Rules form equivalence classes under certain transformations: conjugation, reflection.
Let’s play
0
NEGATION
1
NOR
23
MINORITY
51
COMPLEMENT
127
NAND
128
AND
170
LEFT SHIFT
204
IDENTITY
232
MAJORITY
240
RIGHT SHIFT
254
OR
255
TAUTOLOGY
FROM CELLULAR AUTOMATA LITERATURE
Dynamics: KAUFMANN BOOLEAN NETWORKS • Each site receives k inputs • Assign Boolean functions to each site at random • = 2 transition from an ordered to a disordered (chaotic) state
Project1.exe
Dynamics: Wolfram Cellular Automata • Regular 1-d and 2-d lattices • All units processing information the same way • No noise • Very regular prescription • Complex dynamical behaviors:
The model • Topology • Dynamics • Noise
– Chaos – Cycles – Fixed point
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Topology of the model
Dynamics • All sites evolve according to its own rule • 3 inputs: leftleft-selfself-right
• Small Small--world: more neighbors neighbors-->average
• Noise in the transmission
Selection of Boolean rules
Rules with trivial dynamics
Characterization of the system
Majority rule (rule 232)
State of the system
ke=0.15
Brownian
ke=0.45
1/f noise
ke=0.90 white noise
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Majority rule
Systematic evaluation of the scaling of the fluctuations • Detrended Fluctuation Analysis (DFA) (Peng et al., www.physionet.org)
Scaling (majority rule)
Majority rule
S(f)=1/fβ F(n)=nα β=2α-1
Robustness
And the other rules? (I)
ki>2 nn, exponential
Preferential attachment: c) outgoing d) incoming
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And the other rules? (II)
And the other rules? (III)
Robustness (232 + 50)
Robustness (232 + random)
Robustness (asynchronous updating)
Robustness (stronger majority)
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Dynamical properties of generic networks • 1/fβ noise – Restrictions on the dynamics – Noise – Random links
• RBN (Kauffmann) - CA (Wolfram) • Robustness of the findings • Aging and disease: changes in connectivity/noise may alter correlation properties
Genetic networks • Attractors = cell types • Dynamical evolution • Genetic networks are hierarchically organized into modules • Robustness of the attractors • Noise:
To study • Relation between the exponent of the fluctuations and the spreading of damage g • Damage needs to be spreaded but in a limited way
– Changes in environment – Influence of the rest of the network
Dynamical rules • Majority makes sense in neural networks • Genes has a precise role (activate and inhibit) • How the network of interactions can be represented? • Difficult to generalize rules to larger number of inputs
Complex Dynamics in a Simple Model of Signaling ...
Nov 15, 2007 - Each site receives k inputs. ⢠Assign Boolean functions to each ... Dynamics. ⢠All sites evolve according to its own rule. ⢠3 inputs: left. 3 inputs: ...
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sired temporal logic properties of the HMGB1 model. The. Boolean network modeling and Model Checking provide an alternative way and new insights into the study of the. HMGB1 signaling pathway in pancreatic cancer. Keywords: Model Checking, HMGB1, Sig
ply Model Checking to the study of a biological system ... of hardware, digital circuits, and software designs. Given .... This is in accord with evidence from cancer.
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Aug 14, 2014 - 39.2%. 17.0%. 7.3%. 36.2%. 37.0%. 34.1%. 33.5%. Figure 4. Image reconstruction examples. We compare reconstructions from two different ...
We address another key dimension of the design spaceâwhether model-based con- .... be found in [19] or [18] with videos of the bicycle and other material).
The remaining errors are all of my own. ... start looking at a stock pollutant instead and therefore we need a model that turns the IEA into a ... Scholars have previously considered Bayesian learning in the literature of the environmental eco-.
Sep 21, 1998 - Current carrier prop- erties are still being debated [4â6], particularly the role played by ... In metals, supercurrent carriers are electronic. Cooper pairs. In cuprates, they are usually pairs of electronic .... the holes created b
