KIII Chaos Characterization using Lyapunov Exponents Mark Myers COMP 7991 - NeuroDynamics

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KIII Chaos Characterization using Lyapunov
ExponentsMark MyersCOMP 7991 - NeuroDynamics

• Abstract KIII combines 3 or more KII populations
  to model functional brain areas such as cortex or
  hippocampus, and are capable of chaotic dynamics
  of the type observed in these regions to, for
  example, derive meaning from perceptual senses
  6. I will discuss how Lyapunov calculations are
  used to find the exponents that correspond to the
  trajectories are in the chaotic region.

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A recipe for chaos involves the following

• a) Nonlinear feedback
• b) High growth rates (? energy)
• c) Many (gt 3) interacting populations.
• In nature, you may find (a) and (c), and
  sometimes (b). In
• order to detect chaos with noise, you need a
  large amount
• of data. Very large biological populations have
  the
• potential for chaotic dynamics. Most biological
  systems are
• too complex to be easily understood. By studying
  chaos in
• models, we may be able to understand the
  characteristics
• of very complex systems.

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Two trajectories with Similar Initial Conditions
X0          X0  ?x0
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Behavior of two adjacent points
Divergence ? gt 0
Stable ?  0
Convergence ? lt 0
Take any arbitrarily set of values, as displayed
above in the space of a chaotic system. Adjacent
points, no matter how close, will diverge to any
arbitrary distance and all points will trace out
orbits that eventually visit every region of the
space 1.
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The KIII Model

• The KIII model is designed to be a dynamic
  computational model that simulates
• the sensory cortex. It is connected by
  feedforward/feedback connections. Using
• the weights described in 7, we find each KII
  model provides the following
• output.

Impulse-Response test on KIII Temporal behavior
of the activation of a KIII set with three layers
6
Embedding dimension of data

• A measure of dimension, called the embedding
  dimension
• is a set has embedding dimension n.
• the embedding dimension of a plane is 2
• the embedding dimension of a sphere is 3
• This function uses time delay vectors with
  dimension dim,
• whereas delay is measured in the input samples.
  The
• result is a n by dim array, each row contains the
  
• coordinates of one point.

7
Power Spectral Density Estimate of KIII Data

• dload('layer2_out')
• xd(1,)
• xd(7519000,2)
• loglog(psd(y,2000))

8
Sine Wave Calculation

• x 0.150
• ysin(x)
• s signal(y')
• t1 amutual(s, 256)
• delay, tmp firstmin(t1)
• t3 embed(s, 1, delay)
• view(t3)

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Sine Wave Calculation Embed Dim 2

• x 0.150
• ysin(x)
• s signal(y')
• t1 amutual(s, 256)
• delay, tmp firstmin(t1)
• t3 embed(s, 2, delay)
• view(t3)

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Diminishing Wave Pattern

• x 0.0160
• a1/16
• y exp(-ax) . sin(x)
• ssignal(y')
• t1 amutual(s, 256)
• delay, tmp firstmin(t1)
• t3 embed(s, 1, delay)
• view(t3)

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Diminishing Wave Pattern Embed Dim 3
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A Slightly Chaotic Display
The following displays a slightly diverging
largest Lyapunov exponent with s(t - 2) on the
y-axis and s(t) on the x-axis.

• delay, tmp firstmin(t1)
• t3 embed(s, 2, delay)
• view(t3)

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Prediction Error of Selected KIII Model

• t4 largelyap(t3, -1, 1000, 40, 10)
• dt4 data(t4)
• view(t4)

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Main Findings

• The data used in this KIII model displays that
  the output is
• slightly chaotic, as shown in the last two
  slides. A
• Lyapunov value 0.15 would be ideal.
• maxposfirstmax(t4) Get y-value of 1st max.
• maxpos 57
• max dt4(maxpos) Get x-value of 1st max.
• max 0.6634 Calculate x/y to derive Lyap.
• max / maxpos .0116 Lyapunov value

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References

• 1 Elert, Glenn Measuring Chaos The Chaos
  Hypertextbook 1995 2003
• 2 Phelan, Steven E. From Chaos to Compexity
  in Strategic Planning Presented at the 55th
  Annual Meeting of the Academy of Management
  Vancouver, British Columbia, Canada August 6-9,
  1995
• 3 Kozma, Robert Freeman, Walter J. Control
  of Mesoscopic /Intermediate-Range
• Spatio-Temporal Chaos in the Cortex Proc.
  2001 American Control Conference ACC01
• 4 Burbanks, Andy Extracting Beauty from
  Chaos
• Plus Magazine Issue 9, September 1999
• 5 Freeman, Walter J Kozma, Robert. Werbos,
  Paul J.Biocomplexity adaptive
• behavior in complex stochastic dynamical
  systems National Science Foundation, 9
• September 2000
• 6 Harter, Derekl Kozma, Robert Nonconvergent
  Dynamics and Cognitive Systems October 23, 2003
• 7 Harter, Derek Kozma, Robert Simulating the
  Principles of Chaotic Neurodynamics
• Memphis, TN 38152 USA
• 8 Li, Haizhon Kozma, Robert A Dynamic Neural
  Network Method for Time Series Prediction Using
  the KIII Model