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Explore using Ising models to analyze data from hundreds of neurons, focusing on functional connectivity and spike pattern distribution.
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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)
Background and basic idea: • New recording technology makes it possible to record from hundreds of neurons simultaneously
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?
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
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
Outline • Data
Outline • Data • Model and methods, exact and approximate
Outline • Data • Model and methods, exact and approximate • Results: accuracy of approximations, scaling of functional connections
Outline • Data • Model and methods, exact and approximate • Results: accuracy of approximations, scaling of functional connections • Quality of the fit to the data distribution
Get Spike Data from Simulations of Model Network Excitatory Population External Input (Exc.) Inhibitory Population 2 populations in network: Excitatory, Inhibitory
Get Spike Data from Simulations of Model Network Excitatory Population External Input (Exc.) Inhibitory Population 2 populations in network: Excitatory, Inhibitory Excitatory external drive
Get Spike Data from Simulations of Model Network Excitatory Population External Input (Exc.) Inhibitory Population 2 populations in network: Excitatory, Inhibitory Excitatory external drive HH-like neurons, conductance-based synapses
Get Spike Data from Simulations of Model Network Excitatory Population External Input (Exc.) Inhibitory Population 2 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.
Get Spike Data from Simulations of Model Network Excitatory Population External Input (Exc.) Inhibitory Population 2 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. Results here for c = 0.1, N = 1000
Tonic input inhibitory (100) excitatory (400) 16.1 Hz 7.9 Hz
Rapidly-varying input Rext Stimulus modulation: t (sec) Filtered white noise = 100 ms
Rasters inhibitory (100) 15.1 Hz excitatory (400) 8.6 Hz
Correlation coefficients Data in 10-ms bins tonic data cc ~ 0.0052 ± 0.0328
Correlation coefficients ”stimulus” data cc ~ 0.0086 ± 0.0278 Experiments: Cited values of cc~0.01 [Schneidmann et al, Nature (2006)]
Modeling the distribution of spike patterns Have sets of spike patterns {Si}k Si = ±1 for spike/no spike(we use10-ms bins) (temporal order irrelevant)
Modeling the distribution of spike patterns Have sets of spike patterns {Si}k Si = ±1 for spike/no spike(we use10-ms bins) (temporal order irrelevant) Construct a distribution P[S] that generates the observed patterns (i.e., has the same correlations)
Modeling the distribution of spike patterns Have sets of spike patterns {Si}k Si = ±1 for spike/no spike(we use10-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): Ising model, parametrized by Jij, hi
An inverse problem: Have: statistics <Si>, <SiSj> want: hi, Jij
An inverse problem: Have: statistics <Si>, <SiSj> want: hi, Jij Exact method: Boltzmann learning
An inverse problem: Have: statistics <Si>, <SiSj> want: hi, Jij Exact method: Boltzmann learning
An inverse problem: Have: statistics <Si>, <SiSj> want: hi, Jij Exact method: Boltzmann learning Requires long Monte Carlo runs to compute model statistics
1. (Naïve) mean field theory Mean field equations: or
1. (Naïve) mean field theory Mean field equations: or Inverse susceptibility (inverse correlation) matrix
1. (Naïve) mean field theory Mean field equations: or Inverse susceptibility (inverse correlation) matrix So, given correlation matrix, invert it, and
2. TAP approximation Thouless, Anderson, Palmer, Phil Mag 35 (1977) Kappen & Rodriguez, Neural Comp 10 (1998) Tanaka, PRE 58 2302 (1998) “TAP equations” (improved MFT for spin glasses)
2. TAP approximation Thouless, Anderson, Palmer, Phil Mag 35 (1977) Kappen & Rodriguez, Neural Comp 10 (1998) Tanaka, PRE 58 2302 (1998) “TAP equations” (improved MFT for spin glasses)
2. TAP approximation Thouless, Anderson, Palmer, Phil Mag 35 (1977) Kappen & Rodriguez, Neural Comp 10 (1998) Tanaka, PRE 58 2302 (1998) “TAP equations” (improved MFT for spin glasses) Onsager “reaction term”
2. TAP approximation Thouless, Anderson, Palmer, Phil Mag 35 (1977) Kappen & Rodriguez, Neural Comp 10 (1998) Tanaka, PRE 58 2302 (1998) “TAP equations” (improved MFT for spin glasses) Onsager “reaction term”
2. TAP approximation Thouless, Anderson, Palmer, Phil Mag 35 (1977) Kappen & Rodriguez, Neural Comp 10 (1998) Tanaka, PRE 58 2302 (1998) “TAP equations” (improved MFT for spin glasses) Onsager “reaction term” A quadratic equation to solve for Jij
3. Independent-pair approximation Solve the two-spin problem:
3. Independent-pair approximation Solve the two-spin problem: Solve for J:
3. Independent-pair approximation Solve the two-spin problem: Solve for J: Low-rate limit:
4. Sessak-Monasson approximation A combination of naïve mean field theory and independent-pair approximations:
4. Sessak-Monasson approximation A combination of naïve mean field theory and independent-pair approximations:
4. Sessak-Monasson approximation A combination of naïve mean field theory and independent-pair approximations: (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 TAP low-rate TAP SM TAP/SM SM TAP/SM
Comparing approximations: N=20 N =200 nMFT ind pair nMFT ind pair low-rate TAP low-rate TAP SM TAP/SM SM TAP/SM the winner!
Error measures SM/TAP SM TAP nMFT ind pair low-rate ind pair low-rate nMFT TAP SM SM/TAP
N-dependence: How do the inferred couplings depend on the size of the set of neurons used in the inference algorithm?