Recurrence analysis methods for the classification of nonlinear systems

Abstract

Recurrence is a common phenomenon in natural systems: A system enters and leaves a state, but after a given period of time, passes near that same state again. Many complex signals, such as weather cycles, heartbeats, or neuron firing patterns, all show recurrence. The recurrence plot (RP) displays all times j where a system returns near a state it has occupied at time i, giving rise to upward-sloping diagonal lines where a system follows a recurrent path, orthogonal lines when the system changes very slowly, or many disconnected points where a system's behavior is unpredictable. Investigation of the RP can then proceed through recurrence quantification analysis (RQA). Three new measures for RQA were developed: diagonality, quantifying diagonal lines, verticality, quantifying vertical lines, and periodicity quantifying the arrangement of recurrence points in periodic structures. These new measures were applied alongside classical recurrence measures to explore trends in random data, identify periodicity and chaotic behavior in the logistic map, estimate the dimensionality of the Lorenz attractor, and discriminate between persistent data signals. In collaboration with biologist Dr. Michael Harris, RQA methods were applied to the discrimination of two neuron types: serotonergic cells are believed to stimulate respiration, while nonserotonergic cells are implicated in respiratory inhibition. Typical discrimination methods compare mean and standard deviation of firing rates to a reference line, which correctly classifies serotonergic cells but incorrectly classifies many nonserotonergic cells. Voltage signals from such cells were converted into inter-spike intervals. Convergence required trials containing over 300 spikes for biological methods, and over 1000 for full investigation using RQA. Whether such cells can be discriminated from baseline firing patterns remains an open question.