Fig 1.
Network dynamics and eigenvalue spectra of two LIF networks.
One network with uniform synaptic connections (left), and one with 20 groups of clustered excitatory connections (right). To create the clustered network, excitatory neurons were partitioned into groups (with in-group connection probability and out-group connection probability ) while keeping the average connectivity constant (see Materials and Methods and Ref. [22]). The ratio controls the strength of the excitatory clustering. A Visualization of the network topologies (top) and exemplars of raster plots (bottom). The dynamics of the clustered network exhibit the banded structure associated with slow-switching group activity. The magnitude of this switching can be characterized statistically a posteriori from the data through the spike-rate variability metric , defined in Materials and Methods Eq (18), as discussed in the text. In this case, the unclustered network has while the clustered network has a much larger value . B Eigenvalue spectra of the network weight matrices W. The weighted connectivity matrix of the clustered network exhibits a clear eigengap Δλ separating the 19 eigenvalues with largest real parts from the cloud of eigenvalues in the bulk. There is no such eigengap for the unclustered network. As indicated by the two arrows, both matrices have a pair of complex conjugate eigenvalues associated with the (damped) global activation modes of the networks characteristic of balanced networks (see text and Ref. [29]).
Fig 2.
The relationship of observed SSA dynamics with the structural connectivity clustering of the LIF network, REE, and the spectral gap, Δλ.
The presence of SSA dynamics is quantified through the spike-rate variability metric (), which measures the variance of the firing rates of the assemblies normalized by a randomly shuffled bootstrap: increases with increasing SSA activity and for completely asynchronous activity (as in the unclustered case in Fig 1A) (see Materials and Methods). A Spike-rate variability is plotted as a function of REE (left) and Δλ (right) for different network sizes (dots: raw data from simulations; line: mean; shading: standard deviation). Above a certain clustering threshold, SSA emerges and increases as REE grows; the intensity of the SSA dynamics is in line with the presence of an eigenvalue gap Δλ in the weight matrix. B Spike-rate variability as a function of REE (left) and Δλ (right) for a network of 2000 neurons with different numbers of clusters, yielding qualitatively similar results (dots: raw data from simulations; line: mean; shading: standard deviation). C Relationship between the clustering strength REE and the spectral gap Δλ. Observe that REE is not sufficient to determine Δλ, i.e. Δλ is influenced by other aspects such as the network size and number of groups.
Fig 3.
Structure of the eigenspectra of stylized and realistic networks that can exhibit SSA activity.
A Schematic of the stylized model of a clustered network Eqs (1)–(2) and illustration of the corresponding eigenvalues Eq (6). B Schematic of a more realistic clustered LIF network with the same wiring scheme on many groups and multiple neurons per group with an illustration of the associated spectrum.
Fig 4.
SSA dynamics can be achieved by changing solely the synaptic strengths even when the topological connections are kept uniform.
To create clustered networks, the ratio was varied, where and refer to the in-group and out-group synaptic weights, respectively. The spike-rate variability measuring the intensity of SSA dynamics is shown as a function of WEE (left; dots: raw data from simulations; line: mean; shading: standard deviation) and the spectral gap Δλ (right). Note that the connection probability is uniform for all the simulations, i.e. REE = 1, and only the clustering of the weights is varied.
Fig 5.
Unraveling the association between the leading Schur vectors of the weight matrix and the observed SSA activity.
A Illustration of a weight matrix of the N = 2000 neuron LIF network with c = 20 groups of excitatory neurons (REE = 3.4) shown together with the heatmap of the real parts of its first 25 Schur vectors. The first 19 Schur vectors exhibit block-uniform patterns relatively constant within each of the 20 groups of neurons, whereas no such pattern is observed for the other Schur vectors. Note that there is no pattern discernible over the inhibitory neurons. The leading Schur vectors correspond to the cloud of ‘slow’ eigenvalues above the gap, as indicated in C, and span a patterned dominant subspace that induces grouped dynamics in the network. B A long simulation (80s) of the LIF network dynamics (only the first 2s shown) was analyzed using PCA and the first 25 principal components (PCs) are shown. Reflecting the banded structure of the simulated dynamics, the leading PCs also show a block-patterned structure consistent with the neuronal groups. C On the spectrum of W, we indicate the group of leading eigenvalues above the gap associated with the dominant subspace. D The alignment between the dominant Schur subspace of the W matrix and the subspace of the strongest principal components is measured by the first principal angle θ Eq (21). Above a threshold of the clustering strength REE, both subspaces become highly aligned in line with the observations in Fig 2 (dots: raw data from simulations; line: mean; shading: standard deviation). E The same effect is observed when the clustering is introduced in the weights by varying WEE, as in Fig 4.
Fig 6.
Effect of large clustering strength on SSA activity.
A Large values of the clustering REE lead to linear instability of the SSA dynamics and localization of the activity on one assembly. As measured by the spike rate variability across time (), the increase of REE leads to SSA (signalled by the increased value of ). If the clustering increases further, decreases, as the dynamics becomes dominated by one assembly only. (dots: raw data from simulations; line: mean; shading: standard deviation). Inset: examples of raster plots for three data points in the three regimes. The analysis corresponds to a clustered LIF network of 1000 neurons. B Plot of the eigenvalue with the largest real component λmax against the clustering strength REE. The linear condition λmax > 1 is a good indicator of the dynamics becoming dominated by one cell assembly.
Fig 7.
SSA dynamics can result from excitatory-to-inhibitory feedback loops.
A Schematic of a stylized network model with two excitatory-to-inhibitory feedback loops corresponding to the model Eq (8). B Illustration of the corresponding LIF network with such a co-clustered feedback mechanism between neuron types. C Raster plot of the dynamics of a co-clustered LIF network with N = 2000 neurons and c = 20 pairs with excitatory-to-inhibitory feedback. Note that the inhibitory neurons also exhibit SSA dynamics here. D Spectrum of the weight matrix exhibiting a spectral gap both on the left and right hand side of the bulk of the spectrum (see text for details).
Fig 8.
The presence of SSA dynamics in LIF networks with excitatory to inhibitory feedback loops.
A The spike-rate variability, which measures the intensity of SSA dynamics, as a function of WEI = WIE (left; dots: raw data from simulations, line: mean, shading: standard deviation) and Δλ (right). See text and Fig 4 for further details. B The first principal angle between the subspaces of the principal firing patterns and the dominant Schur vectors of the weight matrix W show high alignment. See text and Fig 5 for further details.
Fig 9.
Small-world networks can exhibit SSA activity.
A Small-world network and the resulting raster plot of its LIF network simulation. SSA activity is observed, yet with less distinct boundaries when compared to the clustered case. B Spectrum of the weight matrix of the small-world network showing a small eigenvalue gap. C The principal angle between the dominant subspace spanned by principal components of the firing rates and the dominant Schur vectors of the weight matrix shows that, as the modularity of the small-world rewiring becomes larger, there is alignment between the observed dynamics and the slow directions of the weight matrix (dots: raw data from simulations; line: mean; shading: standard deviation).
Fig 10.
SSA activity in hierarchical LIF networks.
A Schematic of a hierarchically clustered LIF network. Each excitatory group consists of two subgroups. B A hierarchical modular LIF network with N = 2000 neurons and the resulting raster plot of its simulation. SSA activity is observed at two time scales, corresponding to the two hierarchical levels embedded in the network structure: slow switching between the large groups and faster switching between the inner subgroups (see inset). C The spectrum of the weight matrix of this hierarchical network exhibits two eigenvalue gaps corresponding to the two slow time-scales the network can support.