Nonlinear Coupling of the Poisson–Nernst Planck System of Equations Using the Discrete Duality Finite Volumes Method: Application to Signal Propagation in Dendritic Spines

How neuronal cells integrate the multitude of signals they receive from interconnected neighbors is a fundamental question in neuroscience, intrinsically linked to the architecture and organization of biological neuronal networks. A first step in addressing this question is to obtain a precise description of voltage and ionic dynamics within small neuronal compartments such as dendritic spines. To this end, we consider in this paper the Poisson–Nernst Planck (PNP) system of equations, which is widely recognized as the standard model for characterizing the electrodiffusion of ions in electrolytes, including ionic and voltage dynamics in the cellular cytosol. This nonlinear system couples a Fokker–Planck equation for the concentrations and a Poisson equation for the potential. It presents challenges for both modeling and simulations, primarily due to its stiff boundary layer, tightly related to the choice of boundary conditions. To realize two-dimensional simulations of the PNP system of equations, we propose a numerical scheme based on the discrete-duality finite volume (DDFV) method. The DDFV method presents the main characteristics of modern numerical methods, allowing for the use of unstructured meshes in complex geometries and giving robust and precise approximate solutions. It is particularly attractive in this context due to its local conservation property and its robustness with respect to mesh distortion. In this paper, we build a numerical scheme coupling two Fokker–Planck equations representing ionic concentrations of two ionic species, with a Poisson equation for the potential. The coupling of the different equations is nonlinear to ensure greater accuracy, and preserves the positivity of ionic concentrations. We prove the existence of the discrete solution using the entropy-dissipation relation and a corollary of Brouwer’s fixed point theorem. Through several simulations, we illustrate the accuracy of our scheme, achieving second-order of convergence in space. Furthermore, using a specific test case, we show that our method can resolve steep gradients. In the final section, we apply our scheme to investigate the propagation and attenuation of an ionic influx in a dendritic spine—a mushroom-like protrusion that receives neuronal inputs. Indeed, the complex ionic and voltage dynamics occurring at dendritic spines are triggering many chemical cascades that are thought to be at the basis of learning and memory. Finally, considering two neighboring dendritic spines, we observe that a signal arriving at one spine invades its neighbor, and that the amplitude of the invading signal decreases with the distance between spines. Surprisingly, while the ionic concentration signal travels only to very close spines, the voltage signal can be sensed by much more distant spines.