Attractor reconstruction from interspike intervals is incomplete

We investigate the problem of attractor reconstruction from interspike times produced by an integrate-and-fire (IF) model of neuronal activity. Suzuki et al. [Biol. Cybernet. 82 (2000) 305–311] found that the reconstruction of the Rössler attractor is incomplete if the IF model is used. We explain this failure using two observations. One is that the attractor reconstruction only reconstructs an attractor of a discrete system, which may be a strict subset of the original attractor. The second observation is that for a set of parameters with non-empty interior numerical simulations demonstrate that the attractor of the discrete system is indeed a strict subset of the Rössler attractor. This is explained by the existence of phase locking in a nearby periodically forced leaky IF model.