Spike Statistics in a HighConductance Cortical Network Model

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Spike Statistics in a High-Conductance Cortical
Network Model

• Joanna Tyrcha
• Stockholm University, Stockholm
• in collaboration with John Hertz, Nordita,
  Stockholm/Copenhagen

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Outline
Cross-correlations between neurons measured in
network simulations (and experiments)
correlation coefficients 0.17 0.07
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Outline
Cross-correlations between neurons measured in
network simulations (and experiments)
correlation coefficients 0.17 0.07
Finding distribution of spike patterns with the
observed cross-correlations
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Outline
Cross-correlations between neurons measured in
network simulations (and experiments)
correlation coefficients 0.17 0.07

• Finding distribution of spike patterns with the
  observed
• cross-correlations
• fit with SK spin glass model with ltJijgt gt 0,

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Outline
Cross-correlations between neurons measured in
network simulations (and experiments)
correlation coefficients 0.17 0.07

• Finding distribution of spike patterns with the
  observed
• cross-correlations
• fit with SK spin glass model with ltJijgt gt 0,
• ltJijgt ? 1/N, std(Jij) ?1/N1/6

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Outline
Cross-correlations between neurons measured in
network simulations (and experiments)
correlation coefficients 0.17 0.07

• Finding distribution of spike patterns with the
  observed
• cross-correlations
• fit with SK spin glass model with ltJijgt gt 0,
• ltJijgt ? 1/N, std(Jij) ?1/N1/6
• in SG phase only for N gt 5000 (?)

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Simulations Model
2 populations in network Excitatory,
Inhibitory
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Simulations Model
2 populations in network Excitatory,
Inhibitory Excitatory external drive
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Simulations Model
2 populations in network Excitatory,
Inhibitory Excitatory external drive HH-like
neurons, conductance-based synapses
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Simulations Model
2 populations in network Excitatory,
Inhibitory Excitatory external drive HH-like
neurons, conductance-based synapses Random
connectivityProbability 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.
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Simulations Model
2 populations in network Excitatory,
Inhibitory Excitatory external drive HH-like
neurons, conductance-based synapses Random
connectivityProbability 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.
Results here for c 0.1, N 1000
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rapidly-varying input
Rext
Stimulus modulation
t (sec)
Flips on and off with exponentially-distributed
on- and off-periods, Mean on/off time 50 ms
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Rasters
inhibitory (100) excitatory (400)
15.5 Hz 8.6 Hz
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Correlation coefficients
Data in 10-ms bins
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Correlation coefficients
Data in 10-ms bins
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Correlation coefficients
Data in 10-ms bins
cc 0.17 0.07
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Modeling the distribution of spike patterns
Have sets of spike patterns Sik 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 Sik Si 1
for spike/no spike (we use 10-ms bins) (temporal
order irrelevant) Construct a distribution PS
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 Sik Si 1
for spike/no spike (we use 10-ms bins) (temporal
order irrelevant) Construct a distribution PS
that generates the observed patterns (i.e., has
the same correlations) Simplest nontrivial
model (Schneidman et al, Nature 440 1007 (2006),
Tkacik et al, arXivq-bio.NC/0611072)
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Modeling the distribution of spike patterns
Have sets of spike patterns Sik Si 1
for spike/no spike (we use 10-ms bins) (temporal
order irrelevant) Construct a distribution PS
that generates the observed patterns (i.e., has
the same correlations) Simplest nontrivial
model (Schneidman et al, Nature 440 1007 (2006),
Tkacik et al, arXivq-bio.NC/0611072)
parametrized by Jij, hi
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Fitting parameters of the distribution (I)
Boltzmann learning
Iterative adjustment of Jij, hi
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Fitting parameters of the distribution (I)
Boltzmann learning
Iterative adjustment of Jij, hi
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Fitting parameters of the distribution (I)
Boltzmann learning
Iterative adjustment of Jij, hi
requires knowing only first- and second-order
spike statistics
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Fitting parameters of the distribution (I)
Boltzmann learning
Iterative adjustment of Jij, hi
requires knowing only first- and second-order
spike statistics A model like would need
3rd-order statistics to find
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Fitting parameters of the distribution (II)
inversion of Thouless-Anderson-Palmer equations
(Tanaka, PRE 58 2302 (1998))
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Fitting parameters of the distribution (II)
inversion of Thouless-Anderson-Palmer equations
(Tanaka, PRE 58 2302 (1998)) TAP free
energy
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Fitting parameters of the distribution (II)
inversion of Thouless-Anderson-Palmer equations
(Tanaka, PRE 58 2302 (1998)) TAP free
energy ?F/?mi 0 gt TAP equations
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Fitting parameters of the distribution (II)
inversion of Thouless-Anderson-Palmer equations
(Tanaka, PRE 58 2302 (1998)) TAP free
energy ?F/?mi 0 gt TAP equations
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TAP inversion (continued) Tanaka procedure

• Measure mi ltSigt and Cij ltSiSjgt - mimj from
  data

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TAP inversion (continued) Tanaka procedure

• Measure mi ltSigt and Cij ltSiSjgt - mimj from
  data
• Invert C and solve for Jij in

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TAP inversion (continued) Tanaka procedure

• Measure mi ltSigt and Cij ltSiSjgt - mimj from
  data
• Invert C and solve for Jij in
• Solve TAP equations for hi

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J distributions
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J distributions
cf Tkacik et al
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Mean J ?1/N
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Standard deviationsfall off slower than 1/N1/2
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Sherrington-Kirkpatrick spin glass model
with

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Sherrington-Kirkpatrick spin glass model
with
Infinite-range (mean field) model

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Sherrington-Kirkpatrick spin glass model
with
Infinite-range (mean field) model
Here,
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J of equivalent SK model increases with N
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Heading for a spin glass state?

• reach SG at N 5000

Almeida and Thouless, J Phys A 1978
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Heading for a spin glass state?
here at N80

• reach SG at N 5000

Almeida and Thouless, J Phys A 1978
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Heading for a spin glass state?
here at N80
here at N160

• reach SG at N 5000

Almeida and Thouless, J Phys A 1978
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Heading for a spin glass state?
here at N80
here at N160

• reach SG at N 5000

reach SG at N5000
Almeida and Thouless, J Phys A 1978
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J distributions are not normal
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Perspective

• Fitting Ising models for PS results look like
  SK models

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Perspective

• Fitting Ising models for PS results look like
  SK models
• -- 2 methods Boltzmann learning and inversion
  of TAP equations

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Perspective

• Fitting Ising models for PS results look like
  SK models
• -- 2 methods Boltzmann learning and inversion
  of TAP equations
• -- J0 independent of N, J ? N1/3

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Perspective

• Fitting Ising models for PS results look like
  SK models
• -- 2 methods Boltzmann learning and inversion
  of TAP equations
• -- J0 independent of N, J ? N1/3
• -- higher-order couplings not necessary

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Perspective

• Fitting Ising models for PS results look like
  SK models
• -- 2 methods Boltzmann learning and inversion
  of TAP equations
• -- J0 independent of N, J ? N1/3
• -- higher-order couplings not necessary

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Perspective

• Fitting Ising models for PS results look like
  SK models
• -- 2 methods Boltzmann learning and inversion
  of TAP equations
• -- J0 independent of N, J ? N1/3
• -- higher-order couplings not necessary
• -- no spin glass phase for Nlt5000.