statistical models of visual neuronside/data/teaching/amsc663/16... · 2017. 3. 8. · need higher...
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Statistical models of visual neurons
Anna Sotnikova Applied Mathematics and Statistics, and Scientific Computation
program
Advisor: Dr. Daniel A. Butts Department of Biology
Update Presentation
Visual system of a neuron
http://www.pc.rhul.ac.uk/staff/j.zanker/ps1061/l2/ps1061_2.htm
spikes
The goal of this field is to identify relationship between the visual stimuli and the resulting neural responses
light electricity
Statistical modeling of neuron’s response
What is the simplest model which describes the computation being performed by the neuron?
Stimulus -> -> Firing rate ?
Prob(n(t)) =r(t)n(t)
n(t)!exp(�r(t))
n(t) - number of spikes at moment t
Linear-Nonlinear-Poisson model
• Moment-based statistical models for linear filter k estimation:
1. First order moment-based model - Spike Triggered Average (STA)
2. Second order moment-based model - Spike Triggered Covariance (STC)
Previous semester models
Maximum Likelihood estimators:
• Generalized Linear Model (GLM)
• Generalized Quadratic Model (GQM)
• Nonlinear Input Model (NIM)
This semester models
• Real data set - Lateral Geniculate Nucleus data (LGN) • Synthetic data set - Retinal Ganglion Cells (RGC)
Data sets description
Both data sets contain : 1. Stimulus vector (S, which depends on time) 2. Spikes vector (n, which depends on time) 3. Time interval of the stimulus update (dt, time step)
0 1 2 3 4 5 6 7 8 9 10time (sec)
-1.5
-1
-0.5
0
0.5
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stim
ulus
0 1 2 3 4 5 6 7 8 9 10time (sec)
0
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num
ber o
f spi
kes
Generalized Linear Model: Idea
Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (2011) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression.
Additional parameters
r(t) = F (k · s(t) + h · robs
(t) + b)
F () = exp()
⌧ = P � 1
P- the time window lengthn - vector of spikesS - stimulus vector b - threshold
GOAL: find optimal k,h and b
s(t) = (S(t� ⌧), . . . , S(t))
robs
(t) = (n(t� ⌧), . . . , n(t))
Generalized Linear Model (GLM): Plan
r(t) = F (k · s(t) + h · robs
(t) + b)
1. GLM: find LogLikelihood LL[k,h,b] -> max 2. Validate the code using simulated data a. without h and b terms b. with h and b terms 3. Validate part of the model(without h and b
terms) comparing with STA filter for RGC data set
4. Find optimal k and h filters for LGN data set
Maximum Likelihood estimation
Poisson distribution
N = {n(t)} ⇥ = {r(t)}
⇥ = {k,h, b}r(t) = F (k · s(t) + h · robs
(t) + b) F () = exp()
LL(⇥) =X
t
n(t)(k · s(t) + h · robs
(t) + b)�X
t
exp(k · s(t) + h · robs
(t) + b)
P (N |⇥) =Y
t
(r(t))n(t)
n(t)!exp(�r(t))
LL(⇥) = log(P (N |⇥)) =X
t
n(t)log(r(t))�X
t
r(t)
Solve log-likelihood using gradient ascent method
Validation: 1. Generate stimulus 2. Chose a simple filter k
3. Use Poisson distribution for spike generation
4. Use generated stimulus and spikes in order to reconstruct the
filter
kj = exp((j � 15)/5)
0 5 10 15time step (j in the formula)
0
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1Created filter vs optimization result
created filteroptimization
GLM code validation:
kj = exp((j � 15)/5) ⇤ cos(2⇡j/7)
0 5 10 15time step (j in the formula)
-0.5
0
0.5
1Created filter vs optimization result
created filteroptimization
GLM code validation:
Try another test k-filter
1 2 3 4 5 6 7 8 9 10time step
-1.2
-1
-0.8
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0h filter and optimization result for generated data
h filter
optimization
0 5 10 15time step
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1k filter vs optimization result for generated data
k filteroptimization
0 5 10 15-0.5
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1k filter used vs optimized
1 2 3 4 5 6 7 8 9 10-1.2
-1
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0h filter used vs optimized
original h filteroptimization result
H1
K2
H2=H1
GLM code validation including history term
K1
Optimal linear filter matches STA for RGC data:
0 5 10 15time step
-3
-2
-1
0
1
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3STA filter vs optimization result for filter k
STAfound filter k
Need higher time resolution to catch history term
0 10 20 30 40 50 60 70time step (time step = dt)
0
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2Spikes for the first 0.5 sec. Low resolution
0 50 100 150 200 250time step (time step = dt/4)
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1Spikes for the first 0.5 sec. Increased resolution
0 100 200 300 400 500 600 700 800 900 1000time step (time step = dt/16)
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1Spikes for the first 0.5 sec. Increased resolution
dt dt/4
dt/16
Regularization is needed at high resolutions
0 5 10 15 20 25 30time step
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-0.3
-0.2
-0.1
0
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0.4k filter without regularization, dt/4
Solve regularized log-likelihood using gradient
ascent method
RLL(⇥) =X
t
n(t)(k · s(t) + h · robs
(t) + b)�X
t
exp(k · s(t) + h · robs
(t) + b)� �
X
i
(ki
� k
i�1)2
GLM implementation: increase the resolution.
Resolution is dt/4, no history term.
max difference between STA and optimization
result is about 1% 0 5 10 15 20 25 30time step
-0.3
-0.2
-0.1
0
0.1
0.2
0.3K filter optimization vs STA, with regularization, dt/4
optimizationSTA
0 5 10 15 20 25 30time step
-0.4
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-0.1
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0.4k filter without regularization, dt/4
LL ~ 104 Regularization term ~ 102
GLM implementation: regularization parameter choice
0 5 10 15 20 25 30time step
-0.3
-0.2
-0.1
0
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0.3Comparison of lambda influense on filter k, dt/4
4000200030002500350010008000500
GLM with k,h and b parameters
Resolution dt/4 Not OK
OK
0 5 10 15 20 25 30time step
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0.3Filter k for dt/4
optimizationSTA
0 5 10 15 20 25 30time step
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0Filter h dt/4
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045time (sec)
-6
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0
1Filter h for dt/16
GLM with k,h and b parameters
Picture reference: Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (2011) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression.
neuron’s history detected for resolution dt/16
• GLM algorithm was implemented and validated on generated data
• The history term was extracted using regularization of log-likelihood
• Optimization of up to 240 parameters simultaneously was implemented
Summary
Updated project scheduleOctober - mid November November
✓ Implement STA and STC models
✓ Test models on synthetic data set and validate models on real data set
November - December December - mid February
✓ Implement Generalized Linear Model (GLM)
✓ Test model on synthetic data set and validate model on LGN data set
January - March mid February - mid April
• Implement Generalized Quadratic Model (GQM) and Nonlinear Input Model (NIM)
• Test models on synthetic data set and validate models on LGN data set
April - May Mid April - May
• Collect results and prepare final report
References1. McFarland JM, Cui Y, Butts DA (2013) Inferring nonlinear neuronal computation based on physiologically
plausible inputs. PLoS Computational Biology 9(7): e1003142.
2. Butts DA, Weng C, Jin JZ, Alonso JM, Paninski L (2011) Temporal precision in the visual pathway through the interplay of excitation and stimulus-driven suppression. J. Neurosci. 31: 11313-27.
3. Simoncelli EP, Pillow J, Paninski L, Schwartz O (2004) Characterization of neural responses with stochastic stimuli. In: The cognitive neurosciences (Gazzaniga M, ed), pp 327–338. Cambridge, MA: MIT.
4. Paninski, L., Pillow, J., and Lewi, J. (2006). Statistical models for neural encoding, decoding, and optimal stimulus design.
5. Shlens, J. (2008). Notes on Generalized Linear Models of Neurons.
Appendix: Gradient in GLM with k,h and b (including regularization term)
for i = 2 to M-1 (except the1st and the last elements ), where M is the number of stimuli = stimulus_length - P +1