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...he spike trains. Furthermore, the computation burden would increase proportional to the length of the spike trains and inversely proportional to the simulation step. However, as shown by Paiva et al. =-=[34]-=-, and utilized in Paiva et al. [25], the van Rossum’s distance can be evaluated in terms of a computationally effective estimator with order O(NiNj), given as d vR(Si, Sj) = 1 2 ⎡ Ni Ni ∑ ∑ ⎣ Lτ (t i ...

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... explore the Hilbertian metrics applied to point processes. Note that our approach is applied to sets of spike trains unlike other spike train metrics that are defined on a pair of spike trains (e.g. =-=[7, 8]-=-). Proposition 1 ([5]). Let 휇1 and 휇2 be two probability measures on 풳 , and 휇 be a measure that dominates both probability measures. Let 푑ℝ+ be a 12 -homogeneous Hilbertian metric on ℝ + . Then 푑ℳ de...

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...ernel smoothing is used as the intensity function estimator in equation (19), this inner product is sensitive to point processes with memory. This statement has been verified empirically for the nCI [=-=Paiva et al. 2009-=-]. In theory, depending on the nonlinearity utilized the memory depth of the inner product can be infinity. However, in practice, the decaying dynamics of the conductance after a spike and the saturat...

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... [9], the intensity function can be estimated from single realization, which transforms the space of spike trains to the space of integrable continuous functions with finite number of discontinuities =-=[10]-=-. According to kernel smoothing intensity estimation, a spike train si comprising of spike times {tim ∈ {m = 1, . . . , Ni}} can be functionally represented by the estimated intensity function: λ̂si(t...

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...eptors, suggesting the history dependence in the interaction of action potential timings. We showed that the van Rossum and Houghton’s distance measures can be derived from the perspective of kernels =-=[82, 84]-=-. 2.3 Reproducing Kernel Hilbert Space for Spike Trains The main advantage of the smoothed spike train approach is that the spike trains can be treated as regular continuous amplitude, continuous time...

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...S [10] to derive an optimal (nonlinear) solution. RKHSs play a major role in statistics [10] and in machine learning [11]. In theoretical neuroscience they have been little used with the exception of =-=[12]-=-. In the latter work, RKHSs have been applied in a probabilistic setting of point process models to study the distance between spike trains of neural populations. Spline models have been used in compu...

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...s state, continuous time models (van Rossum 2001; Houghton and Sen 2008; Houghton 2009), and a number of others (Schreiber et al. 2003; Kreuz et al. 2007; Quiroga et al. 2002; Hunter and Milton 2003; =-=Paiva et al. 2009-=-a, b). However, these authors have seldom sought summary statistics or ensuing statistical models under these metrics. In fact, these distances may not always provide interesting intuition or tractabl...

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...s state, continuous time models (van Rossum 2001; Houghton and Sen 2008; Houghton 2009); and a number of others (Schreiber et al. 2003; Kreuz et al. 2007; Quiroga et al. 2002; Hunter and Milton 2003; =-=Paiva et al. 2009-=-a, b). The applications of these methods have mainly focused on the clustering or classification of spike trains with respect to different stimuli. Their relevance in deriving statistical summaries re...