Population Codes in the Retina Michael Berry Department of Molecular Biology Princeton University.

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