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Neural Coding: Integrate-and-Fire Models of Single and Multi-Neuron Responses. Jonathan Pillow HHMI and NYU http://www.cns.nyu.edu/~pillow Oct 5, Course lecture: “Computational Modeling of Neuronal Systems” Fall 2005, New York University. x stimulus. model. spike response.
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Neural Coding: Integrate-and-Fire Models of Single and Multi-Neuron Responses Jonathan Pillow HHMI and NYU http://www.cns.nyu.edu/~pillow Oct 5, Course lecture: “Computational Modeling of Neuronal Systems” Fall 2005, New York University
x stimulus model spike response General Goal: understand the mapping from stimuli to spike responses with the use of a model y • Model criteria: • flexibility (captures realistic neural properties) • tractability (for fitting to data)
Example 1: Hodgkin-Huxley Na+ activation (fast) spike response Na+ inactivation (slow) stimulus K+ activation (slow) + flexible, biophysically realistic - not easy to fit
Example 2: LNP K f x y (receptive field) + easy to fit (spike-triggered averaging) +not biologically plausible
LNP model stimulus filter K filter output spike rate spikes time (sec)
x stimulus model spike response Linear Filtering Nonlinear Probabilistic Spiking more realistic models of spike generation “cascade” models y
K h Generalized Integrate-and-Fire Model x(t) y(t) Inoise Istim Ispike related: “Spike Response Model”, Gerstner & Kistler ‘02
K h Generalized Integrate-and-Fire Model + powerful, flexible + tractable for fitting
0 0 Model behaviors: bistability
The Estimation Problem Learn the model parameters: K = stimulus filter g = leak conductance s 2 = noise variance h = response current VL = reversal potential K h From: stimulus train x(t) spike times ti Solution: Maximum Likelihood - need an algorithm to compute Pq(y|x)
P(spike at ti) = fraction of paths crossing threshold at ti ti Likelihood function hidden variable:
ti Likelihood function hidden variable: P(spike at ti) = fraction of paths crossing threshold at ti
1 t Diffusion Equation: P(V,t) P(V,t+Dt) fast methods for solving linear PDE efficient procedure for computing likelihood ti Computing Likelihood • linear dynamics • additive Gaussian noise
1 t Diffusion Equation: (Fokker-Planck) fast methods for solving linear PDE efficient procedure for computing likelihood Computing Likelihood • linear dynamics • additive Gaussian noise reset ISIs are conditionally independent likelihood is product over ISIs
Main Theorem: The log likelihood is concave in the parameters {K, t, s, h, VL} , for any data {x(t), ti} Maximizing the likelihood • parameter space is large ( 20 to 100 dimensions) • parameters interact nonlinearly gradient ascent guaranteed to converge to global maximum! [Paninski, Pillow & Simoncelli. Neural Comp. ‘04
Application to Macaque Retina • isolated retinal ganglion cell (RGC) • stimulated with full-field random stimulus (flicker) • fit using 1-minute period of response t (Data: Valerie Uzzell & E.J. Chichilnisky)
IF model simulation Stimulus filter K Iinj V time (ms)
Noise IF model simulation Stimulus filter K Iinj h V time (ms)
ON cell RGC LNP IF 74% of var 92 % of var
0 time (ms) 200 P(spike) Accounting for spike timing precision
Stim 1 Stim 2 Resp 1 ? Resp 2 Decoding the neural response
Stim 1 Stim 2 a Resp 1 ? Resp 2 Solution: use P(resp|stim) P(R1|S1)P(R2|S2) P(R1|S2)P(R2|S1)
Stim 1 Stim 2 Discriminate each repeat using P(Resp|Stim) Resp 1 Resp 2 ? P(R1|S1)P(R2|S2) P(R1|S2)P(R2|S1)
Stim 1 Stim 2 Discriminate each repeat using P(Resp|Stim) Resp 1 Resp 2 ? 94 % correct Compare to LNP model P(Resp|Stim) LNP: 68%correct
Decoding the neural response IF model % correct LNP model % correct
Part 2: how to characterize the responses of multiple neurons? • Want to capture: • the stimulus dependence of each neuron’s response • the response dependencies between neurons.
cell 1 cell 2 2 types of correlation: • stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r1,r2| stim) = P(r1|stim)P(r2|stim) stimuli responses
cell 1 cell 2 2 types of correlation: • stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r1,r2| stim) = P(r1|stim)P(r2|stim) 2. “noise” correlation: arises if responses are not conditionally independent given the stimulus, i.e. P(r1,r2| stim) P(r1|stim)P(r2|stim) Noise stimuli responses
y1 y2 Modeling multi-neuron responses K x h11 h12 coupling h currents: h21 K x h22
Methods spatiotemporal binary white noise (24 x 24 pixels, 120Hz frame rate) simultaneous multi-electrode recordings of macaque RGCs • Model parameters fit to five RGCs using 10 minutes of response to a non-repeating binary white noise stimulus
OFF cells cell 1 cell 2 cell 3 ON cells cell 4 cell 5 Fits
cell 1 ON + OFF cells cell 2 cell 3 cell 4 cell 5 Fits
cell 1 cell 2 cell 3 cell 4 cell 5 Fits
spikes s s t t i i m m u u l l u u s s f f i i l l t t e e r r IF IF novel stim spikes novel stim hij cell j spikes Pairwise coupling analysis • Compare likelihoods: • The single-cell model for cell i: • vs. • 2. The pairwise model for i with coupling from cell j
Pairwise coupling analysis Coupling Matrix likelihood ratio Functional Coupling
Accounting for the autocorrelation RGC simulated model post-spike current O F F c e l l s 1 2
RGC coupled model RGC, shuffled uncoupled model Accounting for cross-correlations ON-ON correlations raw (stimulus + noise) stimulus-induced 2 1 5 1 0 1 5 0 0 - 1 - 5 - 1 0 0 - 5 0 0 5 0 1 0 0 - 1 0 0 - 5 0 0 5 0 1 0 0 t i m e ( m s ) t i m e ( m s ) 4 to 5 5 to 4
RGC coupled model RGC, shuffled uncoupled model raw (stimulus + noise) stimulus-induced OFF-OFF cell correlations 1 6 4 0 1 vs 3 2 - 1 0 - 2 - 2 1 6 4 2 vs 3 0 2 1 vs 3 0 - 1 - 2 - 1 0 0 - 5 0 0 5 0 1 0 0 t i m e ( m s ) 3 to 2 2 to 3
RGC coupled model RGC, shuffled uncoupled model raw (stimulus + noise) stimulus-induced OFF-ON cell correlations 4 2 1 vs 4 0 - 2 - 4 - 6 1 to 4 4 to 1
5 0 - 5 2 0 - 2 2 0 - 2 - 4 4 2 0 - 2 - 4 1 5 1 0 5 0 - 5 - 1 0 0 - 5 0 0 5 0 1 t i m e ( m OFF-ON cell correlations raw (stimulus + noise) stimulus-induced 1 vs 5 2 vs 4 2 vs 5 3 vs 4 3 vs 5 - 1 0 0 - 5 0 0 5 0 1 0 0 0 0 s ) t i m e ( m s )
Conclusions 1. generalized-IF model: flexible, tractable tool for modeling neural responses 2. fitting with maximum likelihood 3. probabilistic framework: useful for encoding (precision, response variability) and decoding 4. easily extended to multi-neuron responses 5. likelihood test of functional connectivity between cells 6. explains auto- and cross-correlations 7. resolves cross-correlations into “stimulus-induced” and “noise-induced”
My collaborators: E.J. Chichilnisky Valerie Uzzell - The Salk Institute Jonathon Shlens Eero Simoncelli - HHMI & NYU Liam Paninski - Columbia U.
5-way coupling analysis Likelihood ratio for fully connected model Functional Coupling
5-way coupling analysis Likelihood ratio for fully connected model Conclusion: the fully connected model gives an improved description of multi-cell responses to white noise stimuli.