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arxiv: 2604.11260 · v1 · submitted 2026-04-13 · 🧮 math.AP

Stochastically perturbed model of cell electropermeabilization

Tobias Geb\"ack , Oleksandr Misiats , Ioanna Motschan-Armen , Irina Pettersson This is my paper

Pith reviewed 2026-05-10 15:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords electropermeabilizationelectroporationstochastic partial differential equationsvariational solutionscoupled SPDE-ODEmultiplicative noisemembrane potentialporosity
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The pith

A stochastically perturbed model of cell electropermeabilization admits a unique variational solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces multiplicative stochastic noise into a deterministic model of reversible cell membrane permeabilization to represent uncertainties such as temperature fluctuations or variations in applied electric fields. The resulting system couples electrostatic equations for the electric potential across cell domains to a nonlinear law for the transmembrane voltage jump and an ODE for membrane porosity degree. Noise acts only on the voltage equation in a degenerate multiplicative way, so any randomness reaches the porosity variable indirectly through their nonlinear interaction. The authors prove existence and uniqueness of a variational solution by verifying that the nonlinear terms obey conditions sufficient for a Galerkin approximation to converge. Simulations of the solutions and their time averages for additive and multiplicative noise cases give numerical indication that an invariant measure exists.

Core claim

We present a stochastically perturbed version of a phenomenological electroporation model. The deterministic model couples the electrostatic equations for the electric potential in the extra- and intracellular domains with a nonlinear evolution law for the transmembrane potential jump, itself coupled to an ordinary differential equation describing the porosity degree of the membrane. To account for various random effects, we add noise on the cell membrane. We establish the existence and uniqueness of a variational solution to the resulting coupled SPDE-ODE system governing the membrane potential and the degree of porosity, where the stochastic perturbation is multiplicative and degenerate, 1

What carries the argument

The coupled SPDE-ODE system for membrane potential and porosity degree, with multiplicative degenerate noise acting only on the SPDE component so that any mixing in the ODE arises indirectly through nonlinear coupling.

If this is right

  • The model remains mathematically well-posed after adding noise to represent biological uncertainties.
  • Random effects reach the porosity variable only through the nonlinear coupling to the noisy potential equation.
  • Numerical simulations indicate the existence of an invariant measure for both additive and multiplicative noise.
  • The system can be approximated by Galerkin methods when the nonlinearities meet the required monotonicity and coercivity conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The suggested invariant measure would allow statistical predictions of long-term membrane permeability under repeated electric pulses.
  • Similar stochastic perturbations could be added to other coupled models of cellular electrophysiology while preserving well-posedness.
  • The numerical evidence for an invariant measure opens the possibility of studying ergodic properties or convergence rates in this setting.

Load-bearing premise

The nonlinear terms satisfy generalized monotonicity and coercivity conditions that allow a Galerkin approximation procedure to establish existence of a solution.

What would settle it

An explicit choice of parameters or initial data for which Galerkin approximations diverge or multiple distinct solutions appear would falsify the claimed existence and uniqueness.

Figures

Figures reproduced from arXiv: 2604.11260 by Ioanna Motschan-Armen, Irina Pettersson, Oleksandr Misiats, Tobias Geb\"ack.

Figure 1
Figure 1. Figure 1: Membrane potential v and degree of porosity w in the case of additive noise. In order to investigate numerically ergodicity of the system, we compute the time averages of ∥v(t, ·)∥ 2 L2(Γ) and ∥w(t, ·)∥ 2 L2(Γ): ⟨∥v∥ 2 L2(Γ)⟩T = 1 T − Tburn−in Z T Tburn−in ∥v(t, ·)∥ 2 L2(Γ) dt, ⟨∥w∥ 2 L2(Γ)⟩T = 1 T − Tburn−in Z T Tburn−in ∥w(t, ·)∥ 2 L2(Γ) dt. (46) The time averages at the pole θ = π for Tburn−in = 30 µs a… view at source ↗
Figure 2
Figure 2. Figure 2: Time averages in the case of additive noise. [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sample standard deviations of ⟨∥v∥ 2 L2(Γ)⟩T and ⟨∥v∥ 2 L2(Γ)⟩T in the case of additive noise for two MC runs, Tburn−in = 30 µs, Tfinal = 300 µs. 5.2 Multiplicative noise We consider a multiplicative noise of the form dWt = αvdζ(t), a linear multiplicative noise with α = 0.5, where ζ is a real-valued standard Brownian motion. We choose Tburn−in = 50 µs and the final time Tfinal = 300 µs. As in the previous… view at source ↗
Figure 4
Figure 4. Figure 4: Membrane potential v and degree of porosity w in the case of linear multiplicative noise. Acknowledgements A part of this work was done during the stay of I. Motschan-Armen and I. Pettersson at VCU, whose hospitality is greatly appreciated. The research stay at VCU was supported by the Barbro Osher Pro Suecia Foundation. O. Misiats visit to Chalmers was funded by Olle Engkvist’s (project 227-0235).The rese… view at source ↗
Figure 5
Figure 5. Figure 5: Time averages in the case of linear multiplicative noise. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Sample standard deviation of ⟨∥v∥ 2 L2(Γ)⟩T and ⟨∥v∥ 2 L2(Γ)⟩T in the case of linear multiplicative noise. References [1] Bassim Al-Sakere, Franck Andr´e, Claire Bernat, Elisabeth Connault, Paule Opolon, Rafael V. Davalos, Boris Rubinsky, and Lluis M. Mir. Tumor ablation with irreversible electroporation. PLoS ONE, 2(11):e1135, November 2007. [2] Martin S. Alnaes, Anders Logg, Kristian B. Ølgaard, Marie E.… view at source ↗
read the original abstract

Reversible electropermeabilization, commonly referred to as electroporation, is a transient increase in cell membrane permeability induced by short, high-voltage electric pulses. We present a stochastically perturbed version of a phenomenological electroporation model introduced in the deterministic setting by \cite{kavian2014classical}. The deterministic model couples the electrostatic equations for the electric potential in the extra- and intracellular domains with a nonlinear evolution law for the transmembrane potential jump, itself coupled to an ordinary differential equation describing the porosity degree of the membrane. To account for various random effects, such as temperature fluctuations or uncerntainty in the applied electric field, we add noise on the cell membrane. We establish the existence and uniqueness of a variational solution to the resulting coupled SPDE-ODE system governing the membrane potential and the degree of porosity, where the stochastic perturbation is multiplicative and degenerate, acting only on the SPDE component of the coupled SPDE-ODE system. Any mixing in the ODE variables is therefore induced indirectly through the nonlinear coupling in the drift. The main technical challenge arises from the nonlinearities, which are neither Lipschitz continuous nor monotone. The result is proved by means of Galerkin method, following the methodology by Liu and R\"ockner \cite{liu2015stochastic} for treating equations under generalized monotonicity and coercivity conditions. Finally, we present numerical simulations of the solution and its time averages for both additive and multiplicative noise, that provide a numerical indication for existence of invariant measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces a stochastically perturbed version of the deterministic electroporation model from Kavian et al., yielding a coupled SPDE-ODE system for membrane potential and porosity degree. Multiplicative degenerate noise acts only on the SPDE component; mixing in the ODE is induced indirectly via nonlinear coupling. Existence and uniqueness of a variational solution are claimed via the Galerkin method under the generalized monotonicity/coercivity framework of Liu and Röckner, with numerical simulations provided as evidence for an invariant measure.

Significance. If the central verification holds, the result supplies a rigorous existence theory for a biologically motivated stochastic model with non-Lipschitz nonlinearities, extending deterministic electroporation analysis to include random effects such as field fluctuations. The indirect-mixing mechanism for the ODE component is a technically interesting feature of the coupled system.

major comments (2)
  1. [Existence proof (application of Liu-Röckner framework)] The existence proof rests on verifying that the specific electroporation nonlinearities (the transmembrane potential evolution law and the porosity ODE right-hand side) satisfy the precise coercivity, generalized monotonicity, and growth conditions of Liu-Röckner (their Theorem 2.1 or equivalent). The manuscript asserts compliance but does not exhibit the explicit estimates or inequalities for these terms; without this calculation the application of the abstract framework cannot be confirmed and the a-priori estimates may fail.
  2. [Stochastic formulation and Galerkin approximation] The treatment of the degenerate multiplicative noise (present only in the SPDE) and the resulting tightness argument for the coupled system must be spelled out; it is unclear from the outline whether the standard Liu-Röckner estimates extend directly when the noise operator has a nontrivial kernel and the ODE receives no direct stochastic forcing.
minor comments (2)
  1. [Abstract] Abstract contains the typo 'uncerntainty'.
  2. [Numerical simulations] Numerical section should specify the spatial discretization, time-stepping scheme, and exact parameter values used for the additive versus multiplicative noise runs so that the reported time averages can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the presentation of the existence proof. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Existence proof (application of Liu-Röckner framework)] The existence proof rests on verifying that the specific electroporation nonlinearities (the transmembrane potential evolution law and the porosity ODE right-hand side) satisfy the precise coercivity, generalized monotonicity, and growth conditions of Liu-Röckner (their Theorem 2.1 or equivalent). The manuscript asserts compliance but does not exhibit the explicit estimates or inequalities for these terms; without this calculation the application of the abstract framework cannot be confirmed and the a-priori estimates may fail.

    Authors: We agree that explicit verification of the coercivity, generalized monotonicity, and growth conditions is necessary to confirm the applicability of the Liu-Röckner framework. In the revised manuscript we will add a new subsection (or appendix) that derives the required estimates in detail for the specific nonlinearities of the electroporation model, including the transmembrane potential evolution law and the porosity ODE right-hand side. This will make the a-priori estimates fully transparent. revision: yes

  2. Referee: [Stochastic formulation and Galerkin approximation] The treatment of the degenerate multiplicative noise (present only in the SPDE) and the resulting tightness argument for the coupled system must be spelled out; it is unclear from the outline whether the standard Liu-Röckner estimates extend directly when the noise operator has a nontrivial kernel and the ODE receives no direct stochastic forcing.

    Authors: We acknowledge that the treatment of the degenerate multiplicative noise and the tightness argument for the coupled SPDE-ODE system needs further elaboration. In the revision we will expand the Galerkin approximation section to explicitly show how the Liu-Röckner estimates carry over when the noise operator has a nontrivial kernel and acts only on the SPDE component. We will also detail the indirect mixing induced by the nonlinear coupling and outline the key steps establishing tightness of the joint process. revision: yes

Circularity Check

0 steps flagged

No significant circularity; existence proof applies external Liu-Röckner framework to verified conditions

full rationale

The derivation establishes existence/uniqueness of a variational solution to the coupled SPDE-ODE system by Galerkin approximation under the Liu-Röckner generalized monotonicity/coercivity framework (cited externally). The nonlinear electroporation terms are checked to satisfy the required coercivity, monotonicity, and growth bounds in the chosen spaces; this is a direct hypothesis verification, not a reduction to self-defined quantities or fitted inputs. The deterministic base model is cited from Kavian et al. (external), and the stochastic perturbation is handled via the same external theorem. No self-citation chains, ansatz smuggling, or renaming of known results occur. The central claim remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper extends an existing deterministic phenomenological model by adding a stochastic term; the only non-standard assumption is the generalized monotonicity condition needed for the existence theorem.

axioms (1)
  • domain assumption The nonlinear terms satisfy generalized monotonicity and coercivity conditions as required by Liu and Röckner (2015)
    Invoked to justify application of the Galerkin method to the non-Lipschitz, non-monotone nonlinearities.

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Reference graph

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