# Help with identifying a transfer function - with neuroscience application

This is a question for you electrical network engineers. The problem has arisen in a neuroscience study I am working on, but I hope to be able to glean some insight from other domains that deal with electrical systems and control networks. The brain is an electrical system after all, so I figure it should be governed by similar processes to large-scale electrical closed loops systems (like power grids for example).

I have devised a transfer function, which I am using to investigate human brain networks. We have recorded electrical signals from region A in the brain and mapped them, via this transfer function, to region B.

The transfer function works as follows.

• Take a recorded time series x which describes the time evolution of some brain process.
• Apply Fourier transform to get X
• Apply a constant phase shift to all components of X to give Y, taking care of the symmetry in the frequency domain (e.g. this could be a 90 deg phase shift).
• Apply the inverse Fourier transform to Y to retrieve the time-shifted signal y

The upshot of this is that lower frequencies are shifted in time MORE than higher frequencies. This appears to fit our brain data very well - which is something of a mystery. We don't actually understand why this works so well, so finding analogous systems in engineering would be a great help.

Therefore, my question is whether any of you can identify the type of transfer function we are using here. Does it have a proper name in engineering, is there a theoretical basis for it? The best I have come up with is that it is a minimum phase transfer function, but I don't yet understand how that might apply to a close loop system.

Even if someone could point me to the relevant textbook, that would also be very helpful.

• I should have added that this transfer function appears to fit the data "very" well indeed. By shifting region A using the zero-order phase shift, we get near perfect correlation of activity at distant locations in the brain. – user13238 May 2 '18 at 16:02
• A pure time delay is represented in the frequency domain as an exponential (transcendental) function. The linear representation of this time delay is typically made using the Pade approximation. This approximation has left-hand plane poles and right-hand plane zeros. The right-hand plane zeros make the delay non-minimum phase. Something that should be clarified is the time/phase delay relationship. Is the time delay constant regardless of frequency? – PICyPICyPICy May 2 '18 at 16:07
• @user13238 Please don't add a comment, edit the original question using the 'edit' button (Lower left, under your post). – Oldfart May 2 '18 at 16:34
• I'm really not sure what your question is. – Andy aka May 2 '18 at 16:51
• Sorry if this was a bit vague, I have tried to make my question a bit clearer. – user13238 May 3 '18 at 15:12

You're just looking at a phase shift. If you're seeing a 90 degree phase lag at all frequencies, you could just be looking at the output of a one-pole high-pass filter at well below the cutoff frequency, in series with a simple inverter. When you're more than a decade below the cutoff, the phase lag is approximately 90 degrees, and the gain is negative.

If you start seeing 180 degree phase shift at higher frequencies than you're looking at, this would provide confirmation.

• Thanks Scott. I will look into that further. Could you suggest the best place to learn more about filter design? I am coming at this from a very different angle and not familiar with all the engineering terminology, such as poles and zeros (although I have a vague idea of what it means). – user13238 May 3 '18 at 22:06
• You could try something like a Schaums outline in Signals and Systems, but a faster approach would be to find a collaborator – Scott Seidman May 3 '18 at 22:58

It may be synthesized in a FIR or IIR filter

• Thank you for the answer. That is true in the first-order delay (pure time delay scenario) but what we are really interested in is the zero-order delay, which amounts to a constant phase shift of all frequency components in the Fourier spectrum. Sorry that wasn't clear. I have updated the question to make that clearer. – user13238 May 3 '18 at 15:17
• Then you would define that transfer function in FIR or IIR filter – Sunnyskyguy EE75 May 3 '18 at 15:38
• No it can't be a pure time delay. that would be linear phase, not constant phase.. – Scott Seidman May 3 '18 at 17:16
• Yes Scott I got that after his edit – Sunnyskyguy EE75 May 3 '18 at 17:20