Cost-based metrics fulfill the above general definition of a metric, and are constructed with the following ingredients:

• a list of allowed elementary steps (allowed transformations of spike trains)
• an assignment of non-negative numerical costs to each elementary step

This is a cost-based metric in which the elementary steps and associated costs are:

• insert a spike: cost = 1
• delete a spike: cost = 1
• shift a spike in time: cost = 0

This is a family of cost-based metrics, parametrized by a "cost per unit time" q (units of 1/sec). The elementary steps and associated costs are:

• insert a spike: cost = 1
• delete a spike: cost = 1
• shift a spike by an amount of time t: cost = qt

If q is very small, this becomes the spike count distance. If q is very large, all spike trains are far apart from each other, unless they are nearly identical. For intermediate values of q, the distance between two spike trains is small if they have a similar number of spikes, occurring at similar times.

The motivation for this construction is that neurons which act like coincidence detectors should care about this metric. The value of q corresponds to the temporal precision 1/q of the coincidence detector.

This distance can be calculated efficiently by a dynamic programming algorithm.

This is a family of cost-based metrics, parametrized by a "cost per unit time" q (units of 1/sec). The elementary steps and associated costs are:

• insert a spike (along with an interspike interval): cost = 1
• delete a spike (along with an interspike interval): cost = 1
• change an interspike interval by an amount of time t: cost = qt

Neurons which show post-tetanic potentiation and similar mechanisms might care about this distance. The optimal value of q corresponds to the temporal precision 1/q of the discrimination between bursts of different carrier frequencies (or interspike intervals).
top introduction cost-based metrics D^count D^spike[q] D^interval[q] generalizations and related dynamic programming algorithm and links to code applications and references

Our algorithm can be applied to these generalizations and extensions of the metrics:

• The spike trains reflect activity of multiple identified neurons. That is, each spike is labelled with its neuron of origin. A new kind of elementary step is introduced, in which the label for the neuron of origin is changed. This step is assigned a cost k. k=0 corresponds to ignoring the neuron of origin; higher values of k (up to 2) correspond to increasing sensitivity to the neuron of origin.
  See an application to real data.
  See a clever algorithm for calculating this metric.
  See a generalization of this algorithm, parallel in the cost parameters.
  See another algorithm (D. Diez et al.), more efficient for 5 or more neurons
• As above, but the cost per unit time of moving a spike in one train to match up with a spike in the second train depends on whether the two spikes have the same neuron of origin, or not. Note that the costs per unit time, q (same neuron of origin) and q' (different neuron of origin) are constrained by the triangle inequality. This metric is best thought of as an assignment of a cost to a scheme for "linking" or "aligning" the two spike trains, rather than as a sum of costs of individual transformations.
• The events to be matched are bursts of spikes, rather than individual spikes. Bursts are described by several parameters, each of which influence the cost of matching two bursts. Reference: Keat, Reinagel, Reid, and Meister (2001) Predicting every spike: a model for the responses of visual neurons. Neuron 30, 803-817. Find it online.
• Time is considered cyclic. This is useful for periodic stimuli.
  An application to real data: simple gratings.
  An application to real data: compound gratings.

• The cost assigned to a timeshift t is a concave-down function of t, such as 1-exp(-q|t|). Here the paths of individual spikes in a minimal-cost transformation can cross. Non-crossing of these paths is an essential feature for the dynamic programming algorithm to work.
• A metric D^motif can be defined, by allowing shifts of a subset of 2 or more (possibly non-contiguous) spikes at the same cost as the shift of a single spike.
• The metrics D^spike and D^interval (and even D^motif) can be combined by including two or three kinds of elementary steps: spikes, intervals, and motifs. Different costs/unit time may be associated with each kind of step.

• Google Search for "edit length distance"
• Google Search for "Levenshtein distance"
• Google Search for "earth-mover's distance"
• Wikipedia link

Euclidean distances that are closely related to the spike time metric but are much easier to compute have been introduced by van Rossum (2001). Houghton and Sen (2007) have generalized the von Rossum distance to the multineuronal context. See van Rossum, M.C.W. (2001) A novel spike distance. Neural Computation 13, 751-763 (download) and Houghton, C., and Sen, K. (2007) A new multi-neuron spike-train metric. Neural Computation, in press. (download).

top introduction cost-based metrics D^count D^spike[q] D^interval[q] generalizations and related dynamic programming algorithm and links to code applications and references

Dynamic programming algorithm for D^spike and D^interval

Hint: it is doubly recursive, and reduces the calculation of the distance between spike trains of length N_a and N_b to consideration of spike trains of lengths N_a-1 and N_b-1. Think about the constraints on the "world lines" of each spike in the above diagram for D^spike.

Spike Train Analysis Toolkit, a user-friendly implementation from the Laboratory of Neuroinformatics that includes algorithms for information estimation via the direct method, the binless method, and the metric space method.

Fortran, matlab, and c code for the metrics

L. Allisons code for edit-length distances

Flynn O'Connell's python code for spike time distance and a sliding variant

top introduction cost-based metrics D^count D^spike[q] D^interval[q] generalizations and related applications and references

Victor, J.D. (2015) Spike train distance. In: Encyclopedia of Computational Neuroscience. Ed: D. Jaeger & R. Jung, pp 2808-2814. Springer New York, Heidelberg ,Dordrecht, London, online.
Summary and download

Houghton, C., and Victor, J.D. (2011) Measuring representational distances – the spike metric approach. In Understanding Visual Population Codes – Towards a Common Multivariate Framework for Cell Recording and Functional Imaging. Eds: Nikolaus Kriegeskorte and Gabriel Kreiman. MIT Press, in press.
Abstract and download

Victor, J.D., and Purpura, K.P. (2010) Spike Metrics. In: Analysis of Parallel Spike Trains. Ed. Stefan Rotter and Sonja Gruen. Springer, pp. 129-156.
Abstract and download

Goldberg, D.H., Victor, J.D., Gardner, E.P., and Gardner, D., (2009) Spike Train Analysis Toolkit: Enabling wider application of information-theoretic techniques to neurophysiology. Neuroinformatics 7, 165-178.
Abstract, download, and links to code

Victor, J.D. (2005) Spike train metrics. Current Opinion in Neurobiology 15, 585–592.
Abstract and download

Use in Topological Data Analysis

Guidolin, A., Desroches, M., Victor, J.D., Purpura, K.P., and Rodrigues, R. (2022) Geometry of spiking patterns in early visual cortex: a topological data analytic approach. Journal of the Royal Society Interface, accepted.
Abstract and download

Theory and algorithms

Diez, D.M., Schoenberg, F.P., and Woody, C.D. (2012) Algorithms for computing spike time distance and point process prototypes with application to feline neuronal responses to acoustic stimuli. J. Neurosci. Meth. 203, 186-192 Download
This describes an alternative algorithm for the multineuronal metric, more efficient than the Aronov (2009) algorithm when there are 5 or more neurons

Dubbs, A.J., Seiler, B.A., and Magnasco, M.O. (2009) A fast L^p spike alignment metric. arXiv:0907.3137v2.
Dubbs, A.J., Seiler, B.A., and Magnasco, M.O. (2010) A fast L^p spike alignment metric. Neural Computation 11, 2785-2808.
Download
These describe an alternative algorithm based on bipartitre graphs for calculating D^spike and related metrics, and also discuss some theoretical advantages for the use of an L² metric, rather than the L¹ metric described here.

Kreuz, T., Haas, J.S., Morelli, A., Abarbanel, H.D.I., and Politi, A. (2007) Measuring spike train synchrony. J Neurosci Methods 165, 151.
Download and code
This describes another interval-based metric for comparing spike trains, and compares six different metrics, including D^spike

Victor, J.D., Goldberg, D., and Gardner, D. (2007) Dynamic programming algorithms for comparing multineuronal spike trains via cost-based metrics and alignments. J. Neurosci. Meth. 161, 351-360.
Abstract and download
This describes dynamic programming algorithms for computing generalizations of D^spike, parallel in one or more cost parameters. The algorithm also extends to L^p-metrics

Aronov, D. (2005) A metric-space approach to the study of information coding by neuronal populations. Senior Thesis, Columbia University.
Download

Aronov, D., and Victor, J. (2004) Non-Euclidean properties of spike train metric spaces. Phys. Rev. E69, 61905.
Abstract, download, and related notes

Aronov, D. (2003) Fast algorithm for the metric-space analysis of simultaneous responses of multiple single neurons. J. Neuroscience Methods 124, 175-179.
Abstract, download, and code

Visual neurophysiology

More references from our lab

Keat, Reinagel, Reid, and Meister (2001) Predicting every spike: a model for the responses of visual neurons. Neuron 30, 803-817. pdf from Neuron
This describes a generalization to sequences of burst events. The authors also describe a novel approach to pruning the dynamic programming algorithm.

Hitt, Dodge, and Barlow (2003) Information content in responses of Limulus optic nerve fibers. Arvo Abstract 3253. Link to ARVO abstracts

Auditory neurophysiology

Machens, C.K., Schutze, H., Franz, A., Kolesnikova, O., Stemmler, M.B., Ronacher, B., and Herz, A.V.M. (2001) Single auditory neurons rapidly discriminate conspecific communication signals. Nature Neurosci. 6, 341-342. pdf from Nature Neuroscience pdf from Ronacher's site

Wohlgemuth, S., and Ronacher, B. (2007) Auditory discrimination of amplitude modulations based on metric distances of spike trains. J. Neurophysiol. 97, 3082-3092. pdf from J. Neurophysiol.

Olfactory neurophysiology

Bäcker, A., MacLeod, K., Wehr, M., and Laurent, G. (1998) Disruption of neuronal synchronization impairs reliable reconstruction of odors by beta lobe cells but not by projection neurons of the antennal lobe of the locust. Soc. Neurosci. Abstr. 24, 911.

MacLeod, K., Bäcker A., & Laurent, G. (1998) Who reads temporal information contained across synchronized and oscillatory spike trains? Nature 395, 693-698.

Somatosensosry neurophysiology

Brasselet, R., Johansson, R.S., and Arleo, A. (2011) Quantifying neurotransmission reliability through metrics-based information analysis. Neural Computation 23, 852-81·
Download

Weber, A.I., Saal, H.P., Lieber, J.D., Cheng, J.-W., Manfredi, L.R., Dammann, J.F. III, and Bensmaia, S.J. (2013) Spatial and temporal codes mediate the tactile perception of natural textures. Proc. Natl. Acad. Sci. USA 110, 17107-17112.
Download

Gustatory neurophysiology

Di Lorenzo, P., Chen, J.-Y., and Victor, J.D. (2009) Quality time: Representation of a multidimensional sensory domain through temporal coding. J. Neurosci. 29, 9227-9238.
Abstract and download

Di Lorenzo, P.M, and Victor, J.D. (2003) Taste response variability and temporal coding in the nucleus of the solitary tract of the rat. J. Neurophysiol. 90, 1418-1431.
Abstract and download

Electric sense

Kreiman, G., Krahe, R., Metzner, W., Koch, C., & Gabbiani, F. (2000) Robustness and variability of neuronal coding by amplitude-sensitive afferents in the weakly electric fish Eigenmannia. J. Neurophysiol. 84, 189-204. Abstract

Birdsong

Huetz, C., Del Negro, C., Lebas, N., Tarroux, P. & Edeline, J.M. (2006) Contribution of spike timing to the information transmitted by HVC neurons. Eur J Neurosci 24, 1091-108. PubMed entry

Le Wang, Narayan, Grana, Shamir, and Sen (2007) Cortical discrimination of complex natural stimuli: Can single neurons match behavior? J. Neurosci. 27,582-589 J. Neurosci. online
The authors use D^spike, D^interval, and the van Rossum metric.

Neural development

Computational studies

Cost-based metrics ("edit length metrics") in EEG classfication

Geophysics: earthquake patterns