Center for Computational Biology

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Analysis and modeling of sensory systemsthrough
Information Distortion
Alexander Dimitrov
Center for Computational Biology Montana State
University
CollaboratorsCCB John Miller, Gwen Jacobs,
Zane AldworthMath Tomas Gedeon, Albert
ParkerCS Brendan Mumey Research supported in
part by NSF BITS grant No. 0129895.Presentation
mad possible by NIH BRIN grant No. P20
RR-16455-01.
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Neural Coding and Decoding.

• The early stages of nervous systems transform
  information about sensory stimuli into a
  representation that is common for the whole
  nervous system a neural coding scheme. There
  have been various ideas about what this
  representation is.
• We describe a method to uncover a coding scheme
  by constructing coarse but highly informative
  approximations through quantization with an
  intrinsic cost function (the information
  distortion).
• This approach can be used as a model of the
  functioning of a sensory system. Then the
  question is what is the animal's distortion
  function?
• A distortion model essentially represents sensory
  perception as a signal discrimination / lossy
  compression problem.
• In reality, apart from signal (object)
  discrimination, there are also many estimation
  problems. We discuss the applicability of this
  distortion approach to this mixed
  discrimination/estimation problem.

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1. Recovering a coding scheme

• Determine the correspondence, P, between
  reproductions (XM,YN) of (M,N) elements, such
  that P preserves as much of the mutual
  information I(XY) as possible. Discard details
  of P that don't matter. Bi-clustering /
  symmetric IB / joint quantization / adaptive
  sieves.

New Goal Find the quantizers q that maximize
I(XM,YN ). MaxEnt formulation q argmax
H(XM,YNX,Y) ßI(XM,YN ).
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2. Sensory processing as a quantization/discrimina
tion problem
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Recovering a coding scheme
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The neural challenge dealing with large spaces.

• It is hard to estimate I(XMYN) directly for
  high-dimensional stimulus and response sets. We
  use models select the quantizers from a
  parametric family of distributions. This produces
  an upper bound to the distortion function. Better
  model tighter bound. Maximize the upper bound.
• Here I will discuss only the stimulus side. We
  model p(xxM) with a Gaussian. This effectively
  represent p(X) as a Gaussian mixture model. Once
  we have the model, we have an analytic expression
  for the estimate of I(XMYN).
• !!! Caveat This class of models impose a metric
  structure on the input space it defines when
  stimuli are close to each other (small
  Mahalanobis distance).
• This mild restriction can be overcome by the
  extra flexibility of M?N. We can have several
  Gaussian clusters represent the stimulus in a
  single stimulus/response class.

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The cricket cercal system(a low-frequency, near
field extension of the auditory system)
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(No Transcript)
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Applying the algorithm to cricket sensory
data.Single cell, unidirectional GWN.
A sequence of refinements in a single cell, along
with the class conditioned mean stimuli.
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Applying the algorithm to cricket sensory data.
Single cell, unidirectional GWN.
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Applying the algorithm to cricket sensory
data.Class 1 of a single cell, unidirectional
GWN.
A case where the flexibility of the Gaussian
mixture quantizer was used A response class
consisting of a single isolated spike required
two Gaussians, not single Gaussian, to explain
the stimulus data. The linear stimulus
reconstruction/single Gaussian conditional mean
is shown in brown.
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Sensory processing as a quantization/discriminatio
n problem
Under the quantization/clustering procedures,
sensory processing is implicitly regarded as a
discrimination problem
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Is sensory processing only a discrimination
problem?
1 2 3 4
neural responses
Y
X
environmental stimuli
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Response Amplitude
Continuous Stimulus Parameter(angle, amplitude,
size)
Yes and no. Originally, the idea of a neural code
was something like the above estimating some
continuous stimulus parameter from responses.
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p(yx)
Continuous Stimulus Parameter(angle, amplitude,
size)
To be honest, we have to formulate the problem
properly, including all the uncertainties in
stimulus and response registration (noise). This
is often avoided when discussing neural coding.
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p(yx)
Continuous Stimulus Parameter(angle, amplitude,
size)
Straight quantization can still resolve such
types of problems, but maybe not in the most
elegant way.
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Behavioral function of the cercal systemboth
discrimination and estimation
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Neural Map of Stimulus Direction
Afferents from both cerci
Afferents from one cercus
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Map of orientation preference in shrew V1.
Fitzpatrick et. al.
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Orientation and spatial frequency map in macaque
V1, along with symmetry model.
Bressloff and Cowan, 02
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Signal Processing with Symmetries(work in
progress)

• Signal discrimination.
• Mod-out nuisance symmetries.
• Estimate orbits of relevant symmetries
• Implicitly quantize the orbit as a series of
  discrimination problems (distortion methods can
  do this).
• Explicitly
• include the symmetry in the discrimination
  algorithm (Pattern theory?)
• Estimate symmetry independently of
  discrimination?(distance, direction, size, etc.)

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Discrimination with symmetries
Object
Symmetry acting on the object
Object
Symmetry acting on the object
Object(symmetry action)
Invariant discrimination? Yes, for nuisance
symmetries.For relevant symmetries - equivariant
discrimination.
Invariant discrimination? Yes, for nuisance
symmetries.For relevant symmetries - equivariant
discrimination.
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Example

• Original Gaussian quantizer
• x N(mi,Ci)
• Symmetry-aware quantizer, e.g., for direction
• x N(g?mi, g?Ci), where
• g? is the action of the symmetry group S1 on the
  model parameters,
• ? p(?)
• Two options for sensory system models
• Marginalize ?, cluster in X only invariant
  discriminators.
• Represent X as X' x S1. Classify x jointly in X'
  x S1 equivariant discriminators.

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Conclusions

• We
• use a new method to quantify a neural coding
  scheme.
• Quantize the response patterns to a smaller
  space.
• Use an information-based distortion measure.
• Minimize the information distortion for a fixed
  size reproduction.
• Refine by increasing the reproduction size (until
  out of data).
• discuss its applicability as a model of sensory
  processing.
• Sensory systems as discriminators.
• Include symmetries in the description.
• Suggest constructing equivariant discriminators
  (implicitly or explicitly).