Computational Issues when
Modeling Neural Coding Schemes
Albert E. Parker
Center for Computational Biology
Department of Mathematical Sciences
Montana State University
Collaborators:
Alexander Dimitrov
John P. Miller
Zane Aldworth
Thomas Gedeon
Brendan Mumey
Neural Coding and Decoding.
Goal: Determine a coding scheme: How does neural ensemble activity
represent information about sensory stimuli?
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.
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 , Yf )   q ( y f , y ) p( x , y ) log

p
(
x
)
p
(
y
)
q
(
y
,
y
)
y,y f
y


f


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 ))
Stimulus and Response Classes
X
stimulus sequences
response sequences
Y
stimulus/response
sequence pairs
distinguishable classes of stimulus/response pairs
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 squashing
(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  Yf
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) -  DI(Y,Yf ).
Small  favor maximum entropy, large  - minimum DI.
 Simplex Algorithm:
maximize I(X,Yf ) over vertices of constraint space
 Implicit solution:
qy f | y  
e
  q DI  q  

  
p

y



y
,
y   e
yf
  q DI  q  

  
p

y



?
?
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
responses
Neural
Responses
(these are
all doublets)
for a 12 ms
window
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.
Yf
1
2
1
Yf
2
3
Y
High Performance Computing
Tools:
 Bigdog: an SGI Origin 2000
 MATLAB 5.3
 Parallel Toolbox
Algorithms:
 Model Building
 Optimization
 Bootstrapping
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.