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TODO List
  • more detail from Q0 (Game of life etc) in order to make more clear this idea of what a model is and what models can tell us



So, what are models?


In the previous sections we have discussed various modeling formalisms. However, can we now define what a model is? The only thing we can really say at this point is that:
A is a model of B when by studying A we (hope to) get to know something about B!

Unfortunately this sounds rather trivial, so is it of any value to us?
Well, we can refine this statement to some extent:

How to make a model of B?

To do this one needs a “mapping” of B in variable/states such that it satisfies requirements of dynamical system unique next state function < I, S,O,SUM,OMEGA >. To this end different model formalisms can play different roles:
  • For cellular automata: (Occupation of) patches of space versus individuals
  • For maps and ODEs: Fixed set of continuous variables versus populations or concentrations

Furthermore, we note the following things:

i) A model should NOT be as general as possible
external image gib4.jpg
"The rock of Gibraltar is a model for my brain." (Ashby). There is a model relation between a brain and the rock in the sense that they both persist for some time.

ii) A model should NOT be as similar as possible
In that case "A is the best model for A."

iii) Models can relate to the things they model in different ways
A model therefore needs to map B into states such that it meets the unique next state function, and this can be done using short cuts (e.g. ordering and/or contracting of possibilities). How the model then relates to the issue it models depends on how concepts are defined for that the modeling formalism chosen is useful. In this light we can have different model relations:
  1. One-to-one model: Matching one thing with a model (modeling a single thing, e.g. weather).
  2. Modeling a class of things: Models with parameters that represent a whole class of cases, and are (to a certain extent) generic
  3. Modeling a class of models: Using specific cases to characterize a class of cases, i.e. using a paradigm system. Often this involves taking extreme cases in order to learn more (as an example of, or a worst or best case scenario).

"As simple as possible, but not more than that" (Einstein)

iv) What needs to be modeled and explained?
How to identify which patterns need explaining?
i) How special is it? -> robustness and comparison model formalisms
ii) How generic is it? -> characterizing generic behaviour (sampling, classification, order parameter)
We discuss this with reference to generic CAs and B-cell nodules.



References
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(CHANGELOG 2014-2015)

- Removed reference to daffodil-page




(NOTES FROM COURSE 2006-2007)

SO WHAT ARE MODELS?

Only thing we can say:
A is a model of B when by studying A we (hope to) get to know something about B

Is this a trivial statement? Well ....

A model should NOT be a general as possible:
Ashby: "The rock of Gibraltar is a model for my brain." (They both persist for some time).

A model should NOT be as similar as possible:
"A is the best model for A."

A model needs to map B into states such that it meets the unique next state function, and this can be done using short cuts (e.g. ordering and contracting possibilities).

(LECTURE 2)
Model relation: define concepts such that model formalism is useful.

a) One-to-one model: matching one thing with a model (modelling a single thing).
b) Modelling a class of things: models with parameters
c) Modelling a class of models: using specific cases to characterize a class of cases => paradigm systems: often take extreme cases in order to learn more (an example of ...).

Einstein: "As simple as possible but not more than that."

How to identify which patterns need explaining?
i) how special is it?
ii) how generic is it?


Review Q1: non-advanced: Classical Growth

Point one: Logistic equation is an arbitrary choice: 1st Taylor expansion of Verhulst: growth should not come out of air and go into air.
Putting in more biological interpretation: other per capita growth curves, difficult to interpret especially concave one.
However this is exactly the one which is most natural in the CA because growth becomes dependent on mesoscale patterns.

Once the field is full one gets chaotic behaviour although constancy globally. This is different from the MAP chaos which with takens reconstruction can be shown to fall within two dimensions (line). In the CA it is highly dimensional chaos.
Message is also that the way we set up experiments determines part of the outcome: e.g. wet growth experiments, and that we often have ODEs in mind, and therefore set up experiments well-mixed.

Also, in global variable, we no longer have unique next state, while we do at the local level, while one might expect it at the global level and not at the local level (i.e. population size is not a predictor for growth rate). Being at the right resolution is therefore very important.

Q2: Classical competiton

Structural changes in SP-M does not change much in relation to MOZ dynamics: structurally stable!

Network Question (homework)

Point: different model formulisms can channel research questions, pushes you in certain directions
ODE: how to get dynamical system which does "cell cycle"?
Network: how are molecules organised to get behaviour that is cycle like?

Tyson: middles out, knows a little bit the rules of system and knows about final behaviour and links them together
It is a bit like reverse engineering.

vs

Need to look what happens: bottom up approach.