This cannot be estimated directly... s 1. s 2. P(spike, stim) P(stim) P(spike stim) =

Transcription

1 LNP cascade model Simplest successful descriptive spiking model Easily fit to (extracellular) data Descriptive, and interpretable (although not mechanistic)

2 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...

3 ML estimation of LNP [on board]

4 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]

5 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]

6 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

7 Computing the STA s1 s2 raw stimuli spiking stimuli

8 STA corresponds to a direction in stimulus space

9 Projecting onto the STA P ( spike(t) ) k s(t) = P ( spike(t) & ) k s(t) /P ( s(t))

10 Solving for nonlinearity nonparametrically

11 Solving for nonlinearity nonparametrically STA response

12 Solving for nonlinearity nonparametrically STA response

13 Projecting onto an axis orthogonal to the STA

14 Projecting onto an axis orthogonal to the STA

15 Projecting onto an axis orthogonal to the STA

16 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

17 !"#$%&& 0 time (s) 1!"# &'( $% '()##*+#,-. /0#)#*+#,-. - Pillow etal, 2004

18 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] Space, X [deg] T=150 ms X X T= 200 ms T T - Ozhawa, etal

19 A B T [ms] C X [deg] Response [spikes/sec] SF [cyc/deg] TF [Hz]

20 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)...

21 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

22 2 2 2 variance variance variance eigenvalue number eigenvalue number eigenvalue number Error Bootstrap error Asymptotic error Number spikes / stimulus dimensions

23 Example 1: sparse noise true STA

24 Example 2: uniform noise true

25 LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells)

26 LNP limitations Symmetric nonlinearities and/or multidimensional front-end (e.g., V1 complex cells) Subspace LNP

27 Classic V1 models Simple cell Complex cell +

28 V1 simple cell time tf STA space sf 1.8 variance STC eigenvalue number [Rust, Schwartz, Movshon, Simoncelli, 05]

29 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

30 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)

31 Linear-Nonlinear-Poisson (LNP) stimulus filter point nonlinearity probabilistic spiking

32 Recursive LNP stimulus filter exponential nonlinearity probabilistic spiking + post-spike waveform [Truccolo et al 05; Pillow et al 05]

33 stimulus & spike train model parameters Critical: Likelihood function has no local maxima [Paninski 04]

34 stimulus & spike train model parameters maximize likelihood Critical: Likelihood function has no local maxima [Paninski 04]

35 28

36 rlnp model stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform

37 rlnp model stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform +

38 Multi-cell rlnp, with model spike coupling stimulus filter exponential nonlinearity probabilistic spiking stimulus + post-spike waveform coupling waveforms +

39 Equivalent model diagram Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n 30

40 Equivalent model diagram Equivalent model diagram x(t) y 1 (t) K h 11 y n (t) h 1n 30

41 stimulus & spike trains model parameters

42 stimulus & spike trains model parameters maximize likelihood

43 stimulus & spike trains model parameters maximize likelihood Again: Likelihood function has no local maxima [Paninski 04]

44 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]

45 Example ON cell Example OFF cell [Pillow, Shlens, Paninski, Chichilnisky, Simoncelli - unpublished]

46 ON cells OFF cells Cross-Correlations RGC rlnp GLM rate (sp/s) time (ms) time (ms)

47 ON cells OFF cells Cross-Correlations RGC GLM rlnp(no coupling) rate (sp/s) time (ms) time (ms)

48

49 Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen )

50 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

51 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

52 Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) spike prediction (bits/sp) LNP

53 Single-trial spike prediction RGC responses single-trial predictions (neighbor spikes frozen ) 2 spike prediction (bits/sp) full model uncoupled model LNP

54 Decoding Stimulus information (bits/s) opt. linear LNP rlnp crlnp