Emergent cortical circuit dynamics contain dense, interwoven ensembles of spike sequences

Neocortical computation occurs largely within microcircuits comprised of individual neurons and their connections within small volumes (<500 μm³). We found evidence for a long-postulated temporal code, the Hebbian assembly phase sequence, by identifying repeated and co-occurring sequences of spikes. Variance in population activity across trials was explained in part by the ensemble of active sequences. The presence of interwoven sequences suggests that neuronal assembly structure can be variable and is determined by previous activity.

Abstract

Temporal codes are theoretically powerful encoding schemes, but their precise form in the neocortex remains unknown in part because of the large number of possible codes and the difficulty in disambiguating informative spikes from statistical noise. A biologically plausible and computationally powerful temporal coding scheme is the Hebbian assembly phase sequence (APS), which predicts reliable propagation of spikes between functionally related assemblies of neurons. Here, we sought to measure the inherent capacity of neocortical networks to produce reliable sequences of spikes, as would be predicted by an APS code. To record microcircuit activity, the scale at which computation is implemented, we used two-photon calcium imaging to densely sample spontaneous activity in murine neocortical networks ex vivo. We show that the population spike histogram is sufficient to produce a spatiotemporal progression of activity across the population. To more comprehensively evaluate the capacity for sequential spiking that cannot be explained by the overall population spiking, we identify statistically significant spike sequences. We found a large repertoire of sequence spikes that collectively comprise the majority of spiking in the circuit. Sequences manifest probabilistically and share neuron membership, resulting in unique ensembles of interwoven sequences characterizing individual spatiotemporal progressions of activity. Distillation of population dynamics into its constituent sequences provides a way to capture trial-to-trial variability and may prove to be a powerful decoding substrate in vivo. Informed by these data, we suggest that the Hebbian APS be reformulated as interwoven sequences with flexible assembly membership due to shared overlapping neurons.

NEW & NOTEWORTHY Neocortical computation occurs largely within microcircuits comprised of individual neurons and their connections within small volumes (<500 μm³). We found evidence for a long-postulated temporal code, the Hebbian assembly phase sequence, by identifying repeated and co-occurring sequences of spikes. Variance in population activity across trials was explained in part by the ensemble of active sequences. The presence of interwoven sequences suggests that neuronal assembly structure can be variable and is determined by previous activity.

despite reliable perception and behavior, a hallmark of neural activity is a relatively high degree of trial-to-trial variability (Arieli et al. 1996; Heggelund and Albus 1978; Vogels et al. 1989). It is not yet known to what degree this variability is related to cortical function or is simply uninformative biological noise. Identifying a complete picture of the statistics describing population dynamics is necessary to distinguish between noisy and informative spiking and in turn understand cortical function and computation. A general feature of neuronal population dynamics is recurring spatiotemporal patterns of neuronal activity occurring in a wide range of preparations and under a variety of conditions (Bathellier et al. 2012; Beggs and Plenz 2004; Johnson et al. 2010; Peters et al. 2014; Villa et al. 1999). Spatiotemporal patterns have been found to correlate with perception, sensory input, and motor behavior (Churchland et al. 2012; Harvey et al. 2012; Luczak et al. 2009), but we still lack the necessary tools to fully capture and describe complex population spike patterns. Noisy spatiotemporal patterns can be summarized as trajectories using dimensionality reduction techniques, facilitating accurate decoding and interpretability (Briggman et al. 2005; Broome et al. 2006; Yu et al. 2009). However, neocortical computation is implemented at the level of spiking neurons, making the mechanisms underlying these reduced-dimension trajectories difficult to resolve. Analytical methods that retain detail relating neurons and their spikes to the information they carry will facilitate an understanding of the network mechanisms supporting meaningful dynamics.

Temporally structured population activity is an expedient mechanism for information representation, particularly for time-varying stimuli. Temporal spike patterns provide a powerful computational substrate (Crowe et al. 2010; Rabinovich et al. 2008; Villa et al. 1999; Wehr and Laurent 1996), but the extent to which local cortical networks can support reliable spike propagation is an open question (Diesmann et al. 1999). Neural networks often exhibit a relatively consistent advancement of activity, which is summarized as a mean global progression (MGP) of activity (Fig. 1E) (Harvey et al. 2012; Luczak et al. 2007; Luczak et al. 2015; Pastalkova et al. 2008), suggesting that neocortical networks have the capacity for reliable spike propagation. The MGP, revealed after sorting neurons by their mean spike time, is consistent with a Hebbian assembly phase sequence (APS), in which functionally related assemblies of neurons propagate activity through the network (Hebb 1949). Cell assemblies facilitate the propagation of spiking, providing an important bridge between the monosynaptic building blocks of a cortical network and trajectories within population dynamics. Whereas spatiotemporal patterns are a robust feature of cortical network activity, neurons are noisy, and consequently MGPs generally exhibit large trail-to-trial variability (Lin et al. 2015; Luczak et al. 2009). The structure of this variability is understudied. Here, we sought to identify subpopulations of the network exhibiting reliable spike propagation due to shared spike-time variability.

Fig. 1.

Structured, emergent dynamics in neocortical populations A: representative field of view showing imaged neurons spanning all cortical lamina. B: 2 sample spontaneous events of activity. C: post-event time histogram (PETH) of population showing a fast rise in activity and slower decay (means ± SD). D: accumulation of variance across the population. Each circle is a neuron (all data sets included; n = 11). For each neuron, spikes were pooled across all events aligned to the start of each event. Overlaid running, means ± SE in blue. E: mean global progression is a product of the population PETH; representative mean global progression (MGP) from data and MGP generated from an inhomogeneous Poisson population. MGPs in each case have unique ordering but similar structure. Normalized activity of neurons sorted by their mean spike time; spikes convolved with a 60-ms Gaussian window for visualization; color map from Niccoli (2012).

To study reliable spike propagation, we employed two-photon calcium imaging to record spontaneous population activity in large neocortical networks ex vivo. This preparation allows us to densely record the local cortical network and observe activity as it propagates undisturbed by top-down modulation or bottom-up sensory drive. The spatiotemporal activity patterns resulted in an MGP that was qualitatively similar to those observed in vivo. We show that in these networks, the population post-event time histogram (PETH) alone can lead to an MGP. To evaluate the capacity of these networks for structured spike timing that could not be explained by the PETH, we identified small ensembles of neurons exhibiting sequential spiking. The large repertoire of spike sequences collectively comprise the majority of all spiking in the network. Viewing network activity from the perspective of these spike sequences helped to capture trial-to-trial variability. Sequences densely manifested within the population dynamics and strongly overlapped in time and in neuronal membership, leading to interwoven ensembles of sequences that characterized each population event. Importantly, the occurrence of any given sequence is not deterministic, and the full ensemble of sequences in any trial is unique. From the perspective of an APS, our results suggest that rather than a concatenated, feedforward group of assemblies, the dynamics are characterized by interwoven assemblies. Structured spatiotemporal activity patterns can encode information and compute (Buonomano and Maass 2009; Durstewitz and Deco 2008; Rabinovich et al. 2008; Sakurai 1999), suggesting that spike sequences may underlie aspects of cortical function, including sensory representations.

MATERIALS AND METHODS

Experimental preparation.

All procedures were performed in accordance with and approved by the Institutional Animal Care and Use Committee at the University of Chicago. Data analyzed here were previously reported in Sadovsky and MacLean (2013). C57BL/6 mice of either sex, P14–17, were anesthetized by intraperitoneal ketamine-xylazine injection. After anesthetic level was confirmed, mice were rapidly decapitated and had their brains removed and placed in ice-cold cutting solution (containing in mM: 3 KCl, 26 NaHCO₃, 1 NaH₂PO₄, 0.5 CaCl₂, 3.5 MgSO₄, 25 dextrose, and 123 sucrose) bubbled with carbogen (95% O₂ and 5% CO₂). Coronal and thalamocortical slices of primary auditory cortex (described in Cruikshank et al. 2002), 500 μm thick, were cut using a vibratome (VT1000S; Leica). These slices were then incubated in 35°C oxygenated artificial cerebrospinal fluid (ACSF; containing in mM: 123 NaCl, 3 KCl, 26 NaHCO₃, 1 NaH₂PO₄, 2 CaCl₂, 6 MgSO₄, and 25 dextrose) for 35–40 min. Slices were then transferred into a small Petri dish containing 2 ml of ACSF with an aliquot of 50 g of Fura-2 AM (Invitrogen), 2 liters of Pluronic F-127 (Invitrogen), and 13 liters of DMSO (Sadovsky and MacLean 2013).

Data acquisition.

Data were acquired with slices submerged in standard ACSF (containing in mM: 123 NaCl, 3 KCl, 26 NaHCO₃, 1 NaH₂PO₄, 2 CaCl₂, 2 MgSO₄, and 25 dextrose), which was continuously aerated with carbogen (95% O₂ and 5% CO₂). Primary auditory cortex was identified visually by internal capsule landmarks under a bright field, and two-photon scanning microscopy was performed using the Heuristically Optimal Path Scanning technique (Sadovsky et al. 2011), resulting in imaging frames lasting 78 ± 14.5 ms. Following data acquisition, action potentials were inferred from fluorescence traces using a modified fast nonnegative inference algorithm (Vogelstein et al. 2010). Data sets with fewer than four identified spontaneous populations events and neurons active in fewer than two events were excluded from analysis.

Experimental design and statistical analysis.

Calcium imaging was performed on 11 mice (P14–17; both sexes). Data from all experiments were included in all analyses. Coronal (n = 5) and thalamocortical (n = 6) slices were pooled, as no statistical differences were found between data sets, as described previously (Sadovsky and MacLean 2013). All statistical tests were performed with MATLAB (MathWorks). Unless otherwise noted, data are shown as means ± SD. Significance of correlation coefficients were computed using a Student’s t-statistic. A difference in the means of Fig. 6C was confirmed using a two-way ANOVA (spike category and animal identity). All other tests assumed nonparametric distributions and used Mann-Whitney U-test.

Fig. 6.

Single trails of MGP are composed of unique subsets of interwoven sequences A: 2 example rasters with overlaid active sequences. Neurons sorted according to MGP; events exhibit unique (top: blue; bottom: orange) and overlapping (green) sequences. B: distribution of shared sequence spikes between event pairs. Low overall overlap, with many events sharing nearly no sequences. C: sequence spikes occur closer to expected MGP position (P = 1.1 × 10⁻³, 2-way ANOVA). Box and whiskers plots displayed as in Fig. 2C. D: all sequences show strong pooling at the end of long sequences (black bar is mean, gray dots are for each data set); active sequences show balanced convergence and divergence throughout event (means ± SE; gray points show each time point across all events).

Normalized count.

Normalized lagged-spiking counts were computed for every pair of neurons as

$$\frac{{{\displaystyle \sum }}_{n=1}^N{{\displaystyle \sum }}_{t=1}^{T-1}{x}_i\left(t\right)\times x_j\hspace{0.17em}\left(t+1\right)}{{{\displaystyle \sum }}_{n=1}^N{{\displaystyle \sum }}_{t=1}^T{x}_i\hspace{0.17em}\left(t\right)}$$

where x_i(t) is the binary spike train of neuron I for each event n and each time frame t. These lagged spike counts quantify the probability of cell j spiking after cell i. The matrix of normalized counts between pairs of neurons served to identify potential sequences before testing for significance (see Fig. 2A).

Fig. 2.

Identification of spike sequences. A: illustration of sequence identification method (details in materials and methods). Candidate sequences are constructed, and a sequence score is computed from weighted spike trains. The sequence was accepted if score was significantly larger than scores from surrogate spike trains. B: spike sequences of each length. Single-example columns: spike trains from 1 event (sequence spikes outlined in green); average columns: normalized spike weight across all sequences and all events showing strong sequential structure. C: no. of sequences identified across all data sets (data), trial-shuffled data sets (trial shuffle), sequences identified with randomly sorted candidate sequences (random) and surrogate spikes replicating population PETH (inhom). Boxes show median and interquartile range (IQR). Whisker length computed with default MATLAB settings (2.7 × SD × IQR).

Candidate sequence construction.

Candidate sequences are constructed from the normalized lagged-spike count matrix. Each sequence length is defined beforehand. We tested 1,000 candidate sequences for sequence lengths of 3–8. For each sequence length, pairs of neurons were deterministically drawn from the highest values in the normalized count matrix. The first pair was drawn starting from the highest value within the entire matrix. A third neuron was added by choosing the neuron with the highest value among all the (2, i) values, excluding the neuron in position 1. Iteratively, neuron n was chosen by the largest (n = 1, i) values, excluding neurons 1,…, n = 1. This guarantees that no neuron repeats within a single candidate sequence. The procedure ends once the predetermined sequence length is reached. To maximize the average normalized count within the candidate sequences, the next candidate was created starting from position k, where the maximum values among all (k, i) are greater than the maximum values among all other starting positions. In this way, the next candidate can branch off the previous candidate, and they would share neurons 1–k.

When random sequences were generated, normalized counts were ignored, but neurons were again not allowed to repeat within individual candidate sequences. After surrogate data sets of inhomogeneous Poisson spiking or trial-shuffled spike trains were generated, a new normalized count matrix was computed, and the identical candidate generation method was applied.

Sequence score.

Each candidate sequence was characterized by its sequence score (S), which was roughly equal to the fraction of spiking in sequence relative to all spiking in the sequence neurons. First, the binary spike rasters of the candidate sequence are transformed by a term frequency-inverse document frequency (TF-IDF) weighting (Carrillo-Reid et al. 2015). The binary spike then becomes the product of its term-frequency weighting and its inverse document frequency weighting:

$$W_{c,e}\left(t\right)=w_{c,e}^{\mathrm{tf}}\left(t\right)\times w_{c,e}^{\mathrm{idf}}\left(t\right),$$

with term-frequency weighting

$$w_{c,e}^{\mathrm{tf}}\left(t\right)=\log \left(\frac{f_e}{1+n_{c,e}}\right)\times X_{c,e}\left(t\right)$$

and the document-frequency weighting

$$w_{c,e}^{\mathrm{idf}}\left(t\right)=\log \left[1+\frac{1}{E}{\displaystyle \sum }_{e=1}^{E}H\left(n_{c,e}\right)\right]\times X_{c,e}\left(t\right),$$

where X_c,e(t) is the binary spike train of cell c in event e, f_e is the number of frames in event e, n_c,e is the number of spikes in cell c in event e, E is the total number of events, and H(x) is the Heaviside function, with the convention H(0) = 0. Term frequency weighting decreases the weight of a spike when a given neuron spikes multiple times within an event (i.e., it may not participate in sequences at a unique time point). Inverse document weighting increases the weight of all spikes in an event when the neuron is active in many different events (i.e., it is more likely to be reliably active in spike sequences).

To compute the sequence score, we first convolve each W_c,e(t) with a five-frame Hamming window. Then, the sequence score (S) and sequence onset time (T_e) is defined as

$$T_e=\underset{1\le t\le f_e}{\mathrm{argmax}}\left[{\displaystyle \sum }_{c=1}^L{W}_{c,e}\hspace{0.17em}\left(t+c-1\right)\right]$$

and

$$S=\frac{1}{E}{\displaystyle \sum }_{e=1}^E{s}_e,$$

where

$$s_e=\frac{{{\displaystyle \sum }}_{c=1}^L{W}_{c,e}\left(T_e+c-1\right)}{{{\displaystyle \sum }}_t{{\displaystyle \sum }}_{c=1}^L{W}_{c,e}\left(t\right)},$$

where L is the sequence length. These equations define the sequence score as the mean fraction of weighted spikes occurring within the identified sequence window. The sequence window is offset by one frame for each subsequent cell in the sequence, and its location is simply that which gives the maximum within-event sequence score.

Surrogate spike trains are then constructed for each candidate sequence. The number of spikes per event is maintained for each cell. In each event, the spike times of all candidate cells are pooled to estimate the multiunit spike time histogram. New spikes are randomly drawn from this distribution for each cell until it has the same number of spikes in the original data. This is done independently for each event until a complete set of new spike trains is generated. This procedure is repeated 1,000 times, leading to 1,000 surrogate sequence scores. This distribution of scores represents how sequentially we can expect this set of neurons to behave given their coactivity and spike timing. Each candidate sequence is accepted as significant if its own sequence score is >99% of the surrogate sequence scores.

MGP analysis.

The MGP is computed by sorting the neurons by their mean spike time across all trials. The percent of spiking within the MGP was calculated by the number of spikes within a spike window centered at each neuron’s mean spike time. All spikes from all trials were convolved with a 60-ms Gaussian window to generate a time-varying firing rate. To define whether a spike occurred within the MGP or not, an MGP window was defined as the nearest local minimum before and after each neuron’s location in the MGP (i.e., its mean spike time). Surrogate MGPs were computed from inhomogeneous Poisson populations for each data set 40 times, with one example shown in Fig. 1E.

Sequence analysis.

Only first-spike times were included to compute the mean and variance of spike times. Cells that were active in fewer than four events were excluded. Spikes were classified as sequence spikes or nonsequence spikes. According to the cell identities, ordering, and event-specific sequence window of each sequence, we expect to see spikes occurring at specific time points in specific neurons. Thus, a sequence spike is a spike that satisfies this condition (i.e., it spikes in the sequence window). Each spike was compared against the expected spike timing of every sequence the given neuron participated in. If the spike matched the expected spike time of any sequences, it was classified as a sequence spike; if it matched none, it was a nonsequence spike. All other analyses are described in results.

RESULTS

Neocortical dynamics exhibit structured progression of neuronal activity.

To understand the intrinsic properties of network activity generated by local circuitry in neocortex, we chose an ex vivo preparation (Fig. 1A). This approach allowed us to study the statistical features of intrinsic dynamics that are produced exclusively by local circuit connectivity, excluding long-range top-down and bottom-up influences. Population activity in this preparation emerged spontaneously as discrete events of coordinated network activity, resulting in elevated membrane potential across the population and spiking in a subset of the population (Cossart et al. 2003; Kruskal et al. 2013; Sadovsky and MacLean 2013;). This activity was brief (1.28 ± 0.46 s) and was separated by long periods of quiescence, making each period of activity a distinct observation of spike propagation through the local network (Fig. 1B). Using two-photon calcium imaging, we recorded 9.4 ± 4.6 of these events from large populations (595 ± 101 neurons), sampling activity in the population at 12.8 ± 2.0 Hz/slice preparation (n = 11). We used a fast nonnegative deconvolution algorithm (Sadovsky and MacLean 2013; Vogelstein et al. 2010) to infer spikes from calcium transients in each neuron. Across the imaged neuronal population, spiking was relatively sparse, encompassing 63 ± 25% of the population, with active neurons spiking 1.5 ± 0.8 times/event (Fig. 1B). On average, the population activity followed a common progression: activity in a small number of neurons rapidly recruiting activity in other neurons, reaching a maximum population firing rate after 512.5 ± 232.2 ms (Fig. 1C). Afterward, spiking gradually decayed until the end of the event. The mean number of active cells per frame was well fit by a difference of exponentials (r² = 0.94). Sorting neurons by their mean spike time revealed mean global progression (MGP) of activity (Fig. 1E), consistent with many reports (Churchland et al. 2012; Harvey et al. 2012; Luczak et al. 2009; Luczak et al. 2015). Alongside this mean progression of activity was a high degree of trial-to-trial variance. At the level of individual cells, variance was correlated with the mean first spike time, resulting in an accumulation of variance throughout the event, as reported previously (Luczak et al. 2015). The few neurons that initiated activity had more reliable latencies, whereas neurons that spiked later did so more variably (r = 0.51, P = 2.2 × 10⁻¹³¹, Student’s t-test; Fig. 1D). We further found that the population PETH was sufficient to replicate the MGP. We generated surrogate spike trains with a population of Poisson processes with inhomogeneous firing rates equal to the PETH. By conserving the number of spikes in each neuron and the mean PETH, the distribution of mean spike times was replicated and in turn generated a qualitatively similar MGP (Fig. 1E). The ordering of the MGP was different, but the percent of spiking within the MGP was similar between the data and the Poisson spiking (data: 47.0 ± 10.8%; Poisson: 50.8 ± 7.9%; P = 0.148, Mann-Whitney U-test), suggesting that the MGP is largely the result of inhomogeneous firing rates across a specific population. There is substantial evidence that MGPs correspond to specific sensory inputs, motor outputs, and behavioral choice, but the high variability of the MGP presents specific challenges from a mechanistic account and an information theoretic perspective. As a result, we sought to determine whether we could identify sequential spiking patterns beyond that described by the global mean.

Identifying reliable spike patterns in global assembly dynamics.

To study whether population activity supports reliable propagation of spiking beyond the MGP, we sought to identify smaller ensembles of neurons with sequential spike times across trials (Abeles and Gerstein 1988; Gansel and Singer 2012; Gourévitch and Eggermont 2010; Reyes-Puerta et al. 2015;). Testing all potential sequences of moderate length in populations of this size is not computationally tractable; therefore, we tested only groups that were most likely to generate sequential spiking (Fig. 2A). Because we were searching for sequential spikes between neurons, we generated potential “candidate” sequences using a normalized lagged-spike count matrix and then tested each candidate sequence for statistical significance. These normalized counts are better conceptualized as a consistent propagation of spikes between two neurons (Villa et al. 1999) rather than simultaneously correlated fluctuations. We systematically identified candidate sequences using the normalized count matrix and extended group size by linking one pair of strongly correlated neurons to another (Fig. 2A; for more details, see materials and methods). All potential sequences were deterministically constructed from the largest elements in the normalized count matrix. We did not allow neurons to repeat in any single sequence to ensure that each neuron had a unique position in the sequence. All candidate sequences were then independently tested to determine whether they exhibited statistically significant sequential spiking.

To determine whether each group of neurons showed reliable, sequential spiking, we used a novel metric to quantify a group of neurons’ sequential firing (Fig. 2). First, spike trains are weighted to highlight spikes more likely to participate in sequences; the metric is then computed as the fraction of weighted spiking occurring in sequence. Binary spike matrices are weighted according to a term frequency-inverse document frequency (TF-IDF) function (Carrillo-Reid et al. 2015). This weighting procedure makes sparse, binary spike trains more amenable to a continuous, rather than discrete, metric of reliability. Briefly, the weighting of a spike was increased when a neuron had few spikes per event and when a neuron spiked in many different events. This served to more strongly weight neurons that were reliably active across events and that spiked at a unique time point within trials.

We then computed a sequence score for each candidate sequence, measured as the in-sequence weight divided by the total weight in the event (i.e., the fraction of sequence spiking compared with all spiking). In-sequence weight was determined by a sequential time window characterizing the expected spike pattern (Fig. 2A) positioned to maximize the sequence score. To allow a small jitter in the sequences, TF-IDF weights were convolved with a five-frame Hamming window so that presence in the window is not all or none. Thus, a neuron not spiking was less costly than a spike out of sequence. Because of the temporal resolution of calcium imaging in these experiments, these sequences cannot be simply the manifestation of a chain of monosynaptic connections (Chambers and MacLean 2015) but rather represent network-dependent propagation of spikes (Chambers and MacLean 2016).

To establish statistical significance, we then compared the sequence score with a distribution of scores using surrogate data. To construct surrogate data, we drew random spike times from the spike time distribution of the candidate cells while maintaining the number of spikes in each cell in each event. Each spike train of the surrogate data has the same distribution of spike times across trials as its corresponding neuron, maintaining its position in the overall MGP. The surrogate data contains the spike features necessary for a population-wide MGP; consequently, differences between real data and surrogate data were due to specific spike-timing relationships between the candidate neurons themselves. We constructed 1,000 surrogate spike trains for each neuron in the candidate sequence and computed a surrogate sequence score for each set of spike trains. If the true sequence score was greater than 99% of the surrogate sequence scores (i.e., P < 0.01), we rejected the null hypothesis that the degree of sequential spiking (i.e., sequence score) occurred by chance. Our approach rejected the vast majority of candidate sequences (85.2 ± 10.1%). Despite this stringent statistical test, we identified many significant sequences after 1,000 candidates from each sequence length to generate a suitable set for quantitative analysis (69.7 ± 69.5 sequences of length 3, 132.8 ± 121.3 sequences of length 4, 182.2 ± 126.1 sequences of length 5, 202.2 ± 108.3 sequences of length 6, 177.1 ± 92.5 sequences of length 7, and 123.0 ± 72.3 sequences of length 8; Fig. 2C). We did not include any sequences of lengths greater than eight because the candidate sequences had a rejection rate of 95.0 ± 6.1%, which was not likely to provide enough statistical power for quantitative analysis. This result suggests a clear limit in the capacity of the local network to support reliable spontaneous sequences on its own. In addition, the total duration of an event acted as an upper bound by limiting the likelihood that we would find a sequence of length nine or longer. Although the significance of a sequence was dependent both on the sequence score and its surrogate distribution, we found that sequences with higher sequence scores had lower P values and were thus more likely to be significant (r = −0.43, P = 0; Student’s t-test).

The sizeable number of sequences with large (6–8) numbers of neurons indicated that there is a high degree of shared variability among large groups of neurons in the population despite the fact that single-cell spike times are highly variable after even a few frames. Illustrative examples of a sequence of each length in a single event show qualitative results of the identification analysis (Fig. 2B). Because of the design of the identification method, missing spikes and misaligned spikes were common in individual events. When we pooled all spikes from all events (aligned to the occurrence of each sequence), we found that a high degree of precision within the neurons with relatively low alignment errors and longer sequences tended to incur more errors than shorter sequences (Fig. 2B).

We performed three control tests for the identification procedure. First, we generated surrogate data in a similar manner to the statistical test but instead used the entire population raster as the spike time distribution, rather than only using spikes from the candidate sequence neurons (see materials and methods). These data are identical to those used in Fig. 1E. The correlation structure in these surrogate spike trains is wholly generated from the population post-event time histogram (PETH), removing all influences of intrinsic reliability, covariability, and higher-order correlations. Thus, one should expect no significant sequential structure from these data. Indeed, the acceptance rate for this control was near the P value threshold of the statistical test (0.4 ± 0.8%; Fig. 2C, right). As a second control, we created surrogate data by shuffling each neuron’s spike trains across trials. This control will destroy sequences that strongly vary in their onset time, but sequences with a reliable onset latency will remain. Indeed, this second class of surrogate data contained fewer significant sequences than the data (147.8 ± 106.8 data; 62.2 ± 52.6 shuffle; Fig. 2C, middle left). Informed by this control, we estimated that 42% of sequences had a consistent onset latency. Finally, we tested the efficacy of the candidate construction method by randomly linking neurons to build candidate sequences. This method was extremely computationally inefficient at identifying sequences (Fig. 2C, middle right). Constructing candidate sequences by chaining normalized count relationships together leadS to a 43-fold increase in the probability of finding a significant sequence. Because of the combinatorial nature of sequences, leveraging the normalized count matrix to generate candidate sequences was necessary for practical tractability.

Because of the deterministic sequence construction and the need for explicit sequence length specification, we found that many of the shortest sequences were themselves subsets of the longer sequences (Fig. 3C). To determine whether long sequences were only comprised of shorter significant sequences, we also tested every possible subsequence from the original significant sequences. We found that some, but not all, subsequences were significant (37 ± 23.4%). Longer subsequences were more likely to be significant than shorter subsequences (length 3,10.8 ± 8.8%; length 4, 24.2 ± 15.5%, length 5, 37.9 ± 16.8%, length 6, 51.3 ± 17.1%, length 7, 61.0 ± 16.6%). Broadly, however, significant sequential spiking within one given group of neurons does not guarantee sequential spiking with a similar group. This suggests that higher-order correlations are responsible for the presence of longer sequences (Chambers and MacLean 2016), and these sequences are not simply composed of chains of pairwise correlations. The abundance of identified sequences motivated a closer analysis of their manifestation in network dynamics.

Fig. 3.

Sequences converge onto high-membership neurons with low-variance spiking. A: sequence membership of a neuron vs. mean sequence position (note log scale on ordinate axis). Majority of neurons initiate few sequences, and a minority participate promiscuously at the end of sequences, leading to broad convergence onto a small subset of neurons. Marginal distributions are at left and bottom; color gradient is from Niccoli (2012). B: promiscuous neurons have more precise spike times relative to event onset. Each neuron variance was z-scored according to its mean spike time quantile (see inset). Mean variance shown with bootstrapped 95% confidence intervals. C: illustrative example of shared neurons between sequences. Directed graph showing sequences that share every neuron (unordered) with another sequence. Each node (n = 855) is a sequence (size corresponds to sequence length); edges lead from 1 sequence to another in which the target contains all neurons of the source.

Neuronal participation within sequences.

Neurons were allowed to have membership in multiple sequences, and among the significant sequences we found a long-tailed distribution of sequence membership across neurons. Most neurons participated in a few sequences, and a few neurons participated in a large proportion of all sequences. Surprisingly, the neurons with highest sequence membership were also those with the latest average sequence position (Fig. 3A). Taken together, this indicated that on average sequences begin in a diversity of neurons, and as sequences grow longer, they pool into fewer and fewer neurons. The broad distribution of sequence membership, spanning two orders of magnitude, emphasizes the amount of convergence at the end of long sequences. This is consistent with the accumulation of spike time variance throughout events, as additional variability would reduce the group of possible neurons to occur at the end of long sequences. With this in mind, we tested the hypothesis that long sequences are converging onto low-variance cells. We needed to control for the accumulation of variance, since neurons at the end of long sequences (i.e., participating in many sequences) are likely to be active later in the events when variance is higher. We separated the distribution of mean spike times into 39 quantiles and computed the z-scored variance of each neuron according to its mean spike time, eliminating the differences in mean variance and heteroscedasticity (Fig. 3B, inset). We found that neurons with higher sequence membership have lower variance in spike times on average (Fig. 3B). Neurons with more precise spike times preferentially participated in many sequences. This finding suggests that to participate in many sequences, it is beneficial to be locked to the population onset, as sequences with variable onset latencies would show higher spike time variability.

Population capacity for spike time reliability places an upper bound on long, reliable sequences.

The accumulation of variance led us to expect sequences to occur near the beginning of events, when spike time precision was highest. We found that on average longer sequences tended to start later than short sequences relative to the onset of activity, but the onset of shorter sequences was more widely distributed than long sequences (Fig. 4A). The consistently delayed onset of long sequences occurred alongside the peak of population activity (see Fig. 1C), possibly suggesting that a requisite level of excitability is necessary for the initiation of stable spike propagation. Onset times of all sequences showed a similar accumulation of variance to individual cells, and this relationship was independent of sequence length [linear fit slope of 3.25 for single cells (see Fig. 1D), 3.40 ± 0.23 across sequence lengths]. Thus, sequence onsets reflected the overall population PETH.

Fig. 4.

Long sequences are bounded by the duration and variance of population activity. A: distributions of onset times for each sequence length. Short sequences occur throughout MGP; long sequences are temporally coupled to peak activity (see Fig. 1C). Only sequences in which 75% of neurons were active in an event were included. Box and whisker plots are displayed as in Fig. 2C. B: running means ± SD of sequence score against sequence onset time. Short sequences are robustly reliable over time; longer sequences lose reliability in conjunction with accumulation of variability (Fig. 1D).

Although every sequence in this analysis is statistically significant, there is a distribution of sequence scores among significant sequences. We next asked how the reliability of a sequence was impacted by the accumulation of variance by analyzing the relationship between sequence score and mean onset time. We found a strong relationship between sequence score and onset time for long sequences, but short sequences showed no such relationship. For short sequences, reliability could be generated by covariability within the small group of neurons and could occur throughout the full duration of the event despite the global accumulation of variability. Conversely, reliability in long sequences was highly dependent on its onset relative to the population (Fig. 4B). Intrinsic covariability was not sufficient to support a high degree of reliability for such long sequences.

Sequence composition of global dynamics.

We next sought to decompose instances of population activity into spikes captured by sequences and spikes unexplained by sequences and consequently categorized each spike as a sequence spike or a nonsequence spike. For each neuron, we compared its spike times to all expected spike times given its sequence membership. If the neuron spiked at the correct time for any sequence, it was categorized as a sequence spike; if the spike belonged to none of its sequences, it was categorized as a nonsequence spike.

On average, 55.1% of all spikes belonged to an identified and statistically significant sequence. The progression of sequence spikes, nonsequence spikes, and total spikes over time is shown in Fig. 5A. On average, the number of sequence spikes is highly correlated with the total number of spikes. Nonsequence spikes were fewer in number and tended to occur later. This is presumably because of increased spike variance later in events and a bias for sequences to start early in an event (see Fig. 4A). We also found that the more coincident spikes in any frame, the higher fraction of them were sequence spikes (r = 0.37, P = 3.49 × 10⁻⁴², Student’s t-test; Fig. 5B). In other words, increased excitability within the population contained more sequences, whereas the sparse activity was composed of very little sequential structure. The moderate correlation coefficient and the concave appearance in Fig. 5B suggest that this is predominately a nonlinear relationship. We next restricted our analyses further by only considering sequences that fully manifested during an event (i.e., every cell in the sequence produced a sequence spike). Complete sequences were deemed “active,” whereas all other sequences were “inactive.” Generally, the presence of active sequences, according to this definition, was uncommon (length 3, 27.0 ± 26.3%; length 4, 14.5 ± 17.8%; length 5, 7.2 ± 10.8%; length 6, 3.5 ± 6.6%; length 7, 1.4 ± 4.1%; length 8, 0.3 ± 1.0%). Unsurprisingly, active shorter sequences on average occurred more often than active longer sequences. Although only a small portion of the population activity was explained by active sequences, its progression was closely tied to the full population (Fig. 5C).

Fig. 5.

Sequences comprise a large fraction of population activity. A: total population spiking and spiking categorized as either sequence or nonsequence spiking (means ± SD). B: sequences track population activity; more spikes belong to sequences when more neurons are spiking (means ± SE). Uniform probability of sequence spikes was generated by shuffling spikes in each event. C: active sequences, within any 1 event, are less common yet show similar evolution in time as the average of events (means ± SD).

Single trial manifestations of active sequences.

We next studied how active sequences were coordinated during events of population activity. We found that distinct but overlapping sequences defined population spiking in single trials (see examples in Fig. 6A). As expected by the distribution of sequence membership, the sequences were highly interwoven with many shared neurons among active sequences. To quantify the degree of shared sequences between events, we computed the Jaccardian index between active sequence spikes in all pairs of events. We found that the median sequence overlap between pairs of events was low (33.7%), and many pairs of events shared nearly no sequences spikes (Fig. 6B). This suggests that although all trials share the same MGP and repertoire of possible sequences, they are differentiated by their unique ensembles of active sequences. We then computed the absolute deviation from a neuron’s mean spike time (i.e., its position in the MGP) for all of its sequence spikes and nonsequence spikes separately. We found that sequence spikes were more similar to the MGP than nonsequence spikes (Fig. 6C), consistent with our result that sequence spikes tracked population activity (see Fig. 5A).

We then tested whether the convergence of all sequences onto a small subset of high membership neurons was also present within single instantiations of active sequences. First, analyzing all sequences aligned to their first neuron (as in Fig. 3A), we computed the number of unique neurons at each sequence position normalized by the expected number of unique neurons. This expectation was calculated assuming a uniform probability of observing each neuron spiking in sequence. The ratio of these two quantities told us how often neurons repeat in sequences as compared with expectations set by chance. Consistent with Fig. 3A, we found relatively diverse neurons at the beginning of sequences but high redundancy later in sequences. Next, we exclusively analyzed active sequences within individual events, computing the number of unique cells participating in each frame normalized by the expected number of unique neurons assuming a uniform distribution. This resulted in a comparable measure of how often neurons repeat in active sequences. We found a constant ratio of unique neurons throughout population activity (Fig. 6D). Because of the specific subsets of active sequences and the distribution of onset times, the overall convergence seen among all sequences was suppressed. Among the overlapping subsets of active sequences, the reliable propagation of spiking was balanced between convergence onto shared neurons and divergence into distinct neurons. Thus sequences were highly interwoven and co-occurred throughout the duration of population activity. Although on average sequences pooled into a small subset of individual neurons, on single trials this was not the case. Interwoven sequences during single events were drawn from a large repertoire of possible sequences and demonstrated that variance about the global mean is not noise but the manifestation of reliable spike times in the population.

DISCUSSION

Consistent trajectories of population activity have been observed in many cortical areas, are qualitatively similar in vitro and in vivo (Sadovsky and MacLean 2013), and have been shown to correlate with behavioral choice (Harvey et al. 2012), motor output (Churchland et al. 2012), and sensory inputs (Luczak et al. 2009). Our analyses provide a more detailed, quantitative description of population spike propagation. Across the population, sorting neurons by their mean spike latency reveals the MGP. However, the population PETH is sufficient to generate an MGP in these data, indicating that a more thorough description of activity is necessary. The most prevalent theoretical framework underlying spatiotemporal transmission of spikes is the Hebbian assembly phase sequence (APS). Although an APS may be implemented in different forms, a common interpretation of Hebb’s (1949) seminal work is that distinct, nonoverlapping assemblies form a propagating sequence of activity, with one assembly recruiting the next, resulting in a temporal progression that reflects a common function (Fig. 7A). An assembly composed of well-defined groups of functionally related neurons is consistent across many disciplines (Carrillo-Reid et al. 2015; Levy et al. 2001; Litwin-Kumar and Doiron 2014; O’Neill et al. 2008; Pastalkova et al. 2008; Wehr and Laurent 1996). Until now, however, there has been little beyond this qualitative description of how the phase sequences manifest within local cortical populations comprised of interconnected neurons. Our data and analytical approach motivate an alternate formulation of the Hebbian APS. Naively, neurons with similar onset latencies might be clustered into distinct assemblies. However, due to shared variability within subpopulations of neurons, our results suggest that the fundamental unit of the APS is the temporal spike sequence rather than the active group of neurons at one specific time. As a consequence of sequences highly overlapping in time and in neuron membership, assemblies are then defined by co-occurring sequences. Different ensembles of sequences manifest in single trials of population activity, resulting in variable sequence coactivity and assemblies. Identifying the repertoire of sequences allowed for description of unique sequence ensembles and helped to explain the large variation about the MGP. Each event is uniquely described by its ensemble of sequences, suggesting that the APS is a propagation of spikes through an interwoven subset of sequences among a common repertoire. As a consequence of sequences with variable onset time and probabilistic sequence activation, different assemblies manifest in each trial, but the MGP and the repertoire of sequences is consistent. We conclude that the MGP is the result of averaging across these interwoven microassemblies rather than a single concatenated chain of assemblies (Fig. 7B).

Fig. 7.

Sequence composition of MGP suggests an interwoven assembly phase sequence framework. A: distinct, nonoverlapping assemblies (neuron color) underlie propagation of activity. Traditionally, the hypothesis is interpreted as dynamics being driven by a concatenated feedforward assembly phase sequence. Assemblies are defined by the group of coactive neurons in a time window, and noise intrinsic to each assembly leads to trial-to-trial variability. B: in our suggested revision, based on our data analysis, neurons belong to many assemblies, and probabilistic activation of sequences underlies propagation of activity. Neuron color denotes single-trial assembly membership. Arrows denote underlying sequences; gray neurons and arrows are inactive in this trial. Assemblies are defined by the ensemble of active sequences.

Previous attempts to quantify temporally structured spiking in local populations of neurons have had mixed results (Abeles and Gerstein 1988; Baker and Lemon 2000; Humphries 2011; Ikegaya et al. 2004; Mokeichev et al. 2007; Roxin et al. 2008; Torre et al. 2016). The combinatorial nature of spatiotemporal spike sequences poses substantial difficulty in identifying spike patterns that cannot be explained by simpler hypotheses. One can attempt to calculate the contributions from every individual neuron and subsequence and then estimate the divergence from this expectation in the data (Martignon et al. 2000; Schneidman et al. 2003;). These approaches require large amounts of data that increase nonlinearly as pattern length increases. A more common approach, which we have taken here, is to compare patterns in the data with surrogate data. Features built into the surrogate data delineate the null hypothesis, and patterns unique to the data are taken as evidence against the null hypothesis (Grün 2009; Harrison et al. 2015; Lee and Wilson 2004;). Our analysis here diverges from many other works in three main ways. First, we do not consider synfire-like patterns with strict millisecond precision (Baker and Lemon 2000; Ikegaya et al. 2004; Lee and Wilson 2004), as our sampling rate and identification method allow for spike jitter. Although many sequence-identification methods allow for spike jitter (Gansel and Singer 2012; Torre et al. 2016), they also require pattern completion. The second difference is that the sequences here may have spikes missing in individual trials. Finally, other identification methods have taken an agnostic approach to the specific spike-patterns that could manifest in neural networks (Abeles and Gerstein 1988; Gansel and Singer 2012; Tatsuno et al. 2006); this work searched exclusively for single-frame lagged spike sequences that have a clear correspondence with APSs and help to better understand how subpopulations contributed to the population-wide MGP. Although this assumption greatly restricts the number of possible patterns, it also limits the scope of this work to a specific form of a Hebbian APS. Furthermore, we did not allow neurons to repeat in sequences. From the perspective of an MGP, a neuron should have a specific time of activation, but highly recurrent networks may produce sequences that reverberate back to an earlier neuron (Hebb 1949). Because of the data acquisition rate and the assumptions built into the method, the sequences identified here are not a complete sampling of spike sequences in the data. Rather, they represent the reliable spike sequences underlying the population MGP. The identification method presented here could be extended to allow for variable lags and repeated neurons in sequences, but other methods are better suited for assumption-free analyses of spatiotemporal spike patterns.

Calcium imaging allows recording of neuronal populations, with single-cell resolution spanning large fields of view (∼1,000 μm). Different data acquisition techniques, such as high-density multielectrode arrays (Berdondini et al. 2009), may be capable of capturing aspects of Hebbian assembly dynamics not observed here and will likely be necessary to reveal generalizable features of spike sequences beyond those measured in these data. Future explorations of sequences propagating at different intervals will help to further delineate the capacity of cortical networks to produce reliable spatiotemporal patterns of activity.

Trajectories of activity in neural networks have attracted attention from theorists and experimentalists alike, as they are capable of encoding information and performing computations (Crowe et al. 2010; Harvey et al. 2012; Klampfl and Maass 2013). Other computational frameworks have shown that time-varying trajectories of activity in neural populations are a powerful computational substrate (Buonomano and Maass 2009; Durstewitz and Deco 2008; Rabinovich et al. 2008;). Experimentally, single-trial deviations of the MGP have been hypothesized to encode stimulus identity (Luczak et al. 2015). Our results show that the population activity deviations are captured by unique subsets of active sequences, demonstrating that local circuitry is able to generate sequences that in turn have the capacity to be stimulus specific. The large repertoire of sequences and their interwoven propagation could even allow for simultaneous encoding of external variables, internal cortical state, and their interactions similar to a coupled hidden Markov model. Synaptic plasticity rules are capable of embedding interwoven spike sequences in a neural network (Izhikevich 2006), and these sequences show a natural capacity to encode stimulus information (Paugam-Moisy et al. 2008). Our analyses summarize complex population dynamics while maintaining individual spikes in specific neurons, leading to a major advantage in interpretation over other methods of representing population activity. Since each sequence is composed of specific neurons, there is a direct biological interpretation of the information represented in sequence activity. In contrast, dimensionality reduction techniques are powerful at harnessing information within population activity, but they project activity onto latent dimensions, making it difficult to attribute the activity of specific neurons to the low dimensional projection (Briggman et al. 2005; Broome et al. 2006; Mante et al. 2013; Yu et al. 2009). The ability to understand how a specific computation or stimulus representation is implemented by local circuitry necessitates an understanding of each neuron’s contribution. In addition, the temporal coding framework of reliably propagating spikes has a clear relationship with the underlying biological substrate. The modulation of activity via synaptic transmission occurs with a delay; therefore, spike propagation has a more direct relationship with network activity than simultaneous modulation of spiking. Neocortex produces both highly variable (Heggelund and Albus 1978; Vogels et al. 1989) and highly precise (Bair and Koch 1996; Reich et al. 1997; Wehr and Zador 2003) activity. By maintaining spike time information and capturing moments of reliable activity, the capacity for information representation is increased (Kayser et al. 2009).

Motivated by observations that neural population activity exhibits a mean global progression, we present a detailed description of the spike sequences underlying this population-wide phenomenon. We decomposed the individual observations of population activity into smaller sequences of spikes that recur across events. Most neurons participated in only a few sequences and tended to initiate sequences. This diversity of initiating neurons converged onto a small subset of neurons that participated in many sequences. Across the population, spiking in sequences was tightly linked to the population firing rate, and each event comprised a specific subset of active sequences. The specific subset of active sequences was highly interwoven, sharing neurons and spike timing. Our analyses capture much of the trial-to-trial variability in the spontaneous activity and suggest that sequences probabilistically converge and diverge. The full set of identified sequences represents the dynamic repertoire of the population, but population activity manifests in the network through unique subsets of these sequences. The probabilistic trajectories of sequences emphasize that neurons belong to multiple interwoven assemblies rather than a feedforward cascade of spike propagation. This method of representing population activity is tied closely to the structure and function of neocortical networks and reveals temporally structured activity with the capacity to aid in information representation.

GRANTS

This work was funded in part by NSF CAREER Award 095286.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

J.B.D. and J.N.M. conceived and designed research; J.B.D. analyzed data; J.B.D. and J.N.M. interpreted results of experiments; J.B.D. prepared figures; J.B.D. drafted manuscript; J.B.D. and J.N.M. edited and revised manuscript; J.B.D. and J.N.M. approved final version of manuscript.