Goals

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Goals

• Looking at data
• Signal and noise
• Structure of signals
• Stationarity (and not)
• Gaussianity (and not)
• Poisson (and not)
• Data analysis as variability decomposition
• Frequency analysis as variance decomposition
• Linear models as variability explanation
• Information-theoretic methods for variability
  decomposition

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Signal and noise

• Data analysis is about extracting the signal that
  is in the noise
• Everything else is details
• But
• What is the signal?
• What is the noise?
• How to separate between them?

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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Example Looking at data
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Signal and noise

• The signal and the noise depend on the
  experimental question
• For sensory experiments, the signal is the
  sensory-driven response and everything else is
  noise
• For experiments about the magnitude of channel
  noise in auditory cortex neurons, sensory
  responses to environmental sounds are noise and
  the noise is the signal

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Signal and noise

• Therefore, we have to know what our signal is
  composed of
• The signal will have a number of sources of
  variability
• The experiment is about some of these sources of
  variability, which are then the signal, while the
  others are the noise

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Sources of variability

• Neurobiological
• Channel noise
• Spontaneous EPSPs and IPSPs
• Other subthreshold voltage fluctuations
• Intrinsic oscillations
• Up and down states
• Sensory-driven currents and membrane potential
  changes
• Spikes

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Sources of variability

• Non-neurobiological
• Tip potentials, junction potentials
• Breathing, heart-rate and other motion artifacts
• 50 Hz interference
• Noise in the electrical measurement equipment
• In the experimental part of the course, you will
  learn how to minimize these.

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Sources of variability

• How to separate sources of variability?
• By measuring them directly
• By special properties
• Rates of fluctuations (smoothing and filtering)
• Shape (spike clipping)
• By their timing
• Event-related analysis

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Direct measurement of sources of variability
Fee, Neuron 2000
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Direct measurement of sources of variability

• Respiration and heart rate (for active
  stabilization of the electrode)
• 50 Hz interference (for removing it using
  event-related analysis see later)
• Identifying the neuron you are recording from, or
  at least approximately knowing the layer
• Video recording of whisker movement during
  recording from the barrel field in awake rats

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Properties of signals

• Rates of fluctuations
• Molecular conformation changes (channel opening
  and closing) (may be lt1ms)
• Membrane potential time constants (1-30 ms)
• Stimulation rates (0.1-10 s)

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Nelken et al. J. Neurophysiol. 2004
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Rates of fluctuations 1 second(1 Hz)
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Rates of fluctuations 100 ms(10 Hz)
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Rates of fluctuations 10 ms(100 Hz)
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Rates of fluctuations 1 ms(1 kHz)
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Rates of fluctuations 0.1 ms(10 kHz)
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Rates of fluctuations

• If you know what are the relevant rates of
  fluctuations, you can get rid of other (faster
  and slower) fluctuations
• When extracting slow rates of fluctuations while
  removing the faster ones, this is called
  smoothing
• More generally, we can extract any range of
  fluctuations (within reasonable constraints) by
  (linear) filtering

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Example smoothing
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Example smoothing
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Example smoothing
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Example smoothing
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Example smoothing
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Example smoothing
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Example smoothing
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Go To Matlab
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Introduction to statistical tests

• We compare two processes with the same means (0),
  but with obviously different variances
• We know that the variance ratio is about 4
• Is 4 large or small?

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Introduction to statistical tests

• 4 is larger than 1 if the noise around 1 is such
  that 4 is unlikely to happen by chance

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Introduction to statistical tests

• In order to say whether 4 is large or small, we
  would really like to compare it with the value we
  will get if we repeat the experiment under the
  same conditions
• Formally, we think of the voltage traces we
  compare as the result of a sampling experiment
  from a large set of potential voltage traces
• When possible, we would like to have a lot of
  examples of these potential voltage traces

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Introduction to statistical tests
Select many pairs from up states and compute
ratios (should be close to 1)
Select many pairs from up and down states and
compute ratios (should be close to 4)
Filter and compute variance
Select many pairs from down states and compute
ratios (should be close to 1)
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Introduction to statistical tests

• When we have a lot of data, this is an
  appropriate approach
• Although note that it pushes the problem one step
  further (how do we know that the 1ish are indeed
  smaller than the 4ish)?
• But often we have only one trace, and we want to
  say something about it
• Need further assumptions!

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Introduction to statistical tests

• Since we have only one trace, we need to invent
  the set from which it came
• We tend to use relatively simple assumptions
  about this set, which are usually wrong but not
  too wrong

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Introduction to statistical tests

• Here we make the following two assumptions
• The two processes are stationary gaussian
• The two processes have identical means
• What does that mean?
• We will see later the details
• We can select an independent subset of samples
  from each process

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Go to Matlab
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Introduction to statistical tests

• Why choosing independent samples is important?
• Many years ago, people showed that ratios of sum
  of squares of independent, zero-mean gaussian
  variables with the same variance have a specific
  distribution, called the F distribution
• So we actually know what the expected
  distribution is if the variances are the same

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Introduction to statistical tests

• The distribution depends on how many samples are
  added together (obviously). These are called
  degrees of freedom, and there are two of them
  one for the numerator and one for the denominator
• Here both numbers are 51

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Introduction to statistical tests

• So our question got transformed to the following
  one
• We got a variance ratio of 5.6, how surprising is
  it when we assume that the variance ratios have
  an F(51,51) distribution?
• In order to do that, lets look at the F
  distribution

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Go to Matlab
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Introduction to statistical tests

• To recapitulate
• We got data
• One trace from an up state, one trace from a down
  state
• We made assumptions about how many repeats should
  look like
• Gaussian stationary with zero mean
• We generated a test for which we know the answer
  under the assumptions
• F test (variance ratio)
• We go to the theoretical distribution and ask
  whether our result is extreme
• Yes!

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Introduction to statistical tests

• A test is as good as its assumptions
• Are our assumptions good?
• How bad are our departures from the assumptions?

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Introduction to statistical tests

• Stationarity means that
• means do not depend on where they are measured
• Variances and covariances do not depend on where
  they are measured
• Gaussian processes are processes such that
• Samples are gaussian
• Pairs of samples are jointly gaussian (and when
  stationary, the distribution depends only on the
  time interval between them)

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When the data is non-stationary?
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Event-related analysis of data

• We select events that serve as renewal points
• Renewal points are points in time where the
  statistical structure is restarted, in the sense
  that everything depends only on the time after
  the last renewal point

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Event-related analysis of data

• Examples of possible renewal points
• Stimulation times
• Spike times (when you believe that everything
  depends only on the time since the last spike)
• Spike times in another neuron (when you believe
  that )

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Event-related analysis of data

• Less obvious renewal points
• Reverse correlation analysis
• The random process for which we look for renewal
  points is now the stimulus
• The renewal points are spike times

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Event-related analysis of data

• Assume that we have renewal points in the
  membrane potential data
• This means that we believe that segments of
  membrane potential traces that start at the
  renewal points are samples from one and the same
  distribution
• We want to characterize that distribution

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Event-related analysis of data

• We usually characterize the mean of the
  distribution
• This is called event-triggered averaging
• When your event is a spike, the result is
  spike-triggered averaging
• If the spike is from the same neuron, the result
  is a kind of autocorrelation
• If the spike is from a different neuron, the
  result is cross-correlation
• When your event is a stimulus, the result is the
  PSTH (or PSTA)

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Go to Matlab
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Event-related analysis of data

• We saw that using the mean does not necessarily
  capture wells the statistics of the ensemble
• Nevertheless, mean is always the first choice
  because in many respects it is the simplest
• Variance and correlations are also used for
  event-related analysis, but this gets us beyond
  this elementary treatment

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When the data is not gaussian?
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Morphological processing

• Identifying signal events by their shape
• Usually based on case-specific methods
• Very little general theory behind it
• Closely linked to clustering

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Morphological processing

• When the shapes are highly repetitive and very
  different from the noise, we can use matched
  filters
• A matched filter is a filter whose shape is
  precisely that of the shape to be detected

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Go to Matlab
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Morphological processing

• When the shapes are not necessarily highly
  repetitive but are still very different from the
  noise, we can use a generalization of matched
  filters
• Principal components are basic shapes whose
  combinations (with different weights) fit our
  shapes, but should poorly fit the noise
• But this takes out beyond this level

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Morphological processing

• Some spike sorting is done using principal
  components or matched filters
• Some EPSP identification is done using principal
  components
• Like all data processing techniques, this is a
  GIGO process and should be checked very carefully

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Morphological processing

• Eventually, much of morphological processing is
  about deciding about classes
• You get a single number and you want to say
  whether it is large or small
• Some standard statistics can be used here, but
  mostly treatment is data-analytic