Beat encoding at mistuned octaves within single sensory neurons

Beats are periodic amplitude modulations resulting from the superposition of two spectrally close periodic signals, where the difference between the two signal frequencies defines the frequency of the beat. Such amplitude modulations are known to be well encoded in corresponding firing rate modulations in auditory fibers as well as in electrosensory afferents. In a field study we showed the behavioral relevance of beat-like amplitude modulations exceeding spectrally close interactions in electric fish. Thus, we here study the encoding of beat-like waveforms over a wide range of difference frequencies in the electrosensory system of Apteronotus leptorhynchus. Contrary to expectations from the previously measured beat tuning, the activity of p-type electroreceptor afferents follows a repetitive pattern with slow modulations of their firing rate reoccurring around multiples of the frequency of the carrier signal. Mathematical reasoning supported by simulations of modified integrate-and-fire models reveals that neither Hilbert transform, squaring, harmonics of the carrier, half-wave rectification, nor the threshold-non-linearity of a spike generator are sufficient to extract slow beating signal envelopes around the octave of the carrier. Rather, a threshold operation smoothed out by exponentiation with a power of three is needed prior to spike generation to explain the repetitive occurrence of slow signal envelopes and electroreceptor responses. Our insights suggest the synapses of inner hair cells as candidate mechanisms underlying the perception of beats at mistuned octaves that has been described by Georg Simon Ohm, Hermann Helmholtz, and others already in the 19th century.