Random Cellular Automata: A case study of Neuropercolation

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Random Cellular Automata A case study of
Neuropercolation

• Dr. R. KozmaPresented by Hui Chen

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Major References

• Balister, P., Bollobas, B., and Kozma, R., Random
  Cellular Automata, to be published, 2002
• Grimett, G., Percolation, New York, Berlin,
  Heidelberg Springer-Verlag, 1989
• Kozma, R., Balister, P., Bollobas, B.,
  Self-Organized Development of Behaviors in
  Spatio-Temporal Dynamical Systems, to appear in
  WCCI 2002 Proceedings, May 12-17, 2002
• Kozma, R., Neuropercolations Dynamical
  Percolation Models of Phase Transition in
  Physical and Biological Systems, Lecture in BISC
  Seminar, University of California at Berkeley,
  February 21, 2002

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Neuropercalations Models of Neurodynamics
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Modeling Mesoscopic Behavior of Neuron Population
Kozma, 2002

• Neuropil
• Densely connected filamentous texture of neural
  tissue (Freeman, since 70s)
• Possible phase transitions from connectivity/gain
  change

• Percolation
• Lattice/Torus
• Randomly initialized
• Rules of Aprousal and Drepression based on
  defined neighborhood

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NEUROPERCOLATION MODELS AS GENERAL (SOFT)
COMPUTATIONAL TOOLSKozma, 2002

• Governed by probabilistic rules
• And not on differential equations
• Emphasis is on connectivity
• Instead of complex functionality of individual
  units
• Relatively simple functionality of nodes
• Motivation
• This approach with simple units can give very
  complex behavior ? can approximate solution to
  ode/pde if needed
• More general than calculus ant it can solve
  problems unreachable by traditional tools

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Random Cellular Automata

• Probability introduced ? Bootstrap Percolation ?
  Random Cellular Automata ? Chaotic behavior
• Computation Model with
• Cell each cell only has limited computational
  power to change its properties
• Topology
• Neighborhoods
• Property updating rule with probability
• Update the properties of the desired cell based
  on the properties of itself and its neighbors

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Current RCA MODEL Model of Simple Neural Network

• Updating/Transition Rule
• if greater than 2 active in neighborhoods,
  activate with probability p
• if less than 3 active in neighborhoods,
  inactivate with probability 1-p

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Density, Probabity and Clusters

• Density (d)
• d (number of active cells)/(total number of
  cells
• Probability (p)
• On this probability the cell is going to change
  its properties according to the properties of
  itself and its neighbors.
• Clustering
• How density is changing with time or different
  probability?
• Critical Probability (pc)

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Critical Prabability pc

• Probability Model
• p0
• d 0 or 1 depends on initial configuration
• p?
• d?0 and d ?1, so d? or d? depends on initial
  configuration.
• p 0.5
• totally random, d?
• Deterministic Model
• there exists a pc
• what happends when p lt pc or p gt pc (many small
  clusters, few big clusters)

?
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RCA runs on 128 x 128 torus
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Probability 0.040
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Probability 0.080
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Probability 0.120
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Probability 0.133
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Probability 0.134
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Probability 0.135
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Probability 0.136
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Probability 0.140
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Probability 0.200
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Critical Probability
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Residence Time Distribution
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p 0.134
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p 0.1338
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p0.1335
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p0.1333
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p0.133
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Summary