Correlation-anti-correlation transition by state-dependent Poisson noise

Noise may induce some degrees of order in many biological and complex systems. We present an example of a noise-dependent transition from correlation to anti-correlation of the states of a bivariate system coupled by state-dependent Poisson noises. By varying the degree of dependence of the rate of jumps on the other variable, the system undergoes different degrees of correlation, from independence to perfect correlation (which in this case coincides with perfect synchronization) in the limit of the deterministic, periodic case.