Higher-Order Correlations in Large Neuronal Populations Stefan - PowerPoint PPT Presentation

Higher-Order Correlations in Large Neuronal Populations Stefan Rotter Computational Neuroscience Lab Bernstein Center Freiburg & Faculty of Biology University of Freiburg, Germany Japan-Germany Joint Workshop on Computational Neuroscience Okinawa 2–5 March 2011 • •

What are “higher-order correlations” (HOCs)? The effect of HOCs on single-neuron dynamics CuBIC: Cumulant-based inference of HOCs Non-stationary spike trains

What are “higher-order correlations” (HOCs)? The effect of HOCs on single-neuron dynamics CuBIC: Cumulant-based inference of HOCs Non-stationary spike trains

Co-activated neuronal groups → Neuron ID − Time − → Question: Are there any neuronal groups that systematically fire together?

Co-activated neuronal groups → Neuron ID − Time − → Question: Are there any neuronal groups that systematically fire together?

Co-activated neuronal groups → Neuron ID − Time − → Question: Are there any neuronal groups that systematically fire together?

Higher-order correlations? spike train = spiking of one particular neuron = activity of patterns this neuron is a member of pairwise correlation = joint spiking of two neurons = activity of patterns that comprise both neurons triplet correlation = joint spiking in three neurons = activity of patterns that comprise all three neurons and so on.

Additive common components A B D E s 3 s 3 y 12 20ms 20ms y 13 s 2 s 2 y 13 y 23 s 1 s 1 F C s 1 ( t ) s 1 ( t ) s 2 ( t ) s 2 ( t ) s 3 ( t ) s 3 ( t ) Staude, Grün, Rotter, 2010

Correlated Poisson processes A y 1 ( t ) y 2 ( t ) y 3 ( t ) y 12 ( t ) y 13 ( t ) y 23 ( t ) y 123 ( t ) s 1 ( t ) s 2 ( t ) s 3 ( t ) B 0 0 1 1 1 0 0 0 1 0 1 1 1 0 σ j 1 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 a j 2 1 1 2 2 1 3 1 1 2 1 1 1 3 m ( t ) s 1 ( t ) s 2 ( t ) s 3 ( t )

Correlated Poisson processes A B 1 1 f A ( ξ ) f A ( ξ ) 0 0 1 10 1 10 ξ ξ fully independent fully correlated

Correlated Poisson processes C D 1 1 f A ( ξ ) f A ( ξ ) 0 0 1 10 1 10 ξ ξ E F 1 1 f A ( ξ ) f A ( ξ ) 0 0 1 10 1 10 ξ ξ pairwise correlation coefficient c = 0 . 4

Problems and issues of HOC inference Combinatorial explosion: A population of N neurons could in principle form 2 N different groups/assemblies: N 10 100 273 2 N 10 3 10 30 1 . 5 × 10 82 Very large samples: Statistical estimation of very many parameters calls for very, very long stationary recordings. Spike sorting: Single-units must be reliably isolated from extracellular spike train recordings. Uneasy mathematics: There seems to be no “natural” parametrization of HOCs, and the results of modeling and data analysis are strongly model-dependent.

Are HOCs relevant at all? Argument 1: Natural stimulation of sensor arrays (e.g. in vision or touch) carries gestalt information, i.e. HOCs. Argument 2: Certain brain theories imply and/or make use of HOCs (Hebb’s neuronal assemblies, Abeles’ synfire chains). Argument 3: Nonlinear synaptic input integration (e.g. via thresholding) turns neurons into sensible HOC detectors.

What are “higher-order correlations” (HOCs)? The effect of HOCs on single-neuron dynamics CuBIC: Cumulant-based inference of HOCs Non-stationary spike trains

Comparing two different input ensembles A B ◮ identical rates and Single Interaction Process Multiple Interaction Process w u w g pairwise correlations ◮ different higher-order x 1 x 1 x 2 x 2 correlations x N x N time time C D SIP MIP 1 1 10 10 spike train index 20 20 30 30 40 40 sum (s −1 ) 10 4 10 4 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 time (s) time (s) Kuhn, Aertsen, Rotter, 2002/2003

Comparing two different input ensembles A B ◮ identical rates and Single Interaction Process Multiple Interaction Process w u w g pairwise correlations ◮ different higher-order x 1 x 1 x 2 x 2 correlations x N x N time time SIP C D SIP MIP 1 1 single rate r = α + β 10 10 correlation c = β/ ( α + β ) spike train index 20 20 cluster rate β = r c 30 30 MIP 40 40 single rate r = α β sum (s −1 ) 10 4 10 4 correlation c = β 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 time (s) time (s) cluster rate α = r/c Kuhn, Aertsen, Rotter, 2002/2003

Higher-order correlations do matter! C D SIP, c = 0.4 MIP, c = 0.4 −50 −50 U (mV) U (mV) −60 −60 −70 −70 100 100 input input 50 50 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 time (s) time (s) correlation coefficient correlation coefficient SIP MIP 140 140 output rate (s −1 ) 120 output rate (s −1 ) 120 100 100 80 80 90 s −1 90 s −1 60 60 50 s −1 50 s −1 40 40 20 s −1 20 s −1 20 20 10 s −1 10 s −1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 correlation coefficient correlation coefficient Kuhn, Aertsen, Rotter, 2002/2003

What are “higher-order correlations” (HOCs)? The effect of HOCs on single-neuron dynamics CuBIC: Cumulant-based inference of HOCs Non-stationary spike trains

The compound Poisson process (CPP) � z ( t ) = δ ( t − t l ) l Carrier rate ν

The compound Poisson process (CPP) 1 1 1 1 1 1 6 1 1 1 1 1 11 1 1 1 1 1 6 1 1 1 � Amplitude Distribution z ( t ) = δ ( t − t l ) · a l l f Probability A Carrier rate ν 1 2 3 4 5 6 Amplitude A

The compound Poisson process (CPP) 1 1 1 1 1 1 6 1 1 1 1 1 11 1 1 1 1 1 6 1 1 1 � Amplitude Distribution z ( t ) z ( t ) = δ ( t − t l ) · a l x1 ( t ) l f Probability A x2 ( t ) Carrier rate ν 1 2 3 4 5 6 . Amplitude A . . xN ( t ) ◮ Spike trains are continuous-time Poisson processes. ◮ “Injecting” simultaneous spikes into ξ spike trains yields correlations of all orders up to ξ . ◮ carrier rate ν → neuronal firing rates amplitude distribution f A → correlation structure.

The compound Poisson process (CPP) 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 � Amplitude Distribution z ( t ) z ( t ) = δ ( t − t l ) · a l x1 ( t ) l f Probability A x2 ( t ) Carrier rate ν 1 2 3 4 5 6 . Amplitude A . . xN ( t ) ◮ Spike trains are continuous-time Poisson processes. ◮ “Injecting” simultaneous spikes into ξ spike trains yields correlations of all orders up to ξ . ◮ carrier rate ν → neuronal firing rates amplitude distribution f A → correlation structure.

The compound Poisson process (CPP) x1 ( t ) x2 ( t ) . . . xN ( t )

The compound Poisson process (CPP) Amplitude Distribution x1 ( t ) f Probability A x2 ( t ) 1 2 3 4 5 6 . Amplitude A . . ? xN ( t ) Task: Infer amplitude distribution f A from measured spike trains

HOCs in population spike trains Instead of tackling the difficult problem Which exact neuronal groups carry HOCs? try to answer a less specific question Are there any HOCs in a given population of neurons?

Measurement x1 ( t ) x2 ( t ) . . . xN ( t ) { h ◮ Segment spike trains into bins of width h , the resulting counting variables X i for each neuron are not binary.

Measurement x1 ( t ) x2 ( t ) . . . xN ( t ) { h Z ( s ) Time ◮ Segment spike trains into bins of width h , the resulting counting variables X i for each neuron are not binary. ◮ Count spikes across the population, yielding the population spike count Z = � N i =1 X i with distribution f Z .

Measurement 1 1 1 1 6 1 1 1 1 1 1 11 1 1 1 1 1 6 1 1 1 1 � Amplitude Distribution z ( t ) = δ ( t − t l ) · a l x1 ( t ) l f Probability A x2 ( t ) 1 2 3 4 5 6 . Amplitude A . Population spike count . f distribution Z Probability ? xN ( t ) { h Z ( s ) 0 5 10 15 20 Complexity Time ◮ Segment spike trains into bins of width h , the resulting counting variables X i for each neuron are not binary. ◮ Count spikes across the population, yielding the population spike count Z = � N i =1 X i with distribution f Z . ◮ Infer amplitudes f A from observed population activity f Z .

Cumulants and correlations 1 1 1 1 6 1 1 1 1 1 1 11 1 1 1 1 1 6 1 1 1 1 � Amplitude Distribution z ( t ) = δ ( t − t l ) · a l x1 ( t ) l f Probability A x2 ( t ) 1 2 3 4 5 6 . Amplitude A . Population spike count . f distribution Z Probability ? xN ( t ) { h Z ( s ) 0 5 10 15 20 Complexity Time ◮ CPPs satisfy: κ m [ Z ] = E[ A m ] νh ( m = 1 , 2 , . . . , N )

Cumulants and correlations 1 1 1 1 6 1 1 1 1 1 1 11 1 1 1 1 1 6 1 1 1 1 � Amplitude Distribution z ( t ) = δ ( t − t l ) · a l x1 ( t ) l f Probability A x2 ( t ) 1 2 3 4 5 6 . Amplitude A . Population spike count . f distribution Z Probability ? xN ( t ) { h Z ( s ) 0 5 10 15 20 Complexity Time ◮ CPPs satisfy: κ m [ Z ] = E[ A m ] νh ( m = 1 , 2 , . . . , N ) ◮ κ 1 [ Z ] = E[ Z ] = � i E[ X i ]