1 Correlations Without Synchrony Presented by: Oded Ashkenazi Carlos D. Brody.
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Transcript of 1 Correlations Without Synchrony Presented by: Oded Ashkenazi Carlos D. Brody.
2
Overview
• Neurological Background
• Introduction
• Notations
• Latency, Excitability Covariograms
• 3 Rules of Thumb
• Conclusion
4
Neurological Background
• neurons (x50 the number of people on earth)
• Each one is connected with synaptic connections
• Total of Synaptic Connections
1110
510
1610
8
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
• 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
• Crosscorrelogram - is a function which indicates the firing rate of one neuron versus another.
• It's pretty simple to compute the crosscorrelogram. The problem is how to interpret it.
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Neurological Background
• The crosscorrelogram provides some indication of the dependencies between the two neurons.
15
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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• On each trial, most spikes in cell 1 have a corresponding, closely timed spike in cell 2.
Introduction
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• The two spike trains were generated independently. But the overall latency of the response varies together over trials.
Introduction
18
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
• 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 co-stimulation of the cells.
22
Notations
R - raw cross-correlogram
K - shuffle corrector (shift predictor)
The covariogram of S1 and S2 is:
R K
24
Notations
• The expected value of V is zero:
• Significant departures of V from zero indicate that the two cells were not independent
25
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.
26
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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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.
• The shuffle corrector K will be affected because the PSTHs are broadened by the temporal jitter introduced by the shifts tr.
R K
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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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Excitability Covariations
Consider a cell whose response can be characterized as the sum of a stimulus-induced 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
37
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 cross-covariogram. Unlike the ones of Excitability or Latency.
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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.
43
conclusion
• For this to work: std << mean
• This is common to gene networks, text searches, network motifs.
• Investigators must interpret means with care !