Fokker Planck and Master equation

Giovanni Marelli

Göttingen 27.01.10

Contents

Connections

Temporal evolution of the probability Hypotesis for the Chapman-Kolmogorov equation Meaning of the master equation The master equation is equivalent to a stochastic differential equation Derivation of the Fokker-Plank from a Langevin equation

Transition Probability

Transition probability at a time τ between two states y_{3} → y_{2} in a τ exspansion

P_{τ} (y_{3}|y_{2}) = δ(y_{3} − y_{2}) + W (y_{3}|y_{2})τ + O(τ^{2})

W (y_{3}|y_{2}) = d_{t}P(y_{3}|y_{2}) transition rate Normalizing: α(y_{2}) = ∫ W (y_{3}|y_{2})dy_{3}

P_{τ} (y_{3}|y_{2}) = (1 − τα(y_{2}))δ(y_{3} − y_{2}) + W (y_{3}|y_{2})τ
= δ(y_{3} − y_{2}) + W (y_{3}|y_{2})τ − W (y_{2}|y_{3})τ + O(τ^{2})

A Chemical reaction: R_{1} + R_{2} ⇌ P_{1} + P_{2} Fermi’s Rule: λ_{if} = \frac{2π}{ℏ} |M_{if} |^{2}ρ_{f}

R_{1} R_{2}P1 P2  \begin{matrix}|n > \\ |m > \end{matrix}
G. Marelli  Fokker-Planck  3 / 15

Chapman-Kolmogorow equation

The probability of a Markov process depends only on the probability of the last process

P(y_{n}|y_{n−1}, , y_{1}) = P(y_{n}|y_{n−1})

We assume a stationary (P(t) = P(t + τ)) and homogenues process (P(t_{1}, t_{2}) = P(t_{1} − t_{2})) (a more general case is the Boltzmann equation) The conditional probability between the state y_{3} and y_{1} can be written:

P(y_{3}|y_{1}) = ∫ P(y_{3}|y_{2})P(y_{2}|y_{1})dy_{2}

One step process: Brownian motion, shot noise, dacay.

Continous master equation

We derivate the conditional probability with respect to the first order ∂_{τ} P(y_{3}|y_{2}) = W (y_{3}|y_{2}) − W (y_{2}|y_{3}) The derivative of the conditional probability is:

∂_{τ} P(y_{3}|y_{1}) = ∫ (W (y_{3}|y_{2})P(y_{2}|y_{1}) − W (y_{2}|y_{3})P(y_{3}|y_{1}))dy_{2}

We remove the stationarity condition, multiply by P(y_{1}, t) and integrate over x_{1}

d_{t}P(y, t) = ∫ W (y, y′)P(y′, t) − W (y′, y)P(y, t)dy′

Einstein coefficients: Spontaneus emission A_{mn} , Absorption: B_{nm} J , Stimulated emission: B_{mn} J

Discrete master equation

A chemical reaction generation/ricombination (discrete, nonlinear) X ⇌_{W^{†}}^{W} 2X :

Ṗ_{n} = W^{†}n(n + 1)P_{n+1} + W (n − 1)P_{n−1} − W^{†}nP_{n} − Wn(n − 1)P_{n}

The master equation says that the probability of a transition in a time t is the sum of the gain in changing from m → n minus the loss between n → m

(n − 1) (n) (n + 1) W W_{†}

In the steady state the lhs of the master equation is zero ∑_{m} W_{nm} P_{m} = (∑_{m} W_{mn} )P_{n} Detailed balance: W_{nm} P_{m} = W_{mn} P_{n} Detailed balance is necessary but not sufficient for the equilibrium (microscopic reversibility)

Fokker-Planck equation

The evolution of a single event P(t) is governed by the master equation which describes all statistical properties. We recast the integro equation of a master equation into the form of a Kramers-Moyal expansion (r = x − x′, small)

∂_{t}P(x, t) = P(x, t) ∫ W (x|r)P(x, t) − W (x| − r)P(x, t)dr
− ∫ r∂_{x}(W (x|r)P(x, t))dr + \frac{1}{2} ∫ r^{2}∂_{x}^{2}(W (x|r)P(x, t))dr ± …

or

Ṗ_{t}(x) = ∑_{n} \frac{−^{n}}{n!} ∂_{x}^{n}D_{KM}^{(n)} P_{t}(x) D_{KM}^{(n)} (x, t) = \frac{1}{n!} lim_{τ→0} \frac{1}{τ} < (x(t + τ) − x(t))^{n} >

From the Pawula Theorem one can either use the first moment, the first and the second or every of them In Most of the case two moments are enough, (e.g. Gaussian noise): Fokker-Planck equation

Ṗ(x, t) = −∂_{x}D^{(1)}P(x, t) + ∂_{x}^{2}D^{(2)}P(x, t)

From master equation to stochastic differential equation

We consider the equation:

ẋ(t) = X (x) + ξ(t)

The moments of the noise ξ are connected with the Kramers-Moyal expansion

lim_{Δ→0} \frac{1}{Δ} ∫_{τ}^{τΔ} ds < ξ(s)|x(t) = x_{0} >= D_{KM}^{(1)} − X (x)
lim_{Δ→0} \frac{1}{Δ} ∫_{τ}^{τ+Δ} dt_{1} … dt_{n} < ξ(t_{1}) … ξ(t_{n})|x(t) = x_{0} >= D_{KM}^{(n)}

The first moment is deterministic, the higher oder stochastic (mesoscopic description) Considering the same moments the following SDEq is equivalent to the previous MEq

A non zero mean of the noise contributes to a drift term

P. Hanggi Z., Physik B, 43, 269-273,(1981)

Flux

From the Fokker-Planck equation we define a flux

J := −D^{(1)}X (x)P(x) + \frac{1}{2}D^{(2)}P(x)

The FPEq can be written as a conservation law:

∂_{t}P(x) − ∂_{x}J (x) = 0

Tipical boundary condition Natural (decay): ∫ P(x) = norm Reflecting (wall): J (x = a, t) = 0 Absorbing (first passage time): P(x = a, t) = 0

Neuron dynamics, role of noise

The dynamics of the voltage in a neuron is ruled by the equation:

V̇(t) = −\frac{V (t)}{τ} + I (t) I (t) = μ + σ_{w}η(t) + σ_{w} \frac{β}{\sqrt{2τ_{c}}} z(t) Where η(t) is a white noise and z(t) is an auxiliary colored noise, V is a potential between H and Θ C (t, t′) =< (I (t)− < I (t) >)(I (t′)− < I (t′) >) >= σ_{w}^{2}δ(t − t′) + \frac{Σ_{2}}{2τ_{c}} e^{− \frac{|t−t′|}{τc}} The Fokker-Planck equation is:

(∂_{V} (f (V ) − μ + \frac{σ^{2}}{2}∂_{V} ) + \frac{1}{τ_{c}} ∂_{z}(z + ∂_{z}) − \sqrt{\frac{2σ^{2}α^{2}}{τ_{c}}}∂_{V} )P = −δ(V − H )J (z)

The firing rate ν is the probability per unit time that the potential cross a threshold Θ. J (z) is the escape probability current.

G. Marelli  Fokker-Planck  10 / 15

Firing rate

P(V , z) is the staedy state probability. We suppose τ_{c} < τ_{ref} (correlation, refractory).

J (z) = \frac{ν_{out}}{\sqrt{2π}}e^{−x^{2}/2}

z after a spike relaxes to the stationary distribution. The Fokker-Planck has to be resolved with the normalisation condition

ν_{out} τ_{ref} + ∫_{−∞}^{Θ} dV ∫_{−∞}^{∞} dzP(V , z) = 1

Finally the output firing rate is give by ν_{out} = ∫ dzJ (z)

G. Marelli  Fokker-Planck  11 / 15

Summary

Transition prob MEq Langevin Kramers-Moyal Fokker-Planck Olstein-Uhlenbeck Neuronal model Stochastic Noise distribution Continuous Two moments Dissipative Non-gaussian noise
G. Marelli  Fokker-Planck  12 / 15

Thank you for your attention

Aknowledgments Tatjana, Mirko (tutors)

References

Response of Spiking Neuros to Correlated Inputs

R. Moreno, J. de la Rocha, A. Renart, N. Parga

PRL 89 288101 (2002)

How Spike Generation Mechanisms Determine the Neuronal Response to Fluctuating

Inputs

N. Foucard-Trocmé, D. Hansel, C. van Vreeswijk, N. Brunel

J. Neuroscience 23 11628

Langevin equation

ẋ = X (x) + ξ(t)

The probability distribution P(y, t) is defined as:

P(y, t) =< δ(y − x(t)) >_{ξ}

Advection (drift) term D^{(1)}: ẋ = X (x)

∂_{t}P(y, t) = −ẋd_{y}δ(y − x) = −X (x)d_{y}δ(y − x) = −d_{y}(δ(y − x)F (x))
= −d_{y}(δ(y − x)F (y)) = −d_{y}(F (y)P(y, t))

Diffusive term D^{(2)}: ẋ = ξ(t)

P(x, t) = \frac{1}{\sqrt{2πΔt}} e^{− \frac{y^{2}}{2Δt}} ∂_{t}P(x, t) = \frac{D^{(2)}}{2}∂_{x}^{2}P(x, t)
∂_{t}P(x, t) = −D^{(1)}∂_{x}P(x, t) + D^{(2)}∂_{x}^{2}P(x, t)

Ornstein-Uhlenbeck process

We start from the dissipative Langevin equation

Δv(t) = −γv(t) + σΔξ(t)

The KM coefficients are D^{(1)} = −γv, D^{(2)} = \frac{σ^{2}}{2} The stationary solution is given by Ṗ(x) =^{!} 0

P(v) = \sqrt{\frac{m}{2πk_{B}T}} e^{− \frac{v^{2}m}{2K_{B} T}}