LNP cascade model Simplest successful descriptive spiking model Easily fit to (extracellular) data Descriptive, and interpretable (although not mechanistic)
For a Poisson model, response is captured by relationship between the distribution of red points (spiking stim) and blue points (raw stim), expressed in terms of Bayes rule: s 1 P(spike stim) = P(spike, stim) P(stim) s 2 This cannot be estimated directly...
ML estimation of LNP [on board]
ML estimation of LNP If f θ ( k x) and log is convex (in argument and theta), f θ ( k x) is concave, the likelihood of the LNP model is convex (for all observed data, {n(t), x(t)} ) [Paninski, 04]
ML estimation of LNP If f θ ( k x) and log is convex (in argument and theta), f θ ( k x) is concave, the likelihood of the LNP model is convex (for all observed data, {n(t), x(t)} ) Examples: e ( k x(t)) ( k x(t)) α, 1 < α < 2 [Paninski, 04]
Simple LNP fitting Assuming: - stochastic stimuli, spherically distributed - mean of spike-triggered ensemble is shifted from that of raw ensemble Reverse correlation gives an unbiased estimate of k (for any f). For exponential f, this is the ML estimate! - Bussgang 52; de Boer & Kuyper 68
Computing the STA s1 s2 raw stimuli spiking stimuli
STA corresponds to a direction in stimulus space
Projecting onto the STA P ( spike(t) ) k s(t) = P ( spike(t) & ) k s(t) /P ( s(t))
Solving for nonlinearity nonparametrically
Solving for nonlinearity nonparametrically STA response
Solving for nonlinearity nonparametrically STA response
Projecting onto an axis orthogonal to the STA
Projecting onto an axis orthogonal to the STA
Projecting onto an axis orthogonal to the STA
Figure 3. Characterization of light response in one ON cell (A, B) and one OFF cell (C, D) simultaneously recorded in salamander retina. A, C, The spike-triggered average L-cone contrast during random flicker stiminput strength - Chander & Chichilnisky 01
!"#$%&& 0 time (s) 1!"# &'( $% '()##*+#,-. /0#)#*+#,-. - Pillow etal, 2004
V1 simple cell Time, T [msec] T=100 ms T= 50 ms T= 50 ms T= 50 ms T=100 ms T=100 ms T=150 ms T=150 ms T= 200 ms T= 200 ms X X X X T T T T Y Y Y Space, X [deg] 300 Space, X [deg] Time, T [msec] Time, T [msec] 0 4 0 4 0 0 300 300 0 0 4 Space, X [deg] T=150 ms X X T= 200 ms T T - Ozhawa, etal
A B T [ms] C 300 0 0 5 700 0 0 3.5 400 0 0 10 X [deg] Response [spikes/sec] 25 20 15 10 5 0 0.1 1 2 25 20 15 10 5 0 0.1 1 2 20 15 10 5 0 0.1 1 2 SF [cyc/deg] 25 20 15 10 5 0 0.1 1 10 20 25 20 15 10 5 0 0.1 1 10 20 15 10 5 0 0.1 1 10 20 TF [Hz]
LNP summary LNP is the defacto standard descriptive model, and is implicit in much of the experimental literature Accounts for basic RF properties Accounts for basic spiking properties (rate code) Easily fit to data Easily interpreted BUT, non-mechanistic, and exhibits striking failures (esp. beyond early sensory/motor)...
STA estimation errors Convergence rate [Paninski, 03] e(ˆk) σ E( k s) d N σ = stim s.d., d = stim dim., N = nspikes Non-spherical stimuli can cause biases Model failures: - symmetric nonlinearity (causes no change in STE mean) - response not captured by 1D projection - non-poisson spiking behaviors
2 2 2 variance 1.5 1 variance 1.5 1 variance 1.5 1 0.5 0.5 0.5 0 10 20 30 40 eigenvalue number 0 10 20 30 40 eigenvalue number 0 10 20 30 40 eigenvalue number 0.1 0.08 Error 0.06 0.04 0.02 Bootstrap error Asymptotic error 0 10 0 10 1 10 2 10 3 Number spikes / stimulus dimensions
Example 1: sparse noise true STA
Example 2: uniform noise true
LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells)
LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells) Subspace LNP
Classic V1 models Simple cell Complex cell +
V1 simple cell time tf STA space sf 1.8 variance 1.6 1 0.6 STC 0.2 0 50 100 150 200 250 eigenvalue number [Rust, Schwartz, Movshon, Simoncelli, 05]
LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells) Subspace LNP Linear front end insufficient for non-peripheral neuraons Cascades / fixed front-end nonlinearities
LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells) Subspace LNP Linear front end insufficient for non-peripheral neuraons Cascades / fixed front-end nonlinearities Responses depend on spike history, other cells Recursive models (GLM)
Linear-Nonlinear-Poisson (LNP) stimulus filter point nonlinearity probabilistic spiking
Recursive LNP stimulus filter exponential nonlinearity probabilistic spiking + post-spike waveform [Truccolo et al 05; Pillow et al 05]
stimulus & spike train model parameters Critical: Likelihood function has no local maxima [Paninski 04]
stimulus & spike train model parameters maximize likelihood Critical: Likelihood function has no local maxima [Paninski 04]
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rlnp model stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform
rlnp model stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform +
Multi-cell rlnp, with model spike coupling stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform coupling waveforms +
Equivalent model diagram Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n 30
Equivalent model diagram Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n 30
stimulus & spike trains model parameters
stimulus & spike trains model parameters maximize likelihood
stimulus & spike trains model parameters maximize likelihood Again: Likelihood function has no local maxima [Paninski 04]
Methods Methods spatiotemporal binary white noise (8 minutes at 120Hz) macaque retinal ganglion cell (RGC) spike responses (ON and OFF parasol) [Pillow, Shlens, Paninski, Chichilnisky, Simoncelli - unpublished]
Example ON cell Example OFF cell [Pillow, Shlens, Paninski, Chichilnisky, Simoncelli - unpublished]
ON cells OFF cells Cross-Correlations 30 20 30 20 RGC rlnp GLM rate (sp/s) 30 20 30 20 30 20 30 20-50 0 50 time (ms) -50 0 50 time (ms)
ON cells OFF cells Cross-Correlations 30 20 30 20 RGC GLM rlnp(no coupling) rate (sp/s) 30 20 30 20 30 20 30 20-50 0 50 time (ms) -50 0 50 time (ms)
Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen )
Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n
Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n
Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) spike prediction (bits/sp) 2 1.5 1 1 1.5 2 LNP
Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) 2 spike prediction (bits/sp) full model 1.5 1 uncoupled model 1 1.5 2 LNP
Decoding Stimulus information (bits/s) opt. linear LNP rlnp crlnp