intermediate updated. section 2 w.i.p.; 2 figures though the example might change. Section 2 still needs more text. Rest unchanged :(
This commit is contained in:
Tom Zuidberg
179
paper.tex
179
paper.tex
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% some packages which may be useful when drafting, but should be removed before submission
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% some packages which may be useful when drafting, but should be removed before submission
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\begin{document}
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\begin{document}
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\title{Influence of update schemes on gene regulatory networks}
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\title{Influence of update schemes on boolean networks}
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@@ -81,7 +84,162 @@ Possible points to mention here:
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\section{Boolean networks}
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\section{Boolean networks}
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Explain Boolean network and what an update scheme is using the synchronous/parallel scheme. Also use figure, e.g. \cref{fig:boolean_example} to further visualize and create a better understanding of boolean networks. Also explain what chaotic behavior is due its relevance for the update schemes.
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A boolean network consists of nodes $x_i(t)$ that have a boolean state, either $0$ or $1$, at a point in time $t$. Each node has a corresponding update function $x_i(t+1) = f_i(x_1(t), x_2(t), \ldots, x_n(t))$ expressing the new state of $x_i(t+1)$. The state of the boolean network can be describe as a boolean number $x_1 x_2\ldots x_n$ where each node is replaced with the corresponding state.
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\begin{figure}
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\centering
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\draw (111.33,207.8) node [anchor=north west][inner sep=0.75pt] {$ \begin{array}{l}
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f_{1} =\neg x_{2}\\
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f_{2} =x_{1}\\
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f_{3} =x_{1} \oplus x_{4}\\
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f_{4} =x_{3}
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\end{array}$};
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\end{tikzpicture}
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\caption{Example of a boolean network with four nodes. Each vertex indicates that a node is part of the update function of the other node. For instance $x_1$ is part of $f_2$ and $f_3$.
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}
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\label{fig:bn_example}
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\end{figure}
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Consider the boolean network shown in \cref{fig:bn_example}. Assume the current state is $0011$, meaning that $x_1, x_2$ are $0$ and $x_3, x_4$ are $1$. The next state when applying the update function all at once is $1011$. Updating the network multiple times creates a trajectory, which is the sequence of the states starting from the initial state. With our example, the trajectory is $0011 \rightarrow 1011 \rightarrow 1101 \rightarrow 0100 \rightarrow \ldots$. Each trajectory in a reasonably small boolean network eventually reaches one of two scenarios. Either it falls into a state that doesn't change when updated, called attractor, or it falls into a cycle of states. In the example, there are 2 cycles of length 8 visible in \cref{fig:bn_ex_state_graph}.
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\end{tikzpicture}
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\centering
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\caption{State graph of the boolean network shown in \cref{fig:bn_example} using synchronous update scheme.}
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\label{fig:bn_ex_state_graph}
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\end{figure}
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\subsection{Notation}
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\subsection{Notation}
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Define clear notation used throughout the paper. Position of this subsection could change to be part of the Introduction instead.
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Define clear notation used throughout the paper. Position of this subsection could change to be part of the Introduction instead.
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@@ -93,16 +251,21 @@ Explain different update schemes including characteristics for behavior especial
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\subsection{Synchronous scheme}
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\subsection{Synchronous scheme}
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The Synchronous (also known as Parallel) update scheme assumes that every node is updated at once.
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The Synchronous (also known as Parallel) update scheme assumes that every node is updated at once.
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\subsection{Sequential scheme}
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\subsection{Sequential scheme}
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close to synchronous. the nodes update in a specific order and take into account the updated input node if that node had been updated before/is positioned earlier in the sequence
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close to synchronous. the nodes update in a specific order and take into account the updated input node if that node had been updated before/is positioned earlier in the sequence
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\subsection{Block-sequential scheme}
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\subsection{Block-sequential scheme}
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mix of synchronous and sequential. predefined blocks update sequential, inside a block the update follows the synchronous scheme
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mix of synchronous and sequential. predefined blocks update sequential, inside a block the update follows the synchronous scheme
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\subsection{Asynchronous deterministic}
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\subsection{Asynchronous deterministic}
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one node is updated per tick following a specific sequence
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one node is updated per tick following a specific sequence
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\subsection{Asynchronous generalized}
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\subsection{Asynchronous generalized}
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same as asynchronous deterministic with the slight change that within the sequence nodes may appear multiple times.
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same as asynchronous deterministic with the slight change that within the sequence nodes may appear multiple times.
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\section{Relevance for Gene Regulatory Networks}
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\section{Relevance for Gene Regulatory Networks}
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\label{sec:relevance_grn}
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\label{sec:relevance_grn}
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Tie the update schemes and their different outcomes or behavior to GRN.
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Tie the update schemes and their different outcomes or behavior to GRN.
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@@ -116,18 +279,6 @@ References: \cite{schwab2020concepts}\cite{aracena2009robustness}\cite{bornholdt
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\begin{figure} % The starred version uses both columns; unstarred only one column
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\centering
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\includegraphics[width=3in]{4bit BN.png}
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% TIP: Ensure the original image file has approximately the right dimensions
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% (if using matplotlib, specify correct figure size) so that the image is not rescaled too brutally.
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\caption{Boolean example (image will be changed!).
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% NB: The '~' inserts a non-breaking space, ensuring 'Ref.' is never separated from its number
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Simple boolean network example; a graph of the network and most likely a table as well for the updated states $x(t) \rightarrow x(t+1)$ (currently missing)
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}
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\label{fig:boolean_example}
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\end{figure}
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\begin{figure*} % The starred version uses both columns; unstarred only one column
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\begin{figure*} % The starred version uses both columns; unstarred only one column
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\centering
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\centering
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Reference in New Issue
Block a user