Neural Coding: IntegrateandFire Models of Single and MultiNeuron Responses

1
Neural Coding Integrate-and-Fire Models of
Single and Multi-Neuron Responses

• Jonathan Pillow
• HHMI and NYU
• http//www.cns.nyu.edu/pillow
• Oct 5, Course lecture
• Computational Modeling of Neuronal Systems
• Fall 2005, New York University

2
General Goal understand the mapping from
stimuli to spike responses with the use of a model
y

• Model criteria
• flexibility (captures realistic neural
  properties)
• tractability (for fitting to data)

3
Example 1 Hodgkin-Huxley
Na activation (fast)
spike response
Na inactivation (slow)
stimulus
K activation (slow)
flexible, biophysically realistic - not
easy to fit
4
Example 2 LNP
K
f
x
y
(receptive field)
easy to fit (spike-triggered averaging)
not biologically plausible
5
LNP model
stimulus
filter K
filter output
spike rate
spikes
time (sec)
6
cascade models
y
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Generalized Integrate-and-Fire Model
x(t)
y(t)
Inoise
Istim
Ispike
related Spike Response Model, Gerstner
Kistler 02
8
Generalized Integrate-and-Fire Model
powerful, flexible tractable for fitting
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Model behaviors adaptation
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Model behaviors bursting
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Model behaviors bistability
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The Estimation Problem
Learn the model parameters
K stimulus filter g leak conductance s 2
noise variance h response current VL
reversal potential
K
h
From stimulus train x(t)
spike times ti
Solution Maximum Likelihood -
need an algorithm to compute Pq(yx)
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Likelihood function
hidden variable
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Likelihood function
hidden variable
P(spike at ti) fraction of paths crossing
threshold at ti
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Computing Likelihood

• linear dynamics
• additive Gaussian noise

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Computing Likelihood

• linear dynamics
• additive Gaussian noise

reset
ISIs are conditionally independent ? likelihood
is product over ISIs
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Maximizing the likelihood

• parameter space is large ( ? 20 to 100
  dimensions)
• parameters interact nonlinearly

? gradient ascent guaranteed to
converge to global maximum!
Paninski, Pillow Simoncelli. Neural Comp. 04
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Application to Macaque Retina

• isolated retinal ganglion cell (RGC)
• stimulated with full-field random stimulus
  (flicker)
• fit using 1-minute period of response

t
(Data Valerie Uzzell E.J. Chichilnisky)
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IF model simulation
Stimulus
filter K
Iinj
V
time (ms)
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IF model simulation
Stimulus
filter K
Iinj
h
V
time (ms)
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ON cell
74 of var 92 of var
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Accounting for spike timing precision
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Accounting for reliability
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Decoding the neural response
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Solution use P(respstim)
P(R1S1)P(R2S2)
P(R1S2)P(R2S1)
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Discriminate each repeat using P(RespStim)
Resp 1
Resp 2
?
P(R1S1)P(R2S2)
P(R1S2)P(R2S1)
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Discriminate each repeat using P(RespStim)
Resp 1
Resp 2
?
94 correct
Compare to LNP model P(RespStim)
LNP 68 correct
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Decoding the neural response
IF model correct
LNP model correct
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Part 2 how to characterize the responses of
multiple neurons?

• Want to capture
• the stimulus dependence of each neurons response
  
• the response dependencies between neurons.

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2 types of correlation

• stimulus-induced correlation persists even if
  responses are conditionally independent, i.e.
• P(r1,r2 stim) P(r1stim)P(r2stim)

stimuli
responses
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2 types of correlation

• stimulus-induced correlation persists even if
  responses are conditionally independent, i.e.
• P(r1,r2 stim) P(r1stim)P(r2stim)
• 2. noise correlation arises if responses are
  not conditionally independent given the stimulus,
  i.e.
• P(r1,r2 stim) ? P(r1stim)P(r2stim)

Noise
stimuli
responses
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Modeling multi-neuron responses
K
x
h11
h12
coupling h currents
h21
K
x
h22
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Methods
spatiotemporal binary white noise (24 x 24
pixels, 120Hz frame rate)
simultaneous multi-electrode recordings of
macaque RGCs

• Model parameters fit to five RGCs using 10
  minutes of response to a non-repeating binary
  white noise stimulus

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Fits
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Fits
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Fits
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Pairwise coupling analysis

• Compare likelihoods
• The single-cell model for cell i
• vs.
• 2. The pairwise model for i with coupling from
  cell j

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Pairwise coupling analysis
Coupling Matrix
likelihood ratio
Functional Coupling
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Accounting for the autocorrelation
RGC simulated model
post-spike current
O
F
F
c
e
l
l
s
1
2
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Accounting for cross-correlations
ON-ON correlations
raw (stimulus noise)
stimulus-induced
2
1
5
1
0
1
5
0
0
-
1
-
5
-
1
0
0
-
5
0
0
5
0
1
0
0
-
1
0
0
-
5
0
0
5
0
1
0
0
t
i
m
e
(
m
s
)
t
i
m
e
(
m
s
)
4 to 5
5 to 4
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OFF-OFF cell correlations
1
6
4
0
1 vs 3
2
-
1
0
-
2
-
2
1
6
4
2 vs 3
0
2
1 vs 3
0
-
1
-
2
-
1
0
0
-
5
0
0
5
0
1
0
0
t
i
m
e
(
m
s
)
3 to 2
2 to 3
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OFF-ON cell correlations
4
2
1 vs 4
0
-
2
-
4
-
6
1 to 4
4 to 1
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OFF-ON cell correlations
stimulus-induced
raw (stimulus noise)
1 vs 5
2 vs 4
2 vs 5
3 vs 4
3 vs 5
-
1
0
0
-
5
0
0
5
0
1
0
0
0
0
s
)
t
i
m
e
(
m
s
)
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Conclusions

• 1. generalized-IF model flexible, tractable
  tool for modeling neural responses
• 2. fitting with maximum likelihood
• 3. probabilistic framework useful for encoding
  (precision, response variability) and decoding
• 4. easily extended to multi-neuron responses
• 5. likelihood test of functional connectivity
  between cells
• 6. explains auto- and cross-correlations
• 7. resolves cross-correlations into
  stimulus-induced and noise-induced

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My collaborators
E.J. Chichilnisky Valerie Uzzell - The Salk
Institute Jonathon Shlens Eero Simoncelli -
HHMI NYU Liam Paninski - Columbia U.
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Basis used for coupling currents
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Extra slides
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5-way coupling analysis
Likelihood ratio for fully connected model
Functional Coupling
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5-way coupling analysis
Likelihood ratio for fully connected model
Conclusion the fully connected model gives an
improved description of multi-cell responses to
white noise stimuli.