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I have a ""which came first: the chicken or the egg?"-kind of question regarding the processing and understanding of speech / non-speech sounds in the human brain. I'm wondering whether speech and generic non-speech sound processing in the human brain are somehow related and whether one of the two is used as a "basis" mathematically speaking or a reference to understand the other.

Most people use "His voice/It sounds like .." to describe sounds/human voices, besides, we are able to tune out and block any non-speech acoustic activity surrounding us if we are not in a conversation which suggests that we have a special sensibility for hearing speech (or maybe our language or just our names if someone is calling us).

I've read an article which states that :

Knowledge of our mother tongue acts as a sort of auditory "template" that influences the way we perceive the sounds of other languages (scientists call this "native listening"). 

Hearing other languages differently than its native speakers is easy to grasp but is there any research that studies the difference in brain activity when speech / other sounds are heard and whether there is something special about hearing speech in particular.

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"I'm wondering whether speech and generic non-speech sound processing in the human brain are somehow related and whether one of the two is used as a "basis" mathematically speaking or a reference to understand the other."

They both have in common acoustic energy - whose rate of change is information in itself. The mathematical basis of choice to compare each against the other, I think, is in how each relate that energy to the brain response - aka the forward model of the stimulus-response relationship.

In auditory studies you often get the (spectro)temporal response function (S/TRF) to estimate an approximate "summary" of this relationship. Note that traditionally, this is done independently, that is, either energy from speech or from other types of sound are separately investigated.

"is there any research that studies the difference in brain activity when speech / other sounds are heard"

There is a history of work contrasting the differential encoding of natural versus artificial sound classes (check out foundational work by Theunissen and colleagues). At these scales (neurons or neuron groups), the differences can be rather stark and happen in birds and mammals alike.

If you are more interested in these coding rules regarding "brain activity" at a larger scale, there are local-field-potential studies (e.g. Gaucher et al. 2012 J Physiol-Paris) that also relate mathematically to the S/TRF. However they are most likely invasive animal studies. Also I don't recall if they contrast speech S/TRFs versus other sound's. If you want something closer to human cognition, probably you might want to look at EEG or MEG studies also.

It turns out that finding similarities can be a bit harder than finding out the differences, maybe because the sounds are so different themselves. The mathematical relationship cannot escape this fact, and may then give distinct S/TRF patterns in return.

In a pre-print that seeks out similarities across stimulus types it is reported that certain information about the acoustic energy changes, in particular temporal onset edges, can be coded by the human auditory cortex, be it from speech, or music, or something else, in a more or less generalizable format across classes, at least qualitatively. https://www.biorxiv.org/content/early/2017/07/27/168997

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