Langevin Equation

The following Ordinary Differential Equation (ODE) can describe the deterministic and stochastic processes of a non-linear, complex system: \[ \begin{equation} \frac{d}{dt}\mathbf{X}(t) = \mathbf{g}(\mathbf{X}(t), t) + \mathbf{h}(\mathbf{X}(t), t)\eta(t) \end{equation} \] where \(\mathbf{g}\) represent the deterministic part and \(\eta\) the stochastic part. \(\mathbf{X}(t)\) is a time-dependent \(d\)-dimensional, stochastic vector and \(\eta\) is Gaussian, white noise, satisfying:

\[ \begin{equation} \langle \eta(t) \rangle = 0 \end{equation} \] and: \[ \begin{equation} \langle \eta(t)\eta(t') \rangle = \delta(t-t') \end{equation} \] Consider the simpler one-dimensional Langevin equation generating a Brownian motion of a highly-damped particle in the potential field \(U(x)\) (Deco et al. 2009):

\[ \begin{equation} \frac{dx}{dt} = \frac{dU(x)}{dx} + \sqrt{2D}\eta(t) \end{equation} \] where \(D\) is a noise intensity constant. The probability density function of \(x\) is a solution to the Fokker-Planck equation.

Linear Growth

The following equation is a simple linear growth model: \[ \begin{equation} x_{t+1} = rx_t + b \end{equation} \] where \(x_t\) is the population at time \(t\), \(r\) is the growth rate and \(b\) is a constant. Such a system can adopt one of three states:

  1. Long-term, steady-state solution (fixed point);

  2. Explosively unstable solution, or;

  3. Periodic dynamics.

Below are six time series plots, showing how \(x_t\) can evolve for six different values of \(r\) ranging from -2 to +2. In each case, \(x(t=1) = 0.1\) and \(b = 0\).

That is a purely deterministic system. Imagine now \(b\) is a stochastic process that randomly takes a value of -1 or 1 at each time-step. When \(r = -2\), the evolution of \(x\) does not change. When \(r = -1\), the evolution looks similar to the cyclic behaviour of the deterministic system but the amplitude varies randomly. For \(-0.5 < r < 1\), the evolution looks drastically different and alot noisier. When \(r = 2\), the evolution is still explosive but negative. This is because the initial value of \(b\) is -1. This is an example of how adding noise to a deterministic system can make it sensitive to its initial conditions. If \(b(t=1) = +1\), the system would have exploded positively.

The sign of \(b\) and its relative magnitude with respect to \(rx_{t}\) determines the evolution of \(x\). The following Stochastic Ordinary Differential Equation (SODE):

\[ \begin{equation} \frac{dx}{dt} = rx + b(t) \end{equation} \] can be written in difference form: \[ \begin{equation} x_{t+1} = (r+1)x_t + b_t \end{equation} \] with a unit time-step. This is said to be an Orstein-Uhlenbeck process if \(b(t)\) is Gaussian noise and \(r < -1\), which describes a particle subjected to random velocity impulses that lead to random changes to its position and a restoring force that tends to bring it back to its origin.

If we solve the SODE for \(x(0) = -b(0)/r\), \(b(0)=0\) and \(r = -1.1\), the evolution looks like:

Deco, Gustavo, Daniel Marti, Anders Ledberg, Ramon Reig, and Maria V. Sanchez Vives. 2009. “Effective Reduced Diffusion-Models: A Data Driven Approach to the Analysis of Neuronal Dynamics.” PLoS Computational Biology. http://link.galegroup.com/apps/doc/A216182198/AONE?sid=googlescholar.