The Rosetta Stone of Neural Models: A Shared Language for Brain Dynamics

From simple oscillators to laminar neural masses, a map for choosing—and translating between—models.

Editor’s note: This post is a brief preview of our forthcoming paper, “The Rosetta Stone of Neural Models,” to be released in the coming months. A preprint and code will be linked here when available.

When neuroscientists talk about brain rhythms—alpha, beta, gamma—they often speak different dialects. Some use phase-only oscillators; others prefer firing‑rate equations or biophysical neural masses. The Rosetta Stone of Neural Models proposes something simple and powerful: treat these choices as translations within one shared dynamical language. Instead of picking a model by tradition or taste, you navigate a principled map—using mathematics to say what you mean and to compare alternatives.

Why a Rosetta Stone?

Modern data span many scales (laminar probes, EEG/MEG, fMRI) and phenomena (bursts, resonance, synchronization). The Rosetta Stone organizes widely used formalisms into a ladder, showing how they relate by limits, approximations, and coordinate transforms. It’s a guide to model intent and provide use guidelines: when you need only phase; when you need amplitude; when you need explicit excitatory–inhibitory (E–I) loops and synapses; and when you need laminar structure or whole‑brain coupling. Also, it provides insight into the biophysical meaning of parameters in these models, so they can be coupled and used in networks to simulate interventions such as brain stimulation– whole brain models, and digital twins.

Figure 1. Oscillatory dynamics across increasing model complexity. Top row: phase-space portraits. Bottom row: qualitative one-parameter bifurcation diagrams. The horizontal arrow indicates increasing model complexity from left to right. Blue traces highlight attracting (or neutrally stable) periodic orbits and their stable cycle branches; gray traces denote unstable invariant sets and illustrative transients. Panels: (a) phase- only oscillator ( no amplitude dynamics); (b) damped oscillator with a globally attracting fixed point; (c) Stuart–Landau oscillator where a Hopf bifurcation creates a stable limit cycle; (d) Wilson–Cowan E–I model with coexistence of equilibria and oscillations; (e–f) two neural-mass models (NMM1, NMM2) showing parameter-dependent onset and growth of oscillations and possible bistability. Sketches are schematic and not to scale.

The push-pull motif. Think of two coupled variables: one pushes away from rest (excitatory), the other pulls back (inhibitory). That simple interplay is enough to generate oscillations. Expressed as a linear differential equation system (ODE) with two variables (x for “eXcitatory” and y for “Ynhibitory”).

Figure 2. Push–pull motif as a harmonic oscillator. Two linearly coupled variables, x(t) (push/excitatory) and y(t) (pull/inhibitory), satisfy dx/dt = −ω y and dy/dt = ω x (left). With weak damping and an amplitude-limiting nonlinearity, this neutral center becomes a stable limit cycle—the minimal scaffold for cortical rhythms.

This system (the harmonic oscillator in disguise! Just substitute the second equation into the first) is reminiscent of other ODEs producing “limit cycles”. Add a bit of friction, add a nonlinearity to keep amplitudes from blowing up, and you’ve got the skeleton of cortical rhythms. 

A tour of the models

Phase‑only & harmonic oscillators. Start with the simplest: motion on a circle, which can be viewed as the simplest instantiation of the push-pull motif. Add forcing and coupling, and you land in the Kuramoto world for network synchrony. Ideal when phases carry the story (entrainment, coherence, phase‑locking).

Damped linear oscillators. Now amplitude matters—ring‑down and resonance appear. Networks of damped oscillators give peaks in spectra, with coupling shaping modes and delays.

Stuart–Landau (Hopf) oscillators. Through a nonlinearity, we can obtain a model with a preferred frequency and amplitude. Near a Hopf bifurcation, amplitude and phase co‑evolve into a stable limit-cycle. A small set of parameters (growth, frequency, nonlinear damping, and frequency shift) captures how non-linear oscillations are born, stabilized, and frequency‑modulated. This is the workhorse for metastable dynamics and whole‑brain Hopf models.

Wilson–Cowan (E–I) firing‑rate models. Need more biophysical realism? Excitation and inhibition form push‑pull loops that can oscillate, amplify, or settle. A biologically inspired nonlinearity is used, the sigmoid. This is a function representing how a population of neurons responds to an external input, heuristically derived from data. Linearization near equilibrium reveals when Hopf behavior emerges—making the bridge back to Stuart–Landau explicit. Biology becomes more tangible, with time scales for neurons and, if desired, synapses.

Neural mass models with second‑order synapses (NMM1). Here, synapses are second-order filters with meaningful delays –natural rise/decay–, while transduction of voltages into firing rates is implemented using the Wilson-Cowan sigmoid. These models, which are often used in multi-population nodes such as Janse-Rit or the LaNMM,  reproduce familiar EEG rhythms and pathological fast activity, and they can be used to provide clear rules (e.g., Barkhausen‑style criteria) for when E–I circuits will oscillate.

Next‑generation, laminar neural masses (NMM2). While the prior models are derived from data and heuristics, we aspire to neural mass models formally derived from first principles: simple models of a large number of individual neurons (the MPR model). This delivers NMM2, a recent development where the sigmoid is a second-order dynamical system. Add second-order synapses, layers, neurons, and interneuron subtypes; connect regions via structural connectivity and delays. This is the right regime for questions about laminar signatures, cross‑frequency coupling, and network‑level interventions.

The thread that ties them together

Two ideas unify the tour. First, coordinate transforms—between phase, Cartesian, and complex forms—reveal the same geometry across models. Second, synapses as filters (via simple differential operators, Laplace transforms, and Green’s functions) make time constants, resonances, and delays explicit. Viewed this way, an E–I loop is a feedback system whose stability and spectral “voice” you can read off from its poles and gains. The Hopf limit (Stuart–Landau) is then the natural normal form of oscillation—what remains after you compress away non‑essential details.

Choosing a model

1. Is the phenomenon near an oscillatory threshold? If yes (bursts, onset/offset, metastability), use Stuart–Landau or the linear damped approximation, depending on whether amplitude nonlinearities matter.

2. Do you need explicit E–I circuitry or biophysical synapses? Choose Wilson–Cowan,  NMM1 or NMM2 (more complex). You’ll gain interpretability (what changes when inhibition weakens? what does synaptic time‑scale do?). 

3. Are laminar structure, cross‑frequency interactions, or region‑to‑region coupling essential? Move to a laminar model using NMM1 or NMM2 with whole‑brain coupling (connectomes + delays).

   
   

Why it matters

For basic science, a common language lets communities compare mechanisms: is a peak a resonance, a limit cycle, or a noise‑sustained quasi‑cycle? For translational work, the map supports model‑based interventions—from DBS and tES entrainment strategies to seizure propagation and cognitive modulation. In practice, we iterate: start minimal to explain the phenomenon; add biological detail only as needed; and keep one eye on the normal forms that make behavior legible.

Take‑home

The Rosetta Stone is not a dogma; it is a set of translation rules and a habit of mind. Start with the simplest model that could plausibly capture your effect; learn what each term buys you; translate parameters into physiology when you need to intervene; and translate physiology into low-dimensional dynamics when you need intuition.

In the end, the reward for working this way is not only better fits but cleaner thinking. The most satisfying moment in modeling is when two different descriptions collapse to the same idea. That is the feeling of a Rosetta Stone doing its job.

References 

Grimbert F, Faugeras O. Bifurcation analysis of Jansen's neural mass model. Neural Comput. 2006 Dec;18(12):3052-68. doi: 10.1162/neco.2006.18.12.3052. PMID: 17052158.

Sanchez-Todo R, Bastos AM, Lopez-Sola E, Mercadal B, Santarnecchi E, Miller EK, Deco G, Ruffini G. A physical neural mass model framework for the analysis of oscillatory generators from laminar electrophysiological recordings. Neuroimage. 2023 Apr 15;270:119938. doi: 10.1016/j.neuroimage.2023.119938. Epub 2023 Feb 11. PMID: 36775081.

Clusella P, Köksal-Ersöz E, Garcia-Ojalvo J, Ruffini G. Comparison between an exact and a heuristic neural mass model with second-order synapses. Biol Cybern. 2023 Apr;117(1-2):5-19. doi: 10.1007/s00422-022-00952-7. Epub 2022 Dec 1. PMID: 36454267; PMCID: PMC10160168.

Ruffini, Giulio. "Analysis and extension of exact mean-field theory with dynamic synaptic currents." bioRxiv (2021): 2021-09.