Ising Models for Neural Data

Ising Models for Neural Data John Hertz, Niels Bohr Institute and Nordita

work done with Yasser Roudi (Nordita) and Joanna Tyrcha (SU)

Math Bio Seminar, SU, 26 March 2009

arXiv:0902.2885v1 (2009)

Background and basic idea:

• New recording technology makes it possible to record from hundreds of neurons simultaneously



• But what to make of all these data?



• But what to make of all these data?• Construct a model of the spike pattern distribution: find

“functional connectivity” between neurons



• But what to make of all these data?• Construct a model of the spike pattern distribution: find

“functional connectivity” between neurons• Here: results for model networks

Outline

Outline

• Data

Outline

• Data

• Model and methods, exact and approximate

Outline

• Data


• Results: accuracy of approximations, scaling of functional connections

Outline

• Data


• Results: accuracy of approximations, scaling of functional connections

• Quality of the fit to the data distribution

Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory

ExcitatoryPopulation

InhibitoryPopulation

ExternalInput(Exc.)


Excitatory external drive



ExternalInput(Exc.)



HH-like neurons, conductance-based synapses



ExternalInput(Exc.)




Random connectivity:Probability of connection between any two neurons is c = K/N, where N is the size of the population and K is the average number of presynaptic neurons.



ExternalInput(Exc.)




Random connectivity:Probability of connection between any two neurons is c = K/N, where N is the size of the population and K is the average number of presynaptic neurons.



ExternalInput(Exc.)

Results here for c = 0.1, N = 1000

Tonic input

inhibitory(100)

excitatory(400)

16.1 Hz

7.9 Hz

Rext

t (sec)

Filtered white noise = 100 ms

Stimulus modulation:

Rapidly-varying input

inhibitory(100)

excitatory(400)

15.1 Hz

8.6 Hz

Correlation coefficientsData in 10-ms bins

22jjii

jijiij

nnnn

nnnncc

cc ~ 0.0052 ± 0.0328

tonic data

Correlation coefficients

cc ~ 0.0086 ± 0.0278

Experiments: Cited values of cc~0.01 [Schneidmann et al, Nature (2006)]

”stimulus” data

Modeling the distribution of spike patterns

Have sets of spike patterns {Si}k Si = ±1 for spike/no spike (we use 10-ms bins)(temporal order irrelevant)



Construct a distribution P[S] that generates the observed patterns (i.e., has the same correlations)



Construct a distribution P[S] that generates the observed patterns (i.e., has the same correlations)

Simplest nontrivial model (Schneidman et al, Nature 440 1007 (2006), Tkačik et al, arXiv:q-bio.NC/0611072):

ij iiijiij ShSSJZSP 2

11 exp][

Ising model, parametrized by Jij, hi

An inverse problem:

Have: statistics <Si>, <SiSj>want: hi, Jij

An inverse problem:


Exact method: Boltzmann learning

An inverse problem:



€

δJij = η SiS j data− SiS j current J ,h[ ]

δhi = η Si data− Si current J ,h[ ]

An inverse problem:



€

δJij = η SiS j data− SiS j current J ,h[ ]

δhi = η Si data− Si current J ,h[ ]

Requires long Monte Carlo runs to compute model statistics

1. (Naïve) mean field theory


€

mi = tanh hi + Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ mi = Si

hi = tanh−1 mi − Jijm j

j

∑

or

Mean field equations:


€


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ mi = Si


j

∑

or

Inverse susceptibility (inverse correlation) matrix

€

Cij−1 =

∂hi

∂m j

=δ ij

1− mi2

− Jij Cij = SiS j − mim j



€


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ mi = Si


j

∑

or

Inverse susceptibility (inverse correlation) matrix

€

Cij−1 =

∂hi

∂m j

=δ ij

1− mi2

− Jij Cij = SiS j − mim j

So, given correlation matrix, invert it, and

€

(i ≠ j) Jij = −Cij−1


2. TAP approximation

2. TAP approximationThouless, Anderson, Palmer, Phil Mag 35 (1977)Kappen & Rodriguez, Neural Comp 10 (1998)Tanaka, PRE 58 2302 (1998)

“TAP equations” (improved MFT for spin glasses)



ijj

ijj

jijii mmJmJhm )1(tanh 221



ijj

ijj


Onsager “reaction term”



ijj

ijj


€

i ≠ j : [C-1]ij =∂hi

∂m j

= −Jij − 2Jij2mim j




ijj

ijj


€

i ≠ j : [C-1]ij =∂hi

∂m j

= −Jij − 2Jij2mim j


A quadratic equation to solve for Jij

3. Independent-pair approximation


Solve the two-spin problem:

€

Zp(S1,S2) = exp h1S1 + h2S2 + J12S1S2( ) S1,S2 = ±1



€


Solve for J:

€

J12 =1

4log

p(1,1) p(−1,−1)

p(1,−1)p(−1,1)

⎛

⎝ ⎜

⎞

⎠ ⎟

=1

4log

1+ S1 + S2 + S1S2( ) 1− S1 − S2 + S1S2( )

1− S1 + S2 − S1S2( ) 1+ S1 − S2 − S1S2( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



€


Solve for J:

€

J12 =1

4log

p(1,1) p(−1,−1)

p(1,−1)p(−1,1)

⎛

⎝ ⎜

⎞

⎠ ⎟

=1

4log

1+ S1 + S2 + S1S2( ) 1− S1 − S2 + S1S2( )

1− S1 + S2 − S1S2( ) 1+ S1 − S2 − S1S2( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

Low-rate limit:

€

S1 , S2 → −1( )

J12 →1

4log 1+

S1S2 − S1 S2

1+ S1( ) 1+ S2( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

4. Sessak-Monasson approximation


A combination of naïve mean field theory and independent-pair approximations:



€

Jij = −Cij−1 +

1

4log

1+ Si + S j + SiS j( ) 1− Si − S j + SiS j( )

1− Si + S j − SiS j( ) 1+ Si − S j − SiS j( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

−Cij

1− mi2

( ) 1− m j2

( ) − Cij( )2



€

Jij = −Cij−1 +

1

4log

1+ Si + S j + SiS j( ) 1− Si − S j + SiS j( )

1− Si + S j − SiS j( ) 1+ Si − S j − SiS j( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

−Cij

1− mi2

( ) 1− m j2

( ) − Cij( )2

(Last term is to avoid double-counting)

Comparing approximations: N=20

nMFT ind pair

low-rate TAP

SM TAP/SM

Comparing approximations: N=20 N =200

nMFT ind pair nMFT ind pair

low-rate low-rateTAP TAP

SM SMTAP/SM TAP/SM

Comparing approximations: N=20 N =200

nMFT ind pair nMFT ind pair

low-rate low-rateTAP TAP

SM SMTAP/SM TAP/SM thewinner!

Error measures

SM/TAP

SM

SM/TAPSM

TAP

TAP

nMFT

nMFT

low-rate

low-rate

ind pair

ind pair

N-dependence:How do the inferred couplings depend on the size of the set of neurons used in the inference algorithm?


N = 20

N=200


N = 20

N=200

10 largest and smallest J’s:


N = 20

N=200

10 largest and smallest J’s:

Relative sizes of differentJ’s preserved, absolute sizesshrink.

N-dependence of mean and variance of the J’s: theory

N-dependence of mean and variance of the J’s: theoryFrom MFT for spin glasses (assumes J’s iid) in normal (i.e., not glassy) state:


€

C =J 1− q( )

2

1− NJ 1− q( ); C2 =

δJ 2S2

1− NδJ 2S

q =1

NSi

2

i

∑ ; S =1

N1− Si

2

( )i

∑2


€

C =J 1− q( )

2

1− NJ 1− q( ); C2 =

δJ 2S2

1− NδJ 2S

q =1

NSi

2

i

∑ ; S =1

N1− Si

2

( )i

∑2

Invert to find statistics of J’s:

€

J =C

1− q( ) 1− q + NC( ); δJ 2 =

C2

S S + NC2( )


€

C =J 1− q( )

2

1− NJ 1− q( ); C2 =

δJ 2S2

1− NδJ 2S

q =1

NSi

2

i

∑ ; S =1

N1− Si

2

( )i

∑2

Invert to find statistics of J’s:

€

J =C

1− q( ) 1− q + NC( ); δJ 2 =

C2

S S + NC2( )

1/(const +N) dependence in mean and variance

N-dependence: theory vs computed

mean

standarddeviation

TAP

TAP

SM/TAP

SM/TAP

SM

SM

theory

theory

Boltzmann

Boltzmann

Heading for a spin glass state?


Tkacik et al speculated (on the basis of their data, N up to 40) that thesystem would reach a spin glass transition around N = 100



Criterion for stability of the normal (not SG) phase: (de Almeida and Thouless, 1978):




€

NδJ 2S <1




€

NδJ 2S <1

In all our results, we always find

€

NδJ 2S ≤ 0.65

Quality of the Ising-model fit

Quality of the Ising-model fitThe Ising model fits the means and correlations correctly, but it does not generally get the higher-order statistics right.


€

dIsing = ptrue(s)logptrue(s)

pIsing(s)s

∑ .

Quality-of- fit measure: the KL distance


€


pIsing(s)s

∑ .


Compare with an independent-neuron one (Jij = 0):

€

dind = ptrue(s)logptrue(s)

pind (s),

s

∑


€


pIsing(s)s

∑ .


Compare with an independent-neuron one (Jij = 0):

€

dind = ptrue(s)logptrue(s)

pind (s),

s

∑

Goodness-of-fit measure:

€

G =1−dIsing

dind

Results (can only do small samples)


dIsing

dind


dIsing

dind

___

___


dIsing

dind

___

___

G


dIsing

dind

increasingrun time

extrapolation

___

___

G


dIsing

dind

increasingrun time

extrapolation

Linear for small N, looks like G->0for N ~ 200

___

___

G


dIsing

dind

increasingrun time

extrapolation

Linear for small N, looks like G->0for N ~ 200

___

___

G

Model misses something essentialabout the distribution for large N

Summary

Summary

• Ising distribution fits means and correlations of neuronal firing

Summary


• TAP and SM approximations give good, fast estimates of functional couplings Jij

Summary



• Spin glass MFT describes scaling of Jij’s with sample size N

Summary




• Quality of fit to data distribution deteriorates as N grows

Summary




• Quality of fit to data distribution deteriorates as N growsRead more at arXiv:0902.2885v1 (2009)

Documents

Ising Models for Neural Data