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Correlations Without Synchrony

Presented by: Oded Ashkenazi

Carlos D. Brody

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Overview

• 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

• neurons (x50 the number of people on earth)

• Each one is connected with synaptic connections

• Total of Synaptic Connections

1110

510

1610

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Neurological Background

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Neurological Background

CPU

inputoutput

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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

• 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

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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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• 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

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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 co-stimulation of the cells.

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Notations

R - raw cross-correlogram

K - shuffle corrector (shift predictor)

The covariogram of S1 and S2 is:

R K

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A B

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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.

• 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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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 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

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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 cross-covariogram. Unlike the ones of Excitability or Latency.

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Autocovariograms

Latency

Excitability

Spike Timing

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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.