MATHEMATICS OF THE BRAIN

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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
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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.
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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 hasnt 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.
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HUMAN ROBOT
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CONTROL SYSTEM
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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
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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
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LTM size
Cerebelum N2,5 107 105 2.51012 B 2.5 TB.
Neocortex N1010 104 1014 B 100 TB.
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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 of B at time t,
where t0 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)

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CONCEPT OF FORCED MOTOR TRAINING
Brain (NS,NM,AM)
External system (W,D)
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).
.
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Turings machine as a system (Robot, World)
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TWO TYPES OF LEARNING
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Mental computations (thinking) as an interaction
between motor control and working memory
(EROBOT.EXE)
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Motor and sensory areas of the neocortex
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Primary sensory and motor areas, association areas
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Association fibers (neural busses)
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SYSTEM-THEORETICAL BACKGROUND
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Fundamental constraint associated with the
general levels of computing power
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
doesnt have such buffer.
Fundamental problem How can the human brain
achieve the highest level of computing power
without a memory buffer?
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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.
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Type 3 Finite-state machines
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Type 0 Turing machines (state machines coupled
with a read/write external memory)
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Basic arcitecture of a primitive E-machine
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The brain as a complex E-machine
D
SUBCORTICAL SYSTEMS
SENSORY CORTEX
S1
AS1
ASk
W
D
MOTOR CORTEX
SUBCORTICAL SYSTEMS
M1
AM1
AMm
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A GLANCE AT THE SENSORIMOTOR DEVICES
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VISION
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EYE
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EYE MOVEMENT CONTOL
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AUDITORY AND VESTIBULAR SENSORS
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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
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OTHER STUFF
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EMOTIONS (1)
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EMOTIONS (2)
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SPINAL MOTOR CONTROL
SENSORY FIBERS
MOTOR FIBERS
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