Complex Dynamics in a Simple Model of Signaling ... - Google Sites

15/11/2007

Complex Dynamics in a Simple Model of Signaling Networks*

Signaling network • Set of units that process signals

Albert DíazDíaz-Guilera Universitat de Barcelona

•rule for processing •set of connections •noise

NWU: L.A.N. Amaral Amaral,, A.A. Moreira Moreira,, L. Guzman HU: A.L. Goldberger, L.A. Lipsitz

http://complex.ffn.ub.es

*PNAS 101 (2004) 15551 J. Stat. Mech. (2007) P01013

Motivation • 1/f fluctuations in physiological signals of healthy individuals • Changes in scaling with disease and aging (Goldberger et al., PNAS 99, 2466; Lipsitz, J Gerontology 57A, B115) • Disease/aging and connectivity (Buchman, Nature 420, 246)

Complex connectivity and physiological maturity 8-11 days

immature

28-33 days

mature

Autonomic maturation in piglets (interbeat intervals) Data from Lipsitz and colleagues

ÎScaling Scaling vs connectivity

Introduction: Boolean rules σ i (t ) = F [σ 1 (t − 1), σ 2 (t − 1),...., σ k (t − 1)]

Boolean functions of 2 inputs 1

inputs 1

outputs 0 1

1

0

0 1

0

1

0 1

0

0

0

i

F: lookup table

22

k

functions of k inputs

16 functions f i

XOR AND OR

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Boolean functions of 3 inputs 1

1

1

b7

1

1

0

b6

1

0

1

b5

1

0

0

b4

0

1

1

b3

0

1

0

b2

0

0

1

b1

0

0

0

b0

Some names of specific rules

decimal expression of the rule

b0 + 2* b1 + 22 * b2 + 23 * b3 + +2 * b4 + 2 * b5 + 2 * b6 + 2 * b7 4

5

6

7

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

Topology • Scale free? • Homogeneous? Poisson? • Experimentally: – Poisson incoming connectivity – Scale-free outgoing connectivity

• Observation in our noisy dynamics

– 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

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