Model

This section is divided into three subsections, each delving into the core principles underpinning the mutual coupling between astrocytic and neuronal networks, which form the foundation of our whole-brain model depicted in Fig 1B. In our model, each node encapsulates a mesoscopic brain region represented through a mass model, detailed in the “Neuron-astrocyte mass model” subsection. These nodes structurally interconnect through a two-layered network architecture, distinguishing between neuronal and astrocytic pathways. The “Network extension for the neuronal compartment” subsection elucidates the neuronal network coupling, while the “Network extension for the astrocytic compartment” subsection describes the astrocytic network coupling.

In the following subsections, temporal derivatives are indicated by overdots. Additionally, we adopt the common practice of setting most model parameters homogeneously across all network nodes [9,10], although it is possible in a more detailed model for any given parameter to vary by region and over time. Tables in S1 File provide a comprehensive overview of the symbols used to represent each variable and parameter introduced.

Neuron-astrocyte mass model

This mass model characterizes the coarse-grained temporal dynamics among four distinct, yet coupled, homogeneous subpopulations of neural cells: glutamatergic pyramidal neurons (Pyr), excitatory interneurons (ExIn; a term used here to encompass both pyramidal neurons that are distinct from the primary Pyr population in their laminar position, as well as spiny stellate cells found in certain sensory cortices), GABAergic inhibitory interneurons (InIn), and astrocytes (Ast). It extends the foundational work presented in reference [11], by offering a nuanced portrayal of the interplay between neuronal and astrocytic subpopulations through electrical firing as well as glutamatergic and GABAergic transmission pathways. This extension critically assumes that astrocytic populations exhibit sufficient functional homogeneity to be biologically plausibly modeled within the mass modeling framework [5]. Notably, our enhanced neuron-astrocyte mass model incorporates the effects of glutamatergic gliotransmission as well as stochastic fluctuations arising from both distant regions and the immediate nodal environments.

At the nodal level, the mass model, indexed by n, articulates two primary types of interactions among the subpopulations: neuron-neuron and neuron-astrocyte.

On the one hand, neuron-neuron interactions are abstracted to the dendro-somatic transformations between subpopulation average postsynaptic potentials (E_Pyr, E_ExIn, and E_InIn) and mean firing rates (F_Pyr, F_ExIn, and F_InIn); assuming that feedforward pyramidal neurons receive feedback from excitatory and inhibitory interneurons, excitatory neuronal network feedback (Q_Pyr), and arbitrary excitatory inputs (q). These interactions are formalized in Eqs (1) and (2), while the neuronal network feedback term is explored in more detail in the subsection “Network extension for the neuronal compartment”. For simplicity, and in contrast to reference [11], we omitted the excitatory self-feedback mechanism on the pyramidal neurons.

Postsynaptic potential dynamics: (1) Neuronal firing rates: (2)

Here S is a sigmoidal function defined as:

On the other hand, neuron-astrocyte interactions are modeled as concurrent synaptic releases and uptakes of neurotransmitters into and from the extracellular space (e), as described in [12]. The model specifically considers the two major neurotransmitters: glutamate (Glu) and GABA, for excitatory and inhibitory signaling, respectively, as detailed in Eqs (3) and (4).

Glutamate dynamics: (3)

Glutamate release (J_Glu) is modulated by the firing activity of pyramidal neurons (F_Pyr), while GABA release (J_GABA) is controlled by the activity of inhibitory interneurons (F_InIn). These release dynamics are also regulated by the astrocytic network, specifically through and , a topic elaborated upon in the subsection “Network extension for the astrocytic compartment”. The uptake of extracellular glutamate (Glu_e) is primarily astrocytic, though neurons contribute to a lesser extent (see also Table B in S1 File). In contrast, the uptake of extracellular GABA (GABA_e) is primarily neuronal, with astrocytes playing a subsidiary role (refer also to Table B in S1 File). Post-uptake, neurotransmitters are degraded within astrocytes, as captured by the state variables Glu_Ast and GABA_Ast [12].

GABA dynamics: (4)

Here H is a Michaelis–Menten function defined as:

Critically, the mass model establishes a relationship between extracellular neurotransmitter concentrations and neuronal firing rates through the excitability levels of targeted neuronal subpopulations, as formulated in Eq (5). This relationship manifests in two ways: an elevation in Glu_e generally leads to a bounded (potentially transient) decrease in the excitability thresholds of both pyramidal cells and inhibitory interneurons, while an elevation in GABA_e typically results in a bounded (potentially transient) increase in the excitability threshold of pyramidal neurons. Focusing on excitability thresholds allows the mass model to effectively capture the immediate, dynamic effects of extracellular neurotransmitter levels on these neuronal subpopulations. This approach specifically targets short-term regulatory processes, without conflating these effects with the more stable, long-term synaptic changes associated with phenomena like long-term potentiation and long-term depression. This distinction is crucial for understanding the rapid, transient interactions between astrocytes and neurons, which are central to our study of spontaneous whole-brain activity. For simplicity, the model assumes that excitatory interneurons remain unaffected by changes in extracellular neurotransmitter levels [11].

Given the concurrent nature of all the nodal processes described so far, complex interactions emerge between neuronal excitatory and inhibitory firings, and neuron-astrocyte uptakes and releases of neurotransmitters, fostering a wide repertoire of dynamics across various (fast–slow) timescales [11,12].

Neuronal excitability levels: (5)

Network extension for the neuronal compartment

Consistent with established practices [9,10], neuronal interconnections across the network’s nodes via white matter tracts are presumed to be excitatory, involving solely the pyramidal cell populations. The corresponding network interaction terms for these connections are detailed in Eq (6) and appear in the state variable E_ExIn in Eq (1). These terms are formulated as a linear combination of incoming firing rates (Q_Pyr), with the weights encapsulated in the parameter matrix Ω_Pyr, and a global coupling parameter, ω_Pyr, modulates the relative impact of Ω_Pyr on nodal dynamics. The detailed methodology to define the matrix Ω_Pyr from empirical diffusion magnetic resonance imaging (MRI) data is provided in the section “Defining structural layers” of “Methods”.

Neuronal network feedback: (6)

Network extension for the astrocytic compartment

Due to the current lack of comprehensive experimental data for whole-brain astrocytic modeling, our network model incorporates insights from various microscale studies. These studies have demonstrated and clarified the activities within astrocytic networks that interconnect through gap junctions [1–4,6,13,14]. For instance, the work referenced in [13] describes a process where a portion of glutamate synaptically released by a neuron into the extracellular space can bind to the glutamate receptors of an astrocyte. This binding may initiate a cascade of events, including the production of inositol 1,4,5-trisphosphate within the astrocyte. This compound can trigger calcium release from the endoplasmic reticulum within the same astrocyte and propagate through gap junctions to stimulate calcium release in adjacent astrocytes. Subsequently, the calcium releases may lead these astrocytes to secrete glutamate into the extracellular space, which can diffuse extrasynaptically and bind to pre-terminal neuronal receptors, potentially inducing neurotransmitter release independently of neuronal firing.

Building on these insights, we have developed a preliminary astrocytic network coupling model. This coupling model extrapolates from the structural concept of astrocytes forming a syncytium connected by gap junctions, and from the functional roles of glutamate neurotransmission in facilitating intercommunication between astrocytes, as well as the impact of excitatory gliotransmission on neuronal pre-terminal receptors [3,4,6].

Structurally, the coupling model posits that astrocytic interconnections across network nodes adhere to a syncytial organization, where an astrocytic population within one region connects exclusively with those in immediately adjacent regions along the cortical mantle. To define this lattice-like network structure, we introduce the parameter matrix Ω_Ast, which uses physical proximity as a surrogate for astrocytic coupling facilitated by gap junctional densities. This modeling approach is designed to simulate the attenuation of intercellular signaling that occurs over distance via gap junctions. The weights within Ω_Ast are determined by the reciprocal of the geodesic distances between the centers of mass of the regions, informed by the brain’s cortical surface geometry. The detailed methodology for constructing the parameter matrix Ω_Ast from empirical MRI data is elaborated in the “Defining structural layers” section within “Methods”. Given the novelty of this framework, to our knowledge, there are no established references for deriving a whole-brain astrocytic connectivity matrix from MRI data [1,3,15]. Our use of the reciprocal of geodesic distances between brain regions represents a new approach, providing a biologically plausible approximation of the pathways through which populations of astrocytes interact, as well as the attenuation of intercellular signaling over these pathways across the cortical surface.

Astrocytic network feedback: (7)

Functionally, the coupling model proposes that the astrocytic network feedback ( and ), as outlined in Eq (7), partially modulate the nodal releases of neuronal glutamate and GABA (J_Glu and J_GABA) through excitatory gliotransmission initiated by nodal glutamate bindings. For simplicity, we assume that astrocytic glutamate binding and uptake share similar sigmoidal kinetics, allowing their sigmoidal parameters to be equated. This modeling choice implies that elevated glutamate levels can intensify astrocytic coupling and network feedback. To express how the nodal neurotransmitter release rates (J_Glu and J_GABA) are modulated, we incorporate linear terms in Eqs (3) and (4), combining local (firing-induced) and distal (astrocytic-network-induced) dynamics, with the latter structurally constrained by Ω_Ast. In these equations, the coupling parameters ω_Glu and ω_GABA are introduced to differentiate the astrocytic network’s impact on glutamate release by pyramidal cells versus GABA release by inhibitory interneurons, and they dictate the relative influence of Ω_Ast on nodal dynamics. This overall approach simulates the diffusion-like influence of distal extracellular glutamate concentrations on local neurotransmitter releases. We further simplify by assuming that astrocytic glutamate release into the extrasynaptic cleft, triggered by local glutamate binding, is comparatively less influential than the effects induced by neighboring astrocytes, thereby maintaining a zero diagonal in Ω_Ast. By omitting astrocytic self-feedback mechanisms, we can focus more precisely on network behaviors, offering new insights into how inter-astrocytic and astrocyte-neuron communications orchestrate whole-brain dynamics. Lastly, as the literature provides less evidence for astrocytic network feedback mediated by GABA [14,16,17], we exclude the considerations of GABA-induced gliotransmission in our current model.

To summarize, the modulation of nodal neuronal glutamate and GABA release rates (J_Glu and J_GABA) by the astrocytic network is represented by linear interaction terms ( and ), with weights determined by the parameter matrix Ω_Ast. These terms, detailed in Eq (7), influence the state variables J_Glu and J_GABA in Eqs (3) and (4), respectively. Additionally, two global coupling parameters, ω_Glu and ω_GABA, govern the impact of Ω_Ast on nodal dynamics.

In essence, this postulated coupling model outlines a large-scale neuron-astrocyte network framework where regional glutamate dynamics encourage neighboring astrocytic populations to synchronize their activities. This synchronization is based on a topology symbolizing gap junctional densities, which is distinct from the neuronal topology determined by axonal densities (Fig 1B). This coordinated astrocytic network activity ultimately influences whole-brain patterns of neuronal excitatory and inhibitory firing rates through gliotransmission.

Analyses overview

Our research aimed to elucidate the contributions of astrocytic networks to the patterns of whole-brain activity and functional connectivity. This investigation involved the systematic manipulation of two key global parameters, ω_Glu and ω_GABA, which regulate the astrocytic network modulation of glutamatergic and GABAergic neurotransmissions, respectively (Fig 2A). We established an exploration grid for these parameters, guided by the objective of aligning our model’s outputs—such as the local field potential (LFP = E_ExIn−E_InIn), and the extracellular concentrations of glutamate (Glu_e) and GABA (GABA_e)—with empirically observed characteristics of normative resting-state dynamics in humans. Specifically, we sought to ensure that the LFP dynamics would exhibit oscillations within the electrophysiological alpha band (8–13 Hz), characterized by waxing and waning patterns that underlie amplitude and phase network synchronizations, while maintaining quasi-stationary slow (less than 0.5 Hz) fluctuations in Glu_e and GABA_e through homeostatic balancing of neurotransmitter uptake and release rates (the section “Constraining dynamical regimes” in “Methods” offers more details). These criteria led us to define 1225 distinct pairs of (ω_Glu;ω_GABA) parameters. Each pair was subjected to ten simulations, with each simulation lasting 120 seconds across a network comprising 216 nodes (see also “Simulation scheme” in “Methods”). The analysis encompassed both whole-brain activity and connectivity, employing a bifurcation-based computational framework to interpret the findings (Fig 2B and 2C).

Neuron-astrocyte network activity analysis

For the activity analysis, detailed in the “Neuron-astrocyte network activity analysis” section under “Methods”, we focused on extracting whole-brain metrics and identifying spatial patterns through regional temporal means and temporal standard deviations.

The relationship between the global coupling parameters (ω_Glu;ω_GABA) and whole-brain levels of empirically concrete state variables, Glu_e and GABA_e, is illustrated in Fig 3A. On the one hand, we observed that increasing ω_Glu, irrespective of ω_GABA levels, was linked to elevated levels of both Glu_e and GABA_e. On the other hand, augmenting ω_GABA, independent of ω_Glu adjustments, was associated with a slight reduction in Glu_e levels while still elevating GABA_e levels. These coherent interaction patterns between the simulation parameters (ω_Glu and ω_GABA) and the two tangible neurophysiological variables (Glu_e and GABA_e) were in accordance with the model’s parameterization and matched the theoretical anticipations discussed in the “Network extension for the astrocytic compartment” section within “Model” (see also section S2.2 in S2 File). Notably, the dynamics of neurotransmitter releases and uptakes were designed to evolve over slow temporal scales (which are below 0.5 Hz, as shown in section S2.4 in S2 File), with all parameter adjustments relative to a common baseline network state, leading to gradual and synchronous alterations in Glu_e and GABA_e levels. To further elucidate, an ω_Glu increase from the baseline state triggers a series of whole-brain events, amplifying glutamate release rates (J_Glu) and subsequently raising Glu_e levels. This Glu_e increase, in turn, lowers excitability thresholds for pyramidal cells and inhibitory interneurons (v_Pyr and v_InIn) which elevates their firing rates (F_Pyr and F_InIn), and simultaneously boosts astrocytic network feedback on glutamate and GABA release rates ( and ). The combination of enhanced neuronal firing rates and astrocytic feedback then leads to further increases in J_Glu and J_GABA, and subsequently Glu_e and GABA_e levels. These processes continue until the uptake rates align upwards with the release rates. With ω_GABA elevation, a similar sequence of events occurs, aiming to balance the neurotransmitter release and uptake dynamics, leading to increased GABA_e and slightly decreased Glu_e. It is crucial to note that the outcomes of the ω_Glu and ω_GABA pathways consistently represent the model’s equilibrium states, when the overall balance between neurotransmitter release and uptake rates remains fundamentally unaltered, despite stochastic disturbances or potential transient fluctuations (such as decreases in Glu_e due to increases in GABA_e along the pathway of ω_Glu elevations, or decreases in GABA_e due to decreases in Glu_e along the pathway of ω_GABA elevations). Furthermore, the funnel-shaped parameter space depicted in Fig 3A, outlined by (ω_Glu;ω_GABA), highlights the highly nonlinear influence of ω_Glu on neuron-astrocyte interactions in comparison to ω_GABA. This differential impact is anticipated, given that extracellular glutamate levels not only affect the excitability thresholds of both pyramidal cells and inhibitory interneurons but also modulate astrocytic network feedback mechanisms.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Global analysis of whole-brain activity.

(a) Links between simulation parameters and empirically concrete state variables. The heatmaps show variations in whole-brain levels of Glu_e (top tile) and GABA_e (bottom tile) as functions of ω_Glu (vertical axis) and ω_GABA (horizontal axis). (b) Links between electrophysiology and neurotransmission. The heatmaps depict whole-brain levels of LFP peak-to-peak amplitudes (top tile) and peak frequencies (bottom tile) as functions of whole-brain concentration levels of Glu_e (vertical axis) and GABA_e (horizontal axis). The black solid curves represent isolines of periodic orbit peak-to-peak LFP amplitudes obtained through bifurcation analyses (refer to Fig 4), traversing specific (Glu_e;GABA_e) coordinates (in μmol×μmol): (8; 15), (9; 15), (10; 15), (11; 15), or (11; 20).

https://doi.org/10.1371/journal.pcbi.1012683.g003

A significant insight from the mappings in Fig 3A, between simulation parameters and state variables, was that each simulation, characterized by specific ω_Glu and ω_GABA values, could be uniquely identified by its Glu_e and GABA_e levels. This bijective correspondence enabled us to directly relate LFP characteristics to Glu_e and GABA_e levels, thereby providing a practical framework to discuss the interplay between electrical and chemical activities in the brain. As depicted in Fig 3B, variations in LFP peak-to-peak amplitudes and peak frequencies predominantly followed monotonic trends along the Glu_e and GABA_e axes. Specifically, both the peak-to-peak amplitude and the peak frequency of LFPs reached their extrema when Glu_e and GABA_e levels were at their collective maximums or minimums. Interestingly, intermediate levels of Glu_e and GABA_e were marked by local extrema in LFP characteristics. We gained further insights into these phenomena by formally characterizing the network’s oscillatory dynamics with bifurcation analysis.

This analysis, focused on the amplitude and frequency properties of stable periodic orbits, as showcased in Fig 4, allowed us to identify a continuum where specific levels of glutamatergic and GABAergic activity enable the network model to transition between physiological and epileptic-like oscillatory behaviors. Specifically, under high Glu_e and GABA_e conditions, the network model verged on spiking behaviors, which are often used to simulate low-frequency (below the alpha band), transient, and large peak-to-peak oscillations characteristic of epileptic-like activity (further outlined in section S2.1 in S2 File). This finding elucidates the trends observed in Fig 3B, where increasing peak-to-peak amplitudes and decreasing peak frequencies were noted. Additionally, the distribution of local extrema across intermediate Glu_e and GABA_e levels was captured by contour lines of periodic orbit peak-to-peak amplitudes (further illustrated in section S5.1 in S5 File), indicating that the interplay between network activity characteristics and the underlying bifurcation landscapes of periodic orbits is intricately linked to the amplitude modulation of neuronal bioelectrical activity. Indeed, our network model was parameterized to persistently display oscillatory behaviors within the neuronal compartment, as evidenced by the bifurcation landscapes of stable periodic orbits depicted in Fig 4. Within these landscapes, the dynamics at each node are driven by Glu_e (mediated through v_Glu),GABA_e (mediated through v_GABA), neuronal network feedback inputs Q_Pyr, and white noise q. The oscillatory drives, except for the white noise, are spatially structured through the astrocytic layer Ω_Ast and the neuronal layer Ω_Pyr. This structuring confines each network node to specific areas of the bifurcation landscapes, where local variations in periodic orbit peak-to-peak amplitudes are pronounced, but changes in periodic orbit mean amplitudes or peak frequencies remain relatively subtle, as evidenced by Fig 3B (see also Figs B and D in S2 File). Furthermore, the model’s parameterization ensures that the neighborhoods covered on the parameter plane by each simulation do not overlap in terms of mean states. This non-overlapping ensures that each simulation explores a distinct zone of the periodic orbit bifurcation landscapes (refer to Fig D in S2 File), and allows the summarization of network behaviors in terms of whole-brain states as done in Fig 3. Consequently, the dynamics driven at each node by Glu_e, GABA_e, neuronal network feedback inputs Q_Pyr, and white noise q give rise to spatially shaped amplitude-modulated oscillations within the neuronal compartment, which vary systematically across a continuum of glutamatergic and GABAergic activity. Moreover, while the periodic orbits in the neuronal compartment are responsible for the fastest oscillations in the model (10–11 Hz), the amplitude modulations of these orbits occur at much lower frequencies (less than 0.5 Hz), as illustrated in section S2.4 in S2 File.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Two-parameter bifurcation landscapes of periodic orbits.

Bifurcation diagram drawn with (a) LFP, (b) E_Pyr, and (c) E_InIn as state variables, and v_Glu and v_GABA as bifurcation parameters. The depicted surface outlines the amplitude extrema of periodic orbits, with color gradients indicating frequencies. Areas left semi-transparent signify frequencies below eight hertz, while the black solid curves delineate contour lines of periodic orbit peak-to-peak amplitudes. The same isolines are graphed on a plane (v_Glu;v_GABA) under the surface, and the thickest black solid curves mark the loci of supercritical Poincaré–Andronov–Hopf bifurcation points, synonymous with a contour height of zero. As detailed in Eq (5), v_Glu and v_GABA are sigmoidal functions of Glu_e and GABA_e, respectively. The green rectangle beneath the surface demarcates the domain correspondence between (v_Glu;v_GABA) and (Glu_e;GABA_e) as drawn in Fig 3B, which further outlines the parameter space explored in the simulations (more details in section S2.2 in S2 File). The isolines (except for the Hopf loci) pass through specific (Glu_e;GABA_e) coordinates (in μmol×μmol): (8; 15), (9; 15), (10; 15), (11; 15), and (11; 20). Notably, these isolines differ slightly across the LFP,E_Pyr, and E_InIn landscapes, suggesting that each variable is sensitive to distinct phenomena within the network. This underscores the need to analyse multiple state variables to fully characterize the diverse neural dynamics within the network.

https://doi.org/10.1371/journal.pcbi.1012683.g004

For instance, Fig 5 presents the relationships between neurotransmitter concentration levels and contour lines of periodic orbit peak-to-peak amplitudes with LFP envelope peak-to-peak amplitude patterns (Fig 5A) and LFP amplitude modulation index patterns (Fig 5B). The variations observed closely align with the trends previously identified in Fig 3B (see section S5.2 in S5 File for complementary results illustrating the correlations between the temporal fluctuations of LFP envelopes, Glu_e, and GABA_e). Importantly, Fig 5 highlights a set of critical isolines where profound changes in the amplitude modulation of LFP occur. These critical zones coincide with areas where more subtle changes in other network characteristics are consistently observed, as seen in Fig 3B. This finding reinforces the narrative developed earlier: that the distinctive amplitude properties of the periodic orbit bifurcation landscapes within the neuronal compartment, along with the resulting amplitude modulation of bioelectrical activity, are the principal factors driving global heterogeneity among network nodes (further discussed in section S5.3 in S5 File). These amplitude properties promote the emergence of chimera states and metastable synchrony, driven by the interaction between independently sampled stochastic noise for each node and the structural layers of the network model.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Global effects of neurotransmission on amplitude modulations of bioelectrical activity.

Whole-brain levels of (a) LFP envelope peak-to-peak amplitudes, and (b) LFP amplitude modulation indices, as functions of whole-brain levels of Glu_e (vertical axis) and GABA_e (horizontal axis). The amplitude modulation index was calculated as the ratio of the LFP envelope peak-to-peak amplitude to the difference between the LFP signal peak-to-peak amplitude and the LFP envelope peak-to-peak amplitude. The black solid curves represent contour lines of periodic orbit LFP peak-to-peak amplitudes, consistent with those in Fig 6A. This alignment facilitates comparisons with the clustering analysis results of network activity patterns, which are presented in Fig 6A. Each isoline passes through specific (Glu_e;GABA_e) coordinates in (μmol)×(μmol): (8; 15), (7; 20), or [9,20].

https://doi.org/10.1371/journal.pcbi.1012683.g005

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Clustering analysis of whole-brain network activity and connectivity patterns.

(a) Activity clustering. This panel showcases the results of the clustering analysis based on the spatial patterns of normalized temporal standard deviations for LFP envelopes and normalized temporal means for Glu_e and GABA_e. Each distinct color represents a cluster, corresponding to a unique partition of the (Glu_e;GABA_e) plane. These clusters can be directly mapped back to the simulation parameters (ω_Glu;ω_GABA) through a one-to-one relationship. The black solid curves illustrate the contour lines of periodic orbit peak-to-peak LFP amplitudes. These isolines pass through specific (Glu_e;GABA_e) coordinates (in μmol×μmol): (8; 15), (7; 20), and (9; 20). Further details on these analyses can be found in section S5.4 in S5 File. (b) Connectivity clustering. This panel displays the clustering results related to the global topological properties of the reconstructed multilayer functional networks. Like panel (a), each cluster is denoted by a specific color, indicating distinct partitions within the (Glu_e;GABA_e) plane. The black solid curves in this panel delineate the contour lines of periodic orbit peak-to-peak E_Pyr amplitudes, crossing through specific (Glu_e;GABA_e) coordinates (in μmol×μmol) at: (8.5; 14.0), (8.50; 18.25), and (9.5; 21.0). Additional insights are provided in section S6.2 in S6 File.

https://doi.org/10.1371/journal.pcbi.1012683.g006

Ultimately, the insights derived from Figs 3, 4, and 5 underscore the interplay between network activity characteristics and the underlying bifurcation landscapes of periodic orbits. We further explored these dependencies through a clustering analysis of the simulation parameter plane. This analysis utilized a Gaussian mixture model with the regional temporal standard deviations of LFP envelopes (serving as a proxy for regional LFP amplitude modulation index patterns), and the regional temporal means of Glu_e and GABA_e as predictors. The results of this clustering analysis are visually represented in Fig 6A, which identifies four distinct clusters within the simulation parameter plane. These clusters, which are mostly spatially contiguous, are primarily delineated by contour lines of periodic orbit peak-to-peak LFP amplitudes. Upon further inspection, we found that while periodic orbit E_Pyr and E_ExIn peak-to-peak amplitude isolines were less effective at delineating cluster boundaries, E_InIn demonstrated a level of effectiveness similar to LFP. These findings emphasize the dual roles of glutamatergic pyramidal neurons and GABAergic interneurons in shaping regional profiles of whole-brain network activity patterns: pyramidal neurons by integrating synaptic inputs, and interneurons by exerting critical inhibitory control over regional dynamics.

To provide a clearer depiction of the defining regional characteristics of each identified cluster, Fig 7 presents the cluster means as fitted by the Gaussian mixture model. This visualization aids in understanding the specific effects that variations in the astrocytic network coupling parameters ω_Glu and ω_GABA have on the network’s behavior.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Cluster means from Gaussian mixture model analysis.

This figure illustrates the cluster means for LFP envelopes, Glu_e, and GABA_e, as determined by a Gaussian mixture model with four components. The means represent the spatial patterns of normalized temporal standard deviations for LFP envelopes and normalized temporal means for Glu_e and GABA_e. This figure complements the clustering analysis shown in Fig 6A. To aid in visualization and maintain simplicity, only patterns from the left hemisphere are shown, as they closely correspond to those in the right hemisphere (refer to Fig F in S5 File for full bilateral views).

https://doi.org/10.1371/journal.pcbi.1012683.g007

In cluster A1, regional patterns are relatively uniform across the brain. In contrast, clusters A2, A3, and A4 exhibit distinct spatial variations, with the precuneus and superior parietal cortices markedly differing from the lateral occipital, middle frontal, and temporal cortices. Additionally, while there is a noticeable shift in patterns between clusters A1 and A2, clusters A2, A3, and A4 demonstrate a more gradual progression of changes (see also section S5.4 in S5 File for detailed illustrations). These spatial patterns in clusters A2, A3, and A4 particularly underscore the influence of the neuronal layer Ω_Pyr. For example, in regions such as the precuneus, which are strongly connected within the neuronal layer Ω_Pyr (see Fig 10 and accompanying text in “Methods”), the highest mean levels of Glu_e and GABA_e were observed. These elevated levels resulted in more diverse periodic orbits, as indicated by a wider range of peak-to-peak amplitude distributions, ultimately leading to the most pronounced amplitude modulations. In contrast, the lateral occipital cortices, which are weakly connected within the neuronal layer and characterized by lower mean levels of Glu_e and GABA_e, exhibited more homogeneous periodic orbit peak-to-peak amplitude distributions and the weakest amplitude modulations (refer also to section S2.4 in S2 File). In general, the spatial patterns observed across all clusters reflect our simulation design and suggest that regions with elevated mean levels of Glu_e also exhibit higher mean levels of GABA_e and stronger LFP amplitude modulation, and conversely. This observation is consistent with the global relationships highlighted in Fig 5.

Moreover, as clusters progress from A2 to A4, regional differences become more subtle, accompanied by increasing modularity. This trend underscores the role of the astrocytic network in shaping dynamics across the simulation parameter plane (see also the properties of the astrocytic layer Ω_Ast in section S4.3 in S4 File). For instance, in clusters A1 and A2, where network nodes display homogeneous behaviors or begin to exhibit mixed patterns akin to bifurcation phenomena, astrocytic network modulation is minimal due to low ω_Glu and ω_GABA values. In contrast, in clusters A3 and A4, where nodes exhibit more heterogeneous behaviors, the astrocytic network enforces consistent glutamatergic and GABAergic release rates across adjacent brain regions due to high ω_Glu and ω_GABA values. This homogenization of neurotransmitter dynamics tends to reduce the contrast between regional values.

Overall, these results from Fig 7 align with the narrative developed thus far, which outlines critical peak-to-peak amplitude values on the periodic orbit bifurcation landscapes where network characteristics undergo profound transformations due to heterogeneity induced by white noise across network nodes. Adjacent to these critical zones (cluster A2 in Fig 6A), the bifurcation landscapes divide into regions characterized by homogeneous network characteristics (cluster A1 in Fig 6A) and those showing a continuous spectrum of changing characteristics (clusters A3 and A4 in Fig 6A). These three distinct zones are reflected in the spatial patterns presented in Fig 7 (see also section S5.4 in S5 File).

In sum, the integration of findings from this section elucidates the nuanced and diverse ways in which spatiotemporal whole-brain dynamics manifest across different types of activity (namely, LFP, Glu_e, and GABA_e) in response to astrocytic network modulation and white noise.

Moving forward, we apply multilayer network theory to capture the heterogeneous functional interactions among brain regions and dissect the simulated spatiotemporal whole-brain dynamics more thoroughly.

Neuron-astrocyte network connectivity analysis

For the connectivity analysis, elaborated in the “Neuron-astrocyte network connectivity analysis” section under “Methods”, we reconstructed a four-layered multiplex functional network from the simulated whole-brain activities. This multiplex network, designed to cohesively capture various facets of neural communication, was structured using the identity matrix to represent inter-layer connections, and different bivariate statistical association measures to define intra-layer connections. Specifically, one intra-layer encoded the alpha-band-limited phase locking values (PLV) of LFP dynamics (LFP-PLV), a metric quantifying the similarity between instantaneous phases. Another intra-layer focused on the alpha-band-limited amplitude envelope Pearson-correlations (AEC) of LFP dynamics (LFP-AEC). The remaining layers were dedicated to capturing the Pearson-correlations of Glu_e or GABA_e dynamics (Glu_e-C or GABA_e-C, respectively).

In Fig 8, we examine the relationships between four key global topological properties of the multilayer functional networks (clustering coefficient, path length, edge overlap, and code length) and the whole-brain levels of (Glu_e;GABA_e). The analysis reveals local extrema and generally monotonic trends that align with the contour lines of periodic orbit peak-to-peak E_Pyr amplitudes, suggesting a key role for glutamatergic pyramidal neurons in shaping whole-brain network synchrony, likely due to their predominant excitatory projections via white matter tracts (Ω_Pyr). Complementary findings, including investigations of Von Neumann entropy, code length savings, and modularity, are provided in section S6.1 in S6 File.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Global topological properties of multilayer functional networks.

(a) Clustering coefficient. This metric serves as an indicator of network segregation, where higher values indicate more segregated networks. (b) Path length. This measure serves as an indicator of network integration, where higher values indicate more integrated networks. (c) Edge overlap. This quantity captures edge redundancy within the network, with higher values reflecting a greater consistency in connection weight patterns across different layers. (d) Code length. This is a quality index of community detection from information theory, with lower values reflecting networks with more optimal data compression of a random walker’s movements on them. (a)–(d) In each panel, the black solid curves represent the same contour lines of periodic orbit peak-to-peak E_Pyr amplitudes as in Fig 6B.

https://doi.org/10.1371/journal.pcbi.1012683.g008

Additionally, a clustering analysis was conducted on the simulation parameter plane using a Gaussian mixture model. This model employed clustering coefficient, path length, edge overlap, and code length as predictors, elucidating the dependencies between network connectivity features and the bifurcation landscapes. Fig 6B showcases four clusters that closely mirror those identified in Fig 6A, which were originally used to inform the contour lines in Fig 8. Notably, cluster C2 predominantly encompasses the local extrema depicted in Fig 8 (further discussed in section S6.2 in S6 File).

To visually represent the variability in functional network architectures, Fig 9 displays an exemplary multilayer network from each identified cluster. Key observations from this figure include the following. (i) Across all clusters, PLV-based layers demonstrated connectivity patterns that often contrasted with those observed in correlation-based layers. This contrast was manifested in preferences for short-range connections over long-range ones; or in the differentiation of specific regional connections, such as those between frontal–cingulate–parietal–insula regions as opposed to parietal–occipital–temporal connections, and vice versa. (ii) A consistent feature across all clusters was the similarity in connection densities between Glu_e-C and GABA_e-C layers, which were subtly different from those observed in LFP-AEC layers and clearly distinct from the patterns in LFP-PLV layers. (iii) The analysis further revealed notable variations in connection densities, centrality measures, and community organizations across all clusters.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Mean multilayer functional networks.

This figure shows the mean multilayer functional networks derived from four distinct simulations, each simulation representing a unique cluster identified in our analysis in Fig 6B. The layers within each network are displayed row-wise, while the column-wise arrangement corresponds to the simulations, differentiated by their whole-brain levels of (Glu_e; GABA_e). A color-coded legend above each column indicates the multilayer communities detected within that particular simulation, with node colors matching their assigned community. Within-community edges adopt the color of their respective community, while between-community edges are drawn in black. It is noteworthy that community alignment across layers is generally consistent, though not invariably so. Node sizes vary to represent their eigenvector versatility (an index of nodal centrality), categorized into three distinct ranges: small (from 0 to 1/3), medium (from 1/3 to 2/3), and large (from 2/3 to 1). To maintain visual clarity, only the top 10% of the most substantial edges by weight are shown in each layer, without specific coding for edge weights. The brain views provided include lateral left, lateral right, and dorsal perspectives, with “L” and “R” markers beneath each view indicating the posterior left and right sides, respectively. For detailed examination, including full adjacency matrices, refer to section S6.3 in S6 File.

https://doi.org/10.1371/journal.pcbi.1012683.g009

The interplay between functional network topologies, as captured by amplitude and phase coupling measures, provides a nuanced portrayal of the dynamics discussed in the earlier section, “Neuron-astrocyte network activity analysis”. In that section, we observed that the amplitude and frequency properties within the periodic orbit bifurcation landscapes of the neuronal compartments manifest as markedly distinct patterns across specific zones of the parameter plane. Within these zones, each property exhibits unique sensitivity to stochastic perturbations. Furthermore, in the whole-brain model, amplitude and frequency modulations become intrinsically coupled. This coupling is enhanced by spatial constraints and network topologies, which promote coherent behaviors within the network’s community organization (see Fig 10 and accompanying text in “Methods”). Indeed, synchrony in both amplitude and phase among nodes emerges simultaneously, depending on the variances in the spatially structured drives experienced by each neuronal compartment and the specific zones of their periodic orbit bifurcation landscapes that these drives sample. These synchrony patterns are primarily shaped by the neuronal layer Ω_Pyr, whose small-world topology and community organization, together with stochastic perturbations from white noise, influence how each node’s dynamics navigate the state space defined by the bifurcation landscapes. Moreover, the astrocytic layer Ω_Ast, with its lattice-like topology that reflects the brain’s geometrical embedding, also shapes these patterns by promoting intra-hemispheric and short-range couplings. Thus, amplitude and phase synchrony offer complementary insights into the network dynamics, at times diverging in topologies (clusters C1 and C2 in Fig 6B) or converging to similar patterns (clusters C3 and C4 in Fig 6B), influenced by the astrocytic network coupling parameters. An in-depth exploration of the relationship between phase-based and amplitude-based connectivity patterns is offered in section S6.3 in S6 File.

In essence, the combined insights from Figs 6B, 8, and 9 underscore a broad spectrum of functional network topologies, shaped by different types of connectivity (LFP-PLV, LFP-AEC, Glu_e-C, and GABA_e-C). These topologies reflect the complex modulatory influences exerted by the astrocytic network on neuronal dynamics.

Lastly, to further elucidate the uniqueness of each connectivity layer, we undertook a structural reducibility analysis across all simulations. This investigation aimed to identify and merge layers that provided redundant topological information, thereby optimizing the architecture of each multilayer network. Interestingly, no merging was required in approximately half of the simulations (48.1%), highlighting the distinct topological contributions of each layer. In the remaining simulations, various merging configurations were observed: all correlation-based layers were merged in 31.7% of simulations, some correlation-based layers in 17.5%, and a combination of PLV-based with some correlation-based layers in 2.7%. Crucially, additional examinations of the reduced networks revealed that the insights derived from these optimized configurations closely mirrored those from the analysis of the original network sets. This consistency underscores the robustness of our findings, regardless of network simplification, as further elaborated in sections S6.2 and S6.4 in S6 File.

Constraining dynamical regimes

This study utilized simulations to explore the contributions of astrocytic networks to whole-brain activity and the emergence of functional connectivity patterns. Central to this investigation was an analysis of how variations in the global astrocytic network coupling parameters, ω_Glu and ω_GABA, influenced the dynamics of the network model. These parameters were pivotal in determining the impact of astrocytic network activity on glutamatergic and GABAergic neurotransmissions, which in turn affected neuronal firing rates.

To establish a biologically plausible exploration plane for the parameters (ω_Glu;ω_GABA), a set of criteria was defined ensuring that key model outputs such as LFP, Glu_e, and GABA_e aligned with characteristics typically observed in empirical resting-state human data. The network parameterization was tailored to produce LFP dynamics with alpha band (8–13 Hz) oscillations [34], marked by waxing and waning patterns that facilitate amplitude and phase network synchronizations, while simultaneously preserving quasi-stationary slow (less than 0.5 Hz) fluctuations in Glu_e and GABA_e through the homeostatic regulation of neurotransmitter uptake and release rates.

The parameterization approach involved implementing a quasi-steady-state approximation to manage the slow temporal changes in Glu_e and GABA_e levels, which are markedly slower compared to the rapid dynamics of the neuronal compartment. This approximation simplified a numerical bifurcation analysis, focusing primarily on the neuronal compartment of the neuron-astrocyte mass model as detailed in section S2.1 in S2 File. The goal was to illustrate that insights gained from this simplified bifurcation diagram could effectively predict the dynamical states of the neuron-astrocyte network model under conditions of weak-to-intermediate global neuronal network coupling and the influence of stochastic noise inputs (see also section S2.2 in S2 File). This predictive capability was particularly relevant given that the structural matrices Ω_Pyr and Ω_Ast of the network model were probabilistically row-normalized. Furthermore, all nodes maintained stable periodic orbits within their neuronal compartments by sharing identical parameters, except each node received independently sampled stochastic noise inputs from the same normal distribution, via their parameter q. Importantly, to maintain a focus on neuron-astrocyte interactions and the structural layers of the model, rather than on the realism of whole-brain (neuronal) dynamics, delays due to axonal transmission were not implemented, and a homogeneous parameterization across network nodes was chosen.

The parameterization strategy was meticulously crafted, incorporating physiologically plausible parameter sets sourced from existing literature (detailed in Table B in S1 File) and a numerical bifurcation analysis of the neuron-astrocyte mass model to assess how variations in v_Glu and v_GABA could induce qualitative changes in neuronal dynamics (further explained in sections S2.1 and S2.2 in S2 File). This strategy enabled the following: (i) establish concentration intervals for Glu_e ([5; 15] μmol) and GABA_e ([5; 35] μmol) within which the neuronal compartment would show heightened sensitivity to modulatory impacts; (ii) constrain the neuronal compartment to display persistent oscillatory behaviors (stable periodic orbits) with frequencies within the [8; 13] Hz range and moderate peak-to-peak amplitudes; (iii) define an initial stable dynamical state proximate to a branch of supercritical Poincaré–Andronov–Hopf bifurcation points, ensuring the network model would exhibit baseline noise-modulated oscillatory activity; (iv) sample baseline neuronal firing rates q, independently for each node, from a normal distribution with a mean ± standard deviation of 240 ± 10 Hz to promote the emergence of chimera states and metastable synchrony, where nodes display transient and partially synchronized behaviors; (v) set ω_Pyr at 7.5 to facilitate amplitude and phase network synchronizations; and (vi) identify optimal pairs of values for ω_Glu ([2.90; 6.47] μmol⁻¹) and ω_GABA ([0.14; 1.94] μmol⁻¹), ensuring that variations in Glu_e and GABA_e would remain within the limits pre-specified in (i).

These methodological choices enabled the network model to generate a diverse array of spatially structured neuronal dynamical states. These states were characterized by a spectrum of stable periodic orbits with varying amplitude and frequency characteristics, alongside distinct profiles of spatially structured excitatory and inhibitory activities, as described in sections S2.1, S2.2 and S2.4 in S2 File. In particular, the interactions involving the heterogeneously specified stochastic noise inputs and the structural layers led to diverse clustered synchronized responses within the network. It is important to note that while the choices regarding the LFP frequency band, the concentration bounds for Glu_e and GABA_e, and the initial state settings were easily generalizable, they came at the cost of reducing the physiological plausibility of certain neuronal compartment parameters, as discussed in section S2.3 in S2 File.

Defining structural layers

Empirical magnetic resonance imaging (MRI) data were utilized to construct the two structural connectivity matrices of the whole-brain network model: Ω_Pyr for neuronal populations and Ω_Ast for astrocytic populations. Both matrices were constructed within the anatomical constraints imposed by the Lausanne-2018 surface-based atlas scale three [100], which divides the cortex into 216 parcels (see also S3 File).

The matrix Ω_Pyr was derived from a state-of-the-art tractography-based connectome reconstruction pipeline applied to minimally preprocessed diffusion and structural MRI data from ten subjects in the Human Connectome Project (Young Adult) dataset [101,102]. This pipeline addressed streamline termination and quantification biases inherent in tractography [103]. It integrated three major components: (i) Tractoflow [104], a fully automated diffusion MRI processing pipeline; (ii) SET, or Surface-Enhanced Tractography [105], which uses cortical surface geometry priors to ensure that streamlines accurately intersect with the cortical surface, while also respecting the fanning patterns of fibers within the gyral blades; and (iii) COMMIT-2, or a variant of convex optimization modeling for microstructure informed tractography [106], which employs microstructural and anatomical priors to quantitatively filter and weight streamlines. Executed individually for each subject, this pipeline yielded quantitative connectomes, with connection weights reflecting an anatomical–microstructural measure of connectivity strength. The final matrix Ω_Pyr represented the average of these connectomes across subjects (detailed further in section S4.1 in S4 File). As illustrated in Fig 10, the precuneus regions demonstrate the most substantial connectivity across this matrix, while the occipital regions display the fewest connections. Moreover, this matrix highlights strong interconnections within and between hemispheres in regions such as the frontal–cingulate–insula and parietal–occipital–temporal areas. Together, these connectivity patterns indicate distinct nodal importance and modular organization within the neuronal interconnection connectome Ω_Pyr (refer also to section S4.3 in S4 File).

The matrix Ω_Ast was generated using a high-resolution tessellation of the mid-surface (the midpoint between the white and pial surfaces) of the ICBM-2009c-asymmetric template [107], processed with FreeSurfer [108]. Weights between adjacent parcels were calculated as the reciprocal of the geodesic distance between their mass centers, forming a lattice-like network that mirrors the brain’s geometrical embedding. This structure, representing immediate neighborhood connections along the cortical mantle, used physical proximity as a surrogate for astrocytic coupling facilitated by gap junctional densities (discussed further in sections S4.2 and S4.3 in S4 File).

Both Ω_Pyr and Ω_Ast matrices, illustrated in Fig 10, were normalized prior to initiating simulations: diagonal elements were set to zero, and each row was normalized so that the sum equals one. This normalization ensured that network inputs were proportionately distributed across each node, thereby facilitating balanced contributions and maintaining consistent input magnitudes within the network’s dynamics.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10. Neuronal and astrocytic structural layers.

The neuronal layer (Ω_Pyr) is shown on the left, and the astrocytic layer (Ω_Ast) on the right. For Ω_Pyr, a thresholded version of the connectome that retains the top 25 percent of connections by weight is displayed on the lower diagonal portion to facilitate the visualization of the strongest connections. This aids in the interpretation of functional connectomes. It is important to note that the neuronal interconnection connectome Ω_Pyr is an average derived from multiple subjects. Individual subject connectomes underwent biological filtering to achieve a target density of approximately 25 percent, whereas the averaged connectome was not filtered, resulting in a higher observed density of about 70 percent. This discrepancy underscores the potential implications of averaging connectomes, a topic that merits further investigation. The development of methodologies for deriving biophysically plausible connectomes, both at individual and group levels, is essential due to the significant role of these connectomes in constraining the main spatial and temporal interactions within dynamical models of whole-brain activity. The parcellation and regions follow the conventions specified in S3 File. Parcels in the left hemisphere are prefixed by “lh” and those in the right hemisphere by “rh”. F: frontal; C: cingulate; P: parietal; O: occipital; T: temporal; I: insula.

https://doi.org/10.1371/journal.pcbi.1012683.g010

Simulation scheme

To accurately model the complex, nonlinear interactions between glutamate and GABA dynamics, the simulation parameter plane defined by (ω_Glu;ω_GABA) was non-uniformly sampled. This strategy ensured that the whole-brain levels of Glu_e and GABA_e remained within the physiological ranges of [5; 15] μmol and [5; 35] μmol, respectively. A total of 1225 unique pairs of (ω_Glu;ω_GABA) were identified, which facilitated the uniform sampling of a grid defined by the whole-brain levels of (v_Glu;v_GABA). Detailed methodology is provided in section S2.2 in S2 File.

The dynamic behavior of the whole-brain network model was governed by a system of coupled stochastic differential equations, consisting of 14 equations per node, resulting in a total of 3024 equations. These equations were numerically integrated using an in-house implementation of the stochastic Heun’s integration scheme, utilizing MATLAB version R2022a [109].

To ensure robustness and minimize biases introduced by transient dynamics, the simulation protocol was meticulously structured. Initially, a single calibration simulation lasting 370 seconds was conducted for each parameter pair (ω_Glu;ω_GABA) to establish a stable baseline. The stability and attainment of persistent oscillatory behaviors were visually confirmed in the final ten seconds of these calibration runs. These calibrated end states then served as the initial conditions for ten subsequent simulation batches, ensuring consistency and reliability across the dataset. For each parameter setting, ten 120-second simulations were conducted under distinct stochastic neuronal inputs (model parameter: q) and initial conditions, resulting in a total of 12250 simulations available for subsequent analyses. Further details are supplied in section S2.2 in S2 File.

Neuron-astrocyte network activity analysis

For each parameter pair (ω_Glu;ω_GABA), whole-brain values were derived from each of the ten simulation batches using the following approach: (i) instantaneous amplitude envelopes for the LFP dynamics were derived using a Hilbert transform, and whole-brain LFP peak-to-peak amplitude, or whole-brain LFP envelope peak-to-peak amplitude, was calculated as the mean of regional amplitudes; (ii) whole-brain LFP peak frequency was determined from the mean spectra of regional Welch’s power spectral density estimates; and (iii) whole-brain levels of Glu_e or GABA_e, and the associated variables v_Glu or v_GABA, were computed as the means of regional temporal means. These whole-brain quantities were chosen for analysis to validate the hypothesis that the network model, depending on the settings of ω_Glu and ω_GABA, exhibits diverse neuronal dynamical states. These states are characterized by stable periodic orbits with varying peak-to-peak amplitudes and peak frequencies, along with distinct profiles of excitatory and inhibitory activities (refer to S2 File for more details).

To visualize the relationship between whole-brain quantities and the parameters (ω_Glu;ω_GABA) or the whole-brain levels of the variables (Glu_e;GABA_e), a two-dimensional natural neighbor interpolation method based on Delaunay triangulations was employed (MATLAB’s scatteredInterpolant function). This interpolation was performed for each simulation batch. Subsequently, the mean of the interpolated graphs across all simulation batches was calculated. Given that the whole-brain values of Glu_e and GABA_e showed minimal variation across batches, the mean of these interpolated graphs was found to be almost identical to individual graphs from each batch.

To perform hard clustering analysis, a Gaussian mixture model was applied to the dataset (MATLAB’s fitgmdist function). The variables analyzed included regional temporal standard deviations of LFP envelopes, and regional temporal means of Glu_e and GABA_e dynamics, with each neurophysiological data type comprising a total of 216 predictors. The dataset consisted of observations from all 12250 simulation runs. Each spatial profile of temporal standard deviations or temporal means was scaled to a range between zero and one, individually for LFP envelopes, Glu_e and GABA_e, and separately for each simulation (see also section S5.4 in S5 File for illustrations). The Gaussian mixture model was chosen for its ability to robustly capture the inherent spatial covariance among graph-based variables, present in the dataset across simulations. A full covariance structure common to all components was selected when fitting the mixture model to accurately model the interconnected nature of the data. This configuration effectively addressed the heterogeneity in cluster shapes and sizes. The model was independently fitted for potential groupings of four, five, or six components (see also section S5.4 in S5 File). To ensure the stability and reliability of the clustering results, ten independent runs of the expectation-maximization algorithm were conducted for each component grouping. Each run was initialized using a k-means clustering heuristic. For each grouping scenario, the model configuration yielding the highest log-likelihood across the ten replicates was selected. Ultimately, the fitted model—either four, five, or six components—was chosen based on its ability to strike an optimal balance between minimizing the Akaike information criterion and maintaining model simplicity.

Visualization of the estimated clusters relative to the whole-brain levels of Glu_e and GABA_e was conducted by creating a heatmap. This heatmap was generated by first graphing the posterior probabilities of each component for each simulation batch using natural neighbor interpolations (MATLAB’s scatteredInterpolant function). The probabilities from these interpolations were then averaged across batches. Subsequently, cluster assignments for each observation were determined based on the highest posterior probability. It was confirmed that the mean interpolated heatmap accurately represented the individual batch graphs.

Neuron-astrocyte network connectivity analysis

To effectively analyze functional connectivity, a multilayer network modeling approach was utilized, ideally suited for network systems with functional units that interact through diverse types of connection channels. For each parameter pair (ω_Glu;ω_GABA), a four-layered interconnected multiplex functional network was reconstructed from each of the ten simulation batches. In this multiplex network structure, the same brain region is represented across various layers, with each layer characterized by unique connectivity patterns that reflect distinct types of interactions. To maintain simplicity, inter-layer connectivity was uniformly defined by identity matrices, while each intra-layer was encoded with specific bivariate measures of functional connectivity [34,87]: (i) phase locking values of alpha-band-limited LFP dynamics (LFP-PLV), measuring consistency of phase differences, ranging from 0 for complete desynchronization to 1 for perfect phase synchronization; (ii) amplitude envelope Pearson-correlations of alpha-band-limited LFP dynamics (LFP-AEC); (iii) Pearson-correlations of Glu_e dynamics (Glu_e-C); and (iv) Pearson-correlations of GABA_e dynamics (GABA_e-C). This multilayer functional network was designed to elucidate the interplay between phase and amplitude couplings in LFP dynamics, driven by the spatially structured, slow fluctuations of Glu_e and GABA_e (further discussed in sections S2.2 in S2 File, S5 in S5 File, and S6.3 in S6 File). Instantaneous phases and amplitude envelopes for the LFP dynamics were derived using a Hilbert transform. Subsequently, the amplitude envelopes and the dynamics of Glu_e and GABA_e were low-pass filtered using a 0.5 Hz cutoff frequency. Additionally, for simplicity, all Pearson-correlations were considered in absolute values. Exploratory analyses indicated that substituting Pearson-correlations with Spearman-correlations yielded similar findings and conclusions.

To enhance the clarity of visualizations, each pair of parameters (ω_Glu;ω_GABA) was analyzed across the ten simulation batches to generate a representative multilayer network, calculated as the mean across these batches. For the phase locking value layers, the arithmetic mean was used directly. In contrast, for the correlation layers, the arithmetic mean of Fisher’s z-transformed coefficients was back-transformed to compute the correlations [110]. Although visualizations were based on these averaged networks, the analyses systematically included both the original dataset, comprising 12250 individual networks, and the mean dataset, encompassing 1225 networks (see also section S6.2 in S6 File). Prior to conducting multilayer network analyses, each functional layer within every multilayer network was thresholded to retain only the top 25 percent of the strongest connections.

The multilayer network analyses aimed to clarify the emerging functional connectivity patterns by quantifying various topological properties across nodal, community, and global scales. (i) At the nodal scale, the multilayer centrality of each node was evaluated using eigenvector versatility, a generalization of eigenvector centrality [111]. (ii) Multilayer community-level organizations were explored using the map equation method, which is grounded in information theory [112]. This method simulates the movement of a random walker and identifies multilayer communities by pinpointing groups of nodes among which the walker’s flow is optimally contained for extended periods. By minimizing the description length of the walker’s pathways, the map equation method effectively isolates groups of nodes that serve as information flow hubs within and across layers. (iii) At the global scale, the global clustering coefficient was calculated to measure network segregation, the average path length was assessed to determine network integration, the global edge overlap was analyzed to evaluate edge redundancy across the network layers, and the code length, derived from the map-equation-based community-level analysis, served as a quality index for assessing the effectiveness of community detection [113,114].

Structural reducibility analysis was implemented on each multilayer network to identify and merge similar layers. The quantum Jensen–Shannon divergence was used to quantify layer similarities, and arithmetic means were employed to aggregate similar layers. The topological properties of the resulting reduced networks were quantified in the same manner as those of the original networks. This approach effectively eliminated redundant or uninformative interactions within each reconstructed multilayer functional network [114], as further detailed in sections S6.2 and S6.4 in S6 File.

Hard clustering analyses utilized a Gaussian mixture model, following the approach outlined in the “Neuron-astrocyte network activity analysis” section, with modifications to the dataset and covariance structure (see also section S6.2 in S6 File). The mixture model incorporated four standardized predictors—global clustering coefficient, average path length, global edge overlap, and code length—with a unique full covariance structure specified for each component.

Visualizations of both global topological properties and clusters, plotted relative to the whole-brain levels of Glu_e and GABA_e, employed methodologies consistent with those applied in the “Neuron-astrocyte network activity analysis” section.

Illustrations

To enhance data interpretation and ensure accessibility for readers with color-vision deficiencies, all visualizations were created using the Scientific Colour Maps package [115]. Brain maps and outlines were standardized using the ICBM-2009c-asymmetric template [107].