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.

An Algorithm Based on a Cable-Nernst Planck Model Predicting Synaptic Activity throughout the Dendritic Arbor with Micron Specificity

Recent technological advances have enabled the recording of neurons in intact circuits with a high spatial and temporal resolution, creating the need for modeling with the same precision. In particular, the development of ultra-fast two-photon microscopy combined with fluorescence-based genetically-encoded Ca²⁺-indicators allows capture of full-dendritic arbor and somatic responses associated with synaptic input and action potential output. The complexity of dendritic arbor structures and distributed patterns of activity over time results in the generation of incredibly rich 4D datasets that are challenging to analyze (Sakaki et al. in Frontiers in Neural Circuits 14:33, 2020). Interpreting neural activity from fluorescence-based Ca²⁺ biosensors is challenging due to non-linear interactions between several factors influencing intracellular calcium ion concentration and its binding to sensors, including the ionic dynamics driven by diffusion, electrical gradients and voltage-gated conductances. To investigate those dynamics, we designed a model based on a Cable-like equation coupled to the Nernst-Planck equations for ionic fluxes in electrolytes. We employ this model to simulate signal propagation and ionic electrodiffusion across a dendritic arbor. Using these simulation results, we then designed an algorithm to detect synapses from Ca²⁺ imaging datasets. We finally apply this algorithm to experimental Ca²⁺-indicator datasets from neurons expressing jGCaMP7s (Dana et al. in Nature Methods 16:649-657, 2019), using full-dendritic arbor sampling in vivo in the Xenopus laevis optic tectum using fast random-access two-photon microscopy. Our model reproduces the dynamics of visual stimulus-evoked jGCaMP7s-mediated calcium signals observed experimentally, and the resulting algorithm allows prediction of the location of synapses across the dendritic arbor. Our study provides a way to predict synaptic activity and location on dendritic arbors, from fluorescence data in the full dendritic arbor of a neuron recorded in the intact and awake developing vertebrate brain.

Physiological insights into electrodiffusive maintenance of gastric mucus through sensitivity analysis and simulations

It is generally accepted that the gastric mucosa and adjacent mucus layer are critical in the maintenance of a pH gradient from stomach lumen to stomach wall, protecting the mucosa from the acidic environment of the lumen and preventing auto-digestion of the epithelial layer. No conclusive study has shown precisely which physical, chemical, and regulatory mechanisms are responsible for maintaining this gradient. However, experimental work and modeling efforts have suggested that concentration dependent ion-exchange at the epithelial wall, together with hydrogen ion/mucus network binding, may produce the enormous pH gradients seen in vivo. As of yet, there has been no exhaustive study of how sensitive these modeling results are with respect to variation in model parameters, nor how sensitive such a regulatory mechanism may be to variation in physical/biological parameters. In this work, we perform sensitivity analysis (using Sobol' Indices) on a previously reported model of gastric pH gradient maintenance. We quantify the sensitivity of mucosal wall pH (as a proxy for epithelial health) to variations in biologically relevant parameters and illustrate how variations in these parameters affects the distribution of the measured pH values. In all parameter regimes, we see that the rate of cation/hydrogen exchange at the epithelial wall is the dominant parameter/effect with regards to variation in mucosal pH. By careful sensitivity analysis, we also investigate two different regimes representing high and low hydrogen secretion with different physiological interpretations. By complementing mechanistic modeling and biological hypotheses testing with parametric sensitivity analysis we are able to conclude which biological processes must be tightly regulated in order to robustly maintain the pH values necessary for healthy function of the stomach.

Electrodiffusion with Calcium-Activated Potassium Channels in Dendritic Spine

We investigate calcium signaling feedback through calcium-activated potassium channels of a dendritic spine by applying the immersed boundary method with electrodiffusion. We simulate the stochastic gating of such ion channels and the resulting spatial distribution of concentration, current, and membrane voltage within the dendritic spine. In this simulation, the permeability to ionic flow across the membrane is regulated by the amplitude of chemical potential barriers. With spatially localized ion channels, chemical potential barriers are locally and stochastically regulated. This regulation represents the ion channel gating with multiple subunits, the open and closed states governed by a continuous-time Markov process. The model simulation recapitulates an inhibitory action on voltage-sensitive calcium channels by the calcium-activated potassium channels in a stochastic manner as a non-local feedback loop. The model predicts amplified calcium influx with more closely placed channel complexes, proposing a potential mechanism of differential calcium handling by channel distributions. This work provides a foundation for future computer simulation studies of dendritic spine motility and structural plasticity.

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

Understanding the relationship between shape and function of dendritic spines is an elusive topic. Several modelling approaches have been used to investigate the interplay between spine geometry, calcium diffusion and electric signalling. We here use a second order finite element method to solve the Poisson-Nernst-Planck equations and describe electrodiffusion in dendritic spines. With this, we obtain relationships between dendritic geometry and calcic as well as electric responses to synaptic events. Our findings support the hypothesis that spine geometry plays a role shaping the electrical responses to synaptic events. Our method was also able to reveal the fine scale distribution of calcium in spines with irregular shapes.

Electrodiffusion models of synaptic potentials in dendritic spines

The biophysical properties of dendritic spines play a critical role in neuronal integration but are still poorly understood, due to experimental difficulties in accessing them. Spine biophysics has been traditionally explored using theoretical models based on cable theory. However, cable theory generally assumes that concentration changes associated with ionic currents are negligible and, therefore, ignores electrodiffusion, i.e. the interaction between electric fields and ionic diffusion. This assumption, while true for large neuronal compartments, could be incorrect when applied to femto-liter size structures such as dendritic spines. To extend cable theory and explore electrodiffusion effects, we use here the Poisson (P) and Nernst-Planck (NP) equations, which relate electric field to charge and Fick's law of diffusion, to model ion concentration dynamics in spines receiving excitatory synaptic potentials (EPSPs). We use experimentally measured voltage transients from spines with nanoelectrodes to explore these dynamics with realistic parameters. We find that (i) passive diffusion and electrodiffusion jointly affect the dynamics of spine EPSPs; (ii) spine geometry plays a key role in shaping EPSPs; and, (iii) the spine-neck resistance dynamically decreases during EPSPs, leading to short-term synaptic facilitation. Our formulation, which complements and extends cable theory, can be easily adapted to model ionic biophysics in other nanoscale bio-compartments.

Sensitivity analysis of the Poisson Nernst-Planck equations: a finite element approximation for the sensitive analysis of an electrodiffusion model

Biological structures exhibiting electric potential fluctuations such as neuron and neural structures with complex geometries are modelled using an electrodiffusion or Poisson Nernst-Planck system of equations. These structures typically depend upon several parameters displaying a large degree of variation or that cannot be precisely inferred experimentally. It is crucial to understand how the mathematical model (and resulting simulations) depend on specific values of these parameters. Here we develop a rigorous approach based on the sensitivity equation for the electrodiffusion model. To illustrate the proposed methodology, we investigate the sensitivity of the electrical response of a node of Ranvier with respect to ionic diffusion coefficients and the membrane dielectric permittivity.

Transport of charged small molecules after electropermeabilization - drift and diffusion

Background

Applications of electric-field-induced permeabilization of cells range from cancer therapy to wastewater treatment. A unified understanding of the underlying mechanisms of membrane electropermeabilization, however, has not been achieved. Protocols are empirical, and models are descriptive rather than predictive, which hampers the optimization and expansion of electroporation-based technologies. A common feature of existing models is the assumption that the permeabilized membrane is passive, and that transport through it is entirely diffusive. To demonstrate the necessity to go beyond that assumption, we present here a quantitative analysis of the post-permeabilization transport of three small molecules commonly used in electroporation research — YO-PRO-1, propidium, and calcein — after exposure of cells to minimally perturbing, 6 ns electric pulses.

Results

Influx of YO-PRO-1 from the external medium into the cell exceeds that of propidium, consistent with many published studies. Both are much greater than the influx of calcein. In contrast, the normalized molar efflux of calcein from pre-loaded cells into the medium after electropermeabilization is roughly equivalent to the influx of YO-PRO-1 and propidium. These relative transport rates are correlated not with molecular size or cross-section, but rather with molecular charge polarity.

Conclusions

This comparison of the kinetics of molecular transport of three small, charged molecules across electropermeabilized cell membranes reveals a component of the mechanism of electroporation that is customarily taken into account only for the time during electric pulse delivery. The large differences between the influx rates of propidium and YO-PRO-1 (cations) and calcein (anion), and between the influx and efflux of calcein, suggest a significant role for the post-pulse transmembrane potential in the migration of ions and charged small molecules across permeabilized cell membranes, which has been largely neglected in models of electroporation.

Electronic supplementary material

The online version of this article (10.1186/s13628-018-0044-2) contains supplementary material, which is available to authorized users.

Computer simulation of voltage sensitive calcium ion channels in a dendritic spine

Membrane current through voltage-sensitive calcium ion channels at the postsynaptic density of a dendritic spine is investigated. To simulate the ion channels that carry such current and the resulting temporal and spatial distribution of concentration, current, and voltage within the dendritic spine, the immersed boundary method with electrodiffusion is applied. In this simulation method a spatially continuous chemical potential barrier is used to simulate the influence of the membrane on each species of ion. The amplitudes of these barriers can be regulated to simulate channel gating. Here we introduce this methodology in a one-dimensional setting. First, we study the current-voltage relationship obtained with fixed chemical potential barriers. Next, we simulate stochastic ion-channel gating in a calcium channel with multiple subunits, and observe the diffusive wave of calcium entry within the dendritic spine that follows channel opening. This work lays the foundation for future three-dimensional studies of electrodiffusion and advection electrodiffusion in dendritic spines.