Abstract
Much of interesting complex biological behaviour arises from collective properties. Important information about collective behaviour lies in the time and space structure of fluctuations around average properties, and two-point correlation functions are a fundamental tool to study these fluctuations. We give a self-contained presentation of definitions and techniques for computation of correlation functions aimed at providing students and researchers outside the field of statistical physics a practical guide to calculating correlation functions from experimental and simulation data. We discuss some properties of correlations in critical systems, and the effect of finite system size, which is particularly relevant for most biological experimental systems. Finally we apply these to the case of the dynamical transition in a simple neuronal model.
Highlights
Much of interesting complex biological behaviour arises from collective properties
There is more to collective behaviour than what is captured by the two-point correlation functions we discuss here
Correlation functions should be in the toolbox of any researcher attempting to understand the behaviour of complex biological systems, and it is probably fair to say that, they have been used and studied for a long time in statistical physics, they have not been exploited enough in biology
Summary
Let x and y be two random variables and p(x, y) their joint probability density. We use . . . to represent the appropriate averages, e.g. the mean of x is x = x p(x)dx (probability distribution of x can be obtained from the joint probability, p(x) = dy p(x, y), and in this context is called marginal probability) and its variance is Varx = (x − x ). The covariance is bounded by the product of the standard deviations (Priestley, 1981, §2.12), Cov2x,y ≤ VarxVary. The variables are said to be uncorrelated if their covariance is null: Covx,y = 0 ⇐⇒ xy = x y (uncorrelated). Absence of correlation is weaker than independence: independence means that p(x, y) = p(x)p(y) (and clearly implies absence of correlation). For uncorrelated variables it holds, because of (4), that the variance of the sum is the sum of the variances, but Covx,y = 0 is equivalent to independence only when the joint probability p(x, y) is Gaussian. The covariance, or the correlation coefficient, are said to measure the degree of linear association between x and y, because it is possible to build a nonlinear dependence of x on y that yields zero covariance (see Ch. 2 of Priestley, 1981)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
