Ornstein-Uhlenbeck Process

ornsteinUhlenbeck.m

%% Ornstein-Uhlenbeck Process
% ...from the paper "FLUCTUATING SYNAPTIC CONDUCTANCES RECREATE IN 
%   VIVO-LIKE ACTIVITY IN NEOCORTICAL NEURONS"
% The stochastic differential equation is given by:
%   dx/dt = 1/tau * (mu - x) + sqrt(D) * chi(t)
% where:
%   x is the random variable
%   D is the amplitude of the stochastic component
%   chi(t) is a normally-distributed (zero-mean) noise source
%   tau is the time constant (tau = 0 gives white noise, tau > 0 gives
%       colored noise)
% x is Gaussian and its variance is given by:
%   sigma^2 = D * tau / 2
% Therefore the SDE can also be expressed as:
%   dx/dt = 1/tau * (mu - x) + sigma * sqrt(2/tau) * chi(t)

% Use a time constant of 5 ms, a variance of 2500, a mean of 250
tau = 5;
sigma = sqrt(2500);
mu = 250;

% Consider dt = 1
tSim = 10000;

x = zeros(tSim, 1);

for i = 1: tSim - 1
    x(i + 1) = x(i) + 1/tau * (mu - x(i)) + sigma * sqrt(2/tau) * randn(1);
end

% The variance seems to be off. Why?
mean(x)
var(x)