Beat encoding at mistuned octaves within single sensory neurons

Abstract

Beats are periodic amplitude modulations resulting from the superposition of two periodic signals of different frequencies. Psychophysical experiments by Georg Simon Ohm, Hermann Helmholtz, and others from the 19th century demonstrate that beats are not only perceived for two close-by frequencies but also at mistuned octaves. However, the physiological mechanisms of this percept are still debated. Motivated by a field study, in which we observed the behavioral relevance of beats at high difference frequencies, we here study the beat encoding over a wide range of difference frequencies in the electric fish Apteronotus leptorhynchus. The activity of P-unit electroreceptor afferents, that share many properties with mammalian auditory fibers, follows a repetitive pattern with slow modulations of their firing rate reoccurring around multiples of the frequency of the carrier signal. By mathematical reasoning and supported by simulations of modified integrate-and-fire models we conclude that neither Hilbert transform, squaring, harmonics of the carrier, nor a threshold operation are sufficient to extract slow beats around the octave. Raising the thresholded signal to a power of three, however, is sufficient to explain the repetitive P-unit responses. Since the threshold-nonlinearity of the afferent's spike generator could be ruled out, it is most likely the transfer function of the electroreceptor synapse that implements such a non-linearity. In the auditory system the hair-cell synapse is known to act as a smooth threshold operation. We thus conclude that this mechanism within each auditory fiber contributes to the perception of beats at mistuned octaves.

Weakly nonlinear responses at low intrinsic noise levels in two types of electrosensory primary afferents

“…Bursting was not explicitly addressed in the previous work, still, the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting (Savard et al, 2011). How bursts influence the second-order susceptibility of P-units will be addressed in the following works (Barayeu et al, 2024). Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in (Savard et al, 2011;Middleton et al, 2007) operates in a regime of strong stimuli, where the firing rate is saturated.…”

“…The frequency of the beat is given by the difference frequency Δ f 1 = f 1 − f EOD between the two fish. P-units encode this beat in their firing rate (Bastian, 1981; Barayeu et al, 2023) and consequently, a peak at this beat frequency appears the the power spectrum of the response (fig. 2 B).…”

“…This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (fig. 10) (Barayeu et al, 2023). The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of trials as in the experiment ( N = 11, compare A and B ii in fig.…”

“…Whether this falls into the weakly nonlinear regime is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use an stochastic leaky-integrate-and-fire (LIF) model (Barayeu et al, 2023), that faithfully reproduces baseline firing properties and responses to step stimuli of a specific electrophysiologically measured P-unit.…”

“…Therefore, stimuli with multiple distinct frequencies are part of the everyday life of wave-type electric fish (Benda, 2020) and interactions of these frequencies in the electrosensory periphery are to be expected. P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, use nonlinearities to extract and encode these AMs in their time-dependent firing rates (Bastian, 1981; Walz et al, 2014; Middleton et al, 2006; Stamper et al, 2012; Savard et al, 2011; Barayeu et al, 2023). P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) (Grewe et al, 2017; Hladnik and Grewe, 2023).…”