Basic Formulas


8 Nov 2018

Harland Harrison

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

Neurons discharge rhythmically if otherwise undisturbed

but a neuron can stimulate other neurons to fire earlier:


(Above represents a synapse, not the logic implemented - See Appendix X)

The energy discharged is larger than the required stimulus

Neurons have a "refractory period" after depolarizing during which

they cannot fire again, even if stimulated

For any neuron X , 

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

The frequency of neurons is incredibly slow for logic systems.

Neural activity is measured in milliseconds; silicon gates

function in nanoseconds.

Neurons are massively parallel.  The "fan out" of a neuron

 is of the order of 10^4 ; the fan out of silicon gate is about 10

Neurons as logic gates could form Turing machines:

The necessary functions are AND/OR, NOT, amplification, delay,

and a logical constant, implemented by eg NAND gates in silicon

(formally: disjunction/conjunction, inversion, ∩ / ∪, )

Multiple synapses,  each stimulating the same cell, provide AND/OR


Depending on sensitivity,  this implements "S1S2->R", where both

S1 AND S2 are necessary, (a coincidence detector), or else,

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

Refractory time as well as inhibitory synapses provide NOT

For a neuron X synchronized at frequency 1

X(t)  = ~ X(t+1)

(If triggered by input, X fires now and so cannot fire a little later, 

but if not stimulated now, it will fire when its full time is up)

Given that basic synapses and cell bodies can implement Turing

logic, that neurons are very slow and parallel rather than serial,

and presuming that rapid response is advantageous to any animal,

it follows that evolution would make that basic structure the

definitive logic.

Hebbian plasticity: 

Successful stimulus-response strengthens synapses

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

And unsuccessful synapses weaken

S(t) ≠ R(t+1) => S ~-> R

"Conditioned reflexes" arise by associating simultaneous stimuli

C(t)=U(t);  U->R =>  C->R

Rule of Fastest Connection (RFC) :

RFC minimizes reflex time, so only the fastest path remains

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

 => S -> B -> R ;  S -> A ~-> R

RFC trains 'lower' reflexes if pathways exist

The time, T(S->R), is inherently less than T(S->P->R)

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

A repeated pattern being detected can be reconstructed

{S}(0) = {S}(4) = {S}(8)...  ;  {S} -> R -> P->C


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

writing {S} -> C ; C -> {S}  as {S} <-> C

 {S} -> R -> P-> C => {S} <-> C

A near match can reconstruct the full pattern

{S} -> R =>  {S} -> R -> {S}

"{s} {S}" // given that s is a subset of S

{s} -> R -> {S}

The speed of near-matching pattern depends on size of the match

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

If there are multiple possibilities,  the best match will fire first

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)

Self 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} -> R => {S}<->R

Conscious thought is a "Global Workspace", GW.  GW is

presumedly in associative cortex and cross-connectable so that

that any element might connect to any other.  Here, it is asserted

that the elements in GW are individual neurons, and that their

interconnections are individual synapses

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

Only about 10**14 synapses exist but the workspace is also quite sparse.

GW contains perceptions, qualia, (see below), and abstractions,

like words, here assigned to individual neurons, designated by

P, Q , and V types, respectively.

Consciousness can retain information across time and circumstance

by looping through the workspace,  regenerating the information

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

RFC trains 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

Input from the senses also enters consciousness


Consciousness normally guides action by integrating sensory information

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

Senses may remain conscious, even while stimulating response

S->P; S->R

Force of Reality, FR, creates "qualia".   A quale is the essence of sensation. 

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

The qualia appear as stubborn, uncontrollable, elements of consciousness

S -> Q -> P ; P <-> P ; P~->Q

"Offline processes" can suspend the FR: REM sleep dreaming,

meditation and hallucination let thoughts cause qualia

P <-> Q

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

"S(t) != S(t+1) ->" S-> Q  // only if S is actively changing, does it set Q

REM sleep is necessary for RFC to train automatic responses to qualia 


P->Q->P->R => P->Q->R


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

physical motion is blocked at the pons during REM sleep:

W= 1; S->Q->P ; P & W -> R

W=0;  P->Q->P ; P & W -> R => R=0

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. 

Token sequencing explains this.  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.  So the patient feels the same way about

moving the paralyzed limb as normally happened and reports anosognosia.

Verbal Process (VP):

Words first arise as sensory-activated tokens for communication


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

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.

Supremacy of VP:

Words must be motivational to be useful, overcoming FR

V -> Q


Words assemble into memes


Memes pass from individual to individual, and evolving for success

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, is limited

to communication and so subject to the limit of active memes:

m(V) = m(M)

Symbolic Threshold, ST:

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


Useful vocabulary is now unlimited because words only need to relate

to each other and not to a meme among multiple individuals



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 (10cm+).  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. 

Evolution of Supremacy of VP makes words potentially motivational

V -> Q

This means that facts described in words can be used for actions, equally with

facts actually perceived. The assertion, here, is that the verbal process uses

exactly the same pathways to trigger actions as a stimulus-response action.

The latter can be either conscious or unconscious. So RFC implies that only

one trigger, Q, will implement perceptual, verbal, or automatic, responses

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 the verbal process,  (usually the left), both

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 also means that the Supremacy of the VP in the

left hemisphere will be mirrored by a similar effect in the right hemisphere.  Tokens

in the non-verbal hemisphere will also be supreme over qualia:


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

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

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

(More to come)