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

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

2

Contents Introduction

Structural Analysis of Networks

Dynamical States

Conclusion

© 2015, SNU CSE Biointelligence Lab., http://bi.snu.ac.kr 3

Introduction


Introduction

Brains and artificial brainlike structures require mathematical tools for the analysis.

How information is processed through dynamicsHow information is processed through dynamics

Types of dynamics that abstracted networks can supportTypes of dynamics that abstracted networks can support

The brain structure abstracted from neuroanatomical results of the arrangement of neurons and the synaptic connections between

them

The brain structure abstracted from neuroanatomical results of the arrangement of neurons and the synaptic connections between

them

3. Information Processing3. Information Processing

2. Dynamical Phenomena2. Dynamical Phenomena

1. Static Structure1. Static Structure


Structural Analysis of Net-works


Structural Analysis of Networks

A neurobiological network has properties that dis-tinguish 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 neu-robiological networks from other networks.

( ) (1

)) .(i j iij

jikk

w tL t x xw

x t ix

jxijw


Spectrum of Networks

Observing the spectrum of networks, classes of networks can be distinguished.

Spectrum of transcription networks Spectrum of neurobiological networks


Dynamical States


Why Study Dynamics?

The neuronal structure is only the static substrate for the neural dynamics.

The relation between dynamic patterns and cogni-tive processes has not been clearly revealed. Dy-namical patterns are where we begin.

Time

Static Structure

Dynamical States


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.


Monotonic vs. Non-Monotonic

The individual elements can be updated by mono-tonic or non-monotonic functions.

A sigmoid

is a monotonic function. The following two functions are non-monotonic.

1( )

1 sf s

e

( ) 4 (1 )f x x x 2

( )2 2

for 0 1/ 2

for 1/ 2 1.

xf x

x

x

x


Independent vs. Coupled

The updates of the elements can be independent or coupled.

The independent update can be done by:

The coupled updates can be done by:

( 1) ( ( )).x t f x t

( ( )).( 1)i jij

j

w f x tx t


Case 1 of Dynamical Systems

The first case: coupled and monotonic

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

Monotonic Non-Monotonic

DependencyIndependent X 2.

Coupled 1. 3.

1( )

1 sf s

e

( ( )),( 1)i jij

j

w f x tx t



The second case: independent and non-monotonic

The system is dynamic, but it has chaotic behavior.

Monotonicity


DependencyIndependent X 2.

Coupled 1. 3.

( 1) ( ( )),x t f x t

( ) 4 (1 )f x x x

2( )

2 2

for 0 1/ 2

for 1/ 2 1.

xf x

x

x

x

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x(0) = 0.10

x(0) = 0.11

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

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0.7

0.8

0.9

1

x(0) = 0.10

x(0) = 0.11



The third case: coupled and non-monotonic

In this case, the synchronization of chaotic behav-ior can occur. As the coupling strength e increases, the solution experiences the state changes:

desynchronized synchronized desynchronized

Synchronization: https://youtu.be/W1TMZASCR-I

Monotonicity


DependencyIndependent X 2

Coupled 1 3

( 1) (11

) ( ))( ( ( ))i i jij

kk

ji

x t tf x w f x tw

ò ò

https://youtu.be/W1TMZASCR-I



Conclusion


Conclusion

Some mathematical tools are required to analyze the brain.

Spectral analysis is helpful to understand the struc-ture of the neurobiological network.

In consideration of the monotonicity of the update function and the dependency of the elements, sev-eral models have been suggested.


Thank You


References

1. Banerjee, A., Jost, J.: Spectral plots and the rep-resentation 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

http://artint.info/figures/ch07/sigmoidc.gif

http://artint.info/figures/ch07/sigmoidc.gif



Appendix: Chaotic Behavior

1 2 3 4 5 6 7 8 9 100.1

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X(0) = 0.1

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X(0)=0.01

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X(0) = 0.111 2 3 4 5 6 7 8 9 10

0

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x(0) = 0.10

x(0) = 0.11

0 5 10 15 20 25 30 35 40 45 500

0.1

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1

x(0) = 0.10

x(0) = 0.11

Documents

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