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Correlations (part I). Mechanistic/biophysical plane: - What is the impact of correlations on the output rate, CV, ... - PowerPoint PPT Presentation
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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:
Quantifying spike correlations
)()( 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:
Quantifying spike correlations
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.)
“mother” train (rate R)
deletion rate = p * R
rate = q * R
How to generate artificial input correlations?3. Thinning & jittering a “mother” train
“mother” train (rate R)
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.
The development of correlations: a minimal model.
2. Afferent correlations
The development of correlations: a minimal model.
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.
Development of correlations: correlated inputs.
Development of correlations: correlated inputs.
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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