A logical constraint-based approach to infer and ...

A logical constraint-based approach to infer and explore diversity and composition in Thresholded Boolean Automaton Networks Quoc-Trung Vuong1, Roselyne Chauvin2, Sergiu Ivanov3, Nicolas Glade1, Laurent Trilling1 1.TIMC-IMAG, CNRS UMR5525, Univ. Grenoble, France / 2.Radboud Univ. Netherland / 3.IBISC Lab. Univ. Evry val d’Essonne, France [email protected] / [email protected]

What's a TBAN ? A

The new state is computed by ν =H (∑ j ∈[1. . D ] w ij ν (t ) i

Thresholded

Boolean

composed

of

1

to

−Θi )

and H (ν)=1 if ν > 0 , 0 otherwise

Automaton

Network is a graph of interactions G=(W,Θ)

(t−1) j

wi1

n1

D

n2

nodes. At a given time, a node ni

wi2 wi3

n3

is active or not. Its Boolean state

the state of node ni at time t ν ∈{0,1 } Θi ∈[−( D+1) , D+1] the threshold of activation of ni w ij ∈[−D , D] the weight of interaction of nj on ni (t ) i

Θi

ni

with

wij

is updated by comparing the sum of

nj

inputs (arcs of weight wij) to its

The dynamics of TBAN always tend to

threshold of activation (Θi).

stationary states (fixed points or limit

TBAN are mainly used in two fields:

cycles). The stationary states or even

- In biology, to model biological regulatory networks.

time patterns on specific nodes could

- In computer sciences, to design artificial neural networks.

match observed behaviors : cell types, oscillators .

Yielding the set of non-redundant TBAN with a specific behavior

Particular behaviors (stationary states viewed on one or several nodes) can be obtained by using particular TBAN having specific initial conditions. Given a number of nodes, many different TBAN can display such behaviors. All may correspond to different “strategies” (in a natural system) to behave in a same way. Such collections of TBAN contain very similar networks as well as very different ones. A challenge would be to study their diversity and the different manners they work, as evolutionists and naturalists do with living species.

To do this, it is necessary to yield the - complete - set of non-redundant TBAN with a specific behavior. Here, we focus on a specific time pattern played by at least one node (ex: (10011)* played on at least node n 1) for at least one initialization.

Results

Methodology: brute force

Number of solutions (#) and

or logic in action ?

computation times for size 3 networks (10011)*

* Brute force : enumeration and simulation of

at

least

one

initialization

* Logic in action : expected TBAN are logical

Perspectives : studying how TBAN are related together in a horizontal or a

models of a – logical – specification.

vertical way. The horizontal analysis of network evolution consists in comparing

different TBAN belonging to a set, i.e. of same size and same function (ex: a

ASP (Answer Set Programming, a logical

TBAN capable of a specific behavior like playing a time pattern). A vertical

programming technology) implementation to

analysis

get the required TBAN.

would

be

to

understand

how

TBAN

can

be

composed

from

more

elementary ones (i.e. to find the rules to split or merge logical functions and adapt the parameters), or can be obtained by folding/unfolding of nodes.

1) To get the consistent TBAN from about the dynamics (time pattern) and the

on

sequence

node for at least a specific

all possible TBAN of size D

constraints defining logically both the knowledge

playing

s t Se

Folding / Unfolding

structure of a TBAN. 2) To get the set of non-redundant TBAN: First: elimination of non-canonical TBAN. network in its equivalent class with parameters nearer to 0. Second: elimination of TBAN equivalent by

label permutation (2 steps). - Building a spanning tree to order nodes. Most redundant networks are eliminated.

- Some others cannot be eliminated due to

an ambiguous ordering. Then, a more costly process called unification is applied.

Composition / Decomposition

A canonical TBAN is such that there is no other

n1: 1001110011... n2: 1000110001… n3: 1100011000...

s e d o n et 3 s

n1: n2: n3: n4:

1001110011... 1000110001… 1100011000… 1100011000...

s e d o n et 4 s

?

n 1: n 2: n 3: n 4: n 5:

1001110011... 1100111001… 1110011100… 1000110001... 1100011000...

s e d o n et 5 s

How to fold big TBAN into smaller ones and maintain the function ? Conversely, how to expand small TBAN ? How to combine two TBAN to

Two equivalent TBAN by permutation. One

of them is eliminated

using a spanning tree.

n1: 1001100100... n2: 1100110011…

Two equivalent TBAN

by permutation. Ambiguity is

solved by unification process.

n1: 1000111000... n2: 1100011100… n3: 1110001110...

obtain a specific function ? How to predict the sets of TBAN pairs that can be combined in this way ?