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
