Neural Coding and Decoding
Albert E. Parker
Center for Computational Biology
Department of Mathematical Sciences
Montana State University
Collaborators:
Alexander Dimitrov
Tomas Gedeon
John P. Miller
Zane Aldworth
Problem: Determine a coding scheme: How does
neural ensemble activity represent information about
sensory stimuli?
Our Approach:
• Construct a model using Probability and Information
Theory
• Optimize the model to cluster the neural responses
which gives an approximation of a coding scheme
given the available data.
• Apply our method to the cricket cercal system
Neural Coding and Decoding.
Goal: What conditions must a coding scheme satisfy?
Demands:
• An animal needs to recognize the same object on repeated
exposures. Coding has to be deterministic at this level.
• The code must deal with uncertainties introduced by the
environment and neural architecture. Coding is by necessity
stochastic at this finer scale.
Major Problem: The search for a coding scheme requires large
amounts of data
How to determine a coding scheme?
Idea: Model a part of a neural system as a communication
channel using Information Theory. This model enables us to:
• Meet the demands of a coding scheme:
o Define a coding scheme as a relation between stimulus and neural
response classes.
o Construct a coding scheme that is stochastic on the finer scale yet
almost deterministic on the classes.
• Deal with the major problem:
o Use whatever quantity of data is available to construct coarse but
optimally informative approximations of the coding scheme.
o Refine the coding scheme as more data becomes available.
• Investigate the cricket cercal sensory system.
A Stochastic Map
Q(Y|X)
input
output
Y
X
The relationship between X and Y is completely described by the conditional
probability Q.
Realizations of X and Y in neural coding
Q(Y=y|X=x)
stimulus sequence X=x
response sequence Y=y
Determining Stimulus/Response Classes
Given a joint probability p(X,Y):
response sequences
1
2
Y
3
4
X
stimulus sequences
response sequences
Stimulus and Response Classes
1
Y
2
3
Distinguishable
stimulus/response
classes
4
X
stimulus sequences
Information Theoretic Quantities
A quantizer or encoder, Q, relates the environmental stimulus, X, to the neural
response, Y, through a process called quantization. In general, Q is a stochastic map
Q( y| x ): X  Y
The Reproduction space Y is a quantization of X. This can be repeated: Let Yf be a
reproduction of Y. So there is a quantizer
q ( y f | y ): Y  Yf
Use Mutual information to measure the degree of dependence between X and Yf.
  q ( y f | y ) p ( x, y ) 
 y

I ( X , Y f )   q ( y f | y ) p ( x, y ) log 

y, y f
 p ( x ) p ( y ) q ( y f | y ) 
y


Use Conditional Entropy to measure the self-information of Yf given Y
H (Yf |Y ) 
 p( y ) q ( y
y,y f
f
| y ) log(q ( y f | y ))
The Model
Problem: To determine a coding scheme between X and Y requires
large amounts of data
Idea: Determine the coding scheme between X and Yf, a clustering
(reproduction) of Y, such that: Yf preserves as much information
(mutual information) with X as possible and the self-information
(entropy) of Yf |Y is maximized. That is, we are searching for an
optimal mapping (quantizer):
q * ( y f | y) : Y  Y f
that satisfies these conditions.
Justification: Jayne's maximum entropy principle, which states that
of all the quantizers that satisfy a given set of constraints, choose
the one that maximizes the entropy.
Equivalent Optimization Problems
 Maximum entropy:
maximize F(q(yf|y)) = H(Yf|Y)
constrained by
I(X;Yf )  Io
Io determines the informativeness of the reproduction.
 Deterministic annealing (Rose, ’98):
maximize F(q(yf|y)) = H(Yf|Y) +  I(X,Yf ).
Small  favor maximum entropy, large : maximum I(X,Yf ).
 Augmented Lagrangian with Newton CG line search
 Implicit solution:
qy f | y  
e
  q DI  q  

  
 p y  
y
,
y   e
  q DI  q  

  
 p y  
yf
 Simplex Algorithm:
maximize I(X,Yf ) over vertices of constraint space
Application to synthetic data
(p(X,Y) is known)
Random clusters
Application to Real Data
(the probabilities p(y) and p(x,y) are NOT known)
•
p(x,y) cannot be estimated directly for rich stimulus sets - there is
not enough data.
•
I(X,Yf )=H(X) - H(X|Yf ). Only H(X|Yf ) depends on the q(y f |y). So an
upper bound of H(X|Yf ) produces a lower bound of I(X,Yf ).
•
•
H X | Y f   E y f H X | Y f  y f 
is bounded by a Gaussian:
1 log 2e  X det C
where
2
X |y
H X | Y f   E y f H G ( X | y f )  E y f
2
C X | y f is the conditional covariance of the stimulus.

  
- E  E
CX | y f   p( x | y f ) x   y f x   y f
x

CX | y f  Ey| y f  CX | y   y  y
T
y| y f
f
T
y
y| y f
 y

T
which is written explicitly as a function of the quantizer (through
p(y|y f )). We estimate the mean and and covariance matrices.
The Optimization Problem for Real Data
Maximum entropy:
maximize F(q(yf|y)) = H(Yf|Y)
constrained by
H(X)-HG(X|Yf )  Io
Io determines the informativeness of the reproduction.
?
?
Modeling the cricket cercal sensory system
as a communication channel
Nervous system
Signal
Communication
channel
Wind Stimulus and Neural Response in the
cricket cercal system
Neural Responses (over a 30 minute recording) caused by white
noise wind stimulus.
X
Time in ms. A t T=0, the first spike occurs
Y
Some of the air current stimuli
preceding one of the neural
Neural
Responses
(these are
all doublets)
for a 12 ms
window
responses
T, ms
Quantization:
A quantizer is any map f: Y -> Yf from Y to a reproduction space Yf with
finitely many elements. Quantizers can be
deterministic
y f   y
or
qy f | y   
probabilistic
refined
Y
Yf
Y
Applying the algorithm to cricket sensory data.
Class 1
Yf
Class 2
Class 1
Class 2
Yf
Class 3
Y
Conclusions
We
• model a part of the neural system as a communication
channel.
• define a coding scheme through relations between
classes of stimulus/response pairs.
- Coding is probabilistic on the individual elements of X and Y.
- Coding is almost deterministic on the stimulus/response
classes.
To recover such a coding scheme, we
• propose a new method to quantify neural spike trains.
- Quantize the response patterns to a small finite space (Yf).
- Use information theoretic measures to determine optimal
quantizer for a fixed reproduction size.
- Refine the coding scheme by increasing the reproduction size.
• present preliminary results with cricket sensory data.