Chapter 4. Formal Tools for the Analysis of
Brain-Like Structures and Dynamics (1/2)
in Creating Brain-Like Intelligence, Sendhoff et al.
Course: Robots Learning from Humans
Cheolho Han
September 25, 2015.
Biointelligence Laboratory
School of Computer Science and Engineering
Seoul National University
http://bi.snu.ac.kr
Contents

Introduction

Structural Analysis of Networks

Dynamical States

Conclusion
2
Introduction
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3
Introduction

Brains and artificial brainlike structures require
mathematical tools for the analysis.
3. Information Processing
How information is processed through dynamics
2. Dynamical Phenomena
Types of dynamics that abstracted networks can support
1. Static Structure
The brain structure abstracted from neuroanatomical results of the
arrangement of neurons and the synaptic connections between them
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4
Structural Analysis of
Networks
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Structural Analysis of Networks



A neurobiological network has properties that
distinguish itself from other networks.
To look for those properties, spectral analysis has
been developed.
The spectral density for the diffusion operator L
yields characteristic features that distinguish
neurobiological networks from other networks.
1
Lx (t ) 
 wik
i
 w  x (t )  x (t ) .
j
ij
j
wij
i
xj
xi
k
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Spectrum of Networks

Observing the spectrum of networks, classes of
networks can be distinguished.
Spectrum of transcription networks
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Spectrum of neurobiological networks
7
Dynamical States
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Why Study Dynamics?

The neuronal structure is only the static substrate
for the neural dynamics.
Static Structure
Dynamical States

Time
The relation between dynamic patterns and
cognitive processes has not been clearly revealed.
Dynamical patterns are where we begin.
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Dynamical Systems

When are systems dynamical?



When the state changes are not monotonic of the
present states
Otherwise, the states would just grow or decrease.
Systems can be dynamical if


the individual elements are dynamical
or the elements are connected.
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Monotonic vs. Non-Monotonic



The individual elements can be updated by
monotonic or non-monotonic functions.
A sigmoid
1
f (s) 
1  e  s
is a monotonic function.
The following two functions are non-monotonic.
for 0  x  1 / 2
 2x
f ( x)  4 x(1  x) f ( x)  
2  2 x for 1 / 2  x  1.
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11
Independent vs. Coupled



The updates of the elements can be independent
or coupled.
The independent update can be done by:
x(t  1)  f ( x(t )).
The coupled updates can be done by:
x i (t  1)   wij f ( x j (t )).
j
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Case 1 of Dynamical Systems

The first case: coupled and monotonic
1
i
j
x (t  1)   wij f ( x (t )), f ( s ) 
 s
1

e
j

If the strength wij of the connection from j to i is
negative, the connection is inhibitory; if positive,
the connection is excitatory. Therefore, we have
dynamical systems.
Monotonicity
Dependency
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Monotonic
Non-Monotonic
Independent
X
2.
Coupled
1.
3.
13
Case 2 of Dynamical Systems

The second case: independent and non-monotonic
x(t  1)  f ( x(t )),

The system is dynamic, but it has chaotic behavior.
1
x(0) = 0.10
x(0) = 0.11
0.9
0.8
0.7
f ( x)  4 x(1  x)
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
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Monotonicity
9
10
1
for 0  x  1 / 2
 2x
f ( x)  
Independent
1 / 2  x  Coupled
1.
2  2 x forDependency
0.8
Monotonic
Non-Monotonic
0.7
0.6
X
2.
0.5
0.4
0.3
1.
3.
0.2
0.1
0
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x(0) = 0.10
x(0) = 0.11
0.9
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Case 3 of Dynamical Systems

The third case: coupled and non-monotonic
1
i
i
j
x (t  1)  (1  т) f ( x (t ))  т
w
f
(
x
(t ))

ij
 wik j
k

In this case, the synchronization of chaotic
behavior can occur. As the coupling strength e
increases, the solution experiences the state
changes:
desynchronized  synchronized  desynchronized
Monotonicity
Monotonic

Non-Monotonic
Independent
X
Synchronization:
https://youtu.be/W1TMZASCR-I
Dependency
Coupled
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2
3
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Conclusion
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Conclusion



Some mathematical tools are required to analyze
the brain.
Spectral analysis is helpful to understand the
structure of the neurobiological network.
In consideration of the monotonicity of the update
function and the dependency of the elements,
several models have been suggested.
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17
Thank You
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References



1. Banerjee, A., Jost, J.: Spectral plots and the
representation and the interpretation of biological
data. Theory Biosci. 126, 15-21 (2007)
2. Fig. http://artint.info/figures/ch07/sigmoidc.gif
3. Video. https://youtu.be/W1TMZASCR-I
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Appendix: Chaotic Behavior
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X(0) = 0.1
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© 2015, SNU CSE Biointelligence Lab., http://bi.snu.ac.kr
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