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Measuring spike train synchrony
Thomas Kreuz, Julie S. Haas, Alice Morelli, Henry D. I. Abarbanel, Antonio Politi
TL;DR
The paper addresses how to measure synchrony or reliability between spike trains when existing methods often depend on a fixed time scale. It proposes the parameter-free ISI-distance, which compares interspike intervals through instantaneous-rate ratios, and finds performance comparable to the best spike-timing measure in simulated-network clustering.
Problem
Synchrony measures are needed for studying response reliability, neuronal coding, and information transfer, while many existing approaches use spike timing at a time scale fixed beforehand.
Method
The ISI-distance compares interspike intervals by evaluating the ratio of instantaneous rates, without binning or a preset time scale.
Results
The ISI-distance performs as well as the best spike-based measure, the Victor-Purpura distance, in reproducing clustering in simulated Hindemarsh-Rose spike trains.
Takeaways & Limitations
The ISI-distance provides a complementary, visualizable measure that is self-adaptive and most independent among the compared similarity measures.
Abstract
from arXiv · showhide
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 fixed beforehand. Here, we propose the ISI-distance, a simple complementary approach that extracts information from the interspike intervals by evaluating the ratio of the instantaneous frequencies. 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 Hindemarsh-Rose network, the ISI-distance performs as well as the best time-scale-optimized measure based on spike timing.
I. INTRODUCTION
Synchrony measures help assess reliability, neuronal coding, and information transfer, but existing approaches often require multiple trials or analyst-defined time scales. The study introduces the ISI-distance as a complementary, parameter-free comparison based on interspike intervals.
- Synchrony measurement supports analyses of response reliability, neuronal coding, and information transfer between coupled neurons.
- Existing synchrony measures include PSTH-based methods, spike-pattern analyses, and bivariate spike-train distances.
- The ISI-distance compares interspike intervals by quantifying the ratio of instantaneous rates rather than directly comparing spike occurrences.
- Without binning, the measure provides up to single-spike time resolution and avoids fixing a time scale beforehand through parameter-free, self-adaptive analysis.
- The study illustrates the method on cortical-cell recordings and evaluates clustering performance using simulated Hindemarsh-Rose neurons with predetermined coupling.
- Spike detection converts continuous recordings into discrete spike times, using a threshold criterion applied to the time series or its derivative.
B. The ISI-distance
The ISI-distance compares spike trains through the ratio of their instantaneous interspike intervals, with time-resolved and averaged forms. It is normalized, interpretable across firing-rate differences, and extendable to coarse-grained or higher-moment analyses.
- Definition: The method assigns each time instant the current interspike interval for each spike train, then compares their ratio.The ratio is evaluated effectively after each new spike in either time series.
- Normalization: The normalized measure is zero for iso-frequent behavior and approaches −1 or 1 when one train fires infinitely faster than the other.The limiting sign indicates which train has the higher firing rate.
- Averaging: Two distance variants average the instantaneous ISI-distance either by integrating its absolute value over time or by evaluating it after each new spike.These are the time-weighted and spike-weighted variants, respectively.
- Extensions: Higher moments of the instantaneous-distance distribution can distinguish random jitter from systematic phase lag when both yield the same ISI-distance.Removing absolute values instead evaluates relative firing rates.
- Examples: In three cortical recording examples, the reported ISI-distances are 0.06, 0.11, and 0.34.The figures visualize detected spikes, interspike intervals, and the renormalized ISI-distance over time.
- Extensions: Averaging over neighboring interspike intervals extends sensitivity to longer time scales and enables coarser comparisons, including burst dissimilarity.The underlying measure adapts to the spikes' time scale, allowing reasonable results for rather short spike trains.
C. Existing measures of spike train distance
The study compares the ISI-distance with five established measures of spike-train similarity. For comparability, all measures are converted into suitably normalized dissimilarities.
- Compared measures: The comparison includes Victor–Purpura and van Rossum spike-train metrics, Schreiber's correlation measure, Hunter–Milton distance, and event synchronization.These methods represent several existing approaches to spike-train similarity and dissimilarity.
- Comparison design: The resulting comparison treats the ISI-distance and existing methods within a common dissimilarity framework.This follows from converting each measure to the same evaluation concept.
- Comparison design: All measures are transformed into suitably normalized dissimilarity measures to align the evaluation with the concept of distance.The normalization is applied before comparing the various methods.
1. Victor-Purpura spike train metric
The Victor–Purpura metric measures spike-train dissimilarity as the minimum transformation cost, with spike-movement cost setting the analysis time scale. Related convolution-based correlation measures likewise depend on a user-selected temporal width.
- The Victor–Purpura distance is the minimum cost of transforming one spike train into another through insertion, deletion, and movement operations.Insertion and deletion each cost one, while movement has adjustable cost cV.
- For small cV, the metric primarily reflects spike-count differences; for high cV, it approaches a timing distance based on non-coincident spikes.Increasing cV shifts the measure from a rate distance toward a timing distance.
- The van Rossum approach convolves each spike with an exponential function before calculating distance between the resulting waveforms.Its exponential time constant τR sets the analysis scale and estimates differences in synaptic effects.
- The van Rossum time constant and Victor–Purpura movement cost are reciprocal: τR = 1/cV.
- Correlation-based measures convolve spike trains with a filter before cross-correlation, with filter width σS determining the interaction time scale.The resulting dissimilarity is obtained as DS = 1 − SS.
4. Hunter-Milton similarity measure
The Hunter–Milton approach compares each spike with its nearest counterpart in the other train and aggregates their coincidence scores into a symmetric similarity. Its free parameter sets the time scale, while event synchronization adapts its coincidence window to local rates.
- The Hunter–Milton method identifies the nearest spike in train y for each spike in train x and quantifies their coincidence with rxy.
- The overall similarity SH is the symmetrized average of directional coincidence scores across the spike trains.
- The parameter τH provides a freely chosen time scale, and dissimilarity is defined as DH = 1 − SH.For identical spike trains, rxy = ryx = 1.
- Event synchronization counts quasi-simultaneous spike appearances and adapts the maximum synchronous lag τij to local spike rates.The method therefore avoids fixing a single time scale beforehand.
- The event-synchronization distance DQ is normalized between 0 and 1, reaching zero exactly when all spikes are synchronous.
D. Assessing clustering quality
Clustering quality is assessed by generating a single-linkage dendrogram from pairwise spike-train distances and evaluating both cluster correctness and separation. The two indices distinguish misclassification from the spacing of correctly recovered clusters.
- Assessing clustering quality: Twenty-nine simulated Hindmarsh–Rose spike trains were organized into three principal clusters, containing 13, 13, and 3 members.Measures were tested for their ability to reproduce this predetermined structure.
- Assessing clustering quality: A hierarchical dendrogram is built from pairwise distance matrices using single linkage, which defines cluster distance as the minimum distance between member sequences.
- Assessing clustering quality: The clustering entropy H quantifies correctness through a confusion matrix whose entries count closest-cluster assignments.A perfect clustering produces a diagonal confusion matrix, whereas misclassifications create non-diagonal entries.
- Assessing clustering quality: H = 1 and F = 0.57 characterize the example dendrogram produced from 29 Hindemarsh–Rose series using the event distance DQ.The colored branches identify the three principal clusters, while black lines mark the separation used for F.
- Assessing clustering quality: The separation index F measures the lengths of the upper branches of the three principal clusters after normalization to the overall distance range.If the clustering is not correct, separation is set to F = 0; otherwise F lies in [0, 1].
E. Correlations between the different measures
The study compares the information carried by spike-train distance measures by optimizing parameterized methods for clustering and correlating their resulting performance-related data. Pairwise distance between measures is then used to construct another hierarchical tree.
- For each time-scale-dependent measure, the parameter value yielding the best clustering result was identified before correlation analysis.
- The data sets were arcsin-transformed and Pearson correlation coefficients were computed across the measures.The transformation was used to better fit a normal distribution.
- A hierarchical cluster tree of the measures was obtained from pairwise distances defined as 1 − correlation using single linkage.
III. RESULTS
The ISI-distance distinguishes spike trains from same and different Hindemarsh-Rose clusters and reproduces the three-cluster structure without parameter optimization. Across comparisons, it performs nearly as well as the best optimized spike-timing distance.
- ISI-distance examples: The ISI-distance shows small deviations for same-cluster trains but larger, long-lasting differences for trains from different clusters.The reported values are DI = 0.019 for same-pattern trains and DI = 0.032 for different-pattern trains.
- ISI-distance clustering: The ISI-distance distance matrix recovers the expected cluster ordering, with smallest within-cluster distances and largest distances between clusters 1 and 2.The 29 neurons were ordered by their known model affiliations.
- ISI-distance clustering: The ISI-distance produces three clearly separated clusters with perfect clustering entropy H = 1 and cluster separation F = 0.66.No misclassifications are reported in the clustering tree.
- Parameter-dependent measures: Parameter-dependent measures yield meaningful clustering only over intermediate parameter ranges; outside those ranges, no clear clustering is recognized.For the Victor-Purpura distance, the optimal separation is F = 0.67 at cV = 0.01.
- Comparison of measures: The ISI-distance performs almost as well as the van Rossum distance and the best Victor-Purpura result, while Hunter-Milton gives the poorest cluster separation.The comparison uses each parameter-dependent measure at its maximum cluster separation and requires no optimization for parameter-free measures.
B. Correlations between the different measures
The six dissimilarity measures are highly correlated overall, but the ISI-distance is the most independent measure in the correlation-based cluster analysis. This difference reflects its use of interspike intervals rather than spike times.
- Correlation analysis: All six spike-train dissimilarity measures show a high overall level of correlation.The minimum reported correlation coefficient is 0.91, between the ISI-distance and Hunter-Milton dissimilarity.
- Measure clustering: The ISI-distance forms the most independent branch, whereas the other measures belong to one large cluster.The corresponding clustering is shown in the measure cluster tree.
- Measure clustering: The ISI-distance differs structurally from the other measures because it is derived from interspike intervals, while they are based on spike times.This distinction is offered as the interpretation of the separate clustering of the ISI-distance.
IV. DISCUSSION
The ISI-distance complements spike-timing measures by comparing interspike intervals, performs as well as the best spike-based measure in simulated clustering, and provides a rate-based perspective. Its application scope includes multivariate use, but phase lags and unassessed practical criteria remain important caveats.
- Discussion: The ISI-distance complements spike-based synchrony measures by comparing relative interspike-interval sizes rather than simultaneous spike occurrences.It also supports visualization of spiking patterns in paired spike trains.
- Discussion: The ISI-distance performed as well as the best spike-based measure, the Victor-Purpura distance, without requiring parameter optimization.It automatically identifies an appropriate time scale, including when firing rates change within a spike train.
- Discussion: The ISI-distance can handle spike trains containing different time scales, whereas parameter-dependent measures may misrepresent regular spiking or bursting.The relevant comparison depends on the parameter selected for the other measures.
- Discussion: The ISI-distance was the most independent of six similarity measures, reflecting its rate-coding basis versus the time-coding basis of the others.The overall correlation level was quite high, indicating that the measures apparently access similar information.
- Discussion: All described measures can also be used in multivariate analyses, including reliability estimates formed by averaging pairwise correlations across trials.The paper notes that phase lags can fool the measures and should be removed by suitably shifting the time series.
- Discussion: Selection for real-data applications should also consider computational cost and noise robustness, which remain unevaluated in this study.The authors identify the ISI-distance combined with a spike-based measure or event synchronization as an appropriate choice based on observed performance and correlation results.
APPENDIX A: DATA
The study illustrates the ISI-distance with cortical-cell recordings from young Long-Evans rats and evaluates it using spike trains generated by a chaotic Hindemarsh-Rose network. The simulated network contains modular neuron groups and uses differential-equation state variables with synaptic weights learned by a Hebbian mechanism.
- Data: The in vitro data came from whole-cell recordings of layer 2 medial entorhinal-cortex neurons in young Long-Evans rats.Cells were selected from a 400-micron slice preparation using superficial position and electrophysiological response characteristics.
- Data: The simulated spike trains were extracted from a larger network of Hindemarsh-Rose neurons operating in the chaotic regime.The network was originally designed to analyze semantic-memory representations using feature-based models.
- Data: Each neuron’s state was described by three first-order differential equations for membrane potential, recovery variable, and slow adaptation current.The variables are denoted X_i, Y_i, and Z_i in the supplied description.
- Data: The network contained 128 neurons organized into 16 modules of 8 neurons each.During learning, inter-neuron synaptic weights were updated using a Hebbian mechanism based on activity variables.