Complexity in Physics: The Living State of Matter

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Music and the Brain interaction between two
complex networks
Paolo Grigolini University
of North Texas
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An example of network
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The Brain as a Complex Network
A set of neurons distributed on Complex networks,
characterized by large clustering and scale free
distribution of links (a) enhances
synchronization and (b) generates non-ergodic
quakes. The discovery of these properties
generates a new approach to the information
transport and the discovery that complex systems
communicate through complexity matching.
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Quakes as abrupt transition from one to another
cluster of synchronized units
Cluster of synchronized neurons in the time
interval ti1, ti2
Cluster of synchronized neurons in the time
interval ti, ti1
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The Minimal Spanning Tree Method
??ij correlation between site i and site j
dij? 2(1 - ??ij)1/2 distance between site
i and site j
N sites yield N! distances
i j
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Configuration 1
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Configuration 2
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Stretched exponential

• is the time distance between two consecutive
  events and ?????is the corresponding probability
  density, yielding the survival probability
  d?????d??????????
• ???????exp???????????????

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Definition of event
An abrupt topological change is an event
Question Do these events have memory of the
earlier events?
Remark The stretched exponential survival
probability is significantly different from the
exponential survival probability, thereby
suggesting that the system departs from the
condition of total randomness.
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Poisson Demon

• Survival probability
• of the Poisson Demon

g N/M ltlt 1
? ?????????g?????e-g?
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Time distances between two consecutive events
t
t0
t1
t2
t3
t4
t5
?(?t) g exp(-g?t)
waiting time distribution
?(?t) exp(-g?t)
Survival probability
The events are independent and the time distance
between two consecutive events is described by
an exponential waiting time distribution
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Aging Experiment
t
t0
t1
t2
t3
t4
t5
?ti ti1 - ti
Is ?ti independent of i?
ta
ta- old histogram
t
??
??
??
With the mobile window of size ta we select
only a fraction of the quiescent region between
two events
The resulting histogram is age dependent
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shuffling
t
t0
t1
t2
t3
t4
t5
t0
t3
t4
t5
t1
t2
t
?exp?t?ta?????exp?t?ta)
?shuffled??t?ta????ren?t?ta?
?exp?t?ta) ?ren?t?ta?
The occurrence of an event resets to zero the
system's memory
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EEG SIGNALS FIT THE RENEWAL CONDITION
This condition indicates that the transition
from a structure A to a structure B does not
depend on A
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Music Compositionas an artificial brain
pitch onset time
note frequency
wave form type
amplitude
articulation
Oscillator number
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The signal produced by music composition yields
stretched exponential SP of renewal kind.
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Demon with a social life
How to explan the Demon's action generating the
events of the brain, and of music composition as
well?
The event occurrence is explained by the
assumption that cooperation forces the Demon to
draw particles from his box only from time to time
The time distance between two consecutive
drawings is given by the subordination function
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Realistic Mittag-Leffler Survival Probability
truncation for t gt 1/G

• truncation for t gt 1/G

?????S???
?S????????????
The demon social life makes??????
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EEG as Mittag-Leffler Survival Probability
a 0.62
0.62
Truncation
Truncation
Stretched Exponential SP
Inverse power law SP
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Interaction between two complex networks
?S(t)
?????????P(t)gt
S
Coupling between the two complex networks
P
?P(t)
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B inherits the complexity of M

• B prepared much earlier
• than t 0
• Perturbation prepared at time t 0, when we
  switch on the interaction between B and M
• (i.e., when we switch on music, the brain has
  been in a
• state of rest for sufficiently long time)
• B inherits the statistics of M if
• ?B gt ?M
• In this case M becomes a basin of attraction for
  the brain

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Response of the brain

• BRAIN COMPLEXITY LEVEL

MUSIC COMPLEXITY LEVEL
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Mirollo and Strogatzssynchronization model
N neurons
Threshold
d x??dt S?????x
x
t
When a given neuron fires, it pulls all the
other oscillators Up by a an amount k and pulls
Them up to firing, whichever is less
d x??dt S?????x f(t)
Random fluctuation
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A complex network high clustering degree and
power law distribution of links

• P. Holme and B. J. Kim, Growing scale-free
  networks with tunable clustering, Phys. Rev. E
  65, 026107, (2002).

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An artificial brain with mB 1.85
lo log(-log(Y(t))
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Increasing the interaction strength makes the
inverse power law show up
T
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Davids harp

• Left ear w D/2
• Right ear w - D/2
• The frequency w undergoes abrupt changes
  automatically determined
• by the MST analysis of the patients brain

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CONCLUSIONS
A set of synchronized neurons generate complex
events, called critical events preliminary
results show that synchronization is improved by
using a complex rather than a regular network.
The brain, as a complex network, responds to
complex perturbations
We expect that these results may contribute to
setting the scientific foundation of therapeutic
techniques ?
Examples?
Neurophysiological biofeedback
EMDR eye movement desensitization and
reprocessing Treatment of Posttraumatic stress
disorder
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Collaborators D.Adams G. Aquino S. Bianco M.
Bologna A. Duggento L. Fronzoni A. Gemignani E.
Geneston M. Ignaccolo M. Latka D. Menicucci M.
Rider B. J. West P. Winsor Funding agencies ARO,
grant W911NF-05-1-005 Welch, grant B-1577 The
project is International USA Poland Italy Taiwan
and Interdisciplinary physics,
neurophysiology,psychology, psychiatry.music
composition and computer science
I
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From Mirollo and Strogatz to a more realistic
model for neuronsynchronization
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Linear response theory

• P. Allegrini, et al. Phys. Rev. Lett., 99,
  010603(2007)
• ??

• HAMILTONIAN APPROACH
• ????(t.t) ? (d/dt) ?(t,t), t gt t
• EVENT DOMINATED THEORY

• ?(t.t) - ? (d/dt) ?(t,t), t gt t

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CORRELATION PHASE SPACE
C?????S?t??P?t???for t -gt infinity
Both systems are prepared at t 0

?P
CORRELATION
NO CORRELATION
2
CORRELATION
CORRELATIcON
1
1
2
infinity
?S
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Kohlraush or Buelfinger? Neither of them
Mittag-Leffler!!!

• stretched exponential survival probability
• ??t???????exp???t???????????????
• Kolraush??????

• asymptotic power law
• ??t??????????????t???????????????
• Buelfinger??????

• Mittag-Leffler (1846-1927)
• ??t???????exp?????t?????????????
• for?t??????
• ??t??????????????t?????????????
• for?t??????

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Central Figure
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Central Figure
PISA EXPERIMENT
PISA MUSIC
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The frontal areavsthe whole brain
Boredom effect The frontal area goes back to the
initial condition
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auditory and frontal part merge into a common
behavior
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Synchronization and stretched exponential
survival probabilities