Short+cuts+or+Modelling+Formalisms

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=Short-cuts or Modeling Formalisms=

(**Concepts**: ordering of states, stable and non-stable equilibria, limit cycle, bistability, deterministic chaos, bifurcation diagram, decomposition into sub-systems, combinatorial effects: "greater than atoms in the universe")

In order to avoid needing to fully specify the next state function (NSF) of a finite state machine (FSM), different model [|formalisms] are used which **short cut** the need for writing down a whole transition table. Such short cuts can take different forms, the most common of which we will discuss here:

//1) Ordering of States//
In this case one considers states which can be ordered, such as real numbers or integers. The states can represent, for example, amounts of something. Then, the next state function of a model is defined by a numeric function which uniquely maps one number to another number. The function takes a variable, which is substituted by a state of the system, and it must be //valid for all// numbers the model is concerned of.

This is a huge short-cut since a transition table of an FSM is replaced by a [|numeric function]. On the one hand, this short-cut imposes a lot of restriction on the model (we will see this later), and the number of variables tends to be minimized. On the other hand, this short-cut allows us to relax the restriction of finiteness of the number of states because the next state function can be valid for an infinite number of values (states). Examples of such modeling formalisms are:
 * **Maps** or [|Difference equations], where time is discrete and the next state function is defined in terms of the current state.
 * **[|Ordinary Differential Equations]** (ODEs), where time is continuous and the next state function is defined in terms of the current state, the direction of change, and the amount of change.

__A bit more on ODEs__
When studying 2D ODE systems we can classify various kinds of equilibrium points (or attractors, see the graph at the right):
 * Stable equilibria**
 * Stable node (3)
 * Stable spiral (6)
 * Unstable equilibria**
 * Unstable node (1)
 * Unstable spiral (4)
 * Saddle point (2)

These attractors can be characterized in terms of the values of the eigenvalues of the matrix of partial derivatives at equilibrium (**Jacobian matrix**). To study nonlinear 2D ODEs, we can use phase-plane analysis (state-space analysis). This gives us a set of observables: (A more elaborate description of the use of phase-plane analysis when studying ODEs is given in the reader [|"Mathematics for Biologists"].)
 * **Nullclines** are sets of states where the time-derivative of the corresponding variable is zero.
 * A **trajectory** is the set of states visited from a specific initial condition at time = 0.
 * A **vector field** gives the direction of change at selected states.
 * **Attractors** are states ór sets of states visited after "enough" time: **Fixed points, Limit cycles**[[image:binf/logist.png width="340" height="345" align="right" caption="Bifurcation diagram of logistic map"]]
 * A **bifurcation diagram** displays attractors as a function of a parameter of the model.

In 2D ODEs we can get interesting phenomena such as: > In 3D-ODEs we get the emergence of a new type of phenomenom: > The main characteristic of deterministic chaos is that a small change of state at a certain time will lead to an arbitrarily large deviation after some ("sufficient") amount of time. In other words, the system is highly sensitive to initial conditions.
 * **Bistability** where we have two stable equilibria and one non-stable equilibrium for a given parameter range
 * **[|Deterministic chaos]** where trajectories follow non-periodic patterns.

Deterministic chaos in MAPS is also possible in a single variable system (1D), most famously in the **[|logistic map]**. This gives the famous bifurcation diagram with **period doubling** (see figure), where for certain values of the intrinsic growth parameter (//r//) one can find an infinite number possible equilibrium densities of N.

//2) Decomposition into sub-systems//
In this case a FSM is composed of a number of simpler FSMs which all have a (small) number of possible states and a fully specified transition table (NSF). While the smaller FSMs have few states, the number of states of the whole FSM can be enormous through **[|combinatorial]** effects. The NSF of the simpler FSMs can be defined easily due to the small number of its states. The large FSM can then be constructed simply through connecting a number of simpler FSMs in such a way that a smaller FSM takes as its input the output of another smaller FSM. In this way, a large FSM can be fully defined without explicitly specifying the NSF for the whole FSM. Note that this is a significant short-cut: If the number of states of a smaller FSM is n, and the number of smaller FSMs is N, then the number of states of the whole FSM is n N, which is the number of combinations of the states of smaller FSMs. The NSF of the whole FSM must specify its next state for each of the n^N states. The number of n N soon becomes large when n>0 and N>>1, and thus explicitly specifying the NSF of the whole FSM would require the construction of a very large transition table, while the above short-cut dispenses this by only defining a transition table of the smaller FSMs. Furthermore, the following should be noted: If the number of states of an FSM is n N, the number of all possible NSFs is (n N )^(n N ), which easily becomes far **greater than the number of [|atoms] in the universe** (thus, it is impossible to study all of them). However, if an FSM is constructed from a number of a smaller FSMs in a certain predefined way, the possible NSFs of the whole FSM is restricted to a subset of all the possibilities. While this restriction would not necessarily enable one to study all the possible NSFs of the subset, the restriction does impose a certain constraint/structure on the behavior of the FSM.

Examples of such modeling formalisms are:
 * **[|Cellular automata] (CA)**
 * **[|Boolean networks]**
 * **[|Neural networks]**

In the following sections we focus on CAs and Boolean networks. We discuss their properties and short-cuts relative to full FSMs and compare them to ODEs and MAPs (we assume that ODEs and MAPs are better known from other courses, e.g. **[|Theoretical Ecology]**).

Next: Intro CA

(CHANGELOG 2014-2015) - Added link to systems biology book (analysis of ODEs) - Added definition of deterministic chaos