In many experiments a stimulus (stimuli) I elicits a measured response(s) u.

The biological, chemical, biochemical, or physical system being considered can be regarded as a "black box". There are several kinds of mathematical constructs which one might consider to model the system. The choice of construct depends on

- whether time t is continuous or discrete
- whether position x is a relevant factor
- the knowledge of the underlying chemistry/physics

There are 5 common types of mathematical constructs which are used:

There are other types of constructs not listed here. Probabilistic, automata and delay models have not been listed (to name a few). Many automata, however, may be regarded as a map model.

In each of the above constructs

- u = the measured quantity
- I = the stimulus (could be constant or none)

and F is some function which describes the biology, chemistry and/or physics of the system being examined. Often, conservation laws are needed to formulate the model, i.e.,

- Conservation of mass (chemistry)
- Conservation of charge (electrophysiology)
- Conservation of momentum (cell locomotion)

These constructs are * "deterministic" * because each
stimuli I(t) elicits exactly one response u(t).

** Forward versus Inverse Problems **

Sometimes stimuli are not known but a response is still measured.
In this scenario one can still adopt one of the mathematical
constructs above to model the experiment and ask what
stimuli elicited the measured response. Often,
such
* "Inverse" *
problems are not well posed (or ill-posed).
Roughly speaking,
* "ill-posed" *
means that more than
one stimuli can elicit the same response.
With each "Inverse" problem there is an associated
* "Forward"*
problem in which the stimulus is known
(and often controllable).

** Controllability **

Sometimes the inverse problem has a solution, i.e., there is a stimulus which will ellicit the desired response. In that scenario, the problem is "controllable". Controllability has very precise mathematical definitions and there are different types of control problems. Often, control problems are stated with constraints. For example, one might be interested in knowing if by adding (dynamically) more substrate to a chemical mix whether a product concentration can be made to reach a target level in some prescribed time. Clearly, physical constraints may prohibit solutions from existing. So, not all control problems have solutions.

** Qualitative versus Quantitative features **

Some models are used to predict
* quantitative * features such as
the maximal concentration, frequency of response,
etc. Sometimes one is less interested in
the exact value of a measured quantity
and more interested in its * qualitative *
features. For instance if the measured quantity
is the transmembrane electrical potential * u(t) *
of an axon or cell, one may be more interested
in whether the stimulus caused an oscillation
whose amplitude decayed or remained constant.

Measured behavior of individual cells and cellular aggregates can have qualitatively different features. These differing behaviors are a consequence of the type of coupling which occurs in the aggregate:

- Electrical synaptic coupling in nervous and sensory systems
- Gap junction coupling in pancreatic Islets of Langerhans
- Extracellular and intercellular diffusion of chemical messengers

Knowing the properties of individual cells is key to understanding
the properties of the cellular aggregate. When coupled, however,
individual cells may exhibit behaviors which they otherwise
could not exhibit indivually when uncoupled. Such
emergent complexity occurs in tissues as well as other systems.
**
**

**
**

- System = Individual units + interactions
- Tissue = Cells + cellular interactions
- Models = compartments + compartmental interactions