Subject: [AUDITORY] Logan's theorem - a challenge From: Alain de Cheveigne <alain.de.cheveigne@xxxxxxxx> Date: Sun, 26 Sep 2021 07:03:32 +0100Hi all, Here=E2=80=99s a challenge for the young nimble minds on this list, and = the old and wise. Logan=E2=80=99s 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)? =E2=80=9CBand-limited=E2=80=9D 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=E2=80=99s theorem suggests = that any information lost due to that non-linearity can be restored, = within each channel. If so, cochlear transduction is =E2=80=9Ctransparent=E2= =80=9D, 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. =20 Logan=E2=80=99s 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=E2=80=99s 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=E2=80=99s worth considering = some issues. First, Logan made clear that his theorem only applies to a = perfectly band-limited signal, and might not be =E2=80=9Capproximately = valid=E2=80=9D for a signal that is =E2=80=9Capproximately = band-limited=E2=80=9D. 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? =20 (3) In what sense can one say that the acoustic signal is "available=E2=80= =9D 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 --- Logan BF, JR. (1977) Information in the zero crossings of bandpass = signals. Bell Syst. Tech. J. 56:487=E2=80=93510. 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=E2=80=93423 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=E2=80=932833. 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 Cheveign=C3=A9, A. (in press) Harmonic Cancellation, a Fundamental of = Auditory Scene Analysis. Trends in Hearing = (https://psyarxiv.com/b8e5w/).=