Reply to "Comment on `Performance of different synchronization measures in real data: A case study on electroencephalographic signals'"

We agree with the Comment by Nicolaou and Nasuto about the utility of mutual information (MI) when properly estimated and we also concur with their view that the estimation based on k nearest neighbors gives optimal results. However, we claim that embedding parameters can indeed change MI results, as we show for the electroencephalogram data sets of our original study and for coupled chaotic systems. Furthermore, we show that proper embedding can actually improve the estimation of MI with the k nearest neighbors algorithm

The Restriction Principle and Commuting Families of Toeplitz Operators on the Unit Ball

On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and their Toeplitz operators with bounded symbols. It is known from our previous work that if a closed subgroup H of \widetilde{\SU(n,1)} has a multiplicity-free restriction for the holomorphic discrete series of \widetilde{\SU(n,1)}, then the family of Toeplitz operators with H-invariant symbols pairwise commute. In this work we consider the case of maximal abelian subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise commutativity of the corresponding Toeplitz operators. To achieve this we explicitly develop the restriction principle for each (conjugacy class of) maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform. In particular, we obtain a multiplicity one result for the restriction of the holomorphic discrete series to all maximal abelian subgroups. We also observe that the Segal-Bargman transform is (up to a unitary transformation) a convolution operator against a function that we write down explicitly for each case. This can be used to obtain the explicit simultaneous diagonalization of Toeplitz operators whose symbols are invariant by one of these maximal abelian subgroups

Event synchronization: a simple and fast method to measure synchronicity and time delay patterns

We propose a simple method to measure synchronization and time delay patterns between signals. It is based on the relative timings of events in the time series, defined e.g. as local maxima. The degree of synchronization is obtained from the number of quasi-simultaneous appearances of events, and the delay is calculated from the precedence of events in one signal with respect to the other. Moreover, we can easily visualize the time evolution of the delay and synchronization level with an excellent resolution. We apply the algorithm to short rat EEG signals, some of them containing spikes. We also apply it to an intracranial human EEG recording containing an epileptic seizure, and we propose that the method might be useful for the detection of foci and for seizure prediction. It can be easily extended to other types of data and it is very simple and fast, thus being suitable for on-line implementations.Comment: 6 pages, including 6 figures, RevTe

Unsupervised spike detection and sorting with wavelets and superparamagnetic clustering

This study introduces a new method for detecting and sorting spikes from multiunit recordings. The method combines the wavelet transform, which localizes distinctive spike features, with superparamagnetic clustering, which allows automatic classification of the data without assumptions such as low variance or gaussian distributions. Moreover, an improved method for setting amplitude thresholds for spike detection is proposed. We describe several criteria for implementation that render the algorithm unsupervised and fast. The algorithm is compared to other conventional methods using several simulated data sets whose characteristics closely resemble those of in vivo recordings. For these data sets, we found that the proposed algorithm outperformed conventional methods