Outline of the algorithm

Suppose we perform spiking measurements from an ensemble of N neurons, and we observe the spiking output of this ensemble in M separate epochs of length T samples (in units of the time bin length). Suppose that there are P distinct activity patterns that tend to reoccur in some of the M epochs. Each pattern generates a set of normalized (to unit mass) cross-correlation histograms among all neurons. Instantiations of the same pattern are different because of noise, but will have the same expectation for the cross-correlation histogram. The normalized cross-correlation histogram is defined as (1) if , and otherwise. Here, s_ij(τ) ≡ ∑_t s_i(t)s_j(t + τ) is the cross-correlation function (or cross-covariance), and s_i(t) and s_j(t) are the spike trains of neurons i and j. In other words, the normalized cross-correlation histogram is simply the histogram of coincidence counts at different delays τ, normalized to unit mass. We take the N × N × (2T + 1) matrix of values as a full representation of a pattern, that is, we consider two patterns to have the same temporal structure when all neuron pairs have the same expected value of for each τ. For simplicity and clarity of presentation, we have written the cross-correlation function as a discrete (histogram) function of time. However, because the SPOTDis, which is introduced below, is a cross-bins dissimilarity measure and requires only to store the precise delays τ at which is non-zero, the sampling rate can be made infinitely large (see Methods). In other words, the SPOTDis computation does not entail any loss of timing precision beyond the sampling rate at which the spikes are recorded.

The SPOTDisClust method contains two steps (Fig 1), which are illustrated for five example patterns (Fig 1A and 1B). The first step is to construct the SPOTDis dissimilarity measure between all pairs of epochs on the matrix of cross-correlations among all neuron pairs. The second step is to perform clustering on the SPOTDis dissimilarity measure using an unsupervised clustering algorithm that operates on a dissimilarity matrix. Many algorithms are available for unsupervised clustering on pairwise dissimilarity matrices. One family of unsupervised clustering methods comprises so called density clustering algorithms, including DBSCAN, HDBSCAN or density peak clustering. Here, we use the HDBSCAN unsupervised clustering method [58–61] (see Methods). To examine the separability of the clusters in a low dimensional 2-D embedding, we employ the t-SNE projection method [62, 63] (see Methods).

The SPOTDis measure is constructed as follows:

1. We compute, for each of the M epochs separately, the cross-correlation function for all pairs of N neurons (see Methods), which yields M matrices of N(N − 1)/2 cross-correlations (Fig 1C and S1 Fig).
2. For each pair of epochs k and m and each pair of neurons i and j, we now want to quantify how similar the temporal correlation of neuron i and j was between epochs k and m, i.e. the similarity of and . To this end, we compute the Earth Mover’s Distance (EMD) between the normalized cross-correlations of each neuron pair, which yields M(M − 1)/2 × N(N − 1)/2 EMD values D_ij,km (see Methods and Fig 1C and 1D, S1 and S2 Figs). We use the L1 norm to measure dissimilarity on the time axis, and we define the cost of transporting the mass in the cross-correlation function between time τ₁ and τ₂ as |τ₁ − τ₂|/(2T + 1), such that the minimum and maximum EMDs are 0 and 1, respectively. The EMD is a metric distance function on probability distributions that determines the minimum transport cost to transform one unit distribution of mass into another unit distribution of mass (Fig 1D). In this case, the mass is the normalized (to a mass of 1) cross-correlation function for a neuron pair.
   The advantage of using the EMD is multi-fold. First, it is a symmetric and metric measure of similarity between two probability distributions (as opposed to e.g. the Kullback-Leibler divergence). Second, as it is a “cross-bins” distance, it can handle jitter in spike timing. In other words, it quantifies not only whether two distributions are overlapping, like the Kullback-Leibler divergence, but also how far they are shifted away, as minimum transport cost, from each other in a metric space (in this case: time). Third, it does not rely on the computation of a measure of central tendency like the center of mass or peak of the probability distribution, but can also compute transport cost between multimodal probability distributions (Fig 1D). It can therefore capture differences between complex patterns (see Figs 3 and 4). Fourth, because our computation of the EMD uses only the exact pairwise delays among pairs of spikes as its input (the computation thus scales with the number of spikes, not bins; see Methods), our implementation does not require any binning or smoothing of the spike trains beyond the sampling rate at which the spikes are recorded, preventing any additional loss in timing precision; this means that the bins can be made infinitely small (see Methods).
3. After computing the EMDs between each pair of epochs for each neuron pair separately, we compute SPOTDis as the average EMD across neuron pairs, (2) (see Methods) (Fig 1C). Here, the weights are defined as (3) where sgn(x) is the sign function, with sgn(x) = 0 for x = 0 and sgn(x) = 1 for x > 0. Thus, only for neuron pairs for which both neuron i and neuron j fired in both epochs k and m will the weight w_ij,km = 1. The rationale behind ignoring the other neuron pairs for computing the SPOTDis is that it avoids assigning an arbitrary value to the EMD in the case where we have no information about the temporal relationship between the neurons (i.e. where we do not have any spikes for one neuron in one epoch). We assume for now, that for all (k, m), , i.e. that for each pair of epochs k and m, there is at least one pair of neurons in which both neurons fired in both epochs k and m. If all the weights equal 1, then we can simplify to (4) From Eq (2) we obtain M(M − 1)/2 SPOTDis values (Fig 1E). These values are then the input to the HDBSCAN clustering algorithm and the t-SNE visualization (Fig 1E) (see Methods).

Ground truth simulations

To test the SPOTDisClust method for cases in which the ground truth is known, we generated P input patterns in epochs of length T = T_epoch = 300 samples, defined by the instantaneous rate of inhomogeneous Poisson processes, and then generated spiking outputs according to these (Fig 1A and 1B) (see Methods). Because the SPOTDis is a binless measure, in the sense that it does not require any binning beyond the sampling frequency, the epochs could for example represent spike series of 3s with a sampling rate of 100Hz, or spike series of 300ms with a sampling rate of 1000Hz. Each input pattern was constructed such that it had a baseline firing rate and a pulse activation firing rate, defined as the expected number of spikes per sample. The pulse activation period (with duration T_pulse samples) is the period in the epoch in which the neuron is more active than during the baseline, and the positions of the pulses across neurons define the pattern. For each neuron and pattern, the position of the pulse activation period was randomly chosen. We generated M/(2 * P) realizations for each of the P patterns, and a matching number of M/2 noise epochs (i.e. 50 percent of epochs were noise epochs). We performed simulations for two types of noise epochs (S3 Fig). First, noise was generated with random firing according to a homogeneous Poisson process with a constant rate (see Fig 1). We refer to this noise, throughout the text, as “homogeneous noise”. For the second type of noise, each noise epoch comprised a single instantiation of a unique pattern, with randomly chosen positions of the pulse activation periods. We refer to this noise as “patterned noise”. For both types of noise patterns, the expected number of spikes in the noise epoch was the same as during an epoch in which one of the P patterns was realized. The second type of noise also had the same inter spike interval statistics for each neuron as the patterns. Importantly, because SPOTDisClust uses only the relative timing of spiking among neurons, rather than the timing of spiking relative to the epoch onset, the exact onset of the epoch does not have to be known with SPOTDis; even though the exact onset of the pattern is known in the simulations presented here, this knowledge was not used in any way for the clustering.

Fig 1 illustrates the different steps of the algorithm for an example of P = 5 patterns. For the purpose of illustration, we start with an example comprising five patterns that are relatively easy to spot by eye; later in the manuscript we show examples with a very low signal-to-noise ratio (Fig 5) or sparse firing (Fig 6). We find that in the 2-D t-SNE embedding, the P = 5 different patterns form separate clusters (Fig 1E), and that the HDBSCAN algorithm is able to correctly identify the separate clusters (S4 Fig). We also show a simulation with many more noise epochs than cluster epochs, which shows that even in such a case t-SNE embedding is able to separate the different clusters (S5 Fig). In S3 Fig we compare clustering with homogeneous and patterned noise. The homogeneous noise patterns have a consistently small SPOTDis dissimilarity to each other and are detected as a separate cluster, while the patterned noise epochs have large SPOTDis dissimilarities to each other and do not form a separate cluster, but spread out rather uniformly through the low-dimensional t-SNE embedding (S3 Fig).

Detectable patterns outnumber recorded neurons

A key challenge for any pattern detection algorithm is to find a larger number of patterns than the number of measurement variables, assuming that each pattern is observed several times. This is impossible to achieve with traditional linear methods like PCA (Principal Component Analysis), which do not yield more components than the number of neurons (or channels), although decomposition techniques using overcomplete base sets might in principle be able to do so. Other approaches like frequent itemset mining and related methods [48–50] require that exact matches of the same pattern occur.

Because SPOTDisClust clusters patterns based on small SPOTDis dissimilarities, it does not require exact matches of the same pattern to occur, but only that the different instantiations of the same pattern are similar enough to one another, i.e. have SPOTDis values that are small enough, and separate them from other clusters and the noise.

Fig 2 shows an example where the number of patterns exceeds the number of neurons by a factor 10 (500 to 50). In the 2-D t-SNE embedding, the 500 patterns form separate clusters, with the emergence of a noise cluster that has higher variance. Consistent with the low dimensional t-SNE embedding, the HDBSCAN algorithm is able to correctly identify the separate clusters (S4 Fig).

When many patterns are detectable, the geometry of the low dimensional t-SNE embedding needs to be interpreted carefully: In this case, all 500 patterns are roughly equidistant to each other, however, there does not exist a 2-D projection in which all 500 clusters are equidistant to each other; this would only occur with a triangle for P = 3 patterns. Thus, although the low dimensional t-SNE embedding demonstrates that the clusters are well separated from each other, in the 2-D embedding nearby clusters do not necessarily have smaller SPOTDis dissimilarities than distant clusters when P is large.

SPOTDisClust can detect complex patterns

Temporal patterns in neuronal data may consist not only of ordered sequences of activation, but can also have a more complex character. As explained above, a key advantage of the SPOTDis measure is that it computes averages over the EMD, which can distinguish complex patterns beyond patterns that differ only by a measure of central tendency. Indeed, we will demonstrate that SPOTDisClust can detect a wide variety of patterns, for which traditional methods that are based on the relative activation order (sequence) of neurons may not be well equipped.

We first consider a case where the patterns consist of bimodal activations within the epoch (Fig 3A). These types of activation patterns might for example be expected when rodents navigate through a maze, such that enthorinal grid cells or CA1 cells with multiple place fields are activated at multiple locations and time points [64–66]. A special case of a bimodal activation is one where neurons have a high baseline firing rate and are “deactivated” in a certain segment of the epoch (Fig 3B). These kinds of deactivations may be important, because e.g. spatial information about an animal’s position in the medial temporal lobe ([67]) or visual information in retinal ganglion cells is carried not only by neuronal activations, but also by neuronal deactivations. We find that the different patterns form well separated clusters in the low dimensional t-SNE embedding based on SPOTDis (Fig 3A and 3B), and that HDBSCAN correctly identifies them (S4 Fig).

Next, we consider a case where there are two coarse patterns and two fine patterns embedded within each coarse pattern, resulting in a total number of four patterns. This example might be relevant for sequences that result from cross-frequency theta-gamma coupling, or from the sequential activation of place fields that is accompanied by theta phase sequences on a faster time scale [68, 69]. These kinds of patterns would be challenging for methods that rely on binning, because distinguishing the coarse and fine patterns requires coarse and fine binning, respectively. We find that the SPOTDis allows for a correct separation of the data in four clusters corresponding to the four patterns and one noise cluster (Fig 4A), and that HDBSCAN identifies them (S4 Fig). As expected, we find that the two patterns that share the same coarse structure (but contain a different fine structure) have smaller dissimilarities to each other in the t-SNE embedding as compared to the patterns that share a different coarse structure.

Finally, we consider a set of patterns consisting of a synchronous (i.e. without delays) firing of a subset of cells, with a cross-correlation function that is symmetric around the delay τ = 0 (i.e., correlation without delays). This type of activity may arise for example in a network in which all the coupling coefficients between neurons are symmetric.

Previous methods to identify the co-activation (without consideration of time delays) of different neuronal assemblies relied on PCA [24], which has the key limitation that it can identify only a small number of patterns (smaller than the number of neurons) Furthermore, while yielding orthonormal, uncorrelated components that explain the most variance in the data, PCA components do not necessarily correspond to neuronal spike patterns that form distinct and separable clusters; e.g. a multivariate Gaussian distribution can yield multiple PCA components that correspond to orthogonal axes explaining most of the data variance.

Fig 4B shows four patterns, in which a subset of cells exhibits a correlated activation without delays. Separate clusters emerge in the t-SNE embedding based on SPOTDis (Fig 4B) and are identified by HDBSCAN (S4 Fig). This demonstrates that SPOTDisClust is not only a sequence detection method in the sense that it can detect specific temporal orderings of firing, but can also be used to identify patterns in which specific groups of cells are synchronously co-active without time delays.

Dependence on the signal to noise ratio

A major challenge for the clustering of temporal spiking patterns is the stochasticity of neuronal firing. That is, in neural data, it is extremely unlikely to encounter, in a high dimensional space, a copy of the same pattern exactly twice, or even two instantiations that differ by only a few insertions or deletions of spikes. Furthermore, patterns might be distinct when they span a high-dimensional neural space, even when bivariate correlations among neurons are weak and when the firing of neurons in the activation period is only slightly higher than the baseline firing around it (see further below). The robustness of a sequence detection algorithm to noise is therefore critical.

We can dissociate different aspects of “noise” in temporal spiking patterns. A first source of noise is the stochastic fluctuation in the number of spikes during the pulse activation period and baseline firing period. In the ground-truth simulations presented here, this fluctuation is driven by the generation of spikes according to inhomogeneous Poisson processes. This type of noise causes differences in SPOTDis values between epochs, because of differences in the amount of mass in the pulse activation and baseline period, in combination with the normalization of the cross-correlation histogram. In the extreme case, some neurons may not fire in an given epoch, such that all information about the temporal structure of the pattern is lost. Such a neural “silence” might be prevalent when we search for spiking patterns on a short time scale. We note that fluctuations in the spike count are primarily detrimental to clustering performance because there is baseline firing around the pulse activation period, in other words because “noisy” spikes are inserted at random points in time around the pulse activations. To see this, suppose that the probability that a neuron fires at least one spike during the pulse activation period is close to one for all M epochs and all N neurons, and that the firing rate during the baseline is zero. In this case, because SPOTDis is based on computing optimal transport between normalized cross-correlation histograms (Eq (5)), the fluctuation in the spike count due to Poisson firing would not drive differences in the SPOTDis.

A second source of noise is the jitter in spike timing. Jitter in spike timing also gives rise to fluctuations in the SPOTDis and in the ground-truth simulations presented here, spike timing jitter is a consequence of the generation of spikes according to Poisson processes. As explained above, because the SPOTDisClust method does not require exact matches of the observed patterns, but is a “cross-bins” dissimilarity measure, it can handle jitter in spike timing well. Again, we can distinguish jitter in spike timing during the baseline firing, and jitter in spike timing during the pulse activation period. The amount of perturbation caused by spike timing jitter during the pulse activation period is a function of the pulse period duration. We will explore the consequences of these different noise sources, namely the amount of baseline firing, the sparsity of firing, and spike timing jitter in Figs 5 and 6.

We define the SNR (Signal-to-Noise-Ratio) as the ratio of the firing rate inside the activation pulse period over the firing rate outside the activation period. This measure of SNR reflects both the amount of firing in the pulse activation period as compared to the baseline period (first source of noise), and the pulse duration as compared to the epoch duration (second source of noise).

We first consider an example of 100 neurons that have a relatively low SNR (Fig 5A). It can be appreciated that different realizations from the same pattern are difficult to identify by eye, and that exact matches for the same pattern, if one would bin the spike trains, would be highly improbable, even for a single pair of two neurons (Fig 5A). Yet, in the 2-D t-SNE embedding based on SPOTDis, the different clusters form well separated “islands” (Fig 5A), and the HDBSCAN clustering algorithm captures them (S4 Fig).

To systematically analyze the dependence of clustering performance on the SNR, we varied the SNR by changing the firing rate inside the activation pulse period, while leaving the firing rate outside the activation period as well as the duration of the activation (pulse) period constant. Thus, we varied the first aspect of noise, which is driven by spike count fluctuations. A measure of performance was then constructed by comparing the unsupervised cluster labels rendered by HDBSCAN with the ground-truth cluster labels, using the Adjusted Rand Index (ARI) measure (see Methods). As expected, we find that clustering performance increases with the firing rate SNR (Fig 5B). Importantly, as the number of neurons increases, we find that the same clustering performance can be achieved with a lower SNR (Fig 5B). Thus, SPOTDisClust does not suffer from the problem of combinatorial explosion as the number of neurons that constitute the patterns increases, and, moreover, its performance improves when the number of recorded neurons is higher. The reason underlying this behavior is that each neuron contributes to the separability of the patterns, such that a larger sample of neurons allows each individual neuron to be noisier. This means that, in the brain, very reliable temporal patterns may span high-dimensional neural spaces, even though the bivariate correlations might appear extremely noisy; absence of evidence for temporal coding in low dimensional multi-neuron ensembles should therefore not be taken as evidence for absence of temporal coding in high dimensional multi-neuron ensembles.

We also varied the SNR by changing the pulse duration while leaving the ratio of expected number of spikes in the activation period relative to the baseline constant. The latter was achieved by adjusting the firing rate inside the activation period, such that the product of pulse duration with firing rate in the activation period remained constant, i.e. T_pulseλ_pulse = c. Thus, we varied the second aspect of noise, namely the amount of spike timing jitter in the pulse activation period. We find a similar dependence of clustering performance on the firing rate SNR and the number of neurons (Fig 5B). Hence, patterns that comprise brief activation pulses of very high firing yield, given a constant product T_pulseλ_pulse, clusters that are better separated than patterns comprising longer activation pulses.

We performed further simulations to study in a more simplified, one-dimensional setting how the SPOTDis depends quantitatively on the insertion of noise spikes outside of the activation pulse periods, which further demonstrates the robustness of the SPOTDis measure to noise (S6 Fig).

In addition, we performed simulations to determine the influence of spike sorting errors on the clustering performance. In general, spike sorting errors lead to a reduction in HDBSCAN clustering performance, the extent of which depends on the type of spike sorting error (contamination or collision) [70] and the structure of the spike pattern (S7 and S8 Figs). This result is consistent with the dependence of the HBDSCAN clustering performance on signal-to-noise ratio and pulse duration shown in Fig 5, as well with the notion that contamination mixes responses across neurons, such that the number of neurons that carries unique information decreases. We further note that in general, SPOTDisClust provides flexibility when detecting patterns using multiple tetrodes or channels of a laminar silicon probe: A common technique employed when analyzing pairwise correlations (e.g. noise correlations) is to ignore pairs of neurons that were measured from the same tetrode or from nearby channels. When the number of channels is large, this will only ignore a relatively small fraction of neuron pairs. Because the SPOTDis measure is defined over pairs of neurons, rather than the sequential ordering of firing defined over an entire neuronal ensemble, this can be easily implemented by, in Eq 2, letting the sum run over neuron pairs from separate electrodes.

Temporal pattern recognition in sparsely firing ensembles

As explained above, an extreme case of noise driven by spike count fluctuations is the absence of firing during an epoch. If many neurons remain “silent” in a given epoch, then we can only compute the EMD for a small subset of neuron pairs (Eq (2)). Such a sparse firing scenario might be particularly challenging to latency-based methods, because the latency of cells that do not fire is not defined. We consider a case of sparse firing in Fig 6 where the expected number of spikes per epoch is only 0.48. Despite the firing sparsity, the low-dimensional t-SNE embedding based on SPOTDis shows separable clusters, and HDBSCAN correctly identifies the different clusters (Fig 6). We also performed a simulation in which patterns consist of precise spike sequences, and examined the influence of temporal jitter of the precise spike sequence, as well as the amount of noise spikes surrounding these precise spike sequences. Up to some levels of temporal jitter, and signal-to-noise ratio, HDBSCAN shows a relatively good clustering performance (S9 Fig). In general, given sparse firing, a sufficient number of neurons is needed to correctly identify the P patterns, but, in addition, the patterns should be distinct on a sufficiently large fraction of neuron pairs. Furthermore, the lower the signal-to-noise ratio, the more neurons are needed to separate the patterns from one another.

Insensitivity to scaling of firing rates

A key aim of the SPOTDisClust methodology is to identify temporal patterns that are based on consistent temporal relationships among neurons. However, in addition to temporal patterns, neuronal populations can also exhibit fluctuations in the firing rate that can be driven by e.g. external input or behavioral state and are superimposed on temporal patterns. A global scaling of the firing rate, or a scaling of the firing rate for a specific assembly, should not constitute a different temporal pattern if the temporal structure of the pattern remains unaltered, i.e. when the normalized cross-correlation function has the same expected value, and should not interfere with the clustering of temporal patterns. This is an important point for practical applications, because it might occur for instance that in specific behavioral states rates are globally scaled [71, 72].

In Fig 7A, we show an example where there are three different global rate scalings, as well as two temporal patterns. The temporal patterns are, for each epoch, randomly accompanied by one of the different global rate scaling factors. The t-SNE embedding shows that the temporal patterns form separate clusters, but that the global rate scalings do not (Fig 7A). Furthermore, HDBSCAN correctly clusters the temporal patterns, but does not find separate clusters for the different rate scalings (Fig 7A and S4 Fig). This behavior can be understood from examination of the sorted dissimilarity matrix, in which we can see that epochs with a low rate do not only have a higher SPOTDis to epochs with a high rate, but also to other epochs with a low rate, which prevents them from agglomerating into a separate cluster (Fig 7A); rather the epochs with a low rate tend to cluster at the edges of the cluster, whereas the epochs with a high rate tend to form the core of the cluster (Fig 7A).

Another example of a rate scaling is one that consists of a scaling of the firing rate for one half of the neurons (Fig 7B). Again, the t-SNE embedding and HDBSCAN clustering show that rate scalings do not form separate clusters, and do not interfere with the clustering of the temporal patterns (Fig 7B and S4 Fig). We conclude that the unsupervised clustering of different temporal patterns with SPOTDisClust is not compromised by the inclusion of global rate scalings, or the scaling of the rate in a specific subset of neurons.

Application to visual responses of V1 populations

We apply the SPOTDisClust method to data collected from monkey V1. Simultaneous recordings were performed from 64 V1 channels using a chronically implanted Utah array (Blackrock) (see Methods). We presented moving bar stimuli in four cardinal directions while monkeys performed a passive fixation task. Each stimulus bar was presented 20 times. We then pooled all 80 trials together, and added 80 trials containing spontaneous activity. Our aim was then to recover the separate stimulus conditions using unsupervised clustering of multi-unit data with SPOTDisClust. The low dimensional t-SNE embedding shows four dense regions that are well separated from each other and correspond to the four stimuli, and HDBSCAN identifies these four clusters (Fig 8). Furthermore, when we performed clustering on the firing rate vectors, and constructed a t-SNE embedding on distances between population rate vectors, we were not able to extract the different stimulus directions from the cluster labels. This shows that the temporal clustering results are not trivially explained by rate differences across stimulus directions, but also indicates that the temporal pattern of population activity might reveal important stimulus information beyond the neuronal firing rates. Thus, SPOTDisClust can be successfully used on real neuronal data to identify different temporal patterns in high-dimensional multi-neuron ensembles.

Optimization of window length

For the simulations and applications above, we have assumed that the temporal window of interest for the application was known, i.e. that we had some a priori knowledge about the length of the spike patterns. For many applications in neuroscience, we want to detect sequences around specific events that are either defined by some external event (e.g. a stimulus onset) or by some “internal” event, e.g. a sharp wave ripple complex, an UP-DOWN state transition, or the cycle of some oscillation, e.g. a hippocampal theta cycle. However, in these cases the precise duration of the spike patterns is not known, and in addition there is most likely some jitter in the onset of the sequence. To handle this onset jitter, SPOTDisClust explicitly defines patterns at the level of cross-correlations rather than on univariate spike trains directly, which endows the measure with time translational invariance. To show the feasibility of determining the window length automatically from the data, we have performed simulations in which sequences occurred around some event with additional temporal jitter. These sequences were flanked by homogeneous noise (i.e. before and after the sequences). We then varied the chosen window length around the event, and measured the cluster quality using the unsupervised Silhouette score, which is based on a comparison of distances within each cluster vs. distances between the clusters (see Methods). As expected, the (unsupervised) Silhouette and ARI (ground-truth) cluster quality scores decreased when we used longer and shorter windows than the true sequence length (S11 Fig). Importantly, however, the Silhouette and ARI score showed a tight correspondence with one another, showing the feasibility of selecting the window length for spike sequence detection in an unsupervised manner. For our application to neuronal data, we optimized the window length used for the clustering using the Silhouette score and show that we can recover the window length that maximizes the cluster quality as compared to ground-truth (ARI) (S12 Fig).

Construction of SPOTDis

The SPOTDisClust method contains two steps. The first step is to construct the pairwise epoch-to-epoch SPOTDis measure on the matrix of cross-correlations among all neuron pairs.

1. We compute, for each of the M epochs separately, the normalized cross-correlation function for all pairs of N neurons, which yields M matrices of N(N − 1)/2 normalized cross-correlations . This function is defined (5) where is the cross-correlation function (or cross-covariance), and s_i,k(t) and s_j,k(t) are the spike trains of neurons i and j in the kth trial, with k = 1, …, M, and i, j = 1, …, N. In other words, the normalized cross-correlation histogram is simply the histogram of expected coincidence counts at different delays τ, normalized to unit mass. We define in Eq (5). S1 and S2 Figs show an example of cross-correlations before and after normalization.
2. We then compute the Earth Mover’s Distance (EMD) between and for each (k,m). While the EMD is usually defined on each entry of the histogram (i.e. for each τ), we can ignore the values of τ for which here since there is no mass to move to or from. We therefore first find all the τ for which s_ij,k(τ) > 0 and s_ij,m(τ) > 0, defining respectively the (in ascending order) sorted vectors and (we omit i and j subscripts here). These have Q = ∑_τ s_ij,k(τ) and R = ∑_τ s_ij,m(τ) elements respectively, and associated mass and . If there exist τ for which s_ij,k(τ) = z with z > 1, , then there will be z elements of τ in and associated mass in , the same for m and . Following these definitions, we have s_ij,k(x_q) > 0 for all q and s_ij,m(y_r) > 0 for all r, i.e. the vectors and contain all the pairwise delays between the spike trains of a pair of neurons in epoch k and m, respectively. Furthermore, x_q ≥ x_p for all q > p, and and y_r ≥ y_z for all r > z, i.e. the vectors contain pairwise delays in ascending order. As an example, if neuron i fired spikes at samples (10, 11, 20, 23) in epoch k and neuron j fired spikes at (14, 15, 20) then and .
   Because the EMD is computed on the precise delay times, we can let Δt → 0 and Δτ → 0, i.e. the sampling rate can be made infinitely large. In practice, we therefore directly find the vectors and and do not compute for all τ.
   The EMD also requires the definition of a moving cost function, which in this case is defined over time points. Let c be the moving cost function which we define as the L1 norm, c(τ₁, τ₂) ≡ |τ₁ − τ₂|/(2T + 1). Note that the normalization of ensures that 0 ≤ D ≤ 1 (the EMD) in Eq (6). Solving the optimal transport problem then amounts to finding a matrix of flows F ≡ [f_q,r], with f_q,r the flow (i.e. the amount of mass moved) from w_q,x to w_r,y, such that the overall cost is minimized, i.e. (6) subject to the constraints that (7) (8) (9) (10) The last Eq (10) ensures that all mass needs to be moved (the total mass equals 1), Eqs (8) and (9) ensure that all the mass from and gets moved, and the first equation ensures that the flow is greater or equal than zero. Here, D is the EMD. This follows the standard definition of the EMD with normalized mass. Note that the EMD with the L1 norm is equivalent to the 1st order Wasserstein distance (which is a special case of the Kantorovich formulation of the optimal transport). However, for simplicity, we use the notation of the EMD here, which uses discrete variables.
   We solve the transport problem algorithmically as follows (S2 Fig). The algorithm can be intuitively interpreted (as reflected in the comments in pseudo-code) as transporting mass from to , in case Q > R, with cost c(x_q, y_r).
   % START ALGORITHM
   SET emd = 0, q = 0, r = 0
   WHILE q < Q DO
   % Move all the remaining mass in w_q,x to w_r,y, but do not move more than w_r,y.
    SET flow = min(w_q,x, w_r,y)
   % The cost equals the amount of mast moved, flow, times the cost of moving from the time point x_q to the time point y_r.
    SET cost = flow × c(x_q, y_r)
   % Add this cost to the total cost.
    SET emd = emd + cost
   % Compute the remaining mass in w_q,x to be moved, which equals the
   previous mass w_q,x minus the mass we just moved, flow.
    SET w_q,x = w_q,x − flow
   % Compute the remaining mass in w_r,y, to which we still need to transport from in the next iterations.
    SET w_r,y = w_r,y − flow
   % If there is no mass left then increment the index, note that the vectors and are sorted.
    IF w_r,y = 0 THEN
     SET r = r + 1
    ENDIF
    IF w_q,x = 0 THEN
     SET q = q + 1
    ENDIF
   ENDWHILE
   % END ALGORITHM
3. After computing the EMDs between each pair of epochs for each neuron pair separately, with computational complexity of order (N² M² n²), where n is the average number of spikes, we compute SPOTDis as the average EMD, i.e. (11) with (12) where sgn is the sign function.

Computing the SPOTDis with the sparse simulation (Fig 6) takes 60 seconds and the computation for the example case (Fig 1) takes approximately 5 minutes, utilizing an Intel Xenon E5-2650 v2 2.60GHz system with 16 cores.

HDBSCAN

HDBSCAN is an automated density clustering algorithm that clusters on the basis of pairwise dissimilarity matrices. An extensive overview of HDBSCAN can be found in [60, 61] and we provide only a brief overview of HDBSCAN here. HDBSCAN comprises the following steps:

1. After pairwise distances have been computed between all data points, HDBSCAN defines a “mutual reachability distance” between each pair of data points (in our case epochs). The mutual reachability distance is an adjustment of the distance measure that effectively acts as a smoother. For each epoch k, the core distance is defined as the SPOTDis dissimilarity, (Eq (2)), to its n_ptsth nearest neighbour m. The mutual reachability distance is then defined as . The mutual reachability distance does not alter the distance between two points that are in dense regions, but it changes the distance for points that are in low-density regions and have a relatively large distance. The purpose of transforming the distance matrix in this way is to make the clustering algorithm more robust to noise.
2. HDBSCAN then defines a minimum spanning tree, in which there is a path between all points (vertices), without any loops (i.e. an acyclic graph), such that the total weight of the edge connections is minimized. Here the edges are the mutual reachability distances.
3. HDBSCAN then constructs an hierarchical cluster dendrogram from the minimum spanning tree as follows: Initially all points are assigned to the “root”, a single cluster containing all points. HDBSCAN then sets a threshold ϵ and cuts edges from the minimum spanning tree whose weight is higher than ϵ. HDBSCAN keeps decreasing the value of ϵ such that more connections are removed and new clusters can appear, forming a cluster dendrogram. Sets of points that have fewer than n_clSize members, the minimum cluster size, at a value of ϵ are deemed noise points at that value of ϵ. Here we take the simplification n_pts = n_clSize [60], such that n_pts is the only hyperparameter, which we set to 10 here, unless specified otherwise. We use the implementation of HDBSCAN developed by [61], and all analyses and simulations were performed in Python. HDBSCAN uses either the “leaf selection” method for selecting clusters, or the “excess of mass” method. In the leaf selection method, the end branches of the hierarchical cluster dendrogram are taken as the selected clusters. In the “excess of mass” algorithm, HDBSCAN chooses the set of clusters that is most stable under a change of ϵ. This is done as follows: at some value of ϵ a cluster is born, ϵ_max, and at some point, the cluster dies, ϵ_min. For each member of the cluster we can define the value of ϵ_k where each kth member fell out, and take the stability as the sum ∑_k1/ϵ_k − 1/ϵ_max over all members. HDBSCAN then selects in the dendrogram the optimal levels at which to cut the tree in order to maximize the stability of the selected clusters, forming a set of clusters. An advantage of this selection procedure is that it allows for clusters of varying density.

t-SNE

T-SNE (t-distributed stochastic neighbor embedding) is a dimensionality reduction technique for high-dimensional datasets [62]. While it typically is computed starting from a high-dimensional dataset that is then converted into a matrix of pairwise Euclidean distances, here we compute it directly on the pairwise dissimilarity matrix. We first outline the algorithm of SNE [63], and after that the adjustments made in t-SNE.

1. For each two data points (k and m), which represent epochs in our case, a measure of similarity is computed. This measure of similarity is taken as the conditional probability of observing m given a Gaussian distribution centered on k, (13) Here, the standard choice for d_km is the L1 norm for a high dimensional dataset x₁, …, x_M, defined as d_km = ||x_k − x_m||, but we take it here as the SPOTDis, . The normalization in Eq (13) simply assures that the probabilities ∑_m p_m|k sum to one. The variance of the Gaussian, , is determined for each data point individually, by finding σ_k to satisfy the equation (14) Here, H is the entropy function using logarithm base 2, P_k is the probability distribution of all the data points given k, P_k = (p_1|k, …, p_M|k), and ψ is the perplexity which is set by the user, which can be interpreted as a smooth measure of the effective number of neighbors each datapoint has. We set the perplexity ψ to 30, which is in the typical range used in the literature. T-SNE is generally quite insensitive to choices of the perplexity, which is usually taken in the range 5-50.
2. SNE then attempts to find a low dimensional set of data points, {y₁, …, y_M}, that have a similar distribution of conditional probabilities (similarities) as the distances derived from the high-dimensional counterparts. In this case the variance of the Gaussian is constant for all data points, and (15)
3. SNE then minimizes a cost function, which in this case is the Kullback-Leibler divergence between p_m|k and q_m|k over all data points. To do that, it starts from a random sample of points (Gaussian distributed) and then performs a gradient descent, in which each point y_k is moved around depending on the attraction or repulsion from other data points (see [62]).

T-SNE makes two main adjustments relative to SNE (the rationale behind these two adjustments is extensively discussed in [62]). First, it uses a symmetric measure of similarity between two data points, as the joint probability (16) Second, it uses a Student’s t-distribution with one degree of freedom instead of a Gaussian for the low dimensional counterparts.

Measures for cluster evaluation

Application to neuronal data

One male macaque monkey performed a passive fixation task while moving bar stimuli (white bars on gray background, 0.25 degrees in visual angle width) were presented. All procedures complied with the German law for the protection of animals and were approved by the regional authority (Regierungspräsidium Darmstadt). Recordings were performed from 64 V1 channels simultaneously, obtained from a chronic Utah array implant (Blackrock). Receptive fields had eccentricities around 3-5 degrees visual angle. We performed band-pass filtering of each channel in the frequency range of action potentials (300-6000Hz) and then thresholded the band-pass filtered signal x(t) according to [83], using the threshold , where med is the median (i.e. effectively three standard deviations). When the signal x(t) crossed this threshold, we denoted a spike. After the detection of a threshold crossing, further threshold crossings were suppressed for 0.75ms. Moving bar stimuli were presented in four cardinal directions. Each stimulus bar was presented 20 times. We then pooled all 80 trials together, and added 80 trials containing spontaneous activity. Our aim was then to recover the separate stimulus conditions using unsupervised clustering with SPOTDisClust. We use n_pts = 3 with leaf selection for the HDBSCAN parameters.