### Research - Measuring synchronization

**Collaborators:**

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.

**Description:**

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. **[6]** below, the Matlab source code of this method can be found
here).

**Publications:**

**[24]** Kreuz T, Satuvuori E, Mulansky M:

SPIKE-order

Scholarpedia, 12(7):42441 (2017).

**[23]** 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 [PDF] and arXiv [PDF] (2017).

**[22]** Kreuz T, Satuvuori E, Pofahl M, Mulansky M:

Leaders and followers: Quantifying consistency in spatio-temporal propagation patterns

New J. Phys., 19, 043028 [PDF] and arXiv [PDF] (2017).

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.

**[21]** Mulansky M, Kreuz T:

PySpike - A Python library for analyzing spike train synchrony

Software X (in press) and arXiv [PDF] (2016).

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.

**[20]**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).

**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.

[19] 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.

[18] Bozanic N, Mulansky M, Kreuz T:

SPIKY

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.

**
[17]** 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.

**[16]**
Kreuz T:

SPIKE-distance.

*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.

**[15]**
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.

**
[14]**
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.

**[13]**
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).

**[12]**
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.

**[11]**
Chicharro D, Kreuz T, Andrzejak RG:

What can spike train distances tell us about the neural code?

*J Neurosci Methods* **199**, 146 (2011).

**[10]** 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.

**[9]** 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.

**[8]** 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.

**[7]** 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.

**[6]** 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.

**[5]** Kreuz T:

Measuring synchronization in model systems and electroencephalographic time series from epilepsy patients.

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 [PDF].

**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.

**[4]** Andrzejak RG, Kraskov A,
Stoegbauer H, Mormann F, Kreuz T:

Bivariate surrogate techniques: Necessity, strengths, and caveats

*Phys Rev E* **68**, 066202 (2003).

**[3]** 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).

**[2]** 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).

**[1]** 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).