Transcript Chaos in the Brain. - Brain Dynamics Laboratory

Chaos in the Brain
Jaeseung Jeong, Ph.D
Department of Bio and Brain Engineering,
KAIST
Nonlinear dynamics and Chaos
Chaos
1. the formless
shape
of matter
that
is conditions
: the tiniest
change
in the
initial
alleged
to have
existed
beforeoutcome,
the
produces
a very
different
even when
Universe
was given equations
order.
the governing
are known exactly
- neither predictable nor repeatable
2. complete confusion or disorder.
3. Physics; a state of disorder and irregularity
that is an intermediate stage between
highly ordered motion and entirely
random motion.
Nonlinear dynamics and Chaos
(2) King Oscar II (1829 – 1907)
offered a prize of 2500 crowns
to anyone solve the n-body problem
 stability of the Solar System
Nonlinear dynamics and Chaos
• N-body problem
The classical n-body problem is that given the
initial positions and velocities of a certain
number (n) of objects that attract one another
by gravity, one has to determine their
configuration at any time in the future..
Nonlinear dynamics and Chaos
(3) Jules Henri Poincarè
(1854 – 1912)
Won the Oscar II’s contest,
 not for solving the problem,
but for showing that even the three-body
problem was impossible to solve.
(over 200 pages )
“…it may happen that small differences in the initial conditions
produce very great ones in the final phenomena. A small error
in the former will produce an enormous error in the latter.
Prediction becomes impossible, and we have the fortuitous
phenomenon”
- in a 1903, essay "Science and Method"
Nonlinear dynamics and Chaos
• N-body problem
This problem arose due to a deterministic way
of thought, in which people thought they
could predict into the future provided they
are given sufficient information. However,
this turned out to be false, as demonstrated
by Chaos Theory.
Nonlinear dynamics and Chaos
Nonlinear dynamics and Chaos
Systems behaving in this manner are now
called “chaotic.”
They are essentially nonlinear, indicating
that initial errors in measurements do not
remain constant, rather they grow and
decay nonlinearly (usually exponentially)
with time.
Since prediction becomes impossible, these systems can
appear to be irregular, but this randomness is only
apparent because the origin of their irregularities is
different: they are intrinsic, rather than due to external
influences.
What is chaos?
• The meteorologist E. Lorenz
He modeled atmospheric convection in terms of three differential
equations and described their extreme sensitivity to the starting
values used for their calculations.
• The meteorologist R May
He showed that even simple systems (in this case interacting
populations) could display very “complicated and disordered”
behavior.
• D. Ruelle and F. Takens
They related the still mysterious turbulence of fluids to chaos and
were the first to use the name ‘strange attractors.’
Nonlinear dynamics and Chaos
Lorenz attractor
Nonlinear dynamics and Chaos
The Logistic equation
• Xn+1=AXn(1-Xn)
The Logistic equations
R
Nonlinear dynamics and Chaos
• Laminar(regular) / Turbulent(chaotic)
• Turbulent of gas flows
Nonlinear dynamics and Chaos
• High flow rate
: Laminar  Turbulent
Department of BioSystems
What is Chaos?
• M Feigenbaum
He revealed patterns in chaotic behavior by
showing how the quadratic map switches from one
state to another via periodic doubling.
• TY Li and J Yorke
They introduced the term ‘chaos’ during their
analysis of the same map.
• A. Kolmogorov and YG Sinai
They characterized the properties of chaos and its
relations with probabilistic laws and information
theory.
Nonlinear dynamics and Chaos
• Taffy – pulling machine
Nonlinear dynamics and Chaos
• The strength of science
It lies in its ability to trace causal relations and so to predict future events.
• Newtonian Physics
Once the laws of gravity were known, it became possible to anticipate
accurately eclipses thousand years in advance.
• Determinism is predictability
The fate of a deterministic system is predictable
• This equivalence arose from a mathematical truth:
Deterministic systems are specified by differential equations that make no
reference to chance and follow a unique path.
Chaos systems
• Newtonian deterministic systems
(Deterministic, Predictable)
• Probabilistic systems
(Non-deterministic, Unpredictable)
• Chaotic systems
(Deterministic, Unpredictable)
Dynamical system and State space
• A dynamical system is a model that determines the evolution
of a system given only the initial state, which implies that these
systems posses memory.
• The state space is a mathematical and abstract construct, with
orthogonal coordinate directions representing each of the
variables needed to specify the instantaneous stae of the system
such as velocity and position
• Plotting the numerical values of all the variables at a given time
provides a description of the state of the system at that time. Its
dynamics or evolution is indicated by tracing a path, or
trajectory, in that same space.
• A remarkable feature of the phase space is its ability to
represent a complex behavior in a geometric and therefore
comprehensible form (Faure and Korn, 2001).
Phase space and attractor
Phase space and attractor
For any phenomena, they can all be modeled as a system
governed by a consistent set of laws that determine the
evolution over time, i.e. the dynamics of the systems.
Linear vs. Nonlinear
Conservative vs. Dissipative
Deterministic vs. Stochastic
• A dynamical system is linear if all the equations describing its
dynamics are linear; otherwise it is nonlinear.
• In a linear system, there is a linear relation between causes and
effects (small causes have small effects); in a nonlinear system
this is not necessarily so: small causes may have large effects.
• A dynamical system is conservative if the important quantities
of the system (energy, heat, voltage) are preserved over time; if
they are not (for instance if energy is exchanged with the
surroundings) the system is dissipative.
• Finally a dynamical system is deterministic if the equations of
motion do not contain any noise terms and stochastic otherwise.
Attractors
• A crucial property of dissipative deterministic dynamical
systems is that, if we observe the system for a sufficiently long
time, the trajectory will converge to a subspace of the total
state space. This subspace is a geometrical object which is
called the attractor of the system.
• Four different types of Attractors:
• Point attractor: such a system will converge to a steady state
after which no further changes occur.
• Limit cycle attractors are closed loops in the state space of the
system: period dynamics.
• Torus attractors have a more complex ‘donut like’ shape, and
correspond to quasi periodic dynamics: a superposition of
different periodic dynamics with incommensurable
frequencies (Faure and Korn, 2001; Stam 2005).
Stam, 2005
Chaotic attractors
• The chaotic (or strange) attractor is a very complex object
with a so-called fractal geometry. The dynamics
corresponding to a strange attractor is deterministic chaos.
• Chaotic dynamics can only be predicted for short time
periods.
• A chaotic system, although its dynamics is confined to the
attractor, never repeats the same state.
• What should have become clear from this description is that
attractors are very important objects since they give us an
image or a ‘picture’ of the systems dynamics; the more
complex the attractor, the more complex the corresponding
dynamics.
Three-dimensional
Lorenz attractors
Characterization of the attractors
I
• If we take an attractor and arbitrary planes which cuts the
attractor into two pieces (Poincaré sections), the orbits which
comprise the attractor cross the plane many times.
• If we plot the intersections of the orbits and the Poincaré
sections, we can know the structure of the attractor.
Characterization of the attractors
II
• The dimension of a geometric object is a measure of its
spatial extensiveness. The dimension of an attractor can be
thought of as a measure of the degrees of freedom or the
‘complexity’ of the dynamics.
• A point attractor has dimension zero, a limit cycle dimension
one, a torus has an integer dimension corresponding to the
number of superimposed periodic oscillations, and a strange
attractor has a fractal dimension.
• A fractal dimension is a non integer number, for instance
2.16, which reflects the complex, fractal geometry of the
strange attractor.
Fractal dimension of the Attractor
Characterization of the attractors III
• Lyapunov exponents can be
considered ‘dynamic’ measures of
attractor complexity.
• Lyapunov exponents indicate the
exponential divergence (positive
exponents) or convergence (negative
exponents) of nearby trajectories on
the attractor.
• A system has as many Lyapunov
exponents as there are directions in
state space.
Characterization of the attractors
IV
• A chaotic system can be considered as a source of
information: it makes prediction uncertain due to the
sensitive dependence on initial conditions.
• Any imprecision in our knowledge of the state is magnified
as time goes by. A measurement made at a later time
provides additional information about the initial condition.
• Entropy is a thermodynamic quantity describing the amount
of disorders in a system.
r
H    p( s) log 2 p(2)
s 1
Control parameters and
multistability
• Control parameters are those system properties that can
influence the dynamics of the system and that are either held
constant or assumed constant during the time the system is
observed.
• Parameters should not be confused with variables, since
variables are not held constant but are allowed to change.
• Multistability: For a fixed set of control parameters, a
dynamical system may have more than one attractor.
• Each attractor occupies its own region in the state space of the
system. Surrounding each attractor there is a region of state
space called the basin of attraction of that attractor.
• If the initial state of the system falls within the basin of a
certain attractor, the dynamics of the system will evolve to that
attractor and stay there. Thus in a system with multi stability
the basins will determine which attractor the system will end
The escape time plot gives the basin of attraction.
Bifurcations
• In a multistable system, the total of coexisting attractors
and their basins can be said to form an ‘attractor
landscape’ which is characteristic for a set of values of
the control parameters.
• If the control parameters are changed this may result in a
smooth deformation of the attractor landscape.
• However, for critical values of the control parameters the
shape of the attractor landscape may change suddenly
and dramatically. At such transitions, called
bifurcations, old attractors may disappear and new
attractors may appear (Faure and Korn, 2001; Stam
2005).
Bifurcations
This EEG time series shows the transition between interictal and ictal brain
dynamics. The attractor corresponding to the inter ictal state is high
dimensional and reflects a low level of synchronization in the underlying
neuronal networks, whereas the attractor reconstructed from the ictal part
on the right shows a clearly recognizable structure. (Stam, 2003)
Route to Chaos
• Period doubling
As the parameter increases, the period doubles: perioddoubling cascade, culminating into a behavior that becomes
finally chaotic, i.e. apparently indistinguishable visually
from a random process
• Intermittency
A periodic signal is interrupted by random bursts occurring
unpredictably but with increasing frequency as a parameter
is modified.
• Quasiperiodicity
A torus becomes a strange attractor.
R
Intermittency
Detecting chaos in experimental data
• Bottom-up approach
We can apply nonlinear dynamical system methods to the
dynamical equations, if we know the set of equations
governing the basic systems variables.
• Top-down approach
• However, the starting point of any investigation in
experiments is usually not a set of differential equations, but
rather a set of observations.
• The way to get from the observations of a system with
unknown properties to a better understanding of the
dynamics of the underlying system is nonlinear time series
analysis.
• Starting with the output of the system, and working back to
the state space, attractors and their properties.
General strategy
of nonlinear dynamical analysis
• Nonlinear time series analysis is a procedure that
consists of three main steps:
• (i) reconstruction of the system’s dynamics in the state
space using delay coordinates and embedding
procedure.
• (ii) characterization of the reconstructed attractor using
various nonlinear measures
• (iii) checking the validity (at least to a certain extent) of
the procedure using the surrogate data methods.
Reconstruction of system dynamics
[problem]
our measurements usually do not have a one to one
correspondence with the system variables we are
interested in.
For instance, the actual state space may be determined
by ten variables of interest, while we have only two time
series of measurements; each of these time series might
then be due to some unknown mixing of the true system
variables.
Delay coordinate and Embedding procedure
• With embedding, one time series are converted to a series or
sequence of vectors in an m-dimensional embedding space.
• If the system from which the measurements were taken has
an attractor, and if the embedding dimension m is sufficiently
high, the series of reconstructed vectors constitute an
‘equivalent attractor’ (Whitney, 1936).
• Takens has proven that this equivalent attractor has the
same dynamical properties (dimension, Lyapunov spectrum,
entropy etc.) as the true attractor (Takens, 1981).
• We can obtain valuable information about the dynamics of
the system, even if we don't have direct access to all the
systems variables.
Takens’ Embedding theorem (1981)
y (t )  [ x j (t ), x j (t   ), x(t  2 ),..., x j (t  (d  1) )]
Takens has shown that, if we measure any single variable with
sufficient accuracy for a long period of time, it is possible to
reconstruct the underlying dynamic structure of the entire system
from the behavior of that single variable using delay coordinates
and the embedding procedure.
Time-delay embedding
• We start with a single time series of observations. From
this we reconstruct the m-dimensional vectors by taking
m consecutive values of the time series as the values for
the m coordinates of the vector.
• By repeating this procedure for the next m values of the
time series we obtain the series of vectors in the state
space of the system.
• The connection between successive vectors defines the
trajectory of the system. In practice, we do not use
values of the time series of consecutive digitizing steps,
but use values separated by a small ‘time delay’ d.
Stam, 2005
Parameter choice
• Time delay d: a pragmatic approach is to choose l equal to
the time interval after which the autocorrelation function (or
the mutual information) of the time series has dropped to 1/e
of its initial value.
• Embedding dimension m: repeat the analysis (for instance,
computation of the correlation dimension) for increasing
values of m until the results no longer change; one assumes
that is the point where m>2d (with d the true dimension of
the attractor).
Spatial Embedding
• The m coordinates of the vectors are taken as the values of
the m time series at a particular time; by repeating this for
consecutive time points a series of vectors is obtained.
• The embedding dimension m is equal to the number of
channels used to reconstruct the vectors. The spatial
equivalent of the time delay d is the inter electrode distance.
• The advantage of spatial embedding is that it achieves a
considerable data reduction, since the dynamics of the whole
system is represented in a single state space.
• The disadvantage is that the spatial ‘delay’ cannot be chosen
in an optimal way.
• Some groups advocated spatial embedding (Lachaux et al.,
1997), whereas others suggested it may not be a valid
embedding procedure (Pritchard et al., 1996, 1999; Pezard et
al., 1999).
Nonlinear dynamical analysis
attractor
x ( ti )




 x ( ti   ) 
 x (ti  2 ) 



Xi  











 x (ti  ( p  1) )
Jeong, 2002
How to quantify dynamical states of physiological systems
Physiological
Topologically
equivalent
system
A deterministic (chaotic)
system
States
Dynamical measures (L1, D2)
Physiological
Time series
Attractor
in phase space
Embedding procedure (delay coordinates)
1-dimensional time series  multi-dimensional dynamical systems
Nonlinear measure: correlation dimension (D2)
1 N
C (r )  2   (r  yi  y j )
N i , j 1
i j
C(r)  rD2
log C (r , N )
D2  lim lim
r 0 N 
log r
D2 algorithm
Nonlinear measure: correlation dimension (D2)
Correlation integral
attractor
Scaling region
Nonlinear measures: The first positive Lyapunov exponent
 i (t )
ln
 i (0)
1 m
L1  
(bits / s)
m i 1 t  ln 2
Why determinism is important?
Whether a time series is deterministic or not decides
our approach to investigate the time series.
• Surrogate data method
This method detects nonlinear determinism.
Surrogate data are linear stochastic time
series that have the same power spectra as
the raw time series. They are randomized to
destroy any deterministic nonlinear structure
that may be present. Statistical differences of
nonlinear measures between the raw data
and their surrogate data imply the presence of
nonlinear determinism in the original data.
Stam, 2005
Bursting as an information carrier of temporal
spiking patterns of nigral dopamine neurons
(a) Dopamine neurons in substantia nigra
Substantia nigra, a region of the basal ganglia that is rich in
dopamine-containing neurons, is thought to be etiologies of
Parkinson’s disease, Schizophrenia, Tourette's syndrome etc.
Electrophysiology of DA neurons in substantia nigra
• Irregular and complex single spiking and bursting states in vivo
• The presence of nonlinear deterministic structure in ISI firing patterns
(Hoffman et al. Biophysical J, 1995)
• Deterministic structure of ISI data produced by nigral DA neurons reflects
interactions with forebrain structures (Hoffman et al. Synapse 2000)
No determinism of non-bursting DA neurons
180
18
160
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140
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Dimensional complexity (D2)
surrogate data
counts/bin
120
100
80
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40
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raw ISI data
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msec
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Embedding dimension
Histogram
Embedding dim. vs. D2
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d=15
d=15
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d=13
d=11
d=9
dlnC(r)/dlnr
dlnC(r)/dlnr
9
d=7
d=5
d=13
d=11
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d=9
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d=7
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d=5
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-4
-3.5
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lnr
D2s of ISI data of DA neurons
0
-4
-3.5
-3
-2.5
-2
-1.5
-1
lnr
D2s of surrogate ISI data
-0.5
0
Nonlinear determinism of bursting DA neurons
180
16
160
14
Dimensional Complexity (D2)
140
counts/bin
120
100
80
60
40
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surrogate data
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raw ISI data
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Histogram
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dlnC(r)/dlnr
dlnC(r)/dlnr
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d=13
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Embedding dim. vs. D2
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Embedding dimension
msec
d=15
d=9
d=11
d=9
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lnr
D2s of ISI data of DA neurons
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-3.5
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-2.5
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-1.5
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lnr
D2s of ISI surrogate data
-0.5
0
The source of nonlinear determinism
in ISI firing patterns of DA neurons
Materials
7 Male Sprague-Dawley rats
anesthetized with chloral hydrate
Original ISI
(a) Non-bursting neurons (3/7)
(b) Bursting neurons (4/7)
Burst time series
(ISI <80ms, 160ms)
Single spike time series
Methods
(a) Estimation of correlation dimension
(b) Surrogate data method
(c) Burst separation method
Nonlinear determinism of burst time series
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dlnC(r)/dlnr
dlnC(r)/dlnr
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d=15
d=13
d=9
d=7
d=5
1
0
-5
-4.5
-4
-3.5
-3
-2.5
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-1.5
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lnr
D2s of ISI burst time series
-0.5
d=13
d=11
d=9
d=7
d=5
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d=11
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2
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d=15
0
1
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-4.5
-4
-3.5
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-2.5
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-1.5
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lnr
D2s of its surrogate ISI time series
0
No determinism of single spike time series
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dlnC(r)/dlnr
dlnC(r)/dlnr
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d=7
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-4.5
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-3.5
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lnr
D2s of ISI single spike time series
-0.5
0
0
-4.5
-4
-3.5
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-2.5
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-1.5
-1
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lnr
D2s of its surrogate ISI time series
0
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d=11
d=13
dlnC(r)/dlnr
dlnC(r)/dlnr
Nonlinear determinism of inter-burst interval data
d=15
d=9
6
d=7
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3
3
2
1
2
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-5
-4.5
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-2
lnr
D2s of IBI data
-1.5
-1
-0.5
0
d=9
d=11
-3.5
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d=7
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d=5
d=15
d=13
d=5
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-4.5
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lnr
D2s of surrogate IBI data
-0.5
0
Computational modeling of
single neurons and small neuronal circuits
Suprachiasmatic nucleus(SCN)
SCN neurons exhibit the circadian rhythm in their mean firing
rates.
Spontaneous Spiking activity of SCN neurons
ISI time series of an SCN neuron
0.6
ISI time (sec)
0.5
0.4
0.3
0.2
0.1
0
0
100
200
300
400
500
Spike event
SCN neurons exhibit irregular spontaneous firing patterns,
accompanied by intermittent bursts, and thus generate complex ISI
patterns, although the average SFR seems to maintain a circadian
rhythm.
Temporal Dynamics Underlying Spiking Patterns of the
Rat Suprachiasmatic Nucleus in vitro.
I. Nonlinear Dynamical Analysis (Jeong et al., 2005)
Correlation dimension
o rig in a l d a ta
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su rro g a te d a ta
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Correlation dimension
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o rig in a l d a ta
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s u rro g a te d a ta
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E m d e d d in g d im e n sio n
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E m b e d d in g d im e n s io n
14
Among 173 neurons, 16 neurons were found to exhibit
deterministic ISI patterns of spikes.
16
Temporal Dynamics Underlying Spiking Patterns of the
Rat Suprachiasmatic Nucleus in vitro.
II. Fractal stochastic Analysis. (Kim et al., 2003)
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Surrogate data
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Surrogate data
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Temporal Dynamics Underlying Spiking Patterns of the
Rat Suprachiasmatic Nucleus in vitro.
II. Fractal stochastic Modeling
A
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Original data
Simulated data
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Nonlinear analysis of the sleep EEG
• In many of these studies it was suggested that sleep EEG reflects
low-dimensional chaotic dynamics (Cerf et al., 1996, Fell et al.,
1993, Kobayashi et al., 1999, Kobayashi et al., 2001, Niestroj et al.,
1995, Pradhan et al., 1995, Pradhan and Sadasivan, 1996, Röschke,
1992, Röschke and Aldenhoff, 1991 and Röschke et al., 1993).
• The general pattern that emerges from these studies is that deeper
sleep stages are almost always associated with a ‘lower complexity’
as exemplified by lower dimensions and lower values for the
largest Lyapunov exponent.
• This type of finding has suggested the possible usefulness of
nonlinear EEG analysis to obtain automatic hypnograms.
Nonlinear analysis of the sleep EEG
• An analysis of an all night sleep recording found an evidence for
weak nonlinear structure but not low-dimensional chaos
(Achermann et al., 1994 and Achermann et al., 1994; Fell et al.
(1996a) using the nonlinear cross prediction (NLCP) to search for
nonlinear structure in sleep EEGs of adults and infants.
• sleep EEG of young infants showed nonlinear structure mostly
during quiet sleep (Ferri et al., 2003).
• The nonlinear measures were better in discriminating between
stages I and II, whereas the spectral measures were superior in
separating stage II and slow wave sleep: Nonlinear structure may be
most outspoken in stage II.
• nonlinear and asymmetric coupling during slow wave sleep in
infants (Pereda et al., 2003).
Nonlinear EEG analysis of Coma and anesthesia
• Matousek et al. (1995) studied the correlation dimension (based
upon a spatial embedding) in a small group of 14 healthy subjects
aged from 1.5 to 61 years. They found an increase of the
dimension during drowsiness as compared to the awake state.
• The usefulness of nonlinear EEG analysis as a tool to monitor
anesthetic depth was suggested (Watt and Hameroff, 1988).
• The correlation dimension correlated with the estimated level of
sevoflurane in the brain (Widman et al., 2000; Van den Broek,
2003).
Epilepsy as a dynamical disorder
• A dynamical brain disorder
• Spatial synchronization of brain electrical / magnetic
activity
• Brain fails to function as a multi-task multi-processing
machine
• Hallmarks of epilepsy:
- Interictal spikes
- Epileptic seizures
(detectable from electroencephalograms – EEGs)
EEGs in Epileptic seizures
• there is now fairly strong evidence that seizures reflect strongly
nonliner brain dynamics (Andrzejak et al., 2001b, Casdagli et al.,
1997, Ferri et al., 2001, Pijn et al., 1991, Pijn et al., 1997 and Van der
Heyden et al., 1996).
• Epileptic seizures are also characterized by nonlinear
interdependencies between EEG channels.
• Other studies have investigated the nature of interictal brain
dynamics in patients with epilepsy. In intracranial recordings, the
epileptogenic area is characterized by a loss of complexity as
determined with a modified correlation dimension (Lehnertz and
Elger, 1995).
• A time dependent Lyapunov exponent calculated from interictal
MEG recordings could also be used to localize the epileptic focus
(Kowalik et al., 2001)
The importance of seizure prediction
• The importance of seizure prediction can easily be appreciated: if a
reliable and robust measure can indicate an oncoming seizure
twenty or more minutes before it actually starts, the patient can be
warned and appropriate treatment can be installed.
• Ultimately a closed loop system involving the patient, a seizure
prediction device and automatic administration of drugs could be
envisaged (Peters et al., 2001).
Controversial about seizure prediction I
• In 1998, within a few months time, two papers were published that,
in restrospect, can be said to have started the field of seizure
prediction. The first paper showed that the dimensional complexity
loss L, previously used by the same authors to identify
epileptogenic areas in interictal recordings, dropped to lower
levels up to 20 min before the actual start of the seizure (Elger and
Lehnertz, 1998 and Lehnertz and Elger, 1998).
• The second paper was published in Nature Medicine by a French
group and showed that intracranially recorded seizures could be
anticipated 2–6 minutes in 17 out of 19 cases (Martinerie et al.,
1998).
• Schiff spoke about ‘forecasting brainstorms’ in an editorial
comment on this paper (Schiff, 1998).
One possible answer for why seizures occur is that:
Seizures have to occur to reset (recover) some
abnormal connections among different areas in
the brain. Seizures serve as a dynamical resetting
mechanism.
Controversial about seizure prediction I
• It was shown that seizure prediction was also possible with
surface EEG recordings (Le van Quyen et al., 2001b). This
was a significant observation, since the first two studies both
involved high quality intracranial recordings.
• Next, it was shown that seizure anticipation also worked for
extra temporal seizures (Navarro et al., 2002).
• This early phase was characterized by great enthusiasm and a
hope for clinical applications (Lehnertz et al., 2000).
Changes in D2 of epileptic patients
Le Van Quyen M et al., Nonlinear interdependencies
of EEG signals in human intracranially recorded
temporal lobe seizures. Brain Res. 792(1):24-40 (1998).
EEG characteristics of seizures
Traditional view: interictal  ictal  postictal
• Seizures’ occurrences are random
• Random occurrence of interictal spikes
• The transition from an interictal to ictal state is very
abrupt (seconds)
• Ictal activity may spread from the epileptogenic focus
to other normal brain areas after seizure’s onset
EEG characteristics of seizures
Emerging view:
interictal  preictal  ictal  postictal
• Seizures or spikes are NOT random events
• Existence of a preictal state
• The transition from the interictal to preictal to ictal
state is progressive (minutes to hours)
• Preictal and ictal spatio-temporal entrainment of the
epileptogenic focus with normal brain sites
• Seizures reset: postictal disentrainment of the
epileptogenic focus from normal brain sites.
Controversial about seizure prediction II
• Aschenbrenner-Scheibe et al. (2003). These authors showed that
with an acceptable false positive rate the sensitivity of the method
was not very high.
• The results of Martinerie et al. were also critically re-examined.
McSharry et al. suggested that the measure used by Martinerie et
al. was sensitive to signal amplitudes and that the good results
might also have been obtained with a linear method (McSharry et
al., 2003).
• Another group attempted to replicate the results of Le van Quyen
et al. in predicting seizures from surface EEG recordings (De
Clercq et al., 2003). These authors could not replicate the results in
their own group.
Online real-time seizure prediction
General features of EEGs in AD
• The hallmark of EEG abnormalities in AD patients is slowing of the
rhythms and a decrease in coherence among different brain regions:
A major promising candidate is the cholinergic deficit. AD is thought
to be a syndrome of neocortical disconnection, in which profound
cognitive losses arise from the disrupted structural and functional
integrity of long cortico-cortical tracts
Correlation dimension analysis
of the EEG in AD patients
• The D2 reflects the number of independent variables that are
necessary to describe the dynamics of the system, and is
considered to be a reflection of the complexity of the cortical
dynamics underlying EEG recordings.
• Thus, reduced D2 values of the EEG in AD patients indicate that
brains injured by AD exhibit a decrease in the complexity of brain
electrical activity (Woyshville and Calabrese (1994) Besthorn et al
., 1995 and Jeong et al., 1998, Stam et al., 1995 and Yagyu et al., 1
997).
EEG dynamics in patients with Alzheimer’s disease
EEG time series recorded from a patient with AD
145
EEG amplitude (mV)
140
135
130
125
120
115
110
105
100
0
500
1000
1500
Time (msec)
2000
2500
3000
Non-linear dynamical analysis of the EEG in Alzheimer's disease
with optimal embedding dimension.
Jeong et al. (1998) Electroencephalogr Clin Neurophysiol
AD patients have significantly lower nonlinear complex measures
than those for age-approximated healthy controls, suggesting that
brains afflicted by Alzheimer's disease show less chaotic
behaviors than those of normal healthy brains.
Pathophysiological implications of the
decreased EEG complexity in AD
• A decrease in dynamic complexity of the EEG in AD patients might
arise from neuronal death, deficiency of neurotransmitters like
acetylcholine, and/or loss of connectivity of local neuronal
networks.
• The reduction of the dimensionality in AD is possibly an expression
of the inactivation of previously active networks. Also, a loss of
dynamical brain responsivity to stimuli might be responsible for the
decrease in the EEG complexity of AD patients.
• AD patients do not have D2 differences between in eyes-open and
eyes-close conditions, whereas normal subjects have prominently
increased eyes-open D2 values compared with eyes-closed D2
values, suggesting a loss of dynamical brain responsivity to external
stimuli in AD patients.
Nonlinear measures
as a diagnostic indicator of AD
• Pritchard et al (1994) assessed the classification accuracy of the
EEG using nonlinear measures and a neural-net classification
procedure in addition to linear methods.
• The combination of linear and nonlinear analyses improves the
classification accuracy of the AD/control status of subjects up to
92%.
• Besthorn et al. (1997) reported that the D2 correctly classified AD
and normal subjects with an accuracy of 70%.
• Good correlations are found between nonlinear measures and the
severity of the disease, a slowing of EEG rhythms, and
neuropsychological performance.
• Furthermore, the global entropy can quantify EEG changes induced
by drugs, suggesting a possibility that nonlinear measures is
capable of quantifying the effect of drugs on the course of the
disease.
Nonlinear dynamical analysis of the EEG in patients
with Alzheimer's disease and vascular dementia.
Jeong et al., J Clin Neurophysiol (2001)
VaD patients have relatively increased values of nonlinear measures
compared with AD patients, and have an uneven distribution of D2
values over the regions than AD patients and healthy subjects.
Controversial on EEG complexity
in Schizophrenia
• The majority of these studies focused upon the question whether
schizophrenia is characterized by a loss of dynamical complexity
or rather by an abnormal increase of complexity, reflecting a
‘loosening of neural networks’.
• Many and especially more recent studies have found a lower
complexity in terms of a lower correlation dimension or lower
Lyapunov exponent (Jeong et al., 1998, Kim et al., 2000, Kotini
and Anninos, 2002, Lee et al., 2001 and Rockstroh et al., 1997).
• However, increases in dimension and Lyapunov exponent have
also been reported in the older studies (Elbert et al., 1992,
Koukkou et al., 1993 and Saito et al., 1998).
Decreased complexity of cortical dynamics
in Schizophrenic patients
• Jaeseung Jeong, Dai-Jin Kim, Jeong-Ho Chae, Soo Yong Kim,
et al. Nonlinear analysis of the EEG of Schizophrenics with
optimal embedding dimension. Medical Engineering and
Physics (1998).
• Dai-Jin Kim, Jaeseung Jeong, Jeong-Ho Chae, et al. The
estimation of the first positive Lyapunov exponent of the EEG
in patients with Schizophrenia. Psychiatry Research (2000).
• Jeong-Ho Chae, Jaeseung Jeong, Dai-Jin Kim, et al. The effect
of antipsychotic medications on nonlinear dynamics of the
EEG in schizophrenic patients. Clinical Neurophysiology
(2003)
• Jaeseung Jeong, Dai-Jin Kim, Soo Yong Kim, Jeong-Ho Chae, et
al. Effect of total sleep deprivation on the dimensional complexity
of the waking EEG. Sleep (2001)
• Jeong-Ho Chae, Jaeseung Jeong, Bradley S. Peterson, Dai-Jin Kim,
Seung-Hyun Jin, et al. Dimensional complexity of the EEG in
patients with Posttraumatic stress disorder. Psychiatry Research:
neuroimaging (2003)
• Dai-jin Kim, Jaeseung Jeong, Kook Jin Ahn, Kwang-Soo Kim,
Jeong-Ho Chae, et al. Complexity change in the EEG in alcohol
dependents during alcohol cue exposure. Alcoholism: Clinical &
Experimental Research (2003)
• Dai-Jin Kim, Won Kim, Su-Jung Yoon, Yong-Ku Kim, Jaeseung
Jeong, Effects of alcohol hangover on cytokine production in
healthy subjects. Alcohol (2003)
Nonlinear dynamics of the EEG
during photic and auditory stimulation
• Jaeseung Jeong, Moo Kwang Joung and Soo Yong Kim.
Quantification of emotion by nonlinear analysis of the chaotic
dynamics of EEGs during perception of 1/f music. Biological
Cybernetics (1998)
• Seung Hyun Jin, Jaeseung Jeong, Dong-Gyu Jeong, Dai-Jin Kim
et al. Nonlinear dynamics of the EEG separated by Independent
Component Analysis after sound and light stimulation. Biological
Cybernetics (2002)
• Jaeseung Jeong, Sangbaek Han, Bradley S. Peterson, and Soo
Yong Kim, "The effect of photic and auditory stimulation on
nonlinear dynamics of the human electroencephalogram," Clinical
Neurophysiology (in press)
Information flow during Tic suppression
of Tourette’s syndrome patients
T3
T4
Resting EEG: normal vs. TS
T3
T4
TS: Normal vs. Tic suppression
The effect of alcohol on the EEG complexity
measured by Approximate entropy
Perspectives
• For the last thirty years, progress in the field of nonlinear
dynamics has increased our understanding of complex systems dy
namics.
• This framework can become a valuable tool in scientific fields
such as neuroscience and psychiatry where objects possess natural
time dependency (i.e. dynamical properties) and non-linear
characteristics.
• Relative estimates of nonlinear measures can reliably characterize
different states of normal and pathologic brain function.
• Nonlinear dynamical analysis provides valuable information for
developing mathematical models of the systems.
Multimodal approach for psychiatric disorders
• For example, the combination of EEG and PET variables results in
approximately 90% of overall correct classification with a
specificity of 100% (Jelic et al., 1999).
• EEG and MRI measurements of the hippocampus obtain the
highest scores of abnormalities in patients with probable AD
( Jonkman, 1997).
• Furthermore, CT- and MRI-based measurements of hippocampal
atrophy provide a useful early marker of AD ( Scheltens, 1999).
• These neuroimaging techniques can offer not only supplementary
information for diagnosis of AD, but also an opportunity to
explore structural, functional, and biochemical changes in the
brain leading to new insights into the pathogenesis of AD.
Coronal MRI slices perpendicular to the long axis of the hippocampus
showing a smaller hippocampus in an MCI patient.
Coronal MRI at the level of the hippocampi showing no significant
atrophy, but FDG-PET SPM indicates posterior cingulate
hypometabolism
(Nestor et al., 2004)
Measures of nonlinear interdependency
• The brain can be conceived as a complex network of coupled and
interacting subsystems. Higher brain functions depend upon
effective processing and integration of information in this network.
This raises the question how functional interactions between
different brain areas take place, and how such interactions may be
changed in different types of pathology.
Mutual information of the EEG
•The MI between measurement xi generated from system X and
measurement yj generated from system Y is the amount of
information that measurement xi provides about yj.
J Jeong, JC Gore, BS Peterson. Mutual information analysis of the EEG
in patients with Alzheimer's disease. Clin Neurophysiol (2001)
Recent MI studies on the EEG
• Schlogl A, Neuper C, Pfurtscheller G. Estimating the mutual
information of an EEG-based Brain-Computer Interface.
Biomed Tech. 2002;47(1-2):3-8.
Na et al., EEG in schizophrenic patients: mutual information
analysis. Clin Neurophysiol. 2002;113(12):1954-60.
Huang L, Yu P, Ju F, Cheng J. Prediction of response to
incision using the mutual information of
electroencephalograms during anaesthesia. Med Eng Phys.
2003;25(4):321-7.
Phase synchronization in chaotic systems
• Coupled chaotic oscillators can display phase synchronization even
when their amplitudes remain uncorrelated (Rosenblum et al., 1996).
Phase synchronization is characterized by a non uniform distribution
of the phase difference between two time series. It may be more
suitable to track nonstationary and nonlinear dynamics.
Phase synchronization
• ‘Synchronization of chaos refers to a process, wherein two (or
many) systems (either equivalent or nonequivalent) adjust a given
property of their motion to a common behavior due to a coupling or
to a forcing (periodical or noisy)’ (Boccaletti et al., 2002).
Nonlinear coupling among cortical areas
Phase synchronization and interdependence
Definition of synchronization: two or many subsystems sharing specific
common frequencies
Broader notion: two or many subsystems adjust some of their timevarying properties to a common behavior due to coupling or common
external forcing
Jansen et al., Phase synchronization of the ongoing EEG and
auditory EP generation. Clin Neurophysiol. 2003;114(1):79-85.
Le Van Quyen et al., Nonlinear interdependencies of EEG signals
in human intracranially recorded temporal lobe seizures. Brain Res.
(1998)
Breakspear and Terry. Detection and description of non-linear
interdependence in normal multichannel human EEG data. Clin
Neurophysiol (2002)
Generalized Synchronization
• Generalized synchronization exists between two interacting systems
if the state of the response system Y is a function of the state of the
driver system X: Y=F(X). Cross prediction is the extent to which
prediction of X is improved by knowledge about Y, which allows the
detection of driver and response systems.
• The nonlinear interdependence is not a pure measure of coupling but
is also affected by the complexity or degrees of freedom of the
interacting systems
Decreased EEG synchronization in MCI
Koenig et al. (2005)
Stam et al. (2003)