Auditory time-interval perception as causal inference on sound sources

Abstract

Perception of a temporal pattern in a sub-second time scale is fundamental to conversation, music perception, and other kinds of sound communication. However, its mechanism is not fully understood. A simple example is hearing three successive sounds with short time intervals. The following misperception of the latter interval is known: underestimation of the latter interval when the former is a little shorter or much longer than the latter, and overestimation of the latter when the former is a little longer or much shorter than the latter. Although this misperception of auditory time intervals for simple stimuli might be a cue to understanding the mechanism of time-interval perception, there exists no model that comprehensively explains it. Considering a previous experiment demonstrating that illusory perception does not occur for stimulus sounds with different frequencies, it might be plausible to think that the underlying mechanism of time-interval perception involves a causal inference on sound sources: herein, different frequencies provide cues for different causes. We construct a Bayesian observer model of this time-interval perception. We introduce a probabilistic variable representing the causality of sounds in the model. As prior knowledge, the observer assumes that a single sound source produces periodic and short time intervals, which is consistent with several previous works. We conducted numerical simulations and confirmed that our model can reproduce the misperception of auditory time intervals. A similar phenomenon has also been reported in visual and tactile modalities, though the time ranges for these are wider. This suggests the existence of a common mechanism for temporal pattern perception over modalities. This is because these different properties can be interpreted as a difference in time resolutions, given that the time resolutions for vision and touch are lower than those for audition.

Keywords: Bayesian inference; causal inference; source identification; time-interval perception.

Figure 1

Perceptual and simulated overestimation of T₂ as a function of T₁ − T₂. Each marker represents a different total duration T₁ + T₂. (A) Perceptual overestimation of T₂. Perceptual overestimation is measured by using the method of adjustment. (From Figure 3B, Miyauchi and Nakajima, , with changes in notations. ©University of California Press Journals. Adapted with permission.) (B) Simulated overestimation of T₂, calculated by subtracting T₂ from the expectation value of P(T₂|s₁,s₂,s₃). The area under the horizontal dashed line indicates the underestimation of T₂, and the area on the left side of the vertical dashed line indicates T₁ < T₂.

Figure 2

(A) A temporal pattern and its observation. The horizontal axis is time. Each solid vertical line indicates an actual sound timing t_i, and each dashed vertical line indicates its observed timing s_i. Each observation is made on the basis of an independent identical distribution. (B) Likelihood function of (T₁,T₂) given observed values of S₁ = 100 ms and S₂ = 140 ms (indicated as “+”). Intensity indicates the degree of likelihood. (C) Prior distribution of (T₁,T₂) with intensity on a logarithmic scale of the probability, illustrating that the prior takes a high value near both axes as well as along the 45° line from the T₁-axis. (D) Posterior distribution of (T₁,T₂) given observed values of S₁ = 100 ms and S₂ = 140 ms (indicated as “+”). Intensity indicates probability. The cross sign (×) corresponds to the peak of the distribution. (E–G) Prior distributions of (T₁,T₂) given that (E) the first and second sounds come from the same source, (F) the second and third sounds come from the same source, and (G) all three sounds come from the same source. Intensity indicates probability.

Figure 3

Schematic figure of the model mechanism. Dashed lines indicate the shape of the prior distribution. Each solid-lined ellipse represents the likelihood function given an observed interval pair marked as a plus sign on the center of the ellipse. Each arrow describes the direction of the perceptual shift.