Ising Models for Neural Data
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Transcript of Ising Models for Neural Data
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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)
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Background and basic idea:
• New recording technology makes it possible to record from hundreds of neurons simultaneously
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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?
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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?• Construct a model of the spike pattern distribution: find
“functional connectivity” between neurons
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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?• Construct a model of the spike pattern distribution: find
“functional connectivity” between neurons• Here: results for model networks
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Outline
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Outline
• Data
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Outline
• Data
• Model and methods, exact and approximate
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Outline
• Data
• Model and methods, exact and approximate
• Results: accuracy of approximations, scaling of functional connections
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Outline
• Data
• Model and methods, exact and approximate
• Results: accuracy of approximations, scaling of functional connections
• Quality of the fit to the data distribution
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Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory
ExcitatoryPopulation
InhibitoryPopulation
ExternalInput(Exc.)
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Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory
Excitatory external drive
ExcitatoryPopulation
InhibitoryPopulation
ExternalInput(Exc.)
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Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory
Excitatory external drive
HH-like neurons, conductance-based synapses
ExcitatoryPopulation
InhibitoryPopulation
ExternalInput(Exc.)
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Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory
Excitatory external drive
HH-like neurons, conductance-based synapses
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.
ExcitatoryPopulation
InhibitoryPopulation
ExternalInput(Exc.)
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Get Spike Data from Simulations of Model Network2 populations in network: Excitatory, Inhibitory
Excitatory external drive
HH-like neurons, conductance-based synapses
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.
ExcitatoryPopulation
InhibitoryPopulation
ExternalInput(Exc.)
Results here for c = 0.1, N = 1000
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Tonic input
inhibitory(100)
excitatory(400)
16.1 Hz
7.9 Hz
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Rext
t (sec)
Filtered white noise = 100 ms
Stimulus modulation:
Rapidly-varying input
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inhibitory(100)
excitatory(400)
15.1 Hz
8.6 Hz
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Correlation coefficientsData in 10-ms bins
22jjii
jijiij
nnnn
nnnncc
cc ~ 0.0052 ± 0.0328
tonic data
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Correlation coefficients
cc ~ 0.0086 ± 0.0278
Experiments: Cited values of cc~0.01 [Schneidmann et al, Nature (2006)]
”stimulus” data
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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)
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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)
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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)
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
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An inverse problem:
Have: statistics <Si>, <SiSj>want: hi, Jij
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An inverse problem:
Have: statistics <Si>, <SiSj>want: hi, Jij
Exact method: Boltzmann learning
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An inverse problem:
Have: statistics <Si>, <SiSj>want: hi, Jij
Exact method: Boltzmann learning
€
δJij = η SiS j data− SiS j current J ,h[ ]
δhi = η Si data− Si current J ,h[ ]
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An inverse problem:
Have: statistics <Si>, <SiSj>want: hi, Jij
Exact method: Boltzmann learning
€
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
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1. (Naïve) mean field theory
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1. (Naïve) mean field theory
€
mi = tanh hi + Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ mi = Si
hi = tanh−1 mi − Jijm j
j
∑
or
Mean field equations:
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1. (Naïve) mean field theory
€
mi = tanh hi + Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ mi = Si
hi = tanh−1 mi − Jijm j
j
∑
or
Inverse susceptibility (inverse correlation) matrix
€
Cij−1 =
∂hi
∂m j
=δ ij
1− mi2
− Jij Cij = SiS j − mim j
Mean field equations:
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1. (Naïve) mean field theory
€
mi = tanh hi + Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ mi = Si
hi = tanh−1 mi − Jijm j
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
Mean field equations:
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2. TAP approximation
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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)
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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
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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
Onsager “reaction term”
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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
€
i ≠ j : [C-1]ij =∂hi
∂m j
= −Jij − 2Jij2mim j
Onsager “reaction term”
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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
€
i ≠ j : [C-1]ij =∂hi
∂m j
= −Jij − 2Jij2mim j
Onsager “reaction term”
A quadratic equation to solve for Jij
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3. Independent-pair approximation
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3. Independent-pair approximation
Solve the two-spin problem:
€
Zp(S1,S2) = exp h1S1 + h2S2 + J12S1S2( ) S1,S2 = ±1
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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( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
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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( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
Low-rate limit:
€
S1 , S2 → −1( )
J12 →1
4log 1+
S1S2 − S1 S2
1+ S1( ) 1+ S2( )
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
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4. Sessak-Monasson approximation
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4. Sessak-Monasson approximation
A combination of naïve mean field theory and independent-pair approximations:
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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
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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
(Last term is to avoid double-counting)
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Comparing approximations: N=20
nMFT ind pair
low-rate TAP
SM TAP/SM
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Comparing approximations: N=20 N =200
nMFT ind pair nMFT ind pair
low-rate low-rateTAP TAP
SM SMTAP/SM TAP/SM
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Comparing approximations: N=20 N =200
nMFT ind pair nMFT ind pair
low-rate low-rateTAP TAP
SM SMTAP/SM TAP/SM thewinner!
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Error measures
SM/TAP
SM
SM/TAPSM
TAP
TAP
nMFT
nMFT
low-rate
low-rate
ind pair
ind pair
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N-dependence:How do the inferred couplings depend on the size of the set of neurons used in the inference algorithm?
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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
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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
10 largest and smallest J’s:
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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
10 largest and smallest J’s:
Relative sizes of differentJ’s preserved, absolute sizesshrink.
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N-dependence of mean and variance of the J’s: theory
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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:
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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
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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
Invert to find statistics of J’s:
€
J =C
1− q( ) 1− q + NC( ); δJ 2 =
C2
S S + NC2( )
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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
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
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N-dependence: theory vs computed
mean
standarddeviation
TAP
TAP
SM/TAP
SM/TAP
SM
SM
theory
theory
Boltzmann
Boltzmann
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Heading for a spin glass state?
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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
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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):
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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
![Page 64: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/64.jpg)
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
In all our results, we always find
€
NδJ 2S ≤ 0.65
![Page 65: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/65.jpg)
Quality of the Ising-model fit
![Page 66: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/66.jpg)
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.
![Page 67: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/67.jpg)
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
![Page 68: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/68.jpg)
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
Compare with an independent-neuron one (Jij = 0):
€
dind = ptrue(s)logptrue(s)
pind (s),
s
∑
![Page 69: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/69.jpg)
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
Compare with an independent-neuron one (Jij = 0):
€
dind = ptrue(s)logptrue(s)
pind (s),
s
∑
Goodness-of-fit measure:
€
G =1−dIsing
dind
![Page 70: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/70.jpg)
Results (can only do small samples)
![Page 71: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/71.jpg)
Results (can only do small samples)
dIsing
dind
![Page 72: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/72.jpg)
Results (can only do small samples)
dIsing
dind
___
___
![Page 73: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/73.jpg)
Results (can only do small samples)
dIsing
dind
___
___
G
![Page 74: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/74.jpg)
Results (can only do small samples)
dIsing
dind
increasingrun time
extrapolation
___
___
G
![Page 75: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/75.jpg)
Results (can only do small samples)
dIsing
dind
increasingrun time
extrapolation
Linear for small N, looks like G->0for N ~ 200
___
___
G
![Page 76: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/76.jpg)
Results (can only do small samples)
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
![Page 77: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/77.jpg)
Summary
![Page 78: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/78.jpg)
Summary
• Ising distribution fits means and correlations of neuronal firing
![Page 79: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/79.jpg)
Summary
• Ising distribution fits means and correlations of neuronal firing
• TAP and SM approximations give good, fast estimates of functional couplings Jij
![Page 80: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/80.jpg)
Summary
• Ising distribution fits means and correlations of neuronal firing
• TAP and SM approximations give good, fast estimates of functional couplings Jij
• Spin glass MFT describes scaling of Jij’s with sample size N
![Page 81: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/81.jpg)
Summary
• Ising distribution fits means and correlations of neuronal firing
• TAP and SM approximations give good, fast estimates of functional couplings Jij
• Spin glass MFT describes scaling of Jij’s with sample size N
• Quality of fit to data distribution deteriorates as N grows
![Page 82: Ising Models for Neural Data](https://reader033.fdocuments.us/reader033/viewer/2022051115/56814871550346895db57ad4/html5/thumbnails/82.jpg)
Summary
• Ising distribution fits means and correlations of neuronal firing
• TAP and SM approximations give good, fast estimates of functional couplings Jij
• Spin glass MFT describes scaling of Jij’s with sample size N
• Quality of fit to data distribution deteriorates as N growsRead more at arXiv:0902.2885v1 (2009)