The fourier domain is just a linear transform of the time domain, i.e. just a matrix multiply away!

Could you expand please? I'm connecting the two via convolution but can't intuitively grasp the link yet.

See “Laplace transform”

https://en.m.wikipedia.org/wiki/Laplace_transform

The fourier transform converts between time domain and frequency domain. The fourier transform is linear, which means it can be implemented as multiplying your time domain signal by a matrix of complex fourier coefficients. Usually we dont do it this way because the fft is faster than matrix multplication, but the result is the same.

Wow, could you point me to other resources to read about this?

There has been a long discussion in the neuroscience community how neuronal firing encodes information. The two extremes of the proposed solutions are temporal coding and rate coding, but there are intermediate positions such as held by the parent poster (and myself, and nowadays most other people that care about this niche).

Wikipedia is decent-ish on the topic: https://en.wikipedia.org/wiki/Neural_coding

Wulfram Gerstner's book has a nice discussion of the theory and limitations of different proposed coding schemes, which might interest the technical audience here. In particular, I think the parent poster alluded to the ideas presented in the last section here: https://neuronaldynamics.epfl.ch/online/Ch7.S6.html

Really and truly this "discussion" should have been put to rest a long time ago. Neurons encode information with the temporal precision needed to encode that information. Since most objects are somewhat permanent, the temporal precision can be very low, yielding effectively a rate code. For tasks for which the (relative) timing is critical, e.g. the encoding of auditory sequences, the temporal precision of the neural code is high, yielding effectively a temporal code. Some problems are neither here nor there, in which case the neuron will encode as much information as possible at a low rate / temporal resolution, and then encode diminishing amounts of information at successively higher temporal resolutions (the Fourier code alluded to by the parent poster). Higher temporal precision typically requires more control, larger circuits, and hence more energy expenditure.

Bill Bialek's group had a paper that quantified the amount of information present in a visual neuron of the fly at different temporal resolutions (/frequencies) as early as 2008: https://journals.plos.org/ploscompbiol/article?id=10.1371/jo...

Is there a certain way to encode typical "continuos" information like a certain tilt of a line in an image?

I ran into this over ten years ago in an experiment on temporary plasticity of sensory input and I didn't find a quick answer before I had to leave the topic (ok, it left me)

Thanks for sharing that book link, btw.