Creating Mathematics
That Describes Brain andInformation(2) the brain mechanism:
the mathematical pprinciples
pof the expression,p, memory,y, andlearning of information inside the brainRIKEN
Brain Science InstituteShunichi Amarispacepodyssey
y y in the brain
Big bang: Energy, matter, astronomical objects,
molecule (order)(13.7 billion years) Physics, Chemistry Energy andEntropyLife : a material form that reproduces itself perpetually(3.6 billion years)
Life scienceh dit molecularheredity,ll mechanism,h iselflf conservationtiBrain and nervous systems：
They act as they utilize,utilize memorize,memorize learn
learn, or judge
surroundings (500 million years)
neuroscience

Human：a life with mind that lives within the society;culture and society (200 thousand years)
Brain sciencescience, information sciencescience, human science1

From NASA
http://en.wikipedia.org/wiki/File:NGC_4414_(NASA-med).jpg

Big bang：Energy, matter – astronomical objects – molecule(order)(13.7 billion years) Physics, Chemistry Energy and Entropy

4.64
6 billionyears ago:
Bi th off theBirththearth

From NASA
http://en.wikipedia.org/wiki/File:The_Earth_seen_from_Apollo_17.jpg

Life : a material form that reproduces itself perpetually(3.6 billion years)
Life science
heredity, molecular mechanism, self conservation

http://ja.wikipedia.org/wiki/ファイル:Egg_Mountain.jpg

A multicellular organismgBrain and nervous system：They act as they utilize, memorize, learn, or judge
surroundings (600 million years)
neuroscienceAnthropoids, and then Human
Human beings: life that living in mind and society;culture and society (100 thousand years) Brain science,
I fInformationti Science,
S i
Human
HScience
S i

http://commons.wikimedia.org/wiki/File:Lightmatter_chimp.jpg

LifeMatter

Information

mind
The law of matter; life and information; brain;
Mind and civilization

The theory of mind
Do Robots Have Consciousness?
Powerful computers
Brain simulation

http://commons.wikimedia.org/wiki/File:Robot_asimo.jpg

Brain
Molecularmachine

Informationo at omachine
Human,
societymolecular machineinformation machine

Molecule
Cell
Circuit module
SystemIndividualSociety
Socety-Study of the Brain
Knowing about the brain … What is human?
from Human to Society
Taking care of the brain … brain diseasedisease, aging
mental disorder
Creating the brain …understanding of the brain
Brain-type
Brainyp information technology
gy
Educating the brain … education and learningHuman thinking

l ilogici ii
intuition

Symbol(language)

pattern(excitation of neurons)

serialization

Parallel interaction

Algorithm(program)Self-organizationneurocomputer
Brain modelcomputer
Artificial Intelligence

FusionTowards an elucidation of the information
principles in the brain

Brain

Neurocomputers
Biological facts Engineering facts

Information principles(parallel, distribution,learning)）

Molecularbiology

Mathematical models

Mathematical theory

HardHddevices

Mathematical Brain science
information principles in the brain
realized thanks to the evolution

Models for neuronal cellsynthesisth iuanswer

M thMathematicalti l models
d l forf neurons

y = ϕ ( ∑ wi xi − h ) = ϕ ( w ⋅ x )xyϕ (u )Majority element
excite/inhibitu

Brain theory and Brain models
Cognitive model percetron

Dispersion expression
Learning/information geometry
Curse of dimensionalityy = ∑ viϕ ( wi ⋅ x )M thMathematicalti l models
d l forf neurons

y = ϕ ( ∑ wi xi − h ) = ϕ ( w ⋅ x )x

yϕ (u )

Majority element
excite/inhibituDynamics and Decision
xt +1 = f ( xt ;θ )x:dx ( t )= f ( x; θ )dtattractor
Stability point
Stability orbit
chaos

memoryydecision
Real number computation
Pulse mode operation

Simple perceptron

yinputxy = sgn ( w ⋅ x − h )w ⋅ x = ∑ wi xi

xnx1

w⋅x−h = 0

Error correction learning

x

yϕ(u)

w → w ′ = w + ex⎧1⎪e=⎨0⎪−1⎩incorrect
answer for
xのxを間違いt answer
xcorrect
を正解
0のxを正解incorrect
answer forP
Perceptrontconvergence theoremthLinear discriminant
x3 = s12 + s22
sx1 = s1xsx2 = s2

Multi-dimensionalizationyModel of a cerebellar perceptron

Multilayery pperceptron
py = ∑ viϕ ( wi ⋅ x ) + nx

ｙx = ( x1 , x2 ,..., xn )ϕ ( wi ⋅ x )f ( x , θ ) = ∑ viϕ ( wi ⋅ x )θ = ( w1 ,..., wm ; v1 ,..., vm )NNeuralld
dynamics
iDynamics of majority decision:
Delphi method
multistability

xi' = sgn ( ∑ wij x j )x ( t + 1) = Tx ( t )Neural economic theory
Ultimatum gameAssociative memory and the hippocampus
x1 = (10110 )x2 = ( 01001)
M
xk = (11100 )x ( t + 1) = Tx ( t )dynamics

xt

xt +1

CA1
hippocampus

CA2

III

dynamics
yof neural fields

si (ξ )

ξ∂ u i (ξ , t )= − u i (ξ , t ) + ∑ ∫ w ij (ξ − ξ ′ ) f ⎣⎡ u j (ξ ′, t ) ⎤⎦ d ξ ′ + s i (ξ , t )τ∂tj

local excitations:travelling wave:
oscillatory:
memorydecision

stimingAmari, Biol. Cybern,1978
Wil
Wilson& Cowan,C1972

∂ u (ξ , t )= − u (ξ , t ) + ∫ w (ξ − ξ ′ ) f ⎡⎣ u (ξ ′, t ) ⎤⎦ d ξ ′ + a
∂t

ξ

( )

r1

r2u ri ,t = 0u (ri + Δri ,t + Δt ) = 0∂ u (r i , t )∂ u (r i , t ) ∂ r i∂ t+∂u (ri ,t )
∂riαi
==−
∂t∂t

∂ ξ

∂ t=0

∫ w(r − ξ ′)dξ ′ = ±W (r − r )r2r1i2

ξ1W (ξ ) = ∫ w (ξ )d ξ0

Multilayer Perceptronψ(x)Neuromanifolds

Function spacey = f ( x, θ )= ∑ vi ϕ ( wi ⋅ x )θ

θ = ( v 1 L , vm ; w1, L , wm )neuromanifolds
 Metric structure Topological structureθSingular point on MLP—anMLP an example

An example of singular models
y = vϕ ( w ⋅ x ) + n

ξ | w |= 0ξＷIf you are interested in themathematics of the brain…
School of Mathematical
Sciences
Department of MathematicalEngineering andInformation Physics
Graduate School ofInformation Science andTechnology

Associativememory

Perceptron

Statistical neurodynamics

Department of Complexity
Science and Engineering,
Graduate School of Frontier
Sciences,
The University of Tokyo

Okada laboratory
http://mns.k.u-tokyo.ac.jp/index.php