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[AUDITORY] Logan's theorem - a challenge
Hi all,
Here’s a challenge for the young nimble minds on this list, and the old and wise.
Logan’s theorem states that a signal can be reconstructed from its zero crossings, to a scale, as long as the spectral representation of that signal is less than an octave wide. It sounds like magic given that zero crossing information is so crude. How can the full signal be recovered from a sparse series of time values (with signs but no amplitudes)? “Band-limited” is clearly a powerful assumption.
Why is this of interest in the auditory context? The band-limited premise is approximately valid for each channel of the cochlear filterbank (sometimes characterized as a 1/3 octave filter). While cochlear transduction is non-linear, Logan’s theorem suggests that any information lost due to that non-linearity can be restored, within each channel. If so, cochlear transduction is “transparent”, which is encouraging for those who like to speculate about neural models of auditory processing. An algorithm applicable to the sound waveform can be implemented by the brain with similar results, in principle.
Logan’s theorem has been invoked by David Marr for vision and several authors for hearing (some refs below). The theorem is unclear as to how the original signal should be reconstructed, which is an obstacle to formulating concrete models, but in these days of machine learning it might be OK to assume that the system can somehow learn to use the information, granted that it’s there. The hypothesis has far-reaching implications, for example it implies that spectral resolution of central auditory processing is not limited by peripheral frequency analysis (as already assumed by for example phase opponency or lateral inhibitory hypotheses).
Before venturing further along this limb, it’s worth considering some issues. First, Logan made clear that his theorem only applies to a perfectly band-limited signal, and might not be “approximately valid” for a signal that is “approximately band-limited”. No practical signal is band-limited, if only because it must be time limited, and thus the theorem might conceivably not be applicable at all. On the other hand, half-wave rectification offers much richer information than zero crossings, so perhaps the end result is valid (information preserved) even if the theorem is not applicable stricto sensu. Second, there are many other imperfections such as adaptation, stochastic sampling to a spike-based representation, and so on, that might affect the usefulness of the hypothesis.
The challenge is to address some of these loose ends. For example:
(1) Can the theorem be extended to make use of a halfwave-rectified signal rather than zero crossings? Might that allow it to be applicable to practical time-limited signals?
(2) What is the impact of real cochlear filter characteristics, adaptation, or stochastic sampling?
(3) In what sense can one say that the acoustic signal is "available” to neural signal processing? What are the limits of that concept?
(4) Can all this be formulated in a way intelligible by non-mathematical auditory scientists?
This is the challenge. The reward is - possibly - a better understanding of how our brain hears the world.
Alain
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Logan BF, JR. (1977) Information in the zero crossings of bandpass signals. Bell Syst. Tech. J. 56:487–510.
Marr, D. (1982) VISION - A Computational Investigation into the Human Representation and Processing of Visual Information. W.H. Freeman and Co, republished by MIT press 2010.
Heinz, M.G., Swaminathan J. (2009) Quantifying Envelope and Fine-Structure Coding in Auditory Nerve Responses to Chimaeric Speech, JARO 10: 407–423
DOI: 10.1007/s10162-009-0169-8.
Shamma, S, Lorenzi, C (2013) On the balance of envelope and temporal fine structure in the encoding of speech in the early auditory system, J. Acoust. Soc. Am. 133, 2818–2833.
Parida S, Bharadwaj H, Heinz MG (2021) Spectrally specific temporal analyses of spike-train responses to complex sounds: A unifying framework. PLoS Comput Biol 17(2): e1008155. https://doi.org/10.1371/journal.pcbi.1008155
de Cheveigné, A. (in press) Harmonic Cancellation, a Fundamental of Auditory Scene Analysis. Trends in Hearing (https://psyarxiv.com/b8e5w/).