Population measures of spike train synchrony

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Conor Houghton (2013), Scholarpedia, 8(10):30635. doi:10.4249/scholarpedia.30635 revision #150870 [link to/cite this article]
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Curator: Conor Houghton

A population measure of spike train synchrony estimates the similarity of a pair of population responses. It is just like a measure of spike train synchrony which estimates the similarity of a pair of spike trains except that it measures similarity between collections of spike trains. Measures of this type are becoming important because of the growing availability of multi-neuronal data sets. Moreover, multi-neuronal data sets are increasingly recorded from neurons that are near each other and correspondingly more likely to participate in the same coding functions. Population measures are proposed as a tool that can be applied to these data and used to assess how neurons cooperate in neuronal coding.

A single-neuron measure of spike train synchrony is a measure in which all spikes from a neuron are treated as identical, so that the spike train is treated as nothing more than a sequence of spike times. Of course, this is not true of real spike trains, in which spike profiles vary from neuron to neuron and from spike to spike; however, typically this variability does not appear to substantially effect the downstream consequences of spike arrival. In a population synchrony measure, spikes do not just have a time, they are also labeled by the neuron that fired them.


Measures of spike train synchrony

There are a number of different approaches to spike train synchrony, which give rise to a corresponding variety of approaches to population measures of synchrony.

To briefly summarize, if \(\textbf{u}\) and \(\textbf{v}\) are two spike trains with spike times \(\{u_1,u_2,\ldots,u_n\}\) and \(\{v_1,v_2,\ldots,v_m\}\) , a measure of spike train synchrony maps the pair to a positive real number. In some measures this number expresses the similarity or synchrony of the pair and, using the notation \(s(\textbf{u},\textbf{v})\), a high value corresponds to two closely related spike trains. It is more common however, to define a measure of distance, or dissimilarity, and using the notation \(d(\textbf{u},\textbf{v})\), very similar spike trains correspond to a low value. Either way, it is common for measures of similarity or dissimilarity to be symmetric, such that \(s(\textbf{u},\textbf{v})=s(\textbf{v},\textbf{u})\) or \(d(\textbf{u},\textbf{v})=d(\textbf{v},\textbf{u})\). In the case of dissimilarity measures, it is also common to choose measures that satisfy all the conditions required for the measure to be a metric: a measure of dissimilarity is a metric if it is positive, symmetric, non-degenerate, so that \(d(\textbf{u},\textbf{v})=0\) implies \(\textbf{u}=\textbf{v}\) and if it satisfies the triangular inequality, so that \(d(\textbf{u},\textbf{w})\le d(\textbf{u},\textbf{w})+d(\textbf{w},\textbf{v})\).

Broadly speaking, population measures come in two varieties, those that measure the over-all synchrony of a set of responses and are calculated by averaging single-neuron measures of synchrony, and those which measure the similarity or dissimilarity between two sets of responses. Since they form the base on which the population measures in this article are built, three different single-neuron distance measures are described here.

The Victor-Purpura metric

The Victor-Purpura metric is an "edit distance", which means that it is a measure of the amount of editing required to transform one spike train into the other (Victor and Purpura, 1996, 1997). In this metric there are three different edit types for changing a spike train, and each has a cost. Spikes can be added or deleted at a cost of one for each operation, and spikes can be moved at a cost of \(q|\delta t|\) for a temporal distance \(\delta t\). The distance is then the least expensive sequence of edits that changes one spike train into the other. The parameter \(q\) determines how costly it is to move a spike. It is never worthwhile to move a spike more than \(2/q\) since doing so would have a higher cost than deleting the spike at one location and adding it at the other. For a small value of \(q\) spikes can be moved with little cost and the distance is substantially determined by the difference in the number of spikes. Conversely, as \(q\) becomes larger, the metric becomes increasingly sensitive to spike times.

The van Rossum metric

The van Rossum metric is an embedding-based metric (van Rossum, 2001). The spike trains are first mapped to continuous functions of time by convolving with a kernel: \[ \mathbf{u}\rightarrow f(t;\mathbf{u})=\sum_{i=1}^{n}h(t-u_{i}) \] where \(h(t)\) is the kernel, usually a causal exponential \( h(t)=\sqrt{\frac{2}{\tau}}\Theta(t)e^{-t/\tau} \) where \(\Theta(t)\) is the Heaviside step function with \(\Theta(t)=0\) for \(t<0\) and \(\Theta(t)=1\) otherwise. The distance between two spike trains \(\textbf{u}\) and \(\textbf{v}\) is then defined as the \(L^2\) distance between the corresponding functions: \[ d(\textbf{u},\textbf{v})=\sqrt{\int_{0}^{T}dt(f(t;\mathbf{u})-f(t;\mathbf{v}))^{2}} \] The timescale \(\tau\), like \(2/q\) in the Victor-Purpura metric, determines the sensitivity of the metric to spike times, with small values giving a metric that is sensitive to spike times and larger values a metric which compares firing rates.

The ISI- and the SPIKE-distance

The ISI- and the SPIKE-distance differ from the two distance measures mentioned above in that they emphasize temporal locality (Quian Quiroga et al., 2002, Kreuz et al., 2007, 2011, 2012). They each define a local measure of the dissimilarity $d(t;\textbf{u},\textbf{v})$ which can be integrated to give an overall distance between the spike trains \[ d(\textbf{u},\textbf{v})=\int_0^T dt |d(t;\textbf{u},\textbf{v})| \] The idea is that the time local profile \(d(t;\textbf{u},\textbf{v})\) can be used to track changes of synchrony during the time course of an experiment. The ISI- and the SPIKE-distance differ in how \(d(t;\textbf{u},\textbf{v})\) is defined, the ISI-distance is calculated using local estimates of firing rates, the SPIKE-distance depends on local differences in spike timing between the two spike trains.

Measures of overall synchrony

Averaging single-neuron similarities or dissimilarities is the most straightforward way to measure the overall synchrony of a population of responses (Kreuz et al., 2009). So, if the spike trains in a population are \(\{\textbf{u}_1,\textbf{u}_2,\ldots,\textbf{u}_N\}\) and \(d(\textbf{u}_i,\textbf{u}_j)\) is a measure of the distance between \(\textbf{u}_i\) and \(\textbf{u}_j\), then \[D=\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{j=i+1}^Nd(\textbf{u}_i,\textbf{u}_j) \] is the average distance. Here, \(d(\textbf{u}_i,\textbf{u}_j)\) could be any of the measures of dissimilarity described in measures of spike train synchrony. If the measure is given by integrating a local dissimilarity profile, as is the case for the ISI- and the SPIKE-distance, then this also gives rise to a local measure of overall population dissimilarity: \[ D=\int_0^T dt D(t) \] where \[ D(t)=\frac{2}{N(N-1)}\sum_{i=1}^{N-1}\sum_{j=i+1}^N|d(t;\textbf{u}_i,\textbf{u}_j)|. \]

A measure of overall synchrony gives a single quantity describing how spread out a set of responses are. These might be multiple responses from a single neuron, for example multiple trials with the same stimulus, in which case overall synchrony quantifies the reliability of the response, or they might be responses to a corpus of stimuli, in which case the overall synchrony measures how strongly modulated the neuron is by the stimulus. Alternatively, the responses might be spike trains from multiple neurons with a single stimulus. In this case, if the role of the population is to reduce noise, the individual neurons will respond in the same way to a single aspect of the stimulus and will be highly synchronized with a small distance $D$ and, conversely, if different neurons respond preferentially to different aspects of the stimulus the neurons will not be synchronized the $D$ will be large. In contrast, the measures of the synchrony of two population responses, which will be examined next, measure a distance between two equally-sized sets of spike trains, most typically, two different population responses or responses from two different sets of neurons where each neuron from one set has an equivalent neuron in the other set, or perhaps, reponses from real neurons and a model of the same network.

Measures of the synchrony of two population responses

Both the Victor-Purpura and van Rossum metrics have extensions to the population case, allowing two sets of populations to be compared. Thus, they map two equal size sets of spike trains \(\mathcal{U}=\{\mathbf{u}_1,\mathbf{u}_2,\ldots,\mathbf{u}_N\}\) and \(\mathcal{V}=\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_N\}\) to a real positive number \(d(\mathcal{U},\mathcal{V})\).

In both cases, the extension to a population metric follows the same framework as the single-neuron metric it is based on, so the extension to the Victor-Purpura metric involves an extra edit corresponding to changing the neuron label of the spike, and the extension to the van Rossum metric involves expanding the space the spike trains are embedded in, to give directions corresponding to the neuron labels. In each case there are extra parameters introduced which specify the significance of the identity of the neuron which fires a spike. This allows the metrics to interpolate between two cases. One extreme is what is called the summed population code where the identity of the neuron is irrelevant to coding so all that is important is that the spike is fired as part of the population response, not which neuron fired it. The other extreme is called the labeled-line code in which there is no population effect and the most appropriate distance between the two population responses is the sum of the distances between the pairs of responses for each neuron.

The population Victor-Purpura metric

The simplest way to describe the Victor-Purpura metric is to think of the population response not as a set of spike trains, but as a single spike train in which the spikes have a label marking which neuron fired them. Just as the time of a spike can be changed at a cost \(q|\delta t|\), the neuron label can also be changed, at a cost $k$, where $k$ is a parameter specifying the significance of a neuron label (Aronov et al., 2003).

If $k=0$, there is no cost for relabeling a spike and the distance is unaffected by which neuron fired each spike. Thus, this represents a summed population code. Conversely, if $k\ge2$ it is at least as expensive to relabel a spike as it is to delete a spike from one spike train and add it to another. In this case, the distance between \(\mathcal{U}\) and \(\mathcal{V}\) is the sum of the distances between each of the individual spike trains and \(k=2\) represents a labeled line code. Here, a single cost, $k$ has been used for the change of spike label. In principle, there could be a different cost for every possible change of label, giving $N(N-1)/2$ $k$-like parameters in all, in practice the metric is simplified, as here, by assuming all label changes have the same cost.

As with the single-neuron Victor-Purpura metric there is a straightforward algorithm for calculating values of the population metric. This algorithm has complexity \(O(Nn^{N+1})\) for \(N\) spike trains with average length \(n\).

The population van Rossum metric

The single-neuron van Rossum metric is defined by a map embedding the spike trains into the space of continuous functions, and then by using the \(L^{2}\) metric on the space of functions to give a distance. To extend this idea to populations the embedding space is enlarged to a space of \(N\)-dimensional functions (Houghton and Sen, 2008). Each of the \(N\) neurons in the set is then associated with a specific direction in this \(N\)-dimensional space, and spike-trains for that neuron are mapped to functions embedded along that direction.

Specifically, for the \(i\)th spike train \[ {\bf u}_{i}\mapsto f(t;\mathbf{u}_{i})\mathbf{e}_{i} \] where \(\mathbf{e}_{i}\) is a $N$-dimensional unit vector associated with the \(i\)th neuron. That is, the unit vectors \(\mathbf{e}_{i}\) determine the direction in which the \(i\)th spike train is mapped. Adding these vectors, one from each single-unit spike train, gives an $N$-dimensional vector of functions of time: \[ \mathcal{U}\mapsto\mathbf{f}(t;\mathcal{U})=\sum_{i=1}^{N}f(t;\mathbf{u}_{i})\mathbf{e}_{i}.\tag{1} \] The population metric is then defined by the norm in this extended space: if \[ \mathbf{g}(t)=\left(\begin{array}{c} g_{1}(t)\\ g_{2}(t)\\ \vdots\\ g_{N}(t)\end{array}\right)\tag{2}\] is a vector of functions of \(t\in[0,T]\), then the norm is \[ \|\mathbf{g}(t)\|=\sqrt{\int_{0}^{T}dt(g_{1}^{2}+g_{2}^{2}+\ldots+g_{N}^{2})}. \] The corresponding metric induced on the space of population responses is given by \[ d(\mathcal{U},\mathcal{V})=\|\mathbf{f}(t;\mathcal{U})-\mathbf{f}(t;\mathcal{V})\|. \]

The directions of the individual unit vectors \(\mathbf{e}_{i}\) serves to parameterize this metric. If the \(\mathbf{e}_{i}\) are all parallel then the metric corresponds to summing the individual response vectors \(f(\mathbf{u}_{i})\mathbf{e}_{i}\). This is equivalent to superimposing the spike trains before mapping them into function space, and thus precisely corresponds to a summed population code. Conversely, if all the vectors are orthogonal, for example if \(\mathbf{e}_{i}\) has one for its \(i\) component and is otherwise zero, then the multineuronal metric is a Pythagorean sum of the individual van Rossum distances between the individual spike trains: this is a labeled line code.

While this description, using an extension of the embedding space to extend the metric, is useful in explaining the population metric in terms similar to the description of the original single-neuron metric, it is possible to reduce the definition to a simpler form. For example, if there are just two neurons, \(N=2\), then \[ d(\mathcal{U},\mathcal{V})= \sqrt{\int_0^\infty dt \left(\sum_i|\delta_i |^2 dt + 2\cos{\theta}\delta_1\delta_2\right)} \] where \(\delta_i=f(t;\mathbf{u}_i)-f(t;\mathbf{v}_i)\). Here \(\theta\) corresponds to the angle between \(\mathbf{e}_1\) and \(\mathbf{e}_2\); these direction vectors are no longer mentioned directly. More generally \[ d(\mathcal{U},\mathcal{V})= \sqrt{\int_0^\infty dt \left(\sum_{i=1}^N|\delta_i |^2 dt + \sum_{i=1}^{N-1}\sum_{j= i+1}^N\cos{\theta_{ij}}\delta_{i}\delta_{j}\right)} \] where \[ \mathbf{e}_1\cdot\mathbf{e}_2=\cos{\theta_{ij}} \] It is possible to reduce the number of parameters by making all the $\theta_{ij}$ as having the same value. This is analogous to the population Victor-Purpura metric where the cost of changing a spike label is always $k$ irrespective of which labels are involved.

In practise the van Rossum metric is calculated using formulas in which the functions are integrated analytically (Schrauwen and Campenhout, 2007, Paiva et al., 2009). The calculation is \(O(n^2N^2)\) which can be reduced to \(O(nN^2)\) using the computational approach discussed in (Houghton and Kreuz, 2012).

Example Applications

In the paper in which it was first introduced (Aronov et al. 2003), the population Victor-Purpura metric was used to analyze neural coding of spatial phase in spike trains recorded simultaneously from multiple neurons in area V1 of Macaque monkeys. A non-zero value of \(k\) was found to be optimal for discriminating responses with different phases, supporting a role for neuron identity in coding. Similarly, in Clemens et al. (2011) the population van Rossum metric was used to provide evidence for a progression from summed population coding to labeled-line coding as signals move along the auditory pathway in crickets.


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Further reading

  • Houghton C, Victor J. Measuring representational distances–the spike-train metrics approach, in Visual population codes: Toward a common multivariate framework for cell recording and functional imaging, editors Kriegeskorte N, Kreima G, MIT Press, Boston MA, 2012.

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