1 Introduction

In cortical circuits, GABA-ergic interneurons constitute approximately 10% to 20% of the total neural population, and can be further divided into subtypes that exhibit distinct characteristics in connectivity, plasticity and tuning properties (Jiang et al. 2015; Tremblay et al. 2016). Fully understanding how these diverse interneuron subtypes influence neural circuit function remains a significant challenge. A key open question is how activity-dependent plasticity of inhibitory synapses in recurrent circuits sculpts this diversity into specific connectivity motifs while maintaining stable network activity.

The dynamic balance between excitation and inhibition is crucial for generating the irregular, Poisson-like firing patterns commonly observed in vivo in cortical circuits (Van Vreeswijk and Sompolinsky 1998; Brunel 2000). This balance enhances the circuit’s sensitivity to input, facilitating rapid information processing (Tsodyks and Sejnowski 1995). Furthermore, it supports essential computations, such as receptive field emergence (Wallenstein and Hasselmo 1997; Froemke et al. 2007; Dorrn et al. 2010; Kleberg et al. 2014; D’amour and Froemke 2015; Rubin et al. 2017), neural assembly formation (Sadeh and Clopath 2021), and the storage of precise temporal firing sequences (Maes et al. 2021; Riquelme et al. 2023; Gong and Brunel 2024; Shao et al. 2024).

Excitatory-inhibitory (E/I) balance can be achieved without any specific inhibitory connectivity structures. Often circuit models consider inhibition simply as a “blanket” that regulates excitation, and incorporate it into a global normalization term or a correction to the synaptic weights between excitatory neurons (Jin 2002; Vogels et al. 2011; Ravid Tannenbaum and Burak 2016; Montangie et al. 2020). While this simplified approach captures fundamental properties of neural dynamics, it overlooks the possible advantages of more specific E/I connectivity patterns. Incorporating structured inhibitory connections can further refine computations and circuit properties such as E/I co-tuning (Clopath et al. 2016; Mackwood et al. 2021; Miehl and Gjorgjieva 2022; Eckmann et al. 2024), memory storage (Mongillo et al. 2018; Bergoin et al. 2023; Miehl et al. 2023; Agnes and Vogels 2024; Meissner-Bernard, Jenkins, et al. 2025; Meissner-Bernard, Zenke, et al. 2025), predictive coding (Schulz et al. 2021), and other aspects of neural dynamics (Mongillo et al. 2018; Dahmen et al. 2020; Sadeh and Clopath 2020; Hafizi et al. 2022).

Experimental findings demonstrate that inhibitory neurons form non-random connectivity patterns that cannot be explained solely by cell distance or neuron morphology. Additional factors, including developmental factors such as spontaneous activity, interneuron subtype, synaptic dynamics and targeting preference significantly shape this connectivity structure (Peng et al. 2021; Udvary et al. 2022; Bollmann et al. 2023; Reimann et al. 2023; Wu et al. 2023; Kuan et al. 2024; Lakhera et al. 2024; Znamenskiy et al. 2024; Harth et al. 2025). In the neonatal mouse cortex, even before the onset of external sensory inputs, inhibitory hub neurons regulate circuit development (Bollmann et al. 2023). Moreover, theoretical studies suggest that spatial organization with a widespread Mexican-hat-like effective connectivity, consisting of local excitation and longer-range inhibition, can give rise to modular activity patterns with long-range spatial correlations propagating throughout the neonatal ferret cortex (Smith et al. 2018). Such correlated activity is thought to serve as a scaffold for the emergence of specific stimulus feature tuning in the early stages of development (Mulholland et al. 2021; Lakhera et al. 2024; Trägenap et al. 2025).

Despite these advances, the mechanisms and principles that enable the formation and maintenance of inhibitory connectivity patterns, especially in the context of excitatory connectivity, remain elusive. Genetic mechanisms play a fundamental role, particularly in early developmental stages (Duan et al. 2020; Hanganu-Opatz et al. 2021; Wu et al. 2023). Equally important is synaptic plasticity driven by neural activity, which refines inhibitory connectivity through a dynamic, activity-dependent process (Sreenivasan et al. 2022). This mechanism enables a closed-loop regulation of neural dynamics, allowing for precise tuning and adaptability in the face of change.

While the majority of experimental and theoretical studies investigate excitatory plasticity (e.g. Kempter et al. 1999; Song et al. 2000; Morrison et al. 2007; Shulz and Feldman 2013, far fewer examine inhibitory plasticity. Although some theoretical work has considered the impact of different forms of iSTDP in feedforward circuit architectures with a single postsynaptic neuron (Kleberg et al. 2014; Luz and Shamir 2014), there is still no general understanding of how pairwise iSTDP might influence the formation of specific connectivity motifs at the level of recurrently connected circuits. Theoretical work has proposed an iSTDP rule that guarantees the homeostatic regulation of neuronal rates (Vogels et al. 2011), becoming the “canonical” choice for plastic circuits. However, this rule tends to over-regulate rates in a homeostatic manner, producing narrow rate distributions in the excitatory population, which contradicts experimental findings Buzsaki and Mizuseki 2014. In other work, inhibitory rules are specifically fit to experimental measurements (Lagzi et al. 2021; Bergoin et al. 2025), derived normatively from computation objectives (Mackwood et al. 2021), or are found through numerical optimization on a large parameter space (Confavreux et al. 2024). Here we ask how different spike timing-based rules at inhibitory synapses shape which excitatory and inhibitory neurons become strongly connected in recurrent circuits, and how the resulting excitatory-inhibitory motifs stabilize network firing rate, while supporting surround modulation and the generation of spatially correlated spontaneous activity.

To address these questions, rather than studying single rules, here we propose an integrated approach by analyzing a broad family of pairwise iSTDP rules and study their impact in small analytically tractable E/I circuits as well as in large-scale recurrent neuronal networks (RNNs) of spiking neurons. We identify mechanisms that stabilize excitatory activity while shaping circuit motifs, and show how, depending on the profile of the rule considered, inhibitory connectivity adjusts to the existing excitatory connectivity in distinct ways.

We first consider a recurrently connected two-neuron circuit of one excitatory and one inhibitory neuron, and derive how the choice of the iSTDP rule determines which features of neural activity drive the plasticity of inhibitory connections. At one extreme, rules dominated by firing rates are weakly influenced by existing connectivity structure in the two-neuron circuit, aiming primarily at the homeostatic regulation of firing rates, and thus promoting “blanket” inhibition. At the opposite extreme, we find covariance-dominated rules, strongly influenced by pairwise spike correlations that, in the absence of external inputs, arise from the underlying circuit connectivity. These rules stabilize neural activity, while promoting specific connectivity motifs, such as a mutually connected E/I pair or lateral inhibition, depending on the iSTDP features. We call the emergence of these specific connectivity motifs “structured stabilization”. This structured stabilization also generalizes to large-scale recurrently connected circuits where the same type of E/I connectivity motifs were observed under the different iSTDP rules. To further explore the functional consequences of covariance-dominated rules, we implement an RNN whose excitatory neurons follow a one-dimensional ring topology, and whose inhibitory neurons follow two distinct iSTDP rules. Here, inhibitory connectivity self-organizes into an effective Mexican-hat profile, generating long-range modular correlations of spontaneous activity. Finally, we demonstrate that the circuit implements contextual modulation, including suppression of responses to matching stimuli and amplification for mismatched stimuli. In sum, our study investigates how distinct inhibitory plasticity rules shape E/I connectivity motifs, from mutual to lateral inhibition, giving rise to network response properties such as surround modulation and long-range activity correlations observed in the adult and developing visual cortex.

2 Results

2.1 Inhibitory plasticity shapes specific E/I circuit motifs

We initially considered a general iSTDP rule that depends exclusively on local spike times applied to a synapse from an inhibitory presynaptic to an excitatory postsynaptic neuron (inh-to-exc) (Kempter et al. 1999; Luz and Shamir 2014; Akil et al. 2021). This rule contains two fixed components triggered by each pre- and post-synaptic firing events, denoted as α_pre and α_post, and a kernel function, L(τ) triggered by the co-occurrence of pre-post (τ > 0) or post-pre (τ < 0) spiking events (see Methods, Section 4.1). When the spiking activity is modeled as a stochastic process, the mean-field equation for the evolution of the average synaptic weight can be decomposed as follows (Kempter et al. 1999):

where r_pre and r_post are the pre- and post-synaptic firing rates, B is the net area of the iSTDP kernel L(τ), and Q captures the effect of the iSTDP kernel on the pre-post pairwise interactions, conveyed by the covariance density curve C(τ) (see Supporting Information, Section S1.1, or Kempter et al. 1999; Morrison et al. 2008; Luz and Shamir 2014; Lagzi and Fairhall 2024). To understand how inhibitory plasticity influences circuit connectivity patterns, we first developed an analytical framework, in which rather than choosing a single rule, we defined a family of iSTDP rules. Using these rules, we aimed to derive the closed-form solution for a two-neuron recurrent E/I circuit, and link the shape of iSTDP to the pattern of learned inhibitory connections.

Several terms in Eq. 1 depend solely on pre- and post-synaptic firing rates. Furthermore, in the general category of rules where B is high, the contribution of the B r_pre r_post term can easily be an order of magnitude above Q, even in the presence of strong pre-post correlations (see Methods, Section 4.3). Therefore, when Q ≪ B r_pre r_post, the plasticity dynamics of Eq. 1 are strongly dominated by the firing rates. We call this plasticity regime “rate- dominated”. A prominent example of this is the previously established and well-studied rate-homeostatic iSTDP rule (Vogels et al. 2011) With α_post = 0 and kernel integral B > 0, this rule drives the inhibitory weight to adapt continuously so that the postsynaptic rate converges to the target value r_post, corresponding to a stable fixed point of Eq. 1, namely: r_post ≈ −α_pre/B.

While a large value of B simplifies the analysis by making covariance-based terms negligible, it only represents a subset of possible plasticity rules. We therefore explored a broader range, including those for which the iSTDP kernel has zero or near-zero net area (B ≈ 0). These rules amplify the contribution of the second-order correlations between pre- and postsynaptic neuron in the Q term. We refer to them as “covariance-dominated”. In choosing specific pairwise interaction kernels, we considered recent experimental findings that reported a symmetric iSTDP rule for parvalbumin-expressing (PV) interneurons, as well as an antisymmetric iSTDP rule for somatostatin-expressing (SST) interneurons, both projecting onto excitatory pyramidal neurons in the orbitofrontal cortex (Lagzi et al. 2021). Consequently, we focused on symmetric kernels with strong short-range potentiation and weak long-range depression (Fig. 1C), or antisymmetric kernels in which pre-post and post-pre interactions balance each other with opposite signs (Fig. 1D).

Figure 1.

Analyzing the full dynamics of Eq. 1 in a recurrently connected circuit is challenging because the covariance C(τ) generally depends on the synaptic weights themselves. We therefore considered a minimal two-neuron E/I recurrent circuit (Fig. 1A) with fixed exc-to-inh connection (w_exc) but plastic inh-to-exc weight (w_inh). We modeled the neurons as time-varying Poisson processes, where each incoming spike changes the instantaneous spiking probability through an excitatory or inhibitory post-synaptic potential (EPSP/IPSP) kernel (internal traces shown in Fig. S1, see also Methods Section 4.4). The inhibitory neuron’s firing rate is kept constant by an external homeostatic mechanism. The excitatory neuron receives an excitatory fixed external input in addition to the inhibitory input. Using this simplified model, we derived analytically a closed-form expression for the fixed-point of the iSTDP dynamics (Methods, Eq. 11; Supporting Information). The condition of fixed r_inh is not required for a stable solution. Allowing r_inh to vary still leads to an analytic expression for the fixed point of the iSTDP dynamics, but the resulting closed-form involves the root of a quadratic equation and is therefore less intuitive (see Supporting Information, S1.3, Fig. S3).

With this analytical framework in hand, we investigated how different iSTDP rules affect the resulting inhibitory synaptic strength and the excitatory postsynaptic activity. We considered two configurations of our E/I circuit: (1) a unidirectional (U) configuration, where the excitatory-to-inhibitory weight w_exc is set to zero, and (2) a mutual (M) configuration, featuring a strong excitatory-to-inhibitory connection. We studied the three distinct iSTDP rules in both configurations: a rate-dominated rule with a symmetric interaction kernel (Fig. 1B), a covariance-dominated rule also with a symmetric kernel (Fig. 1C), and a covariance-dominated rule with an antisymmetric kernel (Fig. 1D). As expected, the rate-dominated rule shows minimal sensitivity to the presence of the mutual connection. In both U and M configurations, the inhibitory synapse adjusts the excitatory firing rate toward the target rate dictated by the rate-dependent components, approximately given by (Fig. 1E). Minor deviations arise from the non-zero contribution of the Q term.

In contrast, the covariance-dominated rules are sensitive to both the circuit’s connectivity structure and the shape of iSTDP kernel. With the symmetric kernel, the final inhibitory weight becomes significantly stronger in the M configuration compared to the U configuration (Fig. 1F, compare brown and light orange). Conversely, with the antisymmetric kernel, the inhibitory weight is lower in the M configuration (Fig. 1G). Such specificity enables these rules not only to regulate excitatory rates, but also to shape specific E/I connectivity motifs—an effect we termed “structured stabilization”. This property emerges from the well-known interaction between the iSTDP kernel and second-order spiking statistics. While this has been previously studied for excitatory plasticity (Kempter et al. 1999; Song et al. 2000; Babadi and Abbott 2013; Shulz and Feldman 2013), here we extend it to inhibitory plasticity in a minimally recurrent two-neuron E/I circuit and hence consider how recurrent inhibition regulates excitation. Specifically, the strong excitatory-to-inhibitory connection in the M configuration generates positively-correlated post-pre events. For the symmetric iSTDP, these events occur within the positive region of the kernel, leading to synaptic potentiation. Conversely, with the antisymmetric iSTDP rule, the same positively-correlated events interact with the negative region of the kernel, resulting in synaptic depression. Additionally, the inhibitory-to-excitatory weight generates a negative covariance density, localized instead in the pre-post region of the curve. As both rules are positive here, this introduces a negative feedback in both cases, limiting the value that w_inh can reach before plasticity switches into depression. Finally, adjusting the fixed components of the iSTDP rule, α_pre and α_post, is also crucial for reaching the desired excitatory firing rate (set to 25Hz) in the preferred circuit configuration.

Hence, our theoretical framework provides a link between different iSTDP rules and inhibitory connectivity in a reduced two-neuron E/I recurrent circuit motif.

2.2 The shape of inhibitory STDP affects inhibitory connectivity

Having established that different covariance-dominated rules can achieve structured stabilization, namely stabilize excitatory firing rates and shape specific E/I connectivity motifs, we next investigated how the specific form of the iSTDP kernel shapes the learned inhibitory connectivity. We considered four kernel shapes: symmetric, antisymmetric, and their “inverted” counterparts, obtained by reversing the sign of the kernel (Fig. 2A). We analyzed the behavior of these kernels in the reduced two-neuron E/I circuit. To explore the parameter space, we varied the rate-dependent parameters B and α_pre, while keeping α_post = 0. With this parametrization, the covariance-dominated regime corresponds to values of B and α_pre close to zero.

Figure 2.

Our analysis found that both symmetric and antisymmetric kernels yield stable fixed-points in the covariance-dominated regime (Fig. 2A,B, top row, and Supporting Information Fig. S2C,D). These kernels, which we term “self-stabilizing” exhibit robustness to parameter fluctuations and reliably converge to a stable inhibitory weight. In contrast, the inverted antisymmetric and the inverted symmetric rule are unstable in the covariance-dominated regime, with inhibitory weights either decaying to zero or saturating depending on the initial conditions. We categorize these as “saturating” kernels (Fig. 2A, bottom panels, and Fig. S2E,F in Supporting Information). Saturating kernels can still reach stability with strong rate-dependent terms, which reduce the overall influence of the kernel shape on the final weight.

This difference in stability arises from how the kernels interact with changes in pre- and postsynaptic correlations. Consider a synapse operating under a covariance-dominated plasticity rule at its fixed point. At equilibrium, potentiation and depression events balance exactly. A perturbation that strengthens the inhibitory weight leads to a reduction in pre-post firing events due to the increased inhibition. For self-stabilizing kernels, which have a positive right branch (corresponding to pre-post events), this reduction in correlation reduces the occurrence of synaptic potentiation events, shifting the balance in favor of synaptic depression. The synapse therefore depresses, eventually compensating the initial perturbation from equilibrium. Conversely, saturating kernels show the opposite effect: since their kernel is negative in the pre-post region, a reduction of pre-post correlation results in fewer synaptic depression events, so the initial balance is further shifted in favor of synaptic potentiation, amplifying the perturbation and leading the synapse towards saturation. Finally, in rate-dominated rules, the specific shape of iSTDP kernel becomes less relevant. Weight fluctuations impact the dynamics through the term rather than the correlation-dependent Q term. Consequently, these rules consistently converge to the classic rate-homeostatic solution (Vogels et al. 2011). In large networks, however, once these rules reach their homeostatic rate target, they can still promote additional structural organization at slower timescales (Lagzi and Fairhall 2024).

To further characterize the self-stabilizing rules, we extended our analysis beyond the U and M circuit configura-tions presented in Fig. 1, exploring a broader range of excitatory-to-inhibitory weights w_exc and input currents to the excitatory neuron. For the symmetric kernel, the learned inhibitory weight increases with w_exc (Fig. 2C), while for the antisymmetric kernel it decreases (Fig. 2D). Increasing the excitatory external input leads to stronger inhibitory weights in both cases (Fig. 2C,D), however the rate of the excitatory neuron still scales linearly with the input strength (Fig. 2F,G). In contrast, the rate-homeostatic rule exhibits a minimal dependence on w_exc (Fig. 2E) and a very tight control on the excitatory rate (Fig. 2H), which restricts the dynamic range of the excitatory neuron, limiting its ability to respond to changes in input. The E/I interactions in the two-neuron circuit allow for the weights to converge without saturation, differently from a two-neuron circuit with only excitatory neurons (Babadi and Abbott 2013).

In summary, our analysis of the reduced E/I circuit reveals a spectrum of iSTDP rules with distinct outcomes on circuit connectivity and dynamics. On the one hand, covariance-dominated rules with self-stabilizing kernels enable structured stabilization (stable activity and specific connectivity motifs). As a consequence, they selectively reinforce connectivity motifs based on pre- and post-synaptic correlations, and allow a broader range of excitatory firing rates. On the other hand, rate-dominated rules prioritize homeostatic control of excitatory activity over the formation of specific connectivity motifs. Thus, these rules ensure robust dynamical stability, potentially at the cost of computational flexibility. These results demonstrate a trade-off between the emergence of specific excitatory-inhibitory motifs and the tight regulation of excitatory rates in the class of iSTDP rules we considered.

2.3 Structured stabilization in randomly connected RNNs

How do our results on structured stabilization from two-neuron E/I circuits translate to large recurrent networks? We simulated a randomly connected RNN of conductance-based leaky integrate-and-fire (LIF) neurons, with 900 excitatory and 100 inhibitory units. We modeled the excitatory connections (exc-to-exc and exc-to-inh) as fixed and sparse, while the inh-to-exc connections were modeled as dense and initially small, but subject to iSTDP (Table 2). In this case the firing rate of each inhibitory neuron was not fixed as in the two-neuron E/I circuit, but was determined by the total amount of input into each neuron from its presynaptic partners, without any external constraint.

Table 1.

Table 2.

We first implemented a covariance-dominated plasticity rule with a symmetric kernel (Fig. 3A). Initially, the network produces a pathological high-firing regime, due to the low inhibitory weights. However, inhibitory plasticity quickly stabilizes the dynamics, generating an asynchronous irregular firing regime characteristic of balanced networks (Fig. 3B). To analyze the resulting inhibitory connectivity, we categorized inh-to-exc synaptic weights as either “mutual” or “unidirectional”. We called an inh-to-exc weight mutual when a reciprocal exc-to-inh connection exists between the same two neurons, reflecting a bidirectional link. Conversely, a weight is unidirectional in the absence of such excitatory reciprocal connection. Following the onset of plasticity, we observed that mutual inhibitory weights increase much more than unidirectional ones, up to the saturation threshold (Fig. 3C). This selectivity can be directly visualized through the matrix of synaptic weights after learning: the pattern of learned inh-to-exc weights mirrors the fixed exc-to-inh weights (Fig. 3E,F; see Supporting Information for full weight matrix).

Figure 3.

Next, we investigated the impact of an asymmetric iSTDP rule within the same random RNN architecture and neural dynamics. As with the symmetric rule, this asymmetric iSTDP stabilizes the network dynamics (Fig. 3G). However, it promotes a markedly different connectivity pattern: purely lateral inhibition. In this configuration, inhibitory neurons strengthen their connections onto excitatory neurons lacking reciprocal connections back to them, thus forming exclusively unidirectional connections (Fig. 3H). This yields an inhibitory-to-excitatory connectivity pattern almost perfectly opposite to the previous configuration, so that existing exc-to-inh synapses carve gaps in the inh-to-exc connections (Fig. 3J,K). Achieving robust lateral inhibition with the asymmetric iSTDP rule requires the introduction of a small rate-dependent component (B = 0.3). This deviation from a purely covariance-dominated regime introduces characteristics of rate-homeostatic plasticity, such as rapid convergence to a narrow distribution of excitatory firing rates (Fig. 3G). This requirement is perhaps unsurprising given the challenges of parameter tuning in large RNNs, where closed-form solutions are unavailable (see Discussion for further considerations). Both the symmetric and the asymmetric rules converged to stable weight distributions, except for a very slow tendency for some mutual weights to reach either the upper bound (symmetric) or zero (asymmetric). These results are also robust to large variations in external input, and are compatible with network models where asynchronous irregular firing is generated intrinsically, without any external source of random noise (SI, Section S2 and Fig. S5; see also Mastrogiuseppe and Ostojic 2017).

In sum, our RNN simulations generalize the analytical predictions from the reduced E/I circuit model to larger networks of recurrently connected integrate-and-fire units. Symmetric iSTDP rules reinforce mutual inhibition, mirroring pre-existing excitatory connectivity, whereas asymmetric kernels favor lateral inhibition.

2.4 Response properties of a one-dimensional ring network

Having shown that different forms of iSTDP shape E/I connectivity in large-scale recurrent networks based on local correlations arising from direct connections, we next explored the functional consequences of structured stabilization in a recurrent network with a one-dimensional ring architecture. Ring networks are well-suited to model continuous attractor dynamics, supporting bumps of persistent activity (Amari 1977; Ben-Yishai et al. 1995; Khona and Fiete 2022). This makes them relevant for studying neural computations such as head-direction encoding (Xie et al. 2002; Kim et al. 2017; Kim et al. 2019; Petrucco et al. 2023) and orientation selectivity in the primary visual cortex (Goldberg et al. 2004).

We built a ring network of 800 excitatory and 200 inhibitory neurons arranged on a circle, with exc-to-exc and exc-to-inh weights decaying exponentially with the angular distance between neurons, following a rescaled Von Mises (Fig. 4A,B). We further divided the inhibitory population into two subclasses, governed by distinct iSTDP rules, but sharing identical neural dynamics. Inspired by experimental results (Lagzi et al. 2021), one population of “PV” interneurons followed a symmetric kernel, and another population of “SST” interneurons followed an antisymmetric one, both rules operating in the covariance-dominated regime. Initially, we set both PV-to-exc and SST-to-exc synapses as all-to-all, weak, and unbiased (see Fig. 4A for schematics, and Methods Table 3 for numerical parameters).

Table 3.

Figure 4.

As the network evolves under the influence of the two plasticity rules, the inhibitory populations develop specialized roles. PV interneurons, governed by symmetric iSTDP, form strong mutual connections with excitatory neurons, mirroring the distance-dependent excitatory connectivity (Fig. 4B and Fig. 4C, red). Conversely, SST interneurons, driven by antisymmetric iSTDP, establish lateral inhibition across the ring (Fig. 4B and Fig. 4C, orange). This structured stabilization, with both localized and broader components, raises the possibility of nontrivial effective interactions between excitatory neurons, potentially mediated by the different roles of the PV and SST interneurons.

To estimate the effective interaction profile between excitatory neurons, we used linear response theory, equivalent to a Taylor expansion of the network dynamics around its operating point (Sadeh and Clopath 2020, see Methods for details). We found an effective connectivity profile resembling a Mexican-hat, with short-range excitation directly inherited from exc-to-exc connectivity and a broader inhibitory surround resulting from the combined action of PV and SST interneurons (Fig. 4D). This effective connectivity profile motivated us to explore the resulting network dynamics and its response to external stimuli. We examined the emergent spontaneous activity and observed a complex spatial correlation structure, with periodic oscillations in correlations over distance (Fig. 4E, see Supporting Information Fig. S8 for full correlation matrix). This emergent dynamics is reminiscent of the spatial correlation maps observed in the developing cortex of ferrets, and is considered a signature of modular cortical organization (Smith et al. 2018; Powell et al. 2024).

After identifying aspects of emergent cortical dynamics in our network consistent with early developmental patterns, we investigated whether it also displays other fundamental computational properties classically measured in the visual system. A well-established nonlinear phenomenon in the visual cortex is surround suppression, in which gradually increasing the stimulus size initially amplifies the neuron’s response up to a peak, but then leads to reduced activation (DeAngelis et al. 1994; Cavanaugh et al. 2002). After inhibitory plasticity self-organized network connectivity, we drove the network with patterned stimuli in the absence of subsequent plasticity. We delivered inputs of different strengths and sizes in the form of excitatory spike trains (Fig. 5A). Each stimulus was presented 100 times in randomized order, with a 0.5 s presentation followed by a 0.5 s blank. We found that larger stimuli elicited lower peak firing rates in their center region, while the response of neurons with receptive field closer to the stimulus’ edge was larger (Fig. 5A, raster plot in SI, Fig. S8). Consequently, the center neuron’s response as a function of stimulus size exhibited a classic surround suppression profile (Fig. 5B, DeAngelis et al. 1994).

Figure 5.

We carried out three control experiments to disentangle the contributions of the two interneuron populations with different plasticity rules to the dynamics of the ring network. A single inhibitory population with rate-dominated iSTDP can stabilize the system, although the emergent inh-to-exc connectivity does not seem patterned (Fig. S9B). This network can still produce surround suppression, however the edge effect is reversed, with neurons on the stimulus edge firing at a lower rate compared to neurons in the center (Fig. S9C, note the convex shape of the curve for large stimuli). Moreover, the correlations do not exhibit any spatial structure. When considering networks with a single covariance-dominated PV-like (symmetric iSTDP kernel) or SST-like (asymmetric iSTDP kernel) plasticity rule, we found only pathological solutions that required careful parameter tuning. The PV-only network displayed pathological population-wide synchrony (Fig. S10E), while the SST-only network produced strongly clustered runaway excitation (Fig. S10J). This suggests that the two interneuron populations with different plasticity rules act synergistically to establish excitatory firing rate stability and specific E/I motifs.

We also investigated whether our model could also reproduce contextual facilitating effects, for example when a stimulus in the preferred orientation is surrounded by an orthogonal stimulus (Sillito and Jones 1996). These effects are mediated by a third subclass of interneurons, the vasoactive intestinal peptide (VIP)-expressing cells, that inhibit the SST population and thereby disinhibit excitatory neurons (Keller et al. 2020). We thus modeled the iso-oriented stimulus as excitatory spike trains delivered to the excitatory neurons, and introduced a contra-oriented stimulus as inhibitory inputs from VIP interneurons to the SST population at the corresponding location (Fig. 5C). We then compared the center-neuron response in three conditions: (1) an iso-oriented stimulus restricted to the neuron’s receptive field; (2) a broader iso-oriented stimulus; (3) an iso stimulus in the receptive field combined with a contra stimulus in the distal regions (Fig. 5D). In agreement with experimental results (Keller et al. 2020), the distal contra stimulus disinhibits the center neuron and facilitates its response in the iso+contra stimulus conditions (Fig. 5,E).

These results demonstrate how structured stabilization shaped by the interplay of symmetric and antisymmetric iSTDP rules can give rise to network connectivity which generates activity consistent with both suppressive and facilitating contextual effects in the primary visual cortex, and to long-range correlation patterns consistent with spontaneous activity in the developing sensory cortex.

3 Discussion

In this work, we have shown that a family of inhibitory STDP rules can simultaneously stabilize excitation at the circuit level, and drive the formation of specific connectivity patterns among excitatory and inhibitory neurons in the form of E/I motifs. In the absence of correlated external inputs, or training signals, circuits self-organize purely through their spontaneous internal dynamics. We found that the shape of the iSTDP rule selectively reinforces different E/I motifs—blanket inhibition, reciprocal E/I connectivity and, lateral E/I connectivity—based on the correlations present in the network.

Prior computational studies addressing the self-organized formation of motifs in recurrent networks through STDP mostly focused on exc-to-exc plasticity (Ocker et al. 2015; Ravid Tannenbaum and Burak 2016; Triplett et al. 2018; Montangie et al. 2020; Manz and Memmesheimer 2023). Previous work on iSTDP has been largely restricted to feedforward circuits and rules dominated by either firing rates or covariance (Kleberg et al. 2014; Luz and Shamir 2014; Akil et al. 2021; Vignoud and Robert 2022). In contrast, our approach combines both: recurrence and iSTDP. This combination makes it possible to characterize how different timing-based rules at inhibitory synapses shape excitatory-inhibitory connectivity in recurrent circuits, and how the resulting E/I motifs influence circuit dynamics, including rate stabilization, surround modulation, and the structure of spontaneous activity.

We implemented a generalized iSTDP framework where individual contributions from spike pairs additively contribute to synaptic change, as in many models of excitatory synaptic plasticity. While many other factors may also contribute to inhibitory plasticity, including multiplicative effects (Morrison et al. 2007; Kunkel et al. 2011), synaptic location (Agnes and Vogels 2024) or plasticity on behavioral timescales (Gonzalez et al. 2023), we found that additive iSTDP can still generate many interesting insights about E/I connectivity in recurrent networks. Our iSTDP framework combines both rate-dependent and covariance-dependent components in inducing inhibitory plasticity. The rate-dependent terms guarantee stable neural activity, whereas the covariance-dependent terms support the emergence of specific E/I motifs, motivated by recent experimental evidence (Znamenskiy et al. 2024). Consequently, our framework provides deeper insight into both the regulatory and structural properties of generalized iSTDP rules, expanding our understanding of plasticity in closed-loop recurrent circuits.

One strength of our framework is that it clarifies how the iSTDP kernel interacts with other plasticity parameters. For example, in rate-dominated regime the weight evolution mainly depends on the net area of the STDP kernel and the rate-dependent terms, and not on the precise kernel shape. In the covariance-dominated regime, the weight evolution depends on specific spike covariance between pre- and postsynaptic spiking and the iSTDP kernel shape. The covariance-dominated regime, however, can pose greater challenges for parameter tuning. A promising way forward is to employ numerical methods, for instance using simulation-based inference and deep learning, which have already been applied to optimize both excitatory and inhibitory STDP rules in recurrent circuits (Confavreux et al. 2024). Our analytical approach can guide and complement these efforts, helping clarify parameter degeneracies and simplifying the parameter search process.

Here we derived analytic solutions for the two-neuron Poisson model, and relied on numerical simulations for larger networks.By contrast, a recent study showed that iSTDP can stabilize recurrent LIF networks and converge to stable solutions through a mean-field approach (Akil et al. 2021). To achieve analytical tractability, that work assumed a regime of weak interactions, where internally generated correlations become negligible as the size of the neural population grows, and plasticity was driven exclusively by externally-induced correlations. In our approach, finite network size, strong interactions, and a dominance of correlation-dependent terms together drive self-organization. It remains an open question whether these two analytic frameworks, based on different assumptions, can be integrated.

While our analytical approach primarily examined how small E/I circuits self-organize under different iSTDP rules, neural circuits can also form larger structures, including feedforward firing sequences and assemblies. Assemblies are groups of strongly connected neurons, often regarded as the fundamental building blocks of neural computations (Buzsáki 2010; Papadimitriou et al. 2020; Miehl et al. 2023). Previous work has examined how various iSTDP rules enable the learning, maintenance and self-repair of neural assemblies (Ocker and Doiron 2019; Lagzi et al. 2021; Bergoin et al. 2023; Bergoin et al. 2025). In particular, recent findings suggest that assemblies are formed and best preserved when inhibitory neurons are divided into subclasses that combine symmetric and antisymmetric iSTDP (Lagzi et al. 2021). Here, the first rule can maintain network stability while the second prevents assemblies from merging through lateral inhibition, as in our work. Related work has also found that a combination of positive symmetric (Hebbian) and negative symmetric (anti-Hebbian) covariance-based rules can form and maintain stable assemblies governed by similar principles (Bergoin et al. 2025). These results on assemblies agree with ours in underscoring the intuition that different rules serve different purposes: some promote E/I balance and internal network stability, while others introduce competition through lateral inhibition.

In this work we further combined multiple iSTDP rules in a network with a ring topology (Ben-Yishai et al. 1995) to demonstrate how their interaction stabilizes activity while also supporting specific network computations, such as contextual input modulation as observed in the visual cortex(DeAngelis et al. 1994; Cavanaugh et al. 2002), and long-range spatial correlations reminiscent of modular spontaneous activity observed in the developing cortex (Smith et al. 2018; Powell et al. 2024). The parametrization of iSTDP rules allowed us to explore the entire space of rules and link them to the emergence of specific E/I connectivity motifs beyond the choice of pre-set, fixed plasticity rules in existing work (Lagzi et al. 2021; Bergoin et al. 2023; Lagzi and Fairhall 2024; Bergoin et al. 2025).

Here, we addressed the spontaneous emergence of circuit connectivity at the sub-assembly level and in the absence of externally patterned inputs. It would be interesting to consider non-stationary inputs, such as input correlations varying in time and across neurons, or highly structured input such as assembly-activations or waves spreading across the circuit as observed during cortical development (Smith et al. 2018). Non-stationary input, jointly with excitatory STDP, might help paint a more complete picture of prolonged cortical development especially after the onset of sensory experience (Trägenap et al. 2025). Excitatory plasticity can enable circuits to learn new representations, refine current ones, and recover after lesions.

Overall, our findings highlight how local inhibitory learning rules can both stabilize global network dynamics and produce specific connectivity motifs in a self-organized manner in recurrent circuits, with implications for understanding neural development and learning under structured stimuli.

4 Methods

4.1 General mean-field solution

We consider a class of pairwise iSTDP rules triggered by spiking events and purely based on spike-timing difference between pre- and postsynaptic neuron firing, defined by the following expression.

where and represent pre- and postsynaptic spike events, respectively. In our numerical simulations the instantaneous change of weight is further regulated by a global learning rate A₀, as in ΔW^{∗ (pre,post)} = A₀ ΔW^(pre,post) . The scaling by A is omitted in our equations for simplicity. Each presynaptic spike triggers a weight change consisting of a fixed component, denoted α_pre, and a pairwise interaction term. Similarly, each postsynaptic spike triggers a weight change with a fixed component α_post and a pairwise interaction term. The pairwise interaction terms capture the cumulative influence of joint spiking events, incorporating the contribution of all preceding spikes (k^′ < k) weighted by a kernel that depends on the time differences between the spikes. The two functions and refer to the post-pre and the pre-post pairwise interaction, and are merged in a single function, known as the iSTDP pairwise interaction kernel, as follows:

By convention, the τ > 0 branch of the curve represents pre-post interactions (presynaptic spike preceding the postsynaptic spike), while the τ < 0 branch represents post-pre interactions (postsynaptic spike preceding the presynaptic spike). In our framework, we consider L(τ) to be independent of α_pre and α_post. This is in contrast to previous formulations that incorporate these terms into the kernel representation (e.g., Vogels et al. 2011 and Confavreux et al. 2024). We assume that synaptic plasticity operates on a much slower timescale than neural dynamics, allowing us to treat synaptic weights as constant when calculating first- and second-order firing statistics (Kempter et al. 1999). We further assume variability in the spike-train statistics, so that it can be effectively described as a stochastic process. Thus, we can write the evolution of the mean synaptic weights as:

where B is the integral of the iSTDP kernel, and Q is the only term that depends on the pairwise covariance density between pre and post-synaptic spike trains, here denoted C(τ). For a complete derivation see Supporting Information or Kempter et al. (1999) and Luz and Shamir (2014).

4.2 Expressions for the symmetric and antisymmetric exponential kernel

Eq. 4 can be applied to any form of iSTDP kernel, including, e.g., kernels with delayed interactions (Babadi and Abbott 2013). Here, we focus on two specific exponentially decaying shapes: symmetric and antisymmetric, defined as follows:

With this parametrization, A₊ = A₋ implies B = 0, corresponding to covariance-dominated rules, whereas for A₋ = 0 we have B = A₊, a rate-dominated regime. This choice of L(τ) allows us to keep track of the entire spiking history through activity traces updated at every spike time and decaying in time with the same constants as the iSTDP kernel. These traces will determine the amount of potentiation (+) or depression (−) triggered by each spike. For the antisymmetric rule, the presynaptic and postsynaptic traces are respectively defined as follows:

The symmetric rule requires, in addition to T_r+(t) and T₀−(t), the traces T_r−(t) and T₀₊ (t), defined analogously. The implementation of the asymmetric iSTDP rule is then given by:

and for the symmetric rule:

4.3 Low impact of correlations in rate-dominated rules

To better understand the role of the net area B of the iSTDP kernel in the weight dynamics (Eq. 4), and how it determines whether a rule is rate-dominated or covariance-dominated, we will consider here a simplified scenario, where L(τ) prescribes a +1 increase in weight for coincident pre-post spikes and no increase otherwise. This choice yields B = A₊ = 1. Consider now a pre and a post-synaptic neuron that fire with Poisson statistics at rates r_pre = r_post = r = 10 Hz. The term B r_pre r_post in Eq. 4 takes a value of 100 s⁻² regardless of the correlation between the two neurons. But how does it compare to the covariance term Q in the presence of strong correlations between pre and post-synaptic activity? We can consider a regime of (unnaturally) high correlations, such that half of the spikes of the two neurons are synchronous (Pearson correlation 0.5). The term Q, corresponding to the spike-count covariance, is 5 s⁻². So even for very strong correlations Q is 20 times smaller than B r_pre r_post. This illustrates that in regimes with strong B, the weight dynamics in Eq. 4 are primarily governed by first-order statistics, with — at best — second-order contributions from correlations in spiking activity.

4.4 The two-dimensional reduced model

Finding a general solution for Eq. 4 can be challenging. For our purposes, we consider a simplified case. First, we model an excitatory and an inhibitory neuron as mutually interacting Poisson processes (Hawkes 1971; Ocker et al. 2017). Each time the presynaptic neuron spikes, the firing probability of the postsynaptic neuron increases (or decreases) proportionally to the excitatory (or inhibitory) post-synaptic potential EPSP (IPSP) kernel:

We further assume that the input to the inhibitory neuron h_inh(t) is regulated homeostatically to keep its rate r_inhconstant. Namely,

In those conditions, Eq. 4 has an approximate fixed-point closed-form solution for the inhibitory connection w_inh:

The coefficients and are found through the linear response theory (Hawkes 1971; Ocker et al. 2017), and depend on the iSTDP kernel and the EPSP / IPSP parameters (refer to the analytic derivation of the SI Text, Section S1.4).

For the symmetric iSTDP:

For the antisymmetric iSTDP instead:

Numerical parameters are in Table 1.

4.5 Randomly connected recurrent neural network

We built a random network of conductance-based leaky integrate-and-fire (LIF) neurons, with 900 excitatory units and 100 inhibitory units. The membrane potential obeys the following dynamics:

With g_AMPA(t) and g_GABA(t) updated for each incoming signal proportionally to the excitatory and inhibitory synaptic weights. We assigned a probability of connection for all non-plastic connections (excitatory to excitatory, excitatory to inhibitory, inhibitory to inhibitory). The inhibitory to excitatory connectivity was all-to-all and plastic. The network received an external input in the form of Poisson spike trains at fixed rate. The excitatory neurons received a small number of shared spikes, along with spikes generated i.i.d. for each neuron, leading to a small overall correlated activity. The network displayed high robustness to variations in input strength (see SI, Fig. S11). Numerical parameters were determined through grid search (Table 2).

4.6 Ring model

The ring model was implemented with the same neural dynamics as the random network (Eq. 14). Here however both excitatory and inhibitory neurons received a baseline external input, independent for each unit. We used 800 excitatory units and 200 inhibitory units, further divided into a PV and an SST subclass. The synaptic weight between an excitatory neuron and its neighbors decayed with distance, following a bell-shaped curve modeled as a rescaled von Mises distribution:

where θ_i and θ_j represent the locations of the post- and presynaptic neuron on the ring, with each population equally spaced in the [0, 2π) interval. The factor k·_←e regulates the connection range. Finally the scaling factor sum w·_←e. All inh-to-inh connections were fixed and dense. We considered two distinct inhibitory populations C·_←e is tuned so that the sum of incoming weight^+-s to either exc, PV or SST population is at a desired value with identical dynamics, but different iSTDP rules. The first population had a symmetric iSTDP kernel, the second an antisymmetric one. The input was different than in the previous model, with a constant external current in the excitatory population, and independent excitatory spike trains for all units.

To measure the network response to external stimuli (Fig. 5), we froze plasticity after learning, and delivered stimuli as uncorrelated spike trains. Iso-oriented stimuli took the form of excitatory spikes delivered to the exc neurons, contra-oriented stimuli were modeled as inhibitory spikes delivered to the SST neurons. Numerical parameters, determined through grid search, are shown in Table 3.

Data availability

Circuit simulations are based on custom code written in Julia (Bezanson et al. 2017, https://julialang.org), also available as separate toolbox at https://github.com/comp-neural-circuits/HawkesSimulator.jl . For the large‐scale network simulations we used Brian2 (Stimberg et al. 2019, https://briansimulator.org). Finally some of the analytic integrals were solved using Maple (https://maplesoft.com). The code is fully open‐source and publicly available on GitHub under the Creative Commons Attribution 4.0 International (CC BY 4.0) License at the following link: https://github.com/comp-neural-circuits/structured-stabilization-in-recurrent-neural-circuits

Supporting Information

S1 Analytic results

S1.1 Mean-field equation for weight dynamics

In this section, we adapt the results introduced by Kempter et al. (1999) to our notation and our specific needs. We first describe a sufficiently general expression for the pairwise STDP rules. Given two neurons, one presynaptic and one postsynaptic, an excitatory or inhibitory STDP rule is a weight update triggered by either a pre- or a postsynaptic spike. The weight update solely depends on the time differences between the previous spike times, relative to the current spike time:

Where and represent pre- and postsynaptic spike events, respectively. We can merge the functions and into a single STDP kernel defined as:

Therefore τ > 0 (the Δ₊ branch) represents pre-post interactions (presynaptic spike preceding the postsynaptic spike), while τ < 0 (the Δ₋ branch) represents post-pre interactions (postsynaptic spike preceding the presynaptic spike).

It is useful to express each spike train as a sum of Dirac delta functions centered at each spike time:

where i denotes the neuron index. With this notation, the number of spikes in the [t_a, t_b] interval is:

Assuming stationary conditions, we can therefore define the firing rate of neuron i as:

Using the definition in Eq. S3, we can then rewrite the pairwise interaction terms in Eq. S1 as:

Where in the last step we performed the change of variable τ ←t − τ ^∗. This leads to the following expression for the instantaneous change in synaptic weights, numerically equivalent to Eq. S1:

Now, using again the same definition of spike train in Eq. S3, an event-based update can be expressed as an integral over time over the pre- and post-synaptic spiking events. The total weight change occurring in an interval [t_a, t_b] is:

We can then express the average weight change per second, considering a time interval [0, T ] large enough to ensure precise sampling in second-order firing statistics, but still small compared to the rate of changes in ΔW. Under this assumption of separation of timescales between neuronal dynamics and plasticity, we can substitute Eq. S7 into Eq. S8, and write:

Since α_pre and α_post are constants, we can factor them out. The first two terms of the r.h.s. of Eq. S9 become:

The pairwise interaction terms instead can be simplified as follows:

and similarly for the last term in Eq. S9. In the above equation, denotes the covariance density between the pre- and the postsynaptic neuron, following the convention that Δ₊ represents the pre-post order and Δ₋ the post-pre order:

An analogous definition holds for , with the pre- and post-temporal order inverted. Therefore

Now using Eq. S2, we can write:

and given that we can perform the following variable substitution:

Finally, substituting Eq. S14 and Eq. S15 in Eq. S13, and defining , we have:

Corresponding to Eq. 1 in the main text.

S1.2 Derivation of the closed-form solution for the E/I motif

In this section, we derive a closed-form solution for a reduced model of one inhibitory and one excitatory neuron (Eq. S16). We model the two neurons as mutually interacting Hawkes processes (Main Text, Fig. 1A, Hawkes 1971; Pernice et al. 2011). The excitatory unit receives a constant input current h_exc, a plastic inhibitory connection w_inh(t) and it projects a fixed excitatory connection w_exc to the inhibitory neuron. For simplicity, we assume that the inhibitory unit is kept at a constant rate r_inh by an adaptive external current h_inh(t) (10). In this framework, the average firing rates behave linearly, therefore, the excitatory rate can be written as:

As we focus on the connection w_inh(t) from the inhibitory to the excitatory neuron, we rename r_inh = r_pre and r_exc = r_post in Eq. S16 and we get:

The only nontrivial quantity to compute is the term Q, as it requires the inhibitory-to-excitatory covariance density curve. To avoid ambiguities, we define the Fourier transform operator acting on a function f (t) and the convolution operator acting on two functions f (t) and g(t) as follows:

where j denotes the imaginary unit.

Note that the total integral Q of the product C(τ) L(τ) (as defined in Eq. S16) is equivalent to its Fourier transform evaluated at ω = 0. But for the inverse convolution theorem, the Fourier transform of the product C(τ) L(τ) corresponds to the convolution of the two transforms C( ω) and . In equations:

The advantage of re-defining Q in the Fourier space is that, for Hawkes processes, the covariance C( ω) can be expressed in a closed form as a function of network connectivity and neural activity (Hawkes 1971; Pernice et al. 2011; Trousdale et al. 2012; Jovanović et al. 2015). More specifically, given the matrix of post-synaptic potential kernels (Eq. 9 in Methods)

and the connectivity matrix

we can first write the vector of firing rates r = [r_exc, r_inh]^⊺ as

where h₀ is the baseline input and G(τ) = E(τ) W is referred to as the matrix of “interaction kernels”. At the equilibrium state,

where the integral in the r.h.s. corresponds to the Fourier transform evaluated at zero, i.e., Ĝ( ω)|_ω=0. Equivalently, the rate vector can be written in an explicit form:

where I represents the identity matrix. Moreover, if we write the covariance matrix (cf. Eq. S12) as

then we have the following expression in the Fourier domain, at the equilibrium state:

where D is a diagonal matrix with the firing rates on its diagonal. Finally, recall that, for a matrix G with spectral radius < 1,

Therefore, the covariance matrix can be written as the following expansion

where the indices α, β indicate the number of connections between any spiking neuron in the network to the post- and to the presynaptic neuron, respectively. This captures the influence of each spike propagated in the network through existing synaptic connections on the correlated firing of the pre- and postsynaptic neurons. Within this formalism, in our reduced model we consider only pairwise interactions, that is, α + β = 1. The inhibitory-to-excitatory covariance is the (1, 2) element of Ĉ( ω). We can compute it as:

By combining Eq. S20 and Eq. S30, we get the following expression for Q:

The , coefficients can be computed analytically depending on the iSTDP rule (see next section). Considering them as constants, we can then write Eq. S18 as:

or equivalently,

with Ψ ∶= α_post + Br_inh + M⁽⁰¹⁾w_exc.

We then find the stationary point (Eq. 11 in Methods)

The equilibrium point is a stable fixed point attractor, and positive, that is , if the following conditions hold:

If a solution for exists, we can also express the steady-state postsynaptic rate as follows:

The solution is valid only for positive rates. This adds one more condition for a stable and positive solution. The constraints are therefore:

S1.3 Circuit motif with unconstrained r_inh

In the 2D E/I motif presented in the main text, the value of r_inh was fixed both for in the numerical simulations and in the analytic expression of the dynamics (Eq. 10). This led to a closed-form expression for the fixed point solution (Eq. 11 in Methods):

In this section, for completeness, we show how the condition of fixed r_inh is not required for a stable solution and can in fact be relaxed at the cost of simplicity in the analytic expression of the fixed-point solution.

Consider the model shown in Fig. S3A. Here both r_exc and r_inh are free parameters, given by the net input. Namely:

where we dropped the time dependence in the notation for simplicity. We can rewrite the equations above as:

Then, plugging the terms above into (S32), we obtain:

We can find a fixed point in the equation above by setting and multiplying by (1 + w_exc w_inh)². The equation becomes a quadratic equation for w_inh:

With

The solution is then given by

as long as b² − 4ac ≥ 0.

Figure S3 shows analytic and numeric results for unidirectional and mutually-connected motif configurations for different iSTDP rules.

S1.4 Derivation of first-order motif coefficients

To calculate the motif coefficients

for the connection W, we need to compute the Fourier transform of the postsynaptic potential E_e, E_i and the Fourier transform of the iSTDP kernel.

The Fourier transforms of the excitatory and inhibitory postsynaptic potential (Eq. 9 in Methods) are respectively given by:

where j is the imaginary unit.

Consider first the Hebbian symmetric STDP kernel, defined via the function

with A₊, A₋ > 0. The Fourier transform is given by

By substituting Eq. S46 and Eq. S48 in Eq. S45, we can calculate the first-order motif coefficients , :

and therefore find an analytical solution for the integral Q in Eq. S31. Consider now the antisymmetric iSTDP rule, with kernel:

whose Fourier transform is given by

Again, we can calculate the first order motif coefficients , for the antisymmetric rule by substituting Eq. S46 and Eq. S51 in Eq. S45:

These expressions can be included into Eq. S34, leading to the final expression for the symmetric rule:

And to the final expression for the antisymmetric rule:

S2 Structured stabilization in spiking networks with intrinsically-generated fluctuating activity

Our main numerical results are based on conductance-based spiking RNNs, where excitatory neurons receive random Poisson inputs, with both a shared and an independent pomponent. To further demonstrate the generality of our results, we consider a somewhat different but important class of spiking RNNs, where fluctuating activity is generated completely intrinsically and does not depend on external inputs (Van Vreeswijk and Sompolinsky 1998; Mastrogiuseppe and Ostojic 2017)

We adapted a current-based spiking RNN model that spontaneously generates irregular activity under constant input currents (Mastrogiuseppe and Ostojic 2017). The single-neuron dynamics is as follows:

(see Table S1 for the full list of parameters). We selected initial conditions that resulted in a high-firing, nearly synchronous firing regime (Fig. S5A,B). In line with our previous results, we found that our choices of iSTDP rules lead to stable dynamics, reducing excitatory firing rates. Moreover, depending on the profile of correlation kernel, the network converged to either a prevalence of reciprocal E/I connections Fig. S5E, or lateral connectivity, expressed by unidirectional connections Fig. S5F.

These observations underscore the robustness of our proposed iSTDP mechanisms in shaping network connectivity beyond specific network architectures. The consistent emergence of either reciprocal or lateral connections, depending on the iSTDP rule, highlights a general principle by which local plasticity can shape E/I circuit structure.

Table S1.

S3 Supporting Figures

Figure S1:

Figure S2:

Figure S3:

Figure S4:

Figure S5:

Figure S6:

Figure S7:

Figure S8:

Figure S9:

Figure S10:

Figure S11:

Acknowledgements

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon Europe research and innovation programme (ERC CoG FeedbackCircuits, grant agreement no. 101170267), and under the Marie Skłodowska-Curie ITN SmartNets (grant agreement No. 860949) under the European Union’s Horizon 2020 research and innovation program, by the International Human Frontier Science Program Organization (RGP0062/2021), and by the Technical University of Munich (TUM). Thanks to the “Computation in Neural Circuits” group for their useful insights and discussions. Special thanks to Alessio Quaresima for his valuable feedback and for independently testing our results.

Additional files

Supplemental material

Additional information

Funding

EC | H2020 | PRIORITY 'Excellent science' | H2020 European Research Council (ERC)

https://doi.org/10.3030/804824

• Claudia Cusseddu

• Julijana Gjorgjieva

• Dylan Festa

EC | Horizon Europe | Excellent Science | HORIZON EUROPE European Research Council (ERC)

https://doi.org/10.3030/101170267

• Claudia Cusseddu

• Julijana Gjorgjieva

EC | Horizon 2020 Framework Programme (H2020)

https://doi.org/10.3030/860949

• Claudia Cusseddu

• Julijana Gjorgjieva

Technische Universität München (TUM)

• Claudia Cusseddu

• Julijana Gjorgjieva

• Dylan Festa

Human Frontier Science Program (HFSP)

https://doi.org/10.52044/hfsp.rgp00622021.pc.gr.163582

• Dylan Festa

• Julijana Gjorgjieva