Mathematical Excitation Propagation Model

Intro

The excitation propagation model describes mathematically how excitation signals are exchanged between nodes.

This model exists mathematically, per se.

The fact that it is linked to a kind of interpretation is incidental.

Its possible algorithmic implementation is discussed elsewhere.

Comments in gray include interpretation notes and algorithmic notes.

The mathematical model is described and organized in 3 successive steps

1. The passive jelly
2. The active unexcited jelly
3. The active and excited jelly

The passive jelly

The description of the passive jelly includes nodes and node links.

Node

A node is just something that may be linked to other nodes. Nothing more to say.

A node represents a concept, apart from any kind of key reference, and apart from any kind of language anchors. These keys and language anchors are other parts of the assothink model, not described here.

The number of nodes is N.

Values for N might be of the magnitude of 10⁵.

A given node is written n_i (0<=i<N).

Two kind of links between nodes are defined: fuzzy links and qualified links.

Fuzzy link

A fuzzy link is defined by 2 nodes identifier (n_i and n_j) and a permeability measure (p_ij).

The permeability values are real values in the [0,1] range.

The zero value of a fuzzy link permeability reflects the absence of fuzzy link.

The permeability indicates a proximity between concepts.

Most permeabilities are 0, and this reflext unlinked concepts.

High permeabilities, close to 1, indicate strong proximity.

There is no reflexive links, thus p_ii=0 (or p_ij=0 when i=j).

A concept has no link to itself.

A fuzzy link is always symmetrical p_ij=p_ji.

To summarize, the fuzzy links may be represented by a square symmetric matrix, with real values in the [0,1] range, and 0 values in the diagonal.

Qualified link

A qualified link is described by 2 linked nodes identifier (n_i and n_j) and a linking node identifier (n_k).

A presence (absence) of such link is described by the values q_ijk (0<=i,j,k<N).

This indicates that two concepts are connected in a way represented by a third concept. For instance If a concept n_k describes the idea known by the english word "subclass of", if a concept n_i describes the idea known by the english word "dog", and if a concept n_j describes the idea known by the english word "animal", then the value of n_ijk should be 1. Most q_ijk values are zero.

Non Insularity

The math model specifies that no node is isolated.

This means that for any node n_i, there is either at least a p_ij non-zero value, or at least a q_ijk value non-zero.

No concept has a recognizable existence if it is not linked to other concepts. No concept is unreachable.

This also means that it is impossible the define a subset of the node set having internal connection, but no connecton to nodes ouside the subset.

No set of concept is isolated from the global mas of concepts. No concept set is unreachable.

This also implies that between any pair of nodes if the node universe, there is at least one link path connecting the first node to the second node.

Active unexcited jelly

To the passive jelly, the following components are added:

• the idea of time (t) evolution. The model becomes dynamic.
• a excitation level for any node (g_i). This excitation level varies in time

The average excitation level is set to 1, and this is true at any time:

Échec de l’analyse (erreur de syntaxe): {\displaystyle \forall \; t \; : \sum_{0 \le i < N} g_i(t) = N }

The active unexcited jelly is described by this key differential equation:

${\dot {g}}(i)=\sum _{0\leq j<N}p_{ij}(g_{j}-g_{i})+\sum _{0\leq j<N}\sum _{0\leq k<N}q_{ijk}g_{k}(g_{j}-g_{i})$

Hence (long but easy to prove):

Échec de l’analyse (erreur de syntaxe): {\displaystyle \forall \; t \; : \sum_{0 \le i < N} \dot g_i(t) = 0 \sum_{0 \le i < N} g_i(t) = N }

Active excited Jelly

To the active unexcited jelly, external excitation is finally added.

The unexcited jelly was a kind of brain jeely without external input. The excited jelly includes external excitations of concepts by any communication means.

The excitation level is now e_i(t) instead of g_i(t) - and generalizing it.

The external signal is described by a time function for any excited node s_i(t).

Concepts in the artificial brain are excited by external signals. Words, images, sounds are likely to be translated into such signals.

If the average excitation level has to be maintained, s_i(t) has to be replaced by r_i(t), using:

$S(t)=\sum _{0\leq i<N}s_{i}(t)$

and
$r_{i}(t)=s_{i}(t)-{\frac {e_{i}(t)S(t)}{N}}$

which implies:
$\sum _{0\leq i<N}r_{i}(t)=0$

The more general differental equation becomes:

${\dot {e}}(i)=r_{i}(t)+\sum _{0\leq j<N}p_{ij}(e_{j}-e_{i})+\sum _{0\leq j<N}\sum _{0\leq k<N}q_{ijk}e_{k}(e_{j}-e_{i})$

And it may be proven again (longer but still easy to prove) that:
Échec de l’analyse (erreur de syntaxe): {\displaystyle \forall \; t \; : 0 \le e_i(t) \le N \quad \sum_{0 \le i < N} \dot e_i(t) = 0 \quad \sum_{0 \le i < N} e_i(t) = N }

Algorithmic translation of the model

Many ways to translate the model described above into artificial intelligence systems may be designed.

This includes software choices, and possibly hardware choices.

This question is discussed elsewhere.