6. I. Gelfand のmathematics as an adequate language
に触発されて、
脳のmathematics as an adequate languageを探す
S. Smaleの第18問題:
自然知能と人工知能の限界を定めよ。
I. Gelfand: C*環、超関数、表現論;1989年京都賞、1979年ウルフ賞数学部門
S. Smale: 可微分多様体、微分可能力学系(馬蹄形写像:カオス)、
実数計算論;1966年フィールズ賞
9. A B
Learning of others’ memories
A B
A B
Recall of own memories
コミュニケーションにおける脳ダイナミクスの数理科学的研究
(新学術領域H21-25:伝達創成機構)
カオス的遍歴の
さまざまな機能
コミュニケーションに伴う神経
機構の数学による解明
⇒ コミュニケーション神経情
報学
9
・ダイナミックな記憶過程
・ミラーニューロン問題
・ダイナミックな推論過程
と想像能力
・共感:類似の力学系と同期
10. Internal state of the brain
via mind
Body/Intention/participation
Body/Sensation/Environmental ChangeEnvironment
人の知覚は離散的;予測をするから連続的に見える
𝑍 𝑛+1 = 𝐹(𝑍 𝑛)
ൗ𝑑𝑥
𝑑𝑡 = 𝑃(𝑥)
Hermeneutic circle (解釈学的循環)
27. (1)数学でニューロンを作る:ニューロンはいかにして生まれたか?
Watanabe, H., Ito, T., Tsuda, I.: Making a neuron model: A mathematical approach.
In: 11th meeting of Mechanisms of Brain and Mind, Niseko, Hokkaido , Japan, Jan. 11−13, (2011)
Watanabe, H., Ito, T., Tsuda, I., to be submitted, 2018.
28. 後期初期
進化のステージ
0
1
2
3
4
5
6
7
8
9 0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 100
0.5
1
xn i
site k
site k
変化・発展する力学系
Mutual information I
( 1)x t
( )x t
iteration t
iteration t
( 1) ( )x t x t 状態変化の規則:
写像で表現
入力時系列が有する情報量
の時間空間変化
入力時系列の時間空間変化
29. 後期初期
進化のステージ
0
1
2
3
4
5
6
7
8
9 0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 100
0.5
1
1.5
2
0
1
2
3
4
5
6
7
8
9
0 20 40 60 80 100
0.5
1
0
1
2
3
4
5
6
7
8
9 0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
xn i
site k
site k
Mutual information I
( )x t
( 1)x t
iteration t
iteration t
( 1) ( )x t x t
32. (2) 機能モジュールの生成
Tsuda, I., Yamaguti, Y., Watanabe, H.:
Modeling the Genesis of Components in the Networks of Interacting
Units. Proc. of ICCN 2013.
Yamaguti, Y., Tsuda, I.: Mathematical Modeling for Evolution of
Heterogeneous Modules in the Brain.
Neural Networks 62: 3-10 (2015)
Yutaka Yamaguti, Ichiro Tsuda, Yoichiro Takahashi
Information flow in heterogeneously interacting systems, Cog.
Neurodynamics 8:17-26, 8 (2014)
33. 脳のモジュール間の結合は非対称結合
Felleman & Van Essen 1991
I-III
IV
V-VI
I-III
IV
V-VI
Ascending (Feed Forward) Projections
I-III
IV
V-VI
I-III
IV
V-VI
Descending (Feed Back) Projections
“Higher” region “Lower” region
視覚野のモジュール間結合
Y.Yamaguti, I.Tsuda, Y. Takahashi,
2014
36. Network model
• The dynamics of each oscillator is defined by
𝜃𝑡+1
(𝑖,𝑘)
= 𝜔
(𝑖,𝑘)
+ 𝜃𝑡
(𝑖,𝑘)
+
𝛼
𝑁𝑝 𝑐
(𝑗,𝑙)∈𝐺
(𝑖,𝑘)
sin 𝜃𝑡
(𝑗,𝑙)
− 𝜃𝑡
𝑖,𝑘
− 𝜓 𝑘𝑙
𝑖𝑗
+ 𝜎𝛽 𝛽𝑡
(𝑖,𝑘)
for k th oscillator in i th module.
• 𝛼 is a coupling strength and 𝛽𝑡
(𝑖,𝑘)
represents an additive Gaussian noise which
applied to each oscillator, independently. Each 𝜓 𝑘𝑙
𝑖𝑗
is assigned one of four
possible values 0,
𝜋
2
, 𝜋,
3𝜋
2
randomly, according to probabilities 𝑝 𝑚
𝑖𝑗
(m=0,…3).
・We use the probability vectors 𝑝 𝑖𝑗 = 𝑝0
𝑖𝑗
, … , 𝑝3
𝑖𝑗
to regulate the mixing ratio of four kinds of connections from
module j to i. 𝑝0
𝑖𝑗
> 0, σ 𝑚 𝑝 𝑚
𝑖𝑗
= 1.
40. 進化したネットワークのダイナミクス
• Differentiation of the dynamics between module 1 and 2 developed under the evolutionary
process that maximizes bi-directional information transmission.
• Oscillations of an averaged relative phase and an amplitude appeared.
• Extremely slow modulation of oscillations appeared.
カオス成分を持つ振動状態の出現は分化(対称性の破れ)をもたらす
Y.Yamaguti, I.Tsuda,
Neural Networks
2015
41. A
B
C
D
E
L1
𝑅 𝑚
or 𝑀
L2 chaotic itinerancy
(Ikeda-Kaneko-Tsuda,
1989,90,91.....)
I. Tsuda, Current Opinion in Neurobiology 2015, 31:67–71
http://www.scholarpedia.org/article/Chaotic_itinerancy
Dynamics between
default modes can be
represented
by chaotic itinerancy
42. Ω
evoked
Instability
via
Interventions
(constraints)
Similar but slightly different invariant manifolds
脳は外部刺激があるときは低次元の
不変多様体上に制限された活動をする
→ intrinsic manifolds (Luczak et al. 2009)
カオス的解釈
D
E
L1
A
B
L2
Chaotic itinerancy
脳は外部刺激がないときは
不変多様体間を遍歴する
Excess pruning via anti-Hebbian
learning
意識U無意識
43. Cooperative and/or
competitive interactions
between elementary units
Emergence of
order
parameters
Macroscopic
constraints
Emergence of
elementary units, or
components
(a) 自己組織化(SO) (b) 拘束条件付き自己組織化(SOC)
cf) For sensorimotor control,
M. Haruno, DM. Wolpert, M. Kawato,
MOSAIC
47. 人の時間認識能力の神経基盤は何か ?
⇒
カントのアプリオリな時間概念の神経科学的解明は可能か?
Circadian rhythms regulated by the suprachiasmatic nucleus,
which are entrained by the earth’s rotation period
Time cells が発見されている:
・temporal sequences (迷路におけるラットの経験)(Eichenbaum 2014)
・time lapse (刺激提示後の経過時間)(Howard et al. 2014).
time lapse に関する理論: Shanker and Howard (2012).
49. (1) Approximate formula of inverse Laplace transform (E. Post, 1930):
delay period (time cells coding)
T(t, t*) =
−1 𝑘
𝑘!
𝑠 𝑘+1 𝑉 𝑘 𝑡, 𝑠 , 𝑤ℎ𝑒𝑟𝑒 𝑠 = − Τ𝑘
𝑡∗ 𝑡 ∗ < 0
𝑇 = ෨𝐿−1
(𝑉) 𝑇 𝑡, 𝑡 ∗ ≅ 𝐹(𝑡 + 𝑡 ∗)
(2) Phase space transformation : spacialization of time information
temporal sequences of events
(Cantor coding: I.Tsuda 1995-2016; Fukushima et al, 2009-2016)
𝑓1(𝑥)= 𝑎1 𝑥 + 𝑏1, ⋯ , 𝑓𝑛 𝑥 = 𝑎 𝑛 𝑥 + 𝑏 𝑛
One of the affine transformations is selected in CA1,
depending on the output pattern of CA3.
50. Cantor coding with chaotic input time series
海馬における時間情報のコーディングの神経機構の解明
海馬CA3, chaotic itinerancy: エピソードの系列の表現
海馬CA1, Cantor sets: 状態空間での時間系列の空間化 (相空間変換).
52. Henry Gustav Molaison (H.M.)
Molaison’ surgery in 1953
Born February 26, 1926, Hartford, Conneticut
Died December 2, 2008, Windsor Locks, Conneticut
He had suffered from
heavy epileptic seizures,
then he was operated upon
removing the hippocampus.
He, then, suffered from heavy
anterograde amnesia, and
moderate retrograde amnesia,
although his procedural
memories and working memory
are intact.
53. A skeleton network
of the hippocampus
● Unidirectional connections: CA3→CA1
(Mishkin 1982)
Experimental
finding on
the dynamics in
the hippocampus
Buzsaki; Toth, et al
⋆ Chaotic activity was observed in rat CA3 pyramidal cells.
(Hayashi & Ishizuka)
⋆ LTP can be enhanced by the input time series of intermittency.
(Tatsuno & Tsukada)
● Recurrent connections in CA3
Many theories and mathematical models have been proposed: Rolls, Traub, Treves,
Tsukada, Hasselmo, McNaughton, Yamaguchi, Hayashi, Erdi, and many others.
Chaos-driven contracting
System:
A skew product
transformation
54. Skew-product Transformations
For ( , ) ,x y M N
Example. →
: , ( ),
: ,( , ) ( )
: , ( , ) ( ( ), ( ))
x x
x
M M x x
M N N x y y
M N M N x y x y
If for
then
is called a direct product
transformation.
, xx M
( , ) ( ( ), ( ))x y x y
is a skew product transformation.
55. • Emergence of affine
transformations (IFS)
y→Ay+b
Return map of each principal component of the membrane
potentials of CA1
1st component
2nd component 3rd component
y’= μy
y’= μy + 1 - μ
Which branch is adopted
is determined, depending
on chaotic variable.
A skew product
transformation
y
y’
60. References on Chaotic Itinerancy and Cantor Coding
Recent works on Chaotic itinerancy
Tsuda I (2015) Chaotic itinerancy and its role in cognitive neurodynamics. Curr. Opin. Neurobio. 31: 67−71.
Tsuda I (2013) Chaotic itinerancy. Scholarpedia 8(1):4459. DOI: 10.4249/scholarpedia.4459
http://www.scholarpedia.org/article/Chaotic_itinerancy
Kaneko K, Tsuda I (2003) Chaotic Itinerancy.Chaos, 13: 926−936
Cantor coding
Yamaguti Y, Kuroda S, Fukushima Y, Tsukada M, Tsuda I ( 2011) A mathematical model for Cantor coding in the
hippocampus. Neur. Net. 24: 43−53
Kuroda S, Fukushima Y, Yamaguti Y, Tsukada M, Tsuda I (2009) Iterated function systems in hippocampal CA1. Cogn.
Neurodyn. 3: 205−222.
Fukushima Y, Tsukada M, Tsuda I, Yamaguti Y, Kuroda S (2007) Spatial clustering property and its self-similarity in
membrane potentials of hippocampal CA1 pyramidal neurons for a spatio-temporal input sequence. Cogn.
Neurodyn. 1: 305−316
Tsuda I, Kuroda S (2001) Cantor coding in the hippocampus. Japan. J. Indus. Appl. Math. 18: 249−258
Tsuda I, Yamaguchi A (1998) Singular-continuous nowhere-differentiable attractors in neural systems.
Neur. Net. 11: 927−937
Tsuda I (1996) A new type of self-organization associated with chaotic dynamics in neural systems.
Int. J. Neural Sys. 7: 451−459
Chaotic itinerancy & Cantor coding
Tsuda I (2009) Hypotheses on the functional roles of chaotic transitory dynamics. Chaos 19: 015113-1−10
Tsuda I (2001) Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behav.
Brain Sci. 24: 793−810; discussions 811−847