Population Codes in the Retina

Michael BerryDepartment of Molecular

BiologyPrinceton University

Population Neural CodesMany ganglion cells look at each point in an image

• Experimental & Conceptual Challenges

• Key Concepts:

Correlation

Independence

Recording from all of the Ganglion Cells

• Ganglion cells labeled with rhodamine dextran

Segev et al., Nat. Neurosci. 2004

Spike Trains from Many Cells

Responding to Natural Movie Clips

14121086Time (s)

Cell J

Cell I

Cell H

Cell G

Cell F

Cell E

Cell D

Cell C

Cell B

Cell A

QuickTime™ and aNone decompressor

are needed to see this picture.

Correlations among Cells

30

20

10

0

Firing Rate of Cell B (spikes/s)

-0.2 -0.1 0.0 0.1 0.2Time Relative to Spike from Cell A (sec)

same trial shuffled trial baseline rate

30

20

10

0-20 -10 0 10 20

Time (msec)

Role of Correlations?

• Discretize spike train: t = 20 ms; ri = {0,1}

• Cross-correlation coefficient:CAB =

pAB 11( ) −pA 1( )pB 1( )pA 1( )pB 1( )

90% of values between

[-0.02 , 0.1]

Correlations are Strong in Larger Populations

N=10 cells:Excess synchrony byfactor of ~100,000!

Combinations of Spiking and Silence

Building Binary Spike Words Testing for Independence

P R( ) = p1 r1( )p2 r2( )L pN rN( ) ? R = r1,r2 ,K ,rN{ }

Errors up to ~1,000,000-fold!

Including All Pairwise CorrelationsBetween Cells

P (2) R( ) =1Z

exp hi rii∑ + J ij ri rj

ij∑

⎧⎨⎩⎪

⎫⎬⎭⎪

• general form:

• setting parameters:

• limits:

Jij =0 ⇒ P R( ) → p1 r1( )p2 r2( )L pN rN( )

Maximum entropy formalism: Schneidman et al. Phys. Rev.Lett. 2003

hi corresponding to ri

Jij corresponding to ri rj

Role of Pairwise Correlations

• P(2)(R) is an excellent approximation!

Schneidman et al., Nature 2006

Rigorous Test• Multi-information:

• Compare:

I R1,R2,K ,RN( ) = H Ri( )i∑ −H R1,R2 ,K ,RN( )

IN sampled vs. I2 assuming P R( ) =P (2 ) R( )

Groups of N=10 cells

Implications for Larger Networks

• Connection to the Ising model

• Model of phase transitions

• At large N, correlations can dominate network states

• Analog of “freezing”?

P(2) R⎛⎝⎞⎠ =1

Zexp hi ri

i∑ + J ij ri rj

ij∑

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

Extrapolating to Large N

• Critical population size ~ 200 neurons

• Redundancy range ~250 µm

• Correlated patch ~275 neurons

Error Correction in Large Networks

• Information that population conveys about 1 cell

CONCLUSIONS

• Weak pairwise correlations lead to

strong network correlations

• Can describe effect of all pairs on network

with the maximum entropy formalism

• Robust, error-correcting codes

Final Thoughts

• Everyday vision: very low error rates

“Seeing is believing”

• Problems: many cells, many objects, detection can occur anytime, anywhere

– assume 1 error / ganglion cell / year

– 106 ganglion cells => error every 2 seconds!

• Single neurons: noisy, ambiguous Perception: deterministic, certain

• Connection to large population, redundancy

Including Correlations in Decoder

• Use maximum entropy formalism:

• Simple circuit for log-likelihood:

• Problem: difficult to find {hi, Jij} for large populations

P(2) R⎛⎝⎞⎠ =1

Zexp hi ri

i∑ + J ij ri rj

ij∑

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

i

j

Readout NeuronhihjJij Voltage ~ lnPR()

Acknowledgments

• Recording All Cells • Natural Movies & Redundancy

Ronen Segev Jason Puchalla

• Pairwise Correlations • Population Decoding

Elad Schneidman Greg SchwartzBill Bialek Julien Dubuis

• Large N Limit

Rava da Silveira (ENS)Gasper Tkachik