Download Wael`s quantum brain - Electrical & Computer Engineering

Studying the “brain” realization
and its simulated quantum
implementation for the Cynthia
robot
Contents
• Introduction to Cynthia robot.
• The goal of the research.
• Examples of different explanations of the brain
system
• The research plan.
• Overview on the previous work.
• Simulation steps
• Current work (Project)
• Future work.
Introduction to Cynthia Robot
Introduction to Cynthia Robot
The goal of the research
• To
build a block which will act as a
brain on top of the MNS and NNS.
• This block will control the behavior of
the robot such that it reflects the learning
process of the robot as well as the
physical phenomena that might
happened and affect the robot behavior
similar to the real brain.
- That requires us to study the biological brain systems
Examples of different
explanations of the brain system
• Gerald Edelman Neural Darwinism:
The Theory of Neural Group Selection.
W11
W12
W21
W31
SUM (Wij)
AF
output
W13
W41
W14
W51
Neural Darwinism:
The Theory of Neural Group Selection.
1
01
10
Examples of different
explanations of the brain system
• Stapp
The Brain as a Quantum Measuring Device
The neural wave function enfolds superposed possibilities, and then
consciousness chooses one classical branch and annihilates the others.
The choice is "unruly," Stapp (1993, p.32) says.
•Some Physical Phenomena couldn't be represented by any low or
mathimatical equations
• Twins spiritual link
It is believed that this could be explained using one of the
quantum mechanics features, called , Entanglement
Examples of different
explanations of the brain system
• Yasue
Quantum Brain Dynamics and
Consciousness: An Introduction (Advances in
Consciousness Research, V. 3
Water Mega molecule
• Ben Goertzel Evolutionary Quantum Computation:
Its Role in the Brain, Its
Realization in Electronic Hardware, and Its
Implications for the Pan psychic, Theory of
Consciousness
Populations of neuronal maps have a quantum aspect as well
as a classical. The brain is an evolving population of quantum
neural networks
Examples of different
explanations of the brain system
•Set up an ensemble of quantum computers, and
allow them to evolve.
• Create criteria for judging QC's, and then, in the
manner of natural selection, allow successful QC's
to survive and (probabilistically) mutate and
combine to form new candidate QC's, whereas
unsuccessful QC's perish.
• The result is that one has quantum computers
fulfilling desired functions via unknown means.
Quantum System &Theory
E24= (0 1 0 0)
• Hilbert Space
E14= (1 0 0 0)
X
E34= (0 0 1 0)
E1 E2 E3 E4 are orthonormal vectors and
called the bases of the Hilbert space
Quantum System &Theory
E24= (0 1 0 0)
• Hilbert Space
r
E14= (1 0 0 0)
X
E34= (0 0 1 0)
| r  1 | E1   2 | E 2   3 | E3   4 | E 4  ...... 
| 1 |  |  2 |  |  3 |  |  4 | ...  1
2
2
2
2
Quantum System &Theory
• Quantum system
• Quantum systems are described by a wave function, r ,
that exists in a Hilbert space.
• The Hilbert space has a set of states, Ei , which is called
the set of bases, and the system is described by a quantum
state, r , which is said to be in a linear superposition of the
basis states Ei , and in general, the coefficients are
complex numbers.
Quantum Theory
• Qubit
• State
• Super position
|  2 |  |  2 | 1
• Uncertainty
|    | 0    | 1 
|  |  |  | 1
2
2
Quantum Theory
0
0
Block Sphere
1
Classical
BIT
1
QuBIT
Quantum Theory
•
Measures the qubit state
The quantum system is said to be collapsed when we make the projection
on one of the basis. That is also called decoherence or the measures.
For example, if we take the projection of |   on the |0> basis then it
will be |    | 0 .
|  2 | is the probability of the qubit to collapse on the state |0>.
Why quantum
• The increasing speed of the computations as well
as reducing the size of the computers will lead to
the quantum mechanics theory will replace the
classical logic theory.
• Implementation of models of the physical
phenomena that could not be implemented before.
• Reduction in time and increase in memory
capacity.
• Parallelism
Quam using Grover
algorithm
Storing Pattern
algorithm
For a set of m binary
patterns with length n,
2n+1 qubits are required
Pattern recall
algorithm
Research algorithm the
speed of researching
is
SPEED
O(
N
).
OF
WHAT??
?
Storing Pattern algorithm
A quantum algorithm for constructing a coherent
superposition of states (bases), that corresponds to the
patterns, with the amplitudes of the states in the
superposition all being equal.
|f> = |X1 X2……Xn >
Where X1,X2,….,Xn are n qubits to represent the n bits
for every binary pattern of the m patterns.
For example
|f > = |10110100> + |11000011> +……..
Storing Pattern algorithm
• To construct this wave function we need to use
n+1 qubits to be used in the process of generation the
function.
|f> = |X1 X2……Xn, G1G2…..Gn-1, C1C2>
G1, G2, Gn-1 as well as C1 C2 are control registers.
Storing Pattern algorithm
Three transformation are used in the process of generating
the function
• S state generation
explain
Where s is the values of the F(z) and s {1,-1} and
1 <= P <=m.
Storing Pattern algorithm
• Control flip transformation
Let’s consider two qubits
|  2   | 0   | 1 
|  1  1 | 0 
0 1 0 0 1 
F
0
1 0 0 0 1 
 1 2  [
][ * ]
0 0 0 1 0 
0 0 1 0 0 
|  2   | 0    | 1 
Storing Pattern algorithm
• Control flip transformation
Let’s consider two qubits
|  2   | 0   | 1 
|  1  1 | 1 
1 0 0 0 0 
F
1
0 1 0 0 0 
 1 2  [
][ * ]
0 0 0 1 1 
0 0 1 0 1 
|  2   | 0    | 1 
Storing Pattern algorithm
• AND transformation
Let’s consider three qubits
|  3   | 0   | 1  |  2  1 | 0 
|  1  1 | 0 
0
1
1
1
B
1
0
1*
1*

1*
0*
B
1*
0*
0*
1*

0*
1*

0*
0*
B
0*
0*

1
00
A
 1 2 3
[
1
][
1
1
1
1
|  3   | 0    | 1 
B
]
Storing Pattern Algorithm
Explain all
symbols
Storing Pattern Algorithm
Step by Step example
To understand the algorithm, let’s assume the following set of learning
patterns
D = {f(01) = -1, f(10) = 1, f(11) = -1}
From D we can deduce the following
Z3 is 01
Z2 is 10
Z1 is 11
n=2
number of qubit to represent the patterns
m=3
number of patterns to be represented
Storing Pattern Algorithm
Step by Step example
1- |f > = |00,0,00 >
2 2- do the for loop
P=3
Z3= 01
Z31=0
Z4= 00
Z41=0
Z32 not equal Z42
C2
X2
F
X1=0 X2=0 g1=0 , C1=0 and C2 =0
Z32=1
Z42=0
flip X2
X2
0
c2 X 2
Then |f > = |01,0,00 >
Storing Pattern Algorithm
Step by Step example
3- Flip C1 state
C2
C1
F
C1
0
Then |f > = |01,0,10 >
c 2 c1
Storing Pattern Algorithm
4- Generate a new state by applying S on C2 C1
C1
C2
1 0
S
13
0
0 1
0
C 1c 2  [
0 0
2/3
0 0  1/ 3
then |f > = -1/
3 |01,0,11 > +
0
0 C 20
0 0 C 21
][ * ]
1 / 3 1 C 20
2 / 3 1 C 21
2/3
|01,0,10>
Storing Pattern Algorithm
• 5- Flip g1 to mark the register
X1
X2
g1
01
A
F 0
01g 1  [
] X 1@ X 2 @ g1
0 I
then |f > = -1/ 3
|01,1,11 > + 2 / 3
|01,1,10>
g1
Storing Pattern Algorithm
26-
Flip C1 which is controlled by g1
1 0 0 0
g1
C1
F
then |f > = -1/
1
0
C10
0 1 0 0 0 C11
g 1c1  [
][ * ]
0 0 0 1 1 C10
0 0 1 0 1 C11
3 |01,1,01 > +
2 / 3 |01,1,00>
C1
Storing Pattern Algorithm
• 5- Flip g1 again to the normal state
X1
X2
01
A
g1
F 0
01g 1  [
] X 1@ X 2 @ g1
0 I
then |f > = -1/ 3
Saved
|01,0,01 > + 2 / 3
g1
|01,0,00>
Go to step 1Again
Storing Pattern Algorithm
• The whole process repeated again with start
|f > = -1/ 3 |01,0,01 > + 2 / 3 |01,0,00>
after the 3rd loop
|f> =
that is what is called storing the pattern.
5 qubit Quam Network
Implemantation
X1
X2
?F
0
A
?F
A
0
g1
0
F
1
C1
C2
F
0
S
0
7 qubit Quam Network
Implemantation
Pattern recall Algorithm
• The idea of pattern recall is collapsing the
function |f > on the required basis (pattern).
•
Grover used his quantum search in data base algorithm in
recalling the pattern. The idea of this search is to change the
phase of the desired state and then rotate the entire |f > around the
average. This process repeated (3.14/4)*
N Where N is the
total possible state.
Pattern recall Algorithm
• The algorithm steps:
1-change the phase of the desired state.
2- compute the average A
3- rotate the entire quantum set around the average.
|f>= 2A-|f >
4- repeat 1-3 for (3.14/4)*
5- Measure the desired state.
N
Pattern recall Algorithm
• Step By Step example
Let’s continue on the same example, used in the
learning phase.
Let’s assume we want to recall the pattern 01.
Since we have only 2 qubits then the possible
combination is 4.
|f> will collapse on the desired state after repeat the
algorithm for (3.14/4)*2, which roughly 1 times.
Pattern recall Algorithm
• Step By Step example
1- |f >
2 2- f > 1/ 3 (0,-1,1,1)
3- 3- Average =1/4
4- 4- |f > 1/2 3
(1,3,-1,-1)
5- 5- Measure the desire state |f >= (
3 /2)
|01>
Pattern recall Algorithm
• It is obvious that the probability of the system to
collapse on the desired state is ¾= 75%.
• The system collapse in the O( N ).
• Which means it is faster than the classical NN
which takes O(N)
Comparison between the Quam and the
NN Hope field Associative memory.
Quam
Max memory
capacity
Number of
neurons
Pattern recall
speed
Phase learning
speed
Hopfield
.15*n
n
2
2n+1
O(
N
O(mn)
n
)
O(N)
O(mn)
Comparison between the Quam and the
NN Hope field Associative memory.
The research plan
•Phase One
Insert a quantum circuit in the
command execution data path, in the
MNS in figure.1. The quantum circuit will
alter the command slightly.
•
•Phase Two
•Phase Three
Phase One
Robot (CRL parcel translator)
MNS ( command initiation)
Quantum Circuit
( Command alteration)
Servos (Motion)
Phase Two
• Study designing the quantum circuit
such that it reflects the learning
process of the robot brain and matches
the behavior, mode and the emotion of
the robot.
Phase Two
Robot (CRL parcel translator)
MNS ( command initiation)
Quantum Circuit
( Design
( Command
the matched
alteration)
Quantum
circuit to the required behavior )
Servos (Motion)
Phase Three
Robot Brain
• Generalize the Idea by built in a
complete block on top of the MNS , which
will act as a brain to the robot.
Overview on the previous work
• A quantum circuit was introduced using the QUASI
quantum simulator.
•The theatre robots communicate using CRL (Common
Robotic Language).
•The inputs for the Quantum circuit will be the data between
the command tags in the CRL file.
• The present version supports only the following command
tags for the recognition of inputs to the Quantum Circuit.
–They are wait, flush, move, normal, smile, frown, cry, look, speak,
speed, accel, open and close.
Simulation steps:
Robot (CRL parcel Command
translator)
Save input data into XML File
Choose Quantum circuit
using Quasi Simulator
Save the circuit into XML data
File
Load the XML files Using Quasi Simulator
Generate the Output sequence from the circuit and
save in a file to be used as an alter command to the
servo
Current work (project)
• Integrate the software which was done in
the previous work. (current)
Robot (CRL parcel Command
translator)
Choose Quantum circuit
using Quasi Simulator
Generate the Output sequence from the circuit and
as an alter command to the servo
Future work
• Design the quantum circuit according to
the learning process of the brain.
• Implement the brain block in the Cynthia
Robot.
END
Introduction to Cynthia Robot
Simulation steps
1.
Generate a circuit:
A new Quantum Circuit can be built using the Quasi simulator. Run
the Quark2.Quark command inside the Quasi folder. (The main
program in the Quark class). Two windows will open up. In the left
window, click on Circuit>New tab and create a new circuit. Save the
circuit (Let us assume you saved it as MyCircuit.xml).
2. Load the input sequence to the circuit:
To load the input sequence, run the following command in the folder
where the program is saved. Ø java CrlAnalysis MyCrlFile.crl
MyCircuit.xml This will calculate the input sequence from the
MyCrlFile.crl and will load it into MyCircuit.xml
Simulation steps:
3. Simulate the circuit:
In the left window of the Quasi program, click circuit>load and
load the MyCircuit.xml file. Then click run to finish button. This will
simulate the circuit and the outputs are displayed in the second
window. The data is stored in the xml format into the file called
QuantumOutput.xml file.
4. Generate new CRL file:
Now, the QuantumOutput.xml file contains the results of the
circuit along with their probabilities alpha2 and beta2. To generate the
new CRL file, we need to run the following command. Ø java
ReadWriteCrl MyCrlFile.crl QuantumOutput.xml The new CRL file
with the name TestOutput.crl file is created.
This file can be used on the robots and the behavior can be
observed.
Defining the Quantum Computer
You don't have to go back too far to find the origins of quantum computing. While computers have been around for the majority of the 20th century, quantum
computing was first theorized just 20 years ago, by a physicist at the Argonne National Laboratory. Paul Benioff is credited with first applying quantum theory to
computers in 1981. Benioff theorized about creating a quantum Turing machine. Most digital computers, like the one you are using to read this article, are based
on the Turing Theory.
The Turing machine, developed by Alan Turing in the 1930s, consists of tape of unlimited length that is divided into little squares. Each square can either hold a
symbol (1 or 0) or be left blank. A read-write device reads these symbols and blanks, which gives the machine its instructions to perform a certain program.
Does this sound familiar? Well, in a quantum Turing machine, the difference is that the tape exists in a quantum state, as does the read-write head. This means
that the symbols on the tape can be either 0 or 1 or a superposition of 0 and 1. While a normal Turing machine can only perform one calculation at a time, a
quantum Turing machine can perform many calculations at once.
Today's computers, like a Turing machine, work by manipulating bits that exist in one of two states: a 0 or a 1. Quantum computers aren't limited to two states;
they encode information as quantum bits, or qubits. A qubit can be a 1 or a 0, or it can exist in a superposition that is simultaneously both 1 and 0 or
somewhere in between. Qubits represent atoms that are working together to act as computer memory and a processor. Because a quantum computer can
contain these multiple states simultaneously, it has the potential to be millions of times more powerful than today's most powerful supercomputers.
This superposition of qubits is what gives quantum computers their inherent parallelism. According to physicist David Deutsch, this parallelism allows a
quantum computer to work on a million computations at once, while your desktop PC works on one. A 30-qubit quantum computer would equal the processing
power of a conventional computer that could run at 10 teraflops (trillions of floating-point operations per second). Today's typical desktop computers run at
speeds measured in gigaflops (billions of floating-point operations per second).
Quantum computers also utilize another aspect of quantum mechanics known as entanglement. One problem with the idea of quantum computers is that if you
try to look at the subatomic particles, you could bump them, and thereby change their value. But in quantum physics, if you apply an outside force to two atoms,
it can cause them to become entangled, and the second atom can take on the properties of the first atom. So if left alone, an atom will spin in all directions; but
the instant it is disturbed it chooses one spin, or one value; and at the same time, the second entangled atom will choose an opposite spin, or value. This allows
scientists to know the value of the qubits without actually looking at them, which would collapse them back into 1's or 0's.
Gerald Edelman's Work
Topobiology; An Introduction to Molecular
Embryology
Neural Darwinism; The Theory of Neuronal Group
Selection
The Remembered Present: A Biological Theory of
Consciousness
Bright Air, Brilliant Fire: On the Matter of the Mind
Once one has committed oneself to looking at groups, the next step is to ask how these groups are organized. A map, in
Edelman's terminology, is a connected set of groups with the property that when one of the inter-group connections in
the map is active, others will often tend to be active as well. Maps are not fixed over the life of an organism. They
may be formed and destroyed in a very simple way: the connection between two neuronal groups may be
"strengthened" by increasing the weights of the neurons connecting the one group with the other, and "weakened" by
decreasing the weights of the neurons connecting the two groups.
Formally, we may consider the set of neural groups as the vertices of a graph, and draw an edge between two vertices
whenever a significant proportion of the neurons of the two corresponding groups directly interact. Then a map is a
connected subgraph of this graph, and the maps A and B are connected if there is an edge between some element of A
and some element of B. (If for "map" one reads "program," and for "neural group" one reads "subroutine," then we
have a process dependency graph as drawn in theoretical computer science.)
This is the set-up, the context in which Edelman's theory works. The meat of the theory is the following hypothesis: the
large-scale dynamics of the brain is dominated by the natural selection of maps. Those maps which are active when
good results are obtained are strengthened, those maps which are active when bad results are obtained are weakened.
And maps are continually mutated by the natural chaos of neural dynamics, thus providing new fodder for the
selection process. By use of computer simulations, Edelman and his colleage Reeke have shown that formal neural
networks obeying this rule can carry out fairly complicated acts of perception.
This thumbnail sketch, it must be emphasized, does not do justice to Edelman's ideas. In Neural Darwinism Edelman
presents neuronal group selection as a collection of precise biological hypotheses, and presents evidence in favor of a
number of these hypotheses.
However, I consider that the basic concept of neuronal group selection is largelyindependent of the biological
particularities in terms of which Edelman has phrased it. As argued in (Goertzel, 1993), I suspect that the mutation
and selection of "transformations" or "maps" is a necessary component of the dynamics of any intelligent system.
Edelman's theory provides half of the argument that the brain is an EQC: it provides evidence that the brain is an evolving
system. Edelman uses nonlinear differential equations on finite-dimensional spaces to model the dynamics of
neuronal groups; he does not consider these groups as quantum systems. There is much evidence, however, that the
brain is not as "classical" a system as Edelman and other more conventional neural net theorists would have it.
Will we ever have the amount of computing power we need, or want? If, as Moore's Law states, the number of transistors on a microprocessor
continues to double every 18 months, the year 2020 or 2030 will find the circuits on a microprocessor measured on an atomic scale. And the logical
next step will be to create quantum computers, which will harness the power of atoms and molecules to perform memory and processing tasks.
Quantum computers have the potential to perform certain calculations billions of times faster than any silicon-based computer
How Quantum Computers Will Work
by Kevin Bonsor
I find Jibu and Yasue's perspective quite appealing. Rather than throwing
out all we have learned about neural networks, in this view, we must
merely accept that there are parallel quantum systems, working together
with neural networks to create thought. In terms of Edelman's theory, we
need not reject the idea of Neural Darwinism -- we must merely accept
that these populations of neuronal maps have a quantum aspect as well as
a classical aspect. In other words, the brain is an evolving population of
quantum neural networks, selected and mutated based on their
functionality in regard to their interaction with perceptual and motor
systems, as determined by needs of the organism. Edelman, plus Jibu and
Yasue, equals the brain as an EQC.
But what is EQC all about? The idea is a very, very simple one. Instead of programming a
quantum computer, set up an ensemble of quantum computers, and allow them to evolve. Create
criteria for judging QC's, and then, in the manner of natural selection, allow successful QC's to
survive and (probabilistically) mutate and combine to form new candidate QC's, whereas
unsuccessful QC's perish. The result is that one has quantum computers fulfilling desired
functions via unknown means.
The neural wave function enfolds superposed possibilities, and then
consciousness chooses one classical branch and annihilates the others.
The choice is "unruly," Stapp (1993, p.32) says, "not individually
controlled by any known law of physics." So the heart of consciousness
is random on Stapp's view. He hopes that some future physics will find a
law (1993, p.216), but it certainly looks like barring an enormous
revolution in quantum physics, Stapp has installed chance deep in his
theoretical framework, where the quantum choices associated with
conscious events take place:
Yasue's Quantum Brain Dynamics
3.1 The brain is remarkable in that it provides a variety of substrates for quantum fields. Different brain substrates for
quantum fields have different functions. The sensory quantum field, for example, supervenes on oscillating biomolecules of
high dipole moment in the neuronal membrane. When the pumping rate reaches a critical value, Froehlich condensation
occurs with macroscopic coherence of quanta (Froehlich, 1968).
3.2 Another quantum field-supporting biosubstrate is a dense nanolevel web of protein molecules which penetrates neuronal
and neuroglial membrane boundaries. I call this filamentous web the "nanolevel neuropil." Inside the neuron the nanolevel
neuropil consists not only of microtubules but also neurofibrils and other structures which connect via protein strands to
proteins floating in the cell membrane. Outside the neuron in the synaptic cleft is the extracellular matrix of collagen and
glyco-conjugates, which are also connected to membrane proteins, so that a pervasive web is formed.
3.3 There are quasi-crystalline water molecules within the microtubules and associated with hydrophylic regions on the web
of protein fila- ments. This ordered water is yet another brain biosubstrate for a quantum
field which supports
super-radiance and self-induced trans- parency within the microtubules (Jibu et al, 1994).
3.4 Jibu and Yasue (1992, 1993) have proposed, following some earlier suggestions by Umezawa (e.g. Ricciardi &
Umezawa, 1967), that vacuum states of this water rotational field record memory. I have suggested that the function of the
nanolevel neuropil is cognitive (Globus, 1995).
3.5 There is a fourth quantum field substrate where an interaction takes place between the sensory quantum field and the
cognition/memory quantum field. This is a plasma of charged particles interacting with the electromagnetic field. The
structure of this bio-plasma is peculiar: it is divided into two very thin layers separated by a permeable membrane. Membrane
channels open and close, and ions rush back and forth between the two layers down electrical and chemical gradients. It is in
this perimembranous bioplasma, whose state is given by the ionic density distribution, that sensory and cognition/memory
quantum fields interact. In this interaction of quantum fields, classical orders may be formed (as when the multiplication of
complex conjugates gives a real number).
An evolutionary quantum computer (EQC) is a physical system that maintains an
internal ensemble of macroscopic "quantum subsystems" manifesting significant
quantum indeterminacy, with the property that the of quantum subsystems is
continually changing in such a way as to optimize some measure of the emergent
patterns between the system and its environment. It seems probable that the brain is an
EQC, and that electronic EQC dissimilar to the brain can also be constructed; a
speculative design in this regard is described, called QELA (Quantum Evolving Logic
Array), involving Superconducting Quantum Interference Devices interfacing with reconfigurable Field Programmable Logic Arrays. EQC has interesting implications for
a quantum pan psychic view of consciousness: it provides an explanation of why, if
everything is conscious to some extent, the human brain is so much more conscious
than most other systems. The explanation is that, via EQC, the brain is able to
maintain significant quantum randomness ("raw awareness") in a way that is
correlated with its structure and behavior. Only EQC provides this kind of correlation,
because only EQC allows uncollapsed quantum systems to interact significantly with
the wave-function-collapsed, classical everyday world. In many- worlds-terms, EQC
allows systems with a broad span over the range of possible universes to interact
significantly with systems existing in narrow regions of universe-space.
4. U/Y v. H/S
4.1 The conception of the brain is far richer in U/Y than H/S; for U/Y, the brain generates second
order quantum fields. A Geiger counter or Schroedinger's cat box has a quantum field
description (as a Bogoliubov transformation of the quantized field) but such ordinary
measurement devices do not sustain quantum fields like the brain does. So reality is described
by wave functions, both microscopic and macroscopic, and among those macroscopic realities
are well- developed human brains which themselves sustain quantum fields and their
interactions.
4.2 We should not think of these second order quantum fields as making measurements but as
offering possibilities to the match. Both sensory input and cognition/memory participate in the
evolution of the state variable by offering possibilities to the match, but the latter is far richer
than the former. I have previously called this rich quantum plenum of superposed possibilities
the "holoworld" (Globus, 1987) and suggested that the probabilities of the various possibilities
are tuned (Globus, 1995). The more limited possibilities of sensory input continually interact
with the tuned holoworld, and a classical order continually unfolds in the perimembranous
bioplasma.
4.3 So instead of a measurement collapsing the wave function of a quantum field to a classical
order, we have a match between quantum cognition/memory and quantum reality, a match in
which classical order is unfolded.
Surprisingly enough, one can argue that this is a viable model of brain
dynamics. Edelman, with his theory of neuronal group selection, has
already made a strong case for the brain as an evolutionary system. And
Jibu and Yasue have made a good case for the brain as a macroscopic
quantum system. Putting these two together, we obtain a strikingly solid
case for the brain as an EQC. The EQC explanation of why the human
brain is so acutely conscious then fits right in.
In order to see that is the inversion about average,
consider what happens when acts on an arbitrary
vector . Expressing D as , it follows that:
. By the discussion
above, each component of the vector is A where A is
the average of all components of the vector . Therefore
the ith component of the vector is given by
which can be written as
which is precisely the inversion about averag