Neural Coding: Integrate-and-Fire Models of Single and Multi-Neuron Responses

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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

General Goal: understand the mapping from stimuli to spike responses with the use of a model y x stimulus model spike response Model criteria: flexibility (captures realistic neural properties) tractability (for fitting to data)

Example 1: Hodgkin-Huxley stimulus Na + activation (fast) Na + inactivation (slow) K + activation (slow) spike response + flexible, biophysically realistic - not easy to fit

Example 2: LNP x K f y (receptive field) + easy to fit (spike-triggered averaging) + not biologically plausible

LNP model stimulus filter K filter output spike rate spikes time (sec)

cascade models y x stimulus model spike response Linear Filtering Nonlinear Probabilistic Spiking more realistic models of spike generation

Generalized Integrate- and- Fire Model x(t) K y(t) h I stim I spike I noise related: Spike Response Model, Gerstner & Kistler 02

Generalized Integrate- and- Fire Model K h + powerful, flexible + tractable for fitting

Model behaviors: adaptation

Model behaviors: bursting

Model behaviors: bistability 0 0

The Estimation Problem Learn the model parameters: K = stimulus filter g = leak conductance σ 2 = noise variance h = response current V L = reversal potential K h From: stimulus train x(t) spike times t i Solution: Maximum Likelihood - need an algorithm to compute P θ (y x)

Likelihood function hidden variable: P(spike at t i ) = fraction of paths crossing threshold at t i t i

Likelihood function hidden variable: P(spike at t i ) = fraction of paths crossing threshold at t i t i

Computing Likelihood Diffusion Equation: linear dynamics additive Gaussian noise P(V,t) 1 τ P(V,t+ t) fast methods for solving linear PDE efficient procedure for computing likelihood t i

Computing Likelihood Diffusion Equation: (Fokker-Planck) linear dynamics additive Gaussian noise 1 τ fast methods for solving linear PDE efficient procedure for computing likelihood reset ISIs are conditionally independent likelihood is product over ISIs

Maximizing the likelihood parameter space is large ( 20 to 100 dimensions) parameters interact nonlinearly Main Theorem: The log likelihood is concave in the parameters {K, τ, σ, h, V L }, for any data {x(t), t i } 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 I inj V time (ms)

IF model simulation Stimulus filter K I inj h V Noise time (ms)

ON cell RGC LNP IF 74% of var 92 % of var

Accounting for spike timing precision 0 time (ms) 200 P(spike)

Accounting for reliability

Decoding the neural response Stim 1 Stim 2 Resp 1 Resp 2?

Solution: use P(resp stim) Stim 1 Stim 2 a P(R1 S1)P(R2 S2) Resp 1 Resp 2? P(R1 S2)P(R2 S1)

Discriminate each repeat using P(Resp Stim) Stim 1 Stim 2 P(R1 S1)P(R2 S2) Resp 1 Resp 2? P(R1 S2)P(R2 S1)

Discriminate each repeat using P(Resp Stim) Stim 1 Stim 2 Compare to LNP model P(Resp Stim) Resp 1 Resp 2? 94 % correct 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.

2 types of correlation: 1. stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r 1,r 2 stim) = P(r 1 stim)p(r 2 stim) cell 1 cell 2 stimuli responses

2 types of correlation: 1. stimulus-induced correlation: persists even if responses are conditionally independent, i.e. P(r 1,r 2 stim) = P(r 1 stim)p(r 2 stim) 2. noise correlation: arises if responses are not conditionally independent given the stimulus, i.e. P(r 1,r 2 stim) P(r 1 stim)p(r 2 stim) cell 1 Noise cell 2 stimuli responses

Modeling multi- neuron responses x K h 11 y 1 coupling h currents: h 12 h 21 x K h 22 y 2

Methods t y x 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

cell 1 Fits OFF cells cell 2 cell 3 cell 4 ON cells cell 5

cell 1 Fits cell 2 ON + OFF cells cell 3 cell 4 cell 5

cell 1 Fits cell 2 cell 3 cell 4 cell 5

Compare likelihoods: Pairwise coupling analysis 1. The single-cell model for cell i: novel stim stimulus filter IF spikes vs. 2. The pairwise model for i with coupling from cell j novel stim stimulus filter IF spikes cell j spikes h ij

Coupling Matrix Pairwise coupling analysis likelihood ratio Functional Coupling

Accounting for the autocorrelation RGC simulated model post-spike current OFF cells 1 2

Accounting for cross-correlations ON-ON correlations raw (stimulus + noise) RGC coupled model 15 10 5 0-5 -100-50 0 50 100 time (ms) stimulus-induced RGC, shuffled uncoupled model 2 1 0-1 -100-50 0 50 100 time (ms) 4 to 5 5 to 4

OFF-OFF cell correlations raw (stimulus + noise) RGC coupled model stimulus-induced RGC, shuffled uncoupled model 6 4-2 02 6 4 2 0-2 -100-50 0 50 100 time (ms) 1 vs 3 2 vs 3 1 0-1 -2 1 0-1 1 vs 3-100 -50 0 50 100 time (ms) 3 to 2 2 to 3

OFF-ON cell correlations raw (stimulus + noise) RGC coupled model stimulus-induced RGC, shuffled uncoupled model 4-2 02-4 -6-100 -50 0 50 100 time (ms) 1 vs 4-100 -50 0 50 100 time (ms) 1 to 4 4 to 1

OFF-ON cell correlations 5 0-5 raw (stimulus + noise) 1 vs 5 stimulus-induced 2 0-2 2 0-2 -4 4 2 0-2 -4 15 10 5 0-5 -100-50 0 50 100 time (ms) 2 vs 4 2 vs 5 3 vs 4 3 vs 5-100 -50 0 50 100 time (ms)

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 Jonathon Shlens Eero Simoncelli - The Salk Institute - HHMI & NYU Liam Paninski - Columbia U.

Basis used for coupling currents time after a spike (ms)

Extra slides:

5-way coupling analysis likelihood ratio Likelihood ratio for fully connected model 3 2 1 1 2 3 4 5 Cell # Functional Coupling

5-way coupling analysis likelihood ratio Likelihood ratio for fully connected model 3 2 1 1 2 3 4 5 Cell # Conclusion: the fully connected model gives an improved description of multi-cell responses to white noise stimuli.

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