Dynamic Modeling, Estimation and Signal Processing with Cortical Waves

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Dynamic Modeling, Estimation and Signal
Processing with Cortical Waves

• Bijoy K. Ghosh
• Center for BioCybernetics and Intelligent Systems
• Department of Systems Science and Mathematics
• Washington University, Saint Louis, Missouri

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• Our goal is to verify that position and velocity
  of a spot of light incident on the retina of a
  turtle are encoded by the spatiotemporal dynamics
  of the cortical waves they generate.

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• This conjecture is verified using a biophysically
  realistic large scale computational model of the
  visual cortex implemented with the software
  package GENESIS.

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• The cortical waves are recorded and analyzed
  using principal component analysis.
• The position and velocity information from the
  visual space is mapped onto an abstract B-space
  using principal component expansion.
• The likely values of the position/velocity are
  estimated using standard statistical detection
  methods.

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Visual cortex of a freshwater turtle
Cross-Section showing the major subdivisions
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• Extra-cellular recordings (Mazurskaya 74) from
  the visual cortex of freshwater turtles show that
  neurons at each cortical locus are activated by
  visual stimuli presented at every point in the
  binocular visual space.
• This suggests that there may not be a simple map
  of the coordinates of the visual space to the
  coordinates of the visual cortex in turtles.

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• Experiments conducted by Senseman using voltage
  sensitive dyes support the view that visual
  information is coded in the spatiotemporal
  dynamics of the cortical waves.
• Ermentrout, Kleinfield, Prechtl, Mitra and others
  have observed traveling electrical waves not only
  in the turtle visual cortex but also across
  olfactory, visual and visuomotor areas of cortex
  in a variety of species.

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Distribution of cells in the visual cortex. The
medial and lateral cells are excitatory. The
stellate and the horizontal cells are inhibitory.
Turtle visual cortex contains at least 11
morphologically distinct types of neurons. We
use only 4 of the well characterized neurons.
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The pyramidal cells are located in the
intermediate layer 2 of the cortex and are
predominantly excitatory. Stellate cells have
somata in the outer layer 1 and are
inhibitory. Horizontal cells have somata in
layer 3 and are also inhibitory.
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Pyramidal and stellate cells receive direct
projections from the Geniculate (LGN)
(feedforward pathway) Geniculate axons
intersect lateral and medial pyramidal cells in
characteristic patterns. Pyramidal cells make
projections to stellate, horizontal and other
pyramidal cells accessing both AMPA and NMDA
receptors.
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Stellate and horizontal cells make feedback
projections to pyramidal cells accessing both
GABAa and GABAb receptors. Stellate cells also
project back to stellate cells via GABAa and
GABAb receptors.
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The geometry of the geniculate afferents and the
distribution of synapses (varicosities) along the
length of the axons are based on anatomical data.
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Each neuron is represented by a 16 compartment
model, based on its anatomy.
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Output of the Visual Cortex Model

• Output of a pyramidal cell

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We record the response of the network to both
stationary (localized) stimuli and stimuli moving
with a constant velocity across visual
field. The stationary stimulus has be simulated
by presenting a 50 msec square current pulse to a
set of adjacent geniculate neurons. A set of 20
equidistant positions has been chosen along the
LGN. The moving stimulus is assumed to be
sweeping across the geniculate complex, from left
to right, and it consists of a sequence of square
pulses equal in amplitude and duration, which
have been delayed with respect to each other. A
set of 20 equally spaced velocity parameters has
been chosen between some minimum and maximum
velocity.
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Latency Maps for Turtle Visual Cortex
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• For each of the two sets of stimuli, stationary
  and moving, a set of 50 cortical samples is
  generated.
• The exact position of the cells within the cortex
  are randomized.
• The response represents the membrane potentials
  of the individual neurons and is a spatiotemporal
  signal in its nature.
• These signals are resampled to a 64 ? 64 uniform
  grid, and the data is color coded and visualized
  as a movie.

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Each movie is represented spatially with respect
to a global basis as follows.
where M1(x,y), M2(x,y) and M3(x,y) are the
principal modes, common to all of the movies, as
they represent the global basis. Therefore the
difference between two movies will be captured in
the time coefficients ?1(t), ?2(t) and ?3(t),
which are shown in a phase space, Alpha space.
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A-space for moving input
A-space for stationary input
Because the positions of the neurons in the model
cortex are randomized with each new simulation,
the vector function ?1(t), ?2(t), ?3(t) can be
viewed as a random process.
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The Alpha-Process can be represented by a
temporal basis function as follows
where ?1(t), ?2(t) and ?3(t) are the principal
components of the global basis. Each data set
I(x,y,t) is therefore represented in a three
dimensional space, Beta-space. For 20 different
parameter values and 50 different samples of the
cortex, a total set of 1,000 movies has been
averaged. The procedure for stationary and moving
stimuli was carried out separately. The points
in the three dimensional subspace are clustered
by the inputs that elicited them.
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Moving Signals
Stationary Signals
Assuming the points within clusters are normally
distributed, a conditional density function for
each cluster is obtained. A standard detection
algorithm yields the following decision space.
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Stationary input
Moving Input
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Data Processing
Beta space
Model output
Alpha space

PCA
PCA
Detection by distance
Detection in white noise
Detection in colored noise
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PCA and Moving Time Window

• Output of the visual cortex model from t 209ms
  to t 219ms

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PCA and Moving Time Window (II)

• Principle components in this particular time
  window

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Beta Space

• Beta generation

• Complete beta strands

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Detection By Distance

• Distance

• Compute the distance between each beta strand and
  the averaged beta strand

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Detection in White Noise (I)

• Assume the vector noise process is white, i.e.
• Three hypothesis
• are their
  ortho-normalization

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Detection in White Noise (II)
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Detection in White Noise (III)

• Choose the biggest

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Detection in Colored Noise

• Actually, the noise process in the received
  signal is colored,
• Solve integral equation
• for and , which are the
  eigenvalues and eigenfunctions of the noise
  process
• Compute KL expansion and likelihood ratio

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Detection window with the same starting point
1-99
1-149
1-199
1-49
1-299
1-399
1-599
1-999
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• Moving detection window

1-99
51-149
101-199
151-249
201-299
301-399
401-499
501-599
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Error Probability for the moving detection window

• Detection error starts from t 310ms.

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Our Immediate Next Goal Fine tune the response
of the cortex by adding an extra feedforward
inhibition. This would be done by adding
subpial cells. These are inhibitory cells and
would be instrumental in slowing down the wave
velocity. Add the Geniculate and the Retinal
Ganglion Cells to the model. Only partial
knowledge exists about the turtle
geniculate. Study the effect of
Cortico-genicular feedback on the cortical waves.

Distribution of cells in the LGN
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Mathematical challenges Replace the pedestrian
distance (Euclidean) by more honorable Kullback
Leibler, when comparing the beta strands. How to
get the beta strands in the beta space by some
form of interpolation (spline perhaps). The beta
space would typically be at least 3 or 4
dimensional. It will be important to understand
the role of Cortical Dynamics as an internal
dynamic representation of the visual space. Does
it make sense to encode visual cues with pattern
generating flows of activity in the cortex as
opposed to, say, features or flow of features on
the retina?