The role of

(power-law) renewal events

in complex systems

Paolo ParadisiInstitute of Information Science and Technologies “A. Faedo” (ISTI-CNR),

Via Moruzzi 1, 56124 Pisa, Italy

In collaboration with:R. Cesari, P. Allegrini, D. Chiarugi, D. Contini, A. Donateo, A. Gemignani, D. Menicucci

Brownian motion

Mesoscopic particlesRobert Brown, 1827:lipid organelles ejected from the pollen grainsJan Ingenhousz, 1785:coal dust particles on alcohol surface

Standard Random Walk(mesoscopic model for Gaussian diffusion)

t0

t1 t2

t3 t4 t5

t6

+1

−1

+1 +1 +1

−1−1

ξn = fluctuating velocity:momentum exchange during a collision event

Diffusion variable:

dX(t)

dt= ξ(t)⇒ X(t) =

∫ t0ξ(s)ds =

n∑i=1

∆xi

Einstein’s theory of Brownian motion (1905)(Equilibrium: Maxwell-Boltzmann distribution of velocities)

Independent collision events ⇒ Independent increments

Central Limit Theorem ⇒⇒ normal (Gaussian) diffusion, Standard Diffusion Equation

〈X2〉(t) = 2Dt

Einstein-Smoluchowski’s relation:

D = m KBT (m = Va/Fext=mobility)

Markovian Master Equation, Time-Continuous Markov Chains(exponential times among collision events)

Summary

• Renewal events in complexity:

- emerging structures and properties

- fractal time intermittency in cooperative systems

• Measuring complexity:

Event-driven diffusion scaling ⇔ complexity index µ

• Some results on real data (human brain, EEG, wake-sleep)

• Why using diffusion scaling ?

• Renewal modeling of complexity: an example from biology

Complexity

m

Emerging structures

in cooperative systems

Coherent structures in turbulence

Relatively long LIFE-TIMES, but NOT infinite

Turbulent transport in wall flows

From Y. Mao, Int. J. Sedim. Res. 18(2), 148-157 (2003).

Particle resuspension and sedimentation(deposition fluxes, aerosol sources)

Competition of sweeps (top-down) and ejections (bottom-up)

Temporal Complexity

m

Fractal time intermittency

in cooperative systems

Dynamical instabilities ⇒ Critical events

Event = SHORT-TIME transition (bursting or decay)

INTER-EVENT TIMES with power-law tails: ψ(τ) ∼ 1/τµ

Fractal Time Intermittency (µ = complexity index)

Weak turbulence in liquid crystals Blinking Quantum Dots

Brain dynamics (human EEG) Earthquakes (Fault dynamics)

Temporal Complexity

m

Critical phenomena

Critical systems

From Pellegrini et al., 2007, ν = 1.8± 0.2

Neural network model: avalanches with scale-free sizePellegrini, et al., Phys. Rev. E 76, 016107 (2007)

Fractal Intermittency in critical systems

Contoyiannis and Diakonos, 2000 (analytical derivation):

3D Ising model at critical point ⇔ type-I intermittent map

(equilibrium, ergodicity)

Grigolini, West and co-workers (numerical derivation):

Simple and complex networks ⇒ fractal intermittency

Turalska et al. (2011) (numerical derivation):

Decision-making model ⇒ fractal intermittency(∼ 2D Ising model)

(non-stationarity, non-ergodicity)

Contoyiannis and Diakonos, Phys. Lett. A 268, 286-292 (2000)Turalska, Phys. Rev. E 83, 061142 (2011)

A prototype for fractal intermittency:the Manneville map (turbulent bursting){

xn+1 = xn + k|xn|z if xn < rxn+1 = (xn − r)/(1− r) if xn > r

z > 1, r : r + k|r|z = 1⇒ xi = 0 marginally unstable (Type-I intermittency )

chaotic zone

0 r 1

laminar zone

1

τ τ τ τ1 2 3 6τ τ4 5

0

0.2

0.4

0.6

0.8

1

0 100 200 300 400 500 600 700 800 900 1000

x_n

n

COMPLEXITY = renewal events + power-law decay

Manneville, J. de Physique 41, 1235-1243, 1980

Summing up ...

Complexity Pure Randomnessm m

Cooperative system Independent sub-systemsSelf-organized structures

m mPower-law (µ) Exponential(Non-Poisson) (Poisson)

↘ ↙Renewal processes

Complexity

⇓

Renewal processes

Definition of Renewal processes

Recurrent events associated with birth and death ofmetastable statesInter-event times → mutually independent random variables

0.01

0.1

1

0.1 1 10 100 1000

Ψ(τ)

τ

Ψ0

ψ(τ) ∼ 1/τµ⇒ µ

µ ⇒ indicator of complexity

Main ingredient of “renewal event”-driven processes

complexity = renewal + power-law

Cooperative systems (emerging structures):

long range correlations, power-law memory,

Non-Poisson events (power-law distribution)

No cooperation:

short-range (exponential) correlations, short-time memory,

Poisson events (exponential distribution)

ψ(τ)⇒ event-driven diffusion process

ψ(τ) ∼ 1/τµ ; µ = complexity index

Correlation : C(t) ∼ 1/tβ

Diffusion : σ(t) = 〈X2〉(t) ∼ t2H

PowerSpectrum : S(f) ∼ 1/fη

PDF Self − Similarity : P (x, t) =1

tδF

(x

tδ

)

β, H, η and δ depend on µ

Different expressions depending on jump statistics

A renewal-based approach in

Time Series (Data) Analysis

Detecting the events ?

Estimating the complexity index ?

RENEWAL events in the brain network ?

Critical Events in the brain: bursts and avalanches

• Observed nonstationarity in EEG signals reflects theswitching among metastable states (formation of neural

assemblies during brain functioning).

• A Neural assembly is a group of neurons for whichcoordinated (coherent) activity persists over substantialtime intervals and underlies basic operations of informationprocessing (quasi-stationary periods in EEG)

• Critical Events: abrupt changes in theElectroEncephaloGrams (EEG) ⇒ abrupt transitionsbetween two quasi-stationary conditions (Rapid TransitionProcesses = RTP).

Martin, Scientist 22, 23 (2008); Plenz et al., Trends Neurosci. 23, 11167 (2003); Chialvo,New Ideas Psychol. 26, 158 (2008); Kaplan et al., Signal Process. 85, 2190 (2005)

Avalanches

α bursts

Event detection (single EEG channels, α wave, 8− 12 Hz)

Kaplan et al., Signal Process. 85, 2190 (2005)Allegrini et al., PRL 99(1), 010603 (2007),Allegrini et al., PRL 103(3), 030602 (2009)

Coincidences between EEG channels (global events)

x xx x

x

x

x

x

x

xx

x

x

x

x

x

x

x

∆tc: time interval defining a coincidence between two EEG channels

Nt: minimum number of coincidences defining a global event

Allegrini et al., PRE 80, 061914 (2009)Allegrini et al., Front. Physio. 1, 128 (2010)

Event-driven diffusion scaling

[Estimating complexity index µ]

Event-driven diffusion: 3 different random walksAsymmetric Jump (AJ)

t0 t2t1

τ 1 τ 2 τ 3

t3

+1 +1+1 +1

Symmetric Velocity (SV)

t0

τ 1 τ 2 τ 3

t1 t2 t3

+1+1

−1 −1

Symmetric Jump (SJ)

t0

t1

t2

t3

τ 1 τ 2 τ 3

+1 +1

−1−1

Allegrini et al., PRE 54, 4760 (1996); Grigolini et al., Fractals 9, 439 (2001)Grigolini et al., PRE, 65, 046203 (2002); Metzler and Klafter, Phys. Rep. 339, 1 (2000)Shlesinger, J. Stat. Phys. 10, 421 (1974)

Scaling δ: Diffusion Entropy

ξ(t) = artificial signal (10011... or +1− 1− 1− 1 + 1 + 1..., etc...) generatedfrom the sequence of events

X(t) =

∫ t

0ξ(t′)dt′

S(t) = −∫dxp(x, t) ln(p(x, t)) = δ ln(t+ T ) + C

p(x, t) is evaluated splitting the time series into overlapping time windows.

Diffusion scaling H: Detrented Fluctuation Analysis

Local trend is evaluated with a least-squares straight line fit

Fluctuations: X(t) = X(t)− at− b:

F 2(t) =1

t

t∑i=1

X2(i)

F (t) ∼ tH

Average over all time windows.

Diffusion Entropy

SJSVAJ

1

0.5

01 2 3

1.5

δ

µ

Detrended Fluctuation Analysis

SJSVAJ

1

0.5

01 2 3 µ

1.5

H

H = δ = 0.5: normal scaling

(Gaussian PDF, short-time correlation, white noise)

Results on human EEG

• 30 healthy subjects

• Relaxed closed-eye condition

• No psychological tasks

• No (visual, ...) external stimuli

Scaling is robust under change of the thresholds ∆tc and Nt

Diffusion Entropy

∆tc = 10ms,Nt = 4∆tc = 10ms,Nt = 2∆tc = 6ms,Nt = 4∆tc = 6ms,Nt = 2∆tc = 2ms,Nt = 4∆tc = 2ms,Nt = 2S(t)

t (ms)

δSJ

= 0.5

104103102101100

5

4

3

2

1

0

DFA

∆tc = 10ms,Nt = 4∆tc = 10ms,Nt = 2∆tc = 6ms,Nt = 4∆tc = 6ms,Nt = 2∆tc = 2ms,Nt = 4∆tc = 2ms,Nt = 2

σ(t)

t (ms)

HSJ

= 0.5

104103102101100

100

10

1

0.1

0.01

Diffusion Entropy

0

1

2

3

4

5

6

100 101 102 103 104

S(t

)

t (ms)

subject 1subject 21subject 29

δAJ=0.9

0

0.5

1

1.5

2

2.5

3

3.5

4

100 101 102 103 104S

(t)

t (ms)

subject 1subject 21subject 29

δSJ=0.5

Detrended Fluctuation Analysis

HSW

= 0.93H

SJ= 0.54

HAJ

= 0.98SW ruleSJ ruleAJ ruleσ(t)

t (ms)(a) Subject 2

104103102101100

104

103

102

101

100

10−1

HSW

= 0.95H

SJ= 0.49

HAJ

= 0.95SW ruleSJ ruleAJ ruleσ(t)

t (ms)

σ(t)

t (ms)(b) Subject 13

104103102101100

104

103

102

101

100

10−1

HSW

= 0.96H

SJ= 0.53

HAJ

= 0.94SW ruleSJ ruleAJ ruleσ(t)

t (ms)

σ(t)

t (ms)

σ(t)

t (ms)(c) Subject 23

104103102101100

104

103

102

101

100

10−1

HSW

= 0.96H

SJ= 0.50

HAJ

= 0.95SW ruleSJ ruleAJ ruleσ(t)

t (ms)

σ(t)

t (ms)

σ(t)

t (ms)

σ(t)

t (ms)(d) Subject 27

104103102101100

104

103

102

101

100

10−1

Diffusion Entropy

Subject label

µAJ

(b)

2621161161

2.35

2.3

2.25

2.2

2.15

2.1

2.05

2

δSJ

δAJ

(a)

10.950.90.850.80.75

0.6

0.58

0.56

0.54

0.52

0.5

0.48

0.46

NO correlation between δAJ and δSJ (NO subject effect)

µAJ = 2.16± 0.16 ; δSJ ' 0.5⇒ µ > 2

Detrented Fluctuation Analysis

HSJ

HAJ

(a)

1.0210.980.960.940.920.90.88

0.6

0.58

0.56

0.54

0.52

0.5

0.48

0.46

0.44

0.42

µAJ

= 2.12± 0.12 ; HSJ ' 0.5⇒ µ > 2

Scaling of EEG during sleep condition (NREM vs. REM):

Towards a diagnostic index for consciousness

1

10

100

103 104 105t (ms)

σ(t)

WAKEREMSWS

H=0.75H=0.5

0.01

0.1

1

10

100 102 103 104 105

t (ms)

PP et al., AIP Conf. Proc. 1510, 151 (2013).Allegrini et al., Chaos Solit. Fract. 55, 32 (2013).

Why using diffusion scaling ?

Effect of Noise

Why do not use direct estimation of µ from WT

distribution ?

Human EEG: Some authors found µ ' 1.6− 1.7 in the brain

(different from µ ' 2.1)

We observe that:

AJ rule: H ' 0.5 in the (relatively) short-time range

⇒ presence of NOISE (short-time, exponentially correlated)

Gong et al., PRE 76,011904, 2007Bianco et al., PRE 75, 061911, 2007

Fractal intermittency + Noise

Erroneous estimation of power exponent directly from the WT

distribution (low statistics and blurring effect of noise)

Superposition of Poisson and Non-Poisson (power-law) events

A model for Poisson + Non-Poisson: 3 Parameters

Complexity index µ: intermittent events emerging from thecooperative (self-organized) system

Time scale T = time necessary to reach the asymptoticalinverse power-law 1/τµ

ψn(τn) = (µ− 1)Tµ−1

(τn + T )µ

Poisson rate r = average number of Poissonian events pertime unit

ψp(τp) = re−rτp

rT = average number of Poisson events over the time scale ofthe Non-Poisson process. Measure of the relative contributionof the Poisson process to the entire process.Allegrini et al., PRE 82, 015103(R) (2010)

Stationary assumption ⇒ you can use P (n) and P (p) in the

computation of probability of having Poisson or Non-Poisson

events ( = time-average of the probability of having a

Non-Poisson or a Poisson event, respectively)

4 complementary situations: (n→ n), (n→ p), (p→ n), (p→ p)

⇒ 4 independent contributions to the total probability:

P [n(t), n(t+ τ)] = P (n)e−rτψn(τ)

P [n(t), p(t+ τ)] = P (n)Ψn(τ)[re−rτ ]

P [p(t), n(t+ τ)] = P (p)e−rτψ∞n (τ)

P [p(t), p(t+ τ)] = P (p)Ψ∞n (τ)[re−rτ ]

Allegrini et al., PRE 82, 015103(R) (2010)

Waiting-time probability density

ψ(τ) ={P (n)[ψn(τ) + 2rΨn(τ)] + P (p)rΨ∞n (τ)

}e−rτ

Survival Probability

Ψ(τ) = e−rτ [P (n)Ψn(τ) + P (p)Ψ∞n (τ)]

Ψn(τ) =(

T

T + t

)µ−1; Ψ∞n (τ) =

(T

T + t

)µ−2

P (n) =τp

τp + τn; P (p) =

τn

τp + τn

τp =1

r; τn =

T

µ− 2(µ > 2)

Allegrini et al., PRE 82, 015103(R) (2010)

Waiting-time distribution density ψ(τ)

Solid-line histogram: events from real EEG data.Solid line: theoretical prediction for ψ(τ) with µ = 2.05, r = 3.0 Hz,T = 6.0s.Dotted line: inverse-power-law best fit to the data using the time interval[0.02,2]s, yielding an erroneous µ = 1.6.

Allegrini et al., PRE 82, 015103(R) (2010)

Modeling complexity

through renewal processes

⇓

An example from biology

Vesicle formation and protein entrapment

No self-organization model:

Entrapment as random sampling ⇒ Poisson expected.

Lipid-Lipid and Lipid-Protein Cooperation:

PP, Chiarugi and Allegrini, BMC Bio-informatics, under revision.

Ψ(τ) = exp

{−∫ τ

0rc(t′)dt′

}; ψ(τ) = −dΨ

dτ(τ)

rc = closure rate of lipid vesicle surface

Lipid-Lipid + Lipid-Protein cooperation

(1) Jamming rc ∝ 1/(1 +N(τ))(2) Semi-permeability (mean input flux): ⇒ 〈N〉(t) = λt ; λ > 0

〈rc〉(t) =r0

1 + λt; N = N(τ) = λτ ; P (N)dN = ψ(τ)dτ

P(N) =µ− 1

(1 + N)µ; µ = 1 +

r0

λPP, Chiarugi and Allegrini, BMC Bio-informatics, under revision.

Conclusions

• COMPLEXITY as RENEWAL + POWER-LAW µ

(Non-Poisson events)

• µ = complexity (or fractal) index (emergent property of

the strong non-linear interactions in the cooperative

system)

• Blurring effect of noise ⇒ estimate of µ and renewal from

diffusion scaling is more reliable (no time distribution)

Paolo Paradisi

E-mail: paolo.paradisi@isti.cnr.it

In collaboration with:

Davide Chiarugi

Dep. Theory and Biosystems, Max Planck Institute of Colloids and Interfaces,Potsdam-Golm Science Park, Am Muhlenberg 1 OT Golm, 14476 Potsdam

R. Cesari, D. Contini, A. Donateo

Institute of Atmospheric Sciences and Climate (ISAC-CNR), Lecce Unit, Strada ProvincialeLecce-Monteroni, km 1.2, 73100 Lecce, Italy.

P. Allegrini1,2, D. Menicucci1,2, A. Gemignani1,2,3,1Institute of Clinical Physiology (IFC-CNR), Via G. Moruzzi 1, 56124 Pisa, Italy2Centro EXTREME, Scuola Superiore Sant’Anna, Piazza Martiri della Liberta 7, 56127Pisa, Italy3Department of Physiological Sciences, University of Pisa, Via San Zeno 31, 56127 Pisa,Italy

THANK YOU FOR YOUR ATTENTION !

RESIDUAL SLIDES

Temporal

Complexity

vs.

Structural, Topological, Spatial

Complexity

⇓

Critical Phenomena

Critical systems I

From Pellegrini et al., 2007, ν = 1.8± 0.2

Neural network model: avalanches with scale-free sizePellegrini, et al., Phys. Rev. E 76, 016107 (2007)

Critical systems II

From Plenz and Thiagarajan, 2007, ν ∼ 1.5

Percolation model: giant cluster in the critical condition

Plenz and Thiagarajan, Trends Neurosci. 30(3), 101-110 (2007)

Critical systems III

From Chialvo, 2010: Is the brain critical ?

Ising model vs. fMRI: scale-free degree distribution

Chialvo, Nature Physics 6, 744-750 (2010)Fraiman et al., Phys. Rev. E 79, 061922 (2009)

An example from turbulence

D0 = 0.3D0 = 0.2

D0 = 0.13D0 = 0.09D0 = 0.04D0 = 0.02

t (s)

(b) WT-SPF

10210110010−110−2

100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

D0 = 0.3D0 = 0.2D0 = 0.13D0 = 0.09D0 = 0.04D0 = 0.02

t (s)

(a) WT-PDF

10110010−110−2

101

100

10−1

10−2

10−3

10−4

10−5

PP et al., Nonlin. Processes Geophys., 19, 113-126, 2012

Is complexity index µ really important ?

Relationships with other fractal indices ?

Fractal time intermittency: why RENEWAL ?

(1)

(2) Minimize entropy increase(min disorder ⇔ max order = self-organization)

West et al., Physics Reports 468 (1-3), 1-99 (2008)Silvestri et al., PRL 102(1), 014502 (2009)

Fractal Dimension vs. intermittencyDF = µ− 1; µ < 2 DF = 1; µ ≥ 2

DF = 1D0 = 0.3D0 = 0.2D0 = 0.13D0 = 0.09D0 = 0.04D0 = 0.02

(b) Data

1/Nb

∆tb (s)10410310210110010−110−2

10−1

10−2

10−3

10−4

10−5

10−6

10−7

DF= 1

DF= 0.8

DF= 0.7

µ = 2.5µ = 2.2µ = 1.8µ = 1.7

(a) Model

1/Nb

∆tb107106105104103102101100

10−1

10−2

10−3

10−4

10−5

10−6

10−7

PP et al., Nonlin. Processes Geophys., 19, 113-126, 2012

Sn =∆X1 + ...+ ∆Xn√

n→ S; G(S) ∝ exp

(− S2

2〈S2〉

); 〈S2〉 = 2D∆t

〈X2〉(t) =n∑i=1

〈∆X2i 〉+

n∑i 6=j

〈∆Xi∆Xj〉 = 〈S2〉n = 2D(∆t n) = 2Dt

Normal scaling: 〈X2〉(t) = 2Dt2H ; H = 0.5 (Hurst exponent)

P (x, t) = 1tδG(xtδ

); Self-similarity: Z = x/t

δ; δ = H = 0.5

∂P

∂t= D

∂2P

∂x2(Standard diffusion equation)

Relative frequency Π of a channel to be recruitedinto a global event (Nc = 2,6,10)

C3 C4

CP3 CP4CPz

Cz

F3 F4F7 F8

FC3 FC4

Fp1 Fp2

FT7 FT8

Fz

O1 O2Oz

P3 P4Pz

T3 T4

T5 T6

TP7 TP8 0.05

0.1

0.15

0.2

0.25

0.3

C3 C4

CP3 CP4CPz

Cz

F3 F4F7 F8

FC3 FC4

Fp1 Fp2

FT7 FT8

Fz

O1 O2Oz

P3 P4Pz

T3 T4

T5 T6

TP7 TP8 0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

C3 C4

CP3 CP4CPz

Cz

F3 F4F7 F8

FC3 FC4

Fp1 Fp2

FT7 FT8

Fz

O1 O2Oz

P3 P4Pz

T3 T4

T5 T6

TP7 TP8 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Spearman’s correlationRΠ,δ = 0.94Rδ,∆δ = −0.8

δ and ∆δ

C3 C4

CP3 CP4CPz

Cz

F3 F4F7 F8

FC3 FC4

Fp1 Fp2

FT7 FT8

Fz

O1 O2Oz

P3 P4Pz

T3 T4

T5 T6

TP7 TP8 0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

C3 C4

CP3 CP4CPz

Cz

F3 F4F7 F8

FC3 FC4

Fp1 Fp2

FT7 FT8

Fz

O1 O2Oz

P3 P4Pz

T3 T4

T5 T6

TP7 TP8 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Avalanche size distribution

10-5

10-4

10-3

10-2

10-1

100

1 2 3 4 5 6 7 8 910 20 30

P(N

)

N

mean histogram0.6/N1.92

10-4

10-3

10-2

10-1

100

1 2 3 4 5 6 78 10 20 30

∆t=0∆t=2ms∆t=4ms∆t=6ms∆t=8ms

∆t=10ms0.6/N1.92

Turing’s conjecture: the working brain needto operate at a critical level, to stay away

from the two extremes, namely the toocorrelated subcritical level and the

explosive supercritical dynamic(intermediate critical condition →

→ phase transitions, critical phenomena,Self-Organized Criticality)

P (n) ∝ 1/nζ ζ = 1.92± 0.12

Beggs and Plenz, J. Neuroscience 23, 11167 (2003)Werner, Biosystems 96, 114-119 (2009)Chialvo and Bak, Neuroscience 90 (4), 1137-1148 (1999)Allegrini et al., PRE 80, 061914 (2009)Allegrini et al., submitted to Frontiers in Fractals Physiology

HAJ

HSV

(c)

0.980.960.940.920.90.880.860.84

1.02

1

0.98

0.96

0.94

0.92

0.9

0.88

µAJ = 2.12± 0.12 ; µSV = 2.13± 0.10

HSV has no ambiguities in passing from µ > 2 to µ < 2.

Linear Response for renewal Non-Poisson events

Events are perturbed by external stimuli

(e.g., brain neural network stimulated by a visual, auditory or

tactile complex signals)

Π(t) = 〈ξs〉(t) = ε∫ t

0dt′χ(t− t′)ξp(t′)

Dichotomous signals ξs = ±1; ξp = ±1:

ψs(τ) ∼ 1

τµs; ψp(τ) ∼ 1

τµp

B.J. West et al., Physics Reports 468, 1-99 (2008)Allegrini et al., Phys. Rev. Lett. 103(3), 030602 (2009)Silvestri et al., Phys. Rev. Lett. 102(1), 014502 (2009)Allegrini et al., Phys. Rev. Lett. 99(1), 010603 (2007)

Analytical result:

〈Π(t)〉p = ε

[Ap(µs, µp)

tµp−1 +As(µs, µp)

tµp+1−µs

]Divergence of As and Ap in the limit µp → µs

0

10

20

30

40

50

60

70

80

90

1.6 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69

A_s

mu_p

mu_s = 1.7

f(x)

Manneville-type stochastic model:

y = α yz; 0 < y < 1; z > 1

����������������������������

����������������������������

��������������������������������

0 1

Random Back InjectionCritical event: y(t−n ) = 1

WT (Exit Time): τn = tn − tn−1

y(t+

n) = ξn: uniform random back injection

Inverse power-law distribution of WTs: y(τn, ξn) = 1⇒ τn = F (ξn)

ψ(τ)dτ = u(ξ)dξ; u(ξ) =

{1 0 < ξ < 10 elsewhere

ψ(τ) =µ− 1

T

1

(1 + τ/T )µ≈ 1

τµ; µ =

z

z − 1> 1; T =

1

α(z − 1)

P. Allegrini et al., Phys. Rev. E 68, 056123 (2003)

Master equation

Two states + or −; Discrete time (∆t = 1):

P(n) =

[P+(n)P−(n)

]; R(n) =

[−r+(n) r−(n)r+(n) −r−(n)

]

P(n+ 1)−P(n) = R(n) ·P(n)

Jump probabilities and transition rates:

p(+→ −, n) = r+(n∆t)∆t ; p(+→ +, n) = 1− r+(n∆t)∆t

p(− → +, n) = r−(n∆t)∆t ; p(− → −, n) = 1− r−(n∆t)∆t

Renewal processes: Cox’s rate of event

production

t0 = 0, t1, t2, ... ; τn = tn+1 − tn

r(t) = limdt→0

1

dtPr {t < τn ≤ t+ dt | τn > t} = − 1

Ψ(t)

dΨ(t)

dt

Ψ(τ) =∫ ∞τ

ψ(s)ds = exp(−∫ τ

0r(t)dt

)

r(t) =r0

1 + r1(t− tn), tn ≤ t < tn+1.

ψ(τ) = (µ− 1)Tµ−1

(T + τ)µ; µ = 1 +

r0

r1; T =

1

r1;

Testing Renewal hypothesis: Aging Analysis

Ψ0

ΨRΨS

τ (sec.)W

T-SPF

(b) PM2.5

104103102101100

100

10−1

10−2

10−3

10−4Ψ0

ΨRΨS

τ (sec.)

WT-SPF

(a) Vertical velocity

103102101100

100

10−1

10−2

10−3

10−4

PP et al., Europ. Phys. J.: Special Topics 174, 207 (2009)

� �

Complex networks Serial intermittency Entropy reduction

Brain at critical point

(a) Intermittency in fluctuations of order parameter

(d) Small Worlds in the brain

(e) Global Workspace (f) Operational Modules

(g) high Φ (integrated information)

(b) Correlational (i. e. functional) connectivity of resting brain MRI is that of a ferromagnet at criticality

(c) At criticality the entropy per volume tends to zero

Features of consciousness explained by the hypothesis that awake resting-state brain works at a critical point

10-6

10-5

10-4

10-3

10-2

10-1

100

1 10 100N

P(N)

REMSWS

WAKEN2

=1.9

                                   

                                   

Event:  integra,ng  excita,ons   Downwards  causa,on  or  SSO-­‐driven  reset  (long  arrow)  

Survival Probability Ψ(τ)

Solid line: theoretical (analytical) prediction with same µ, r and T as before.Effect of poor fit in the short time range propagate in the large time rangewhen considering Survival Probability. Open squares stem from data. Opencircles stem from a numerical simulation of the model (same parameters asbefore, same statistics as the data).Dotted line is a stretched exponential A exp[(t/B)α], with A = 0.5,B = 45s, α = 0.6 (compatible with µ = α+ 1 = 1.6).

Allegrini et al., PRE 82, 015103(R) (2010)