Basic Formulas


5 Mar 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)

A charge gradually accumulates across the cell membrane.

Reaching a certain trigger level,  the energy is released in an

avalanche. Then the cycle repeats.

e = (1/D) k ; e < T // phase 0

e = -(1/D)  ; e > 0  // phase 1


e = cell membrane voltage

T = threshold voltage to trigger transition to phase 1

k = metabolic rate for gradual voltage increase, (k<1)

1/D = integral over dt ( t being time)

Neurons thus discharge rhythmically,  if left undisturbed.  A neuron

can fire earlier, however, if stimulated by other sources of energy,

such as light, chemical reaction, pressure, electric discharge, etc.

A discharging neuron, connected to another neuron by a synapse,

can stimulate the other neuron to fire soon after:

S->R // 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 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 gate is 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 by using many

more connections in order to achieve a useful 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 cell body would delay

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".  The

effect depends on relative timing,  not just energy.   The synapse can

strengthen 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 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.

They could do so by simple diffusion, depending on the electric field

of each cell to attract the polarized molecules.)

"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 


Trehub defines a similar structure as a "synaptic matrix"

with input X and output Y, arranged orthogonally, in a

"crossbar" pattern.  Algebraically, the register |S<->P|  is:

 {S0}<->P0...{Sn}<->Pn |S<->P|

|S<->P| ; {Sn}(t)  = Pn(t+1)

Since neurons are relaxation oscillators,  the speed of each

matching response will depend on the magnitude of its 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 when only partial data is available.


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

Here it is asserted that Auto selection is the process for learning,

neonatal development, healing after injury, and plasticity. Chemical

signals define the class of cells, R0..Rn, to which the axon of S

may connect, and activity selects the particular cell, R

The register detects the best match for an input because the proper

neuron responds. If another neuron with a partial match is more sensitive

an inaccurate response will occur.  The register is most accurate when

all the neurons are equally sensitive. Sensitivity, as described above,

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

so more O2 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 constant levels. 

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

I = (1/D)(R(t)-M(t)) // integrate diffusion & 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 // counterintuitively,  activity increases external O2 


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.  The 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 symbolized, here,  by

P, Q , and V, respectively.

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

We are aware of perceptions,  but not aware of much, (or most) sensation.


Senses can become conscious, even while stimulating reflexive responses


Consciousness normally guides action by integrating sensory information

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

A conscious response will be inherently slower than a reflex

T(S->P->R) > T(S->R)

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 & 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->R0,P0 ; P0->P1...Pn->R1

S(t) = R0(t+1) = Pn(t+n) = R1(t+n+1)

R0(t) = S(t-1)

R1(t) = S(t-1-n)

R(t) = S(t-1) + S(t-1-n) // "Conscious" response also arrives,  but later

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

(P  & ~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=0  // if S is not changing at all, it stops setting Q

Mammals need REM sleep. After dreaming is deprived, "REM Rebound"

occurs, replacing lost time. REM is theorized to "consolidate memory".

Here it is asserted that REM sleep trains faster, automatic, responses to

qualia  by RFC.  During "consolidation", the stimulus-response chain must

be activated to make any change.  If the original response was conscious,

the subject will experience dreaming.  The entire stimulus-response chain

becomes active so that the actual quale neurons, Q, can form synaptic

connections with their 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

Body movement in response to imagined stimuli could be dangerous,

however, physical motion is blocked at the pons during REM sleep.

(Failure of this mechanism is Sleep Behavioral Disorder.)

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

W= 0; Q(t), Q->P ; P & W -> R; R(t) = 0 // dreams, but 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 ideographs and finally 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 relates more to other words than to an original perception which the

word symbolizes. Terrence Deacon describes the Symbolic Threshold, (ST),

which seems to correspond to this phase. 

V <-> V // Words are responses to other words when the ST is reached

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.

Upon crossing the ST, useful vocabulary becomes almost unlimited.

Because words only need to relate to each other,  and do not need to belong

to popular memes which many individuals share, 


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 only one hemisphere.  This is because

the corpus callosum which connects the hemispheres is slow because it is

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

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

for a token set predominated by their input and output which would arise in

different hemispheres.  A set of words beyond the ST are tokens which

most commonly link to other words in the set and so move to one side.

Writing L and R for any token in left or right hemispheres:

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, with only a small initial bias.

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 that the left and right develop as

mirror images, like the two kidneys.

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

the non-verbal cortex is not so 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,  with different results than the verbal hemisphere. 

 {s} -> R -> {S}  //  A near-match in subdominant 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

consequent 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 effects describe by

Sigmund Freud as the "omnipotence of mind".  These effect follow the laws of

Sympathetic Magic and Contagious Magic defined by Sir James George Frazer

as the basis of primitive belief in "The Golden Bough".  A symbol can influence

perception, and so appear to control reality, by its association with desired


{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)