Integrate-and-fire-based neural

encoding models

Liam Paninski

Department of Statistics and Center for Theoretical Neuroscience

Columbia University

http://www.stat.columbia.edu/∼liam

liam@stat.columbia.edu

June 27, 2007

Neural encoding models

“Encoding model”: pθ(r|s).

— Fit parameter θ instead of full p(r|s)

Main theme: we want model to be flexible but not overly so

Flexibility vs. “fittability”

— Also want model to reflect underlying biophysics

Multiparameter HH-type model

— highly biophysically plausible, flexible

— but very difficult to estimate parameters given spike times alone(figure adapted from (Fohlmeister and Miller, 1997))

Cascade (“LNP”) model

— easy to estimate: spike-triggered averaging

(Simoncelli et al., 2004)

— but not biophysically plausible (fails to capture spike timing

details: refractoriness, burstiness, adaptation, etc.)

Two key ideas

1. Use likelihood-based methods for fitting.

— well-justified statistically

— easy to incorporate prior knowledge, explicit noise

models, etc.

2. Use models that are easy to fit via maximum likelihood

— concave (downward-curving) functions have no

non-global local maxima =⇒ concave functions are easy to

maximize by gradient ascent.

Recurring theme: find flexible models whose loglikelihoods are

guaranteed to be concave.

Filtered integrate-and-fire model

dV (t) =

(

−g(t)V (t) + IDC + ~k · ~x(t) +0∑

j=−∞

h(t − tj)

)

dt+σdNt;

(Gerstner and Kistler, 2002; Paninski et al., 2004b)

Model flexibility: Adaptation

Model flexibility: Burstiness

Model flexibility: Bistability

The estimation problem

(Paninski et al., 2004b)

First passage time likelihood

P (spike at ti) = fraction of paths crossing threshold for first time at ti

(computed numerically via Fokker-Planck or integral equation methods)

Method 2: the “method of images”

If Vth is “far away” (negligible), then Gθ(y, t|x, s) is computable

(Gaussian): probability of V (t) = y, given V (s) = x

— Every time a particle hits threshold, we have to subtract its

contribution away:

P (V, t) = G(V, t|Vreset, 0) −

∫ t

0

G(V, t|Vth, s)p(s)ds

p(t) = P (spike at time t)

Integral equation

Since P (Vth, t) = 0,

Gθ(Vth, t|Vreset, 0) =

∫ t

0

Gθ(Vth, t|Vth, s)p(s)ds

Discretizing time into d points, we get a lower-triangular linear

system: O(d2) (Paninski et al., 2007)

Can compute gradient p(t) w.r.t. θ via matrix perturbation:

efficient maximization

Method 3: Hidden Markov model approach

V (t) is Markovian; spikes are instantaneous observations of V :

hidden Markov model

Compute likelihood via

p(O0:T ) =

∫

p(O0:T , VT )dVT

p(O0:t, Vt) via standard forward probability recursion

EM algorithm

Need to compute Ep(V|O,θ(i)) log p(V,O|θ).

Model:

V (t + dt) = V (t) + dt(

I − gV (t))

+ ǫt, ǫt ∼ N (0, σ2dt)

Complete log-likelihood:

log p(V,O|θ) = −1

2σ2dt

T∑

t=0

[

V (t + dt) − V (t) − dt(

I − gV (t))]2

E-step: standard forward-backward algorithm for E(V (t)|O)

M-step: simple quadratic optimization in (I, g)

(Nikitchenko and Paninski, 2007)

Maximizing likelihood

Maximization seems difficult, even intractable:

— high-dimensional parameter space

— likelihood is a complex nonlinear function of parameters

Main result: The loglikelihood is concave in the parameters,

no matter what data {~x(t), ti} are observed.

=⇒ no non-global local maxima

=⇒ maximization easy by ascent techniques.

Proof of log-concavity theorem

Based on probability integral representation of likelihood:

L(θ) =

∫

1(V ∈ C)G~x,θ(V)dV

G~x,θ(V) = Gaussian probability on voltage paths V

C = set of voltage paths V (t) consistent with spike data:

V (t) ≤ Vth; V (ti) = Vth; V (t+i ) = Vreset

Now use fact that marginalizing preserves log-concavity

(Prekopa, 1973): if f(~x, ~y) is jointly l.c., then so is

f0(~x) ≡

∫

f(~x, ~y)d~y.

Application: retinal ganglion cellsPreparation: dissociated salamander and macaque retina

— extracellularly-recorded responses of populations of RGCs

Stimulus: random “flicker” visual stimuli (Chander and Chichilnisky, 2001)

Spike timing precision in retina

0 0.25 0.5 0.75 1

RGC

LNP

IF

0

200

rate

(sp

/sec

) RGCLNPIF

0 0.25 0.5 0.75 10

0.5

1

1.5

varia

nce

(sp2 /b

in)

0.07 0.17 0.22 0.26

0.6 0.64 0.85 0.9

(Pillow et al., 2005)

Spike timing precision in retina

0

1

2

stimulus filter k

−1

0

1

2

3

spike response r

−150 −100 −50 0

−2

−1

0

1

t (ms)0 10 20

−1

0

1

2

3

t (ms)

−2 0 20

5

10

VL

coun

t

0 10 20

τ (ms)0.25 0.5 0.75

σn

0

1

2

STA

k

0

100

200

nonlinearity

firin

g ra

te (

sp/s

ec)

−150 −100 −50 0

−2

−1

0

1

t (ms)−2 0 2

0

100

firin

g ra

te (

sp/s

ec)

STA projection

(Pillow et al., 2005)

Linking spike reliability and

subthreshold noise

(Pillow et al., 2005)

Likelihood-based discrimination

Given spike data, optimal decoder chooses stimulus ~x according

to likelihood: p(spikes|~x1) vs. p(spikes|~x2).

Using correct model is essential (Pillow et al., 2005)

Application 2: Decoding subthreshold activity

Given extracellular spikes, what is most likely intracellular V (t)?

5.1 5.15 5.2 5.25 5.3 5.35 5.4 5.45 5.5 5.55

−80

−70

−60

−50

−40

−30

−20

−10

0

10

time (sec)

V (

mV

)

?

Computing VML(t)Loglikelihood of V (t) (given LIF parameters, white noise Nt):

L({V (t)}0≤t≤T ) = −1

2σ2

∫ T

0

[

V̇ (t) −

(

− gV (t) + I(t)

)]2

dt

Constraints:

• Reset at t = 0:

V (0) = Vreset

• Spike at t = T :

V (T ) = Vth

• No spike for 0 < t < T :

V (t) < Vth

Quadratic programming problem: optimize quadratic function

under linear constraints. Concave: unique global optimum.

Most likely vs. average V (t)

0 0.5 10

0.5

1

V

0

0.5

1

0 0.5 10

0.5

1

V

0

0.5

1

0 0.5 10

0.5

1

t

V

t0 0.5 1

0

0.5

1

σ=0.4

σ=0.2

σ=0.1

P (V (t)|{ti}, θ̂ML, ~x) via forward-backward algorithm (Paninski, 2006a).

The doublet-triggered average

forw

ard

−1

0

1

back

war

d

−3

−2

−1

0

1fu

ll

−1

0

1

−0.5

0

0.5

1

V

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

time

V

samp statrue staML V

(Paninski, 2006b)

The spike-triggered average

0

0.5

1

V

0

0.5

1

V

0

0.5

1

V

−2 −1 0 1 20

0.5

1

τ

V

µ = 2; σ = 0.2

µ = 2; σ = 0.4

µ = 2; σ = 1.0

µ = 2; σ = 1.5

(Paninski, 2006b)

Application: in vitro dataRecordings: rat sensorimotor cortical slice; dual-electrode whole-cell

Stimulus: Gaussian white noise current I(t)

Analysis: fit IF model parameters {g,~k, h(.), Vth, σ} by maximum likelihood

(Paninski et al., 2003; Paninski et al., 2004a), then compute VML(t)

Application: in vitro data

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 1.13

−60

−40

−20

0

V (

mV

)

1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.11 1.12 1.13−75

−70

−65

−60

−55

−50

−45

time (sec)

V (

mV

)

true V(t)V

ML(t)

Extension to nonlinear caseGeneral case: nonlinear subthreshold noisy dynamics:

dV (t) = f(V, t)dt + σdNt

e.g., linear case:

f(V, t) = −gV (t) + I(t)

Constraints:

• Reset at t = 0:

V (0) = Vreset

• Spike at t = T :

V (T ) = Vth

• No spike for 0 < t < T :

V (t) < Vth

Computing VML(t)

Loglikelihood of V (t) (under white noise Nt):

L({V (t)}0≤t≤T ) = −1

2σ2

∫ T

0

(

V̇ (t) − f(V (t), t))2

dt

Under linear dynamics f(V, t): standard quadratic optimization

More generally: calculus of variations, Kuhn-Tucker conditions

=⇒ VML(t) solves second-order ODE or inequality

Example: quadratic integrate-and-fire cell

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.2

0.4

V

f(V

)

0 1 2 3 4 5 6 7 8 9 10

−0.3

−0.2

−0.1

0

t

NM

L

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

t

VM

L

(Paninski, 2006a)

Next

— decoding full stimulus, not just two alternatives

— analyzing population codes, not just one neuron at a time

— closer connections to biophysical models

ReferencesChander, D. and Chichilnisky, E. (2001). Adaptation to temporal contrast in primate and salamander retina.

Journal of Neuroscience, 21:9904–16.

Fohlmeister, J. and Miller, R. (1997). Mechanisms by which cell geometry controls repetitive impulse firing in

retinal ganglion cells. Journal of Neurophysiology, 78:1948–1964.

Gerstner, W. and Kistler, W. (2002). Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge

University Press.

Nikitchenko, M. and Paninski, L. (2007). An expectation-maximization Fokker-Planck algorithm for the noisy

integrate-and-fire model. COSYNE.

Paninski, L. (2006a). The most likely voltage path and large deviations approximations for integrate-and-fire

neurons. Journal of Computational Neuroscience, 21:71–87.

Paninski, L. (2006b). The spike-triggered average of the integrate-and-fire cell driven by Gaussian white

noise. Neural Computation, 18:2592–2616.

Paninski, L., Haith, A., Pillow, J., and Williams, C. (2007). Integral equation methods for computing

likelihoods and their derivatives in the stochastic integrate-and-fire model. Journal of Computational

Neuroscience, In press.

Paninski, L., Lau, B., and Reyes, A. (2003). Noise-driven adaptation: in vitro and mathematical analysis.

Neurocomputing, 52:877–883.

Paninski, L., Pillow, J., and Simoncelli, E. (2004a). Comparing integrate-and-fire-like models estimated using

intracellular and extracellular data. Neurocomputing, 65:379–385.

Paninski, L., Pillow, J., and Simoncelli, E. (2004b). Maximum likelihood estimation of a stochastic

integrate-and-fire neural model. Neural Computation, 16:2533–2561.

Pillow, J., Paninski, L., Uzzell, V., Simoncelli, E., and Chichilnisky, E. (2005). Prediction and decoding of

retinal ganglion cell responses with a probabilistic spiking model. Journal of Neuroscience,

25:11003–11013.

Prekopa, A. (1973). On logarithmic concave measures and functions. Acad Sci. Math., 34:335–343.

Simoncelli, E., Paninski, L., Pillow, J., and Schwartz, O. (2004). Characterization of neural responses with

stochastic stimuli. In The Cognitive Neurosciences. MIT Press, 3rd edition.