Measures of complexity derived from ergodic theory of dynamical systems were developed and applied to an exemplary data set describing locomoter movements of rats in a bounded space. A symbolic dynamical system was obtained by partitioning the event space into equally probable partition elements, using a k-dimensional tree. The measures calculated from the symbolic sequences included the topological entropy (h(t)) - i.e., the rate of increase of all possible sequences with increasing sequence length - and the metric entropy (h(m)) - i.e., the rate of increase of all likely sequences with increasing sequence length. These measures were used to assess changes in rat locomoter behavior as recorded in the behavioral pattern monitor (BPM) that were induced by amphetamine (0.25, 0.50, 1.0, or 2.0 mg/kg) and 3,4-methylenedioxymethamphetamine (MDMA; 1.25, 2.5, 5.0, or 10.0 mg/kg). Amphetamine increased the mean activity, h(t), and h(m). MDMA resulted in a monotonic dose-response curve for activity but exhibited a biphasic dose response in h(t) and h(m). In particular, some animals in the higher dose groups showed a h(t) in the range of the saline controls, whereas other animals exhibited a significantly reduced h(t) and a greater decrease in h(m), suggesting that two different behavioral reactions coexist within the same higher dose range of MDMA. An important implication of our method is that, in applied ergodic measure-theoretic approaches, the partition that determines the elements of the symbolic dynamical system should not be specified a priori on abstract mathematical grounds but should be chosen relative to its significance with respect to the data set in question. Here, the animal constructs its own spatiotemporal partition in behavioral phase space.
