Deterministic Mathematical Models
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 other types of constructs not listed here. Probabilistic, automata and delay models have not been listed (to name a few).
In each of the above constructs
u = the measured quantity
I = the stimulus (could be constant)
and F is some function which describes the biology, chemistry and/or physics of the system being examined.
These constructs are "deterministic" because each stimuli I(t) elicits exactly one response u(t).
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. In general such an "Inverse"problem is 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).
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.
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
- Second messenger coupling via a chemical which diffuses in the extracellular spaces
Knowing the properties of individual cells is key to understanding the properties of the cellular aggregate. First, we define and illustrate some key concepts and qualitative features in individual cells (and uncoupled systems).
- System = Individual units + interactions
- Tissue = Cells + cellular interactions
- Models = compartments + compartmental interactions