Abstract

The distance between a pair of spike trains, quantifying the differences between them, can be measured using various metrics. Here we introduce a new class of spike train metrics, inspired by the Pompeiu-Hausdorff distance, and compare them with existing metrics. Some of our new metrics (the modulus-metric and the max-metric) have characteristics that are qualitatively different from those of classical metrics like the van Rossum distance or the Victor and Purpura distance. The modulus-metric and the max-metric are particularly suitable for measuring distances between spike trains where information is encoded in bursts, but the number and the timing of spikes inside a burst do not carry information. The modulus-metric does not depend on any parameters and can be computed using a fast algorithm whose time depends linearly on the number of spikes in the two spike trains. We also introduce localized versions of the new metrics, which could have the biologically relevant interpretation of measuring the differences between spike trains as they are perceived at a particular moment in time by a neuron receiving these spike trains.

1.  Introduction

In recent years, several spike train distances, some inspired by existing mathematical distances and others not, have been proposed and used to measure the variability of neural activity (Victor & Purpura, 1996, 1997; van Rossum, 2001; Schreiber, Fellous, Whitmer, Tiesinga, & Sejnowski, 2003; Victor, 2005; Schrauwen & Van Campenhout, 2007; Kreuz, Haas, Morelli, Abarbanel, & Politi, 2007; Kreuz, Chicharro, Andrzejak, Haas, & Abarbanel, 2009; Naud, Gerhard, Mensi, & Gerstner, 2011; Kreuz, Chicharro, Greschner, & Andrzejak, 2011; Kreuz, Chicharro, Houghton, Andrzejak, & Mormann, 2013). The distance between two spike trains reflects their similarity or dissimilarity. Because how information is represented in the spatiotemporal patterns of spike times exchanged by neurons is still a heavily debated topic in neuroscience, metrics based on different neural codes are available. Traditionally it was thought that the mean firing rate of neurons encapsulated all the relevant information exchanged by neurons. This idea dates back to the work of Adrian (1926), who showed that the firing rate of motor neurons is proportional to the force applied. More recently, scientists have revealed increasing evidence of the importance of precise spike timings to representing information in the brain (Bohte, 2004; VanRullen, Guyonneau, & Thorpe, 2005; Tiesinga, Fellous, & Sejnowski, 2008). For example, temporally structured multicell spiking patterns were observed in the hippocampus and cortex and were associated with memory traces (Nádasdy, Hirase, Czurkó, Csicsvari, & Buzsáki, 1999; Ji & Wilson, 2007), while the coding of information in the phases of spikes relative to a background oscillation has been observed in many brain regions (Lee, Simpson, Logothetis, & Rainer, 2005; Jacobs, Kahana, Ekstrom, & Fried, 2007; Fries, Nikolić, & Singer, 2007; Montemurro, Rasch, Murayama, Logothetis, & Panzeri, 2008; Siegel, Warden, & Miller, 2009). This change in viewpoints from rate codes to spike time codes is also reflected in spike train metrics.

The most basic metrics are the ones that rely on counting the total number of spikes within a spike train. A major drawback of such an approach is that all the temporal structure is lost. Although binning techniques were introduced as a way to overcome this loss by dividing the spike train into discrete bins, the temporally encoded information within a bin was neglected (Geisler, Albrecht, Salvi, & Saunders, 1991). Other, more complex spike train metrics can be obtained by focusing on the precise spike timing instead of their total count. One example is the van Rossum (2001) distance, which is calculated by filtering the time series corresponding to the raw spike train with a smoothing kernel, typically an exponential one, and then using the standard Euclidean distance. Another metric is the Schreiber et al. (2003) correlation-based measure, which uses a symmetric (gaussian) filter. In both cases, the choice of the kernel's parameters is arbitrary and has a high influence on the properties of the metric.

Another metric was introduced by Victor and Purpura (1996, 1997). According to this metric, the distance between two spike trains is given by the minimum cost of basic operations needed to transform one spike train into the other. The basic operations are insertion or deletion of spikes, with a cost of 1, and the shifting of a spike, with a cost of , where q is a parameter and the shifting interval. The parameter q significantly influences the behavior of the metric: for q = 0, the metric counts the difference in the total number of spikes, while for large values of q, the metric returns the number of noncoincident spikes.

Kreuz et al. (2007, 2009, 2011, 2013) more recently introduced a series of parameter-free and timescale-independent measures of spike train synchrony.

Here we introduce a new class of spike train metrics inspired by the Pompeiu-Hausdorff distance between two nonempty compact sets (Pompeiu, 1905; Hausdorff, 1914). Preliminary results have been presented in abstract form in Rusu and Florian (2010).

Section 2 describes the new metrics. Section 3 introduces the localization of spike train metrics. Section 4 presents the application of the new metrics in several simulations that explore the features of the metrics and compare the new metrics with existing ones. After sections that discuss the results and present the conclusions, the letter ends with a series of appendixes that include detailed mathematical proofs that the introduced metrics are finite and obey the mathematical requirements for metrics, as well as computer algorithms that allow fast computation of the modulus-metric.

2.  A New Class of Spike Train Metrics

We consider bounded, nonempty spike trains of the form
formula
2.1
where are the ordered spike times and is the number of spikes in the spike train. We consider spike trains with no overlapping spikes. If n>1, then t(i−1)<t(i), . We denote by a and b some bounds of the considered spike trains, , with , finite, and a<b. We denote by the set of all possible such spike trains bounded by a and b. We study metrics that compute the distances between two spike trains T and from .
The new metrics that we introduce are inspired by the Pompeiu-Hausdorff distance (Pompeiu, 1905; Hausdorff, 1914). When applied to a pair of spike trains, the Pompeiu-Hausdorff distance h returns the largest difference in absolute value between the timing of a spike in one train and the timing of the closest spike in the other spike train,
formula
2.2
or, equivalently, the minimal number such that the closed -neighborhood of T includes and the closed -neighborhood of includes T:
formula
2.3
Another equivalent form of the Pompeiu-Hausdorff distance is the following (Papadopoulos, 2005; Rockafellar & Wets, 2009; Deza & Deza, 2009):
formula
2.4
We introduce a distance d between an arbitrary timing and a spike train T:
formula
2.5
Equation 2.2 can then be rewritten as
formula
2.6
and equation 2.4 as
formula
2.7
Because the global supremum is achieved on the interval [a, b], we also have
formula
2.8
The Pompeiu-Hausdorff metric has quite poor discriminating power; for many variations of the spike trains, the distances will be equal, and any spike train space endowed with this metric would be highly clustered. Our new metrics generalize the form of the Pompeiu-Hausdorff distance given in equation 2.8 by introducing features that are more sensitive to spike timings.

2.1.  The Max-Metric.

We consider to be the space of all continuous, positive functions that satisfy the condition ,
formula
2.9
Because
formula
2.10
formula
2.11
and because is continuous, a sufficient condition for satisfying equation 2.9 is that .
On compact sets, continuous functions are bounded (Protter, 1998). We denote by m the upper bound of on the interval [0, ba]:
formula
2.12
In typical applications, has a maximum for x=0 and is a decreasing function of x, for example, an exponential,
formula
2.13
or a gaussian,
formula
2.14
with a positive parameter.
We introduce the max-metric as
formula
2.15
The max-metric integrates, through the variation of s along the interval [a, b] that contains the two spike trains, the maximum of the difference in absolute value between the distances from a point x in that interval to the two spike trains, weighted by the kernel , which focuses locally around s. Figure 1 shows how the distance between two spike trains is computed using the max-metric.
Figure 1:

The modulus-metric and the max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The distances between a timing x and the spike trains, d(x, T) and , as a function of x. (D) The difference as a function of x. The modulus-metric distance is the area under this curve. (E) The kernel as a function of s with a fixed x=250 ms. is an exponential (see equation 2.13) with a decay constant ms. (F) The weighted difference as a function of s for discrete values of x. (G) The supremum of the weighted difference, , as a function of s. The max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

Figure 1:

The modulus-metric and the max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The distances between a timing x and the spike trains, d(x, T) and , as a function of x. (D) The difference as a function of x. The modulus-metric distance is the area under this curve. (E) The kernel as a function of s with a fixed x=250 ms. is an exponential (see equation 2.13) with a decay constant ms. (F) The weighted difference as a function of s for discrete values of x. (G) The supremum of the weighted difference, , as a function of s. The max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

The max-metric is a generalization of the Pompeiu-Hausdorff distance, since in the particular case that , we have .

In appendix  A we show that dm is finite and that it satisfies the properties of a metric. We also show that regardless of the kernel , all the max-metrics are topologically equivalent to each other (O'Searcoid, 2007) because they are equivalent to the Pompeiu-Hausdorff distance. Each metric will generate the same topology, and thus any topological property is invariant under a homeomorphism. This means that the metrics generate the same convergent sequences in the space of spike trains . Thus, learning rules derived from these metrics will converge in the same way regardless of the choice of .

2.2.  The Modulus-Metric.

We define the modulus-metric as
formula
2.16
Figures 1A to 1D show how the distance between two spike trains is computed using the modulus-metric. The modulus-metric is a particular case of the max-metric in the limit that is
formula
2.17

The modulus-metric does not depend on any kernels or parameters, and it also allows a fast computer implementation with linear complexity. This is because the graph of the function is made out of line segments that join or end in the following points: all timings of spikes in the two spike trains T and , the time moments that lie at the middle of the interval between two neighboring spikes from the same spike train, the time moments that lie at the middle of the interval between a pair of neighboring spikes where the two spikes belong to different spike trains, and the bounds a and b. This is exemplified in Figures 1D and 3D. We denote by the set of these points. In order to compute the integral of this function , it is sufficient to compute the function at the points from . Since between these points the function is linear, the integral can be then computed exactly.

Algorithm 1 presents an implementation of the do metric in pseudocode. (All algorithms are in appendix  E.) In this algorithm, the function is computed in a set of points that includes but also other points. In Algorithm 2, the set as well as the value of in the points of is computed with a single pass through the spikes in the two spike trains. The algorithms’ duration depends linearly on the number of spikes in the two spike trains, . (Implementations in Python and C++ of the two algorithms are available at https://github.com/modulus-metric/.)

It can be shown that the distance do is finite and that it satisfies the properties of a metric by particularizing the proofs in appendix  D with , .

2.3.  The Convolution Max-Metric.

The max-metric can also be given in a convolution form. To construct this form of the metric, we consider an arbitrary continuous, smooth, positive kernel , with the properties that it is strictly increasing for x<0 and strictly decreasing for x>0, with finite and positive. We thus have . Typically is an exponential,
formula
2.18
with being a positive parameter. We convolve the two spike trains T and with the filtering kernel to obtain
formula
2.19
formula
2.20
We denote by the set of all possible filtered spike trains from .

We also consider a function that is strictly positive, that is derivable on (0, ba), and has a bounded derivative.

The convolution max-metric is defined as
formula
2.21
Figure 2 shows how the distance dc between two spike trains is computed. In appendix  B, we show that dc is finite and that it satisfies the properties of a metric.
Figure 2:

The convolution max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The spike trains T and filtered with an exponential kernel (see equation 2.18) with a decay constant ms. (D) The difference as a function of x. (E) The kernel as a function of s with a fixed x=250 ms. is an exponential (see equation 2.13) with a decay constant ms. (F) The weighted difference as a function of s for discrete values of x. (G) The supremum of the weighted difference, . The convolution max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

Figure 2:

The convolution max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The spike trains T and filtered with an exponential kernel (see equation 2.18) with a decay constant ms. (D) The difference as a function of x. (E) The kernel as a function of s with a fixed x=250 ms. is an exponential (see equation 2.13) with a decay constant ms. (F) The weighted difference as a function of s for discrete values of x. (G) The supremum of the weighted difference, . The convolution max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

3.  Localized Metrics

In the case of the max metric, with or without convolution, the use of the kernel served the purpose of providing a local perspective, around each point within [a, b], of the distance between the spike trains. These local perspectives were then integrated in the final distance. In this section, we introduce different metrics that also depend on a kernel , but for which the kernel has a different purpose. Here, the kernel may be regarded as a magnifying glass to be used to focus on one specific area of the spike trains. The kernel should be a continuous, positive function, . Similarly with , because is a continuous function with bounded support, it is bounded:
formula
3.1
Such a metric is biologically relevant if, for example, we take into consideration how a neuron responds to input spikes. Recent spikes influence the neuron more than old ones do. If we would like to measure the distance between two spike trains according to how the differences between them influence the activity of a neuron at a particular moment of time, recent differences should be taken into account with a greater weight than differences in the distant past. For the localized metrics, could thus model the shape of postsynaptic potentials (PSP) that reflects the dynamics of the effect of one presynaptic spike on the studied neuron. Thus, could typically be an exponential, (see equation 2.13), an alpha function,
formula
3.2
a difference between two exponentials,
formula
3.3
or, if we model the postsynaptic potential generated in an integrate-and-fire neuron by a synaptic current that is a difference between two exponentials,
formula
3.4
where , , and are positive parameters.

3.1.  Localized Max-Metric.

We introduce the localized max-metric as
formula
3.5
Figure 3 shows how the distance dl between two spike trains is computed. The differences between the spike trains that account the most for the distance are those that are close to b. The shape of has a high impact on the behavior of the metric. In appendix  C, we show that the distance dl is finite and that it satisfies the properties of a metric.
Figure 3:

The localized max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The distances between a timing x and the spike trains, d(x,T) and , as a function of x. (D) The difference as a function of x. (E) The supremum, , as a function of s. (F) The kernel as a function of s, an exponential (see equation 2.13) with a decay constant ms. (G) The supremum weighted by the kernel, , as a function of s. The localized max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

Figure 3:

The localized max-metric. (A) Spike train ms. Each spike time is represented as a vertical bar. (B) Spike train ms. (C) The distances between a timing x and the spike trains, d(x,T) and , as a function of x. (D) The difference as a function of x. (E) The supremum, , as a function of s. (F) The kernel as a function of s, an exponential (see equation 2.13) with a decay constant ms. (G) The supremum weighted by the kernel, , as a function of s. The localized max-metric distance is the area under this curve. In panels C, D, and G, the dashed vertical lines represent the timing of spikes in T and .

3.2.  Localized Modulus-Metric.

The modulus metric can also be given in a localized form:
formula
3.6
In appendix  D, we show that dn is finite and that it satisfies the properties of a metric.

3.3.  Localizing the van Rossum Metric.

A localization by a kernel similar to the one we applied in equations 3.5 and 3.6 can also be applied to existing metrics. Let . Consider the van Rossum (2001) distance defined as
formula
3.7
where
formula
3.8
formula
3.9
H is the Heaviside step function, H(x)=0 if x<0 and H(x)=1 if , and is defined in equation 2.18. When localized with , the distance becomes
formula
3.10
Here, may be chosen to have the same qualitative properties as the kernels used in equations 3.2 to 3.4.

4.  Simulation Results

We analyzed the behavior of the introduced metrics through computer simulations using simple setups. Across all simulations, was an exponential (see equation 2.13) with ms. For the localized metrics dl and dn, also was an exponential (see equation 2.13) with ms. The convolution kernels for the dc and van Rossum distances were chosen as exponentials (see equation 2.18) with ms. The width of the gaussian filter for the Schreiber et al. distance was 10 ms. For the Victor and Purpura distance, we set q=0.2 ms−1, except in Figures 7 to 9 where we also used q=0.001 ms−1 as in Dauwels, Vialatte, Weber, and Cichocki (2009). We set a=0 ms and b the maximum length of spike trains (either 200 or 500 ms, except in Figure 12, where the length was variable).

4.1.  Inserting or Shifting One Spike.

We computed the distances between a particular spike train T and a spike train obtained from it by either inserting or shifting one spike. In the insertion case, was generated by inserting a spike into T at various timings. In the shifting case, was generated by shifting a particular spike of T. The distance was plotted against the time of the inserted spike or of the shifted spike to see how the change is reflected by the metrics. To compute the distance, we used the introduced metrics, a simple spike count distance (c), the van Rossum (2001) (dR), Victor and Purpura (1996, 1997) (dVP), Schreiber et al. (2003) (s), and Pompeiu-Hausdorff (h) distances, as well as the ISI distance (ki) and the improved SPIKE distance (ks) by Kreuz et al. (2007, 2013). The spike count distance is defined as
formula
4.1
where n and are the number of spikes in each train. The spike trains were 200 ms long.

The results for the insertion case are presented in Figure 4. The Victor and Purpura distance was constant since the cost of adding and removing a spike is fixed at 1 regardless of its timing. Similarly, the van Rossum metric was insensitive to the time of the inserted spike, a result that can be also shown analytically (van Rossum, 2001). The spike count distance remained constant regardless of where the spike was inserted. The results were qualitatively different in the case of our newly introduced distances, with the exception of the convolution max-metric, and in the case of the Kreuz et al. and Schreiber et al. metrics. In the case of the Pompeiu-Hausdorff distance, max-metric, modulus-metric, of the localized variants of the max-metric and the modulus-metric, and of the Schreiber et al. and Kreuz et al. metrics, the insertion time of the spike had a significant impact on the outcome (see Figure 4). When the inserted spike overlapped an existing spike, the Schreiber et al. distance had a low value but remained nonzero, while our new metrics, with the exception of the convolution max-metric, as well as the Kreuz et al. metrics, returned a zero distance. It can also be seen that the localized distances were strongly influenced by the shape of the kernel.

Figure 4:

Metric comparison: inserting a spike. We computed the distance between the spike train ms and one obtained from this spike train by inserting a spike at different locations. At each time ms, a spike was inserted to generate , and the distance between T and was computed and plotted against x. (A) The spike train T. (B) The van Rossum distance. (C) The Victor and Purpura distance. (D) The spike count distance. (E) The Schreiber et al. distance. (F) The Kreuz et al. improved SPIKE distance. (G) The Kreuz et al. ISI distance. (H) The Pompeiu-Hausdorff distance. (I) The max-metric. (J) The convolution max-metric. (K) The modulus-metric. (L) The localized max-metric. (M) The localized modulus-metric.

Figure 4:

Metric comparison: inserting a spike. We computed the distance between the spike train ms and one obtained from this spike train by inserting a spike at different locations. At each time ms, a spike was inserted to generate , and the distance between T and was computed and plotted against x. (A) The spike train T. (B) The van Rossum distance. (C) The Victor and Purpura distance. (D) The spike count distance. (E) The Schreiber et al. distance. (F) The Kreuz et al. improved SPIKE distance. (G) The Kreuz et al. ISI distance. (H) The Pompeiu-Hausdorff distance. (I) The max-metric. (J) The convolution max-metric. (K) The modulus-metric. (L) The localized max-metric. (M) The localized modulus-metric.

The results for the shifting case are presented in Figure 5. When the spike at t(4) was shifted, the Victor and Purpura and van Rossum distances were dependent only on the width of the shifting interval. These results are confirmed by analytical derivations (Victor & Purpura, 1996; van Rossum, 2001). As in the previous case, the spike count distance was insensitive to the shift operation and remained zero since the number of spikes did not change. In contrast, our newly introduced distances, with the exception of the convolution max-metric, and the Schreiber et al. and Kreuz et al. metrics, showed a dependence not only on the width of the shifting interval but also on the particular timing of the shifted spike. The results are similar to the ones in Figure 4 of Kreuz et al. (2011).

Figure 5:

Metric comparison: shifting a spike. We computed the distance between the spike train ms and one obtained from this spike train by shifting the spike at t(4)=125 ms. The spike was shifted at timings ms to generate , and the distance between T and was computed and plotted against x. (A) The spike train T. (B) The van Rossum distance. (C) The Victor and Purpura distance. (D) The spike count distance. (E) The Schreiber et al. distance. (F) The Kreuz et al. improved SPIKE distance. (G) The Kreuz et al. ISI distance. (H) The Pompeiu-Hausdorff distance. (I) The max-metric. (J) The convolution max-metric. (K) The modulus-metric. (L) The localized max-metric. (M) The localized modulus-metric.

Figure 5:

Metric comparison: shifting a spike. We computed the distance between the spike train ms and one obtained from this spike train by shifting the spike at t(4)=125 ms. The spike was shifted at timings ms to generate , and the distance between T and was computed and plotted against x. (A) The spike train T. (B) The van Rossum distance. (C) The Victor and Purpura distance. (D) The spike count distance. (E) The Schreiber et al. distance. (F) The Kreuz et al. improved SPIKE distance. (G) The Kreuz et al. ISI distance. (H) The Pompeiu-Hausdorff distance. (I) The max-metric. (J) The convolution max-metric. (K) The modulus-metric. (L) The localized max-metric. (M) The localized modulus-metric.

4.2.  Bursts.

We generated a template spike train T containing three bursts and one isolated spike. We computed, in six setups, using various metrics, the distance between T and another spike train obtained from T by removing one spike from a burst, inserting one spike into a burst, or removing or inserting one spike not belonging to the bursts. After computing the distances for each of the setups, the distances for each metric were normalized to the maximum distance for that particular metric among the setups. The normalized distances are shown, for each setup, in Figure 6. When a spike is added to a burst or removed from a burst, the max-metric and modulus-metric distances, as well as the Kreuz et al. distances, are close to zero. Those distances become nonnegligible when a solitary spike, far from a burst, is removed or a new spike is added far from a burst. The normalized convolution max-metric and the van Rossum distances are close to 1 and the Victor and Purpura distance is exactly 1 in all setups. The Schreiber et al. distance exhibited an intermediate behavior, but it also remained nonnegligible in all setups.

Figure 6:

Dealing with bursts. We measured distances between a template spike train that includes bursts and a spike train obtained from the template by adding or removing one spike. Arrows indicate where the spike was added or removed. Left: The pair of spike trains. Right: The normalized distances (do, modulus metric; dm, max metric; dc, convolution max-metric; dR, van Rossum; dVP, Victor and Purpura; s, Schreiber et al.; ks, Kreuz et al., improved SPIKE distance; ki, Kreuz et al. ISI distance). (A, B) One spike removed from a burst. (C, D) One spike inserted into a burst. (E) One isolated spike removed. (F) One isolated spike inserted.

Figure 6:

Dealing with bursts. We measured distances between a template spike train that includes bursts and a spike train obtained from the template by adding or removing one spike. Arrows indicate where the spike was added or removed. Left: The pair of spike trains. Right: The normalized distances (do, modulus metric; dm, max metric; dc, convolution max-metric; dR, van Rossum; dVP, Victor and Purpura; s, Schreiber et al.; ks, Kreuz et al., improved SPIKE distance; ki, Kreuz et al. ISI distance). (A, B) One spike removed from a burst. (C, D) One spike inserted into a burst. (E) One isolated spike removed. (F) One isolated spike inserted.

Figure 7:

Normalized average distances between a template spike train and another one generated from it as a function of the probability p of removing spikes from the template spike train and of jitter added to spikes. (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

Figure 7:

Normalized average distances between a template spike train and another one generated from it as a function of the probability p of removing spikes from the template spike train and of jitter added to spikes. (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

4.3.  Discriminating Timing Precision versus Event Reliability.

Timing precision and event reliability are distinct characteristics of the variation of a spike train (Tiesinga et al., 2008). In computational neuroscience, there is a need for spike train metrics to both characterize the overall variability and parse out the precision and reliability separately (Toups, Fellous, Thomas, Sejnowski, & Tiesinga, 2012). We investigated how our new metrics and existing metrics perform in discriminating precision versus event reliability (Dauwels et al., 2009). We generated a 200 ms template spike train T using a Poisson process with a rate of 100 Hz. We also generated various spike trains obtained from T by both applying to spikes from T a gaussian jitter with zero mean and variance in the range of 0 and ms, and removing spikes with a probability p ranging from 0 to pm=80%. For each pair, we ran 100 trials, where in each trial we generated randomly a new . For each trial, we computed the distances between T and using various metrics and then averaged the results over the trials. For each metric, average distances were then normalized to the maximum average distance across the parameters. The results are presented in Figures 7 to 10. Figure 7 represents the normalized average distances as a function of and p. Figures 8 and 9 represent sections trough the graphs in Figure 7, for illustrating more clearly the dependencies of distances on reliability and precision.

Figure 8:

Normalized average distances as a function of the probability of removing spikes p, for several levels of jitter (sections through the panels in Figure 7). (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

Figure 8:

Normalized average distances as a function of the probability of removing spikes p, for several levels of jitter (sections through the panels in Figure 7). (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

Figure 9:

Normalized average distances as a function of jitter , for several levels of the probability of removing spikes p (sections through the panels in Figure 7). (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

Figure 9:

Normalized average distances as a function of jitter , for several levels of the probability of removing spikes p (sections through the panels in Figure 7). (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

If Figure 10, we tried to illustrate how reliability compares to the precision in determining the variability of the distances. For a metric , it is not possible to compare directly the partial derivatives and because they are variables of different physical dimensions: p is dimensionless, while represents a time interval. In order to compare them, we take into consideration that the relevant intervals on which p and vary are practically bounded. Choosing and pm is theoretically arbitrary, but in practice, pm is something slightly less than 1 and the relevant is constrained by the timescales of the considered spike trains and the distances that depend on timescale-like parameters. We use these practical bounds so that the variances with respect to the two parameters can be brought to the same dimension and compared. By dividing the intervals [0, pm] and into the same number of bins N, we get two-dimensional pixels of size and on which we can consider that the variation of d along the two axes can be compared. We computed
formula
4.2
where was replaced with the considered metrics. In areas of the p, space where is positive, we may say that the metric is more sensitive to reliability, while in areas where it is negative, the metric is more sensitive to the precision of spikes. The interpretation of results should take into consideration the caveat that changes of the ratio may change the sign of .
Figure 10:

Illustration of the dominance of reliability over precision for determining the variability of distances (see the main text). In the white areas, the distance varies more with respect to reliability than with respect to precision. In the dark areas, the dominance is reversed. (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

Figure 10:

Illustration of the dominance of reliability over precision for determining the variability of distances (see the main text). In the white areas, the distance varies more with respect to reliability than with respect to precision. In the dark areas, the dominance is reversed. (A) Modulus-metric. (B) Max-metric. (C) Victor and Purpura, q=0.2 ms−1. (D) Victor and Purpura, q=0.001 ms−1. (E) Van Rossum. (F) Schreiber et al. (G) Kreuz et al. improved SPIKE distance. (H) Kreuz et al. ISI distance.

As expected, the Victor and Purpura distance with q=0.001 ms−1 does not depend at all on the precision, just on reliability, since it basically counts the difference in the number of spikes between the spike trains. Our modulus-metric and max-metrics have a stronger dependence on reliability than on precision on most of the considered range, except for high reliability (very low p) where the dependence on precision is still dominant. For the van Rossum and Schreiber et al. distances, the dependence on reliability and precision is somehow balanced. The Kreuz et al. distances and the Victor and Purpura distance with q=0.2 ms−1 have a stronger dependence on precision than on reliability. For all distances, except for the Victor and Purpura distance with q=0.001 ms−1, the sensitivity to reliability increases with the unreliability (with the probability p of spikes not being fired). These results may be different for a different choice of the ratio, of the timescales of the considered spike trains and of the parameters of the parameter-dependent distances. Our results are similar to those of Dauwels et al. (2009) for the Victor and Purpura distance q=0.001 ms−1, but different for the Schreiber et al. distance, probably because Dauwels et al. (2009) used a different approach for modeling the unreliability of spikes.

4.4.  Correlations.

We explored the correlation between the newly introduced metrics and the classical Victor and Purpura and van Rossum distances. We generated a 500 ms Poisson spike train with a firing rate of 20 Hz. From this spike train, we generated a new one by adding a gaussian jitter with zero mean and 20 ms variance. We considered only generated and jittered spike trains that contained 10 spikes. We then measured the distance between the original and the jittered spike train using various metrics. We repeated this in 1000 trials, where for each trial, the original spike train and the jitter were generated randomly. For each metric, distances were normalized to the mean value across samples. The results are displayed in Figure 11. Table 1 shows the correlation coefficients between the max-metric, the modulus-metric, the convolution max-metric, the Schreiber et al. distance, the Kreuz et al. distances, and, respectively, the van Rossum and Victor and Purpura distances.

Figure 11:

Correlations between the Victor and Purpura (left column) and the van Rossum (right column) metrics and, respectively, the distances computed with our newly introduced metrics, the Schreiber et al. and the Kreuz et al. metrics. We computed distances between Poisson spike trains and a jittered version of them. Figures represent the distribution of pairs of the normalized values of distances measured with the different metrics. (A) Max-metric. (B) Convolution max-metric. (C) Modulus-metric. (D) Schreiber et al. (E) Kreuz et al. improved SPIKE distance. (F) Kreuz et al. ISI distance. (G) Victor and Purpura/van Rossum.

Figure 11:

Correlations between the Victor and Purpura (left column) and the van Rossum (right column) metrics and, respectively, the distances computed with our newly introduced metrics, the Schreiber et al. and the Kreuz et al. metrics. We computed distances between Poisson spike trains and a jittered version of them. Figures represent the distribution of pairs of the normalized values of distances measured with the different metrics. (A) Max-metric. (B) Convolution max-metric. (C) Modulus-metric. (D) Schreiber et al. (E) Kreuz et al. improved SPIKE distance. (F) Kreuz et al. ISI distance. (G) Victor and Purpura/van Rossum.

Table 1:
Correlation Coefficients between the van Rossum and Victor and Purpura Distances and, Respectively, Other Distances.
DistancedR Correlation CoefficientdVP Correlation Coefficient
dm 0.54 0.48 
do 0.54 0.47 
dc 0.84 0.81 
s 0.87 0.88 
ks 0.51 0.70 
ki 0.43 0.44 
dR 1.00 0.78 
dVP 0.78 1.00 
DistancedR Correlation CoefficientdVP Correlation Coefficient
dm 0.54 0.48 
do 0.54 0.47 
dc 0.84 0.81 
s 0.87 0.88 
ks 0.51 0.70 
ki 0.43 0.44 
dR 1.00 0.78 
dVP 0.78 1.00 

Note: Computed from data presented in Figure 11.

4.5.  Computation Speed.

We computed the distances between pairs of randomly generated spike trains, the spike trains within a pair having the same number of spikes n. We varied n from 5 to 500 while keeping constant the firing rate of the spike trains. The trains were generated by randomly choosing the n firing times from a uniform distribution between 0 and nT, where T = 35 ms. We measured the average time needed to compute distances as a function of the number of spikes. Performance was measured using C++ implementations of the metrics, running on an Intel Core 2 processor. The Victor and Purpura metric was computed using the algorithm in Victor and Purpura (1996). The van Rossum metric was implemented using the exact, optimized algorithm presented in Houghton and Kreuz (2012) (A1) and a discrete-time integration with a time step of 1 ms (A2), which turned out to be slightly faster than the optimized one. The Schreiber et al. and the Kreuz et al. metrics have also been computed using a 1 ms integration time step. The modulus-metric was implemented using algorithms 1 and 2. (The code used for all metrics is available at https://github.com/modulus-metric/.) The results are presented in Figure 12. In Figure 12A, the results were averaged over 1000 trials; in the other panels, the results were averaged over 10,000 trials.

Figure 12:

Time needed for computing various distances, as a function of the number of spikes in the spike trains. Note the different timescales. (A) dm, max-metric. s, Schreiber et al. (B) dVP, Victor and Purpura; ks, Kreuz et al. improved SPIKE distance. (C) dR, van Rossum; A1, algorithm by Houghton and Kreuz (2012); A2, discrete time integration; ki, Kreuz et al. ISI distance. (D) do, modulus metric; A1, algorithm 1; A2, algorithm 2.

Figure 12:

Time needed for computing various distances, as a function of the number of spikes in the spike trains. Note the different timescales. (A) dm, max-metric. s, Schreiber et al. (B) dVP, Victor and Purpura; ks, Kreuz et al. improved SPIKE distance. (C) dR, van Rossum; A1, algorithm by Houghton and Kreuz (2012); A2, discrete time integration; ki, Kreuz et al. ISI distance. (D) do, modulus metric; A1, algorithm 1; A2, algorithm 2.

The simulations showed that the max-metric and the Schreiber et al. metric are relatively slow to compute. Those two metrics and the Victor and Purpura metric require a computation time that grows more than linearly with the number of spikes. The other metrics have an approximately linear dependence on the number of spikes; we fitted them with a line and computed the proportionality coefficients in Table 2. The fastest distances or algorithms were, in order, modulus-metric, algorithm 2; van Rossum, A2; Kreuz et al. ISI distance; van Rossum, A1; modulus-metric, algorithm 1; and Kreuz et al. improved SPIKE distance. It should be noted that while the modulus-metric algorithms and the van Rossum A1 algorithm compute the distances exactly (within machine numerical precision), the other linear-time algorithms compute numerical approximations of the distances through discrete time integration, with a precision that depends on the integration time step.

Table 2:
Average Computing Time per Spike in a Spike Train, for Various Distances.
Distance (and Algorithm)Computing Time per Spike (ms)
do A2 0.001570 
dR A2 0.001831 
ki 0.001965 
dR A1 0.002437 
do A1 0.003323 
ks 0.010910 
Distance (and Algorithm)Computing Time per Spike (ms)
do A2 0.001570 
dR A2 0.001831 
ki 0.001965 
dR A1 0.002437 
do A1 0.003323 
ks 0.010910 

Note: Notation and data as in Figure 12.

5.  Discussion

The max-metric and the modulus-metric behave in a qualitatively different way from the classical van Rossum and Victor and Purpura distances (see Figures 4 to 6), but similar to the Kreuz et al. distances. Within a set of spike trains that are considered by the van Rossum and Victor and Purpura metrics to be at equal distance from a reference spike train, the max-metric, the modulus-metric, and the Kreuz et al. metrics can distinguish a range of distances that reflect similarities in the structure of the compared spike trains (see Figures 4 and 5). When comparing spike trains that include bursts, the max-metric, the modulus-metric, and the Kreuz et al. metrics ignore differences in the number and position of spikes inside the bursts, while these kinds of differences are considered by the van Rossum and Victor and Purpura metrics as significant as in the case that differences exist outside the bursts (see Figure 6). This makes the max-metric, the modulus-metric, and the Kreuz et al. metrics particularly suitable for measuring distances in spike trains where information is encoded in the timing of bursts or solitary spikes, but not in the internal structure of bursts. This is the case in some experiments, such as Reinagel, Godwin, Sherwin, and Koch (1998), where bursts regarded as unitary events encoded more information per event than otherwise, or such as Kepecs and Lisman (2003), where the timing of the first burst spike carried 70% of the information and the spike count only 22% of the information. However, in other cases, the internal structure of bursts does carry information (Krahe & Gabbiani, 2004). The Schreiber et al. metric has a behavior that is intermediate between the one of the max-metric, the modulus-metric, and the Kreuz et al. metrics, on one side, and the van Rossum and Victor and Purpura metrics, on another side.

While the max-metric depends on a kernel that can be particularized to cause distinct behaviors and the van Rossum, Victor and Purpura, and Schreiber et al. distances also depend on parameters that must be chosen by their users, the modulus-metric does not depend on any parameters, similar to the Kreuz et al. (2009, 2011, 2013) metrics. The lack of parameters allows one to start analyzing data immediately, without the need to preprocess them in order to find the appropriate parameters. A parameter-free distance also gives a more objective measure that does not depend on any assumptions to be made by the experimenter. In some cases, when the timescales vary during an experiment, no single timescale may characterize the spike trains, and thus a timescale-independent measure may be preferable.

We have also shown that the modulus-metric can be computed faster than any of the other considered metrics through an algorithm that operates in a time that depends linearly on the number of spikes in the considered spike trains. This fast algorithm computes the distance exactly (within machine precision), not as a numerically approximated discrete time integration, as some algorithms for other metrics do.

The convolution-metric that we introduced, although analytically similar to the max-metric, is qualitatively similar to the van Rossum distance. A qualitative difference between the convolution-metric and the van Rossum distance appears when the differences between the spike trains are localized near the ends of the integration interval; this is a simple consequence of the difference between the bounded integration interval for the convolution-metric and the infinite integration interval for the van Rossum distance.

We have considered only spike trains having nonoverlapping spikes. If we relax this constraint, for our newly introduced distances, with the exception of the convolution max-metric, we get a zero distance between a spike train and a second one generated from the first by adding an extra spike to the first, overlapping an existing spike (see Figure 4). This is due to the distance d between an arbitrary timing and a spike train in equation 2.5 that does not distinguish between overlapping spikes in a train. Thus, if we relax the constraint of not allowing nonoverlapping spikes, these distances become pseudometrics because there might be a zero distance between two spike trains that differ through overlapping spikes. However, the case of overlapping spikes is biologically implausible if we consider spike trains fired by single neurons. If it is enough that the distances are pseudometrics, we may also relax some of the conditions of the kernels, such as the requirement for to be strictly positive on (0, ba] or the conditions on .

For our metrics, when the integrating interval [a, b] extends beyond the interval covered by extremes of the spike trains, for example, for a pair of spike trains, the result of the integration in the area not covered by extremes of the spike trains adds to the distance without contributing information about the spike trains. Thus, the integrating interval should preferably be chosen as the interval covered by the extremes of the considered set of spike trains. Alternatively, one may artificially add to all considered spike trains two extra spikes at the extremities a and b of the integrating interval, a procedure that is also used by the Kreuz et al. metrics (Kreuz et al., 2013).

We have introduced localized versions of our metrics, which, depending on the localization kernel , could have a biologically relevant interpretation of measuring the differences between two spike trains as they are perceived at a particular moment in time by a neuron receiving these spike trains.

6.  Conclusion

We have introduced here a new class of spike train metrics, inspired by the Pompeiu-Hausdorff distance. The max-metric and the modulus-metric behave in a qualitatively different way from classical metrics and are particularly suitable for measuring distances in spike trains where information is encoded in the identity of bursts as unitary events. The modulus-metric does not depend on any parameters and can be computed faster than other distances in a time that depends linearly on the number of spikes in the compared spike trains.

Appendix A:  Analysis of the Max-Metric

Proposition 1. 

.

Proof. 
From equation 2.8, for every , we have
formula
A.1
Multiplying by , which is positive, we obtain, ,
formula
A.2
By taking the supremum and integrating, equation A.2 becomes
formula
A.3
The left side of equation A.3 is the max-metric dm (see equation 2.15). Because is independent of s and x,
formula
A.4
Since and , (see equation 2.12), we obtain
formula
A.5
Proposition 2. 

is a metric.

Proof. 

In order to show that is a metric, we need to prove that it is nonnegative, that for any , that it is symmetric, and that it satisfies the triangle inequality.

It is trivial to show that is nonnegative and symmetric, and that if .

In order to prove that , we use a reductio ad absurdum argument. Assume that with . Then there must be at least one spike in one of the two spike trains that is not in the other, since we do not allow overlapping spikes within the spike trains. Consider that this spike belongs to T; in the case that it belongs to , the proof is analogous. Let u be the timing of this spike, . Because , we have, ,
formula
A.6
Because , d(u, T)=0. Thus,
formula
A.7
By integrating equation A.7, we obtain
formula
A.8
Because , . Also, considering equation 2.9, we have
formula
A.9
Thus, from equations A.8 and A.9, . Since we considered that , this cannot be true. Hence, . Likewise, one can show that , and so .
In order to prove the triangle inequality, consider . We have, ,
formula
A.10
Because for any two functions f and g, it follows that
formula
A.11
After integration, it results that
formula
A.12

With this final equality, we have shown that the distance is indeed a metric, and the proof ends.

Proposition 3. 

The metricis topologically equivalent to the Pompeiu-Hausdorff distance.

Proof. 

In order to show that the metrics dm and h are topologically equivalent, it is sufficient to prove that the identity function , and its inverse are both continuous (O'Searcoid, 2007, p. 229; Deza & Deza, 2009).

We first show that is continuous, which is equivalent to  we have . We choose with
formula
A.13
From equation 2.9, we have that . From equation 2.15, for all ,
formula
A.14
formula
A.15
formula
A.16
and thus
formula
A.17
formula
A.18
Analogously, for all ,
formula
A.19
Taking the max value in equations A.18 and A.19, we obtain
formula
A.20
From equation 2.6, it follows that
formula
A.21
Because , from the equation A.21 we have that . Thus, since .
It remains to show that is continuous, which is equivalent to . We choose with
formula
A.22
We have
formula
A.23
Considering equation 2.9, we have B(a, b)>0. From equation 2.15, we have
formula
A.24
formula
A.25
From equations 2.8 and A.23, we have
formula
A.26
Since and B(a, b)>0, we get and, finally, .

Appendix B:  Analysis of the Convolution Max-Metric

Lemma 1. 
Let be a continuous function and be a continuous function that is derivable on (0, ba) and has a bounded derivative. Then the function ,
formula
B.1
is continuous on [a, b].
Proof. 
Consider . We need to show that , such that , we have
formula
B.2
We have
formula
B.3
formula
B.4
formula
B.5
The function g is bounded since it is continuous on a compact interval (Protter, 1998). We denote by M the bound of the absolute value of g: , . We denote by L the bound of the absolute value of the derivative of h: , . Let and . Then for all ,
formula
B.6
formula
B.7
From equations B.5 and B.7,
formula
B.8
Because , we have , it follows that
formula
B.9
Because ,
formula
B.10
Applying supremum to equation B.10, we obtain
formula
B.11
It follows that
formula
B.12
Analogously, by switching s and s0, in equation B.3 and the ensuing equations, we get
formula
B.13
Thus, we have proved equation B.2, and the proof ends.
Proposition 4. 

.

Proof. 
Because is positive, we have and . Moreover, ,
formula
B.14
Since , ,
formula
B.15
From equation 2.21,
formula
B.16
formula
B.17
Because (see equation 2.12), it follows that
formula
B.18
Proposition 5. 

is a metric.

Proof. 
It is trivial to show that is nonnegative and symmetric, and that if . In order to prove that , we use a reductio ad absurdum argument. Assume that with . For , let
formula
B.19
Because is continuous, from the properties of and lemma 1, we obtain that q is continuous. Because of the properties of , such that ; and because is strictly positive, it follows that q is not zero everywhere, such that q(x)>0. Because q is continuous, it follows that
formula
B.20
which contradicts the hypothesis that . Hence, .

The triangle inequality can be proven in a similar way to the proof in the max-metric case.

Appendix C:  Analysis of the Localized Max-Metric

Proposition 6. 

.

Proof. 
For every we have
formula
C.1
formula
C.2
By integrating equation C.2, we obtain
formula
C.3
Since and , (see equation 2.22), it follows that
formula
C.4
Proposition 7. 

is a metric.

Proof. 

It is trivial to show that is nonnegative and symmetric, and that if .

In order to prove that , we use a reductio ad absurdum argument. Assume that with . Then there must be at least one spike in one of the two spike trains that is not in the other. Consider that this spike belongs to T; in the case it belongs to , the proof is analogous. Let u be the timing of this spike: .

First, we consider the case u>a. We have, ,
formula
C.5
Because , d(u, T)=0 and , thus,
formula
C.6
Multiplying by and integrating equation C.6, we obtain
formula
C.7
We also have
formula
C.8
By adding the last two inequations, C.7 and C.8, and considering equation 3.5, we have
formula
C.9
Because is strictly positive on (0, ba] and continuous, we have . Because we have , we get
formula
C.10
Since we have considered , it follows that equation C.10 cannot be true. Hence, .
Second, we consider the case u=a. Let v be the timing of the first spike in either T or , other than u. Since , v>u. Because
formula
C.11
from equation 3.5, we have
formula
C.12
formula
C.13
For all , we have d(x, T)=xu, , and because on this interval vx>xu, we have . Because is strictly positive on (0, ba], we get
formula
C.14
and thus
formula
C.15
Since we have considered , it follows that equation C.15 cannot be true. Hence, .

Thus, we have shown that in both the case u>a and the case u=a, we have . Likewise, we can show that , and so if .

The triangle inequality can be proven in the same way as the proof in the max-metric case.

Appendix D:  Analysis of the Localized Modulus-Metric

Proposition 8. 

.

Proof. 
From equation 2.8, we have, ,
formula
D.1
Multiplying by , which is positive, we obtain
formula
D.2
By integrating equation D.2, we obtain
formula
D.3
Since and , , it follows that
formula
D.4
Proposition 9. 

is a metric.

Proof. 
Let . It is trivial to show that is nonnegative and symmetric, and that . In order to prove the converse we use a reductio ad absurdum argument. Assume with . For , let
formula
D.5
Because is continuous and is continuous, q is continuous. Because such that ; because is strictly positive on (0, ba], it follows that q is not zero everywhere, such that q(s)>0. Because q is continuous, it follows that
formula
D.6
which contradicts the hypothesis that . Hence, .

The triangle inequality can be proven in the same way as the proof in the max-metric case.

Appendix E:  Algorithms

Algorithm 1: An Algorithm for Computing the Modulus-Metric Distance do, between Two Spike Trains T1 and T2.

The text surrounded by represents comments.
formula

Algorithm 2: Another Algorithm for Computing the Modulus-Metric Distance do, between Two Spike Trains T1 and T2.

The text surrounded by represents comments.
formula
formula
formula

Acknowledgments

We acknowledge the useful suggestions of Adriana Nicolae and Ovidiu Jurjuţ. This work was funded by the Sectorial Operational Programme Human Resources Development (POSDRU, contract 6/1.5/S/3, ”Doctoral Studies: Through Science towards Society”), a grant of the Romanian National Authority for Scientific Research (PNCDI II, Parteneriate, contract 11-039/2007), and the Max Planck–Coneural PartnerGroup.

References

Adrian
,
E. D.
(
1926
).
The impulses produced by sensory nerve endings
.
Journal of Physiology
,
61
,
49
72
.
Bohte
,
S. M.
(
2004
).
The evidence for neural information processing with precise spike-times: A survey
.
Natural Computing
,
3
(
2
),
195
206
.
Dauwels
,
J.
,
Vialatte
,
F.
,
Weber
,
T.
, &
Cichocki
,
A.
(
2009
).
On similarity measures for spike trains
. In
Advances in Neuro-Information Processing. 15th International Conference, ICONIP 2008, Auckland, New Zealand, November 25–28, 2008, Revised Selected Papers, Part I, volume 5506 of Lecture Notes in Computer Science
(pp.
177
185
).
New York
:
Springer
.
Deza
,
M. M.
, &
Deza
,
E.
(
2009
).
Encyclopedia of distances
.
New York
:
Springer
.
Fries
,
P.
,
Nikolić
,
D.
, &
Singer
,
W.
(
2007
).
The gamma cycle
.
Trends in Neurosciences
,
30
(
7
),
309
316
.
Geisler
,
W. S.
,
Albrecht
,
D. G.
,
Salvi
,
R. J.
, &
Saunders
,
S. S.
(
1991
).
Discrimination performance of single neurons: Rate and temporal information
.
Journal of Neurophysiology
,
66
,
334
362
.
Hausdorff
,
F.
(
1914
).
Grundzüge der Mengenlehre
.
Leipzig
:
Verlag von Veit & Comp
.
Houghton
,
C.
, &
Kreuz
,
T.
(
2012
).
On the efficient calculation of van Rossum distances
.
Network: Computation in Neural Systems
,
23
,
48
58
.
Jacobs
,
J.
,
Kahana
,
M. J.
,
Ekstrom
,
A. D.
, &
Fried
,
I.
(
2007
).
Brain oscillations control timing of single-neuron activity in humans
.
Journal of Neuroscience
,
27
(
14
),
3839
3844
.
Ji
,
D.
, &
Wilson
,
M. A.
(
2007
).
Coordinated memory replay in the visual cortex and hippocampus during sleep
.
Nature Neuroscience
,
10
(
1
),
100
107
.
Kepecs
,
A.
, &
Lisman
,
J.
(
2003
).
Information encoding and computation with spikes and bursts
.
Network: Computation in Neural Systems
,
14
(
1
),
103
118
.
Krahe
,
R.
, &
Gabbiani
,
F.
(
2004
).
Burst firing in sensory systems
.
Nature Reviews Neuroscience
,
5
,
13
24
.
Kreuz
,
T.
,
Chicharro
,
D.
,
Andrzejak
,
R.
,
Haas
,
J.
, &
Abarbanel
,
H.
(
2009
).
Measuring multiple spike train synchrony
.
Journal of Neuroscience Methods
,
183
(
2
),
287
299
.
Kreuz
,
T.
,
Chicharro
,
D.
,
Greschner
,
M.
, &
Andrzejak
,
R.
(
2011
).
Time-resolved and time-scale adaptive measures of spike train synchrony
.
Journal of Neuroscience Methods
,
195
(
1
),
92
106
.
Kreuz
,
T.
,
Chicharro
,
D.
,
Houghton
,
C.
,
Andrzejak
,
R. G.
, &
Mormann
,
F.
(
2013
).
Monitoring spike train synchrony
.
Journal of Neurophysiology
,
109
(
5
),
1457
1472
.
Kreuz
,
T.
,
Haas
,
J. S.
,
Morelli
,
A.
,
Abarbanel
,
H. D.
, &
Politi
,
A.
(
2007
).
Measuring spike train synchrony
.
Journal of Neuroscience Methods
,
165
(
1
),
151
161
.
Lee
,
H.
,
Simpson
,
G.
,
Logothetis
,
N.
, &
Rainer
,
G.
(
2005
).
Phase locking of single neuron activity to theta oscillations during working memory in monkey extrastriate visual cortex
.
Neuron
,
45
(
1
),
147
156
.
Montemurro
,
M. A.
,
Rasch
,
M. J.
,
Murayama
,
Y.
,
Logothetis
,
N. K.
, &
Panzeri
,
S.
(
2008
).
Phase-of-firing coding of natural visual stimuli in primary visual cortex
.
Current Biology
,
18
(
5
),
375
380
.
Nádasdy
,
Z.
,
Hirase
,
H.
,
Czurkó
,
A.
,
Csicsvari
,
J.
, &
Buzsáki
,
G.
(
1999
).
Replay and time compression of recurring spike sequences in the hippocampus
.
Journal of Neuroscience
,
19
(
21
),
9497
9507
.
Naud
,
R.
,
Gerhard
,
F.
,
Mensi
,
S.
, &
Gerstner
,
W.
(
2011
).
Improved similarity measures for small sets of spike trains
.
Neural Computation
,
23
,
3016
3069
.
O'Searcoid
,
M.
(
2007
).
Metric spaces
.
New York
:
Springer
.
Papadopoulos
,
A.
(
2005
).
Metric spaces, convexity and nonpositive curvature
.
Zürich
:
European Mathematical Society
.
Pompeiu
,
D.
(
1905
).
Sur la continuité des fonctions de variables complexes
.
Annales de la Faculté des Sciences de Toulouse
,
2/7
(
3
),
265
315
.
Protter
,
M. H.
(
1998
).
Basic elements of real analysis
.
New York
:
Springer-Verlag
.
Reinagel
,
P.
,
Godwin
,
D.
,
Sherman
,
S. M.
, &
Koch
,
C.
(
1998
).
Encoding of visual information by LGN bursts
.
Journal of Neurophysiology
,
81
(
5
),
2558
2569
.
Rockafellar
,
R. T.
, &
Wets
,
R.J.-B.
(
2009
).
Variational analysis
.
New York
:
Springer
.
Rusu
,
C. V.
, &
Florian
,
R. V.
(
2010
).
A new spike train metric
.
BMC Neuroscience
,
11
(
Suppl. 1
),
P169
.
Schrauwen
,
B.
, &
Van Campenhout
,
J.
(
2007
).
Linking non-binned spike train kernels to several existing spike train metrics
.
Neurocomputing
,
70
(
7–9
),
1247
1253
.
Schreiber
,
S.
,
Fellous
,
J.
,
Whitmer
,
D.
,
Tiesinga
,
P.
, &
Sejnowski
,
T.
(
2003
).
A new correlation-based measure of spike timing reliability
.
Neurocomputing
,
52–54
,
925
931
.
Siegel
,
M.
,
Warden
,
M.
, &
Miller
,
E.
(
2009
).
Phase-dependent neuronal coding of objects in short-term memory
.
Proceedings of the National Academy of Sciences of the USA
,
106
(
50
),
21341
21346
.
Tiesinga
,
P.
,
Fellous
,
J.
, &
Sejnowski
,
T.
(
2008
).
Regulation of spike timing in visual cortical circuits
.
Nature Reviews Neuroscience
,
9
(
2
),
97
107
.
Toups
,
J. V.
,
Fellous
,
J.-M.
,
Thomas
,
P. J.
,
Sejnowski
,
T. J.
, &
Tiesinga
,
P. H.
(
2012
).
Multiple spike time patterns occur at bifurcation points of membrane potential dynamics
.
PLoS Computational Biology
,
8
(
10
),
e1002615
.
van Rossum
,
M. C. W.
(
2001
).
A novel spike distance
.
Neural Computation
,
13
,
751
763
.
VanRullen
,
R.
,
Guyonneau
,
R.
, &
Thorpe
,
S. J.
(
2005
).
Spike times make sense
.
Trends in Neurosciences
,
28
(
1
),
1
4
.
Victor
,
J. D.
(
2005
).
Spike train metrics
.
Current Opinion in Neurobiology
,
15
,
585
592
.
Victor
,
J. D.
, &
Purpura
,
K. P.
(
1996
).
Nature and precision of temporal coding in visual cortex: A metric-space analysis
.
Journal of Neurophysiology
,
76
,
1310
1326
.
Victor
,
J. D.
, &
Purpura
,
K. P.
(
1997
).
Metric-space analysis of spike trains: Theory, algorithms and application
.
Network: Computation in Neural Systems
,
8
,
127
164
.