MIDTERM 3

Question 1

What are 3 uses that Fourier transforms and power spectra have in neuroscience.

Answer 1

• Identifying biologically important signals in EEG or LFP (local field potential) recordings that may indicate e.g. brain state or sleep stage.
• Analysis of spike trains
• Filtering (often done before/after data acquisition) i.e. Narrow-band filter may be used to remove noise from signals

Question 2

What is the difference between the transform and the power spectrum?

Answer 2

• The transform is a linear operation that characterises data in terms of frequencies. It includes the real and imaginary components, and preserves all the information in the original signal.
• The power spectrum represents the frequencies present in the signal that is computed from the transform and doesn’t preserve the original signal.

Question 3

What does a power spectrum show, including the DC component, and how do we interpret the meaning of peaks at specific frequencies?

Answer 3

The power spectrum shows the strength of the frequency present from the signal. The peaks indicate a frequency that was found in the signal. The DC component shows the mean value of the signal summed over ALL the samples.

Question 4

How do both cosine and sine terms (= real and imaginary components) contribute to the calculation of power.

Answer 4

Power Formula: P(f) = A^2(f) + B^2(f)
Real part is cos term, imaginary part is sin term.
Plotted as a vector, the angle gives that phase of each frequeny component, which is thrown away if we take the power (length squared of the vector).

Question 5

How can filtering be done in the frequency domain?

Answer 5

Filtering can be done in the frequency domain after transforming the raw data and the weighting function, then multiplying them and then transforming back to get the result.

Question 6

What is the convolution theorem and give an example?

Answer 6

• Convolution theorem: multiplication in the fourier domain.
• Example: convolution with a Gaussian can be done by multiplying the signal’s transform with the transform of the Gaussian. The multiplication removes high frequencies and preserves lower ones depending on the width of the Gaussian.

Question 7

What are the 4 basic types of frequency domain filter and what are some situations they might be used in?

Answer 7

• Low-pass: High frequencies are removed, preserve low ones. Can be used to detect local field potential.
• High-pass: Low frequencies are removed, preserve high ones. Can be used to detect spikes.
• Band-pass: remove BOTH low and high frequencies from the signal, preserving a range in the middle. Can make signals easier to detect if they’re confined to specific frequency ranges.
• Notch filtering: Will remove a narrow range of frequencies from the signal, often 60 Hz due to electrical interference (removes noise).

Question 8

What is the sampling theorem and the Nyquist limit?

Answer 8

Sampling theorem: Must sample a signal at a frequency at least twice as high as the highest frequency in the signal.
- Nyquist limit: value that is two times the highest frequency of the signal.

Question 9

What is aliasing and what leads to it?

Answer 9

• Aliasing is when artefacts are created by sampling at a too low of a frequency.
• If there are frequencies in the signal that are higher than ½ the sampling frequency, you will see difference frequencies that are not really there.

Question 10

What are 4 ways to avoid aliasing?

Answer 10

• Ignore it on biological grounds
• Sample at a rate much higher than you probably need
• Analogue filter the signal before analogue to digital conversion to remove frequencies above 0.5 times the sampling frequency.
• If the sampled spectrum has little or no energy close to the Nyquist limit, deduce that the original signal probably did not either

Question 11

How do simple forms of signal smoothing, Gaussian or 1:2:1, can be done in the signal domain (i.e. time or space) and how do the terms convolution and kernel relate to this process?

Answer 11

Signal smoothing takes the weighted sum of the signal values, with the weights given by the weighting/smoothing function. The signal is replaced by the summation, as it goes from point to point.
- Convolution relates to the process, because as the weighting function goes through the data, the data and the weighting function convolve, computing a weighted sum as it passes. The weighting function is also sometimes termed the kernel, because it is small and remains the same.

Question 12

What are the 5 steps for Fourier domain filtering?

Answer 12

1. Get the FFT of the signal
2. Multiply by the desired filter function, including +ve and -ve frequencies
3. Transform in the reverse direction
4. The result is the filtered signal
5. When finished you could check the power spectrum of the filtered signal - the filtered parts of the signal would no longer be there.

Question 13

Describe in words the basic receptive field properties of centre-surround, ON and OFF retinal ganglion cell types.

Answer 13

• Receptive field: region of space in which a visual stimulus can change the firing rate of a cell.
• Best at detecting changes in space and time. Meaning they do not respond well to uniform illumination and constant stimulus exposure.

Question 14

What is lateral inhibition and what gives rise to it in the retina?

Answer 14

Lateral inhibition is what arises from the different spatial distribution of excitatory and inhibitory inputs to the cell. The different spatial distributions can affect the retinal ganglion cell’s firing pattern.

Question 15

What is Hermann’s grid and how might you explain it?

Answer 15

• Hermann’s grid involves black squares that are organised in a grid like fashion, with white gaps evenly separating the squares throughout.
• It presents an optical illusion of black dots at the intersection of white gaps in the periphery of vision, which then disappears when you focus on the intersection.
• This illusion occurs due to lateral inhibtion, where the RFs at the intersections are receiving more inhibition than RFs elsewhere, and thus those neurons fire less.

Question 16

What mathematical function has been used to model the behaviour of retinal ganglion cells? What part of it relates to lateral inhibition?

Answer 16

See slides for equation.
The part relating to lateral inhibition is Gi, which represents the strength and size of the surrounding inhibition.

Question 17

What is an LNP model? What are the 3 stages?

Answer 17

LNP stands for Linear-Nonlinear-Poisson. It is a useful class of model that can applied to modelling visual RFs.

1. Make a linearly weighted sum of the RF (DOG function) and the image. Meaning that every point in the RF is multiplied with the corresponding image intensity at that point, and then added together.
2. Nonlinear: apply a threshold to the output in stage 1.
3. Uses the output of stage 2, which generates spikes with a probability that gives rise to randomly occurring spikes, similar to the behaviour of real neurons.

Question 18

Explain how the Fourier transform of the receptive field predicts the responses of retinal ganglion cells to different spatial frequencies.

Answer 18

• Fourier transform of the DOG shows attenuated response to low and high spatial frequencies.
• This predicts the responses of retinal ganglion cells, because these cells are more sensitive to spatial frequency gratings that have frequencies that are not very high, or very low.

Question 19

Describe Olshausen and Field’s sparse coding model for visual cortex.

Answer 19

• Sparse coding involves activating a minimal subset of neurons from a large population to represent parts of an image.
• Neurons have random receptive fields that are matched to an image patch
• Neuronal activity becomes sparse through kurtosis, so that a minimal number of them are activated for the best representation of the image patch.

Question 20

What is the evidence that some neural structures have no redundancy?

Answer 20

One piece of evidence is that dendritic arborisations of retinal ganglion cells tile the surface of the retina with little overlap, meaning that if one of the cells die, there will be a hole left in the retina.

Question 21

What is Bayes’ Rule (write the formula)?

Answer 21

P(A|B) = P(A) * ((P(B|A)) / P(B))
The posterior is equal to the prior multiplied by the division of the likelihood and marginal.

Question 22

Give 2 examples of the role played by Bayes’ rule in perception.

Answer 22

• The blue and black dress: Where some people perceived the dress as white and gold, vs. blue and black. This is due to people’s prior of the lighting being natural, which then affects their actual perceptions of the dress.
• Change blindness: If your attention is elsewhere, you don’t see things, in spite of the impression that you see everything. Your attention is the prior that is combined with the sensory evidence to work out the most probablye cause.

Question 23

Distinguish between labelled line and population codes.

Answer 23

• Labelled line is the idea that perception is linked to specific neurons/pathways, where unusual stimulation of a specific pathway would lead to the perception associated with the pathway.
• Population codes is the idea that information is distributed among a population of neurons.

Question 24

How might rate codes be used in the brain? To what extent is it a) feasible and b) supported by evidence?

Answer 24

• Involves the number of spikes that occur during a period of time. Can determine what information is being coded
• Well supported by evidence and is found to be used by many brain structures
• However, an experiment on visual perception shows that processing in the visual system may be done in a faster way, potentially related to timing.

Question 25

How might timing codes be used in the brain? To what extent is it a) feasible and b) supported by evidence?

Answer 25

• Cortical neurons have been found to fire in a precise manner
• However it is unclear which brain structures use this method.
• Sound localization has been found to utilise this coding however, as sound can hit the ears at different times because of the distance between the two ears.

Question 26

How might oscillatory codes be used in the brain? To what extent is it a) feasible and b) supported by evidence?

Answer 26

• Arguably a basic neural function that has been discovered in the cerebral cortex using EEG.
• Oscillations in the brain range between 1-100 Hz.

Question 27

How might phase codes be used in the brain? To what extent is it a) feasible and b) supported by evidence?

Answer 27

Has been found in the hippocampus to encode an animal’s location in space by having neurons vary the phase of their firing relative to the theta rhythm of the local field potential depending on their position in space.

Question 28

Describe 3 situations in neuroscience in which PCA can be useful.

Answer 28

PCA can be useful for analysing data with multiple dimensions and determine which dimensions are important.
- Can be used to analyse behavioural tests and determine what are common factors affecting responses to questions or if subjects can be classified into groups.
- Can be used to sort spike train data
- Can be used to simplify neuronal response data

Question 29

What is PCA and how can it help in data analysis?

Answer 29

• Simplifies data through dimension reduction. It takes a data matrix and then provides another matrix that can be used to transform the data values.
• Can help with data analysis by demostrating the existence of clusters in the data and by also determining what factors are affecting the data.

Question 30

What is covariance, and what is a covariance matrix?

Answer 30

• Covariance is similar to correlation, but it is not limited to values between -1 to +1.
• A covariance matrix arises from taking the covariance between every pair of columns in the raw data matrix, including a column with itself. The covariance matrix is used to determine the eigenvalues and eigenvectors

Question 31

What are eigenvalues and eigenvectors?

Answer 31

• Eigenvectors represent the dimensions of the raw data.
• Eigenvalues represent how important an eigenvector is, and indicates the amount of variance. Each eigenvector has its own eigenvalue.

Question 32

How are eigenvalues and eigenvectors used?

Answer 32

• Eigenvalues are used to determine how important an eigenvector is and allows eigenvectors to be ordered based on their highest to lowest eigenvalue.
• Eigenvectors are used to transform the data by taking the dot product of each data vector with a given eigenvector.

Question 33

What issues govern the choice of which eigenvectors and eigenvalues to use and which to ignore?

Answer 33

If the eigenvalues decline rapidly, then that would mean that the data can be transformed using just the first few eigenvectors, since most of the variance is captured by the first few transformed values per point. The rest of the eigenvectors can be discarded.
- However if your eigenvalues are slow, potentially your data doesn’t have much meaning.

Question 34

What are the 7 steps of PCA?

Answer 34

1. Construct the data matrix from your raw data
2. Calculate the covariance matrix
3. Determine the eigenvectors and eigenvalues from the covariance matrix
4. If needed, order the eigenvectors based on their eigenvalues in decreasing magnitude
5. Examine the eigenvalue distribution and decide how many you need
6. Transform each data vector (sample value) by multiplying (dot product) with the relevant eigenvectors
7. If looking for clusters, a good strategy is to plot the first two transformed data values and inspect visually

Question 35

What is 1:2:1 smoothing?

Answer 35

• Each data point is replaced by 0.5 of its value plus 0.25 of the value of each of the neighbouring points
• This is a weighted sum (0.25 + 0.25 +0.5 = 1). As a result the average value of the data points remains the same.

Question 36

What is Gaussian smoothing?

Answer 36

• Each data point is replaced by the weighted sum of points that is determined by the width of the Gaussian.
• The weights are given by the Gaussian.

Question 37

What is one strategy for reducing energy use by the retina?

Answer 37

Signals from the cones are broken down into ON-cell and OFF-cell outputs in the retinal ganglion cells to reduce the total number of spikes initially used, and thus reduce energy use in the retina.