Fractal character of the auditory neural spike train

Long-counting-time pulse-number distributions (PND's) were measured from a broad variety of cat primary auditory fibers using different tone and noise stimuli, counting times T, and number of samples NT. Whereas short-counting-time PND's (T approximately 50 ms) manifest the presence of spike pairs (an enhancement of even over odd-count probabilities), the irregular shapes of the long-counting-time PND's (T approximately greater than 0.1 s) reveal that the underlying sequence of action potentials consists of spike clusters when viewed on a longer time scale. For all units measured, the count variance-to-mean ratio (Fano factor) F(T) varied little over some 90 dB change in the stimulus level. On the other hand, F(T) increased substantially as T and/or NT were increased, corresponding to the capture of larger and larger spike clusters in the counting time. A relationship is developed between the Fano-time function F(T) and the normalized coincidence rate function, g(tau) versus delay time tau. A plausible form for g(tau) leads to a Fano-time function in good accord with the data. The observed power-law growth of the Fano factor for large counting times F(T) approximately T alpha where 0 less than alpha less than 1 is accompanied by a power-law decay of the coincidence rate for large delay times g(tau) approximately tau alpha -1 and a power-law form for the power spectral density at low frequencies S(f) approximately f -alpha. The behavior of the PND's and the scale invariance implicit in these fractional-power-law relationships suggest that the neural events on all primary auditory fibers exhibit fractal behavior for sufficiently large times (sufficiently low frequencies). The spike pairs and spike clusters in the PND's are natural consequences of this behavior. The fractal dimension D identical to alpha is estimated to be in the range of 0.3 approximately less than D approximately less than 0.9 for counting times in the range 0.1-10 s. The fractal dimension provides a measure of the degree of event clustering, or irregularity of a sequence of events, that is preserved over different time scales. PND's from low-skew vestibular units, in contrast, do not exhibit fractal behavior. It is suggested that auditory neural-firing patterns may serve to efficiently sample natural fractal noises.