Post on 29-Jul-2020
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.