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Lecture 1

MATHEMATICS OF THE BRAIN

with an emphasis on the problem

of a universal learning computer (ULC)

and a universal learning robot (ULR)

Victor Eliashberg

Consulting professor, Stanford University,

Department of Electrical Engineering

Slide 0

WHAT DOES IT MEAN TO UNDERSTAND THE BRAIN?

1. User understanding.

2. Repairman understanding.

3. Programmer (educator) understanding.

4. Systems developer understanding.

5. Salesman understanding.

Slide 1

TWO MAIN APPROACHES

1. BIOLOGICALLY-INSPIRED ENGINEERING (bionics)

Formulate biologically-inspired engineering / mathematical problems. Try to solve these problems in the most efficient engineering way.

This approach had big success in engineering: universal programmable computer vs. human computer , a car vs. a horse, an airplane vs. a bird.

It hasn’t met with similar success in simulating human cognitive functions.

2. SCIENTIFIC / ENGINEERING (reverse engineering = hacking)

Formulate biologically-inspired engineering or mathematical hypotheses. Study the implications of these hypotheses and try to falsify the hypotheses. That is, try to eliminate biologically impossible ideas!

We believe this approach has a better chance to succeed in the area of brain-like computers and intelligent robots than the first one. Why?

So far the attempts to define the concepts of learning and intelligence per se as engineering/mathematical concepts have led to less interesting problems than the original biological problems.

Slide 2

HUMAN ROBOT

Slide 3

CONTROL SYSTEM

Slide 4

OUR MOST IMPORTANT PERSONAL COMPUTER

12 cranial nerves ; ~1010 neurons in each hemisphere

~1011 neurons

31 pairs of nerves; ~ 107 neurons

8 pairs

12 pairs

5 pairs

6 pairs

Slide 5

The brain has a very large but topologically simple circuitry

The shown cerebellar network has ~1011 granule (Gr) cells and ~2.5 107 Purkinje (Pr) cells. There are around 105 synapses between T-shaped axons of Gr cells and the dendrites of a single Pr cell.

Pr

Memory is stored in such matrices

Slide 6

LTM size:

Cerebelum: N=2,5 107 * 105= 2.51012 B= 2.5 TB.

Neocortex: N=1010 * 104= 1014 B= 100 TB.

Big picture: Cognitive system (Robot,World)

External system (W,D)

Computing system, B, simulating

the work of human nervous system

Sensorimotor devices, D

B

D

W

Human-like robot (D,B)

External world, W

B(t)is a formal representation ofBat time t, where t=0 is the beginning of learning. B(0) is an untrained brain.B(0)=(H(0),g(0)), where

H(0) = H is the representation of the brain hardware,

g(0) is the representation of initial knowledge (state of LTM)

Slide 7

CONCEPT OF FORCED MOTOR TRAINING

M

SM

External system (W,D)

Brain (NS,NM,AM)

D

AM

S

NS

Motor control:

W

associations

M

NM

Teacher

During training, motor signals (M) can be controlled byTeacher or by learner (AM) . Sensory signals (S) are received from external system (W,D).

.

Slide 8

Working memory and

mental imagery

M

AS

D

S

MS

S

associations

S

NS

Motor control

W

S

AM

M

SM

M

associations

M

NM

Teacher

TWO TYPES OF LEARNING

Slide 10

Mental computations (thinking) as an interaction between motor control and working memory (EROBOT.EXE)

Slide 11

Motor and sensory areas of the neocortex

Working memory, episodic

memory, and mental imagery

Motor control

AM

AS

Slide 12

SYSTEM-THEORETICAL BACKGROUND

Slide 15

Type 0: Turing machines (the highest computing power)

Type 0

Type 1

Type 1: Context-sensitive grammars

Type 2

Type 2: Context-free grammars

(push-down automata)

Type 3

Type 4

Type 3: Finite-state machines

Type 4: Combinatorial machines

(the lowest computing power)

Fundamental constraint associated with the general levels of computing power

Traditional ANN models are below thered line. Symbolic systems go above the red line but they require a read/write memory buffer.The brain doesn’t have such buffer.

Fundamental problem: How can the human brain achieve the highest level of computing power without a memory buffer?

Slide 16

Type 4: Combinatorial machines

X={a,b,c}

PROM

a b c b a c

Y={0,1}

G

010011

f: X×G→Y

y

x

f

General structure of universal programmable

systems of different types

PROMstands forProgrammable Read-Only Memory.

In psychological terms PROM can be thought of as a Long-Term Memory (LTM). Letter G implies the notion of synaptic Gain.

Slide 17

Type 3: Finite-state machines

PROM

x

X={a,b,c}

a

b

c

a

b

c

a

a

s

S=Y={0,1}

0

0

1

1

G

0

1

snext

0

1

0

1

1

1

f: X×S×G→S×Y

a

a

y

0

0

0

1

1

1

y

x

f

snext

s

register

Slide 18

Type 0: Turing machines(state machines coupled with a read/write external memory)

PROM

f: X×S×G×M→S×M×Y

G

y

M

x

f

s

Memory buffer, e.g, a tape

snext

register

Slide 19

Basic arcitecture of a primitive E-machine

Association inputs

Data inputs to ILTM

INPUT LONG-TERM MEMORY (ILTM)

DECODING, INPUT LEARNING

Similarity function

Control inputs

E-STATES (dynamic STM and ITM)

MODULATION, NEXT E-STATE PROCEDURE

Modulated (biased) similarity function

CHOICE

Data inputs to OLTM

Control outputs

Selected subset of active locations of OLTM

OUTPUT LONG-TERM MEMORY (OLTM)

ENCODING, OUTPUT LEARNING

Association outputs

Data outputs from OLTM

Slide 20

The brain as a complex E-machine

D

SUBCORTICAL SYSTEMS

SENSORY CORTEX

S1

AS1

ASk

W

D

MOTOR CORTEX

SUBCORTICAL SYSTEMS

M1

AM1

AMm

Slide 21

VISION

Slide 22

EYE

Slide 23

EYE MOVEMENT CONTOL

Slide 24

AUDITORY AND VESTIBULAR SENSORS

Slide 25

AUDITORY PREPROCESSING

~100,000,000 cells

~580,000 cells

~4,000 inner hair cells ~12,000 outer hair cells

~390,000 cells

~90,000 cells

~30,000 fibers

Slide 26

OTHER STUFF

Slide 27

EMOTIONS(1)

Slide 28

EMOTIONS(2)

Slide 29

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