Abstract
The hypothesis that cortical networks employ the coordinated activity of groups of neurons, termed assemblies, to process information is debated. Results from multiple singleunit recordings are not conclusive because of the dramatic undersampling of the system. However, the local field potential (LFP) is a mesoscopic signal reflecting synchronized network activity. This raises the question whether the LFP can be employed to overcome the problem of undersampling. In a recent study in the motor cortex of the awake behaving monkey based on the locking of coincidences to the LFP we determined a lower bound for the fraction of spike coincidences originating from assembly activation. This quantity together with the locking of single spikes leads to a lower bound for the fraction of spikes originating from any assembly activity. Here we derive a statistical method to estimate the fraction of spike synchrony caused by assemblies—not its lower bound—from the spike data alone. A joint spike and LFP surrogate data model demonstrates consistency of results and the sensitivity of the method. Combining spike and LFP signals, we obtain an estimate of the fraction of spikes resulting from assemblies in the experimental data.
Keywords
LFP Synchrony Oscillations Network dynamics Motor cortex1 Introduction
A common hypothesis concerning the processing of information by cortical networks involves the propagation of activity through synchronously firing groups of neurons, termed assemblies (Hebb 1949). A hallmark signature of an activated assembly is the repeated coincident firing of neurons in relation to a specific stimulus or behavioral demand resulting in coincidence counts that exceed the chance level estimated from the firing rates (Gerstein et al. 1989). Despite the inherent undersampling of state of the art multiple singleunit recordings, experimental studies indirectly substantiate the assembly idea with findings of such behaviorrelated significant synchronous spiking (e.g., Riehle et al. 1997; Kilavik et al. 2009). Independently thereof, a signal on the population level, like the mesoscopic local field potential (LFP), typically exhibits temporally structured oscillations commonly interpreted as correlated network activity. Synaptic transmembrane currents have been identified as the primary contributor to LFP generation (Mitzdorf 1985; Logothetis and Wandell 2004). Counterintuitively, although synchronized membrane potential oscillations of neurons in the vicinity (Katzner et al. 2009) of the recording electrode show strong correlations with the LFP (Poulet and Petersen 2008; Okun et al. 2010), these synchronized potentials do not induce the same degree of coincident spiking between the same neurons. Correlated spiking appears largely independent of synchrony on the level of membrane potentials (Tetzlaff et al. 2008; Poulet and Petersen 2008).
In a recent study (Denker et al. 2010) we were able to demonstrate in data of the motor cortex of the awake behaving monkey the missing link between significant spike synchrony and the LFP. A conceptual model enabled us to derive lower bounds for the fraction of spike coincidences β originating from observed assembly activity identified by periods containing excess spike synchrony, and the fraction of spikes of neurons γ caused by assembly activity whether observed or not. The results were obtained by comparing the locking of spike coincidences to the LFP in time periods with significant synchrony to the locking outside of these periods.
In the remainder of this section we first introduce the Unitary Events analysis method (Section 1.1) used to quantify significant excess spike synchrony. We then review our earlier findings (Denker et al. 2010) on the phase locking between spikespike coincidences and the LFP (Section 1.2) and the modelbased interpretation of the data (Section 1.3). In the present work we remove the limitation to lower bounds of β and γ by evaluating a stochastic model of the composition of spike coincidence counts. Section 2 introduces the model and derives an estimator for β only based on the configuration of spikes in the respective time interval. In Section 3 we explain that the probability of spikes γ to be part of an assembly—in contrast to β—cannot be extracted from the spike trains alone. However, equipped with the estimate of β we are in the position to compute γ using the previously found relationships between the phase distributions of the spikeLFP coupling (Section 1.3). The possibility to extract a parameter that relates to macroscopic features of the network dynamics illustrates the relevance of the LFP as a population signal in addressing the undersampling problem.
We demonstrate the consistency of the concept with the help of a joint spikeLFP toy model. In the physiological range, the parameters can reliably be determined and the lower bounds obtained from the phase locking are compatible with the parameter estimates yielded if the spike statistics is considered in addition. Finally, we utilize our new tool to reanalyze the experimental data (Section 4) for β and γ. Although highly simplified the toy model enables us to discuss (Section 5) various aspects of the distribution of parameters obtained from the experimental data set.
1.1 Analysis of spike synchrony
1.2 Analysis of phaselocking
In the next step we investigated the relation of spikes to the LFP. In particular, we were interested if spikes in synchronous events, i.e., chance coincidences (CC) or UE, have a different relation to the LFP than spikes that are not involved in a coincidence, i.e. isolated spikes (ISO). Therefore we classified each spike of a data set into one of these three classes and analyzed the classes separately (see Fig. 1(a) for an illustration; color codes for ISO, CC, and UE are retained throughout this article). Spike triggered averages (STA) of the LFP revealed for all classes an oscillatory structure at about 17 Hz with spikes occurring preferentially at the decaying amplitude. However, the amplitude of the STAs derived for the different spike classes were strikingly different: consistent across the whole data set the STA triggered on UE spikes exhibited the largest amplitude, for CC spikes a smaller amplitude and for ISO the smallest.
However, the STA analysis cannot uncover the reason for the differences in STA amplitude, since it may be due to differences in the LFP amplitudes or due to different degrees of phase locking between spikes and the LFP. In order to disentangle these aspects we performed a phaseamplitude analysis of the LFP. Using a Hilbert transform of the LFP we gained the instantaneous phase and amplitude as time dependent functions, and extracted the respective measures at spike times (the detailed procedures will be explained in Section 4.1). The phase distribution exhibits the phase preferences by nonuniformity of the phase histograms (see Fig. 1(b)), and we observe that UE spikes express the strongest degree of phase locking, CC less (at the predicted chance level) and even less for ISO. A resampling procedure ensures that histograms are compared with comparative numbers of spikes: The phase histogram of CC (UE) is compared to resampled versions of the ISO (CC) phase histogram, each containing the same number of data points as the CC (UE) histogram (bands in Fig. 1(b), middle and right graph). The differences in phase locking were found to be consistent across the whole population of recorded neurons, and 68% of the neurons (n = 123 neuron pairs; see Section 4.1 for details) exhibit a stronger phase locking of UE than CC. In contrast we found much smaller or even negligible differences of the LFP amplitude measured as its envelope at spike times for the different spike classes.
1.3 Interpretation and conceptual model
Our analysis of the relation of spikes to the LFP revealed several unexpected results. Firstly, we found a difference in the locking degree for spikes involved in different categories of coincidences, i.e. UE vs CC. Following the hypothesis that active assemblies are expressed by coordinated spiking activity, UE coincidences are interpreted as a signature of such active assemblies. In the following, we will define the assembly as the set neurons exhibiting coordinated activity that is reflected in the excess synchronous events during UE periods. Our new finding on the phase locking of UE spikes to the LFP leads to the interpretation that assembly activity occurs in a pulsed fashion, locked to the LFP oscillation. The fact that spikes of chance coincidences also occur phase locked seems contradictory. However, their degree of locking is fully explained by the (weak) locking of single spikes, which trivially leads to an enhanced locking of such spikes if occurring coincidently (cf., predictor in Fig. 1(b)). Thus the question is rather, why isolated spikes outside UE periods exhibit phase locking at all.
As a result, the enhanced locking of UE as compared to CC is a consequence of the presence of assembly coincidences i.e. β > 0.
2 Estimation of assembly activations from spike statistics
In order to capture coincidences with a temporal jitter larger than the resolution of the data h, the multipleshift method (Grün et al. 1999) sums the coincidences over a range of s = 2b + 1 displacements − b,..., + b of one spike train with respect to the other. At each shift only some of the n _{c} coincidences are detected and a particular coincidence is by definition only detected at a single displacement. Under the assumption that in the s displacements the spike counts n _{1}and n _{2} are conserved, the relation (Eq. (8)) between n _{emp} and n _{c} holds if we replace the number of spikes available for chance coincidences by sn _{ i } − n _{c} and the number of available bins by sT _{ h } − n _{c}. The estimate of the number of coincidences originating from assembly activation (Eq. (9)) then reads \(n_{\mathrm{c}}=\left(s{T_{h}}n_{\mathrm{emp}}s^{2}n_{1}n_{2}\right)/\left(sT_{h}+n_{\mathrm{emp}}s\left(n_{1}+n_{2}\right)\right)\).
2.1 Correspondence of models of assembly activations
2.2 Accuracy of the count of assembly activations
Before we turn to the experimental data in Section 4 we need to assess the accuracy of our estimator of n _{c}. To this end we construct a surrogate data set with parameters adapted to the experimental data. The data are organized into 32 blocks containing an increasing number of assembly activations from n _{c} = 0 to 31. A block is composed of M = 2,700 time windows, consistent with the number of UE windows found in the experimental data. A time window has a duration (Eq. (6)) of T _{ h } = 5,000 at a resolution of h = 1 ms, corresponding to the 100 ms segments of UE analysis covering 50 trials. Each time window contains surrogate spike trains of two neurons with totals of exactly n _{1} = n _{2} = 100 spikes (corresponding to a rate of 20 Hz). In each spike train the n _{ i } − n _{c} nonassembly spikes are uniformly distributed over the time window.
3 Estimating the assembly participation probability in a joint spikeLFP model
In the following section, we extend the simple spike model defined in Section 2 to include a representation of the LFP locking and verify numerically that the results obtained in the previous section yield a reliable estimate of our model parameters β and γ.
3.1 Combined spikeLFP model
First, we choose a parameter γ ^{set} as the assumed probability that any spike in the network is part of an assembly activation, in agreement with our conceptual model. In the following we simulate for each choice of n _{c} the set of M = 2,700 UE time windows as combinations of an injection process and a background process for two neurons as described in Section 2.2. In addition, CC time windows are generated only by a background process with equal spike counts n _{1} = n _{2} = 100 (see Fig. 2(b) for an illustration). Again, only significant UE windows (n _{emp} > n _{ α }) and nonsignificant CC windows (n _{emp} ≤ n _{ α }) are retained. To model the experimental results, we assign a label ’a’ to all spikes that originate from an assembly activation (Fig. 2(b)). By definition of our assembly process, every spike that originates from the injection process receives a label. In addition, a random proportion γ ^{set} of spikes from the background process is labeled (in both, UE and CC time windows). In our simulations, the overall probability for a spike to belong to an assembly is set to γ ^{set} = 0.1. Next, we define two distributions p _{n}(ϕ) and p _{a}(ϕ), where the latter has a larger modulation depth, which describe the locking of nonassembly and assembly spikes, respectively (Fig. 2(a)). Here, p _{n}(ϕ) is modeled as a uniform distribution, whereas for p _{a}(ϕ) the distribution is modeled as a Gaussian as an approximation of a von Mises distribution. The modulation of the Gaussian was chosen to mimic that of the experimentally observed distributions p _{CC}(ϕ), and p _{UE}(ϕ) (compare Figs. 1(b) and 2(a)). Classification of spikes into the groups ISO (taken here as the single spikes in CC windows), CC and UE allows us to calculate the simulated phase distributions p _{ISO}(ϕ), p _{CC}(ϕ), and p _{UE}(ϕ) as mixtures of p _{n}(ϕ) and p _{a}(ϕ).
3.2 Estimating the minimal β from phase distributions
3.3 Estimation of γ from phase distributions
We are now prepared to extract the parameter γ from the simulated phase distributions using the estimate of β ^{UE} derived in Section 2. First, using β = β ^{UE} we extract \(p_{\mathrm{a}}^{2}(\phi)\) from Eq. (3) by solving the equation separately for each bin of \(p_{\mathrm{a}}^{2}(\phi)\). Thus, p _{a}(ϕ) is determined independently of the distribution p _{n}(ϕ) describing the general entrainment of spikes to the LFP. Taking the square root p _{a}(ϕ), we renormalize the distribution to unit area, and insert it in Eq.(1). By variation of the parameter γ, we find the value γ ^{ϕ,UE} that minimizes the sum of the absolute binbybin differences between the distribution p _{ISO}(ϕ) and the right side (1 − γ)·p _{ n }(ϕ) + γ·p _{ a }(ϕ) of Eq. (1).
Using this method, we first analyze separately each of the 32 blocks of data with a fixed number of injected coincidences n _{c}. Let \(\tilde{\beta}\) and \(\tilde{\gamma}\) be the estimated values of β ^{UE} and γ ^{ϕ,UE} within a single block. In Fig. 3(e) \(\tilde{\beta}\) and \(\tilde{\gamma}\) are compared to their theoretical means \(\bar{\beta}\) (calculated as the mean of n _{c}/n _{emp} over all significant time windows from one set of simulations with fixed n _{ c }) and γ ^{set}. The estimates are in good agreement for datasets where n _{c} was set above the significance threshold n _{ α }. Below this threshold, the number of injected coincidences, and hence \(\tilde{\beta}\) and \(\tilde{\gamma}\), is overestimated. Clearly, in this regime contributing UE periods become significant only due to an unusually high number of coincidences in the background process. Therefore, the mean approximation (Eq. (9)) does no longer hold. In principle it is possible to correct for the bias due to the selection of significant periods, although it is not possible to arrive at an expression in closed form for n _{c}. Nevertheless, in fact only a small fraction of windows that were created with low values of n _{c} actually become significant due to such exceptionally high coincidences counts from the background (the shaded graph in Fig. 3(e) shows the probability for a window to become significant for given n _{c}). Therefore, in practice, where the true n _{c} is unknown, only few windows enter the analysis when n _{c} is low. For this reason, the bias does not affect the estimate when resorting the data according to the observed n _{emp} (Fig. 3(a)–(c)), and the original value of γ ^{set} is well estimated as \(\mbox{\ensuremath{\gamma^{\phi\mathrm{,UE}}}}=0.098\) from the complete dataset spanning all n _{c} (black cross in Fig. 3(e)). Consistently, the lower bounds \(\beta_{\mathrm{min}}^{\phi}\) and \(\gamma_{\mathrm{min}}^{\phi}\) are a lower bound on the respective estimates \(\tilde{\beta}\) and \(\tilde{\gamma}\) for all values of n _{c}.
To quantify whether our procedure works for other values for the total spike counts n _{1} and n _{2} we introduce two error measures. First, we note that our prediction of β will be least accurate when the coincidence count n _{emp} is low. Thus, we numerically quantify the absolute error of β for the smallest value of n _{emp} to become significant (Fig. 3(f)). Moreover, we quantify the absolute error of our final estimate γ obtained from the complete spikeLFP model (Fig. 3(g)). Both measures indicate that the parameters are reliably extracted for a wide range of activity levels in both neurons, even in the case of large rate differences between the two. In summary, these calibrations using a simple combined spikeLFP model demonstrate that our method and the approximation (Eq. (9)) are well suited to estimate the parameter γ from the phase distributions in a dataset with a realistic choice of parameters.
4 Analysis of experimental data
In this section we estimate the percentage of coincident spike events reflecting assembly activity (model parameter β) and the percentage of spikes that are part of an assembly activation (model parameter γ) from neuronal data of primary motor cortex of monkeys.
4.1 Experimental procedures
Two rhesus monkeys were trained to perform arm movements in two different tasks involving an instructed delay. One of these tasks requires a selfinitiated movement after one of two preset delays, while the other requires the discrimination between two such delays. In this study we exclusively analyze the delay activity during the preparatory period for the upcoming arm movement. Only correct trials were considered, in which the monkey responded within a predefined time window and in which movements were performed in the required movement direction. LFPs and spikes were recorded simultaneously in primary motor cortex using a multielectrode device of 2–4 electrodes. Spikes of single neurons were detected by an online sorting algorithm. The interelectrode distance was on the order of 400 μm. LFPs were sampled at a resolution of 250–500 Hz and hardware filtered (band pass, 1–100 Hz). In total, we analyzed 53 recording sessions, which yielded 143 single neurons or 570 combinations of neurons and behavioral conditions. Each of these eight possible behavioral conditions is a combination of the task during which the neuron was recorded, the length of the delay (short or long), and the direction of the arm movement (left or right). This selection included only those neurons which exhibited an average firing rate of 5 Hz or more and exhibited at least 25 spikes in total. On average 33±11 (mean±standard deviation) trials were recorded per experimental condition. In analyses that combine spikes and LFP, each neuron enters only once, and we never combined LFP and spikes that were recorded on the same electrode to exclude the possibility of spike artifacts in the signal. We confirmed that simultaneously recorded LFPs are highly synchronous in the frequency regimes of interest. For experimental details see Roux et al. (2006).
The Unitary Events analysis was applied to all simultaneously recorded pairs of neurons recorded on different electrodes thus totaling 123 pairs of neurons. Data were analyzed separately for the possible behavioral conditions and each experimental session. Defining a neuron by the combination of its identity and the behavioral condition (see above) during which it was recorded, data from the same neuron may enter a population average up to eight times. Based on the results of the UE analyses each individual spike was marked as either ISO, CC or UE (see Section 1.1 and Denker et al. (2010) for details). We observed a dominant component of the βband (in both monkeys around 17 Hz) in the LFP during the preparatory period (see, e.g., Murthy and Fetz 1996a) and therefore filtered the LFP accordingly before applying the Hilbert transform to extract its instantaneous phase and amplitude (i.e. the envelope). LFPs of both monkeys are filtered with a zerophase 10–22 Hz band pass filter (Butterworth, 8pole). Finally, we derived the phase histogram pooled over the whole population for each class of spikes (see Fig. 1(b)). The histogram was pooled after averaging, such that each spike enters the distribution unweighted (as is required for our analysis).
4.2 Estimating the minimal β from phase distributions
4.3 Estimate of β ^{UE} from coincidence counts
Based on Eq. (9) we estimate the number of coincidences resulting from the active assembly n _{c} for each n _{exp}, n _{emp} combination. We use a version of the UE analysis that also allows to detect temporally imprecise coincidences by employing the multipleshift method. This method detects coincidences of systematically shifted spike trains up to a predefined shift, and sums the coincidence counts from all shifts. For the UE evaluation, the expected number of coincidences then has to be adjusted by a factor of 2s + 1 with s being the maximal shift (cf. Section 1.1; for details see Grün et al. 1999). Thus to estimate the number of coincidences n _{c} that generically result from active assemblies, Eq. (9) was adjusted correspondingly (Section 2). Figure 5(b) shows n _{c} versus its corresponding n _{emp} and we find an average number of assembly coincidences of \(n_{c}^{\mathrm{UE}}=6.70\).
Next, we can derive for each pair of (n _{c}, n _{emp}) an estimate of the percentage of coincidences reflecting assembly activity β (Fig. 5(c)). We find these values peaked around a mean of β ^{UE} = 0.51 with a standard deviation of 0.123. The obtained distribution of β is in good agreement with the minimum \(\beta_{\mathrm{min}}^{\phi}\) obtained in Section 4.2, as \(\beta_{\mathrm{min}}^{\phi}\) lies well below the mean β ^{UE} of the distribution of β as expected for an estimate of a lower bound. Although \(\beta_{\mathrm{min}}^{\phi}\) in fact seems to be a bound for the complete distribution of β, this must not necessarily be the case: As a measure derived from the population estimate of phase distributions, it can only have predictive power on the population mean β ^{UE}, but not the estimate β of single windows (compare Fig. 3(d)).
4.4 Estimation of γ from phase distributions
5 Discussion
Despite the complex mechanisms that contribute to the formation of the local field potential, it is well established that a primary contribution to the oscillatory LFP dynamics arises from the superposition of synchronized, slow transmembrane currents of cells close to the recording site (Mitzdorf 1985; Logothetis and Wandell 2004). Nevertheless, how rhythmicity in the LFP is linked to synchrony on the spiking level has remained an open question (Poulet and Petersen 2008; Tetzlaff et al. 2008) due to the unspecificity of the LFP signal and lack of a clear global oscillatory spiking activity. Several authors have interpreted the LFP as reflections of the specific synchronous synaptic activity responsible for the coactivation of neurons in the context of cell assemblies (Eckhorn et al. 1988; Murthy and Fetz 1996b) which, in the simplest case, time their activations to the network rhythm revealed by the LFP (Singer 1999). Our recent experimental findings (Denker et al. 2010) demonstrate that indeed only assembly activity, which is identified as transient periods of significant excess spike synchrony between two neurons (UE analysis, Grün et al. 2002a, b), shows an exceptional phase relationship to LFP oscillation that exceeds expectation (Fig. 1(b)), thus confirming the hypothesis.
This link provides a handle on characterizing the spike synchronization dynamics in the context of the network oscillations. Here, we introduced a simple model that captures the main experimental findings on the spikeLFP relationship in such a way that it includes a parameter γ that measures the overall participation probability of individual spikes in an assembly—independent of whether it is observed or not. However, the estimation of this network parameter from the data requires knowledge of a second model parameter β, the expected relative number of (excess) coincidences stemming from an active assembly (as opposed to chance coincidences) during UE periods. Estimating this number from the spike data alone and integrating it into the conceptual model yields an average participation of single spikes to assembly activity of γ = 22%, a parameter that describes precise spike synchronization on the network level.
Our model assumes that the observed excess synchrony is the result of the specific activation of the observed neurons. An alternate hypothesis states that spike synchronization is solely caused by fluctuations of the spiking likelihoods due to the entrainment of neurons to a common LFP oscillation. In this case, no UEs would be observed due to the lack of a spike coincidence count exceeding the expectation based on the rate time course. Only if the analysis would unsuccessfully correct for nonstationarities of the firing rates falsepositive detections of UE periods emerged. Consequently, these falsepositives would trivially correlate with the LFP. To exclude this possibility, we repeated the analysis presented in this study by replacing the parametric distribution of coincidences used for evaluating the significance of observed coincidence counts by numerically derived distributions based on surrogate data. The employed surrogate method (spike train dithering, see Grün 2009) considers further statistical features of the experimental spike trains (in particular nonstationarity of rates on a short time scale and the interspike interval distributions) while destroying precise spike coincidences. Simulations showed that this method is more conservative (Louis et al. 2010) than the analysis used here, but despite the decreased sensitivity confirms the differences in the phase distributions for ISO, CC, and UE. These findings are the essential experimental foundation of our study.
The combination of measurements of synchrony on the local and mesoscopic scales enables us to access parameters of the network dynamics that remain hidden on the two individual levels of observation. The nature of the estimation process via the population phase distributions obtained from the LFPspike locking requires a vast pooling of the experimental data in different sessions and two different monkeys in order to obtain a large sample of neurons as an appropriate representation of the network activity in motor cortex. The resulting value for γ must therefore be seen as a coarse estimate of the degree of assembly activity. Despite the finding that the parameter β shows a rather narrow distribution (Fig. 5(c)), the extent to which an individual neuron contributes to the assembly dynamics will fluctuate around the network mean γ. By quantifying the quantiles for 2 standard deviations of the experimental distribution of β we find corresponding γ values between γ = 0.14 and γ = 0.39 as an estimate of the range of γ across neurons.
In understanding the underlying dynamical structure, more interesting than the precise value of γ itself are two simple observations: first, γ < 1. Therefore, not all spikes are part of an assembly activation, and some spikes must be attributed to a complementary mechanism. Intuitively, this is clear from the observation that in individual UE periods, it is always the falling phase of the LFP oscillation where increased locking is observed. Second, γ > 0, specifically about one fourth of all spikes must originate from assembly activity (with a nonzero lower bound \(\gamma_{\mathrm{min}}^{\phi}\)). Thus, under the assumptions of our model, the activation of assemblies is an ubiquitous phenomenon in the network, providing compelling evidence for the presence of an assembly coding scheme in addition to the correlation of synchrony with behavior (Kilavik et al. 2009). Indeed, we typically observe a fraction of about 26% of neurons that show UEs during a given task. Therefore, combined with the large value of γ this suggests that typically any neuron is part of one or several assemblies. Nevertheless, information about the existence of higherorder synchrony between neurons is required for a further characterization of the assembly, such as an estimate of the number of participating neurons. Moreover, such an analysis could better disambiguate whether spikes that do not originate from assembly activation show an intrinsic degree of phase locking that deviates from our model assumption of a uniform p _{n}(ϕ).
A crucial part of our analysis is the estimation of the average relative amount of excess synchrony β during UE periods directly from the synchrony analysis. By design, all currently available methods to detect the presence of an active assembly activation, such as the UE analysis, rely on a significance test (Grün 2009). Therefore, we must assume that the amount of excess synchrony β is influenced by the explicit choice of the significance level α used for the significance test: For a very restrictive significance criterion, only UE periods with very high excess synchrony n _{c} will be detected, resulting in higher values of β. Nevertheless, by choosing the same α level for the phase locking analysis, we will in turn also obtain a correspondingly more modulated distribution for p _{UE}(ϕ), reflecting that it contains a higher proportion of assembly spikes. Therefore, β yields a consistent phase distribution p _{a}(ϕ) of the assembly spikes independent of α.
The approximate formula (Eq. (9)) does not make any assumptions on preselecting significant periods of synchrony in the first place. However, we apply this estimate specifically in periods that display a significant surplus of synchrony, i.e. UE periods. If the number of coincidences n _{c} coming from the injection process is small, significance is only reached with an unexceptionally high amount of coincidences originating from the background. Therefore, in our stochastic spikeLFP model, in those very few trials that are significant for a given small n _{c}, the average background rate will be underestimated and the average injection rate overestimated (Fig. 3(c)). However, restructuring the model data to the realistic case where only n _{emp} is measured (pooling across the data sets with different n _{c}) automatically incorporates the low frequency of significant windows with small n _{c}. Therefore the overall estimate of β is not significantly affected.
Our spikeLFP model (Section 3) is created in the spirit of providing a simple abstraction of the experimental findings in order to test the analysis under controlled conditions (equal counts per spike train with fixed injections). Due to this intentional lack of experimental detail in the model, the distribution of β that is obtained in the spikeLFP model naturally deviates from the one found in the data (compare Figs. 3(d) and 5). The model assumes an equal probability for a large range of n _{c} (number of injected coincidences). In real data, n _{c} likely follows a much more narrow distribution that does not exploit this range, resulting in lower values of β (compare Fig. 3(e)). Moreover, we assumed fixed counts n _{1}, n _{2} for all neurons in the model. In reality, rates vary considerably across neurons. Especially for higher rates, where a higher proportion of coincidences can be assumed to originate from the background, we would expect a tendency towards lower β values.
In generating the spikeLFP model in Section 3 we did not explicitly model an LFP oscillation to place spikes at specific points of the field potential. In extreme situations this might be an oversimplification, where the constraints placed on the spikes due to the LFP locking influence the probability to detect coincidences. In our model, however, nonassembly spikes are associated with a uniform phase distribution, such that p _{n}(ϕ) does not impose any constraints on the spiking probability in time. The Poissonian spike interval statistics of nonassembly spikes thus remain unaffected. In contrast, the assembly spikes from the background process must in principle be adjusted with respect to a hypothetical LFP so that their phase distribution matches the assumed the distribution p _{a}(ϕ) of assembly spikes. However, as p _{n}(ϕ) is uniform, doing so will only influence the probability of finding a coincidence between two spikes that are both assembly spikes, i.e. that participate by chance in two different assemblies at the same time. Nevertheless, the probability (γ ^{set})^{2} = 0.01 of this to happen is negligibly small. Finally, assembly spikes from the injection process are synchronous by definition, and therefore remain unaffected by the choice of p _{a}(ϕ). Due to their small number they can always be freely placed in the time window T in accordance with p _{a}(ϕ). Taken together, the simple model introduced in Fig. 2 to represent the experimental findings in the context of the spike model includes sufficient detail without the need to explicitly model the actual positions of individual spikes with respect to an artificial LFP.
List of symbols
 T

Duration of each simulated time window
 h

Temporal resolution of spike trains
 p_{n}(ϕ)

Phase distribution of spikes not participating in an assembly
 p_{a}(ϕ)

Phase distribution of spikes participating in an assembly
 p_{ISO}(ϕ)

Measured phase distribution of ISO spikes
 p_{CC}(ϕ)

Measured phase distribution of CCs
 p_{UE}(ϕ)

Measured phase distribution of UEs
 β

Fraction of coincidences part of the observed assembly during a UE period
 γ

Fraction of spikes in the network activated in assemblies
 \(\beta_{\mathrm{min}}^{\phi}\)

Minimal value of β extracted from the phase distributions alone
 \(\gamma_{\mathrm{min}}^{\phi}\)

Value of γ that corresponds to the lower bound \(\beta_{\mathrm{min}}^{\phi}\)
 β ^{UE}

Average value of β obtained from the UE analysis
 γ ^{ϕ,UE}

Final estimate of γ obtained from phase distributions using β ^{UE}
 α

Significance level of the Unitary Event analysis
 n _{ α }

Number of coincidences required to reach significance at the αlevel
 n _{ i }

Spike count of neuron i in a given window across trials
 n _{emp}

Empirical number of coincidences in a given time window
 n _{exp}

Expected number of coincidences in a given time window based on the firing rates
 n _{c}

Number of spikes part of the observed assembly (i.e., number of injected coincidences in the stochastic spikeLFP model)
 M

Number of windows simulated in the spikeLFP model per choice of n _{c}
 γ ^{set}

Predefined value of γ in the spikeLFP model
Acknowledgements
Partially funded by the Helmholtz Alliance on Systems Biology, the French National Research Agency (ANR05NEUR04501), EU grant 15879 (FACETS), and the NextGeneration Supercomputer Project of MEXT, Japan. Part of the work was carried out while SG and MDi enjoyed a scientific stay at the Norwegian University of Life Sciences, Ås in January, 2009.
Open Access
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