Basic Formulas


8 Feb 2019

Harland Harrison

(See Appendix X for a description of symbols)

Neurons are slow relaxation oscillators, pulsing periodically:

X(0)=1; X(1)=0; X(t)=X(t+2) // Time is normalized for period of 2

X(t) * X(t+1) = 0

X(t)  = ~ X(t+1)

Neurons discharge rhythmically if undisturbed, but one neuron

connected by a synapse, can stimulate another neuron to fire:


(This represents a synapse, axon of S connected to dendrite of R)

Neurons have a "refractory period" after depolarizing during which

they do not fire again. Here it is asserted that the sensitivity to a

stimulus depends on the charge accumulated over time as well

as the magnitude of the stimulus.

The frequency of neurons is incredibly slow for logic systems.

Neural activity is measured in milliseconds; silicon gates function in

nanoseconds.   Accordingly,  the organization of neurons must differ

from that of von Neumann machines.  Here it is asserted that

minimizing delay is an end product of evolution and development.

The brain is massively parallel.  The "fan out" of a cerebral neuron

is near 10^4,  but  the fan out of a silicon gate is only about 10.  The

same silicon gates are cycled over and over again for processing

involving thousands of terms.  Here it is asserted that a neuron

accomplishes similar computations in a single cycle using many

more terms in order to achieve an acceptable response time.

The structure consisting of cell bodies and synapses implements

some logic processing,  at least by controlling the entry of signals 

into the cells. Here it is asserted that this structure is the basis of

all computational logic in the brain, in a manner described by

Arnold Trehub.  Additional logic within the neuron would delay its

response and would increase the information that the synapses

would be required to transmit, further delaying responses.  

Neurons used as logic gates could form a Turing machine.

A Turing machine is a simple computer that can simulate any

computer or similar logical device.  Consequentially,  any Turing

machine can simulate any other Turing machine. 

The necessary functions for Turing machine are AND/OR, NOT,

amplification, and delay, which can all be implemented by NAND

gates in silicon (formally: disjunction/conjunction, inversion, / , )

Multiple synapses stimulating the same cell, provide AND/OR


Depending on sensitivity,  it implements "S1S2->R", S1 AND S2,

where both are necessary, (a coincidence detector), or else,

"S1S2->R", S1 OR S2, in the case of more sensitive synapses

Inhibitory synapses provide an inversion, NOT, function, but the

phasing of refractory time can also do so

X(t)  = ~ X(t+1)

The energy released by the post-synaptic neuron is greater than

the required stimulus, providing amplification.  (The extra energy

was stored in the post-synaptic cell body.)

There is no suggestion here that the brain is organized as a Turing

machine, but only that it can perform the same kind of processing. 

Hebbian plasticity: 

Successful stimulus-response can gradual strengthen a connecting


S(n)=1,R(n+1)=1,S->R  =>  S +-> R

S(t) = R(t+1) => S-> R

And unsuccessful synapses can be weakened

S(n)=1,R(n+1)=0,S->R  =>  S --> R

S(t) ≠ R(t+1) => S ~-> R

It is assumed that the Hebbian effect  is "spike-timing-dependent"

and so can strengthen a synapse even if the pre-synaptic cell is not

the major cause of the subsequent response of the post-synaptic cell.

The "success" of the synapse is the fact of a post-synaptic response

at the  appropriate time, regardless of the magnitude or contribution

of the pre-synaptic input. (Neurotransmitters enter the synaptic

cleft to strengthen the synapse and could do so by diffusion, 

depending on the electric field of each cell.)

"Conditioned reflexes" arise by associating simultaneous stimuli

through Hebbian effects.  

C(t)=U(t);  U->R =>  C->R

Rule of Fastest Connection, (RFC) :

RFC minimizes response time, by Hebbian effect, so that only

the fastest path remains

S -> A -> R ;  S -> B -> R ;  T(B->R) < T(A->R)

    => S -> B +-> R ;  S -> A --> R

    => S -> B -> R; ~ (A -> R)

RFC trains 'lower' reflexes if pathways exist.  The direct pathway

is inherently faster but omits unnecessary logic. The time, T(S->R),

of the reflex is inherently less than T(S->P->R)

S->P->R => S->R

A repeated pattern being detected can be reconstructed:

{S}(t) = {S}(t+4) = {S}(t+8)...  ;  {S} -> R -> P->C


C(t+3) = {S}(t+3+1) => C-> {S}

by RFC the pattern {S} can now also connect to C directly

 {S} -> R -> P-> C => {S} -> C

To simplify, write the complete pattern detecter,

 {S} -> C ; C -> {S}  


 {S} <-> C

A near match can reconstruct the full original pattern

{S} <-> C

{s} {S} //  let s be a subset of S

{s}(t) = C(t+1)   //   and assume M(s) is sufficient
{s}(t) = C(t+1) = {S}(t+2) // all of S will be activated


An array of possible patterns to match becomes a

"register", written as  |X<->Y|

 {S0}<->P0...{Sn}<->Pn |S<->P|

|S<->P| ; {Sn}(t)  = Pn(t+1)

Since neurons are relaxation oscillators,  the speed of each matching

response depends on the magnitude of the matching stimuli:

T({s}->C) 1/M({s})

When there are multiple possibilities,  the best match will tend to fire

first.  This allows the register to detect and latch the best  complete

known pattern from partial data

Auto selection for new input is implemented by amplifying a small,

random, match among otherwise unused elements, to select one

{s} -> R => {s}->R

"{s} {S}" //   subset s is a random 'seed' to select a match for S

{S}(t), {s} -> R=> R-> {S} => {S} <-> R

The register detects the best match if the proper neuron responds.

An inaccurate response might occur if another neuron with a partial

match is more sensitive.  The register is most accurate when all the

neurons are equally sensitive. Sensitivity depends on the phase as

well as the chemical state of a neuron. 

The phases are only equal when all of the cells in the register have

fired simultaneously.  Since relaxation oscillators will synchronize

when they are  coupled, however loosely, simultaneous action

should be expected.  Here it is asserted that synchronization serves

the important purpose of preparing registers to identify incoming data.

Accordingly,  synchronization is observed during resting states,

sensory deprivation, and epileptic seizures.

The chemical activity of the cell depends on internal concentrations

of oxygen and CO2, which diffuse across the cell membrane. Their

internal concentrations must remain constant for constant sensitivity.  

Yet as activity increases, more O2 is consumed and more must come

in. It is known that astrocytes dilate the capillaries, increasing blood

flow, as neural activity increases.  This causes the well-studied BOLD

signal.  Here it is asserted that the increase in blood flow keeps the

concentrations within the cells at a constant level. 

R = k(E-I) // diffusion is proportional to the difference in concentrations

I = (1/D)(R(t)-M(t)) // integrate diffusion and metabolism for internal O2

(1/D)R(t) = (1/D)M(t) // eventually, diffusion will transport all O2 used

I = 1 // asserting that internal concentration always remains constant

E = M/k + 1 // shows that external O2 concentration varies with activity


E = external O2 concentration

I = internal O2 concentration

R = rate of diffusion

M = rate of metabolism

Consciousness as a Global Workspace

Conscious thought is a "Global Workspace", (GW), as shown by

Bernard Baars.  The GW is  presumedly in associative cortex and

cross-connectable so that any element can connect to any other. 

Here it is asserted that the elements in GW are, again, individual

neurons, and that their interconnections are individual synapses.

|Ci <-> Cj|  // i...  j...  = 10^10 for about 10^20 possible connections

Only about 10^15 synapses exist but conscious content is also quite

sparse.  GW contains three classes of elements, perceptions, qualia,

and abstractions.  Words are a kind of abstraction.  Qualia will be

discussed below.  Each element is assigned to individual neurons.

The neurons of each class will be designated by P, Q , and V,


Some input from the senses, but not all, enters conscious "awareness".


Senses can become conscious, even while stimulating reflexive responses


Consciousness normally guides action by integrating sensory information

S0...Sn -> P0...Pn -> P' -> R

The conscious response, "awareness", will be inherently slower than a reflex

S(t) = R(t+1) = P(t+1+n)

R(t) = S(t-1) = P(t+n)

RFC trains the "smarter" conscious action into automatic, reflexive, behavior

S->P->R  =>  S->R

RFC also supports learning by study, reducing steps between thoughts

P0 -> P1 -> ... -> Pn => P0 -> Pn

Consciousness can retain information across time and circumstance

by looping through GW,  regenerating the information

P0 -> ...->Pn -> P0

Here it is asserted that "attention" consists of maintaining such a loop,

and "ignoring" a thought consists of blocking its recirculation.

P0 -> ...->Pn; (Pn AND A) -> P0

Conscious attention to well-trained ability can degrade performance.

Here it is asserted that the conscious path is too slow and interferes.

S->R,P ; P->R

R(t) = S(t-1) + P(t+n+1) // "Conscious" response also arrives,  but it is late

A quale is the essence of a sensation, eg the "redness" of the color red.

Normal, waking, thought cannot reproduce the sensory input completely.

The qualia appear as stubborn, uncontrollable, elements of consciousness.

Here it is asserted that qualia occur at the point where senses enter GW.

Because a quale neuron is driven by sensory nerves, as well as by other

GW neurons, it cannot be fully controlled by processes within GW.  The

control of qualia by sensory nerves will be called Force of Reality, (FR).

S -> Q -> P ; P <-> P ; ~(P->Q) // A quale is conscious but only as sensed

Offline Processes:

Certain states can suspend the FR. Dreaming during REM sleep,

meditation, hallucination, etc can all allow thoughts to cause qualia.

Letting W=1 if conditions are REM etc, and otherwise W= 0

(P  AND W) -> Q  // P <-> Q during REM, sensory deprivation, etc

Presumedly, sensory deprivation is all that is required to suspend FR

S(t) = S(t+1);W=1  // if S is not changing at all, it stops setting Q

REM sleep is theorized to be useful for "consolidating memory".  Here

it is asserted that REM sleep is necessary to train automatic responses

to qualia because training depends on RFC. The entire response chain

must be activated so that the actual quale neurons, Q, can form synaptic

connections with the response neurons, R.

S->Q->P->R // Initial response path goes through consciousness, (slowly)

Q->P->R => Q->R  // A direct unconscious path created during REM sleep

S->Q->P->R => S->Q->R // Now, fast reflexes will occur when awake

Since physical motion in response to imagined stimuli could be dangerous,  

physical motion is blocked at the pons during REM sleep:

W= 1; S(t), S->Q->P ; P & W -> R; R(t) = S(t) // awake response

W= 0; S(t), S->Q->P ; P & W -> R; R(t) = 0 // sleep paralysis

Since chains of conscious thought are slow, "tokens" arise by  RFC

S->P0 -> P1->P2...->Pn-1->Pn -> R => S->P0->K->Pn->R

Tokens can predict physical motion for coordination:

A sequence of positions is normally followed during motion.

The body supplies the proprioception as the muscles move.

The slow conscious practice relies on the continuous feedback

loop through each Rn to the next Sn+1: 

S0(t) = S1(t+1) = S2(t+2)... Sn(t+n)

S0->R0, S1->R1... Sn->Rn 

The trained response can presume a token sequence, simulating

the movement without waiting for long nerves to relay each position.

Responses so programmed by the token sequence will be much faster: 

S0->R0, S1->R1... Sn->Rn =>


K1->R1 ... Kn->Rn

Note that the spinal chord is very slow,  but the commissures connecting

the cerebral hemispheres,  eg corpus callosum, are also long and slow


Stroke victims are sometimes unable to believe that a limb is paralyzed. 

Here it is asserted that token sequencing explains anosognosia.  A person

is usually not aware of the actual sensory sequence, S1...Sn, of a motion

which arrive after the motion is complete.  Instead, the sequence of tokens,

K1...Kn, predicts the movement whether it actually happens or not.  The

patient feels the same way about moving the paralyzed limb as normally

happens with unaffected limbs,  and so displays anosognosia.

Verbal Process (VP):

Words first arise as sensory-activated tokens for communication


The first words would be imitative gestures or onomatopoeia. Since words

are tokens, they can link to anything in GW.  Words evolved into arbitrary

sounds, signs, and symbols.  The evolution of the written words can be

seen from pictures, to symbolic strokes, to signs and letters.


Words assemble into memes


Memes pass from individual to individual, and evolve for success in


M -> P => M' -> P'

Useful vocabulary appears limited to symbols passing through the memes

m(V) = m(M)

Here it is asserted that the VP in non-human species, if any, is limited

to communication and so is subject to this limit of active memes

Symbolic Threshold, (ST):

In humans VP further evolves to internal communication ie thinking in words

A word often relates more to other words than to an original perception which

the word symbolizes. Terrence Deacon describes the attainment of this stage

the Symbolic Threshold, (ST). 

V <-> V 

The ST is a revolutionary event in human evolution.  High intelligence and

brain size is justifiable for a symbolic species,  but for "lower" animals, who

cannot use symbols in the same way, increasing brain size and maturity time

does not necessarily increase fitness.  Over 25% of the blood supply must go

to the human brain. Human infants are born helpless and take years to even

walk.  Human intelligence must be compensating for these disadvantages.

Crossing the Symbolic Threshold, useful vocabulary  becomes unlimited

because words only need to relate to each other and not to a meme among

many individuals


Supremacy of Verbal Process (SVP):

Words must be motivational to be useful; facts known only from words must

cause a similar response as a sensory experience.  Here it is asserted that

the verbal process uses exactly the same pathways to trigger actions as the

senses.  If so, the VP can stimulate perception and even qualia ; the VP can

overcome the FR. Thus, the VP functions exactly like the dreams of REM

sleep, creating the equivalent of sensory experience by activating the very

same cells as the incoming sensory neurons.

V -> Q  // In humans, words can overcome the perception of reality


Although the cerebral hemispheres start out about equal,  a sequence of tokens

will be fastest if confined to one hemisphere.  This is because the corpus callosum

is relatively long and slow (10 cm+).  RFC would shift a set of tokens which

predominately relate to each other, toward one hemisphere.  This would not happen

for a token set predominated by input and output from different hemispheres.

Writing L & R for any token in left or right:

L->R->L => L->L->L

Of course, the inverse sequence tends to shift back:

R->L->R => R->R->R

Given that the tokens mostly lead to each other, essentially randomly, the likelihood

of any given set of X->Y->X going one way or the other, depends on the size of the

existing populations of X and Y.  The result, (tested by simulation in Listing 1), moves

almost all tokens into one side or the other, quite rapidly. 

Left-brain vs Right-brain

Crossing the ST separates human from non-human species. The increase of

m(V) must have been quite sudden. The cerebral cortex enlarged quickly,

(on an evolutionary time scale), to support exponential growth in the VP.

Although only one hemisphere is required for a verbal process,  the hemispheres

remain about equal in size and structure.  A likely cause is the structure of the

gene sequence, mandating the left and right to be mirror images like two kidneys.

While the VP is mainly sequenced by grammar and descriptive requirements, the

non-verbal cortex is not restricted. This allows near matches to control sequencing.

"Free word association" is an example of near-match sequencing in the dominant

hemisphere. Here it is asserted that the subdominant hemisphere normally functions

in the same, "near-match", mode,  {s} -> R -> {S} , with different results than the

verbal hemisphere. 

Magic, Spirituality, etc , the Spiritual Process, SP

The symmetry of the hemispheres implies that the SVP in the left hemisphere

might be mirrored by a similar effect in the right hemisphere.  Here it is asserted

that tokens in the non-verbal hemisphere can also overcome FR,  and so be

supreme over qualia, as well:


The partial matches in the non-verbal hemisphere create their own symbolic process

so that totems, archetypes, icons etc can affect perception and behavior. These

partial matches not only recall memory,  they can create perception and memory,

(falsely), and cause emotion and belief without the usual "rationality" provided by

the VP in the opposite hemisphere.

The evolutionary success of VP beyond the ST is so great, however, that intelligence

and vocabulary increased rapidly by natural selection, despite these drawbacks


The archetypes, communicated between individuals, evolve along with verbal memes

into Magic, Art, Spirituality etc

I -> P => I' -> P'

The supremacy over qualia of the SP creates the laws of Sympathetic Magic and

Contagious Magic defined by Sir James George Frazer in the Golden Bough.  A

symbol can influence perception, and so appear to control reality, by its association

with desired results:

{s} -> {S} -> I -> Q

Here it is asserted that no other process is necessary to create the experience

of conscious awareness of reality and of the supernatural.

The next section will discuss volition,  and the perception of souls and spirits as

entities which can be independent or separable from a functional body and brain.

(More to come)