Modelling dendritic spines with the finite element method, investigating the impact of geometry on electric and calcic responses

Voltage mapping in subcellular nanodomains using electro-diffusion modeling

Voltage distribution in sub-cellular micro-domains such as neuronal synapses, small protrusions, or dendritic spines regulates the opening and closing of ionic channels, energy production, and thus, cellular homeostasis and excitability. Yet how voltage changes at such a small scale in vivo remains challenging due to the experimental diffraction limit, large signal fluctuations, and the still limited resolution of fast voltage indicators. Here, we study the voltage distribution in nano-compartments using a computational approach based on the Poisson-Nernst-Planck equations for the electro-diffusion motion of ions, where inward and outward fluxes are generated between channels. We report a current-voltage (I-V) logarithmic relationship generalizing Nernst law that reveals how the local membrane curvature modulates the voltage. We further find that an influx current penetrating a cellular electrolyte can lead to perturbations from tens to hundreds of nanometers deep, depending on the local channel organization. Finally, we show that the neck resistance of dendritic spines can be completely shunted by the transporters located on the head boundary, facilitating ionic flow. To conclude, we propose that voltage is regulated at a subcellular level by channel organization, membrane curvature, and narrow passages.

Ion-concentration gradients induced by synaptic input increase the voltage depolarization in dendritic spines

The vast majority of excitatory synaptic connections occur on dendritic spines. Due to their extremely small volume and spatial segregation from the dendrite, even moderate synaptic currents can significantly alter ionic concentrations. This results in chemical potential gradients between the dendrite and the spine head, leading to measurable electrical currents. In modeling electric signals in spines, different formalisms were previously used. While the cable equation is fundamental for understanding the electrical potential along dendrites, it only considers electrical currents as a result of gradients in electrical potential. The Poisson-Nernst-Planck (PNP) equations offer a more accurate description for spines by incorporating both electrical and chemical potential. However, solving PNP equations is computationally complex. In this work, diffusion currents are incorporated into the cable equation, leveraging an analogy between chemical and electrical potential. For simulating electric signals based on this extension of the cable equation, a straightforward numerical solver is introduced. The study demonstrates that this set of equations can be accurately solved using an explicit finite difference scheme. Through numerical simulations, this study unveils a previously unrecognized mechanism involving diffusion currents that amplify electric signals in spines. This discovery holds crucial implications for both numerical simulations and experimental studies focused on spine neck resistance and calcium signaling in dendritic spines.

Sensitivity of the electrical response of a node of Ranvier model to alterations of the myelin sheath geometry

Nodes of Ranvier play critical roles in the generation and transmission of action potentials. Alterations in node properties during pathology and/or development are known to affect the speed and quality of electrical transmission. From a modelling standpoint, nodes of Ranvier are often described by systems of ordinary differential equations neglecting or greatly simplifying their geometric structure. These approaches fail to accurately describe how fine scale alteration in the node geometry or in myelin thickness in the paranode region will impact action potential generation and transmission. Here, we rely on a finite element approximation to describe the three dimensional geometry of a node of Ranvier. With this, we are able to investigate how sensitive is the electrical response to alterations in the myelin sheath and paranode geometry. We could in particular investigate irregular loss of myelin, which might be more physiologically relevant than the uniform loss often described through simpler modelling approaches.