Modeling robust transitions between tonic and burst firing in a biophysical network model

We simulate the transition from tonic to burst firing in a heterogeneous network of conductance-based model neurons (Materials and methods), with the network architecture shown in Fig 1A. The network is composed of presynaptic excitatory neurons projecting to postsynaptic excitatory neurons with membrane potentials and , respectively. These neurons all receive GABA currents from a single inhibitory neuron (), the activity of which controls whether the network is in a tonic firing or bursting regime. This modulation in inhibitory levels mimics the variation of neuromodulators such as acetylcholine, dopamine, serotonin, or histamine [26], known to control brain state transitions [5,27–31]. Each neuron also receives an individual applied current () to set its firing rate for learning specified input patterns. For details on equations governing neural dynamics, see Materials and methods.

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Fig 1. Collective burst firing drives synaptic weights towards attractor in the weight space.

A. The network model consists of one inhibitory neuron () projecting to all the excitatory neurons, connected in a feedforward configuration with presynaptic neurons () to postsynaptic neurons (). An external current applied to the inhibitory neuron controls the network state (). Each excitatory neuron is subjected to an individual external current mimicking an external input. B. Hyperpolarizing the current applied to the inhibitory neuron () switches the network state from input-driving tonic firing to collective burst firing. Raster plot activity of the excitatory neurons and local field potential (LFP) traces are displayed. C. Evolution of the synaptic weights during switches from input-driven tonic firing to collective burst firing. The network comprises 50 presynaptic neurons connected to 50 postsynaptic neurons for 100 traces among 2500 (gray lines) with 5 randomly highlighted lines (colored lines). and indicate the end of the -th tonic and burst firing state, respectively. D-E. Comparison of the synaptic weights at the end of the third and fourth tonic firing states (dark blue) or the third and fourth burst firing states (light blue), normalized between the minimal and maximal values shown on a scatter plot. The histograms highlight the weight distribution. The weights converge to a fixed point in the weight space, we call burst-induced attractor (, ).

https://doi.org/10.1371/journal.pcbi.1014001.g001

Fig 1B illustrates the transition between the two firing regimes in a network of presynaptic and postsynaptic neurons: input-driven tonic firing and collective burst firing, controlled by the current applied to the inhibitory neuron.

During input-driven tonic firing, a depolarizing current applied to the inhibitory neuron induces regular tonic spiking activity, as commonly observed in inhibitory pacemaking cortical neurons [1]. Consequently, this pacemaking inhibitory neuron sets the resting membrane voltage of all the excitatory neurons. Upon receiving an applied current (), these excitatory neurons generate action potentials. For details on protocol inducing tonic firing, see Materials and methods.

To trigger a network transition from input-driven tonic firing to collective burst firing, a hyperpolarizing current is applied to the inhibitory neuron.

This bursting activity, in which each neuron generates a rapid succession of action potentials followed by a quiescent period [32–34], is generated endogenously by intrinsic ion channel dynamics and recurrent inhibition: once in this regime, we silence external inputs in and and retain only a small noisy baseline current to the excitatory neurons, which does not by itself drive spiking. Thus, the timing and occurrence of bursts are not imposed by external patterned input, but emerge from the intrinsic properties of the neurons, such as T-type calcium channels that deinactivate when the neuron is hyperpolarized. Sufficiently strong or temporally structured external inputs could modulate the burst pattern, but the burst regime characterized here does not require such inputs. The term “collective” refers to the synchronization of individual bursting neurons despite heterogeneity in intrinsic properties and synaptic connections, resulting in slow large-amplitude population activity shown in the LFP oscillation, contrasting the fast small-amplitude observed in input-driven tonic firing (Fig 1B, LFP traces) [5]. For details on LFP calculation, see Materials and methods.

Modeling synaptic plasticity

Synapses between presynaptic and postsynaptic excitatory neurons undergo activity-dependent plasticity, a fundamental process underlying learning and memory. In biological systems, plasticity arises from a combination of biochemical, structural, and electrical mechanisms that respond to intracellular calcium dynamics.

To capture these dynamics quantitatively, we use a state-of-the-art calcium-based model established by Graupner et al. [25], which was fit to experimental data obtained through a frequency pairing protocol [35]. The model implements two opposing calcium-triggered pathways that lead to either an increase or a decrease in synaptic strength: potentiation or depression is triggered when calcium levels exceed specific thresholds. Although we illustrate our results with this model, we later show that the same principles apply across a wide range of plasticity rules, highlighting the generality of our results. In this framework, the synaptic weight from the presynaptic neuron to the postsynaptic neuron is denoted and its evolution is governed by Eq (3). For details on equations governing synaptic plasticity, see Materials and methods.

Collective burst firing drives synaptic weights towards an attractor in the weight space

To study how synaptic plasticity operates across tonic and burst firing regimes, we track the evolution of the weights over the course of eight successive tonic and burst epochs (Fig 1C).

During tonic firing (Fig 1C, dark blue), each neuron receives a random pulse train input, with the stimulation frequency selected uniformly at random between 0.1 and 50 Hz. For details on the computational experiment, see Section Computational experiment related to Fig 1. The temporal evolution of the synaptic weight is determined by the correlation between the activity of presynaptic neuron and postsynaptic neuron , which drives plasticity at those synapses. As a result, at the end of each tonic firing period (time , ), the weight converges to a value determined by the specific external input received. Since different neurons receive different input statistics, the weights across the network settle at a broad range of values (see Fig 1D, histograms). Furthermore, because the input statistics vary across tonic periods, the synaptic weights also differ from one tonic episode to the next, as illustrated by plotting against (Fig 1D, scatter plot; see also S1 Fig for comparison across other states).

During burst firing (Fig 1C, light blue), the network is driven only by an inhibitory current, with no external input to the excitatory neurons. Collective bursting activity drastically alters synaptic weight dynamics: previously potentiated connections tend to depress, and vice versa. Each initial weight converges to a unique steady-state value within a narrow range (Fig 1E, histograms), and repeats from one burst episode to the next one independent of its initial condition, as illustrated by plotting against (Fig 1E, scatter plot; see S1 Fig for other pairs). We call this phenomenon a burst-induced attractor and define it as a narrow region in weight space to which synaptic weights converge under collective bursting dynamics.

The burst-induced attractor emerges in a wide range of synaptic plasticity models

We next investigate whether the choice of synaptic plasticity rule influences the emergence of a burst-induced attractor. We repeat the same experiment in the same network using various synaptic plasticity models and observe that the effect persists across a wide range of rules (S2 Fig), including different variants of calcium-based models [21–23], phenomenological models based on pairwise [18] or triplet [20] spike timing, weight-dependent rules [36,37], and models with parameters fitted to cortical frequency-based protocols [35] or hippocampal spike-based protocols [38]. We also compare the influence of weight dependence by testing both soft and hard bounds in the synaptic rule. During collective burst firing, synaptic weights converge towards an attractor in the weight space no matter the choice of the synaptic plasticity rule.

Analytical insight into the burst-induced attractor

Why does collective burst firing constrain synaptic weights into an attractor, whereas tonic firing produces a broad spread? To address this question, we derive the fixed points of synaptic weight dynamics under calcium-based and spike-timing–based plasticity rules for a given firing regime and then verify these predictions in network simulations.

We first focus on calcium-based plasticity rules, and we derive the fixed point for one of the rules, [22]. Two thresholds determine the outcome of calcium dynamics: a depression threshold and a potentiation threshold . Each calcium regime is associated with a steady-state value (potentiation or depression) and a characteristic relaxation timescale. When calcium remains below (), the weight does not change. For intermediate calcium levels (), the weight drifts toward the depression steady state with time constant . When calcium exceeds (), the weight relaxes toward the potentiation steady state with time constant . As shown in Fig 2A, calcium fluctuations can therefore be decomposed into potentiation, depression, and neutral intervals. By averaging the fast calcium dynamics, we obtain the fixed point of the synaptic weight:

(1)

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Fig 2. Burst firing constrains synaptic weights toward an attractor via shared calcium dynamics.

A. Calcium trace decomposition into potentiation, depression, and neutral zones, based on pre- and postsynaptic spike timing. Red and orange shading indicate time spent above the potentiation threshold or below the depression threshold, respectively. Right: effective time in each region, weighted by corresponding time constants (, ). B-C. Example spike trains and calcium traces for two synapses during tonic and burst firing. Tonic activity produces diverse calcium dynamics, while burst firing induces synchronized fluctuations across synapses. D-E. Distribution of effective time allocated in different plasticity regions during tonic and burst firing. Tonic firing shows broad distributions, while burst firing produces narrower, more consistent distributions across synapses. F-H. Evolution of synaptic weights over time during tonic (F) and burst (H) firing, starting from the same initial conditions. Weights diverge toward distinct values under tonic input, whereas they converge to a narrow range under burst firing, indicating the emergence of a burst-induced attractor. G-I. Predicted steady-state weight using analytical equations Eq (1) compared with actual simulated weights at different time steps. The linear regression between the predicted values and the actual final values at the end of the stimulation is closer in burst firing compared to tonic firing ( in burst and in tonic).

https://doi.org/10.1371/journal.pcbi.1014001.g002

Here, and denote the effective times spent in the depression and potentiation regimes, respectively. Eq (1) is valid for synapses that spend non-zero time in the depression or potentiation regimes, i.e., when . When there is neither depression nor potentiation (), synaptic plasticity is inactive and Eq (1) does not apply; in that case the synaptic weight remains unchanged and , where is the weight at the onset of the firing state (or equivalently at the switch between firing states). This derivation is based on previous works [37,39]. For more details on the derivation, see S1 Text D.

A similar principle holds for STDP rule, where correlations between pre- and postsynaptic spikes reflect how much time is spent in potentiation versus depression window. For the STDP model of [19], the fixed point is:

(2)

where and are potentiation and depression parameters of the STDP rule, and and are measures of the synaptic potentiation and depression that stems from all input–output correlations with positive and negative time lag, respectively. For more details on the derivation, see S1 Text E. The results for hard-bounds can be found in S1 Text F.

In both cases, the fixed point depends on the plasticity parameters and on activity statistics, time spent in calcium regimes or spike correlations [37,40].

We test these predictions in the generic network shown in Fig 1A, consisting of 50 presynaptic and postsynaptic neurons. The network operates either in tonic mode, with random inputs (0.1–50 Hz), or in burst mode, induced by hyperpolarizing the inhibitory neuron. For details on the computational experiment, see Section Computational experiment related to Fig 2.

During tonic firing, inputs are uncorrelated, so each synapse experiences its own calcium pattern. As shown in Fig 2B-C, calcium traces differ widely across synapses. The resulting effective times in potentiation and depression are broadly distributed (Fig 2D), and weights converge to a wide range of fixed points (Fig 2F). In contrast, burst firing synchronizes activity, producing similar calcium traces and correlations across the network (Fig 2E). Consequently, all synapses converge toward a narrow set of fixed points—the burst-induced attractor (Fig 2H).

Analytical predictions closely match simulated weights (Fig 2G–I), with stronger correspondence in bursts () compared to tonic activity (). The lower in tonic reflects both finite-time and stochastic effects. Simulations are run for 50 s per state to keep tonic and burst epochs comparable and within biologically and computationally plausible timescales; within this window, some synapses are analytically predicted to converge toward low values but do so only slowly, which increases residuals. In addition, tonic firing is driven by distinct noisy input spike trains, leading to heterogeneous calcium dynamics and variability in the effective time spent in potentiation and depression across synapses.

In contrast, collective burst firing synchronizes spike trains and calcium fluctuations across the network, producing more homogeneous plasticity drive and faster convergence toward the analytically predicted attractor. The STDP model shows the same pattern, lower in tonic () and higher in burst (), for analogous reasons. Notably, the convergence demonstrated here characterizes the direction and structure of synaptic drift under burst firing; in shorter burst epochs, weights would be expected to move partially toward the same attractor, potentially interacting with slower consolidation mechanisms. In other neuromodulatory or biophysical regimes, transitions into burst firing could additionally rescale plasticity parameters (and thus the effective learning rate), so that synapses are still drawn toward the same burst-induced attractor but reach it on faster or slower timescales than those illustrated here.

Together, these results show that burst firing creates an attractor in weight space because shared activity patterns synchronize the effective plasticity drive across synapses, forcing them toward similar fixed points. Tonic firing, by contrast, disperses weights because each synapse follows its own independent activity statistics.

Burst-induced attractor sensitivity to rhythmic state modulation

We next apply our analytical result to show how the burst-induced attractor can be modulated in practice, turning the apparent “reset” into a flexible mechanism for consolidation. Our framework predicts that altering the collective bursting dynamics modifies the effective time spent in potentiation and depression regimes (in calcium-based rules) or the spike-time correlations (in spike-based rules). As a result, the location of the attractor can be shifted without any other alteration.

To illustrate this, we return to the network described in Fig 1A. We emulate changes in the neuromodulatory drive by varying the external current applied to the inhibitory neuron across several values (Fig 3A, ; the values used in each epoch are indicated above the pink trace and along the -axis in Fig 3B). This manipulation mimics changes in neuromodulatory drive and produces distinct patterns of collective bursting activity, as seen in the raster plots and LFP profiles (Fig 3A). For details on the computational experiment, see Section Computational experiment related to Fig 3.

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Fig 3. Modulation of burst-induced attractor in synaptic weights.

A. Evolution of the network activity and synaptic weights when the network is driven from tonic to burst firing epochs with three variations of the collective burst firing (achieved by modulating the external current applied to the inhibitory neuron). indicates the end of the tonic firing state, and indicates the end of the -th burst firing state. 100 traces are randomly displayed among 2500 synaptic connections. The steady-state values associated can be modulated by the collective burst firing, moving the attractor in the weight space as a function of burst structure (e.g., spike content and duration). B. Summary of synaptic weights at the end of each burst epoch as a function of . Vertical segments denote the standard deviation across all synapses, while horizontal bars indicate the mean weight. Tonic firing yields a broad spread of synaptic weights, whereas collective bursting compresses weights toward a narrow attractor whose location shifts smoothly with .

https://doi.org/10.1371/journal.pcbi.1014001.g003

Consistent with the analytical prediction, the burst-induced attractor shifts slightly depending on the bursting regime, yet synaptic weights still converge within a narrow region of weight space (Fig 3A, bottom). The magnitude and direction of this shift reflect changes in the structure of collective bursting rather than the external current itself: regimes with stronger or longer bursts generate calcium transients that more frequently cross potentiation thresholds, biasing the attractor toward higher values. Because intrinsic neuronal heterogeneity and network interactions shape burst synchrony and spike content, the attractor position emerges from the resulting burst dynamics rather than from a simple linear dependence on the inhibitory drive.

Fig 3B summarizes this effect by plotting, for each burst epoch, the mean (horizontal segments) together with their standard deviation (vertical bars) as a function of . Tonic firing leaves weights broadly spread between the minimum and maximum values (Fig 3A), whereas collective bursting compresses them toward a narrow attractor whose position shifts smoothly with , showing that bursting reorganizes synaptic weights in a structured and controllable way.

Neuromodulated and tag-dependent synaptic plasticity rules can split and shift the burst-induced attractor for memory consolidation

We can show the potential of the burst-induced attractor as a mechanism for synaptic consolidation and selection by applying our analytical result (Eq (1) and Eq (2)) to show how the burst-induced attractor can be modulated, in practice, by altering the plasticity parameters (e.g., or for calcium-based rule or or for STDP-like rule).

Recent evidence shows that synaptic plasticity is modulated by neuromodulators such as acetylcholine, dopamine, noradrenaline, serotonin, and histamine [31,41–51]. These neuromodulators influence plasticity at multiple stages of induction and consolidation [45,47,52,53]. Several computational models have incorporated neuromodulated plasticity rules [54,55]. For example, in sleep-dependent memory consolidation, the classical STDP kernel observed during wakefulness can be replaced with a purely depressive kernel during sleep, promoting synaptic down-selection [56]. Other models have shown that acetylcholine or dopamine can dynamically shift the STDP kernel to explain receptive field plasticity and reward-driven navigation [57,58]. Additional frameworks introduce neuromodulation via eligibility traces or third-factor models [43,50,51].

Therefore, we model a simplified circuit of one inhibitory neuron connected to two excitatory neurons. Two consecutive 30-second periods of tonic and burst firing were simulated. Six circuit instances were tested: three with highly correlated excitatory firing (25–60 Hz) and three with low correlation ( 10 Hz). For details on the computational experiment, see Section Computational experiment related to Fig 4. During the second burst epoch, the potentiation level was reduced from 0.65 to 0.2, producing a global shift in the burst-induced attractor (Fig 4B). By modulating these parameters, neuromodulators can shift the burst-induced attractor in the synaptic weight space. This effect can be replicated in spike-based rules by decreasing the amplitudes of potentiation () and depression () in the STDP kernel (S3 Fig). Dynamically, the same shifted attractors would be obtained if one started from an arbitrary heterogeneous weight distribution and integrated the burst-regime plasticity with these neuromodulated parameters; here, the preceding tonic phase simply provides a plausible way to generate such heterogeneity.

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Fig 4. Neuromodulated and tag-dependent synaptic plasticity rules exploit the burst-induced attractor for memory consolidation.

A. Calcium-based (left) and spike-based (right) plasticity rules, in which potentiation and depression parameters (, , ) are modified by neuromodulation. B. Evolution of synaptic weights in six circuits subjected to two tonic–burst epochs. In the last burst epoch, the plasticity rule is down-neuromodulated (yellow bar), leading to a different lower burst-induced attractor. Half of the circuits display correlated excitatory activity (gray) and half uncorrelated activity (black). C. Calcium-based (left) and spike-based (right) plasticity rules with tag-dependent neuromodulation. Tagged synapses (orange) receive up-neuromodulation, while untagged synapses (yellow) receive down-neuromodulation. D. Same protocol as in B, but with tag-dependent neuromodulation during the last burst epoch (yellow–purple hatched bar). Synapses tagged during the tonic state (weights above threshold) converge to a higher burst-induced attractor, while untagged synapses converge to a lower one, resulting in bimodal weight consolidation.

https://doi.org/10.1371/journal.pcbi.1014001.g004

We next apply weight-dependent parameter changes only to synapses that exceed a tagging threshold at the end of the preceding tonic phase. Tagged synapse experience enhanced potentiation parameters (), while untagged synapses are downscaled ().

This selective modulation stabilizes tagged synapses near the higher values attractor and drives untagged synapses toward the lower one, leading to a bimodal distribution (Fig 4D). This tag-dependent burst-induced attractor implements selective stabilization for strong synapses while weakening others, a computational realization consistent with synaptic tagging and capture [59–62]. In principle, the same bimodal consolidation could be obtained by assigning tags directly to an initial weight distribution and then applying burst-regime plasticity, but in our protocol the tonic phase provides a natural setting for tag formation.

Importantly, this process is not limited to bimodal outcomes. Because the attractor depends directly on the chosen parameters, any mapping from synaptic weight to parameter values can produce different target distributions during bursting. The resulting distribution can therefore be unimodal, bimodal, multimodal, or graded.

Together, these results show that neuromodulation and tagging make the apparent reset during bursting a flexible consolidation stage. Bursting compresses weights into an attractor, but the attractor itself can be shifted or split into several stable states depending on parameter changes.

Computational models

Neuron and network model.

All neurons are modeled using a single-compartment conductance-based using Hodgkin-Huxley formalism [68]. The membrane voltage of a neuron evolves according to the equation described by [5,30]:

where represents the membrane capacitance, is the leak current, are the intrinsic ionic currents with the set of all ionic channels, are the synaptic currents with the set of all synaptic neurotransmitter types and the set of all presynaptic neurons, and denotes the applied current. Details about the ionic and synaptic currents and their associated dynamics and parameters are provided in S1 Text.

The neuronal network comprises an inhibitory neuron () projecting onto all excitatory neurons through GABA_A and GABA_B connections (refer to Fig 1B). Excitatory neurons are interconnected via a feedforward AMPA synapse, where the presynaptic neurons () influence the postsynaptic neurons (). The number of excitatory neurons varies across different computational experiments. The excitatory synaptic current perceived by the postsynaptic neuron from presynaptic neuron is characterized by:

where represents the synaptic weight, represent the maximal conductance of the AMPA postsynaptic receptor (AMPAr). The variable denotes the gating variable of the AMPAr, dynamically modulated by the presynaptic membrane voltage () and is the reversal potential of AMPAr.

Switch from tonic to burst firing

The inhibitory neuron dynamically influences the network activity. An external current applied to the inhibitory neuron () is used to control the firing regime of the network, mimicking the effect of neuromodulatory signals. A depolarizing current induces tonic firing activity in this inhibitory neuron, causing all excitatory cells to remain at their resting potential until stimulated. A sufficiently large external depolarizing pulse can evoke action potentials in excitatory neurons. Conversely, a hyperpolarizing current applied to the inhibitory neuron switches the entire network into a synchronized collective bursting activity throughout the network [5,69]. In each computational experiment, the depolarizing current is equal to 3 nA/cm² for tonic state and -1.2 nA/cm² for burst state.

The use of a single inhibitory neuron to control network state is not intended as a literal microcircuit model, but as a coarse-grained representation of state-dependent inhibitory or neuromodulatory control of network activity. Similar low-dimensional control motifs are commonly used in models of rhythmic bursting and brain-state transitions, where inhibition determines whether excitatory populations operate in tonic or bursting regimes [70–72]. In our setting, hyperpolarizing this inhibitory neuron unmasks intrinsic bursting dynamics in the excitatory population, rather than imposing bursts through externally patterned input.

External applied pulse train current

In tonic firing states, each excitatory neuron is triggered by an applied current () to make the neuron fire at a nominal frequency . To generate this input-driven tonic activity, a neuron receives a pulse train current, where each pulse lasts 3 ms and has an amplitude that is independently sampled from a uniform distribution on an interval between 50 nA/cm² and 60 nA/cm². The interpulse intervals (i.e., the time between two successive pulses) are independently sampled from a Normal distribution with a mean equal to and a standard deviation equal to .

In burst firing states, each excitatory neuron undergoes a noisy baseline such as the applied current () randomly fluctuating between the interval 0 and 1 nA/cm². This amplitude is chosen to remain well below the levels required to elicit spikes during tonic firing, so it acts as background noise rather than a driver of bursting; the rhythmic burst pattern therefore arises from intrinsic conductances and network interactions, not from an externally imposed oscillatory input.

Homogeneous and heterogeneous network

We define a homogeneous network (or circuit), where each neuron has the same intrinsic ion channel maximal conductances (numerical values provided in S1 Text). We define a heterogeneous network (or circuit), where the maximal conductance for each neuron is randomly chosen within a ± 10 % interval around its nominal value , such as with .

Local field potential

The local field potential (LFP) measures the average behavior of interacting neurons. It reflects the collective excitatory synaptic activity received by the postsynaptic neuron population. The overall synaptic activity is measured by the mean of the individual synaptic currents:

where is the number of postsynaptic neurons and is the number of presynaptic neurons [5,30].

Synaptic plasticity

Calcium-based model.

The change in the synaptic weight between a presynaptic neuron and a postsynaptic neuron is governed by the calcium-based model proposed by [25]:

(3)

Here, represents the time constant, defines the stable state (equal to 0.5), is the potentiation rate, is the depression rate, is the potentiation threshold, and is the depression threshold. The function is the Heaviside function, which returns 1 when the argument is positive and otherwise.

The change in synaptic weight depends on the calcium concentration , which is the sum of the calcium caused by the activity of the presynaptic neuron and the activity of the postsynaptic neuron . A pre- or postsynaptic spike translates into a calcium exponential decay. Further explanations are provided in S1-B Text. The calcium-based rule defined by [25] implements a soft-bound, where the perceived potentiation rate is smaller for high than for low . The same reasoning applies to the depression rate . Detailed parameter values are provided in S1 Text C.

In Fig 2–4, the calcium-based model is modified according to the version of [22]. The model is similar except that the stable state is removed and the fitted plasticity parameters are adapted:

(4)

The only difference between the two calcium-based models is that when calcium levels fall below the depression threshold, the synaptic weight in the 2016 model is no longer subject to the cubic term and therefore remains unchanged. Both rules can be applied, and in either case the burst-induced attractor is present. However, the 2016 formulation is more convenient for demonstrating this phenomenon.

Computational experiment related to Fig 1

Fig 1A-B illustrate the activity in the feedforward network, comprising 50 presynaptic neurons connected to 50 postsynaptic neurons. The network is heterogeneous. The network is in tonic firing mode during 1.5 s. Each excitatory neuron receives a train of current pulses with nominal frequencies randomly sampled from a uniform distribution between 0.1 and 50 Hz. Then, the network switches to synchronized collective bursting during 2.5 s.

Fig 1C illustrates the evolution of the weights, consisting of presynaptic neurons connected to postsynaptic neurons, along with one inhibitory neuron projecting GABA current onto all of them. It comprises a total of weights. This computational experiment consists of states, interleaving tonic and burst firing, each lasting 20 s. During tonic firing, neurons are stimulated with a pulse train current, where the pulse frequency for each neuron is randomly chosen from a uniform distribution between 0.1 Hz and 50 Hz.

Fig 1D demonstrates the values of the normalized weights at the end of the fourth tonic state (vertical axis) with the values of the normalized weights at the end of the third tonic state (horizontal axis). The weights are normalized using the formula

where and denote the maximum and minimum values obtained by comparing the values at the end of the third and fourth tonic states. The distribution of the normalized weights is represented for bins of width 0.025 between 0 and 1. Similarly, for burst firing (Fig 1E), the same data analysis protocol is used for the synaptic weights at the end of the third burst state and the end of the fourth burst state. The scatter plots computed for the other states are shown in S1 Fig.

Computational experiment related to Fig 2

Fig 2A-C illustrate the calcium fluctuations between two presynaptic neuron j connected to one postsynaptic neuron i. The calcium dynamics are following [25] model, (iii) [22,25] (see S1 Text). Fig 2D-E are the time spent in each calcium region (see The burst-induced attractor emerges in a wide range of synaptic plasticity models for calculation). Fig 2F-H illustrate the evolution of the synaptic weights during two states of 50 s within the same network of 50 pre- and 50 postsynaptic neurons for 100 weights among 2500. The duration of 50 s per state was chosen as a compromise between biological plausibility and computational cost, and to allow a direct comparison between tonic and burst regimes under matched time windows. The network has the same set of intrinsic and synaptic conductances as in Fig 1. The two states are initialized with the same set of initial conductances chosen randomly within 0 and 1. Fig 2G-I are the comparison of the predicted value given by the Eq (1) and the simulated value directly extracted during the simulation.

Computational experiment related to Fig 3

Fig 3A (bottom) illustrates the evolution of the synaptic weights during four states of 20 s within the same network of 50 presynaptic and 50 postsynaptic neurons, together with one inhibitory neuron, maintaining identical maximal conductances as in Fig 1. The first state corresponds to a tonic state, where the inhibitory neuron is depolarized at 3 nA/cm² during 20 s. The excitatory neurons receive a pulse train current with the pulse frequency randomly chosen from a uniform distribution between 0.1 Hz and 50 Hz for each neuron. This tonic state is succeeded by three bursting states, each lasting 20 s. The hyperpolarizing current was varied across multiple values in a hyperpolarized range (from approximately -1.7 nA/cm² to -0.9 nA/cm²), as indicated in Fig 3A and 3B, thereby producing distinct collective bursting regimes.

Fig 3B displays the distributions of the synaptic weights at the end of each burst state. The synaptic weights are not normalized, and the distribution is computed for bins of width 0.025 between 0 and 1.

Computational experiment related to Fig 4

Fig 4B shows the evolution of synaptic weights over four 30 s states in six homogeneous circuits. Each circuit consists of one inhibitory neuron connected to two excitatory neurons. During the tonic firing states, three circuits are configured with correlated excitatory activity, where the pulse frequency is selected between 30 Hz and 60 Hz, and three circuits are configured with uncorrelated activity, with pulse frequencies between 0 Hz and 10 Hz. This setup allows us to sample a broad range of initial weight trajectories, from potentiation to depression. In the second burst state, the potentiation level is reduced from 0.65 to 0.2.

Fig 4D uses the same circuit configuration and parameters, except that at the end of the second tonic state, synapses are tagged according to their weight values. Synapses with at the end of the tonic state are assigned an up-neuromodulated calcium-based rule with , while the remaining synapses follow a down-neuromodulated rule with .