Oscillatory Dynamics for a Coupled Membrane-Bulk Diffusion Model with Fitzhugh--Nagumo Membrane Kinetics
Bibliographic record
Abstract
Oscillatory dynamics associated with the coupled membrane-bulk PDE-ODE model of Gomez-Marin, Garcia-Ojalvon, and Sancho [Phys. Rev. Lett., 98 (2007), 168303] in one spatial dimension is analyzed using a combination of asymptotic analysis, linear stability theory, and numerical bifurcation software. The mathematical model consists of two dynamically active membranes with Fitzhugh--Nagumo kinetics, separated spatially by a distance $L$, that are coupled together through a diffusion field that occupies the bulk region $0<x<L$. The flux of the diffusion field on the membranes at $x=0$ and $x=L$ provides feedback to the local dynamics on the membranes. In the absence of membrane-bulk coupling the membrane kinetics have a stable fixed point. The effect of bulk diffusion is to trigger either synchronous and asynchronous oscillations in the two membranes. In the singular limit of slow-fast membrane dynamics, and with only one diffusing species in the bulk, phase diagrams in parameter space showing where either synchronous or asynchronous oscillations occur, together with the corresponding Hopf frequencies at onset, are provided analytically. When the membrane kinetics is not of slow-fast type, a numerical study of the stability problem together with the numerical bifurcation software XPPAUT [G. B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002] is used to to construct global bifurcation diagrams of steady-states and the bifurcating periodic solution branches for the case of either one or two diffusing species in the bulk. Overall, our results show the existence of a wide parameter range where stable synchronous oscillatory dynamics in the two membranes can occur. Predictions from the analytical and bifurcation theory are confirmed with full numerical simulations of the PDE-ODE system.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".