Detecting determinism from point processes

The detection of a nonrandom structure from experimental data can be crucial for the classification, understanding, and interpretation of the generating process. We here introduce a rank-based nonlinear predictability score to detect determinism from point process data. Thanks to its modular nature, this approach can be adapted to whatever signature in the data one considers indicative of deterministic structure. After validating our approach using point process signals from deterministic and stochastic model dynamics, we show an application to neuronal spike trains recorded in the brain of an epilepsy patient. While we illustrate our approach in the context of temporal point processes, it can be readily applied to spatial point processes as well.

Figure 1

Results for ${\mathcal{S}}_{\mathrm{SPIKE}}$ and ${\mathcal{S}}_{\mathrm{ISI}}$ for the original spike trains are shown in green (light grey) and red (dark grey), respectively. Results for the corresponding surrogates are shown in black [lower bands in (a)–(e), crosses in (f)]. (a)–(e) dependence of results on the noise percentage $\gamma$ for models A–E, respectively. For each model and every $\gamma$ value we generated 300 independent spike trains realizations. From each realization we generated one surrogate spike train. Bands in the two upper parts show averages $\pm$ one standard deviation of $\mathcal{S}$ across the 300 original spike trains and across the 300 surrogates. Solid lines in the lower parts show the fraction $f$ of realizations for which the $\mathcal{S}$ for the original spike train was higher than the one for its surrogate. (f) results for the neuronal spiking data (pentagons) and 199 surrogates (crosses). Filled pentagons indicate cases for which the $\mathcal{S}$ values of the original exceeds all surrogates.