Flexoelectricity

Flexoelectricity in Neuronal Membranes

Overview

Flexoelectricity is a universal property of dielectric materials by which a spatially non-uniform mechanical deformation (curvature gradient) induces an electric polarization, and conversely, a non-uniform electric field induces mechanical curvature. In the context of biological membranes, flexoelectricity arises from the asymmetric redistribution of charged lipid headgroups under bending — a direct consequence of the bilayer's dielectric and geometric properties. The application of this principle to ultrasound-neuron interaction proposes that ultrasound-induced membrane curvature oscillations generate oscillating transmembrane polarization currents sufficient to modulate neuronal excitability, without requiring intramembrane cavitation.

1. Physical Basis

The flexoelectric constitutive relationship for a membrane element relates the induced polarization $P$ to the local curvature gradient $\nabla² u$ (where $u$ is the membrane displacement normal to the bilayer plane):

\[P = \mu_f \nabla^2 u\]

Where $\mu_f$ is the flexoelectric coefficient — the central material parameter of the framework, with units of C/m (or equivalently nC/m).

For a planar membrane patch oscillating under acoustic pressure, the curvature-induced transmembrane voltage $\Delta V{flexo}$ and the associated flexoelectric current density $J{flexo}$ are:

\[\Delta V_{flexo} = \frac{\mu_f}{\varepsilon_0 \varepsilon_r} \cdot \kappa\]

\[J_{flexo} = \mu_f \frac{\partial \dot{\kappa}}{\partial x}\]

Where $\kappa$ is the local membrane curvature, $\varepsilon_0 \varepsilon_r$ is the membrane permittivity, and $\dot{\kappa}$ is the time derivative of curvature. The current is therefore proportional to both the flexoelectric coefficient and the rate of change of curvature — meaning that oscillating acoustic fields, which drive periodic curvature changes, produce oscillating flexoelectric currents that can depolarize or hyperpolarize the membrane depending on the phase of oscillation and the sign of $\mu_f$.

2. Relationship to NICE Capacitive Current: Sassaroli & Vykhodtseva (2016)

A critical theoretical contribution establishing the flexoelectric framework in the FUS context was made by Sassaroli & Vykhodtseva (2016), who showed that the NICE capacitive current and the flexoelectric polarization current are not independent phenomena but two terms of a single generalized expression for the total membrane current under mechanical deformation:

\[I_{total} = \underbrace{V_m \frac{dC_m}{dt}}_{\text{NICE capacitive term}} + \underbrace{\mu_f \frac{\partial \dot{\kappa}}{\partial x}}_{\text{flexoelectric polarization term}}\]

The NICE term dominates when the membrane undergoes large-amplitude volumetric oscillations (cavitation regime); the flexoelectric term dominates when deformation is primarily curvature-based at sub-cavitation amplitudes. This unification implies that flexoelectricity is relevant even in the absence of cavitation — providing an alternative or complementary excitation pathway at lower acoustic intensities where sonophore oscillation may be insufficient to generate the NICE capacitive current alone.

3. Chen & Jerusalem (2019, 2021): MECAXON Framework

Chen & Jerusalem developed the MECAXON model, a finite element-based framework that couples acoustic wave propagation, membrane mechanics, and flexoelectric polarization in a spatially extended axon geometry. This represents the most mechanistically complete flexoelectric model of ultrasound-neuron interaction.

Chen & Jerusalem (2019) — Static and quasi-static flexoelectric response

The 2019 paper established the baseline MECAXON framework, modeling a myelinated axon as a cylindrical elastic shell with flexoelectric properties. Key modeling choices:

• Axon modeled as a multi-layer cylindrical shell: myelin layers, axolemma, intracellular fluid
• Acoustic pressure applied as a spatially uniform oscillating pressure boundary condition
• Flexoelectric polarization computed from local curvature via $P = \mu_f \kappa$
• HH-type ionic currents (NaV, KV, leak) at nodes of Ranvier, passive myelin internodes
• Mechanical and electrical subsystems coupled bidirectionally: deformation → polarization → membrane voltage → ion channel gating → forces on membrane

The central finding was that flexoelectric polarization at nodes of Ranvier, where membrane curvature is highest due to geometric transitions between myelin and node, is sufficient to generate action potentials at experimentally relevant acoustic pressures (100–400 kPa). The model predicted frequency-dependent threshold behavior consistent with experimental observations, with lower thresholds at frequencies matching the mechanical resonance of the axon geometry.

Key parameters used:

Chen & Jerusalem (2021) — Dynamic flexoelectric response and nonlinear effects

The 2021 extension introduced nonlinear membrane mechanics and dynamic acoustic loading, addressing limitations of the quasi-static 2019 model:

• Nonlinear elastic constitutive law for the bilayer under large deformations
• Full dynamic FEM simulation of acoustic wave propagation through the axon geometry
• Frequency-dependent mechanical resonance of the axon shell
• Prediction of harmonic generation: nonlinear membrane mechanics under acoustic forcing produce current components at integer multiples of the carrier frequency, potentially contributing to neuronal excitation at sub-threshold carrier amplitudes through rectification effects

The 2021 model confirmed that flexoelectric currents at physiologically reasonable $\mu_f$ values (_$10^{-10}$ C/m) are sufficient for AP generation, and identified the node of Ranvier geometry as a flexoelectric hotspot due to the curvature discontinuity at the node-myelin junction.

Key limitation of MECAXON: full FEM spatial integration makes the model computationally incompatible with the SONIC/MorphoSONIC lookup table approach. MECAXON cannot be directly merged with the SONIC framework — the two operate at fundamentally different levels of spatial and temporal resolution.

4. Badawe & Khraiche (2023): Ca²⁺-Dependent Flexoelectric Sign Reversal

Badawe & Khraiche (2023) introduced a critical biological dimension absent from purely physical flexoelectric models: the dependence of the effective flexoelectric response on intracellular Ca²⁺ concentration. This study moved beyond the assumption of a fixed $\mu_f$ and demonstrated that the membrane's flexoelectric behavior is dynamically modulated by the cell's physiological state.

Core finding: the sign and magnitude of the net flexoelectric polarization current depend on intracellular Ca²⁺ levels. At low $[\text{Ca}^{2+}]i$, acoustic-driven curvature produces a depolarizing (excitatory) flexoelectric current; at elevated $[\text{Ca}^{2+}]i$, the same acoustic stimulus produces a hyperpolarizing (inhibitory) current. This Ca²⁺-dependent sign reversal provides the first flexoelectricity-based mechanistic account of ultrasound-induced silencing that does not require network-level inhibition or depolarization block.

Mechanistic basis: elevated intracellular Ca²⁺ alters the charge distribution of the inner membrane leaflet through Ca²⁺-lipid interactions (particularly with phosphatidylserine and phosphatidylinositol headgroups), shifting the asymmetry of the bilayer charge profile. Since flexoelectric polarization depends on this charge asymmetry, a Ca²⁺-driven shift changes the effective $\mu_f$ — potentially reversing its sign.

Modeling approach: Badawe & Khraiche (2023) implemented a modified HH framework in which:

\[I_{flexo} = \mu_f([\text{Ca}^{2+}]_i) \cdot \frac{\partial \dot{\kappa}}{\partial x}\]

With $\mu_f([\text{Ca}^{2+}]_i)$ described as a sigmoid function of Ca²⁺ concentration:

\[\mu_f([\text{Ca}^{2+}]_i) = \mu_{f,max} \cdot \left(1 - \frac{2}{1 + \exp(-k_{Ca}([\text{Ca}^{2+}]_i - [\text{Ca}^{2+}]_{1/2}))}\right)\]

Where $[\text{Ca}^{2+}]{1/2}$ is the Ca²⁺ concentration at which the sign reversal occurs and $k{Ca}$ governs the steepness of the transition.

Implications for modeling: this finding introduces a state-dependent nonlinearity into the flexoelectric current that couples the acoustic drive to the neuron's recent firing history (since Ca²⁺ accumulates during activity). Neurons that have been recently active — and therefore have elevated $[\text{Ca}^{2+}]_i$ — will respond differently to the same acoustic stimulus than quiescent neurons, providing a mechanism for activity-dependent bidirectionality of ultrasound neuromodulation.

5. Badawe et al. (2024): Thermal-Flexoelectric Interaction

Badawe et al. (2024) extended the framework further by incorporating thermal effects into the flexoelectric model, addressing the interaction between sonication-induced heating and membrane flexoelectric properties.

Core contributions:

• Demonstrated that the flexoelectric coefficient $\mu_f$ is itself temperature-dependent: small temperature rises (ΔT = 0.5–2°C, within FDA-compliant LIFU range) alter lipid bilayer fluidity and headgroup charge distribution, modifying $\mu_f$
• Incorporated Q₁₀-dependent kinetic shifts for HCN and Kv channels alongside the flexoelectric current, showing that thermal and flexoelectric contributions are additive but non-linearly coupled
• At moderate sonication intensities, thermal effects (Q₁₀ kinetics) and flexoelectric effects (curvature-driven currents) act synergistically to produce excitation; at higher intensities or longer durations, thermal upregulation of K⁺ conductances counteracts flexoelectric depolarization, shifting the balance toward inhibition
• Established that prolonged sonication (> a few seconds) at sub-ablative intensities produces net inhibition primarily through the thermal pathway (Kv upregulation, AHP enhancement) even when the instantaneous flexoelectric current remains depolarizing

Temperature-dependent flexoelectric coefficient:

\[\mu_f(T) = \mu_{f,0} \left(1 + \alpha_{flexo}(T - T_{ref})\right)\]

Where $\alpha{flexo}$ is a thermal sensitivity coefficient and $T{ref} = 37°C$ for mammalian CNS applications.

Relevance to the unified model: this work motivates the thermal module in the unified model as a mandatory component for long-duration sonication protocols, not merely an optional correction. The thermal-flexoelectric coupling also suggests that parameters optimized for excitation at short durations may produce inhibition at longer durations — a critical consideration for protocol design.

6. Critical Parameter: The Flexoelectric Coefficient $\mu_f$

Across all flexoelectric models, $\mu_f$ is the central and most uncertain parameter. A comprehensive survey of published values reveals:

The scatter spans approximately 9 orders of magnitude, with no CNS-specific measurements available. This is the single most critical limitation of the flexoelectric framework — the model is highly sensitive to $\mu_f$, but its physiological value in cortical or thalamic neurons is entirely unknown. Numerical unification of published values is inadvisable; instead, $\mu_f$ must be treated as a free parameter bounded by theoretical and experimental constraints from non-CNS preparations.

7. Unified Current Equation

Integrating the NICE capacitive term, the direct flexoelectric polarization current, Ca²⁺-dependent modulation, and the thermal module, the proposed unified membrane current balance for a single compartment is:

\[\frac{dQ_m}{dt} = -\sum_k I_k(V_m, \text{gates}) + \underbrace{V_m \frac{dC_m}{dt}}_{\text{NICE/SONIC}} + \underbrace{\mu_f([\text{Ca}^{2+}]_i, T) \frac{\partial \dot{\kappa}}{\partial x}}_{\text{flexoelectric}} + I_{stim}\]

With:

• $C_m(t)$ from SONIC lookup tables (cavitation-driven oscillation)
• $\mu_f([\text{Ca}^{2+}]_i, T)$ from Badawe 2023/2024 Ca²⁺ and thermal modulation functions
• $[\text{Ca}^{2+}]_i$ from a Ca²⁺ dynamics sub-model (L-type channel influx + buffering + extrusion)
• Thermal module providing $T(t)$ and Q₁₀-scaled kinetics at $T_{ref} = 37°C$

8. Strengths and Limitations

Strengths

• Provides an excitation pathway independent of cavitation, operative at sub-cavitation intensities
• Ca²⁺-dependent sign reversal gives a single-neuron mechanistic account of silencing — a major gap in NICE/SONIC
• Thermal-flexoelectric interaction provides a unified account of duration-dependent excitation/inhibition transitions
• Mathematically compatible with HH formalism — can be added as an additional current term
• Node of Ranvier geometry identified as a flexoelectric hotspot, consistent with MorphoSONIC node selectivity predictions

Limitations

• $\mu_f$ is entirely unconstrained for CNS neurons: 9-orders-of-magnitude scatter makes quantitative predictions unreliable without CNS-specific measurements
• No direct experimental confirmation of flexoelectric currents in neurons under ultrasound
• MECAXON incompatibility with SONIC lookup table architecture prevents direct integration
• Curvature computation requires spatial resolution: the flexoelectric current depends on $\partial \dot{\kappa}/\partial x$, which requires spatially resolved membrane deformation profiles unavailable in point neuron frameworks
• Ca²⁺ model adds coupled complexity: the sign-reversal mechanism requires a validated intracellular Ca²⁺ dynamics sub-model, introducing additional uncertain parameters
• Temperature dependence of $\mu_f$ not characterized for CNS membranes: $\alpha_{flexo}$ is estimated from non-neuronal preparations