Abstract
We present two graphical model-based approaches to analyse the distribution of neural activities in the prefrontal cortex of behaving rats. The first method aims at identifying cell assemblies, groups of synchronously activating neurons possibly representing the units of neural coding and memory. A graphical (Ising) model distribution of snapshots of the neural activities, with an effective connectivity matrix reproducing the correlation statistics, is inferred from multi-electrode recordings, and then simulated in the presence of a virtual external drive, favoring high activity (multi-neuron) configurations. As the drive increases groups of neurons may activate together, and reveal the existence of cell assemblies. The identified groups are then showed to strongly coactivate in the neural spiking data and to be highly specific of the inferred connectivity network, which offers a sparse representation of the correlation pattern across neural cells. The second method relies on the inference of a Generalized Linear Model, in which spiking events are integrated over time by neurons through an effective connectivity matrix. The functional connectivity matrices inferred with the two approaches are compared. Sampling of the inferred GLM distribution allows us to study the spatio-temporal patterns of activation of neurons within the identified cell assemblies, particularly their activation order: the prevalence of one order with respect to the others is weak and reflects the neuron average firing rates and the strength of the largest effective connections. Other properties of the identified cell assemblies (spatial distribution of coactivation events and firing rates of coactivating neurons) are discussed.
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Acknowledgments
This work is a follow-up of a previous study in collaboration with F.P. Battaglia and U. Ferrari (Tavoni et al. 2015), to whom we are very grateful.
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Action Editor: Alain Destexhe
This work was funded by the [EU-]FP7 FET OPEN project Enlightenment 284801.
Appendices
Appendix A: Comparison with Principal Component Analysis–based methods
In this section we compare our method to identify coactivated groups of neurons in data with techniques based on Principal Component Analysis (PCA). We consider again session 1, for which (Peyrache et al. 2009) have shown the presence of reactivation during Sleep Post of the first principal component of the Pearson Correlation matrix of the activity in Task. Despite this result, we show below that the identification of cell assemblies in single epochs is in general difficult with PCA.
We bin the neuron spike trains into time windows of 10 ms (as in our model inference) and 100 ms (as used by Peyrache et al. in their analysis) and compute the Pearson correlation matrix of the activity for these two choices of the time bin. For Δt=10 ms we identify six signal (principal) components, and five for Δt=100 ms. Our criterion for identification of the signal eigenvectors is as follows: we select the modes whose corresponding eigenvalues are larger than the upper bound of the Marcenko-Pastur eigenvalue distribution, \(\lambda ^{+}=(1+\sqrt {N/B})^{2}\), where B= number of time bins in the recording, N= number of recorded neurons. The largest entries of the first component in the Task epoch correspond to the replay group 1-9-20-21-26, which is also represented (at least partially) in the two principal components of Sleep Post. This result explains the agreement between the replay group identified in (Peyrache et al. 2009) and in our analysis (see also (Tavoni et al. 2015)).
We have then tried to use clustering procedures to identify neural groups. We represent each neuron as a point in the space of the signal components, with coordinates \(v_{m}(i) \sqrt {\lambda _{m}}\) (where v m (i) is the entry corresponding to neuron i in the m th signal eigenvector, and λ m is the m th eigenvalue). We then apply the classical k-means clustering algorithm to these N points, where the number k of clusters is arbitrarily chosen, as we expect groups of closely correlated neurons to be represented by points far from the origin and close to each other in this dimensionally-reduced space. Unfortunately, this method applied to our data does not seem to be able to identify significant clusters, well separated from noisy clusters, as shown in Fig. 19, where the identified clusters are projected onto the bi-dimensional space of the first two signal components. Each panel shows the clustering (optimized over 104 random initial conditions) in each epoch, for a particular choice of k=2,3,4. Neurons in the same cluster are represented by the same symbol; full symbols show the farthest clusters from the origin (distance d c >0.3), empty symbols correspond to the closest ones ( d c <0.3). With a few exceptions, e.g. the cluster of upward full triangles in Task, which is rather robust with respect to the choice of k, signal clusters are in general not clearly separated from each other and from the noisy clusters. In addition no obvious choice for the value of k seems to be optimal to extract the groups of maximally coactivated neurons in each epoch. Finally, this method assigns each neuron to one cluster, and does not allow us to identify overlapping cell assemblies.
Appendix B: Analysis of sessions 2, 3 and 4
The analysis described in Sections 3.1 and 3.2 for session 1 is here illustrated for other three representative sessions (called 2, 3 and 4), with the purpose of validating the method and showing the phenomenology of coactivated groups that can be found in the data. We first compute the neuron susceptibilities as a function of drive H (Methods). For all sessions we find bell-shaped susceptibilities similar to those of session 1, with several susceptibilities showing maxima or minima comprised between 0 and 0.25 (indicating neurons which are substantially independent); some susceptibilities have maxima or minima at very close values of H, and correspond to neurons that are coactivated or coinhibited. For each epoch we rank the susceptibility maxima \(\chi _{i}^{+}\) and show, through computation of the coactivation ratio (17), that neurons having large \(\chi _{i}^{+}\) are strongly coactivated in the real data, thus validating our method to identify cell assemblies.
Session 2 has highly similar coactivated neurons in the Task and Sleep Post Task, and these neurons are not coactivated in the Sleep Pre Task (Fig. 20), similarly to the replay group found for session 1. This is in agreement with the identification of a replay group for this session from comparison of the inferred interaction networks in (Tavoni et al. 2015).
Differently from session 2, session 3 shows large susceptibility maxima also in the Sleep Pre Task (Fig. 21) indicating that cell assemblies have been sampled in all epochs, as confirmed by the large coactivation ratios of the identified groups. These groups are substantially different across the three epochs and no large replay group has been recorded in this session.
Session 4 is characterized by the presence of a group of neurons with high \(\chi _{i}^{+}\), large and positive couplings and a large coactivation ratio in all experimental epochs (Fig. 22).
Appendix C: Minimal number of coactivation events required for detection
The resolving power of our approach in detecting rare coactivation events can be estimated analytically. To fix notations we consider a set of N neurons, with uncorrelated spiking activities and average firing rate f. On top of this activity a simple firing pattern (spikes of neurons 1 and 2 separated by less than ΔT) is repeated K times; note that the choice of a pattern involving two neurons only can be considered as a worst case, as patterns involving a higher number of neurons would produce more correlations and would be easier to detect. The total recording time is T and the time-bin duration is ΔT. Within this framework we can compute (for simplicity we assume that K is small compared to the expected number of spikes, f T):
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the expected largest apparent correlation between any two neurons resulting from limited sampling in the absence of any pattern repetition ( K=0). This value can be estimated from extreme value theory to be (for f ΔT≪1):
$$ C_{noise-max} \simeq 2\, f\, \frac{({\Delta} T)^{3/2}}{T^{1/2}}\, \sqrt{\log N} $$(29) -
the minimal number of repetitions, K m i n , such that the correlation between neurons 1 and 2 produced by the K repetitions exceeds C n o i s e−m a x :
$$ K_{min} \simeq \frac T{\Delta T}\, C_{noise-max} $$(30) -
the susceptibility curve of neurons 1 and 2 (identical) for K repetitions. To compute this curve, we first infer the Ising model for neurons 1 and 2 (all neurons being independent), which involves two local inputs h 1 = h 2 and one coupling J 12. The parameters can be analytically inferred as functions of K. We then compute the average activity 〈σ 1〉(H) as a function of the drive H and its derivative with respect to H, that is, the susceptibility χ 1(H).
Results are reported in Fig. 23. For the Task epoch of session 1, we have T≃1400 sec, f≃5.8 Hz; with these parameters the number of repetitions is K m i n ≃70. This corresponds to a minimum frequency of repetition of the pattern equal to f m i n = K m i n /T≃0.05 Hz, less than 1 % of the average firing frequency of neurons. As K exceeds K m i n the maximum of the susceptibility curve increases beyond 0.25 and moves to the left, in agreement with results (for experimental data) shown in Section 4 (see in particular Fig. 18, left).
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Tavoni, G., Cocco, S. & Monasson, R. Neural assemblies revealed by inferred connectivity-based models of prefrontal cortex recordings. J Comput Neurosci 41, 269–293 (2016). https://doi.org/10.1007/s10827-016-0617-5
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DOI: https://doi.org/10.1007/s10827-016-0617-5