Local Existence for the Cauchy Problem of a Reaction-Diffusion System with Discontinuous Nonlinearity.

Abstract

The most famous model for nerve conduction is due to Hodgkin and Huxley. However, a mathematical analysis of their model has proven very difficult. The complexity of the Hodgkin and Huxley model has led a number of other authors to introduce simpler models. In this report we consider one such simplification. It has been demonstrated (experimentally) that impulses in the nerve axon travel with constant shape and velocity. Mathematically, this corresponds to traveling wave solutions. A number of authors have proven that the mathematical equations considered here do possess traveling wave solutions. Another property of impulses in the nerve axon is the existence of a threshold phenomenon. This corresponds to the biological fact that a minimum stimulus is needed to trigger an impulse. Here we prove some preliminary results which will be used in a later report when it is demonstrated that the equations under study do indeed exhibit a threshold phenomenon.

Open PDF

Document Details

Document Type
Technical Report
Publication Date
May 01, 1981
Accession Number
ADA100564

Entities

People

  • David Terman

Organizations

  • University of Wisconsin–Madison

Tags

Communities of Interest

  • C4I

DTIC Thesaurus Topics

  • Cauchy Problem
  • Complex Variables
  • Differential Equations
  • Diffusion
  • Equations
  • Integral Equations
  • Mathematical Analysis
  • Mathematics
  • North Carolina
  • Numerical Analysis
  • Real Variables
  • Sequences
  • Step Functions
  • Theorems
  • Traveling Waves
  • United States

Fields of Study

  • Mathematics

Readers

  • Finite Element Method (FEM) for solving Partial Differential Equations (PDEs)
  • Neuroscience
  • Theoretical Analysis.