Assemblies, synapse clustering, and network topology interact with plasticity to explain structure-function relationships of the cortical connectome

Learning and memory are orchestrated by synaptic plasticity, the ability of synapses to change their efficacy in an activity-dependent manner. Donald O. Hebb’s postulate about how synaptic plasticity might manifest was paraphrased with the well-known mantra: ‘cells that fire together, wire together’ (Hebb, 1949; Shatz, 1992). The first proof of coincident pre- and postsynaptic population activity leading to potentiation (an increase in efficacy) came from pathway stimulation in hippocampal slices (Bliss and Lomo, 1973). It was later confirmed at the neuron pair level (Markram et al., 1997; Bi and Poo, 1998), and spike-time dependent plasticity (STDP) became a quintessential protocol to study Hebbian plasticity in vitro. In the early 2000s, a plethora of cortical pathways were studied and plasticity proved to be synapse location—and therefore pathway—dependent (Sjöström and Häusser, 2006; Letzkus et al., 2006; Froemke et al., 2010). The molecular substrate of Hebbian coincidence detection is the N-methyl-D-aspartate (NMDA) receptor, which upon removal of the Mg²⁺ block by depolarization, conducts Ca²⁺(Mayer et al., 1984). The calcium-control hypothesis, put forward by Lisman, 1989 postulates that prolonged, moderate amounts of Ca²⁺ lead to depression (a decrease in efficacy) while large transients of Ca²⁺ lead to potentiation. By putting these together, it became evident that it is not necessarily the timing of the postsynaptic spike, but rather the depolarization induced-entrance of calcium into the postsynaptic dendrite that is important to evoke changes in synaptic efficacy (Goldberg et al., 2002; Lisman and Spruston, 2005).

In parallel with slice electrophysiology, Hebbian plasticity was also studied through its effect on behavior via fear conditioning experiments (McKernan and Shinnick-Gallagher, 1997) and this line of research led to numerous new techniques for tagging and re-activating cells that participate in newly formed memories (Tonegawa et al., 2015). While these studies highlighted the need to study plasticity at the network level, most changes are expected to happen at the synapse level. Therefore, high-throughput methods tracking synaptic proteins like PSD95 (Ray et al., 2023) and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionate (AMPA) subunit GluA1 (Graves et al., 2021; Kim et al., 2023) are currently being developed. While readily applicable to monitor synaptic efficacy in vivo, currently these techniques cannot be supplemented with recordings of neural activity, thus the reason for the changes in efficacy can only be speculated.

The bridge between in vitro pairs of neurons and in vivo behavior is often provided by complementary, simulation-based approaches. Early theoretical work explored the potential link between memories and cells that fire and therefore wire together, concentrating on the storage and retrieval of memories in strongly recurrent networks (Hopfield, 1982), which remained an active topic of research (Fusi and Abbott, 2007; Krotov and Hopfield, 2016; Widrich et al., 2020). Inspired by STDP experiments, modelers have postulated diverse plasticity rules that fit the most recent experimental findings (Gerstner et al., 1996; Kempter et al., 1999; Song et al., 2000; Pfister and Gerstner, 2006; Clopath et al., 2010). Models that take into account the crucial role of calcium in driving plasticity outcomes have also been proposed (Shouval et al., 2002; Graupner and Brunel, 2012; Rubin et al., 2005; Jędrzejewska-Szmek et al., 2017; Rodrigues et al., 2023). The calcium-based model of Graupner and Brunel, 2012 describes the evolution of intracellular calcium concentration (${\displaystyle [C{a}^{2+}{]}_i}$) given the pre- and postsynaptic spike trains and updates the efficacy of the synapse, upon ${\displaystyle [C{a}^{2+}{]}_i}$ crossing thresholds for depression and potentiation. Bioplausible plasticity rules have been shown to bring about Hebbian cell assemblies, i.e., groups of neurons that fire together to enable long-term memory storage (Litwin-Kumar and Doiron, 2014; Zenke et al., 2015; Fauth and van Rossum, 2019; Kossio et al., 2021). A common theme in these models is the necessity for homeostatic plasticity to keep the networks stable. While experimental evidence exists for homeostatic plasticity (Turrigiano and Nelson, 2004), this has been shown to be too slow to induce stability (Zenke et al., 2017a) and alternative mechanism have been proposed (Vogels et al., 2011; Delattre et al., 2015). While these studies provide mechanistic explanations of learning and memory, they used point-neuron models, therefore neglecting the structural and functional importance of dendrites in plasticity (with the exception of Bono et al., 2017; Kastellakis and Poirazi, 2019). The compartmentalized nature of dendritic trees gives rise to spatial clustering of synapses (Kastellakis and Poirazi, 2019; Farinella et al., 2014; Iacaruso et al., 2017; Tazerart et al., 2020) and local, non-linear voltage events (Poirazi et al., 2003; Stuart and Spruston, 2015) both of which are thought to contribute to the removal of the Mg²⁺ block from NMDA receptors and therefore gate plasticity.

To investigate the effect of dendrites and multi-synaptic interactions for plasticity at the network level, in this study we equipped the biophysically detailed, large-scale cortical network model of Isbister et al., 2023 with our recently developed model of functional plasticity (Chindemi et al., 2022) between excitatory cells (Figure 1). At this level of detail, a calcium-based plasticity model is the natural choice since calcium dynamics are already part of the simulation. Moreover, it allowed us to model in vivo-like conditions, specifically the low extracellular calcium concentration (${\displaystyle [C{a}^{2+}{]}_o}$, Chindemi et al., 2022), which has been experimentally shown to reduce plasticity (Inglebert et al., 2020, Figure 1E). As we had access to the pre- and postsynaptic activity and efficacy of millions of synapses, including their dendritic location, we could characterize the rules governing plasticity at the microcircuit level. During simulated evoked activity, plasticity acted sparsely and specifically, although it was still able to reorganize the network’s dynamics, manifesting in more pattern-specificity after plasticity. Potentiation dominated in amplitude and depression counteracted it in frequency, which led to stable firing rates without explicitly introducing any homeostatic terms (Zenke et al., 2017a; Turrigiano and Nelson, 2004). We found plasticity was driven by the network topology beyond pairwise connectivity. Specifically, by co-firing cell assemblies, spatial clustering of synapses on dendrites, and the centrality of a connection in the network. Furthermore, we confirmed our testable predictions in the (MICrONS, 2021) dataset, an openly available electron microscopic reconstruction.

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To study how plasticity and various network level features interact to explain structure-function relationships, we used a bio-realistic, large-scale cortical model of the rat non-barrel somatosensory cortex (nbS1). By doing so, we had a continuous readout of both the activity of all neurons and the efficacy of the synapses between them under in vivo-like, low ${\displaystyle [C{a}^{2+}{]}_o}$ conditions (∼1 mM instead of 2-2.5 mM in vitro). The current model improves on Markram et al., 2015 in terms of both anatomical, e.g., atlas-based cell composition and placement (described in Reimann et al., 2024a), and physiological properties, e.g., improved single-cell models, multi-vesicular synaptic release, and layer-wise compensation for missing synapses (described in Isbister et al., 2023). For this study, we used a seven column subvolume comprising 211,712 neurons in 2.4 mm³ of tissue (Figure 1A). In line with the biological variability, excitatory cells are modeled as a diverse set of morphologies (Reimann et al., 2024a; Kanari et al., 2019; Figure 1B) equipped with conductances distributed across all electrical compartments (Reva et al., 2023; Figure 1—figure supplement 1A). The connectivity and synaptic physiology of these cells were extensively validated (Isbister et al., 2023; Reimann et al., 2024a; Figure 1C; Figure 1—figure supplement 1C). To deliver input with spatio-temporal precision to the circuit, we used thalamic fibers from the ventral posteriomedial nucleus of the thalamus (VPM) and the high-order posteriomedial nucleus of the thalamus (POm; Figure 1D; Meyer et al., 2010).

To simulate long-term plasticity, we integrated our recently published calcium-based plasticity model that was used to describe functional long-term potentiation and depression between pairs of pyramidal cells (PCs) (Chindemi et al., 2022). In short, the model follows the formalism of Graupner and Brunel, 2012, where pre- and postsynaptic spikes led to changes in synaptic ${\displaystyle [C{a}^{2+}{]}_i}$ (Figure 1E). Calcium entering through NMDA receptors and voltage-dependent calcium channels (VDCCs) contributes to ${\displaystyle [C{a}^{2+}{]}_i}$ (Equation 2 in Methods). When the integrated calcium trace of a synapse crosses the threshold for depression (${\displaystyle {\theta }_d}$) or the higher one for potentiation (${\displaystyle {\theta }_p}$), synaptic efficacy (${\displaystyle \rho }$) is updated (Figure 1E left; Equation (1) in Methods). Changes in ${\displaystyle \rho }$ are then converted into changes in the utilization of synaptic efficacy (${\displaystyle U_{SE}}$), a variable of the Tsodyks-Markram model of short-term plasticity describing the baseline probability of vesicle release (Tsodyks and Markram, 1997), and the peak AMPA receptor conductance (${\displaystyle {\hat{g}}_{AMPA}}$; Equations (7) and (8) in Methods, respectively). As a result of updating ${\displaystyle U_{SE}}$, short- and long-term plasticity are tightly coupled in the model (Markram and Tsodyks, 1996; Costa et al., 2015; Deperrois and Graupner, 2020). In the absence of ${\displaystyle [C{a}^{2+}{]}_i}$ influx ${\displaystyle \rho }$ exhibits bistable dynamics (Lisman, 1985; Graupner and Brunel, 2012) and at initialization, synapses are assumed to be at one of the two fixed points (fully depressed ones at ${\displaystyle \rho =0}$ and fully potentiated ones at ${\displaystyle \rho =1}$) and their assignment to these states is pathway-specific (Figure 1—figure supplement 1C3).

We calibrated layer-wise spontaneous firing rates and evoked activity to brief VPM inputs matching in vivo data from Reyes-Puerta et al., 2015 (Figure 1—figure supplement 3). Spontaneous activity was driven by somatic injection of a layer- and cell-type-specific noisy conductance signal (Isbister et al., 2023, see Methods). Evoked activity was driven by a thalamocortical input stream already described in Ecker et al., 2024. In short, 10 VPM input patterns were repeatedly presented in random order with a 500 ms inter-stimulus interval, together with a non-specific POm input. These VPM patterns were defined with varying degrees of overlap in the sets of activated fibers (Figure 2A, Figure 1—figure supplement 4; see Methods). An exemplary raster plot, characterizing the evoked state of the plastic network is shown in Figure 2B.

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Plasticity induces sparse and specific changes driven by the stimulus and the network’s topology

We have shown that the changes in synaptic efficacy stabilize after a transient period, which raises the question: What predictions does the model make about the structure of the plastic changes in vivo? First of all, the changes were sparse, implying that the configuration of synaptic efficacy of the overall network remained largely unchanged. Indeed, we found that the number of connections changing was below 18% across the whole network and remained low when restricted to individual layers (Figure 3A1). Moreover, the propensity of changes increased as the pairwise firing rates increased (Figure 3A2), in line with previous modeling insights (Litwin-Kumar and Doiron, 2014; Graupner et al., 2016). On the level of individual synapses changes were even sparser (Figure 3B1), and in both cases, depression was more common than potentiation. Layer 5 (L5) PCs contributed mostly to changes on the basal dendrites, while apical changes happened mostly on L6 PCs (Figure 3B2).

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So far, we characterized plastic change in terms of ${\displaystyle \rho }$, as this parameter is bounded to the ${\displaystyle [0,1]}$ interval and thus easy to interpret across pathways. Another parameter affecting the function of the synaptic connection is ${\displaystyle {\hat{g}}_{AMPA}}$. We found a minimal decrease (–2 pS, Figure 3C1) in its mean value due to plasticity, which was explained by frequent depression and a heavy tail of potentiation amplitudes (Figure 3C2). The distribution of ${\displaystyle {\hat{g}}_{AMPA}}$ remained lognormal, in line with biology (Buzsáki and Mizuseki, 2014; Rößler et al., 2023, Figure 3C1, bottom). The plasticity model employed was bistable around ${\displaystyle {\rho }_{\ast }=0.5}$, i.e., in the absence of further Ca²⁺ influx, values above 0.5 would converge towards 1 and values below towards 0 (see Graupner and Brunel, 2012 and Methods). The fraction of changes in ${\displaystyle \rho }$ that crossed this threshold was slightly higher for depression (26%) than for potentiation (23%) (compare Figure 3D1-B1). Furthermore, the vast majority of these crossings occurred early, with 60% of potentiation and 40% of depression happening within the first minute (Figure 3D2).

Additionally, we hypothesized that both the amount, i.e., the level of sparsity, and the structure of plastic changes are non-random and instead shaped by the stimuli and the underlying network’s topology. We confirmed our hypothesis first for the amount of plastic changes. We ran control simulations where we delivered random Poisson spikes on the same VPM fibers with the same mean rate, but without the spatio-temporal structure of the stimuli. We found that this reduced the number of connections undergoing plasticity by 25%, demonstrating the importance of stimulus structure over simple firing of pre- and postsynaptic neurons (Figure 3—figure supplement 1). Additionally, without synaptic transmission, plasticity was reduced even further: In simulations of stimulus streams where intrinsic connectivity between the simulated neurons was cut, we found occasional changes in ${\displaystyle \rho }$, but the number of changing synapses was an order of magnitude below baseline conditions (Figure 3—figure supplement 1). Thus, our plasticity model is not strictly Hebbian, since the effect of postsynaptic firing alone could change synaptic efficacy, although presynaptic release was required for most of the observed changes (Graupner and Brunel, 2012; Graupner et al., 2016). When the external stimuli were also left out, negligible plastic changes occurred (Figure 3—figure supplement 1).

We then confirmed that the network’s topology-shaped plastic changes as well. In particular, we found layer-to-layer pathway specificity in the amount of plastic changes observed. That is, the amount of changes between layers differed from those expected in a random control with the same pre- and postsynaptic populations (Figure 4A2 vs. Figure 4A3). We further quantified the difference in the subnetwork of changing connections from the random control by counting directed simplices (Figure 4B). These are motifs that have previously shown to be linked to network function (Reimann et al., 2017) as well as quantify the complexity of the network’s topology (Kahle, 2009; Bobrowski and Kahle, 2018). Succinctly, a directed simplex of dimension ${\displaystyle k}$, is a motif on ${\displaystyle k+1}$ neurons with all-to-all feed-forward connectivity (Figure 4A1 right, see Methods). We found strong overexpression of simplices in the subgraph of changing connections compared to the control (Figure 4B). Furthermore, the maximal simplex dimension found in the subgraph exceeded the value expected by chance by two. This result shows that the connections changing are not determined only by their pre- and postsynaptic populations but also depend on their embedding in the whole network. We will explore this further in the Network-level metrics predict plasticity and separate potentiation from depression section.

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Finally, we showed that the input stimuli shaped plastic changes by running 2-min-long plastic simulations in which we only presented a single pattern (several times, with the same 500 ms inter-stimulus interval as before) and compared the changes in mean ${\displaystyle \rho }$ matrices. By using an input - output distance correlation analysis, we found that the distances between these mean ${\displaystyle \rho }$ matrices were highly correlated with the input distances of the associated VPM fiber locations (${\displaystyle r=0.716}$, ${\displaystyle p<0.0001}$, Figure 4C2, Figure 4—figure supplement 1A). Furthermore, this required using Euclidean distance between the ${\displaystyle \rho }$ matrices on the output side, implying that both the mean ${\displaystyle \rho }$ values themselves and their position in the network are relevant. When either Hamming distance (taking only the identity of changing connection into account) or Earth mover’s distance (taking only the distribution of mean ${\displaystyle \rho }$ values into account) were used instead, we found weaker and non-significant correlations, respectively (Figure 4C3 and C4, respectively).

In summary, we observed that ∼7% of synapses undergo long-term plasticity under realistic in vivo-like conditions in 10 min of biological time, and most of these synapses are on L5 PC’s basal dendrites. Moreover, the changes are not random, but depend not only on the firing rates but also on the recurrent connectivity and input stimuli. Potentiation dominated in amplitude, while depression counteracted it in frequency, keeping the network stable without needing to model homeostatic plasticity (Zenke et al., 2017a; Turrigiano and Nelson, 2004).

More frequent plastic changes within and across cell assemblies

We have shown that plastic changes are sparse and highly specific. We, therefore, tried to understand the rules determining which synapses changed. From the parametrization of the plasticity model, we learned that presynaptic spikes contribute orders of magnitude higher ${\displaystyle [C{a}^{2+}{]}_i}$ than postsynaptic ones if the NMDA receptors are fully unblocked (Figure 1—figure supplement 2B). Thus, in order to effectively depolarize the dendrites and unblock NMDA receptors, spikes at low, in vivo-like rates must be synchronized in time. Therefore, we specifically hypothesized that plasticity of connections may be structured by the membership of participating neurons in Hebbian cell assemblies (Hebb, 1949; Harris, 2005). In our previous analysis (Ecker et al., 2024), and in line with experimental results (Harris, 2005; Song et al., 2005; Perin et al., 2011), we observed that the number of afferents from an assembly is a great predictor of a neuron’s membership in an assembly. Strong positive interactions were also found across assemblies, but only when the direction of innervation reflected the temporal order of assembly activation. These results, combined with the biophysics of the plasticity model, suggest that connections within an assembly and the ones between temporarily ordered assemblies, are expected to undergo plastic changes with a higher probability.

To test this hypothesis, we detected cell assemblies, based on their co-firing function, from the in silico spiking activity of the 10-min-long plastic simulation using methods established by experimentalists (Carrillo-Reid et al., 2015; Herzog et al., 2021, see Methods). In short, spikes were binned and bins with significantly high firing rates were hierarchically clustered (Figure 5A). These clusters correspond to the functional assemblies of neurons (Figure 5C), with a neuron being considered a member if its spiking activity correlates with the activity of an assembly significantly stronger than chance level. Since time bins and not neurons, were clustered in the first place, this method yields one assembly per time bin and single neurons can be part of several assemblies (Figure 5B and D). To foster effective discussions in the field (Miehl et al., 2023) urged to call co-firing neurons, detected from purely functional activity ensembles, and strongly interconnected ones - usually by design in modeling studies - assemblies. While we agree with the distinction, we prefer to call the groups of neurons used for analysis in the study functional assemblies as a mix of the two terms, since we have shown in our previous investigation using the same circuit model that they not only co-fire but are interconnected more than expected (Ecker et al., 2024). Note, that, in contrast to classical modeling studies, where assemblies are usually defined based on their strong internal connectivity (i.e. their structure Litwin-Kumar and Doiron, 2014; Zenke et al., 2015; Fauth and van Rossum, 2019; Kossio et al., 2021) our functional assemblies are detected from the activity of the network only, eliminating a potential bias arising from links between structural properties of changing synapses and the assembly detection method.

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Functional assemblies were activated in all stimulus repetitions and a series of three to four assemblies remained active for 190 ± 30 ms (Figure 5B). Pattern C elicited the strongest response, while pattern H the weakest, and the responses of patterns H and I were the most similar to each other, as expected, since they share 66% of the VPM fibers (Figure 2A). Functional assembly activations had a well-preserved temporal order - with some of them always appearing early during a stimulus, while others later - and from now on we will refer to them as early, middle, and late assemblies, and will order them in the figures accordingly.

When checking the propensity of changes within and across functional assemblies, we indeed found more synapses undergoing long-term plasticity (Figure 5F). While only 5% of synapses depressed in the whole dataset, we found up to 13.8% when restricting the analysis to assemblies. Similarly, compared to 2% of all synapses potentiating, we observed up to 5.2% when restricting to functional assemblies. Interestingly, large values were found in the off-diagonal entries (Figure 5F), i.e., synapses across assemblies underwent more plastic changes than the synapses within these assemblies. In our model, the initial ${\displaystyle \rho }$ values are pathway-specific and highest in L4 pathways (Brémaud et al., 2007, Figure 1—figure supplement 1C3). Therefore, early assemblies, with large numbers of L4 cells have a higher than average initial ${\displaystyle \rho }$ (Figure 5C and E, respectively), thus their synapses are more likely to depress (Figure 5F left). As early assemblies are stimulus-specific, and thus not part of the same assembly sequences, synaptic depression between these cells can be seen as some kind of stimulus separation. On the other hand, late assemblies, that are predominantly composed of cells from the deep layers, have a low initial ${\displaystyle \rho }$ (Figure 5E; Figure 1—figure supplement 1C3) and synapses towards them are more likely to potentiate (Figure 5F right). These functional assemblies are mostly non-specific and participate in all assembly sequences, thus the potentiation of their efferents means a general strengthening of the stimulus response as a whole.

We have shown above that connections between pairs of neurons with high firing rates are more likely to undergo plastic changes (Figure 3A2). Thus, we were wondering if firing rate alone explains the increased propensity of changes we observed when restricting our analysis to assembly synapses. To this end, for each assembly, we randomly sampled the same number of neurons with the same firing rate distribution from the whole circuit (Figure 5G1) and studied the changes of their synapses. We observed fewer synapses undergoing plastic changes and also a loss of specificity in these rate-based controls (Figure 5G2, Figure 5—figure supplement 1A). In conclusion, co-firing in a Hebbian cell assembly is a better predictor of plastic changes than just firing at a high rate.

Together, these results indicate that, in line with 70-y-old predictions, cells that fire together wire together (Hebb, 1949). Our contribution lies in making the qualitative statement above into a quantitative one: Under in vivo-like conditions, cells that fire together more than expected have three times higher chances of changing the efficacy of their synapses.

Synapse clustering interacts with assembly formation and plastic changes in complex ways

In addition to co-firing, a group of presynaptic neurons is more effective in depolarizing a given dendritic branch if they all innervate the same branch, i.e., they form a spatial synapse cluster (Kastellakis and Poirazi, 2019; Farinella et al., 2014; Iacaruso et al., 2017; Tazerart et al., 2020; Kirchner and Gjorgjieva, 2021; Kirchner and Gjorgjieva, 2022). To quantify this, we selected the 10 most innervated L5 TTPCs (thick-tufted pyramidal cells) within a cell assembly and then detected spatial clusters of synapses, defined as at least 10 synapses within a 20 μm stretch of a single dendritic branch (Methods; ∼15k clustered synapses per assembly on average). Next, we grouped all synapses on these 10 selected neurons per assembly into four categories based on assembly membership of the presynaptic neuron and whether the synapse was part of a cluster or not (see exemplary clustered assembly synapses in Figure 6A). While some of the ${\displaystyle \rho }$ values of clustered synapses within an assembly, underwent small constant changes, many of them changed at the same time (vertical stripes on Figure 6B); indicating the importance of spatial co-localization for plasticity.

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To systematically quantify this effect, we calculated the likelihood of plastic change in each category: (not) in an assembly and (non-) clustered, by contrasting the conditional probability of observing it in a given category with the probability of observing any change irrespective of the category (see Equation 15 in Methods; Figure 6C). Surprisingly, clustered synapses within an assembly were only likely to undergo plastic changes in the larger, but less general late assembly, A11, but not in early and middle assemblies (Figure 6C black arrows top). In fact, the only category that was likely to change across all assemblies is that of non-clustered within-assembly synapses. Therefore, we concluded that synapses within the other three categories tend to keep their initial arrangement of ${\displaystyle \rho }$ values. To study the initial arrangements as well, we repeated the same analysis on the probability of the initial value of ${\displaystyle \rho }$ to be either 0 or 1. Initial ${\displaystyle \rho }$ values in early and middle assembly synapses, especially the clustered ones, were very likely to be initialized as fully potentiated, while synapses within the late assemblies were likely to be initialized in the fully depressed state (Figure 6D black arrows bottom). However, when comparing not the likelihood, but the amplitude of changes across categories with a two-way ANOVA, we found that clustered, within-assembly synapses depress to a significantly smaller degree than all other categories (Figure 6—figure supplement 1A). In contrast, non-clustered, within-assembly synapses (the ones that are most likely to change) depress the most. On the other hand, non-assembly synapses usually tended to be initialized depressed and to stay that way. Thus, the picture emerging is as follows: early and middle assemblies are partially defined by clustered (both spatial and functional) synapses that are initialized as fully potentiated. These synapses are unlikely to change, but when they do, they depress less than the others, and would converge back to ${\displaystyle \rho =1}$ in absence of activity, as they do not cross the ${\displaystyle {\rho }_{\ast }=0.5}$ unstable fixed-point. These stable early assemblies can therefore function as a stable backbone amid ongoing synaptic plasticity.

In our previous investigation, we found that most changes happened across assemblies (Figure 5F), so we extended the analysis described above to cross-assembly synapses (Figure 6—figure supplement 1B). Here, the picture was reversed: cross-assembly synapses that were part of a spatial cluster were likely to be initiated as fully depressed and then had a high chance of undergoing potentiation (Figure 6—figure supplement 1C black arrow). Together with the previous results, this suggests that synapses between assemblies are more likely to change, which is even more pronounced if these synapses form a cluster on the postsynaptic dendrite.

Network-level metrics predict plasticity and separate potentiation from depression

So far, we have found co-firing and synapse clustering to be good predictors of plastic changes, in line with previous work (Zenke et al., 2015; Kastellakis and Poirazi, 2019; Harris, 2005). Additionally, we have shown that changes are also driven by the underlying network’s connectivity (Figure 4A and B). We now unfold the effect of network connectivity on plastic changes further, by studying the effect of the presynaptic population of the source (presynaptic) and target (postsynaptic) neurons of a connection on the ${\displaystyle \rho }$ dynamics. Note first that the indegree of a neuron, i.e., the size of its source population, is weakly correlated to its average firing rate (Figure 7A; ${\displaystyle r=0.13}$). Furthermore, the indegree of the pre- and more strongly of the postsynaptic neurons of a connection are predictive of plastic changes (Figure 7B1). However, just like for assemblies, a purely rate-based control highlighted that the picture is more complex, firing rate alone is not the best predictor of plastic changes (Figure 5—figure supplement 1B).

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Moreover, the correlation of the pre- and postsynaptic indegree (${\displaystyle r=0.18}$) complicates the (Figure 7B2). Thus, we took the natural choice of considering common innervation (see below), instead of making a choice of how to combine independent innervation of both neurons mediating the connection. Additionally, it has been shown that not only the size of the presynaptic population but also its internal connectivity affects co-firing and assembly formation (Ecker et al., 2024). We explore these effects using ${\displaystyle k}$-edge indegree, a novel yet simple network metric of edge centrality that considers how the edges (connections) are embedded in the entire network. More precisely, the ${\displaystyle k}$-edge indegree of a connection is the number of ${\displaystyle k}$-simplices that the connection is innervated by, where we call ${\displaystyle k}$ the dimension (Figure 7C1, see Methods). In particular, for ${\displaystyle k=0}$ it is the number of neurons innervating both the pre- and post-synaptic neurons of the connection, i.e., a common neighbor as in Perin et al., 2011. Indeed, for our circuit, in the lower dimensions, ${\displaystyle k}$-edge indegree is correlated to the indegree of its source and target populations (Figure 7—figure supplement 1A; ${\displaystyle r=0.48,0.35}$ for ${\displaystyle k=0}$ and ${\displaystyle r=0.35,0.27}$ for ${\displaystyle k=1}$). We expected these motifs to have an effect on plasticity for two main reasons. First, pairs of connected neurons have higher spike correlations if the connection has higher ${\displaystyle k}$-edge indegree (Reimann et al., 2017), and the strength of this correlation depends on the overall higher-order structure of the network (Nolte et al., 2020). Second, simplex motifs innervating a cell have been shown to be a good predictor of assembly membership (Ecker et al., 2024), which in turn have an effect on plasticity as demonstrated above.

Across dimensions, as k-edge indegree increased so did the probability of undergoing plastic change (Figure 7C2). Furthermore, among the connections that change, the ones with the highest k-edge indegree tend to get potentiated while those with the lowest favor depression (Figure 7C3). Thus, we hypothesized k-edge indegree to be a good predictor of synaptic strength. We verified this both in our model as well as in the MICrONS, 2021 mm³ dataset, an electron microscopic reconstruction of cortical tissue that combines synaptic resolution with the scale needed to calculate meaningful k-edge indegree. Both networks had similar maximal ddimensionsof simplices as well as ranges of k-edge indegree values (Figure 7—figure supplement 1B), making the results of this analysis comparable. For our model ,we used the sum of ${\displaystyle {\hat{g}}_{AMPA}}$ values as a proxy for connection strength. For MICrONS, we used total synapse volume of a connection (see Methods), as a measure of connection strength (Harris and Stevens, 1989). Indeed, high k-edge indegree led to higher connection strength in MICrONS, 2021 as predicted by our modeling insights; moreover, the curves across dimensions are qualitatively the same for both datasets (Figure 7D).

To gain insights into pattern-specificity, we analyzed the 2-min-long single pattern simulations and considered k-edge indegree within the assembly-specific subgraphs (Figure 7E1-Sub). When doing so, we found an even higher predictive power, which was additionally pattern-specific (Figure 7E2). More precisely, the probability of changes only increased with k-edge indegree in the assembly sub-graph associated with the presented pattern or to a late assembly and that it was the strongest for the former (see Figure 7E2 left for pattern A for an example, right for a summary of the maxima attained for all patterns, and Figure 4—figure supplement 1B for consistency across repetitions of the same simulation).

Finally, pattern-indegree, i.e., the number of VPM fibers belonging to a pattern that innervate a neuron, is also a good predictor of the probability of a connection to change i.e., an increase of either the pre- or postsynaptic pattern-indegree of a connection leads to more frequent plastic changes (Figure 7—figure supplement 1C11). However, given the highly non-random structure of the full network, the values of pre- and postsynaptic pattern-indegree are not independent and thus their effect in driving plastic changes is hard to disentangle (Figure 7—figure supplement 1C2 and D; ${\displaystyle r=\mathrm{0.230.}}$ on average).

In summary, k-edge indegree in the full network allowed us to distinguish potentiation from depression given change, as well as predict connection strength, which we confirmed in the (MICrONS, 2021) dataset. Moreover, k-edge indegree within assembly subgraphs is a pattern-specific predictor of plastic changes.

Increased stimulus-specificity and more reliable assembly sequences characterize the network after plasticity

As the changes are sparse, one might wonder if they cause any functional change. To study this, we compared the network’s activity before and after plasticity (see Methods). Firing rates decreased slightly as a result of plasticity (Figure 8A left), while spike correlations remained stable, in line with recent findings (Oby et al., 2019; Feulner et al., 2022, Figure 8A right). Both of these effects were more pronounced in the deeper layers (Figure 8—figure supplement 1A). Correlations of connected pairs of neurons decreased slightly after plasticity following the decrease in firing rate (Figure 8—figure supplement 1A). However, the highest correlations both before and after plasticity were found in the most central edges in the network i.e, those with highest edge participation in high-dimensional simplices (Figure 8B1). We further verified this prediction in the MICrONS dataset (Figure 8B2, see Methods). It has been previously shown that assemblies have many edges that are central in the network, therefore, we hypothesized that they remain functional and reliably active (Fauth and van Rossum, 2019; Kossio et al., 2021; Pérez-Ortega et al., 2021). We showed this on the level of assembly sequences: plastic changes promote stimulus-specific and reliable responses. While we found considerable overlap between assemblies detected before and after plasticity, an early assembly responding to patterns B and E before plasticity, split into two stimulus-specific assemblies (Figure 8C). Note that this is not an artifact of the thresholding applied during assembly detection as forcing the pipeline to yield the same number of assemblies instead splits a middle assembly first (Figure 8—figure supplement 1E). Moreover, the assembly sequences responding to repetitions of a given pattern generally became more reliable after plasticity (Figure 8—figure supplement 1B). To quantify this, we measured the Hamming similarity of assembly sequences between repetitions and found significant increases for most patterns (see Methods, Figure 8D). Interestingly, this increase on the assembly level lied in contrast with a slight decrease in the reliability of the neuronal activity on the single cell level (Figure 8E).

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The plastic changes we observed at the network level are in contrast with in vitro observations at the connection level. In particular, with the redistribution of synaptic efficacy towards earlier spikes during high-frequency firing caused by changes in ${\displaystyle U_{SE}}$ (Chindemi et al., 2022; Markram and Tsodyks, 1996; Costa et al., 2015; Selig et al., 1999; Sjöström et al., 2003), which under our in vivo-like, low firing rates and ${\displaystyle [C{a}^{2+}{]}_o}$ is not in the ranges where this connection-level redistribution is relevant. Specifically, at in vitro levels of ${\displaystyle [C{a}^{2+}{]}_o}$ the potentiated ${\displaystyle U_{SE}}$ shifts the connection further into the depressing regime, causing the redistribution (Figure 8F left). In vivo the lower ${\displaystyle [C{a}^{2+}{]}_o}$ counters this by moving the connection into the pseudo-linear regime; additionally, the low firing rates make this phenomenon less relevant in general (Figure 8F right and G, respectively). Thus, while Markram and Tsodyks, 1996 showed a redistribution of synaptic efficacy after plasticity at the single connection level in vitro (Figure 8F), we found a redistribution at the network level under in vivo-like conditions (Figure 8A-D, Figure 8—figure supplement 1A).

In summary, we observed a network-level redistribution of synaptic efficacy that maintained spike correlations globally, while enhancing stimulus-specificity and the reliability of the subplopulations in terms of assembly activation. Furthermore, we derived and confirmed in the (MICrONS, 2021) dataset a prediction stating that the most central connections of the network drive the correlated activity.

In this work, we aimed to understand how network structure, function, and dendritic processing interact to drive synaptic plasticity at the population level in the neocortex. Specifically, we used a plasticity model, whose parameters were determined by pairwise STDP experiments but modeled at the level of dendrites. We studied its effect at the population level under in vivo-like conditions to determine network-level rules that best predict plastic changes by simulating it in a biophysically detailed model of a cortical microcircuit that was subjected to 10 different stimuli under a variety of protocols. Our principal observations are as follows: (1) Plastic changes were sparse, affecting only 7% of the synapses. A balance between occasional, large-amplitude potentiation and more frequent depression stabilized the network without explicitly modeling homeostatic plasticity. Moreover, the changes were non-random and stimulus-specific. (2) Plastic changes were largely determined by the anatomical structure of synaptic connectivity and its relation to functional units, i.e., changes were most likely between co-firing cell assemblies, at clustered synapses, between neurons highly innervated by the same input, and in central edges in the network. (3) While assemblies remained fairly stable, they became more stimulus-specific and reliable which has important implications for coding and learning.

The first observation (1) is quite significant considering that we did not design the learning rule to be sparse and stable. In previous models of plastic networks, changes were not sparse and additional mechanisms were needed to keep the network’s activity stable (Litwin-Kumar and Doiron, 2014; Delattre et al., 2015; Zenke et al., 2015; Fauth and van Rossum, 2019; Kossio et al., 2021; Zenke et al., 2017a; Turrigiano and Nelson, 2004). The machine learning community is also aware of the importance of sparse changes, as in continual learning one has to balance plasticity and stability of existing weights to avoid catastrophic forgetting (McCloskey and Cohen, 1989; Ratcliff, 1990). In recent years, they have come up with impressive techniques that mask or slow down weight updates (e.g. by a regularizer on the loss) to improve the performance of deep neural networks (Zenke et al., 2017b; Kirkpatrick et al., 2017; Mallya and Lazebnik, 2018; Frankle and Carbin, 2019), whereas in our model it emerged without any mechanisms explicitly enforcing it. Note, however, that despite the comparison to machine learning, our model is not designed to solve any specific task. Instead, it is a model of sensory cortex which only learns to represent stimulus features in an unsupervised manner. Nonetheless, demonstrating the stimulus-specificity of changes as well as an increase in the reliability of assembly sequences, was a crucial validation that our results are not just a description of a biophysical process, i.e., plasticity, but have implications for learning in general.

For our second observation (2), we described three different kinds of plasticity rules that emerge from the underlying plasticity model based on: activity, subcellular structure, and network structure. While all of these are powerful predictors of plastic changes, none of them fully determines them. It is rather the interplay between them (and potentially additional unknown rules) that brings about non-homogeneous changes across the full microcircuit. The first two can be explained from the biophysics of the plasticity model and links our results to the classical work of Hebb, 1949 as well as the recent literature on synapse clustering (Kastellakis and Poirazi, 2019; Farinella et al., 2014; Iacaruso et al., 2017; Tazerart et al., 2020). With respect to synapse clustering, we would highlight that our synapses are stochastic and the release probability between PCs is ∼ 0.04 at the simulated low ${\displaystyle [C{a}^{2+}{]}_o=1.05}$ mM (Markram et al., 2015; Ecker et al., 2020; Jones and Keep, 1988; Borst, 2010). Therefore, care should be taken when comparing our results with either glutamate uncaging experiments, which bypass the presynaptic machinery (Pettit et al., 1997; Losonczy and Magee, 2006), or with other modeling studies that use deterministic synapses (Farinella et al., 2014; Poirazi et al., 2003; Ujfalussy and Makara, 2020). With respect to network-based rules, previous simulation approaches have characterized the learned connectomes as having an overexpression of reciprocal connectivity and densely connected neuron motifs (Brunel, 2016; Zhang et al., 2019). We expanded on this body of previous work, by introducing cell and connection-specific metrics that directly yield the probability of observing depression or potentiation of any given connection. Most of our predictors behaved similarly with respect to depression and potentiation. Only a metric based on the embedding of individual edges in the global (or assembly-specific) network was able to separate them. This edge-based prediction was confirmed by analyzing an electron microscopic reconstruction of cortical tissue in a comparable way (MICrONS, 2021).

Our third observation (3) is that assembly sequence becomes more reliable and stimulus-specific may sound counter-intuitive considering that we observed reduced firing rates and less reliable single neurons. This shows that the plasticity mechanism can contribute to the emergence of a robust population code with unreliable neurons. Crucially, this means that plasticity can make circuit responses more distinct (between stimuli by increasing specificity) but also less distinct (for a given stimulus by increasing reliability). This has important computational implications that may be explored in a future study.

To facilitate further research, we are open-sourcing our model alongside detailed instructions to launch simulations and analyze the results (see Data and code availability). Simulating the model requires a performant hardware and software infrastructure. With respect to the latter, we are continuously improving the performance and efficiency of the simulator (Kumbhar et al., 2019). The model has several assumptions and limitations, which will ideally be improved upon iteratively and in a community-driven manner. First, by using the circuit model of Isbister et al., 2023, we inherit all assumptions listed therein; one of the most important one being on source data. A common limitation of data-driven modeling is the lack of high-quality source data, thus while we refer to our model as juvenile rat non-barrel somatosensory cortex, sometimes we used data from the barrel cortex, from adult animals and from mouse. Second, for our simulations we used the model of Chindemi et al., 2022 with parameters obtained in a previous version of the circuit. Due to various improvements of synapse parameters (e.g. ${\displaystyle U_{SE}}$ of L5 PCs) since fitting plasticity-specific parameters, the results for pairing protocols at various frequencies shift slightly while maintaining the presence of STDP (Figure 1—figure supplement 2A). When exploring slightly different parameters of the plasticity model, we found that while the exact numerical values changed, our main conclusions about nonrandom changes and various features predicting them were not affected (Figure 8—figure supplement 2.) Third, the extracellular magnesium concentration of 1 mM used in in vitro preparations was assumed to be representative of the in vivo level. Fourth, experimental evidence indicates that firing bursts affects plasticity (Letzkus et al., 2006; Williams and Stuart, 1999; Larkum, 2013). However, L5 TTPC bursts were rare in our simulations, as the model is based on an early developmental stage (P14-16: juvenile rats) and firing bursts only becomes prominent as the animals mature (Zhu, 2000). Furthermore, bursts can be triggered by synaptic input onto apical dendrites (Larkum, 2013; Hay et al., 2011). Inputs to apical dendrites from high-order thalamus have been shown to gate plasticity in L23 PCs in vivo, via disinhibiting the distal dendrites (Gambino et al., 2014; Williams and Holtmaat, 2019). While POm input was present in our model and targeted apical dendrites, we activated them in random and non-specific ways. On the other hand, top-down inputs represent context or brain state and are thought to serve as an error or target signal for learning and thus are likely very specific (Makino, 2019). Feedback signals from high-order cortical areas also innervate apical dendrites (Harris et al., 2019), but as our model is of a single cortical area only, these inputs were absent. Missing excitatory input from other areas were compensated by somatic conductance injections (Isbister et al., 2023), which could be replaced in the future by simulating additional dendritic synapses instead. Finally, learning in the brain is orchestrated by several mechanisms in unison and some of these are beyond the scope of this paper, e.g., inhibitory plasticity, metabotropic glutamate receptors, neuromodulation, slow homeostatic plasticity, structural plasticity, and calcium-induced calcium release (Magee and Grienberger, 2020).

Bottom-up modeling is often criticized for being arbitrarily complex. In our simulations, the mean value of synaptic efficacies of an over-excited network reduced, comparable to Turrigiano and Nelson, 2004 without introducing rate homeostasis explicitly. In the machine learning field, mechanisms are added to classical algorithms to make updates sparser, overcome catastrophic forgetting, and allow the learning of tasks sequentially (Kirkpatrick et al., 2017; Zenke et al., 2017b). The calcium-based plasticity rule employed here results in sparse updates without additional mechanisms, unlike STDP rules, which update efficacies at each spike arrival. Thus, we showed that adding biological complexity allowed us to reduce the number of mechanisms explicitly modeled. Calcium-based plasticity models existed for a long time (Shouval et al., 2002; Rubin et al., 2005), have been shown to be stable (Higgins et al., 2014; Graupner et al., 2016; Wang and Aljadeff, 2022), have been modified to mimic in vivo-like low ${\displaystyle [C{a}^{2+}{]}_o}$ (Graupner et al., 2016; Chindemi et al., 2022), and versions more complex than ours exist (Mäki-Marttunen et al., 2020; Rodrigues et al., 2023). Our contribution lies in integrating all these different aspects in network simulations at scale with subcellular resolution. Using such simulations allowed us to explore not only the temporal aspects of plasticity but also their link to spatial and network aspects, such as clusters of synapses and cell assemblies. Future work could explore the constellation of changing connections when stimuli change or cease, providing a more concrete examination of continuous learning paradigms within this model. Furthermore, the analysis methods developed on large-scale functional and structural data empowered us to make and test novel predictions in the (MICrONS, 2021) dataset, which while pushing the boundaries of big data neuroscience, was so far only analyzed with focus on either single cells or first-order network metrics (e.g. node degree) (Ding et al., 2024; Wang et al., 2023). We extend this work by considering the higher-order structure of the network.

The 2.4 mm³ subvolume of the juvenile rat somatosensory cortex, containing 211,712 neurons and 312,709,576 plastic synapses in SONATA format (Dai et al., 2020) is freely available at: https://doi.org/10.5281/zenodo.8158471. Exemplary Jupyter notebooks using the packages listed in the Key resources table above were deposited in the same repository on Zenodo.

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Author details

1. András Ecker

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Conceptualization, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing – original draft, Writing – review and editing

   Contributed equally with

   Daniela Egas Santander

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-9635-4169

2. Daniela Egas Santander

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Conceptualization, Software, Formal analysis, Validation, Methodology, Writing – original draft, Writing – review and editing

   Contributed equally with

   András Ecker

   Competing interests

   No competing interests declared

3. Marwan Abdellah

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Visualization

   Competing interests

   No competing interests declared

4. Jorge Blanco Alonso

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Resources

   Competing interests

   No competing interests declared

5. Sirio Bolaños-Puchet

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Software, Validation, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-4049-6488

6. Giuseppe Chindemi

   1. Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland
   2. Department of Basic Neurosciences, University of Geneva, Geneva, Switzerland

   Contribution

   Conceptualization, Investigation, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

7. Dhuruva Priyan Gowri Mariyappan

   1. Department of Neurosciences, Faculty of Medicine, Université de Montréal, Montreal, Canada
   2. Centre de Recherche Azrieli du CHU Sainte-Justine, Montréal, Canada
   3. Mila Quebec AI Institute, Montréal, Canada

   Contribution

   Validation

   Competing interests

   No competing interests declared

8. James B Isbister

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Validation, Methodology, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1013-3013

9. James King

   Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

   Contribution

   Software

   Competing interests

   No competing interests declared

10. Pramod Kumbhar

    Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

    Contribution

    Software

    Competing interests

    No competing interests declared

11. Ioannis Magkanaris

    Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

    Contribution

    Resources

    Competing interests

    No competing interests declared

12. Eilif B Muller

    1. Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland
    2. Department of Neurosciences, Faculty of Medicine, Université de Montréal, Montreal, Canada
    3. Centre de Recherche Azrieli du CHU Sainte-Justine, Montréal, Canada
    4. Mila Quebec AI Institute, Montréal, Canada

    Contribution

    Conceptualization, Methodology, Writing – review and editing

    Competing interests

    Reviewing editor, eLife

    "This ORCID iD identifies the author of this article:" 0000-0003-4309-8266

13. Michael W Reimann

    Blue Brain Project, École polytechnique fédérale de Lausanne (EPFL), Campus Biotec, Geneva, Switzerland

    Contribution

    Conceptualization, Software, Formal analysis, Supervision, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

    For correspondence

    mwr@reimann.science

    Competing interests

    No competing interests declared

    "This ORCID iD identifies the author of this article:" 0000-0003-3455-2367

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