  Motivated by neuroscience applications, we introduce the concept of
qualitative detection, that is, the problem of determining on-line the current
qualitative dynamical behavior (e.g., resting, oscillating, bursting, spiking
etc.) of a nonlinear system. The approach is thought for systems characterized
by i) large parameter variability and redundancy, ii) a small number of
possible robust, qualitatively different dynamical behaviors and, iii) the
presence of sharply different characteristic timescales. These properties are
omnipresent in neurosciences and hamper quantitative modeling and fitting of
experimental data. As a result, novel control theoretical strategies are needed
to face neuroscience challenges like on-line epileptic seizure detection. The
proposed approach aims at detecting the current dynamical behavior of the
system and whether a qualitative change is likely to occur without
quantitatively fitting any model nor asymptotically estimating any parameter.
We talk of qualitative detection. We rely on the qualitative properties of the
system dynamics, extracted via singularity and singular perturbation theories,
to design low dimensional qualitative detectors. We introduce this concept on a
general class of singularly perturbed systems and then solve the problem for an
analytically tractable class of two-dimensional systems with a single unknown
sigmoidal nonlinearity and two sharply separated timescales. Numerical results
are provided to show the performance of the designed qualitative detector.

Optimization and Control