Correlations (part I)

Correlations (part I)

1. Mechanistic/biophysical plane:- What is the impact of correlations on the output rate, CV, ...

Bernarder et al ‘94, Murphy & Fetz ‘94, Shadlen & Newsome ’98, Stevens & Zador ’98, Burkit & Clark ’99, Feng & Brown ‘00, Salinas & Sejnowski ’00, Rudolf & Destexhe ’01, Moreno et al

’02, Fellows et al ‘02 , Kuhn et al ’03, de la Rocha ’05

- How are correlations generated? Timescale? Shadlen & Newsome ’98, Brody ’99, Svirkis & Rinzel ’00, Tiensinga et al ’04, Moreno & Parga,

2. Systems level:- do correlations participate in the encoding of information?Shadlen & Newsome ’98, Singer & Gray ’95, Dan et al ’98, Panzeri et al ’00, DeCharms & Merzenich ’95, Meister et al, Neimberg & Latham ‘00

- linked to behavior?

Vaadia et al, ’95, Fries et al ’01, Steinmetz et al ’00, …

Type of correlations:

1.- Noise correlations:

stimulus s response: r1, r

2, ... , r

n

2.- Signal correlations:

Quantifying spike correlations

)()( , k

rik

ri tttS

itr

ri tS

,)(

jitr

rj

riijij tStSttCttC

,)'()()'()',(

1. Stationary case:

t1i,r t2

i,r t3i,r ...

trial index

cell index

)()(),( YYXXYXCov

• Corrected cross-correlogram:


)()( ttS ir

ri

)'()()'()()',( tttStSttC jir

rj

riij

2. Non-stationary case:

tijij ttCC ),()(

tjitr

rj

ri tttStS )()()()(

,

• Joint Peristimulus time-histogram:

• Time averaged cross-correlogram:


dAdA

dC

jjii

ij

ij

)()(

)(

• Correlation coefficient:YX

XY

YXCov

),(

T

o

ri

ri dttSTN )()(

)()(

))(),((

TNTN

jiij

ji

TNTNCov

• Spike count:

if T >>

How to generate artificial correlations?1. Thinning a “mother” train.

“mother” train (rate R)

deletion

rate = p * R

How to generate artificial input correlations? 2. Thinning a “mother” train (with different probs.)


deletion rate = p * R

rate = q * R

How to generate artificial input correlations?3. Thinning & jittering a “mother” train


deletion + jitter

random delay: exp(-t/c) /

c

rate = p * R

How to generate artificial input correlations?4. Adding a “common” train

“common” train (rate R)

summationrate = r + R

How to generate artificial input correlations?: 5. Gaussian input

1

Diffusion approximation

)(),( tDt

White input:

1

2

)(),( tDt

)(),( tDt

)(t

How to generate artificial input correlations?: 5. Gaussian input

Impact of correlations

Impact of correlations on the input current of a single cell:

1

2

NE

…

1

2

NI

…EE

II

EI

ExcitationRate = E

,

Poi

sson

inpu

ts; Z

ero-

lag

sync

hron

y

InhibitionRate = I

IIIEEE JNJN

EEEEEE NJN )1(122

IIIIII NJN )1(12

EIIEIEIE JJNN 2

Impact of correlations on the output rate:

Balanced state

Unbalanced state

The development of correlations: a minimal model.

1. Morphological common inputs.


2. Afferent correlations


3. Connectivity

Development of correlations: common inputs

Shadlen & Newsome ‘98

p

p

p

p

R(1-x)

R(1-x)

Rx

Rp(1-x)

Rp(1-x)

Rp2x+Rpx(1-p)

• Independent: Rp(1-x)+Rpx(1-p)= Rp(1-xp)= Reff (1-xeff)

• Common: Rp2x = Reff xeff

where we have defined:

•Effective rate: Reff = R p

•Effective overlap: xeff = x p

Development of correlations: common inputs & synaptic failures

Development of correlations: common inputs & synaptic failures

Development of correlations: common inputs.Experimental data.

Development of correlations: correlated inputs.




Biophysical constraints of how fast neurons can synchronize their spiking activity

Rubén Moreno Bote

Nestor Parga, Jaime de la Rocha and Hide Cateau

Outline

I. Introduction.

II. Rapid responses to changes in the input variance.

III. How fast correlations can be transmitted? Biophysical constrains.

Goals:

1. To show that simple neuron models predict responses of real neurons.

2. To stress the fact that “qualitative”, non-trivial predictions can be made using mathematical models without solving them.

I. Introduction. Temporal changes in correlation

Vaadia el al, 1995 deCharms and Merzenich, 1996

I. Temporal modulations of the input.

mean

variance

Diffusion approximation:

white noise process withmean zero and unit variance

I. Temporal modulations of the input.

commonvariance

I. Problems

Problem 1: How fast a change in and can be transmitted?

Problem 2: How fast two neurons can synchronize each other?

Leaky Integrate-and-Fire (LIF) neuron

common source of noise

II. Probability density function and the FPE

Description of the density P(V,t) with the FPE

V

P(threshold)=0P(V)

Firing rate:

II. Non-stationary response. Fast responses predicted by the FPE

V

P(threshold, t)=0P(V,t)

1. Firing rate response

time

Mean input current

2. Firing rate response

time

Variance of the current

Described by the equation

II. Rapid response to instantaneous changes of (validity for more general inputs)

Silberberg et al, 2004

Moreno et al, PRL, 2002.

Change in the mean

Changein the inputcorrelations for dif.correlation statistics

Changein inputvariance

II. Stationary rate as a function of c

>0

<0

=0

Moreno et al, PRL, 2002.

III. Correlation between a pair of neurons

0time lag = t1 - t2

Cross-correlation function = P( t1 , t2 )

Moreno-Bote and Parga, submitted

III. Transient synchronization responses. Exact expression for the cross-corr.

The join probability density of having spikes at t1 and t2 for IF neurons 1 and 2 receiving independent and common sources of white noise is:

Total input variances at indicated times for neurons 1 and 2.

Joint probability densityof the potentials of neurons1 and 2 at indicated times

III. Predictions

1. Increasing any input variance produces an “instantaneous” increase of the synchronization in the firing of the two neurons.

2. If the common variance increases and the independent variances decrease in such a way that the total variance remains constant, the neurons slowly synchronize.

3. If the independent variances increase, there is a sudden increase in the synchronization, and then it reduces to a lower level.

… biophysical constraints that any neuron should obey!!

III. Slow synchronization when the total variances does not change.

III. Fast synchronization to an increase of common variance.

III. Fast synchronization to an increase of independent variance, and its slow reduction.

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Correlations (part I)