COMPUTATIONAL MODELING IN ELECTRONICS AND BIOMATHEMATICS 096659

Objectives and contents of the course

Electronic and biological systems share unexpected structural similarities that make them amenable to a unified mathematical and numerical treatment. As a matter of fact, transmembrane ion flow regulating the functional response of a neuronal or a cardiac cell, as well as the motion of electric charge transporting current in a nanoscale-sized transistor, obey the same phenomenological description, the so-called Nernst-Planck transport model(in Biology) and the Drift-Diffusion transport model (in Electronics). Under this unifying perspective, the course has the objective of providing the common mathematical and computational foundations to modeling and simulation of specific problems in cellular biology and solid-state electronics, the final scope being to couple the two classes of problems in the study of bio-hybrid devices in which both components (cellular and solid-state) coexist in a dynamically interacting operating mode. This final goal reflects the state-of-the-art in modern Neuroscience and represents an original attempt of this course to confront the student with the frontiers of research in Life Sciences of our times.

Description of course topics

Introduction to cellular biology and ion electrodiffusion. ODE models for transmembrane ion flow in cellular physiology. The Kirchhoff current law: capacitive and resistive transmembrane currents. The linear resistor model; the Goldman-Hodgkin-Katz (GHK) model; the Hodgkin-Huxley (HH) model. PDE models for transmembrane ion flow in cellular physiology: the Poisson-Nernst-Planck (PNP) system for M ionic species. Introduction to semiconductor device electronics. Multiscale structure of integrated circuits. Micro/nanoscale view: the Maxwell equation system and the quasi-static approximation. Atomic/macroscale view: charge transport in solids; Ohm's law in metal conductors; the Drift-Diffusion (DD) model in semiconductor materials. The Poisson-DD (PDD) PDE model for semiconductor device simulation at the micro/nanoscale. Model analogies: PDD = PNP with M=2. Scaling. Functional iterations: Newton's method and Gummel's map. Diffusion-advection-reaction linear model problem (with gradient advective field): well-posedness analysis and numerical approximation with a stabilized Finite Element Method (FEM). Convergence analysis. Conservation properties. Numerical stability of the FEM: continuous and discrete maximum principles. Examples in cellular biology: excitable cells. Nernst potential of a ionic species; cellular homeostatis: the GHK potential. Action potential propagation: the Cable Equation model coupled with the HH ODE system. Simplified treatment of intracellular and extracellular compartments: the electroneutral PNP model. Examples in device electronics: the p-n junction. Thermal equilibrium, reverse and forward bias. I-V curves and the law of ideal diode. The full depletion approximation: analytical solution of the PDD system. 1D models for the Metal-Oxide-Semiconductor (MOS) transistor. The n-MOS capacitor. The n+ - n – n+ structure for the n-MOS channel. Example of bio-hybrid devices: interface between cells and a MOS transistor. Cell-to-chip stimulation. Cell-to-cell stimulation. Reduced-order models: area-contact and lumped parameter models.

Organization of the course

The course is organized into class lectures accompanied by laboratory sessions (labs). Seminars will also be scheduled during the course to introduce the students to applications of relevant industrial and/or biological impact. Class lectures provide the theoretical foundations of models and numerical methods. Labs are carried out in a computer room using the Matlab environment and allow the students to verify and compare on simple exercises and more advanced examples the physical accuracy of the models and the numerical performance of the methods.

The final examination is divided into two parts: a written exam and an oral exam. The written exam consists of a number of exercises to be solved with the support of the Matlab codes used in the labs. The oral exam consists of the critical discussion of the written exam, plus theoretical questions regarding the topics treated in the class lectures.

This file contains the lecture notes of the PhD Course Multiscale Modeling of Interface Phenomena in Biology. A substantial part of the material is object of the course.

This file contains the lecture notes of the course Computational Electronics.