Transcript presentation

Reading population codes: a
neural implementation of ideal
observers
Sophie Deneve, Peter Latham,
and Alexandre Pouget
encode
Stimulus (s)
neurons
decode
Response (r)
Tuning curves
• sensory and motor
info often encoded in
“tuning curves”
• neurons give a
characteristic “bell
shaped” response
Difficulty of decoding
• noisy neurons create
variable responses to
same stimuli
• brain must estimate
encoded variables
from the “noisy hill” of
a population response
Population vector estimator
• assign each neuron
a vector
• vector length is
proportional to activity
• vector direction
corresponds to
preferred direction
Sum vectors
Population vector estimator
• Vector summation is
equivalent to fitting a
cosine function
• peak of cosine is
estimate of direction
How good is an estimator?
• need to compare variance
of estimator after repeated
presentations to a lower
bound
• the maximum likelihood
estimate gives the lower
variance bound for a given
amount of independent
noise
VS
encode
Stimulus (s)
neurons
decode
Response (r)
Maximum Likelihood Decoding
Maximum likelihood estimator
Decoding
Encoding
Goal: biological ML estimator
• recurrent neural network
with broadly tuned units
• can achieve ML estimate
with noise independent of
firing rate
• can approximate ML
estimate with activitydependent noise
Pλ
General Architecture
• units are fully connected
and are arranged in
frequency columns and
orientation rows
• weights implement a 2-D
Gaussian filter:
Preferred Frequency
20
Preferred
orientation
PΘ
20
Input tuning curves
• circular normal functions with some
spontaneous activity:
• Gaussian noise is added to inputs:
Unit updates & normalization
• units are convolved with
filter (local excitation)
• responses are normalized
divisively (global inhibition)
Results
• Rapidly converges
•strongly dependent on
contrast
Results
• sigmoidal response curve
after 3 iterations, becomes
a step after 20
• actual neuron
Noise Effects
Flat Noise
• Width of input tuning curve
held constant
• width of output tuning
curve varied by adjusting
spatial extent of the weights
Proportional Noise
Analysis
Flat Noise
Q1: Why does the optimal
width depend on noise?
Q2: Why does the network
perform better for flat noise?
Proportional Noise
Analysis
Smallest achievable variance:
= inverse of the covariance matrix of the noise
= vector of the derivative of the input tuning curve with respect to
For Gaussian noise:
Trace term is 0 when R is independent of Θ
(flat noise)
Θ
Summary
• network gives a good
approximation of the optimal
tuning curve determined by
ML
• type of noise (flat vs
proportional) affected
variance and optimal tuning
width