Motivated by neuroscience applications, we introduce the concept ofqualitative estimation as an adaptation of classical parameter estimation tononlinear systems characterized by i) large parameter variability andredundancy, ii) a small number of possible robust, qualitatively differentbehaviors and, iii) the presence of sharply different characteristictimescales. These properties are omnipresent in neurosciences and hamperquantitative modeling and fitting of experimental data. As a result, novelestimation strategies are needed to face neuroscience challenges like onlineepileptic seizure detection. In this context, the objective is no longer toseek for the exact value of the unknown parameters as traditionally done in thecontrol literature. Instead, we propose to estimate the distance of the unknownparameters to (unknown) critical values at which a change of activity occurs,we talk of qualitative estimation. We introduce these ideas on a class ofnonlinear systems with a single unknown sigmoidal nonlinearity and two sharplyseparated timescales. This class of systems is shown to either have a globallyexponentially stable fixed point, corresponding to the resting activity, orexhibit relaxation oscillations, depending on a single ruling parameter andindependently of the exact shape of the nonlinearity. We then design andanalyze a qualitative estimator, which estimates the distance between theruling parameter and the unknown critical value at which theresting/oscillation transition happens without using any quantitative fittingof the measured data. The designed estimator therefore provides onlineinformation about the actual activity of the system and how close it is to achange of activity.
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