Unraveling the interplay between connectivity and spatio-temporal dynamics in neuronal networks is a key step to advance our understanding of neuronal information processing. Here we investigate how particular features of network connectivity underpin the propensity of neural networks to generate slow-switching assembly (SSA) dynamics, i. e. , sustained epochs of increased firing within assemblies of neurons which transition slowly between different assemblies throughout the network. We show that the emergence of SSA activity is linked to spectral properties of the asymmetric synaptic weight matrix. In particular, the leading eigenvalues that dictate the slow dynamics exhibit a gap with respect to the bulk of the spectrum, and the associated Schur vectors exhibit a measure of block-localization on groups of neurons, thus resulting in coherent dynamical activity on those groups. Through simple rate models, we gain analytical understanding of the origin and importance of the spectral gap, and use these insights to develop new network topologies with alternative connectivity paradigms which also display SSA activity. Specifically, SSA dynamics involving excitatory and inhibitory neurons can be achieved by modifying the connectivity patterns between both types of neurons. We also show that SSA activity can occur at multiple timescales reflecting a hierarchy in the connectivity, and demonstrate the emergence of SSA in small-world like networks. Our work provides a step towards understanding how network structure (uncovered through advancements in neuroanatomy and connectomics) can impact on spatio-temporal neural activity and constrain the resulting dynamics.
Neuronal ensembles exhibit a broad repertoire of activity patterns. Such dynamics are governed by a time-evolving network of synaptic connections with an intricate, yet structured, organization. Due to the advancement of connectomics, our knowledge about such networks is rapidly growing, and increasingly detailed maps of neuronal wiring are becoming available. In parallel, modern recording techniques, such as calcium imaging and multi-electrode arrays, allow neuroscientists to monitor the activity from thousands of neurons simultaneously, with recordings from entire brains at single neuron resolution becoming technologically feasible [1–3]. The observed dynamics of neural networks exhibit an interplay of structured spatio-temporal scales, which underpin a wide range of cognitive functions [4]. The idea that neuronal group activity induced by network structure is at the core of neural computation dates back at least to the work of Hebb [5], who hypothesized that the transient activity of groups of neurons (so called cell assemblies) is the currency of information processing [4,6]. This notion is supported by recent experiments showing that reciprocal connections between neurons occur above chance level [7,8], especially if neurons receive common inputs [9,10]. In the case of the visual system, for instance, excitatory neurons with similar response features tend to be more connected to each other [11,12]. Moreover, studies have demonstrated that neurons exhibit layer-specific connectivities within rodent sensory cortex [13] and neocortex [14]. In addition, organized architectures have been observed to occur at multiple hierarchical scales [15–18] and in non-mammalian organisms [19]. These findings suggest that cortical regions contain well-defined subnetworks. However, the underlying question is whether given a network topology, we can predict the potential of the network to sustain structured spatio-temporal activity. Such questions are not only of interest for network dynamics, but have also implications for memory formation and learning, since neural networks undergo topological changes over time due to plasticity [20,21]. Recently, it has been shown computationally (see e. g. Ref. [22]) that leaky-integrate-and-fire (LIF) networks with equal excitatory and inhibitory connection net strengths (i. e. balanced [23]) yet with clustered excitatory connections, can exhibit prolonged heightened group activity, with the activity transitioning between groups in the network (Fig 1A). Here we characterize the emergence of such slow-switching segregated dynamics in balanced LIF networks as a result of the network connectivity. Specifically, we find that the spectral properties of the synaptic weight matrix (i. e. , the existence of an eigenvalue gap and a block-localized dominant subspace) provide a criterion to predict the appearance of such activity in the network. We then use simple linear rate models to gain insight into the mechanisms underpinning the origin of such dynamics in structurally clustered LIF networks. Using these insights, we construct novel LIF topologies that display slow-switching group activity with distinct properties: involving both inhibitory and excitatory neurons; exhibiting multiple slow time-scales; as well as demonstrating the possibility of such dynamics in networks with no obvious clustered connectivity, such as small-worlds. Finally, we discuss briefly possible implications of the different wiring schemes for neural computation.
Clustered excitatory topologies in a balanced LIF network can lead to dynamics in which localized high activity states transition between assemblies of neurons within the network [22]. This is illustrated in Fig 1A, where the dynamics of an unstructured and a balanced clustered network with 20 groups are shown side by side (see Materials and Methods for a description of the networks). Hereafter, we will refer to such activity as slow-switching assembly (SSA) dynamics. Visually, SSA dynamics manifests itself as bands of increased activity in the raster plots, and can be statistically quantified from the resulting spike-train dynamics a posteriori (see Eq (18) in Materials and Methods). Ideally, however, we would like to establish a priori, solely from the given connectivity, the possibility of such dynamical patterns emerging. The full dynamics of LIF networks are notoriously difficult to analyze due to their inherent non-linearity; hence an exact analytical treatment of the dynamical evolution for an arbitrary clustered topology is essentially intractable. However, two concepts from spectral graph theory and linear systems provide valuable insights: (i) for symmetric, non-negative connectivity matrices, it can be shown that a modular network structure implies a gap in the spectrum of the graph (i. e. , in the set of eigenvalues of the weight matrix) [24,25], as well as the block-localization of the associated eigenvectors on the modules of the network (noting that isolated eigenvalues may also be the result of other features [26]); (ii) for linear systems, a gap in the spectrum of the graph results in a separation of time scales in the dynamical process [27–29]. This relation between the modular structure, the eigenvalues and associated eigenvectors, and linear network dynamics can be used to discover modular structures in graphs from a dynamical perspective [30–33]. In fact, the weighted connectivity matrices of unclustered and clustered LIF networks [22] display different spectral characteristics. Fig 1B shows the spectra of two networks with 1600 excitatory and 400 inhibitory neurons with unclustered (left) and clustered (right) topologies. In the unclustered case, we find the expected circular distribution of eigenvalues, which follows from the properties of random graphs [34,35], although the presence of groups of neurons with different cardinalities and variances means that the eigenvalue distribution is not completely uniform on the circle [34]. Note also that the balanced construction of the LIF network, with a marginally larger inhibitory input for each neuron in order to keep the network stable, leads to the existence of one pair of complex conjugate eigenvalues which lies separate from the main bulk of the spectrum (black arrows in Fig 1B). This eigenvalue pair is associated with the global activation mode of the network (as explained in the following sections and in Ref. [29]). As shown in Fig 1A, this unclustered network exhibits the expected asynchronous, unstructured neuronal spiking dynamics. In contrast, the LIF network with clustered excitatory neurons displays banded SSA dynamics. Spectrally, its weight matrix exhibits a clear gap Δλ along the real axis of its spectrum (Fig 1B, right) and, as shown in a later section, the associated Schur vectors also exhibit a measure of structural block-localization. We remark that, in general, this should not be expected a priori, since the weight matrices are asymmetric and include both positive (excitatory) and negative (inhibitory) couplings. To ascertain the dynamical relevance of the spectral gap, we examined the relation between the clustering strength in the network, defined as the ratio of probabilities of connections inside and outside the neural assemblies (R E E = p in E E / p out E E), the spectral gap (Δλ), and the magnitude of the numerically observed SSA dynamics. To quantify the assembly spike-rate variability, we have defined two complementary metrics. First, the metric S ̂ measures the heterogeneity in the firing rates of the putative cell-assemblies averaged over the simulation (see Eq (18) in Materials and Methods). A large value of S ̂ indicates that the average firing rates of the assemblies are diverse, whereas a low S ̂ indicates that all groups have very similar firing rates at all times and no group shows elevated firing. Under SSA dynamics, the variability of firing rates across groups increases in time as the assemblies transition between high and low firing rates. As discussed below, it is possible that the heightened firing activity is localized in a particular assembly and does not switch between assemblies. This non-switching dynamics where only a group of neurons exhibits elevated firing over the entire simulation time can be detected using the second variability measure S ̂ T, defined in Eq (20). For SSA dynamics, both S ̂ and S ̂ T give similar results (see S1 Fig). In the rest of the paper, we focus on SSA dynamics and mainly use S ̂, except when discussing the transition to non-switching behavior. Definitions of these quantities are given in Materials and Methods. Fig 2 shows that SSA becomes observable above a threshold of the clustering strength REE, although the dependence of this threshold on the size of the network and number of clusters does not seem to follow an obvious pattern. On the other hand, our numerics show that the spectral gap provides a more direct indicator of the presence of SSA dynamics in the network. The relationship between REE and Δλ (Fig 2C) shows that knowing REE is not sufficient to determine Δλ as the spectral gap depends on other factors such as the network size and the number of groups in the network. Together with the examination of the structure of the associated orthogonal subspace (see Section “The block-localization of the dominant linear subspace”), such spectral characterization may be used to establish the potential of networks to sustain SSA activity, even if there is no obviously clustered network model known a priori. The observed spectral properties of clustered LIF networks lead us to consider a stylized linear rate model for neuronal activity in the next section, as a means to gain insights into the defining factors of the wiring diagram leading to SSA. So far we showed that a linear rate model can provide valuable insights into the mechanisms underpinning SSA dynamics in networks with clustered excitatory neurons. The crucial point for the emergence of SSA is the separation of time scales, dictated by the splitting of the leading eigenvalues of the weight matrix, together with the block-localization of the associated dominant subspace. These spectral properties can be introduced into LIF networks by entirely different synaptic couplings, and we now consider a mechanism in which the inhibitory neurons are involved more centrally in generating SSA dynamics. Consider the wiring diagrams presented in Fig 7A and 7B, representing the simple rate model and its corresponding LIF schematic. In this case, the wiring does not involve clustered coupling between groups of excitatory neurons, but rather relies on preferential connectivity patterns between inhibitory and excitatory neuron groups only. Each group of excitatory neurons activates preferentially an associated group of inhibitory neurons and, in turn, this group of inhibitory neurons feeds back more weakly to its associated group of excitatory neurons (see Materials and Methods for a full description of the network). Therefore the effective functional circuitry consists of both excitatory and inhibitory neurons embedded in a feedback loop and, as a consequence, the inhibitory neurons play an integral role in generating the spatio-temporal dynamics and display SSA dynamics too, as we show below. This wiring mechanism was suggested by the stylized firing rate model with two coupled pairs of inhibitory/excitatory feedback loops in Fig 7A. This system is described by Eq (1) with a vector of firing rates for the four groups r = (re1, re2, ri1, ri2) T and a synaptic weight matrix defined as: W = (w w - k ϵ - k s w w - k s - k ϵ s ϵ - k w - k w ϵ s - k w - k w), (8) s > ϵ, k ≥ 1, w = (s + ϵ) / 2. (9) Similarly to Eq (2), the system is balanced and s−ϵ captures the clustering strength within the excitatory-inhibitory pairs. The Schur decomposition of W leads to the following Schur form Q = (- w + w ff 0 k (s - ϵ) w ff, 2 - k (s - ϵ) ), w + = (k - 1) (s + ϵ), w ff = (k + 1) (s + ϵ), w ff, 2 = - (k - 1) (s - ϵ), (10) where the eigenvalues of W are on the diagonal. When the leak term is considered, the largest eigenvalue of Eq (1) is (− 1 + k (s − ϵ) ). Hence, to keep the system stable we need to constrain 0 < k (s - ϵ) < 1, (11) and the spectral gap is again controlled by s−ϵ. In the associated orthonormal Schur basis, the first two modes of the firing rate dynamics u 1 = 1 2 (1 1 1 1), u 2 = 1 2 (1 1 - 1 - 1) (12) are again global ‘sum’ and ‘difference’ modes, which interact via a balanced amplification mechanism [29]. However, there are also two localized modes u3 and u4 that can lead to slow structured activity: u 3 = 1 2 k + 2 (k - k 1 - 1), u 4 = 1 2 k + 2 (1 - 1 - k k). (13) Of these, u3 is associated with the largest eigenvalue and describes the slow (est) dynamics. This mode corresponds to a firing pattern of correlated activity within the pairs (re1, ri1) and (re2, ri2), and anti-correlated activity across the pairs. As before, this analysis extends to networks with more groups and/or stochastic coupling between groups (for related arguments see supplementary material of Ref. [29]). In Fig 7C and 7D, we show that the implementation of this wiring mechanism into a full LIF network displays SSA dynamics with a spectral gap, as predicted by our simple rate model, and the paired excitatory/inhibitory neurons act as a functional circuit displaying synchronous firing behavior. Importantly, note that, in contrast to the excitatory clustered scenario, there are no groups with dense reciprocal couplings in this network topology. Our numerics also show that varying the strength of the functional grouping (WEI = WIE) leads to the emergence of SSA dynamics, linked to the presence of the spectral gap Δλ (Fig 8A), and to the alignment of the leading Schur vectors with the ‘correct’ functional circuits, i. e. , pairs of excitatory and inhibitory groups together (Fig 8B). It is interesting to note that in this topology there is also a group of eigenvalues bounded away in the negative direction (see Fig 7D). These modes are the quickest decaying and correspond to ‘anti-correlated’ firing states, in which excitatory neurons act in synchrony with the inhibitory groups to which they are not functionally associated. Such fast decay reinforces the survival of synchrony within the functional groups in the network. Interestingly, these modes were already present in our linear rate model: u4 in Eq (13) shows the same firing pattern with the fastest eigenvalue λ 4 = − k (s − ε). Although perhaps the most intuitive way of generating SSA dynamics follows from clustering the connectivity of the LIF network, our analysis above suggests that alternative types of structural organization support SSA dynamics in LIF networks, as long as they are characterized by a gap in their eigenvalue spectrum and a measure of block-localization of the Schur vectors. More generally, one could conjecture that any low rank perturbation of the weight matrix of a balanced network which leads to an eigenvalue gap might be a valid candidate for a mechanism to generate SSA in neuronal dynamics. A few such network architectures are worth highlighting, as they have been considered of particular relevance in a neuroscience context.
Answering the question of how the wiring of neural circuits governs neural dynamics is a key step in understanding neuronal computations. Here, we showed how knowledge of certain spectral features of the matrix of neuronal connectivity can help understand what dynamics a network can support. In this paper we focused on LIF networks exhibiting slow-switching assembly (SSA) dynamics, where groups of neurons show a sustained and coherent increase in their firing rates, with slow switching between epochs of localized firing in different groups across the network. We found that the presence of the slow switching time scale is reflected in the spectral properties of the synaptic weight matrix: a gap separating the leading eigenvalues together with a block-localization of the associated Schur vectors on groups of neurons is a key indicator of the presence of SSA dynamics in the network. In line with this observation, multiple gaps in the eigenvalue spectrum are indicative of further time scales in the network dynamics (see Fig 10). Moreover, when the leading eigenvalue becomes larger than one, the SSA dynamics becomes localized on one of the assemblies (see Fig 6). First, we revisited the case of balanced LIF networks with clustered excitatory neurons [22] and observed that only when there is an eigenvalue gap and Schur block-localization does the network display SSA dynamics. Further analytical understanding from a stylized firing-rate model allowed us to determine that the clustering strength drives the development of the eigenvalue gap responsible for the slow switching between localized firing modes. We remark that clustered excitatory connectivity leads to a Hebbian amplification regime [29]: the more stable the pattern formation, the slower the dynamics of any banded activity. Fast switching between well-defined, stable patterns is thus only achievable for strong input changes. As suggested from our spectral characterization, and confirmed by simulations, we showed that SSA dynamics can be achieved through the structured tuning of synaptic strengths, rather than through the clustered rewiring of the connections. While the equivalence between topological and weight organizations is clear for a linear system, such a direct correspondence is not guaranteed a priori for a non-linear system. In fact, the clustering of weights appears to have a slightly larger influence compared to topological organization in our simulations of non-linear LIF systems. More importantly, this dynamical equivalence between clustered topology and clustered synaptic weights has potential ramifications for learning and synaptic plasticity; it indicates that the alteration of the weight structure can lead to the emergence of grouped activity without the need for structural rewiring of physical synaptic connections, thus suggesting a cost-effective adaptation to stochastically encode different firing patterns [47]. Within LIF networks with clustered excitatory connectivity [22], inhibitory neurons only play a balancing background role, whereas experimental studies have revealed a vast diversity of interneuron subtypes that play key roles in computation [48,49]. We thus investigated mechanisms in which inhibitory neurons have an active, functional role in generating SSA dynamics. We demonstrated both analytically and via numerical simulations how such functional circuits can be constructed with the help of a feedback loop between excitatory and inhibitory neurons, so that inhibitory neurons themselves exhibit SSA dynamics and become an integral functional part of the dynamics. As fewer inhibitory neurons are present, adapting the weights according to this functional co-clustering scheme may also provide a cost effective alternative to generate SSA activity in neuronal networks. The emergence of SSA dynamics can be achieved not only by clustering excitatory neurons but also by shaping the network structure in several ways. For instance, a small-world network topology of the excitatory neurons is also able to support SSA dynamics due to the spectral properties of small-worlds, which induce a separation of the slow eigenvalues and the localization and switching of firing activity associated with slowly spatially-varying dominant Schur vectors. In contrast, scale-free networks lack the block-localization of the Schur vectors on multiple groups which is necessary to consistently induce SSA dynamics in LIF networks. Further topologies will be explored in the future. In particular, inhibitory neurons that inhibit other inhibitory neurons (so-called disinhibition patterns [48,50–54]) provide an interesting example. A different avenue might be provided by balanced amplification mechanisms [29], which could be potentially used for creating grouped activity, yet without the introduction of a slow time scale. Other interesting questions in this respect are: which wiring mechanisms provide the most economical, or evolutionarily fit, variant [47,55] to induce a given dynamics; and how does the observed diversity of interneurons (and the multiple roles they can play) relate to this dynamical picture. Our work emphasizes the importance of the spectral characterization of the weight matrix for the dynamics taking place on these topologies. Although spectral properties are also key in characterizing the dynamical response of networks operating at criticality [56], our observations here correspond to a different phenomenon. The emergence of a dominant assembly when λmax ≳ 1 leads to a saturation of the network and a decrease of its dynamical heterogeneity, in contrast to systems of coupled non-modular excitatory units at criticality, which maximize their dynamic range for λmax = 1 [56,57]. Our work links up with experimental findings that cortical states can arise spontaneously and exhibit switching behavior. Recordings from anesthetized cat visual cortex have revealed activity states that dynamically switch, and these states match closely to recorded orientation maps [58]. Such cortical states likely arise due to similarly tuned neurons having higher intra-cortical connectivity [11,12], as is also predicted through models of orientation maps [59–61]. These observations are similar to our theoretical outcomes which indicate that dynamic transitioning between different network states can be driven and sustained based only on the underlying network topology. Other experiments have identified states in CA3 network activity of the hippocampus [62] where active cell assemblies exist for tens of seconds before sharply transitioning to a new state. However, these were metastable states (hence unlikely to be reactivated) and included a core population of cells consistently active in all states. While such complex dynamics are beyond the simple models presented here, it may be feasible to model such metastable dynamics through more elaborate network topologies, which may include synaptic plasticity [63]. Finally, let us remark that our reduced model descriptions effectively focused on networks implementing a rate-based coding mechanism and do not include spike-timing of cell assemblies, which has been identified to be an important component in neuronal computation, e. g. in rodents [64,65], monkeys [66], songbirds [67], and grasshoppers [68]. Whether our results can be translated to a time-coding regime would be an interesting question for future work. As connectomics continues to advance the mapping of connections and synaptic strengths in wide areas of the brain, experimentally obtained weight matrices can be analyzed spectrally as described above to determine if SSA dynamics can be supported by the networks under study. While we focused here on the simplest mechanisms underlying SSA activity, future work could also consider the construction of biophysically realistic models that take into account the growing literature on the distribution of synaptic strengths, synaptic contacts, firing rates, and other relevant cortical parameters that tend to show a lognormal distribution [7,69–71]. Although knowledge of the network structure (connectomics) is not sufficient to predict the dynamics a network circuit will display (for instance, the network dynamics can be dominated completely by a strong input), it can still give valuable insights on the firing patterns the network can support. Conversely, the observation of neuronal dynamics alone may not be sufficient to understand neural computation in detail, since different network topologies can yield similar dynamics. Hence, our work hints at how connectomics and neuronal dynamics data can provide complementary and intertwined routes for systems neuroscientists to study the computational principles implemented by the brain. Finally, it is important to remark that in order to get a fuller picture of the relation between structure and dynamical network properties many other factors not considered here will be of interest, including the distribution of inputs, the precise location of synapses on the post-synaptic neuron, and the plasticity rules that govern the evolution of the network over time [72]. Linking the insight gained from our work to experimental observations or to highly detailed computational models [3,73] would thus be a fruitful next step in order to bridge the gap between neuronal structure, dynamics, and ultimately function.
We simulated leaky-integrate-and-fire (LIF) networks where the non-dimensionalized membrane potential of each neuron (Vi (t), i = 1, …, N) was modeled by: d V i (t) d t = 1 τ m (μ i - V i (t) ) + ∑ j W i j g j E / I (t), (14) with a firing threshold of 1 and a reset potential of 0. The constant input terms μi were chosen uniformly in the interval [1. 1,1. 2] for excitatory neurons, and in the interval [1,1. 05] for inhibitory neurons. The membrane time constants for excitatory and inhibitory neurons were set to τm = 15 ms and τm = 10 ms, respectively. The refractory period was fixed at 5 ms for both excitatory and inhibitory neurons. Note that although the constant input term is supra-threshold, balanced inputs guaranteed that the average membrane potential is sub-threshold [22]. The network dynamics is captured by the sum in Eq (14), which describes the input to neuron i from all other neurons in the network. The topology of the network is encoded by the weight matrix W, i. e. , Wij denotes the weight of the connection from neuron j to neuron i, where Wij is zero if there is no connection. Synaptic inputs are modeled by g j E / I (t), which is increased step-wise instantaneously after a presynaptic spike of neuron j (g j E / I → g j E / I + 1) and then decays exponentially according to: τ E / I d g j E / I d t = - g j E / I (t), (15) with time constants τE = 3 ms for an excitatory interaction, and τI = 2 ms if the presynaptic neuron is inhibitory. Eq (14) can be rewritten in matrix notation as: d V (t) d t = T - 1 [ μ - V ] + W g (t), (16) where T = diag (τi) and V = [V1, …, VN]T and the vectors V, μ, and g are N × 1 vectors. The spectral analyses in the main text (eigenvalues and Schur vectors) refer to the weight matrix W. Different network topologies correspond to different weight matrices W, as explained below. In all simulations, the ratio of excitatory to inhibitory neurons was fixed to be 4: 1. Unless otherwise stated, the networks comprise N = 2000 units (1600 excitatory, 400 inhibitory) and were simulated over 20 seconds to calculate the statistics reported. The time step for all simulations was 0. 1 ms. All simulations were run in MATLAB (version 2011b or later). Different network topologies were simulated using different W matrices but maintaining a general balanced network. If not indicated differently, all parameters below correspond to the case N = 2000. To evaluate the extent to which a network displays slow-switching assembly dynamics, we used the following two spike-rate variability measures. Given a partition of the neurons into c groups, we compute the average spiking frequency fi (t) of the neurons in each cluster i ∈ {1, …, c} over non-overlapping windows of 100ms. For each time window, we obtain the firing rate vector f (t) = [f1 (t), …, fc (t) ]T of which we compute the standard deviation σ (t). The standard deviations are then averaged over the duration of the simulation: S = 1 T ∑ t = 1 T σ (t), (17) where T is the total number of time windows in the simulation. We then obtain a boot-strapped expectation ⟨Sshuff⟩ computed by reshuffling neurons at random into groups of the same sizes as those in the partition. The spike rate variability score is then defined as: S ^ = S - ⟨ S shuff ⟩, (18) where the average ⟨Sshuff⟩ is obtained over 10 random reshufflings of the neurons. If the network fires uniformly (with no localized patterns), S ̂ is low; whereas S ̂ increases if the network displays heterogeneous activity aligned with the partition under investigation. As explained in the text, we introduced a second spike-train variability measure to quantify the variations of the group firing-rates across time. This complementary measure allows us to discern scenarios in which there is a group of neurons dominating the firing (thus leading to a large variation across groups), but no switching between groups. To quantify these effects we use an analogous measure to S ̂ above. Given a partition of the neurons into c groups, we compute the average spiking frequency fi (t) of the neurons in each cluster i ∈ {1, …, c} over non-overlapping windows of 100ms. For each group i, we compute the standard deviation σT (i) of the vector of coarse-grained firing rates across time fi = [fi (t1), …, fi (tN) ], where tk stands for the kth time bin and i = 1, …, c. We then average over groups to obtain: S T = 1 c ∑ i = 1 c σ T (i). (19) As above, we obtain a boot-strapped expectation ⟨ S T shuff ⟩ by reshuffling neurons at random into groups of the same sizes as those in the partition. The spike rate variability score over time is then defined as: S ^ T = S T - ⟨ S T shuff ⟩, (20) where the average ⟨ S T shuff ⟩ is obtained over 10 random reshufflings of the neurons. Again, if the network fires homogeneously (with no localized patterns in time), S ̂ T is low; whereas S ̂ T increases if the network displays heterogeneous firing rates over time. In S1 Fig, we show the behavior of S ̂ T for the examples discussed in the main manuscript. As expected, for all cases where SSA dynamics is present, both measures S ̂ and S ̂ T behave consistently. On the other hand, S ̂ T detects the end of the SSA region when, through increased clustering, the dynamics gets localized on one cell assembly, as shown in Fig 6. First, we find the dominant firing patterns in the network dynamics. We perform simulations of the LIF network and obtain the firing rates of every neuron in 250 ms bins to generate an N × T matrix, where N is the number of neurons and T is the number of bins. On this matrix, we perform a principal component analysis (PCA) and select the first c−1 principal components {pi}i=1c−1. This set of N-dimensional vectors captures most of the variability observed in the simulated dynamics. We then assess how aligned the c−1 principal components are with the dominant Schur vectors of the weight matrix of the network. More precisely, we compute the Schur decomposition of the weight matrix W; keep the c−1 dominant Schur vectors {ui}i=1c−1 associated with the eigenvalues with largest real part; and then compute the (first) principal or canonical angle θ between the subspaces spanned by the two sets of vectors 𝓟 = span{pi} and 𝓤 = span{ui} [38,39]: cos (θ) = max { u T p ∥ u ∥ ∥ p ∥ | u ∈ 𝓤 p ∈ 𝓟 }, (21) The first principal angle measures how ‘close’ the observed firing patterns are to the dominant modes computed solely from the weight matrix. If cos (θ) ≈ 1, there is a large alignment between the span of both subspaces.
Neural networks display a wide range of spatio-temporal behaviors. Understanding how this complex orchestration of neuronal firing activity is determined by the structure of the network (i. e. , its wiring) is an important step towards comprehending how neural computation is manifested, especially given the growing experimental access to temporal recordings and connectomics. Here we investigate the link between network structure and the dynamics of neuronal assemblies in the context of leaky-integrate-and-fire (LIF) networks. We show how structural features in the wiring of the network can introduce additional time-scales to the dynamics, and how such structured wiring can lead to spatio-temporally segregated, coherent activity of groups of neurons. Using rate models we gain insight into how such spatio-temporal dynamics emerge as a direct consequence of the spectral properties of the network, and use this understanding to examine further circuit topologies that enable phenomenologically similar behavior, yet rely on fundamentally different connectivity and functional patterns.
Emergence of Slow-Switching Assemblies in Structured Neuronal Networks