Our contribution in this paper is threefold. First, we present a framework for characterizing spike synchrony in neuronal spike-train recordings that is based on the identification of spikes (also called electrical impulses or action potentials) with what we call influence maps: real-valued functions that describe an influence region around the corresponding spike times within which a continuous and possibly graded notion of synchrony among spikes is defined. Second, we provide a model of synchrony within our framework that is based on a continuous, two-valued (i.e., bivalent) measure of synchrony, aimed at overcoming the drawbacks of time discretization in the bin-based one (the almost exclusively applied model in the field), which we also describe within our framework. Third, in connection with the assessment of synchrony in our continuous model, we provide methodology and algorithms for the identification of frequent parallel episodes in sequences of events (i.e., sets of items, normally required to occur within a certain time span in the sequence). Special attention is given to the notion of frequency of parallel episodes and its computation. . , k}. We want to show that H k+1 J k+1 . We know that J k ≺ J k+1 and, therefore, that H k ≺ J k+1 (i.e., H k and J k+1 are independent). We can proceed as above (assuming that H k+1 < J k+1 ; if not the contradiction is straightforward), by reductio ad absurdum, with J k+1 in place of J 1 and H k+1 in place of H 1 , to show that the assumption that H k+1 J k+1 leads us to a contradiction.
There is no established formal framework for expert systems based on weighted IF-THEN rules. We discuss three mathematical models that have been recently proposed by the authors for CADIAG-2-a well-known system of this kind. The three frameworks are based on fuzzy logics, probability theory and possibilistic logic, respectively. CADIAG-2 is used here as a case study to evaluate these frameworks. We point out their use, advantages and disadvantages. In addition, the described models provide insight into various aspects of CADIAG-2.
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