## 6 Finding Modes (Compartments) Visible in Output Data Signals through Graphical Inspection
An input signal entering compartment \(1\) (an impulse input in this example) will first stimulate the first mode represented by that state variable--dynamic system compartment \(1\) itself. Then, if \(k_{21}\) is nonzero, the signal travels to \(2\) via \(k_{21}\), where it stimulates the second distinct mode, represented by that state variable--dynamic system compartment \(2\). Finally, with \(k_{12}\) nonzero, material travels back to \(1\) via \(k_{12}\), with a dynamic reflecting both the stimulus from \(1\) along \(k_{21}\)_and_ the corresponding dynamic due to stimulation of the distinct mode in \(2\). So we see _two_ modes in both compartments. They go back and forth, feeding their signals forward and back, because they are connected reversibly. _Other Parameterizations:_ Now, using the same reasoning and inspection of the graph, we see that if \(k_{12}=0\) and \(k_{21}\) is nonzero, we find only one mode in compartments \(1\) and \(2\) in compartment \(2\), for the same input and ICs, because the mode associated with compartment \(2\) is not visible in \(1\) because of the lack of connection back to \(1\) (\(k_{12}=0\)). We discovered this using the mathematics above, but this graphical solution is easier. _Nonzero ICs:_ Going one step further, if we started with _zero_ input, \(k_{12}\
eq 0\), \(k_{21}\
eq 0\) and nonzero ICs in both compartments, both would have _two_ modes visible in them because the ICs are equivalent to impulse inputs. And, unlike the zero IC case above, when \(k_{21}=0\), compartment \(1\) would still show \(2\) modes because mode \(2\) would be stimulated by the nonzero IC in \(2\) and travel to \(1\) along \(k_{12}\). So nonzero ICs can be as important as inputs in establishing model complexity because--in essence--they are equivalent to impulse inputs and generate the same modal responses. _Example 3:_ **3-Compartment Candidate Model**  
We add a third compartment, the only one measured in this example, and evaluate the number of modes visible in the output of compartment \(3\) in Figure 5, using only graphical reasoning. We again assume zero ICs everywhere, an impulse input and all \(k_{ij}>0\). In the previous example, we established that two modes would be visible in compartment \(2\) if it were measured. Despite the fact that compartment \(2\) is not measured in this problem, there are still two \"excited\" modes in \(2\), and they travel to compartment \(3\) along nonzero path \(k_{32}\). So, at least two modes are visible in compartment \(3\) measurements. But, compartment \(3\) is itself a dynamic state variable compartment, and when it is stimulated, it generates its own mode. Stimulated by two modes coming from \(2\), it therefore exhibits _three_ modes in its output. No math needed!_Different Inputs_: Going one step further, if the input \(u_{1}\) were an _exponential function_ distinct from the fundamental modes of the model, e.g., a single exponential input, then _four_ modes would be visible at \(y_{3}\) because the input \"mode\" would also be visible in the output. Inputs are external to the model; therefore, exponential inputs can be thought of as the output of a compartment external to the model (another model!) and so on...  
_Remarks_: In principle, the results above do not depend on parameter values, only structure, assuming all \(k_{i\!j}>0\) in the models. In reality, the relative magnitudes of the parameter values matter because depending on parameter values, some modes may be very small and difficult to detect in real data. This is a problem for model parameter sensitivity analysis, beyond our scope here. Impulse inputs (approximate brief-pulse inputs in practice) are in principle maximally informative for modal analysis. If the input were instead a step, or ramp, or any integral of the impulse, the modal responses would be qualitatively the same--still exponential, with the same frequency \(\lambda_{i}\) (eigenvalues), but with different shapes (eigenvectors).