Physica A 188 (1992) 206-209 North-Holland PflYSICA

Mechanisms for propagation failure in discrete reaction-diffusion systems

V i c t o r i a B o o t h a n d T h o m a s E r n e u x

Department o f Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, 1L 60208, USA

We model a chain of coupled bistable reactors by the discrete Nagumo reaction-diffusion equation: du,,/dt = f(u,,) + d(u, +~ + u,, 1 - 2u,,), where f(u~) = u~(u,, - 1)(a - u,,) and a, d are parameters. We determine a propagating front initiated by the following boundary and initial conditions: u~ = 1 (t >/0) and u,, = 0 (n/> 2 and t = 0). We show that the front fails to propagate (1) if d is too small (d< d*) or (2) if d = d(n) is not uniformly distributed in the system. We investigate both cases by analyzing the limit a--~0 and d(a)---~O.

1. Introduction

Evidence of p ropaga t ion failure as well as other conduc t ion irregulari t ies

(delay, partial reflection) has been found exper imenta l ly with cardiac tissues

and o ther excitable m e m b r a n e systems. These anomal ies may have impor t an t

medica l consequences , and various mechanisms leading to p ropaga t ion failure

have been invest igated by compute r s imulat ions of nerve conduc t ion [1-3].

Expe r imen ta l observat ions of propagat ing fronts in arrays of coupled bi-

s table chemical reactors are available recent ly [4]. Analyt ica l work on bistable

waves in discrete r eac t ion -d i f fus ion systems has been l imited to the fol lowing

equa t i on [5, 6]:

du,, d t - f ( u , , ) + d ( u , , + , + u,, j - 2 u , , ) , 1)

where f ( u , , ) = u , , (u , , - 1)(a - u , ) and a, d are fixed parameters . K e e n e r [5]

invest igated the limit d--+ w and d--+ 0 (a fixed) and concluded that there exists

a va lue d = d* ~ 0 below which propaga t ion fails. E r n e u x and Nicolis [6]

analyzed the limit a--+ 0 and d = d(a) - -+ 0 and found that

d*(a) = ' 2 aa + U(a3) . (2)

In this no te , we consider a second mechan i sm for p ropaga t ion failure, namely

the case of a variable coupl ing constant d = d ( n ) .

0378-4371/92/$05.00 © 1992-Elsevier Science Publishers B.V. All rights reserved

V. Booth, T. Erneux / Propagation failure in reaction-diffusion systems 207

2. Nonuniform distribution of a parameter

The effects of a nonuniform distribution of the coupling constant have been investigated in detail by numerous computer simulations. It models, for example, the change in axon diameter along a segment of the nerve fiber or a variation of the exchange rate between coupled chemical reactors. We analyze this case by considering a modification of eq. (1) given by

d u n d t = f ( u , ) + d(un+ , + u._, - 2 u , ) ( 2 ~ < n ~ < N - 2 ) ,

dUN 1 d t -- f ( U N - I ) Ac d ( U N 2 -- UN 1) ~- D(UN - UN-I ) ,

d u N dt - f ( U N ) -~ D(UN 1 -- U N ) "

(3)

(4)

(5)

In eqs. (3 ) - (5 ) , d is defined as the exchange rate between cell n and cell n - 1 (2 ~< n < N - 2) including the exchange between cell N - 2 and cell N - 1. D is defined as the exchange rate between cell N and cell N - 1 (see fig. 1). A propagating wave is initiated by the following boundary and initial conditions:

u l = l ( t />0) and u ~ = 0 ( 2 ~ < n ~ < N a n d t = 0 ) . (6)

We solve eqs. (3 ) - (6) in the limit a--e0 assuming both d(a) and D(a) small. Specifically, we assume that d = ~'(a 2) > d* where d*(a) is given by (2). This implies that propagation is possible from cell 2 to cell N - 2 . However , propagation from cell N - 1 to N depends on the value of D. We have found two cases: (1) D = G(a 2) and (2) D = 0(a). Case (1) is similar to the problem

d d D

1 I 2 N

Fig. 1. N cel ls are coup l ed l inear ly . In each cell o p e r a t e s a b i s t ab le reac t ion , d d e n o t e s the e x c h a n g e ra te b e t w e e n cell n and cell n - 1 (n = 2 , . . . , N - 1) and D is the exchange ra te b e t w e e n

cel l N and N - 1.

208 V. Booth, T. Erneux / Propagation failure in reaction-diffusion systems

of a uni form coupl ing constant . We find that p ropaga t ion is possible if D > D~ where

O l ( a ) = l 2 za + C(a3) . (7)

Case (2) is more interesting. Assuming that u n ~-1 (2 ~< n ~< N - 2) as t > t*,

we concent ra te on the equat ions for u N ~ and u N. We first in t roduce

d ( a ) = a2do + ~'(a 3) and D = a D o + G(a 2) (8)

into eqs. (4) and (5) and seek a solution of the form

uj(~-, a) = avj(z) + C'(a 2) ( j = N - 1 and N ) , (9)

where ~- = a ( t - t*). The leading problem for v x i and v N is given by

dVN 1 dr - d° + D°(UN -- ON I) -[- ON I(UN 1 -- 1 ) , ( 1 0 )

d o N d-r - D " ( v N , - - V N ) + V N ( V N - - 1 ) , V N , (O)=vN(O ) = 0 . (11)

P ropaga t ion appears if v N ~ and v N both jump f rom zero to a large value. This

UN

I I

D1 D2 D Fig. 2. Bifurcation diagram of the small amplitude steady states if d * < d < d**. We represent u;,, as a function of D. D~ and D 2 denote two limit points. Propagation (jump transition) is possible if D t < D < D 2. The diagram has been computed from eqs. ( i0) and (11) with dvu/dT-= do N j~ d~- = 0 and d o = 0.4.

V. Booth, T. Erneux / Propagation failure in reaction-diffusion systems 209

is possible if eqs. (10) and (11) do not admit a stable steady state solution. An analysis of the steady state equations then shows that the conditions for

propagat ion are

(i) d > d * * ( a ) = ½ a 2 + C ( a 3) and D > D , ( a ) , (12)

o r

(ii) d*(a)< d< d**(a) and D~(a)< D < D2(d, a). (13)

The second case is illustrated in fig. 2. In this figure, we represent the bifurcation diagram of the small amplitude steady states. It can be shown from the steady state equations that D2(d, a)--~ as d - d * * - - - > 0 .

We conclude f rom case (2) that propagat ion failure is possible if D is large provided that d is sufficiently small. We have verified numerically this result by direct integration of eqs. (3 ) - (6 ) with the following values of the parameters: N = 1 0 , a = 0 . 1 , d = 0 . 0 4 and D = 0 . 0 3 6 . (Note that d * = 2 . 5 x 1 0 3 and d** = 5 × 10 -3. Thus the condition d* < d < d** has been verified.) Propaga- tion failure for a large value of D is also possible if we consider a system of N cells with exchange rate d for cells 1 to M < N and exchange rate D for cells M to N. This can be demonstra ted by investigating the limit D large of the steady state equations.

Acknowledgements

T E was suppor ted by the US Air Force Office of Scientific Research under Gran t no. AFOSR-90-0139 and the National Science Foundat ion under Grant no. DMS-9001402. VB was supported by an NSF Graduate Fellowship.

References

[1] J.P. Keener, J. Theor. Biol. 148 (1991) 49. [2] J. Bell, Excitability behavior of myelinated axon models, in: Reaction-Diffusion Equations,

K.J. Brown and A.A. Lacey, eds. (Clarendon, Oxford, 1990) pp. 95-116. [3] J. Rinzel, Ann. N.Y. Acad. Sci. 591 (1990) 51. [4] J.P. Laplante and T. Erneux, Propagation failure in arrays of coupled bistable chemical

reactors, J. Phys. Chem. (1992), in press. [5] J.P. Keener, SIAM J. Appl. Math. 47 (1987) 556. [61 T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction-diffusion systems,

submitted (1991).