Neuroscience: LFP at fixed point with different simulataneous frequencies

[nobahar]

Hello!

I have a neuroscience question about local field potenitals (LFPs).
I was reading a jounral article where the LFP in a given location was recorded overtime, and then the LFP at ecah time point was divided into different frequency groups: i.e. the LFP at a given point in space oscillates at different frequencies simultaneously (or at least a number of different frquencies can be obtained from an LFP at a given pint in space at a given point in time). This is what I need help understanding.

My understanding of the LFP is that it is a measure of the electric potential at a given point in space. It can oscillate over time, but it has a single value at any given point in time.If this is true, how can it be separated out into different simultaneous frequencies (i.e. it has a low frequency component over the duration of the recording, and a high frequency component over the duration of the recording, etc)? I though that it might have a variable frequency over time, but it nonetheless has only a single frequency value (at time t).

I was wondering if it is perhaps analogous to superposition of sound waves: sound waves from different sources can interact with constructive and destructive interference to produce a 'total' sound wave, the properties of which are determined by the contributing sound waves; properties being such things as amplitude, frequency, etc. I am guessing it is possible (to some extent) to look at a sound wave that is composed of sound waves from a number of sources, and determine the properties of the contributing waves. For an electric field, the frequency of oscillation in the potential at some point is then determined by local sources that influence the potential at that point; sources such as neurons. Their different contribution combine to produce the LFP variation over time, and the contributing sources can be determined by some process.

Is this in anyway accurate? I don't understand how a recording of a LFP at a given point can oscillate at different frequencies.

Any help much appreciated.

 

[Pythagorean]

Hello!
My understanding of the LFP is that it is a measure of the electric potential at a given point in space. It can oscillate over time, but it has a single value at any given point in time.If this is true, how can it be separated out into different simultaneous frequencies (i.e. it has a low frequency component over the duration of the recording, and a high frequency component over the duration of the recording, etc)? I though that it might have a variable frequency over time, but it nonetheless has only a single frequency value (at time t).

It has no frequency at time t. Frequencies always require a duration: a length of time $\Delta t$. Your lowest frequency available will be defined by that duration (you call the duration a period and divide 1 by it for your lowest frequency: $T = 1/f$). Your highest frequency is determined by your sample rate.

 

[nobahar]

It has no frequency at time t. Frequencies always require a duration: a length of time $\Delta t$. Your lowest frequency available will be defined by that duration (you call the duration a period and divide 1 by it for your lowest frequency: $T = 1/f$). Your highest frequency is determined by your sample rate.

Hi Py, thanks for the response.

I am fairly new to sinusoidal waves and so on (as you can tell), so thanks for the input.

I thought that the 'particles' that make up the wave do not actually move along the wave, but are simply displaced from an equilibrium position: so if the wave is 'travelling' in the x direction, then the individual points along x are displaced in the y-direction and oscillate about an equilibrium position, this displacement is then communicated - or propagated - along (this is the case for a transverse wave). The frequency is then a measure of how far the particle gets through a single cycle per unit time; a cycle being from some displacement, through to either + or - the maximum amplitude, through to the other extreme, and then back to the initial displacement. The displacement being 'up-and-down' at a point x - in the y-direction in this case.
This frequency, which is caused by the source, and the velocity, caused by the medium through which the wave travels, are what determine the wave length. I.e. the wavelength is the consequence of the frequency and the velocity.
Since the frequency is a measure of how far through one cycle 'up-and-down' a particle at a fixed point x gets, then I figured this could have an instantaneous value in a similar way to something like velocity: if $v(t) = t^2$ at $v(t_{1})$, an object might have a velocity of 5m/s, and at $v(t_{2})$ it might have a velocity of 7m/s. This is its instantaneous velocity, it doesn't actually go 5m/s or 7m/s, except perhaps instantaneously... It's just that if it did have a constant velocity of, say, 5m/s, then it would go 5 metres after 1 second if it was a constant velocity. It's not a fixed constant that needs to take place over a fixed [/tex]\Delta[/itex]t, the object can have an instantaneous velocity.
I figured the frequency could be the same, if it has a frequency of f(t) = t^2, then it might go f(2) = 4 cycles per second at time=2, and f(4) = 16 cycles per second at t=4. This means that if the source was to produce a wave of frequency f(2)=4 then it would complete 4 cycles per second if it was constant; but, like velocity in the above case, this is just its instantaneous frequency. I kind of imagine the particle being displaced at some speed up-and-down, and then this changes over time, so at a later time, it might displace up-and-down at a faster rate, which would be reflected in its frequency.
If the medium remains the same, then the wavelength would change, also.
Sorry if this makes no sense. I think I have confused myself...

Any more help appreciated.

 

[atyy]

The LFP (V) is a has a different value at each place and time, so it's V(x,t).

If we agree to measure it at a fixed place, then since x is fixed, then we can omit the x and write V(t).

If V(t)=sin(2*pi*f1*t) + cos(2*pi*f2*t), then we say that it is composed of two (temporal) frequencies f1 and f2.

In general V(t) can be decomposed into many frequencies via Fourier decomposition. Often it will be useful to chop it up into tiny time pieces and Fourier decompose each time piece separately - this is called the short time Fourier transform. It is also very common to use other decompositions such as one of the many wavelet transforms or the Hilbert transform. Which decomposition is most useful depends on where you are recording from, what the task being performed is, and what you want to use it for.

Anyway, the basic idea is Fourier decomposition - an arbitrary waveform can be written as a sum of sinusoids.

http://www.dsprelated.com/dspbooks/mdft/Fourier_Transforms_Continuous_Discrete_Time_Frequency.html

In contrast to consider an LFP in space and time V(x,t) then we require a 2D Fourier transform, and each sinusoidal component will be a wave V(t)=sin(kx-wt) with temporal frequncies w/(2*pi) and spatial frequencies (k/2*pi), but you don't have to worry about that yet, since you are considering the LFP at one point in space.

 

[atyy]

I was wondering if it is perhaps analogous to superposition of sound waves: sound waves from different sources can interact with constructive and destructive interference to produce a 'total' sound wave, the properties of which are determined by the contributing sound waves; properties being such things as amplitude, frequency, etc. I am guessing it is possible (to some extent) to look at a sound wave that is composed of sound waves from a number of sources, and determine the properties of the contributing waves.

Yes, this is called the Fourier transform.

 

[nobahar]

Thanks for the response atyy.

In general V(t) can be decomposed into many frequencies via Fourier decomposition. Often it will be useful to chop it up into tiny time pieces and Fourier decompose each time piece separately - this is called the short time Fourier transform.

This is because the LFP is non-stationary, right? And a non-stationary signal is one which has a frequency component that changes with time?

This was one of my points of confusion above - that the frequency can be described as a function of time, and it can have an instantaneous frequency. Having said that, 'chopping' up the time series for an LFP would imply that the frequency is constant for that time interval? Or is it treated as being approximately constant? Does that make sense?

Anyway, the basic idea is Fourier decomposition - an arbitrary waveform can be written as a sum of sinusoids.

I'm currently working through Fourier series, it'll probably take some time. I am having difficulty understanding why a FT can't be used for a non-stationary signal, when an FT can be applied to an arbitrary function that isn't periodic... Not sure if that makes sense either... My brain hurts!

 

[ManFrommars]

This is because the LFP is non-stationary, right? And a non-stationary signal is one which has a frequency component that changes with time?

This was one of my points of confusion above - that the frequency can be described as a function of time, and it can have an instantaneous frequency. Having said that, 'chopping' up the time series for an LFP would imply that the frequency is constant for that time interval? Or is it treated as being approximately constant? Does that make sense?

I'm currently working through Fourier series, it'll probably take some time. I am having difficulty understanding why a FT can't be used for a non-stationary signal, when an FT can be applied to an arbitrary function that isn't periodic... Not sure if that makes sense either... My brain hurts!

Sure. A Fourier transform can be used for a non-stationary signal, but you will lose any information about when the frequencies change, and just get an overall representation of all the frequencies over the whole recorded signal. So say in your LFP, for 5 seconds it has a dominant frequency of 15Hz, then for the next five seconds this frequency stops and a new frequency of 30Hz appears, then when you Fourier transform the whole signal, your Fourier spectrum will show peaks at both 15Hz and 30Hz (there isn't necessarily an implication that the signals are constant, just that the Fourier transform doesn't contain this information). If you instead split your signal up into shorter time chunks and take the Fourier transform of each of those, as suggested by atyy, in the first few chunks you will only see a peak at 15Hz and in the next few chunks you will only see a peak at 30Hz. You can plot this as a spectrogram. There are, naturally, tradeoffs to splitting the signal into shorter chunks: generally what you gain in temporal resolution, you lose in frequency resolution. So, if you are interested in when a frequency change occurs, but not so interested in exactly what that frequency is, you can use very short chunks, and vice versa.

Regarding your initial post about dividing the LFP into different frequency bands, this is fairly common as different frequency bands in the LFP represent different types of neural activity. For a variety of reasons, the general frequency content f of the LFP tends to go as 1/f^k, (with k a number usually between 1 and 3 or so). They contain a broad spectrum of frequencies, with low frequencies dominating. But your basic analogy with superposition of sound waves is spot on - instead of sound sources, the LFP is caused by current sources (and sinks) that are distributed within the brain tissue.