Subject: Re: [AUDITORY] Logan's theorem - a challenge From: PIerre DIVENYI <pdivenyi@xxxxxxxx> Date: Sun, 26 Sep 2021 10:09:02 -0700Dear Alain, Quite a challenge, indeed!=20 I have only a non-scientific observation to add, one that seems to jibe with= Logan=E2=80=99s proposition. Just last week my 82-year old piano tuner came= to tune my piano. Beside his 440-Hz tuning fork he uses only his ears to do= his job and up to now the results have always been OK, as judged by this ot= her octogenarian=E2=80=99s ears. This time, however, the tuning of the upper= 2 octaves, especially the top one, was so flat that I had to call him back t= o do an adjustment. Re/Logan: high-frequency hearing loss makes the peripheral trace of a harmon= ic signal essentially band-limited, making it hard for the old guy to zero i= n on the f0 > 2.5 kHz. Since the duration of the signal is practically unlim= ited, Logan=E2=80=99s theorem seems to have held, with the result that the p= oor guy had to come back and, because I wanted to be fair, I felt I had to p= ay him at least for his gas. Your thoughts? Pierre=20 Sent from my autocorrecting iPad > On Sep 26, 2021, at 02:46, Alain de Cheveigne <alain.de.cheveigne@xxxxxxxx= eu> wrote: >=20 > =EF=BB=BFHi all, >=20 > Here=E2=80=99s a challenge for the young nimble minds on this list, and th= e old and wise. >=20 > 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 s= ignal is less than an octave wide. It sounds like magic given that zero cro= ssing information is so crude. How can the full signal be recovered from a s= parse series of time values (with signs but no amplitudes)? =E2=80=9CBand-l= imited=E2=80=9D is clearly a powerful assumption. >=20 > Why is this of interest in the auditory context? The band-limited premise= is approximately valid for each channel of the cochlear filterbank (sometim= es characterized as a 1/3 octave filter). While cochlear transduction is no= n-linear, Logan=E2=80=99s theorem suggests that any information lost due to t= hat non-linearity can be restored, within each channel. If so, cochlear tran= sduction is =E2=80=9Ctransparent=E2=80=9D, which is encouraging for those wh= o like to speculate about neural models of auditory processing. An algorithm= applicable to the sound waveform can be implemented by the brain with simil= ar results, in principle. =20 >=20 > Logan=E2=80=99s theorem has been invoked by David Marr for vision and seve= ral authors for hearing (some refs below). The theorem is unclear as to how t= he original signal should be reconstructed, which is an obstacle to formulat= ing 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 th= at it=E2=80=99s there. The hypothesis has far-reaching implications, for ex= ample it implies that spectral resolution of central auditory processing is n= ot limited by peripheral frequency analysis (as already assumed by for examp= le phase opponency or lateral inhibitory hypotheses). >=20 > Before venturing further along this limb, it=E2=80=99s worth considering s= ome issues. First, Logan made clear that his theorem only applies to a perf= ectly 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 p= ractical signal is band-limited, if only because it must be time limited, an= d thus the theorem might conceivably not be applicable at all. On the other= hand, half-wave rectification offers much richer information than zero cros= sings, so perhaps the end result is valid (information preserved) even if th= e theorem is not applicable stricto sensu. Second, there are many other imp= erfections such as adaptation, stochastic sampling to a spike-based represen= tation, and so on, that might affect the usefulness of the hypothesis. >=20 > 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 practic= al 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 a= uditory scientists? >=20 > This is the challenge. The reward is - possibly - a better understanding o= f how our brain hears the world. >=20 > Alain >=20 > --- > Logan BF, JR. (1977) Information in the zero crossings of bandpass signals= . Bell Syst. Tech. J. 56:487=E2=80=93510. >=20 > Marr, D. (1982) VISION - A Computational Investigation into the Human Repr= esentation and Processing of Visual Information. W.H. Freeman and Co, republ= ished by MIT press 2010. >=20 > Heinz, M.G., Swaminathan J. (2009) Quantifying Envelope and Fine-Structure= Coding in Auditory Nerve Responses to Chimaeric Speech, JARO 10: 407=E2=80=93= 423 > DOI: 10.1007/s10162-009-0169-8. >=20 > Shamma, S, Lorenzi, C (2013) On the balance of envelope and temporal fine s= tructure in the encoding of speech in the early auditory system, J. Acoust. S= oc. Am. 133, 2818=E2=80=932833. >=20 > Parida S, Bharadwaj H, Heinz MG (2021) Spectrally specific temporal analys= es of spike-train responses to complex sounds: A unifying framework. PLoS Co= mput Biol 17(2): e1008155. https://doi.org/10.1371/journal.pcbi.1008155 >=20 > de Cheveign=C3=A9, A. (in press) Harmonic Cancellation, a Fundamental of A= uditory Scene Analysis. Trends in Hearing (https://psyarxiv.com/b8e5w/). >=20