Estimating the contribution of assembly activity to cortical dynamics from spike and population measures

In terms of spike coincidences, β is the ratio of the number of coincidences resulting from assembly activation and the total number of coincidences in a given time window The task therefore is to construct an estimate of n _c on the basis of the known properties of the observed spike trains. For two independently spiking neurons with n ₁ spikes in one spike train and n ₂ spikes in the other, the probability to observe k coincidences in a time window T is given by the hypergeometric distribution (Grün et al. 2003) where we introduced the shorthand and h is the resolution of the discretized time axis. We define \(\mathcal{H}=0\) outside of \(\left[0,T_{h}\right]\). The expected number of coincidences is The expectation value (Eq. (7)) is just the probability to observe a coincidence in a given bin \(\left(n_{1}/T_{h}\right)\left(n_{2}/T_{h}\right)\) multiplied by the number of available time bins T _h . In the presence of n _c deterministic coincidences, however, only a reduced number of spikes of the two neurons is available to form chance coincidences The construction of the respective spike trains is illustrated in Fig. 2(b). Let us now assume that we measure the number of coincidences \(\left\langle n_{\mathrm{emp}}\right\rangle \) averaged over many repetitions of the experiment with exactly the same values for n _c, n ₁, n ₂. In this case we can equate the empirical average with the expectation value \(\left\langle n_{\mathrm{emp}}\right\rangle =n_{\mathrm{exp,c}}\). In this expression n _c is the only unknown variable. Hence, we can express n _c in terms of n ₁, n ₂, and \(\left\langle n_{\mathrm{emp}}\right\rangle \). Given only the triplet of a single realization \(\left(n_{1},n_{2},n_{\mathrm{emp}}\right)\) we can still hope that the measured n _emp is typical and write Multiplication with \(\left(T_{h}-n_{\mathrm{c}}\right)\) shows that the quadratic terms in n _c cancel and collecting terms with n _c gives Using Eq. (9) we can already compute β for all experimental UE periods with their individual triplets \(\left(n_{1},n_{2},n_{\mathrm{emp}}\right)\) and obtain the distribution of β as well as its mean β ^UE (see Section 4). Clearly Eq. (9) is an approximation, the expression is negative for small n _emp, in particular when no coincidence has been observed (n _emp = 0).

In order to capture coincidences with a temporal jitter larger than the resolution of the data h, the multiple-shift 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 ₁and 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 \(n_{\mathrm{c}}=\left(s{T_{h}}n_{\mathrm{emp}}-s^{2}n_{1}n_{2}\right)/\left(sT_{h}+n_{\mathrm{emp}}-s\left(n_{1}+n_{2}\right)\right)\).

Next we consider a slightly more realistic model with a variable number of assembly activations n _c. The purpose of the model is to help us to understand the conditions under which we can reliably extract β. In the absence of any knowledge about the process generating the additional coincidences n _c we assume that each value consistent with the observed number of coincidences is equally likely. Thus, each possible value of n _c occurs with probability 1/(n _emp + 1). For a particular n _c, however, the probability to be consistent with the n _emp observed coincidences is again given by the hypergeometric distribution (Eq. (5)) The probability that a particular n _c underlies the observation therefore is and the total probability to observe n _emp coincidences is Consequently, the expected n _c given the observed triplet \(\left(n_{1,}n_{2},n_{\mathrm{emp}}\right)\) is which reduces to Note that the denominator is not unity as the sum extends over probabilities from different hypergeometric distributions.

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.

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.

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 spike-LFP 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 non-stationarities of the firing rates false-positive detections of UE periods emerged. Consequently, these false-positives 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 non-stationarity of rates on a short time scale and the inter-spike 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 LFP-spike 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 non-zero 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 higher-order 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 spike-LFP 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 spike-LFP 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 spike-LFP 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 spike-LFP 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, non-assembly 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 non-assembly 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.

The analysis we provide in this manuscript focuses on estimating a global parameter that characterizes the network dynamics. Given the overall consistency of our conceptual assembly model, the results suggest that oscillations and assemblies are tightly coupled. However, one may still speculate whether LFPs are in fact the direct cause of the assembly process, or whether they act as a supporting mechanism that coordinates synchronized firing. Whichever the initial cause of the LFP oscillation, a promising next step is to return to the level of the single neuron and integrate our knowledge on the dependence of spike synchrony on the LFP in order to discriminate which of the spikes are likely candidates to originate from the assembly process, and which are not. Two pieces of information aid us in this task: first, the knowledge of the observed UE phase distributions in relation to the ISO and CC distributions determines the likelihood of a single spike to be part of an assembly activation. Second, the parameter γ is informative of the relative number of spikes to assign to the assembly dynamics. Indeed, a third parameter not discussed in this manuscript is the dependence of spike synchrony on the LFP amplitude (Denker et al. 2010). Evaluating these three factors on a spike-by-spike basis, there is reason to believe that analyzing the LFP in relation to the spiking activity may well provide an independent indicator to identify the probabilities with which recorded neurons become transiently synchronized as part of an assembly activation. Moreover, the pulsed activation of assemblies triggered on the LFP oscillations cycle suggested by the experimental data stimulates the idea that maybe even assemblies, where only one neuron is recorded, may be inferred from the phase locking statistics in a probabilistic manner. Thus, incorporating a mesoscopic signals may serve to overcome problems related to the undersampling in multiple single-neuron recordings. The success of the method presented here not only corroborates the assembly hypothesis of neuronal processing, but offers a promising vista on reinterpreting the dynamical implications of observed LFP signals.