Slide 1

Mathematics of the brain:

A. Overview of the general approach

B. Promising directions of research

Victor Eliashberg

Consulting professor, Stanford University, Department of Electrical Engineering

A. Overview of the general approach

Slide 2

This is our most important “personal computer” – the pinnacle of biological system engineering. What does this computer compute?

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 3

12 cranial nerves ; ~1010 neurons in each hemisphere

8 pairs

12 pairs

5 pairs

6 pairs

31 pairs of nerves; ~ 107 neurons

~1011 neurons

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), call it Brain 0 (Brain Zero), is an “unprogrammed” brain.

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 4

Slide 5

Processor

LTM

RAM

OtherW,D

Conventional computer versus the brain

Processor

LTM

E-states

OtherW,D

1 encoded (symbolic) states 2 analog (dynamic) states

There is a big intuitive difference between 1 and 2 that is difficult to express formally.

Time vs. space: computational universality

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

Slide 6

NOTE. We can perform, in principle, any mental computations. This means that the human brain is a system of type 0 – the limited size of our working memory is of no principle importance. Importantly, an attempt to represent a behavior of higher type in terms of a system of a lower type leads to a combinatorial explosion of the size of such an inadequate representation!

The mystery of human learning

How big is B(0)? How big is B(t)?

B(0) B(t), t >20 years

MEGABYTE(S)? TERABYTES?

learning

The main part of brain’s software must be created in the course of learning

Slide 7

General approach to learning

HYPOTHESIS. The results of learning depend only on the SMI-sequence and not on the way this sequence is produced.

NI –internal observable centers such as, e.g., the centers of emotion, etc.

Teacher

BD

NS

NM

NI

W

Concept of observable behavior

NS –sensory centers

NM –motor centers

S

M

I

S(1) S(2) . . . S(i) . . . S(n)

M(1) M(2) . . . M(i) . . . M(n)

I(1) I(2) . . . I(i) . . . I(n)

SMI – sequence or the brain’s complete experience =

Slide 8

G-states and E-states: the brain as an E-machine

where

gt+1 = fg(x*) fg: X* G (learning

procedure)

et+1 = fe(xt, et, gt ,qt) fe: X×E×G×Q E (next E-state

procedure)

yt = fy(xt, et, gt , qt) fy: X×E×G×Q Y (interpretation

procedure)

X and Y are the input and output sets, respectively, G is the set of states of

“symbolic” LTM, E is the set of states of “dynamic” STM and ITM, X* is the

set of SMI-sequences (see slide 7), t is discrete time. Q all other states.

BDW

xS

yS

xM

yM

xI yI

x =(xS,xM,xI)

y =(yS,yM,yI)

NS

NM

NI

Slide 9

Wolfgang Amadeus Mozart

Henri Poincaré John von Neumann

People who had eidetic memory

Slide 10

Savant Kim Peek Dustin Hoffman and savant Stephen Wiltshire

Eidetic memory vs. creativity

Slide 11

Working memory and mental imagery as a simulation of external system (W,D). System (W,D) is the “Teacher” for system AS

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 12

Motor and sensory areas of the neocortex

Slide 13

Working memory, episodic memory, and mental imagery

ASAM

Motor control

Slide 14

Primary sensory and motor areas, association areas

Association fibers (neural busses)

Slide 15

Basic structure of a primitive E-machine

Slide 16

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

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

Slide 17

Mental computations (thinking) as an interaction between motor control and working memory (EROBOT) , (PEM)

Slide 18

~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

From vectors to symbols: auditory pathways

Slide 19

Slide 20

Eye movements: oculomotor, trochlear, and abducense nerves

From symbols to vectors: voluntary movements

Slide 21

Internal signals: from emotions to symbols and vice versa

Slide 22

Slide 23

W

AS1

ASk

AM1

AMm

S1

SENSORY CORTEX

MOTOR CORTEX

SUBCORTICAL SYSTEMS

SUBCORTICAL SYSTEMS

M1

D

D

The brain as a complex E-machine

B. Promising directions of research

Slide 24

1. Investigating and simulating computational resources of a single neuron

The metaphor the brain as an E-machine promotes the hypothesis that a large portion of the brain hardware computations – especially the MODULATION and the NEXT E-STATE procedures -- is performed at the level of individual cells.

Slide 25

Typical neuron

Neuron is a very specialized cell. There are several types of neurons with different shapes and different types of membrane proteins. Biological neuron is a complex functional unit. Where does this complexity come from?

Slide 26

Computational machinery of a cell

Nucleus

Membrane proteins

Membrane

It took evolution much longer to create individual cells than to build systems containing many cells, including the human brain. Different cells differ by their shape and by the types of membrane proteins.

Nucleus

Membrane proteins

Membrane

18nm

3nm

Slide 27

Protein molecule as a probabilistic molecular machine (PMM)

i

Slide 28

Ensemble of PMMs (EPMM)

E-states as occupation numbers

Slide 29

EPMM as a statistical mixed-signal computer

Slide 30

2. Investigating and simulating neural networks with reciprocal inhibition

The metaphor the brain as an E- machine promotes the hypothesis that a neural layer with reciprocal inhibition can perform the procedure of random equally probable choice. It is interesting to study different possible implementations of such a procedure in the case of a large number of neurons.

Slide 31

Simple “3-neuron” associative neural network (WTA.EXE)

Slide 32

DECODING

ENCODING

RANDOM CHOICE

Input long-term memory (ILTM)

Output long-term memory (OLTM)

addressing by content

retrieval

S21(I,j)

N1(j)

S21(i,j)

3. Studying possible neural implementations of Input and output LTM in the case of a very large number of neurons.

Slide 33

Example of a possible implementation of a very big ILTM

Slide 34

Example of a possible implementation of a very big OLTM

Slide 35

5. Studying possible neural implementations of associative connections in the case of a very large number of neurons.

NOTE. It is physically impossible to have a crossbar connectivity between, say, 108 and 108 neurons – the number of required synapses is 1016. The hypothesis of hash-recoding offers a possible solution. Is this hypothesis correct? What other solutions are possible?

Slide 36

6. Studying the possibilities of primitive E-machines with different DECODING, MODULATION, CHOICE, ENCODING, NEXT E-STATE, and LEARNING procedures.

NOTE. It is particularly interesting to study the possibilities of primitive E-machines with universal learning algorithms – the algorithms that don’t lose training information.

Slide 37

7. Studying the possibilities of universal learning E-machines (type 0) consisting of two primitive E-machines

NOTE. The program EROBOT.EXE provides a simple example of such a universal learning E-machine.

Slide 38

8. Studying the possibilities of complex E-machines with hierarchical structure of associative memory

NOTE. Eliashberg 1979 contains a simple example of such an E-machine.

Slide 39

9. Using natural language analysis for reverse engineering the basic hardware mechanisms of a hypothetical brain-like E-machine.

The problem of a natural language is the most important and interesting problem that seems to be consistent with the metaphor the brain as an E-machine. Therefore, the analysis of a natural language provides a rich source of reliable psychological facts for developing this metaphor.

NOTE. So far I’ve been unable to find either psychological or neurobiological facts that would force me to reject this general metaphor. It is interesting to try to find such facts .

Slide 40

10. Understanding the possibilities of dynamically reconfigurable associative software.

This is the most promising and practically unlimited area of research. An E-machine with a given knowledge (G-state) can be dynamically reconfigured into a combinatorial number of different symbolic machines by changing the state of its dynamic STM and ITM (E-state). It is my belief that understanding the possibilities associated with such a context-dependent dynamic reconfiguration of knowledge (software) is a key to understanding the work of the human brain and to creating really intelligent autonomous robots.

It is especially promising and challenging to try to understand how increasingly complex dynamically reconfigurable associative software can be created in the course of learning.

NOTE. Interesting as they are for specific applications, traditional ANN models have a limited general level of computing power (not higher than type 3) and cannot address the problem of nontrivial brain software – more so, the problem of context-dependent dynamically reconfigurable software.

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