Neural chaos
(Chapter 11 of Wilson 1999)
Klaas Enno Stephan
Laboratory for Social & Neural Systems Research
Dept. of Economics
University of Zurich
Wellcome Trust Centre for Neuroimaging
Institute of Neurology
University College London
Computational Neuroeconomics and Neuroscience
University of Zurich, 4 May 2011
Recap: Oscillations & limit cycles
•
Oscillation:
A trajectory X(t) of a dynamical system with any
number of dimensions is an oscillation if:
X(T+t) = X(t) for some unique T > 0 and all t.
•
Limit cycle:
An oscillatory trajectory in the state space of a
nonlinear system is a limit cycle if all trajectories in a
sufficiently small region enclosing the trajectory are
spirals.
If these neighboring trajectories spiral towards the
limit cycle as t, then the limit cycle is
asymptotically stable.
If neighboring trajectories spiral away from the limit
cycle as t, the limit cycle is unstable.
•
Stable limit-cycles imply self sustained oscillations.
Any (small) perturbation from the closed trajectory
would cause the system to return to the limit cycle,
making the system stay on the limit cycle.
stable limit cycle
unstable limit cycle
Recap: Poincaré-Bendixon Theorem
• PB Theorem:
If an annulus can be constructed in a two-dimensional system such that all
trajectories enter it, yet it contains no steady states, then a limit cycle must
exist within the annulus.
• Why not simply extend this to ≥3 dimensions by replacing the annulus with a
torus?
• In a deterministic autonomous system, trajectories can never cross, and this
provides a definitive constraint in two dimensions.
• However, trajectories in a system with ≥2 dimensions can pass by without
intersecting in an infinite number of ways.
– quasiperiodic trajectories
– aperiodic (chaotic) trajectories
Definition: quasiperiodic trajectories
• system with ≥3 dimensions that has at least two frequencies, at least one
being irrational
• the trajectory of such a system never rejoins or intersects itself
• however, quasiperiodic trajectories are similar to periodic trajectories and two
quasiperiodic trajectories starting from similar initial conditions will remain
close to each other
• demo: quasiperiodic.m
Definition of chaos
• A deterministic nonlinear dynamical system in three or more
dimensions exhibits chaos if all of the following conditions are
satisfied for some range of parameters:
1. Trajectories are aperiodic (not quasiperiodic).
2. These aperiodic trajectories remain within a bounded volume of the
state space but do not approach any steady states.
3. There is sensitivity to initial conditions such that arbitrarily small
differences in initial conditions between nearby trajectories grow
exponentially in time.
• NB: this refers to systems which are deterministic, nonlinear,
and with ≥ 3 dimensions
Chaos =
deterministic behaviour that is very sensitive
to initial conditions
• NB: chaotic systems are deterministic systems!
• they are perfectly predictable given knowledge of the initial conditions
• however, in real life, we never know the initial conditions precisely; even tiny
errors in measuring initial conditions will produce trajectories which wildly
differ
• example: weather forecasting
Example of a chaotic system: The Lorenz oscillator
• Inspection of isoclines reveals three equilibrium
points:
0,0,0 ,
72, 72, 27 ,
72, 72, 27
• Analyzing the Jacobian at the equilibria indicates:
– one saddle node
– two unstable spiral points
• Therefore, no trajectory will ever reach an
equilibrium point!
• demo: Lorenz_mod_1.m
(see also lorenz.m demo provided by Mathworks)
The Lorenz oscillator
20
15
10
5
x
0
-5
-10
50
-15
-20
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
40
45
50
30
40
20
30
10
y
20
0
-10
-50
-20
10
-30
0
50
0
-20
45
-15
-10
40
-5
0
5
10
15
20
50
35
z
30
25
20
15
strange attractor = the orbit is bounded, but neither periodic nor
quasiperiodic and not convergent
10
5
0
Demo: dependence on initial conditions
50
45
40
35
30
25
20
15
10
5
0
-40
-20
0
20
40
20
15
10
5
0
-5
red and blue trajectories: change in the initial conditions by 10-5
demo: Lorenz_mod_2.m
-10
-15
-20
Testing for the presence of chaos
• Qualitative criteria that can be applied to measured time series
– Fourier power spectrum
– first return map (Poincaré map)
• Quantitative criterion that requires knowing the system’s differential equation
– Lyapunov exponent
Fourier transform of non-chaotic systems
Fourier Power Spectrum
Fourier Power Spectrum
Fourier Power Spectrum
0
-4
10
0
10
-2
10
10
-2
-6
10
Power
Power
Power
10
-4
-8
10
-4
10
-6
10
10
-6
-10
10
10
2000
4000
random
6000
Frequency
8000
10000
12000
500
1000
1500
Frequency
2000
2500
periodic
f (t )  2sin(4 t )
3sin(10 t )
500
1000
1500
Frequency
2000
2500
quasiperiodic
f (t )  2sin(4 t )
3sin(2 23 t )
Fourier transform of the Lorenz oscillator
• qualitative signature of chaos:
very complex spectrum with all
frequencies present and with power
decreasing with frequency
Fourier Power Spectrum
0
10
Power
-2
10
-4
10
-6
10
100
200
300
Frequency
400
500
600
Poincaré map
• intersection of a periodic orbit with a lower
dimensional subspace (Poincaré section) that
is transversal to the flow of the system at a
point p
• can be interpreted as discrete dynamical
system with a state space that is one
dimension smaller than the original continuous
dynamical system
• preserves many properties of the original
system and is thus sometimes used for
analyzing the original system
• conceptual basis for computing what Wilson
refers to as a “first return map” in one
dimension
P : Z U  S
P 0 : p
P n 1 : P P n
P  n 1 : P 1 P  n
First return map of the Lorenz oscillator
• First return times wrt. to z0:  | z (t )  z (t   )  z0 
• Descriptively:
one chooses a value z0 and measures all the
intervals between the times when z(t) returns
to this same value
• First return map:
A two-dimensional graph which plots
function of i
i+1 as a
• For pure oscillations, the first return map
consists of a single point (equal to the period).
First return map of the
Lorenz oscillator for z0=25
Lyapunov exponent (LE)
• expresses the dependency on initial
conditions:
average exponent  of rate of
divergence et of neighboring
trajectories
• requires knowing the system’s
equations
• one LE for each dimension of the
system; due to exponential growth,
only the largest one needs to be
considered
• if largest LE is positive, then the
system is chaotic
1 N  i 

ln  

N  t i 1   
Chaotic behaviour of Hodgkin-Huxley neurons
1
0.9
• parameter A controls amplitude of injected
sinusoidal current
0.8
0.7
0.6
R
• implicitly a four-dimensional system
– sine term is the solution of two linear 1st order
ODEs
0.5
0.4
0.3
0.2
0.1
0
-1
-0.8
-0.6
-0.4
-0.2
V
0
0.2
0.4
0.6
Fourier Power Spectrum
Chaotic behaviour of
HH neurons
-2
10
-4
10
chaotic behaviour for A=0.007 (and
I0=0.075 and =264.6 Hz)
•
Lyapunov exponent = 0.16
10
•
this will produce divergence by a factor of
e16=107 over 100 ms
10
Power
•
-6
10
-8
-10
1
2
3
Frequency
4
5
6
4
x 10
First Return Map
1400
V0 = -0.7
V0 = 0
1200
Return(T+1)
1000
800
600
400
200
200
400
600
800
Return(T)
1000
1200
1400
Implications for modelling neuronal systems
• if a neural system is in a chaotic regime, we cannot predict details of spike
trains beyond very brief intervals (determined by the Lyapunov exponent)
• this cannot be overcome by including ever more biophysical details in the
model!
• only two options:
– be content with predicting only the immediate future
– predict coarser states (e.g. mean firing rate over time periods or over
neurons)
• cf. weather prediction: detailed predictions impossible beyond a few days,
but monthly averages can be predicted accurately far into the future
Implications for modelling neuronal systems
• “Neuroscientists who believe that it is necessary to understand in detail all of
the ionic currents of every neuron in a network to make predictions should
recognize that greater descriptive detail in a chaotic neural system will not
generally lead to greater predictive power.”
H.R. Wilson, p. 183
Some philosophical thoughts
• On a more philosophic level, what does neural chaos have to say about our
own brains and thought processes? ... First, chaos may provide a limitation
on each individual's ability to predict his/her own behavior in detail... Although
we know very little about the neural processes involved in thinking or
consciousness, we do know that they involve neurons that might sometimes
operate in chaotic regimes. Even though the brain may be totally
deterministic, therefore, we may sometimes have no idea why we suddenly
perform an unexpected act or make a snap decision! Thus, the old free-will
versus determinism controversy in philosophy may have its resolution in
neural chaos: a totally deterministic brain may nevertheless produce
behaviors that are not predictable either by that brain itself or by any other
brain on the planet!”
H.R. Wilson , p. 184
Thank you