## 6 Finding Modes (Compartments) Visible in Output Data Signals through Graphical Inspection  
Not all modes and compartments (state variables) they represent are visible in a particular output \(y(t)\), in response to a particular input \(u(t)\)--i.e., an input-output port pair. But they may be visible from other I-O data ports. We take advantage of the fact that modes are intrinsically responses to initial conditions (ICs) or, equivalently, _impulse_ inputs. The procedure is to systematically test each I-O port pair of a _candidate model structure_, counting the number of modes in each designated output compartment in response to an impulse input into the designated input compartment. Candidate model structures are derived from available mechanistic biological system domain knowledge. _Alternative_ candidates typically account for unknown or incomplete knowledge of compartmental connectivities. Checking all possible compartment I-O pairs in this manner then gives all possible modes in the system, i.e., all possible compartments in the candidate model. This, incidentally, reveals new experiment designs for finding otherwise hidden compartments. Analysis can be performed mathematically, but it is easier through systematic inspection of the compartmental model graph. This graphical approach exposes visually why the full dimensionality of a model and the system it represents is difficult to establish. In the process, it also provides a more facile tool for discovering the hidden structure in the underlying system. The following two simple examples illustrate the procedure step-by-step. Additional, more complex examples can be found in [6]. **Example 2**: **2-Compartment Candidate Model with One Input and Two Outputs**  
In the simple model of Figure 4, we assume zero ICs and a single impulse input. We relax these assumptions, later, to discover how ICs and other inputs affect results. The \(k_{ij}\)s are non-negative rate constants (time\({}^{-1}\)), and \(V_{i}\)s are volumes. We are interested in the number of modes visible in outputs 1 and 2 under various conditions. We evaluate this number first mathematically and then more simply using graphical inspection. _The Math Solution:_ It can be shown [6] that both outputs are a sum of two real and distinct exponential modes:  
Figure 4: Example 2-compartment model, with one input and two outputs, for illustrating visible mode identification mathematically and by visual inspection. \[y_{1}(t)=\frac{1}{V_{1}(\lambda_{1}-\lambda_{2})}\Big{[}(k_{02}+k_{12}+\lambda_{1 })e^{\lambda_{1}t}-(k_{02}+k_{12}+\lambda_{2})e^{\lambda_{2}t}\Big{]},\ \ \ y_{2}(t)=\frac{k_{21}}{V_{1}(\lambda_{1}-\lambda_{2})}\Big{(}e^{ \lambda_{1}t}-e^{\lambda_{2}t}\Big{)}\]  
and the two exponents (eigenvalues) are:  
\[\lambda_{1,2}=\left(-(k_{02}+k_{12}+k_{21})\pm\sqrt{(k_{02}+k_{12}+k_{21})^{2}- 4k_{02}k_{21}}\right)/2\leq 0\ \text{for all}\ k_{ij}\ \geq\ 0\]  
We have exactly two distinct (exponential) modes measurable in each compartment for an impulse input for all \(k_{ij}>0\). Now suppose \(k_{21}=0\) (and ICs remain zero), then \(\lambda_{2}=-(k_{02}+k_{12})\), which nulls \(\lambda_{1}\) and the second exponential term in the equation for \(y_{1}\), i.e., the solution for \(y_{1}\) reduces to only _one mode_, and that for \(y_{2}\) reduces to _zero modes_ because the coefficient goes to zero (by L'Hopital's rule). Thus, the model with \(k_{21}=0\) will exhibit one mode in \(1\) and none in \(2\). That is what the math tells us. _The Simpler Graphical Solution:_ We redo the problem by reasoning physically about each case, \(k_{21}\) nonzero and \(k_{21}\) zero, i.e., with and without the forward connection from compartment \(1\) to \(2\). The principle here is that each compartment, when it is stimulated by a signal entering it, elicits a distinct mode associated with that compartment in its dynamic response. Whether this effect is visible in an output depends on the locations of the I-O probes and the compartment interconnections.