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Motivated by neuroscience applications, we introduce the concept ofqualitative detection, that is, the problem of determining on-line the currentqualitative dynamical behavior (e.g., resting, oscillating, bursting, spikingetc.) of a nonlinear system. The approach is thought for systems characterizedby i) large parameter variability and redundancy, ii) a small number ofpossible robust, qualitatively different dynamical behaviors and, iii) thepresence of sharply different characteristic timescales. These properties areomnipresent in neurosciences and hamper quantitative modeling and fitting ofexperimental data. As a result, novel control theoretical strategies are neededto face neuroscience challenges like on-line epileptic seizure detection. Theproposed approach aims at detecting the current dynamical behavior of thesystem and whether a qualitative change is likely to occur withoutquantitatively fitting any model nor asymptotically estimating any parameter.We talk of qualitative detection. We rely on the qualitative properties of thesystem dynamics, extracted via singularity and singular perturbation theories,to design low dimensional qualitative detectors. We introduce this concept on ageneral class of singularly perturbed systems and then solve the problem for ananalytically tractable class of two-dimensional systems with a single unknownsigmoidal nonlinearity and two sharply separated timescales. Numerical resultsare provided to show the performance of the designed qualitative detector.

On-line detection of qualitative dynamical changes in nonlinear systems: the resting-oscillation case