The neural code�Visualizing population codes

Kenneth D Harris, UCL

Exploratory vs. confirmatory analysis

• Exploratory analysis
• Helps you formulate a hypothesis
• End result is usually a nice-looking picture
• Any method is equally valid – because it just helps you think of a hypothesis

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• Confirmatory analysis
• Where you test your hypothesis
• Multiple ways to do it (Classical, Bayesian, Cross-validation)
• You have to stick to the rules

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• Everything we are doing today is exploratory

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Two types of raster plot

One cell�Rows: repeats

Allen ephys session 791319847: unit 180 (primary visual cortex)

One repeat�Rows: cells

Sorting cells by correlation with stimuli/behavior

ALWAYS use a validation set when sorting cells

If you don’t use a separate set of trials/times for sorting cells than what you display, you will see artifactual order

Sorting cells by correlation with each other

Sorted by first principal component

Sorting can reveal unexpected structure

Sorting cells by latency

Dimensionality reduction

• The firing rates of N neurons define a vector in N-dimensional space

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• Dimensionalilty reduction methods reduce that to 2- or 3-dimensional
• So you can visualize the results
• While keeping some structure from the original data

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• Remember these are exploratory analyses

Principal component analysis

Mathematics of PCA

PCA features usually reflect smoothness of data

Structure of correlation matrix depends on timescale

• At short timescales, all correlations are non-negative (at least in cortex)
• Reflects common fluctuations (up- and down-phases)
• What will the 1^st PC vector look like?

PCA: auditory cortex population vectors

Other methods for dimensionality reduction

• There are many other ways to do dimensionality reduction
• They are usually optimal by some criterion
• None of them are “correct” – they are all exploratory analyses

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• If dimensionality reduction suggests a result, you must test it with confirmatory analysis
• Usually this happens in the high-dimensional space, not the reduced one

Discriminant analysis

Discriminant analysis: auditory cortex

• Projections chosen to maximally separate sustained responses
• Looks completely different to PCA!

Many more methods…

jPCA: Churchland, Cunningham et al, Nature 2012

Mante, Sussillo et al, Nature 2013

Nonlinear dimensionality reduction

• PCA, discriminant analysis are linear methods
• Nonlinear methods are more flexible
• But harder to interpret

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• Examples:
• UMAP
• Isomap (->)
• Local linear embedding
• tSNE

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Summary

• When you have population data, visualizations help you understand it
• Use sorted rasters and dimensionality reduction to form your hypotheses
• But remember these are exploratory analyses
• And you need to test your hypotheses with confirmatory analysis