slide 0 lecture 1 mathematics of the brain with an emphasis on the problem of a universal learning...

Lecture 1

MATHEMATICS OF THE BRAIN

with an emphasis on the problemof a universal learning computer (ULC)and a universal learning robot (ULR)

Victor Eliashberg

Consulting professor, Stanford University, Department of Electrical Engineering

WHAT DOES IT MEAN TO UNDERSTAND THE BRAIN?

1. User understanding.

2. Repairman understanding.

Slide 1

3. Programmer (educator) understanding.4. Systems developer understanding.

5. Salesman understanding.

TWO MAIN APPROACHES

1. BIOLOGICALLY-INSPIRED ENGINEERING (bionics)

2. SCIENTIFIC / ENGINEERING (reverse engineering = hacking)

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. 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

Frontal Lobe

Temporal Lobe

Parietal Lobe

Occipital Lobe

Cerebellum

Cervical Spinal Cord

Thoracic Spinal Cord

Lumbar Spinal Cord

Cauda Equina

Our brain still lives in a sea!

Dura mater

Slide 5

12 cranial nerves ; ~1010 neurons in each hemisphere

8 pairs

12 pairs

5 pairs

6 pairs

31 pairs of nerves; ~ 107 neurons

~1011 neurons

The brain has a very large but topologically simple circuitryThe 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.

Cerebelum: N=2,5 107 * 105= 2.51012 B= 2.5 TB. Neocortex: N=1010 * 104= 1014 B= 100 TB.

Pr

Memory is stored in such matrices

LTM size:Slide 6

Big picture: Cognitive system (Robot,World)

B(t) is a formal representation of B at 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)

BDW

Human-like robot (D,B)External world, W

External system (W,D)

Sensorimotor devices, D

Computing system, B, simulatingthe work of human nervous system

Slide 7

CONCEPT OF FORCED MOTOR TRAINING

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

D

NS

AM

Teacher

W

S

M

associations

Motor control:SM M

NM

Brain (NS,NM,AM)External system (W,D)

.

Slide 8

Turing’s machine as a system (Robot, World)

Slide9

TWO TYPES OF LEARNING

AS

D

Working memory and mental imagery

associations

NS

AM

Teacher

SM M

Motor controlW

associationsMS S

S

S

M

M

S

M

NM

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

Slide 12

Working memory, episodic memory, and mental imagery

ASAM

Motor control

Primary sensory and motor areas, association areas

Association fibers (neural busses)

Slide 14

SYSTEM-THEORETICAL BACKGROUND

Slide 15

Fundamental constraint associated with the general levels of computing power

Type 1: Context-sensitive grammars

Type 4: Combinatorial machines (the lowest computing power)

Type 0

Type 1

Type 2

Type 3

Type 4

Type 0: Turing machines (the highest computing power)

Type 2: Context-free grammars (push-down automata)

Type 3: Finite-state machines

Traditional ANN models are below the red 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

General structure of universal programmable systems of different types

PROM stands for Programmable 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.

Type 4: Combinatorial machines

f: X×G→Y

G

yxf

X={a,b,c}

Y={0,1}

PROM

a b c b a c

0 1 0 0 1 1

Slide 17

Type 3: Finite-state machines

PROM

f: X×S×G→S×Y

register

snext

G

yx

fs

aa 1 0 1

0 1 0 0 1 1

11

aaa c a c

0 0 0 1 1 1

bb

0

X={a,b,c}

S=Y={0,1}

x

y

s

snext

Slide 18

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

register

f: X×S×G×M→S×M×YPROM

snext

G

yx

fsMemory buffer, e.g, a tape

M

Slide 19

Basic arcitecture of a primitive E-machine

Slide 20

Control outputs

Association outputs

E-STATES (dynamic STM and ITM)MODULATION, NEXT E-STATE PROCEDURE

CHOICE

Data inputs to ILTM

Data inputs to OLTM

Control inputs

INPUT LONG-TERM MEMORY (ILTM)DECODING, INPUT LEARNING

OUTPUT LONG-TERM MEMORY (OLTM)ENCODING, OUTPUT LEARNING

Association inputs

Data outputs from OLTM

Modulated (biased) similarity function

Similarity function

Selected subset of active locations of OLTM

W

AS1

ASk

AM1

AMm

S1

SENSORY CORTEX

MOTOR CORTEX

SUBCORTICAL SYSTEMS

SUBCORTICAL SYSTEMS

M1

D

D

The brain as a complex E-machine

A GLANCE AT THE SENSORIMOTOR DEVICES

Slide 21

VISION

EYE

Slide 23

EYE MOVEMENT CONTOL

AUDITORY AND VESTIBULAR SENSORS

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

~30,000 fibers~90,000 cells

~390,000 cells

~580,000 cells

~100,000,000 cells

AUDITORY PREPROCESSING

Slide 26

slide 0 lecture 1 mathematics of the brain with an emphasis on the problem of a universal learning...

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