Transcript שקופית 1 - Tel Aviv University

Correlations Without
Synchrony
Carlos D. Brody
Presented by:
Oded Ashkenazi
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Overview
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Neurological Background
Introduction
Notations
Latency, Excitability Covariograms
3 Rules of Thumb
Conclusion
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Neurological Background
The Human Brain:
A Complex
Organism
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Neurological Background
• 1011 neurons (x50 the number of people on
earth)
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10
• Each one is connected with
synaptic
connections
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• Total of 10 Synaptic Connections
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Neurological Background
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Neurological Background
input
CPU
output
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Neurological Background
Spike Trains – plots of the spikes of each neuron
as a function of time
Raster Plot – a plot of a few Spike Trains
simultaneously
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Neurological Background
• Histograms - a plot of the binned data as a
function of time.
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Neurological Background
• PSTH - peri-stimulus time histogram
is a Histogram of stimulated neurons lined
up by the stimulus marker. (marks the
beginning of the stimulus).
• The PSTHs give some measure of the
firing rate or firing probability of a neuron
as a function of time.
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Neurological Background
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Crosscorrelogram - is a function which
indicates the firing rate of one neuron
versus another.
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It's pretty simple to compute the
crosscorrelogram. The problem is how
to interpret it.
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Neurological Background
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Neurological Background
• The crosscorrelogram provides some indication of
the dependencies between the two neurons.
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Introduction
• Peaks in spike train correlograms are usually
taken as indicative of spike timing
synchronization between neurons.
• However, a peak merely indicates that the two
spike trains were not independent .
• Latency or excitability interactions between
neurons can create similar peaks.
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Introduction
• On each trial, most spikes in cell 1 have a
corresponding, closely timed spike in cell 2.
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Introduction
• The two spike trains were generated independently.
But the overall latency of the response varies
together over trials.
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Introduction
• The spikes for the two cells were generated
independently. but the total magnitude of the
response varies together over trials
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Notations
The spike trains of two cells will be represented by
two time-dependent functions, S1(t) and S2(t).
The cross-correlogram of each trial (r) is:
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Notations
• Let
represent averaging over trials r.
The PSTH of Si is:
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Notations
• When you stimulate the cells that you're
recording from, you increase their firing rates.
• If you do this simultaneously in both cells you've
introduced a relationship between the firing
probabilities of the cells.
• The Covariogram removes the peak in the
original correlogram that was due to costimulation of the cells.
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Notations
The covariogram of S1 and S2 is:
R
K
R - raw cross-correlogram
K - shuffle corrector (shift predictor)
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Notations
• The expected value of V is zero:
• Significant departures of V from zero indicate
that the two cells were not independent
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Notations
• Estimating the significance of departures
of V from 0 requires some assumptions:
– S1 is independent of S2.
– Different trials of S1 are independent of
each other.
– Different bins within each trial of S1 are
independent of each other.
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Notations
The variance of V is:
Where:
and
are the mean and variance
of
over r trials.
and
are the number of trials in the
experiment.
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Spike Timing Covariations.
• How Spike Timing covariations lead to a
peaked covariogram.
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Latency Covariations
• Lets consider the responses of two
Independent neurons.
• For each trial r, take the responses of
both neurons and shift both of their spike
trains, together by some amount of time tr
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Latency Covariations
How will it affect the covariogram ?
• The raw correlogram R will not be
affected.
R
K
• The shuffle corrector K will be affected
because the PSTHs are broadened by
the temporal jitter introduced by the shifts
tr.
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Latency Covariations
• The latency shifts will make K broader,
and therefore shallower, while having no
effect on R.
• The width and shape of the peak in V are
largely determined by the width and
shape of the peak in R.
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Latency Covariations
• How latency covariations lead to a
peaked covariogram.
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Excitability Covariations
Consider a cell whose response can be
characterized as the sum of a stimulusinduced response plus a background
firing rate.
• Z(t) is the typical stimulus-induced firing
rate.
• “gain” factors, and , represent possible
changes in the state of the cell.
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Excitability Covariations
• Suppose the 2 cells only interaction is
through their gain parameters.
• What is their covariogram ?
• Reminder:
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Excitability Covariations
• The shape of V will be the shape of the
corrector K:
• K has a width determined by the width of
peaks in the cell’s PSTHs
• The amplitude of V will be given by:
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Excitability Covariations
• An easily computable measure of
excitability covariations is the integral
(sum) of the covariogram:
• It is proportional to the covariation in the
mean firing rates of the two cells
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Excitability Covariations
• How Excitability covariations lead to a
peaked covariogram.
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Rules of Thumb
• There are three major points in
comparison to latency and excitability
covariations:
– Autocovariograms
– Covariogram shapes
– Covariogram integrals
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Autocovariograms
• Autocovariograms: This function lets
you discern the fine time structure, if any,
in the spike train of a single neuron
• Spike Timing autocovariograms are flat
and not at all similar to the crosscovariogram. Unlike the ones of
Excitability or Latency.
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Autocovariograms
Latency
Spike Timing
Excitability
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Covariogram shapes
• Spike timing covariogram shapes are
much more arbitrary than Latency or
Excitability covariogram shapes.
• Latency and Excitability shapes are tied
to the shapes of the PSTHs
• Spike timing shapes are not.
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covariogram integrals
• Large, positive covariogram integrals
imply the presence of an excitability
covariations component.
• In the spike timing case, the integral, if
positive, will often be small.
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conclusion
• We want to analyze neuron
synchronization by sampling only a small
number of trials.
• This is a special case of a more general
problem:
Taking the mean of a distribution as
representative of all the points of the
distribution.
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conclusion
• For this to work: std << mean
• This is common to gene networks, text
searches, network motifs.
• Investigators must interpret means
with care !
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The End
Questions ?
Thank you for listening.
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