Numerical Simulations on the Biophysical Foundations of the Neuronal Extracellular Space

Abstract

The electric activity of neurons creates extracellular potential fields. Recent findings show that these endogenous fields act back onto the neurons, contributing to synchronization of population activity. The influence of extracellular potential fields is also relevant for understanding therapeutic approaches such as transcranial direct current stimulation, transcranial magnetic stimulation and deep brain stimulation. The mutual interaction between fields and the neuronal membrane is not captured by today's modeling tools of neuronal electrophysiology, as those are based on isolated membranes in an infinite, isopotential extracellular space. Even the direct influence of the field is not correctly represented by the commonly used ``activating function''. While a reduced set of Maxwell's equations can be used to couple membrane currents to extra- and intracellular potentials, this approach is rarely taken, most likely because adequate computational tools are missing. This thesis presents a computational method that implements this set of equations. The fundamentals of the method are thoroughly described starting from first principles, passing by the discretization procedure, and up to the solution algorithms. By introducing an implicit solver, numerical stability is attained even with large time-steps: this allows simulation times of tens of minutes instead of weeks, even for complex problems. The method was implemented as an open source software package which is now freely available to the neuroscience community. This tool allows simulation of cells under realistic conditions: a conductive, non-homogeneous space, sub-micron cell morphology, mixed boundary conditions and various ion channel properties and distributions. The extracellular fields are accurately represented, including secondary fields, which originate at inhomogeneities of the extracellular space and can reach several millivolts. Example applications of this method and the tool are also presented.