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

Time − → Neuron ID − →

Question: Are there any neuronal groups that systematically fire together?

Co-activated neuronal groups

Time − → Neuron ID − →

Question: Are there any neuronal groups that systematically fire together?

Co-activated neuronal groups

Time − → Neuron ID − →

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

s1(t) s2(t) s3(t) s1(t) s2(t) s3(t) s2

A B C D E F

y13 y12 y23 y13 s1 s3 s2 s1 s3

20ms 20ms

Staude, Grün, Rotter, 2010

Correlated Poisson processes

m(t) y1(t) y2(t) y12(t) s1(t) s2(t) σj aj 1 1 2 1 y3(t) y123(t) y13(t) y23(t) s3(t) 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 3 1 1 2 1 1 2 2 s1(t) s2(t) s3(t)

A B

Correlated Poisson processes

1 10 1 ξ fA(ξ )

A

1 10 1 ξ fA(ξ )

B

fully independent fully correlated

Correlated Poisson processes

1 10 1 ξ fA(ξ )

C

1 10 1 ξ fA(ξ )

D

1 10 1 ξ fA(ξ )

E

1 10 1 ξ fA(ξ )

F

pairwise correlation coefficient c = 0.4

Problems and issues of HOC inference

Combinatorial explosion: A population of N neurons could in principle form 2N different groups/assemblies: N 10 100 273 2N 103 1030 1.5 × 1082 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

◮ identical rates and

pairwise correlations

◮ different higher-order

correlations

0.1 0.2 0.3 0.4 0.5 time (s) 104 1 10 20 30 40 MIP x1 x2 xN x1 x2 xN 1 10 20 30 40 spike train index SIP 0.1 0.2 0.3 0.4 0.5 time (s) sum (s−1) 104 Single Interaction Process Multiple Interaction Process time time

A C D

wu wg

B Kuhn, Aertsen, Rotter, 2002/2003

Comparing two different input ensembles

◮ identical rates and

pairwise correlations

◮ different higher-order

correlations

SIP single rate r = α + β correlation c = β/(α + β) cluster rate β = r c MIP single rate r = α β correlation c = β cluster rate α = r/c

0.1 0.2 0.3 0.4 0.5 time (s) 104 1 10 20 30 40 MIP x1 x2 xN x1 x2 xN 1 10 20 30 40 spike train index SIP 0.1 0.2 0.3 0.4 0.5 time (s) sum (s−1) 104 Single Interaction Process Multiple Interaction Process time time

A C D

wu wg

B Kuhn, Aertsen, Rotter, 2002/2003

Higher-order correlations do matter!

−70 −60 −50 U (mV) 0.1 0.2 0.3 0.4 0.5 50 100 time (s) input −70 −60 −50 U (mV) 0.1 0.2 0.3 0.4 0.5 50 100 time (s) input

C D

SIP, c = 0.4 MIP, c = 0.4 correlation coefficient correlation coefficient 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 correlation coefficient

• utput rate (s−1)

10 s−1 50 s−1 90 s−1

SIP 0.2 0.4 0.6 0.8 1 20 40 60 80 100 120 140 correlation coefficient

• utput rate (s−1)

10 s−1 50 s−1 90 s−1 20 s−1

MIP

20 s−1

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)

Carrier rate ν z(t) =

• l

δ(t − tl)

The compound Poisson process (CPP)

Carrier rate ν 6 6 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

The compound Poisson process (CPP)

x2 (t ) xN (t ) x1 (t ) . . . z (t ) Carrier rate ν 6 6 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ 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 fA → correlation structure.

The compound Poisson process (CPP)

x2 (t ) xN (t ) x1 (t ) . . . z (t ) Carrier rate ν 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ 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 fA → correlation structure.

The compound Poisson process (CPP)

x2 (t ) xN (t ) x1 (t ) . . .

The compound Poisson process (CPP)

x2 (t ) xN (t ) x1 (t ) . . .

?

f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Task: Infer amplitude distribution fA 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

x2 (t ) xN (t ) x1 (t ) . . .

{

h

◮ Segment spike trains into bins of width h, the resulting

counting variables Xi for each neuron are not binary.

Measurement

x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

◮ Segment spike trains into bins of width h, the resulting

counting variables Xi for each neuron are not binary.

◮ Count spikes across the population, yielding the population

spike count Z = N

i=1 Xi with distribution fZ.

Measurement

x2 (t ) xN (t ) x1 (t ) . . . 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f

?

f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ Segment spike trains into bins of width h, the resulting

counting variables Xi for each neuron are not binary.

◮ Count spikes across the population, yielding the population

spike count Z = N

i=1 Xi with distribution fZ. ◮ Infer amplitudes fA from observed population activity fZ.

Cumulants and correlations

x2 (t ) xN (t ) x1 (t ) . . . 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f

?

f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ CPPs satisfy:

κm[Z] = E[Am]νh (m = 1, 2, . . . , N)

Cumulants and correlations

x2 (t ) xN (t ) x1 (t ) . . . 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f

?

f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ CPPs satisfy:

κm[Z] = E[Am]νh (m = 1, 2, . . . , N)

◮ κ1[Z] = E[Z] =

i E[Xi]

Cumulants and correlations

x2 (t ) xN (t ) x1 (t ) . . . 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f

?

f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ CPPs satisfy:

κm[Z] = E[Am]νh (m = 1, 2, . . . , N)

◮ κ1[Z] = E[Z] =

i E[Xi]

◮ κ2[Z] = Var[Z] =

i Var[Xi] + i=j Cov[Xi, Xj]

Cumulants and correlations

x2 (t ) xN (t ) x1 (t ) . . . 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f

?

f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al

◮ CPPs satisfy:

κm[Z] = E[Am]νh (m = 1, 2, . . . , N)

◮ κ1[Z] = E[Z] =

i E[Xi]

◮ κ2[Z] = Var[Z] =

i Var[Xi] + i=j Cov[Xi, Xj]

◮ κm[Z] = mth order correlations

CuBIC

Raster plot

Pairwise correlations imply higher-order correlations!

Amplitude distribution

1 2 3 4

m neurons with identical spike trains ⇔ all pairwise correlation coefficients 1 ⇒ correlations of order m

CuBIC

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

CuBIC

κ∗

2,2

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

1 2

Raster Plot Amplitude Distribution

◮ κ∗

2,2 = max

• κ2[Z]
• fA(i) = 0 for i > 2 and κ1[Z] = ˆ

κ1

CuBIC

κ∗

2,2

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

1 2

Raster Plot Amplitude Distribution

◮ κ∗

2,2 = max

• κ2[Z]
• fA(i) = 0 for i > 2 and κ1[Z] = ˆ

κ1

• ◮ ˆ

κ2 > κ∗

2,2: pairwise correlations imply correlations of order > 2

CuBIC

κ∗

2,2

κ∗

2,3

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

1 2 1 2 3

Raster Plot Amplitude Distribution

◮ κ∗

2,2 = max

• κ2[Z]
• fA(i) = 0 for i > 2 and κ1[Z] = ˆ

κ1

• ◮ ˆ

κ2 > κ∗

2,2: pairwise correlations imply correlations of order > 2

◮ κ∗

2,3 = max

• κ2[Z]
• fA(i) = 0 for i > 3 and κ1[Z] = ˆ

κ1

• ◮ ˆ

κ2 > κ∗

2,3: pairwise correlations imply correlations of order > 3

CuBIC

κ∗

2,2

κ∗

2,3

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

1 2 1 2 3

Raster Plot Amplitude Distribution

◮ κ∗

2,2 = max

• κ2[Z]
• fA(i) = 0 for i > 2 and κ1[Z] = ˆ

κ1

• ◮ ˆ

κ2 > κ∗

2,2: pairwise correlations imply correlations of order > 2

◮ κ∗

2,3 = max

• κ2[Z]
• fA(i) = 0 for i > 3 and κ1[Z] = ˆ

κ1

• ◮ ˆ

κ2 > κ∗

2,3: pairwise correlations imply correlations of order > 3

◮ κ∗

3,15 = max

• κ3[Z]
• fA(i) = 0 for i > 15 and κ1[Z] = ˆ

κ1 and κ2[Z] = ˆ κ2

• ◮ ˆ

κ3 > κ∗

3,15: triplet correlations imply correlations of order > 15

CuBIC

κ∗

2,2

ˆ κ1 ˆ κ2

κ2[Z]

κ1[Z]

1 2

Raster Plot Amplitude Distribution

Hypothesis testing Hm,ξ

• bserved cumulants ˆ

κ1[Z], . . . , ˆ κm[Z] are statistically compatible with correlations bounded by ξ pm,ξ < α

• bserved cumulants imply correlations of order

strictly larger than ξ

CuBIC: illustration

◮ 100 simulated spike trains

100 s, 10 Hz, bin size 5 ms

◮ 70 independent and 30

correlated neurons

◮ independent background plus

patterns of order 7

◮ pairwise correlation c = 0.01 in

correlated subgroup

◮ only low-order cumulants are

measured: ˆ κ1[Z], ˆ κ2[Z], ˆ κ3[Z]

Amplitude distribution fA

1 2 3 4 5 6 7 8 9 10 11 0.005 0.01 Probability Amplitude ξ 50 100 Neurons

Raster Plot and Population Histogram

500 1500 5 10 15 Counts Time [ms] 10 20

Complexity Distribution fZ

Counts Complexity Z 10 20 1 2 3 4 5 6 7 8 9 10 11 ξ p3,ξ

CuBIC: illustration

◮ 100 simulated spike trains

100 s, 10 Hz, bin size 5 ms

◮ 70 independent and 30

correlated neurons

◮ independent background plus

patterns of order 7

◮ pairwise correlation c = 0.01 in

correlated subgroup

◮ only low-order cumulants are

measured: ˆ κ1[Z], ˆ κ2[Z], ˆ κ3[Z]

Amplitude distribution fA

1 2 3 4 5 6 7 8 9 10 11 0.005 0.01 Probability Amplitude ξ 50 100 Neurons

Raster Plot and Population Histogram

500 1500 5 10 15 Counts Time [ms] 10 20

Complexity Distribution fZ

Counts Complexity Z 10 20 1 2 3 4 5 6 7 8 9 10 11 ξ p3,ξ

Amplitude distribution fA

1 2 3 4 5 6 7 8 9 10 11 0.005 0.01 Probability Amplitude ξ 50 100 Neurons

Raster Plot and Population Histogram

500 1500 5 10 15 Counts Time [ms] 10 20

Complexity Distribution fZ

Counts Complexity Z 10 20 1 2 3 4 5 6 7 8 9 10 11 ξ p3,ξ

ξ ^

ˆ ξ = lower bound on the order of correlation

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

Non-stationary firing rates

50 100 500 1500 5 10 15 Time [ms]

◮ CuBIC assumes spike trains with stationary firing rates

Non-stationary firing rates

50 100 500 1500 5 10 15 Time [ms]

Luczak et al 2007, rat somatosensory cortex, up/down states

◮ CuBIC assumes spike trains with stationary firing rates ◮ Spike trains observed in experiments often exhibit spontaneous

and stimulus/behavior induced non-stationarities

Non-stationary firing rates

50 100 500 1500 5 10 15 Time [ms]

Luczak et al 2007, rat somatosensory cortex, up/down states

◮ CuBIC assumes spike trains with stationary firing rates ◮ Spike trains observed in experiments often exhibit spontaneous

and stimulus/behavior induced non-stationarities

◮ Co-varying firing rates induce correlations, which may lead

CuBIC to overestimate the correlation order

Non-stationary CPP

x2 (t ) xN (t ) x1 (t ) . . . Carrier rate ν 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 6 6 Z (s)

Time

{

h Population spike count distribution

5 10 15 20

Complexity Probability

Z f f Amplitude Distribution

Amplitude A Probability

1 6 5 4 3 2

A z(t) =

• l

δ(t − tl) · al ν (t )

Non-stationary CPP

Population spike count distribution Z f x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

5 10 15 20

Complexity Probability

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 6 1 1 6 ν (t ) f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Carrier rate ν(t) z(t) =

• l

δ(t − tl) · al

◮ Time-varying carrier rate ν(t), constant correlation structure fA

Non-stationary CPP

Population spike count distribution Z f x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

5 10 15 20

Complexity Probability

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 6 1 1 6

?

ν (t ) f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Carrier distribution

fν ◮ Time-varying carrier rate ν(t), constant correlation structure fA

Non-stationary CPP

Population spike count distribution Z f x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

5 10 15 20

Complexity Probability

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 6 1 1 6

?

ν (t ) f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Carrier distribution

fν ◮ Time-varying carrier rate ν(t), constant correlation structure fA ◮ κ1[Z] = E[A]κ1[ν]h

Non-stationary CPP

Population spike count distribution Z f x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

5 10 15 20

Complexity Probability

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 6 1 1 6

?

ν (t ) f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Carrier distribution

fν ◮ Time-varying carrier rate ν(t), constant correlation structure fA ◮ κ1[Z] = E[A]κ1[ν]h ◮ κ2[Z] = E[A2]κ1[ν]h + E[A]2κ2[ν]h2

(“Law of total variance”)

Non-stationary CPP

Population spike count distribution Z f x2 (t ) xN (t ) x1 (t ) . . . Z (s)

Time

{

h

5 10 15 20

Complexity Probability

1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 6 1 1 6

?

ν (t ) f Amplitude Distribution

Amplitude A Probability

A

1 6 5 4 3 2

Carrier distribution

fν ◮ Time-varying carrier rate ν(t), constant correlation structure fA ◮ κ1[Z] = E[A]κ1[ν]h ◮ κ2[Z] = E[A2]κ1[ν]h + E[A]2κ2[ν]h2

(“Law of total variance”)

◮ κ3[Z] = E[A3]κ1[ν]h + 3 E[A] E[A2]κ2[ν]h2 + E[A]3κ3[ν]h3

Non-stationary CuBIC

5 10 15 10

−5

10 k Pr{Z=k} 2 4 8 10 0.1 ξ Pr{a=ξ} 2 10 Time [s] Z(s) 2 4 6 8 10 Neuron ID z(t) 50 ν(t) fR 2 10 Time [s] 5 10 15 20 25 50 fR 2 10 Time [s] 10 20 30 40 50 50 fR

A B C D E

relevant: carrier rate distribution fR (resp. fν) irrelevant: assignment of spikes to individual neurons

Staude, Grün, Rotter, 2010

“Stimulus-driven” non-stationarities

1000 2000 ν [Hz] Pure non−stationarity 25 50 Neuron ID 2 10 Z(s) Time [s] 20 0.1 0.2 0.3 Pr{Z=k} k 20 1 3 5 7 0.5 1 ξ p3,ξ Pure correlation 2 Time [s] 20 k 20 1 3 5 7 ξ Non−stationarity & correlation 2 Time [s] 20 k 20 10

−4

10

−2

10 1 3 5 7 ξ

cosine-like modulation of carrier rate assuming stationary rates, cosine modulation, or two rate levels

Staude, Grün, Rotter, 2010

“Internally generated” non-stationarities

1600 3200 ν [Hz] Pure non−stationarity 25 50 Neuron ID 2 10 Z(s) Time [s] 20 0.1 0.2 Pr{Z=k} k 20 1 3 5 7 0.5 1 ξ p3,ξ Pure correlation 2 Time [s] 20 k 20 1 3 5 7 ξ Non−stationarity & correlation 2 Time [s] 20 k 20 10

−4

10

−2

10 1 3 5 7 ξ

gamma-distributed modulation of carrier rate assuming stationary rates, uniform modulation, or gamma modulation

Staude, Grün, Rotter, 2010

Conclusions

◮ Higher-order correlations in the presynaptic population strongly

influence single-neuron spike trains, in particular firing rates.

◮ For the point process models in continuous time considered

here, additive spike patterns and cumulants are tightly related. Therefore, cumulants provide intuitive measures of correlation.

◮ Cumulants and log-linear parameters establish quite different

views on higher-order correlations. In particular, zero log-linear interactions do NOT imply the absence of synchronous patterns.

◮ CuBIC can infer the presence of high-order patterns from

measured low-order cumulants of population spike counts. The necessary sample sizes are fully compatible with state-of-the-art in vivo multi-neuron spike train recordings.

◮ Non-stationary rates are smoothly integrated into hypothesis

testing, with reliable performance. Non-Poissonian spiking as found in real neurons can be tolerated to some degree.

Benjamin Staude Imke Reimer Sonja Grün Clemens Boucsein Werner Ehm Robert Gütig Alexandre Kuhn Ad Aertsen

Thanks!

October 4–6, 2011 Freiburg, Germany

http://www.bccn-2011.uni-freiburg.de

Analysis of Parallel Spike Trains

SPRINGER SERIES IN COMPUTATIONAL NEUROSCIENCE

Sonja Grün Stefan Rotter Editors

Grün • Rotter Editors

Analysis of Parallel Spike Trains

Action potentials, or spikes, are the most salient expression of neuronal processing in the active brain, and they are likely an important key to understanding the neuronal mechanisms of behavior. However, it is the group dynamics of large networks of neurons that are likely to underlie brain function, and this can only be appreciated if the action potentials from multiple individual nerve cells are observed simultaneously. Techniques that employ multielectrodes for parallel spike train recordings have been available for many decades, and their use has gained wide popularity among neuroscientists. To reliably interpret the results of such electrophysiological experiments, solid and comprehensible data analysis is crucial. The development

• f data analysis methods, however, has not really kept pace with the advances in recording technology. Nei-

ther general concepts, nor statistical methodology seem adequate for the new experimental possibilities. Promising approaches are scattered across journal publications, and the relevant mathematical background literature is buried deep in journals of different fields and compiling a useful reader for students or col- laborators is both laborious and frustrating. This situation led us to gather state-of-the-art methodologies for analyzing parallel spike trains into a single book, analysis which will serve vantage point for current techniques and a launching point for future development. Sonja Grün, born 1960, received her MSc (University of Tübingen and Max-Planck Institute for Biological Cybernetics) and PhD (University of Bochum, Weizmann Institute of Science in Rehovot) in physics (theo- retical neuroscience), and her Habilitation (University of Freiburg) in neurobiology and biophysics. During her postdoc at the Hebrew University in Jerusalem, she performed multiple single-neuron recordings in behaving monkeys. Equipped with this experience she returned back to computational neuroscience to further develop analysis tools for multi-electrode recordings, first at the Max-Planck Institute for Brain Research in Frankfurt/Main and then as an assistant professor at the Freie Universität in Berlin associated with the local Bernstein Center for Computational Neuroscience. Since 2006 she has been unit leader for statistical neuroscience at the RIKEN Brain Science Institute in Wako-Shi, Japan. Her scientific work focuses

• n cooperative network dynamics relevant for brain function and behavior.

Stefan Rotter, born 1961, holds a MSc in Mathematics, a PhD in Physics and a Habilitation in Biology. Since 2008, he has been Professor at the Faculty of Biology and the Bernstein Center Freiburg, a multidis- ciplinary research institution for Computational Neuroscience and Neurotechnology at Albert-Ludwig Uni- versity Freiburg. His research is focused on the relations between structure, dynamics, and function in spiking networks of the brain. He combines neuronal network modeling and spike train analysis,

• ften using stochastic point processes as a conceptual link.

SPRINGER SERIES IN COMPUTATIONAL NEUROSCIENCE

Analysis of Parallel Spike Trains

BIOMEDICINE 9 7 8 1 4 4 1 9 5 6 7 4 3 ISBN 978-1-4419-5674-3

Publications on higher-order correlations

Staude B, Grün S, Rotter S Higher-order correlations In: Grün S, Rotter S (eds) Analysis of Parallel Spike Trains Springer Series in Computational Neuroscience, Vol. 7, 2010 Staude B, Grün S, Rotter S Higher-order correlations in non-stationary parallel spike trains: statistical modeling and inference Frontiers in Computational Neuroscience 4: 16, 2010 Staude B, Rotter S, Grün S CuBIC: cumulant based inference of higher-order correlations in massively parallel spike trains Journal of Computational Neuroscience, epub ahead of print, 2009 Ehm W, Staude B, Rotter S Decomposition of neuronal assembly activity via empirical de-Poissonization Electronic Journal of Statistics 1: 473-495, 2007 Gütig R, Aertsen A, Rotter S Analysis of higher-order neuronal interactions based on conditional inference Biological Cybernetics 88(5): 352-359, 2003 Kuhn A, Aertsen A, Rotter S Higher-order statistics of input ensembles and the response of simple model neurons Neural Computation 15(1): 67-101, 2003 Kuhn A, Rotter S, Aertsen A Correlated input spike trains and their effects on the response of the leaky integrate-and-fire neuron Neurocomputing 44-46: 121-126, 2002

Correlated Poisson processes

Consider N neurons with spike trains s1(t), s2(t), . . . , sN(t).

◮ For every non-empty group G ⊆ {1, 2, . . . , N} of neurons

(“pattern”), the joint spiking (some jitter allowed) of all its group members is described by a Poisson process yG(t).

◮ The spike train si(t) of neuron i is comprised of all spikes

• f all groups neuron i is a member of

si(t) =

• G∋i

yG(t).

◮ Alternatively, one may consider the carrier process

comprising all spikes y(t) =

• G

yG(t) and assign a mark to each of its events, namely the pattern produced at that point in time.

Correlated Bernoulli variables

Time Neurons

Raster Plot

(adapted from Grün et al. 2002)

x(t)

P(x) = exp  θ +

• i

θixi +

• i<j

θijxixj +

• i<j<k

θijkxixjxk + . . .   . The θij represent pairwise, the θijk triplet correlations, . . .

Martingnon et al 1995, 2000; Nakahara & Amari 2002; Shlens et al 2006; Schneidman 2006; Roudi et al 2009 and many others

Cumulants vs. log-linear parameters

1 2 3 4 5 6 7 10

−20

10

−10

10 k Pr[ΣSi = k] Population spike counts

SIP MaxEnt Independent

A N = 7, λ = 1 Hz, c = 0.01

1 2 3 4 5 6 7 10

−20

10

−10

10

Cumulant correlations κ

σ(M )[S]

k = |M | B

1 2 3 4 5 6 7 −40 −20 20 40

Exponential parameters θM k = |M | C

Staude, Grün, Rotter, 2010

Non-Poissonian spike trains

✛ ✲

Poisson

CV=1.00

coefficient of variation CV = √

Var[ISI] E[ISI]

Non-Poissonian spike trains

✛ ✲

Poisson

CV=1.00

CV < 1

regular

CV=0.40

rat somatosensory cortex

(Boucsein et al.)

Non-Poissonian spike trains

✛ ✲

Poisson

CV=1.00

CV < 1

regular

CV=0.40

rat somatosensory cortex

(Boucsein et al.)

CV > 1

irregular

CV=1.31

rat visual cortex

(Boucsein et al.)

Robustness of CuBIC

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 0.75 1 1.25 1.5 1.75 2 C V

ˆ ξ C V / ˆ ξ C P P

1ms 3ms 5ms 7ms 9ms 11ms 13ms 15ms

Correlated log-normal processes generated via thinning:

N = 50, λ = 10 Hz, c = 0.05, T = 500 s, ˆ ξ = mean over 100 simulations