BIT Numerical Mathematics (2005) 45: 725–741 c  Springer 2005 DOI: 10.1007/s10543-005-0023-2 A MAXIMUM PRINCIPLE FOR AN EXPLICIT FINITE DIFFERENCE SCHEME APPROXIMATING THE HODGKIN-HUXLEY MODEL ⋆ MONICA HANSLIEN 1 , KENNETH H. KARLSEN 2 and ASLAK TVEITO 1,⋆⋆ 1 Department of Scientific Computing, Simula Research Laboratory, P.O.Box 134, N-1325 Lysaker, Norway. email: {monicaha,aslak}@simula.no 2 Center of Mathematics for Applications, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway. email: kennethk@math.uio.no Abstract. We analyze an explicit finite difference scheme for the general form of the Hodgkin- Huxley model, which is a nonlinear partial differential equation coupled to a set of ODEs. The system of equations describes propagation of an electrical signal in ex- citable cells. We prove that the numerical solution is bounded in the L ∞ -norm and L 2 converges to a unique solution. The L ∞ -bound, which is the key point of our analysis, is proved by showing that the discrete solutions are invariant in a physically relevant bounded region. For the convergence proof we use the compactness method. AMS subject classification (2000): 65F20. Key words: explicit scheme, maximum principle, convergence, compactness. 1 Introduction. Cell signalling has been an important subject of research over the past decades, particularly that of excitable media including neurons and cardiac cells. Hodgkin and Huxley are well known for their experimental studies on the squid giant axon [4], which led to a mathematical model to be investigated in the present paper. The system is a parabolic equation coupled to a set of three ordinary differential equations, and reads (1.1) v t = μv xx - g Na m 3 h(v - E Na ) - g K n 4 (v - E K ) - g L (v - E L ), dm dt = (1 - m)α m (v) - mβ m (v), dh dt = (1 - h)α h (v) - hβ h (v), dn dt = (1 - n)α n (v) - nβ n (v), (x, t) ∈ Ω × [0, ∞). (1.2) ⋆ Received September 2004. Revised accepted August 2005. Communicated byPer L¨otstedt. ⋆⋆ This work has been supported by the BeMatA program of The Research Council of Norway.