Henri D.I. Abarbanel, Ralph G. Andrzejak, Nebojsa Bozanic, Daniel Chicharro, Peter Grassberger, Martin Greschner, Julie Haas, Conor Houghton, Alexander Kraskov, Klaus Lehnertz, Alice Morelli, Florian Mormann, Mario Mulansky, Antonio Politi, Rodrigo Quian Quiroga, Andreea Sburlea, Harald Stoegbauer.
This includes the development, analysis and comparison of different approaches to quantify the synchronization between two continuous time series. The measures are applied to coupled model systems as well as electrophysiological data (mostly EEG). This is complemented by measures that estimate the synchronization between discrete events within the time series (such as spikes in neuronal recordings). While the ISI-distance quantifies local dissimilarities based on covariances of the
neurons firing rate profiles, both the SPIKE-distance and SPIKE-synchronization capture the relative timing of local spikes. However, whereas the SPIKE-distance weights and normalizes the differences between nearest neighbor spikes, SPIKE-synchronization acts as a binary coincidence detector, i.e. there is a cutoff at the (adaptive) time lag relative to which two neighboring spikes are either con-
sidered coincident or not and all detailed information both within or outside this coincidence window is discarded.
The figures below show two examples in which the ISI-Distance is applied first to very similar and then to rather different neuronal time series (for details see Ref.  below, the Matlab source code of this method can be found here).
 Kreuz T, Houghton C, Victor JD:
Spike Train Distance
Encycl Comp Neurosci, DOI: doi.org/10.1007/978-1-4614-7320-6_409-2 (2020) [PDF]
 Satuvuori E, Mulansky M, Daffertshofer A, Kreuz T:
Using spike train distances to identify the most discriminative neuronal subpopulation
JNeurosci Methods, 308, 354 and arXiv [PDF] (2018) [PDF]
 Satuvuori E, Kreuz T:
Which spike train distance is most suitable for distinguishing rate and temporal coding?
JNeurosci Methods 299, 22 and on the arXiv [PDF] (2018) [PDF]
 Malvestio I, Kreuz T, Andrzejak RG:
Robustness and versatility of a nonlinear interdependence method for directional coupling detection from spike trains
Physical Review E 96, 022203 (2017) [PDF]
 Kreuz T, Satuvuori E, Mulansky M:
Scholarpedia, 12(7):42441 (2017)
 Satuvuori E, Mulansky M, Bozanic N, Malvestio I, Zeldenrust F, Lenk K, Kreuz T:
Measures of spike train synchrony for data with multiple time-scales
JNeurosci Methods 287, 25 and arXiv [PDF] (2017) [PDF]
 Kreuz T, Satuvuori E, Pofahl M, Mulansky M:
Leaders and followers: Quantifying consistency in spatio-temporal propagation patterns
New J. Phys., 19, 043028 and arXiv [PDF] (2017) [PDF]
Abstract: Repetitive spatio-temporal propagation patterns are encountered in fields as wide-ranging as climatology, social communication and network science. In neuroscience, perfectly consistent repetitions of the same global propagation pattern are called a synfire pattern. For any recording of sequences of discrete events (in neuroscience terminology: sets of spike trains) the questions arise how closely it resembles such a synfire pattern and which are the spike trains that lead/follow. Here we address these questions and introduce an algorithm built on two new indicators, termed SPIKE-Order and Spike Train Order, that define the Synfire Indicator value, which allows to sort multiple spike trains from leader to follower and to quantify the consistency of the temporal leader-follower relationships for both the original and the optimized sorting. We demonstrate our new approach using artificially generated datasets before we apply it to analyze the consistency of propagation patterns in two real datasets from neuroscience (Giant Depolarized Potentials in mice slices) and climatology (El Ni˜no sea surface temperature recordings). The new algorithm is distinguished by conceptual and practical simplicity, low computational cost, as well as flexibility and universality.
 Mulansky M, Kreuz T:
PySpike - A Python library for analyzing spike train synchrony
Software X 5, 183 and arXiv [PDF] (2016) [PDF]
Abstract: Understanding how the brain functions is one of the biggest challenges of our time. The analysis of experimentally recorded neural firing patterns (spike trains) plays a crucial role in addressing this problem. Here, the PySpike library is introduced, a Python package for spike train analysis providing parameter-free and time-scale independent measures of spike train synchrony. It allows to compute similarity and dissimilarity profiles, averaged values and distance matrices. Although mainly focusing on neuroscience, PySpike can also be applied in other contexts like climate research or social sciences. The package is available as Open Source on Github and PyPI.
 Mulansky M, Bozanic N, Sburlea A, Kreuz T:
A guide to time-resolved and parameter-free measures of spike train synchrony
IEEE Proceeding on Event-based Control, Communication, and Signal Processing (EBCCSP), 1-8 and arXiv [PDF] (2015) [PDF]
Abstract: Measures of spike train synchrony have proven a valuable tool in both experimental and computational neuroscience. Particularly useful are time-resolved methods such as the ISI- and the SPIKE-distance, which have already been applied in various bivariate and multivariate contexts. Recently, SPIKE-Synchronization was proposed as another time-resolved synchronization measure. It is based on Event-Synchronization and has a very intuitive interpretation. Here, we present a detailed analysis of the mathematical properties of these three synchronization measures. For example, we were able to obtain analytic expressions for the expectation values of the ISI-distance and SPIKE-Synchronization for Poisson spike trains. For the SPIKE-distance we present an empirical formula deduced from numerical evaluations. These expectation values are crucial for interpreting the synchronization of spike trains measured in experiments or numerical simulations, as they represent the point of reference for fully randomized spike trains.
 Kreuz T, Mulansky M, Bozanic N:
SPIKY: A graphical user interface for monitoring spike train synchrony
JNeurophysiol 113, 3432 (2015) [PDF]
Abstract: Techniques for recording large-scale neuronal spiking activity are developing very fast. This leads to an increasing demand for algorithms capable of analyzing large amounts of experimental spike train data. One of the most crucial and demanding tasks is the identification of similarity patterns with a very high temporal resolution and across different spatial scales. To address this task, in recent years three time-resolved measures of spike train synchrony have been proposed, the ISI-distance, the SPIKE-distance, and event synchronization. The Matlab source codes for calculating and visualizing these measures have been made publicly available. However, due to the many different possible representations of the results the use of these codes is rather complicated and their application requires some basic knowledge of Matlab. Thus it became desirable to provide a more user-friendly and interactive interface. Here we address this need and present SPIKY, a graphical user interface which facilitates the application of time-resolved measures of spike train synchrony to both simulated and real data. SPIKY includes implementations of the ISI-distance, the SPIKE-distance and SPIKE-synchronization (an improved and simplified extension of event synchronization) which have been optimized with respect to computation speed and memory demand. It also comprises a spike train generator and an event detector which makes it capable of analyzing continuous data. Finally, the SPIKY package includes additional complementary programs aimed at the analysis of large numbers of datasets and the estimation of significance levels.
 Bozanic N, Mulansky M, Kreuz T:
Scholarpedia 9(12), 32344 (2014)
Abstract: SPIKY is a graphical user interface written in Matlab that facilitates the application of time-resolved measures of spike train synchrony to both simulated and real data. It contains several approaches to analyze spike train synchrony: the standard Peri-Stimulus Time Histogram (PSTH), the ISI-distance, SPIKE-distance, and SPIKE synchronization. For a given data set SPIKY calculates the measures of choice and allows the user to switch between many different visualizations such as dissimilarity profiles, pairwise dissimilarity matrices, or hierarchical cluster trees. SPIKY also includes a spike train generator and an event detector which makes it capable of analyzing continuous data. Finally, the SPIKY-package includes complementary programs for the automatized analysis of a large number of datasets and for the evaluation of the statistical significance of the results.
 Kreuz T, Chicharro D, Houghton C, Andrzejak RG, Mormann F:
Monitoring spike train synchrony
J Neurophysiol 109, 1457 (2013) [PDF]
Abstract: Recently, the SPIKE-distance has been proposed as a parameter-free and time-scale independent measure of spike train synchrony. This measure is time-resolved since it relies on instantaneous estimates of spike train dissimilarity. However, its original definition led to spuriously high instantaneous values for event-like firing patterns. Here we present a substantial improvement of this measure which eliminates this shortcoming. The reliability gained allows us to track changes in instantaneous clustering, i.e., time-localized patterns of (dis)similarity among multiple spike trains. Additional new features include selective and triggered temporal averaging as well as the instantaneous comparison of spike train groups. In a second step, a causal SPIKE-distance is defined such that the instantaneous values of dissimilarity rely on past information only so that time-resolved spike train synchrony can be estimated in real-time. We demonstrate that these methods are capable of extracting valuable information from field data by monitoring the synchrony between neuronal spike trains during an epileptic seizure. Finally, the applicability of both the regular and the real-time SPIKE-distance to continuous data is illustrated on model electroencephalographic (EEG) recordings.
 Kreuz T:
Scholarpedia 7(12), 30652 (2012)
Abstract: The SPIKE-distance is an estimator of the dissimilarity between two (or more) spike trains. In contrast to most other spike train distances (such as the Victor-Purpura distance) it is time-resolved and is able to track changes in instantaneous clustering, i.e., time-localized patterns of (dis)similarity among two or more spike trains. Additional features include selective and triggered temporal averaging as well as the instantaneous comparison of spike train groups. The SPIKE-distance can also be formulated as a causal measure which is defined such that the instantaneous values of dissimilarity rely on past information only so that time-resolved spike train synchrony can be estimated in real-time.
 Houghton C, Kreuz T:
On the efficient calculation of van Rossum distances
Network: Computation in Neural Systems 23, 48 (2012) [PDF]
Abstract: The van Rossum metric measures the distance between two spike trains. Measuring a single van Rossum distance between one pair of spike trains is not a computationally expensive task, however, many applications require a matrix of distances between all the spike trains in a set or the calculation of a multi-neuron distance between two populations of spike trains. Moreover, often these calculations need to be repeated for many different parameter values. An algorithm is presented here to render these calculation less computationally expensive, making the complexity linear in the number of spikes rather than quadratic.
 Kreuz T:
Measures of neuronal signal synchrony
Scholarpedia 6(12), 11922 (2011)
Abstract: Measures of neuronal signal synchrony are estimators of the synchrony between two or sometimes more continuous time series of brain activity which yield low values for independent time series and high values for correlated time series. A complementary class of approaches comprises measures of spike train synchrony which quantify the degree of synchrony between discrete signals.
Synchronization of continuous time series can manifest itself in many different ways. The simplest case of complete synchronization (Fujisaka and Yamada, 1983) can be attained if identical systems are coupled sufficiently strongly so that their states coincide after transients have died out. The concept of generalized synchronization (Afraimovich et al., 1986) introduced for uni-directionally coupled systems, describes the presence of some functional relation between the states of the two systems. Finally, phase synchronization, first described for chaotic oscillators (Rosenblum et al., 1996), is defined as the global entrainment of the phases while the amplitudes may remain uncorrelated.
Following this variety of concepts many different approaches to quantify the degree of synchronization between two continuous signals have been proposed. These approaches comprise linear ones like the cross correlation or the spectral coherence function as well as nonlinear measures like mutual information (Gray, 1990), transfer entropy (Schreiber, 2000), Granger causality (Granger, 1969), or the nonlinear interdependence (Arnhold et al., 1999; Quian Quiroga et al., 2002; Andrzejak et al., 2003). Furthermore, different indices of phase synchronization such as the mean phase coherence (Kuramoto, 1984; Mormann et al., 2000) have been introduced.
 Kreuz T:
Measures of spike train synchrony
Scholarpedia 6(10), 11934 (2011)
Abstract: Measures of spike train synchrony (or inversely spike train distances) are estimators of the (dis)similarity between two or sometimes more spike trains. Here spike train refers to a sequence of neuronal action potentials. Under the assumption that neither the shape of the action potential nor the background activity carry relevant information, neuronal responses are reduced to a spike train where the only information maintained is the timing of the individual spikes. A complementary class of approaches comprises measures of neuronal signal synchrony.
Measures that estimate the degree of synchrony between spike trains are important tools for many applications. Among others, they can be used to quantify the reliability of neuronal responses upon repeated presentations of a stimulus (Mainen and Sejnowski, 1995) or to test the performance of neuronal models (Jolivet et al., 2008).
 Andrzejak RG, Kreuz T:
Characterizing unidirectional couplings between point processes and flows
Eur Phys Lett 96, 50012 (2011) [PDF]
Abstract: Experimental data comprising both time-continuous flows and point processes are recorded in many scientific disciplines. The characterization of causal interactions from such signals is key to an advanced understanding of the underlying dynamics. We therefore introduce a unified approach to characterize unidirectional couplings between point processes, between flows, as well as between point processes and flows. For this purpose we show and exploit the generality of the asymmetric state similarity conditioning principle. We use Hindmarsh-Rose neuron models and Lorenz oscillators to illustrate the high sensitivity and specificity of our approach.
 Chicharro D, Kreuz T, Andrzejak RG:
What can spike train distances tell us about the neural code?
J Neurosci Methods 199, 146 (2011) [PDF]
 Kreuz T, Chicharro D, Greschner M, Andrzejak RG:
Time-resolved and time-scale adaptive measures of spike train synchrony
J Neurosci Methods 195, 92 (2011) [PDF]
Abstract: A wide variety of approaches to estimate the degree of synchrony between two or more spike trains have been proposed. One of the most recent methods is the ISI-distance which extracts information from the interspike intervals (ISIs) by evaluating the ratio of the instantaneous firing rates. In contrast to most previously proposed measures it is parameter free and time-scale independent. However, it is not well suited to track changes in synchrony that are based on spike coincidences. Here we propose the SPIKE-distance, a complementary measure which is sensitive to spike coincidences but still shares the fundamental advantages of the ISI-distance. In particular, it is easy to visualize in a time-resolved manner and can be extended to a method that is also applicable to larger sets of spike trains. We show the merit of the SPIKE-distance using both simulated and real data.
 Haas JS*,Kreuz T*, Torcini A, Politi A, Abarbanel HDI:
Rate maintenance and resonance in the entorhinal cortex
Eur J Neurosci 32, 1930 (2010) [PDF]
Abstract: Throughout the brain, neurons encode information in fundamental units of spikes. Each spike represents the combined thresholding of synaptic inputs and intrinsic neuronal dynamics. Here, we address a basic question of spike train formation – how do perithreshold synaptic inputs perturb the output of a spiking neuron? We recorded from single entorhinal principal cells in vitro and drove them to spike steadily at 5 Hz (theta range) with direct current injection, then used a dynamic-clamp to superimpose strong excitatory conductance inputs at varying rates. Neurons spiked most reliably when the input rate matched the intrinsic neuronal firing rate. We also found a striking tendency of neurons to preserve their rates and coefficients of variation, independently of input rates. As mechanisms for this rate maintenance, we show that the efficacy of the conductance inputs varied with the relationship of input rate to neuronal firing rate, and with the arrival time of the input within the natural period. Using a novel method of spike classification, we developed a minimal Markov model that reproduced the measured statistics of the output spike trains and thus allowed us to identify and compare contributions to the rate maintenance and resonance. We suggest that strength of rate maintenance may be used as a new categorization scheme for neuronal response and note that individual intrinsic spiking mechanisms may play a significant role in forming the rhythmic spike trains of activated neurons; in the entorhinal cortex, individual pacemakers may dominate production of the regional theta rhythm.
 Kreuz T, Chicharro D, Andrzejak RG, Haas JS, Abarbanel HDI:
Measuring multiple spike train synchrony
J Neurosci Methods 183, 287 (2009) [PDF]
Abstract: Measures of multiple spike train synchrony are essential in order to study issues such as spike timing reliability, network synchronization, and neuronal coding. These measures can broadly be divided in multivariate measures and averages over bivariate measures. One of the most recent bivariate approaches, the ISI-distance, employs the ratio of instantaneous interspike intervals. In this study we propose two extensions of the ISI-distance, the straightforward averaged bivariate ISI-distance and the multivariate ISI-diversity based on the coeffcient of variation. Like the original measure these extensions combine many properties desirable in applications to real data. In particular, they are parameter free, time scale independent, and easy to visualize in a time-resolved manner, as we illustrate with in vitro recordings from a cortical neuron. Using a simulated network of Hindmarsh-Rose neurons as a controlled configuration we compare the performance of our methods in distinguishing different levels of multi-neuron spike train synchrony to the performance of six other previously published measures. We show and explain why the averaged bivariate measures perform better than the multivariate ones and why the multivariate ISI-diversity is the best performer among the multivariate methods. Finally, in a comparison against standard methods that rely on moving window estimates, we use single-unit monkey data to demonstrate the advantages of the instantaneous nature of our methods.
 Kreuz T, Haas JS, Morelli A, Abarbanel HDI, Politi A:
Measuring spike train synchrony
J Neurosci Methods 165, 151 (2007) [PDF]
Abstract: Estimating the degree of synchrony or reliability between two or more spike trains is a frequent task in both experimental and computational neuroscience. In recent years, many different methods have been proposed that typically compare the timing of spikes on a certain time scale to be optimized by the analyst. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous firing rates. The method is parameter free, time scale independent and easy to visualize as illustrated by an application to real neuronal spike trains obtained in vitro from rat slices. In a comparison with existing approaches on spike trains extracted from a simulated Hindmarsh-Rose network, the ISI-distance performs as well as the best time-scale-optimized measure based on spike timing.
 Kreuz T, Mormann F, Andrzejak RG, Kraskov A, Lehnertz K, Grassberger P:
Measuring synchronization in coupled model systems: A comparison of different approaches
Phys D 225, 29 (2007) [PDF]
Abstract: The investigation of synchronization phenomena on measured experimental data such as biological time series has recently become an increasing focus of interest. Different approaches for measuring synchronization have been proposed that rely on certain characteristic features of the dynamical system under investigation. For experimental data the underlying dynamics are usually not completely known, therefore it is difficult to decide a priori which synchronization measure is most suitable for an analysis. In this study we use three different coupled model systems to create a controlled setting for a comparison of six different measures of synchronization. All measures are compared to each other with respect to their ability to distinguish between different levels of coupling and their robustness against noise. Results show that the measure to be applied to a certain task can not be chosen according to a fixed criterion but rather pragmatically as the measure which most reliably yields plausible information in test applications, although certain dynamical features of a system under investigation (e.g., power spectra, dimension) may render certain measures more suitable than others.
 Kreuz T:
Measuring synchronization in model systems and electroencephalographic time series from epilepsy patients [PDF]
Interdisciplinary PhD thesis in physics, University of Wuppertal, Research Center Juelich (2003)
Supervisors: Prof. P. Grassberger, Research Center Juelich, Germany; Dr. K. Lehnertz, University of Bonn, Germany
Abstract: The main aim of this dissertation is the comparative investigation of different measures of synchronization derived from various approaches and concepts. These include both measures for estimating the degree of dependence between two time series as well as measures which quantify the directionality of this dependence. The first group comprises the linear cross correlation, mutual information, six different indices for phase synchronization (based either on the Hilbert or on the wavelet transform) as well as symmetrized variants of two nonlinear interdependence measures and of event synchronization. The anti-symmetrized variants of the last three measures form the group of measures of directionality.
In the first part of this dissertation the symmetric measures are tested in a controlled setting by means of various model systems. Using the coupling strength as a first control parameter it is investigated to which extent the different measures are able to distinguish between different degrees of dependence. Furthermore, the robustness of the measures against external noise is estimated by varying the signal-to-noise ratio as the second control parameter.
Subsequently, all measures are employed to analyze electroencephalographic recordings from epilepsy patients. This application part consists of two single studies. First a comprehensive comparison on the predictability of epileptic seizures is carried out. Object of investigation is the capability of the different measures to reliably distinguish between the intervals preceding epileptic seizures and the intervals far away from any seizure activity. Already in this study a great deal of attention is paid to the statistical validation of seizure predictions. This issue is particularly addressed in the last part of this dissertation in which the method of measure profile surrogates is introduced as an appropriate tool to distinguish between measures and algorithms unsuited for the prediction of epileptic seizures, and more promising approaches. Two of the measures of synchronization are used to illustrate this new approach.
 Andrzejak RG, Kraskov A, Stoegbauer H, Mormann F, Kreuz T:
Bivariate surrogate techniques: Necessity, strengths, and caveats
Phys Rev E 68, 066202 (2003) [PDF]
 Quian Quiroga R, Kraskov A, Kreuz T, and Grassberger P:
Reply to "Comment on 'Performance of different synchronization measures in real data: A case study on electroencephalographic signals.'"
Phys Rev E 67, 063902 (2003) [PDF]
 Quian Quiroga R, Kreuz T, and Grassberger P:
Event Synchronization: A simple and fast method to measure synchronicity and time delay patterns
Phys Rev E 66, 041904 (2002) [PDF]
 Quian Quiroga R, Kraskov A, Kreuz T, and Grassberger P:
Performance of different synchronization measures in real data: A case study on electroencephalographic signals
Phys Rev E 65, 041903 (2002) [PDF]