ S is called the action J ϕ = J( t)ϕ( t)d d t e.g. φ 4 theory S[ϕ( t)] = ϕ( t)k 1 ( t, t )ϕ( t )d d td d t + g ϕ 4 ( t)d d t Must compute perturbatively (asymptotic solutions of integrals in infinite dimensions)

21 Application to stochastic differential equations dx Ito interpretation dt = f(x)+g(x)η(t) dx = f(x)dt + g(x)db t Discretization x i+1 x i = f(x i )dt + g(x i )w i dt db t = w t t

22 Probability density P [x i w i ]=δ[x i+1 x i f(x i )dt g(x i )w i dt] = δ(x 1 x 0 f(x 0 )dt g(x 0 )w 0 dt)δ(x2 x 1 Use Fourier transform representation δ(z) = 1 2π e ik iz dk i P [x i w i ]= i dk i 2π e i P i k i(x i+1 x i f(x i )dt g(x i )w i t)

23 P [x i w i ]= i dk i 2π e i P i k i(x i+1 x i f(x i )dt) e ik ig(x i )w i dt wi is Normal or Gaussian N(0,1) with density P [w i ]= 1 2π e 1 2 w2 i P [x i ]= i dw i P [x i w i ]P [w i ] = i dk P i 2π e i i k i(x i+1 x i f(x i )dt) i dw i 2π e ik ig(x i )w i dt e 1 2 w2 i

24 Complete the square P [x i ]= i dk i 2π e Pi (ik i)( x i+1 x i dt f(x i ))dt+ P i 1 2 g2 (x i )(ik i ) 2 dt Make a change of variables ki = ik i Take the continuum limit P [x(t)] = D ke R dt k(ẋ f(x(t))+ 1 2 k 2 g 2 (x(t)) action S[x(t), k(t)] = dt k(ẋ f(x(t)) 1 2 k 2 g 2 (x(t))

25 From SDE to action ẋ = f(x)+g(x)η(t) P [x] = P [x] = P [x] = Dηδ[ẋ f(x) g(x)η(t)]e R η 2 (t)dt DηD ke R k(ẋ f(x))+ kg(x)η(t) η 2 (t)dt D ke R k(ẋ f(x))+ 1 2 k 2 g 2 (x)dt S[x(t), k(t)] = dt k(ẋ f(x(t)) 1 2 k 2 g 2 (x(t))

26 Ornstein-Uhlenbeck ẋ = ax + Dη(t) P [x] = D ke R k(ẋ+ax)+ D 2 k 2 dt G 1 (t t ) = ( d dt + a)δ(t t ) P [x] = D ke R dtdt k(t)g 1 (t t )x(t )+ R dt D 2 k 2 S[x, k] = dtdt k(t)g 1 (t t )x(t ) dt D 2 k 2

27 Generating functional Z[J(t), J(t)] = DxD ke R dtdt kg 1 x+ R D 2 k 2 dt+ R Jxdt+ R J kdt = DxD ke R dtdt kg 1x (1 + µ + 1 2! µ ! µ3 + ) D where µ = 2 k 2 dt + Jxdt + J kdt Expansion in moments of complex Gaussian

28 Z CG [J(t), J(t)] = DxDke R dtdt ikg 1 x+ R ik(t)j(t)dt+ R x(t) J(t)dt Restore k = ik for convenience Discretize and diagonalize dt G 1 x λ(ω)ˆx(ω) Expand in ˆx(ω) Z CG [Ĵ, ˆ J] = k, J, J DˆxDˆke P ω ˆk(ω), Ĵ(ω), ˆ J(ω) iˆkˆxλ+iˆkĵ+ˆx ˆ J = ω dˆxdˆk 2π e iˆkˆxλ+iˆkĵ+ˆx ˆ J

29 = ω = ω dˆxdˆk 2π e iˆk(ˆxλ Ĵ)+ˆx dˆxδ(ˆxλ Ĵ)eˆx ˆ J ˆ J = ω e λ 1 Ĵ ˆ J Transform back to get Z CG [J, J] =e R J(t)G(t,t ) J(t )

30 Propagator ( d dt + a)x = G 1 (t, t )x(t )dt G 1 (t, t )=( d dt + a)δ(t t ) G 1 (t, t )G(t,t )dt = δ(t t ) ( d dt + a)g(t 1,t 2 )=δ(t 1 t 2 ) Green s function

31 Moments ij x(t i ) k(t j ) = ij δ δ J(t i ) δ δj(t j ) er J(t)G(t,t )J(t )dtdt J= J=0 Only surviving moments have equal numbers of x and k Wick s theorem still applies but only for contractions between x and k e.g. x(t 1 )x(t 2 ) k(t 3 ) k(t 4 ) = G(t 1,t 3 )G(t 2,t 4 )+G(t 1,t 4 )G(t 2,t 3 )

32 Back to the OU problem Z[J(t), J(t)] = DxD ke R dtdt kg 1 x+ R D 2 k 2 dt+ R Jxdt+ R J kdt = DxD ke R dtdt kg 1x (1 + µ + 1 2! µ ! µ3 + ) D where µ = 2 k 2 dt + Jxdt + J kdt µ 1 + µ 2 + µ 3

33 Can see that only some combinations of the μ terms will survive i.e. μ 2 μ 3 and μ 1 μ 2 2 Z[J(t), J(t)] = J(t)J(t )x(t) k(t ) dtdt + D 4 J(t) J(t )x(t)x(t ) k 2 (t ) dtdt dt +... x(t 1 )x(t 2 ) = D 2 x(t 1 )x(t 2 ) k 2 (t ) dt = D G(t 1,t )G(t 2,t )dt

34 Feynman diagrams Each term in action can be represented by a diagram Turn perturbation theory into rules for assembling diagrams Diagrams consist of edges and nodes, representing terms in the action

35 Every factor of k is given an outgoing edge Every factor of x is given an incoming edge Convention is time goes from right to left Propagators are lines, other terms are vertices

36 S[x, k] = dtdt k(t)g 1 (t, t )x(t ) dt D 2 k 2 Assign a diagram to each term in the action Propagator G(t, t ) t t Other diagrams are called vertices Vertex D 2 t sign negative from action

37 Feynman rules Surviving terms in perturbation series of generating functional consist of equal numbers of k and x factors For vertices, this means that outgoing edges are joined to incoming edges Propagators are then attached to all edges in all possible ways (accounts for Wick s theorem) Vertices are integrated over

38 Moment rules Moments of x are given by derivatives over J, moments of k are given by derivatives over J Thus moments can also be expressed as diagrams Each factor of each factor of x k is given an outgoing edge and is given an incoming edge Cumulants are connected diagrams

39 Examples Propagator: x(t 1 ) k(t 2 ) = = G(t t,t 2 ) 2nd moment: x(t 1 )x(t 2 ) = =2 G(t 1,t )G(t 2,t ) D 2 dt

40 Doing the integrals ( d dt + a)g(t 1,t 2 )=δ(t 1 t 2 ) G(t 1,t 2 )=e a(t 1 t 2 ) H(t 1 t 2 ) H(t) = { 1 t>0 0 t 0 D G(t 1,t )G(t 2,t )dt = D t2 0 e a(t 1 t ) e a(t 2 t ) dt t 2 >t 1 = D e2a(t 1 t 2 ) e 2a(t 1+t 2 ) 2a = D 2a (1 e 2at ) t 1 = t 2 = t

41 Nonlinear terms ẋ = ax + bx 2 cx 3 + A + Bx + Cx 2 η(t)+x 0 δ(t t 0 ) Using the action formula S[x(t), k(t)] = dt k(ẋ f(x(t)) 1 2 k 2 g 2 (x(t)) S[x(t), k(t)] = dt k(ẋ + ax bx 2 + cx 3 x 0 δ(t t 0 )) k 2 A + Bx + Cx2 2

42 S[x(t), k(t)] = dt k(ẋ + ax bx 2 + cx 3 x 0 δ(t t 0 )) k 2 A + Bx + Cx2 2 Propagator G(t, t ) Vertices x 0 δ(t t 0 ) b c A 2 B 2 C 2

43 x(t 1 )x(t 2 ) = Tree level 2 B 2 x 0 G(t 1,t )G(t 2,t )G(t,t 0 )dt One loop

44 Acknowledgments and References I will be writing a review that will be posted on my blog at sciencehouse.wordpress.com Michael Buice Zinn-Justin, Quantum Field Theory and Critical Phenomena Kardar, Statistical Physics of Fields