Next-Generation Neural Mass Models

Updated 4 December 2025

NG-NMMs are mathematically rigorous models derived via exact mean-field reduction of spiking neuron dynamics using Lorentzian-distributed inputs.
They capture complex neural phenomena such as oscillations, bursting, cross-frequency coupling, and partial synchrony in low-dimensional ODE systems.
Model extensions incorporate synaptic dynamics, adaptation, and multi-population networks, enhancing predictions in large-scale brain simulations.

Next-generation neural mass models (NG-NMMs) constitute a mathematically rigorous framework for modeling the collective dynamics of large populations of spiking neurons, grounded in exact mean-field reductions from microscopic dynamics. Departing from classical neural mass constructs that rely on heuristic firing-rate transfer functions and phenomenological closures, NG-NMMs exploit the statistical and dynamical properties of quadratic integrate-and-fire (QIF) or θ-neuron networks, typically with Lorentzian-distributed heterogeneities, enabling analytic closure of the macroscopic equations. This approach affords tractable, low-dimensional ODE systems that capture both the firing rate and voltage dynamics, and extend seamlessly to more complex microcircuit architectures, synaptic dynamics, adaptation, and plasticity.

1. Mathematical Foundations and Model Construction

The core principle underlying NG-NMMs is the exact mean-field reduction of all-to-all coupled, heterogeneous neuron populations. When inputs or intrinsic currents are Lorentzian distributed, the formulation yields a closed system for the macroscopic firing rate $r(t)$ and mean membrane potential $v(t)$ : $\dot r = \frac{\Delta}{\pi} + 2 r v,\qquad \dot v = v^2 - (\pi r)^2 + \bar\eta + I(t)$ where $\Delta$ is the Lorentzian half-width (heterogeneity), $\bar\eta$ the center (mean input), and $I(t)$ external drive. The Ott–Antonsen ansatz collapses the population density onto a two-parameter dynamics, exploiting the analytic continuation in the complex plane of the Lorentzian distribution (Coombes et al., 2016, Pietras et al., 26 Sep 2024). This derivation generalizes to synaptic filtering (with additional variables for exponential or second-order kinetics), spike-frequency adaptation, short-term synaptic plasticity, and multi-population networks.

The mathematical tractability is a unique feature of Lorentzian heterogeneity. For general (e.g., Gaussian) distributions, the mean-field dynamics becomes higher-dimensional and less analytically transparent, although qualitative agreement is often maintained away from bifurcations (Pietras et al., 26 Sep 2024).

2. Dynamical Repertoire: Oscillations, Bursting, and Cross-Frequency Phenomena

NG-NMMs exhibit a rich dynamical landscape. They support stationary states, limit cycles via (supercritical) Hopf bifurcation, relaxation oscillations, mixed-mode oscillations, bursting via slow–fast mechanisms, and even low-dimensional chaos, as demonstrated by detailed bifurcation analyses (Ceni et al., 2019, Taher et al., 2021, Ferrara et al., 2022). When equipped with realistic synaptic or adaptation kinetics, these models capture gamma (30–100 Hz), beta, theta (4–8 Hz), and even cross-frequency coupling regimes, including theta–gamma phase–amplitude and phase–phase coupling (Ceni et al., 2019, Segneri et al., 2020, Ferrara et al., 2022).

In models with explicit slow–fast separation, bursting and spike-adding transitions are organized by geometric singular perturbation structures such as canards, folded saddles, and torus canards. For working memory and synaptic facilitation/depression, four-dimensional MF equations reproduce reactivation, multi-item juggling, and gamma/beta oscillations directly observed in cortex (Taher et al., 2020, Taher et al., 2021).

3. Model Extensions: Microcircuit Topologies, Synaptic and Cellular Mechanisms

The NG-NMM framework is extensible to multiple neural populations (E–I, PING/ING), adaptation, plasticity, and multi-scale hierarchies:

PING/ING models with coupled excitatory/inhibitory populations admit exact low-dimensional closures that support gamma oscillations and cross-frequency dynamics (Delicado et al., 3 Dec 2025, Segneri et al., 2020).
Second-order and short-term synaptic dynamics introduce additional macroscopic variables (e.g., synaptic gating $s(t)$ , depression $x(t)$ , facilitation $u(t)$ ), faithfully reproducing the effects of plasticity and supporting complex bursting (Taher et al., 2021, Taher et al., 2020, Clusella et al., 2022).
Spike-frequency adaptation (SFA) generates multi-timescale patterns, including theta–gamma nested oscillations and symmetry-broken chimeras (Ferrara et al., 2022).
Energy-like and near-Hamiltonian structure: In the homogeneous or strong-coupling regime, time-rescaled NG-NMMs exhibit exact or near-conserved Hamiltonians, providing insight into orbit geometry, spike amplitudes, and connections to energy landscape theories used in macroscopic brain data analysis (Andrean et al., 12 Sep 2025).
Canonical microcircuit models allow formal perturbative analysis, phase–amplitude reduction, gradient flows on multi-well potentials, and analytically solvable model inversion via variational Bayes, making possible the identification of multi-stable state itinerancy in real neural data (Cooray et al., 2022).

4. Network and Large-Scale Brain Modeling

Interconnecting NG-NMM nodes with empirically derived connectivity (e.g., structural connectomes) scales the framework to whole-brain models. Each region (node) employs a local NG-NMM (usually a PING-type E–I pair). Stability analyses (linearization, Floquet theory, Master Stability Function/dispersion relations) identify the conditions for homogeneous oscillatory or resting states and their loss of synchrony via transverse modes. The resulting spatiotemporal complexity (traveling waves, cluster states, chaos) and cross-frequency couplings are linked analytically to the spectrum of the connectivity Laplacian and local dynamical properties (Delicado et al., 3 Dec 2025).

Unlike classical rate models, where spatial patterns and multistability are not governed by exact bifurcation geometry, NG-NMMs provide direct mechanistic predictions for the emergence and stabilization of functionally relevant patterns, including gamma amplitude modulated by slow rhythms, without exogenous drives (Delicado et al., 3 Dec 2025).

5. Comparison with Classical Neural Mass Models

Classical neural mass models (e.g., Wilson–Cowan, Jansen–Rit) employ heuristic sigmoidal transfer functions and cannot generally be derived from the exact dynamics of spiking neurons. Consequently, they often fail to capture phenomena depending on instantaneous synchrony, chaos, resonance, or fast oscillatory phenomena (gamma, beta). By contrast, NG-NMMs are exact in the thermodynamic limit and replicate the full dynamical spectrum of the underlying spiking networks, including event-related (de)synchronization, adaptation-driven bursting, and cross-frequency coupling (Coombes et al., 2016, Clusella et al., 2022, Taher et al., 2020).

A key distinction is that the voltage variance (synchrony) is a dynamical variable in NG-NMMs (entering as $(\pi r)^2$ or similar terms), conferring explicit access to regimes of partial synchrony, clustering, and energy-like quantities. Analytical bifurcation theory, based on parameters traceable to neuronal and synaptic biophysics, is thus practical within NG-NMMs but generally unavailable in classical frameworks.

6. Limitations, Universality, and Analytical Tractability

The analytic closure in NG-NMMs relies on the assumption of Lorentzian input heterogeneity, leveraging Cauchy’s residue theorem. For other distributional forms (Gaussian, uniform), the closure leads to higher-dimensional systems; in most regimes, these produce qualitatively similar macrodynamics, but exceptions arise. For example, under gap-junction coupling, Lorentzian heterogeneity may fail to capture diversity-induced synchrony that occurs with Gaussian inputs (“non-universality”) (Pietras et al., 26 Sep 2024). Thus, Lorentzian-based models trade universal physiological representativeness for analytical parsimony.

In stochastic or sparse network regimes, or when incorporating realistic synaptic, dendritic, or molecular mechanisms, model generalizations may lose mathematical tractability or predictive accuracy. Finite-size corrections and structured heterogeneity are not generically handled in closed-form.

7. Practical Applications and Impact

NG-NMMs have been applied to the modeling and interpretation of working memory, seizure dynamics, cross-frequency coupling phenomena in hippocampus and cortex, and the design of neurostimulation protocols (Taher et al., 2020, Ersöz et al., 2020, Ceni et al., 2019, Delicado et al., 3 Dec 2025). Their exact linkage between microscopic neural dynamics and macroscopic observables (LFP, EEG proxies) makes them suitable for inference and data-driven model inversion, including variational Bayesian approaches to dynamic causal modeling (Cooray et al., 2022).

The framework has demonstrated enhanced sensitivity for predicting the effects of weak or uniform external fields (e.g., tES, tACS), supports analytic design of entrainment protocols, and provides direct computation of resonant responses and amplitude gains as functions of synaptic and network parameters (Clusella et al., 2022).

In summary, NG-NMMs are a mathematically rigorous, biophysically faithful class of neural population models that provide low-dimensional, closed-form macroscopic dynamics directly derived from spiking neuronal networks. Their explicit treatment of heterogeneity, synaptic filtering, adaptation, and multiscale network structure underpins their unmatched descriptive power for oscillations, bursting, synchrony, chaos, and cross-frequency phenomena in large-scale brain circuits (Coombes et al., 2016, Pietras et al., 26 Sep 2024, Delicado et al., 3 Dec 2025, Andrean et al., 12 Sep 2025).