Search Results for “boolean network”

Source: Boolean network

A Boolean network consists of a discrete set of Boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those variables and output that determines the state of the variable it is assigned to. This set of functions in effect determines a topology (connectivity) on the set of variables, which then become nodes in a network. Usually, the dynamics of the system is taken as a discrete time series where the state of the entire network at time t+1 is determined by evaluating each variable's function on the state of the network at time t. This may be done synchronously or asynchronously.
Boolean networks have been used in biology to model regulatory networks. Although Boolean networks are a crude simplification of genetic reality where genes are not simple binary switches, there are several cases where they correctly convey the correct pattern of expressed and suppressed genes.
The seemingly mathematical easy (synchronous) model was only fully understood in the mid 2000s.

Classical model

Boolean

= Attractors

Boolean

network

Stability

network

Boolean

network

, critical if

K
=

K

c

{\displaystyle K=K_{c}}

, and unstable if

K
>

K

c

{\displaystyle K>K_{c}}

.
The state of a given node

n

i

{\displaystyle n_{i}}

is updated according to its truth table, whose outputs are randomly populated.

p

i

{\displaystyle p_{i}}

denotes the probability of assigning an off output to a given series of input signals.
If

p

i

=
p
=
c
o
n
s
t
.

{\displaystyle p_{i}=p=const.}

for every node, the transition between the stable and chaotic range depends on

p

{\displaystyle p}

. According to Bernard Derrida and Yves Pomeau
, the critical value of the average number of connections is

K

c

=
1

/

[
2
p
(
1
−
p
)
]

{\displaystyle K_{c}=1/[2p(1-p)]}

.
If

K

{\displaystyle K}

is not constant, and there is no correlation between the in-degrees and out-degrees, the conditions of stability is determined by

⟨

K

i
n

⟩

{\displaystyle \langle K^{in}\rangle }

The network is stable if

⟨

K

i
n

⟩
<

K

c

{\displaystyle \langle K^{in}\rangle
, critical if

⟨

K

i
n

⟩
=

K

c

{\displaystyle \langle K^{in}\rangle =K_{c}}

, and unstable if

⟨

K

i
n

⟩
>

K

c

{\displaystyle \langle K^{in}\rangle >K_{c}}

.
The conditions of stability are the same in the case of networks with scale-free topology where the in-and out-degree distribution is a power-law distribution:

P
(
K
)
∝

K

−
γ

{\displaystyle P(K)\propto K^{-\gamma }}

, and

⟨

K

i
n

⟩
=
⟨

K

o
u
t

⟩

{\displaystyle \langle K^{in}\rangle =\langle K^{out}\rangle }

, since every out-link from a node is an in-link to another.
Sensitivity shows the probability that the output of the Boolean function of a given node changes if its input changes. For random Boolean networks,

q

i

=
2

p

i

(
1
−

p

i

)

{\displaystyle q_{i}=2p_{i}(1-p_{i})}

. In the general case, stability of the network is governed by the largest eigenvalue

λ

Q

{\displaystyle \lambda _{Q}}

of matrix

Q

{\displaystyle Q}

, where

Q

i
j

=

q

i

A

i
j

{\displaystyle Q_{ij}=q_{i}A_{ij}}

, and

A

{\displaystyle A}

is the adjacency matrix of the network. The network is stable if

λ

Q

<
1

{\displaystyle \lambda _{Q}<1}

, critical if

λ

Q

=
1

{\displaystyle \lambda _{Q}=1}

, unstable if

λ

Q

>
1

{\displaystyle \lambda _{Q}>1}

.

Variations of the model

= Other topologies

=
One theme is to study different underlying graph topologies.

The homogeneous case simply refers to a grid which is simply the reduction to the famous Ising model.
Scale-free topologies may be chosen for Boolean networks. One can distinguish the case where only in-degree distribution in power-law distributed, or only the out-degree-distribution or both.

= Other updating schemes

=
Classical Boolean networks (sometimes called CRBN, i.e. Classic Random Boolean Network) are synchronously updated. Motivated by the fact that genes don't usually change their state simultaneously, different alternatives have been introduced. A common classification is the following:

Deterministic asynchronous updated Boolean networks (DRBNs) are not synchronously updated but a deterministic solution still exists. A node i will be updated when t ≡ Qi (mod Pi) where t is the time step.
The most general case is full stochastic updating (GARBN, general asynchronous random Boolean networks). Here, one (or more) node(s) are selected at each computational step to be updated.
The Partially-Observed Boolean Dynamical System (POBDS) signal model differs from all previous deterministic and stochastic Boolean network models by removing the assumption of direct observability of the Boolean state vector and allowing uncertainty in the observation process, addressing the scenario encountered in practice.
Autonomous Boolean networks (ABNs) are updated in continuous time (t is a real number, not an integer), which leads to race conditions and complex dynamical behavior such as deterministic chaos.

Application of Boolean Networks

= Classification

=
The Scalable Optimal Bayesian Classification developed an optimal classification of trajectories accounting for potential model uncertainty and also proposed a particle-based trajectory classification that is highly scalable for large networks with much lower complexity than the optimal solution.

References

Dubrova, E., Teslenko, M., Martinelli, A., (2005). *Kauffman Networks: Analysis and Applications, in "Proceedings of International Conference on Computer-Aided Design", pages 479-484.

External links

Analysis of Dynamic Algebraic Models (ADAM) v1.1
bioasp/bonesis: Synthesis of Most Permissive Boolean Networks from network architecture and dynamical properties
CoLoMoTo (Consortium for Logical Models and Tools)
DDLab
NetBuilder Boolean Networks Simulator
Open Source Boolean Network Simulator
JavaScript Kauffman Network
Probabilistic Boolean Networks (PBN)
RBNLab
A SAT-based tool for computing attractors in Boolean Networks