Evidence for chaos in spike trains of neurons that generate rhythmic motor patterns

The findings presented here of work on the opisthobranch mollusc Pleurobranchaea californica indicate that some of the variability that has been observed in the activity of neurons during patterned motor activity may be attributable to low-dimensional chaos. We obtained long trains of action potentials (spikes) from these neurons, scanned them using adjacent temporal windows having equal widths, and converted the counts into frequency time series. These series were passed through a low-pass filter and detrended when necessary. The resulting time series gave a view of the envelopes of high-frequency bursts of spikes relating to the repetitive motor activity rather than of the intervals between spikes. Where applicable, we also compared analyses of smoothed data with the unprocessed spike intervals and found similar results for each type of time series. Autocorrelation functions of the processed data quickly decreased to zero, indicating that the long-term evolution of the time series could not be predicted from information at some given time. The first zero crossing of the autocorrelation function was used to define the lag for mapping the series into multidimensional phase space. These constructions were then used to examine the dynamics of the motor patterns directly from the state parameters of the time series: 1-D maps obtained from Poincaré slices of 2-D phase portraits, principal Lyapunov exponents, and correlation dimensions all indicated that the activity may be attributable to low-dimensional chaos. The present findings are similar to those of previous work in which equal-interval time series were obtained by interpolation of the unequal-interval spike trains. We discuss the implications of chaos and the difficulties in the application of extant dynamical tools to spike trains. An accompanying paper inquires into the ability of neural networks to read and transmit chaotic activity.