Stability properties of a model of parallel nerve fibers

JOURNAL OF DIFFERENTIAL 40, 303-315 EQUATIONS Stability (1981) Properties of a Model Parallel Nerve Fibers JONATHAN of BEI.I. AND CHRIS C~XNER Departmenr of Mathematics, Texas A&M Uniuersity, College Station, Texas 77843 Kcccived January 10, 1980; revised June 30, 1980 1. INTRODUCTION AND PRELIMINARIES Iiodgkin and Huxley 1141 were the first to introduce a mathematical model for the conduction process in nerve axons which gave reasonable agreement to an extensive variety of experimental evidence. This model consists of three first-order ordinary differential equations coupled to a reac tion-diffusion equation. Evans and Shenk [6] proved the existence and uniqueness of solution of equations of generalized lo a system Hodgkin-Huxley type (see also 123, 11). There have also been a number of papers concerned with the existence of traveling wave solutions to systems related to the Hodgkin-Huxley model; see for example [ 13, 4 ]. Because of the analytic complexity of the Hodgkin-Huxley model, most work on the model, particularly qualitative behavior of the solutions, has been numerical. To gain some insight into the mathematical processes involved in the nerve condution phenomenon, FitzHugh [7] and Nagumo et al. [ 15 ] introduced a simpler prototype system of the form (1.1) where u represents the membrane potential and y represents a “recovery” process. Here the current-voltage relation f(u) has a cubic behavior displayed in Fig. 1. This system is known to give pulse-like solutions qualitatively similar to those of the Hodgkin-Huxley model. For background FIG. 1. Qualitative behavior of the f appearing in system (1.1). 303 OO22-039618 1/060303-l 3SOZ.OOjO Copyright :C; 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. 304 BELL AND COSNER on the behavior of (1.1) compared to the Hodgkin-Huxley equations, see [5, 181. Questions of existence and behavior of solutions of various kinds have been pursued very actively; see, for example, [ 10-12, 16, 19, 171. Much less attention has been given to the modeling of interactions that might take place between neighboring excited axons even though experimental work concerned with such questions appears to be quite old. For a review of some aspects of interaction phenomena and subsequent modeling efforts, see [20, 3 ] and references therein. In particular, in 121, a model is developed and the question of existence of traveling waves is examined. Such a model for two parallel axons consists of two reaction-diffusion systems coupled through diffusion terms. In this paper we are interested in examining a model of parallel nerve fibers of the structure mentioned above, but where we use single cell dynamics represented by the FitzHugh-Nagumo formulation (1.1). That is, in this paper, we consider a system of the form =(JIuI Ylf u2/ +P2~lxx -~2Gr --Y1Y,v = -J2@2> =02u2 Y2l (1.2) - Y2 9 - YZY,, for (x, t) E 10, co) X 10, co), where h(u) = U(U - u~)(u - bj) with 0 < aj < bj for j = 1, 2, and /i , : A 2, p, , p2, u, , c2, y, , and y2 are positive constants such that (1.3) Condition (1.3) ensures that the matrix is positive definite. We will consider the system (1.2) with the following u,x(O, t) =P,O), ~z,r(O, t) = P2Oh I t >o, (1.4) U,(X> 0) =4,(x), u2(x, 0) = Qz(xh y,(x,O)=!&(x), Y2cG 0) = @4(x): I boundary conditions: x > 0 ’ * A MODEL OF PARALLEL NERVE FIBERS 305 It will be assumed that i,,(O) ==p,(O) and #z,(O) =,+(O) and that the functions p,(t) and pr(t) have compact support. Further technical assumptions will also be made on the boundary data. The main object of the present article is to discuss the stability of the rest state for the system. It turns out that the rest state is locally stable, but a form of threshold behavior occurs which shows that the rest state is not globally stable. In trying to show the global existence and stability of solutions, it is natural to look for invariant regions, as in (171. However, the coupling of the system (1.2) in the diffusion terms puts restrictions on the, possible forms of invariant regions which are incompatible with the structure of the nonlinearity. Hence, the tool of invariant regions appears not to be applicable, and must be replaced with Lyopunov methods. Before analyzing the behavior of solutions as time goes to infinity? a brief discussion of the existence of global solutions is in order. Equation (1.2) may be rewritten as U,,--n,~,,,+P,~zxx+~,=-Sr(U,)+~r-YI~ U2r + PZUlXX - ~2u2xx $ u, =; -f,(u2) + l.42- J-2, Y,, + Y, = d,u, + (1 - Y,)Y,, (1.5) Yzc+Yz=~*U2+(1-y2)Y2. Let A = M(a*/aX*)+ I, 0 ( 0 I, 1 ; then (1.5) may be rewritten as VI +Av =.7(v), (1.6) where v = (u, , u2, y,, ;1*) andTis the right side of (1.5). Equation (1.6) may be regarded as an operator equation in the appropriate space. Let H,, denote the L*-Sobolev space of functions with two derivatives in L*[O, co], and let Z = {U E H, : u,(O) = O/ G H,. To ensure the existence of global smooth solutions of (1.2), (1.4), we assume the following: Hypothesis S. The functions ~1,and ,u~in (1.4) are such that there exists a vector function 8(x, t) = (0, ,6,, 0,O) such that 6j,(O, t) =pj(t)l j = 1,2, with 0 and 8, uniformly bounded in H, x H, and Lipschitz in t, and 0(x, t) z 0 for all t > t, for some t, > 0. Also, if h(r) = 8, + AB, we require that h(t) E 2 x % x (0)’ x (0) h(0) = h,(O) = 0, and h,(l) is Lipschitz in t. Finally, if Q,= (4,) ti2, d3, #4), then Q - 0(x, 0) and A (a - 0(x, 0)) belong to ZXZXL2XL2. 306 BELLANDCOSNER Hypothesis S will be true if 0, ,u, and ,u, are smooth, ~1, and ,uz have compact support in t, @ and the derivatives of order less than or equal to four of 4, and 0, decay rapidly enough as x-+ co, and the following compatibility condition is satisfied: In fact, we can obtain local existence of solutions to (1.2), (1.4) under slightly weaker hypotheses, but we need some additional smoothness of the solutions to obtain global existence. To obtain existence results, we use semigroup theory. A realization of A as a closed unbounded operator on L* x L* x L* x L* is obtained by choosing dam(A) = 2 x 2 x L* x L*. It follows by standard computations that for some constant c, A satisfies IU -W’II<c/(l +PO (1.7) for all 1 E Cc such that n/2 -E < arg 2 < 3rr/2 + E. Thus, A generates analytic semigroup. Hence, we can solve Tr + Al- = -h(t), I-(O) = @ - qx, 0). Let Y = r + 8; then let w = v - Y; then if w E dom A, and w satisfies w,+Aw=f(w+Y); w(0) = 0, (1.8) the function v < w + \y satisfies (1.6) and (1.4). The local existence of solutions to (1.8) can be established via the following result, which is originally due to Sobolevski: THEOREM 1. Suppose that A is a closed operator on a Banach space Y, such that (1.7) holds. Suppose that g(t, w) is such that there exist constants a, q E (0, 1) so that for any R > 0, there is a C(R) for which lI~~~,~-“~~-~(~,~-“~~lly~~~~)[l~-~l~+Il~-~ll,l for all t,zE [O,t,], v, w E Y with IIvllv, llwlly &R. Then for any w0 E dom A, and each R > [[A”w,,[(~, there exists a t * = t*(R, (IAawOII,,) > 0 such that the problem wt + Aw = g(t, w) w(0) = wg (1.9) A MODEL OF PARALLEL NERVE 307 FIBERS has a unique solution in [0, t*]. Furthermore, if there exist constants R’ and C’ such that for any solution w of (1.9) in (0,7] with 0 < t < t, , then R may be chosen so that R > R’, and the local existence assertion of the theorem may be applied again on [t*, 2t*], and so on until 10, t,] is exhausted. Remarks. A more general version of Theorem 1 is proved in (81 as Theorems 11.16.1-11.16.5. Equation (1.8) satisfies the hypotheses of Theorem 1 up to (1.10); that can be established in a straightforward manner following the discussion in 181 for a single parabolic equation. The discussion following Theorem 2 of [9] and Hypothesis S ensure that the local solution of (1.8) (and thus of (1.2), (1.4)) will have two continuous time derivatives in Y = L2 X L2 X L2 X L2. To show that the local solution extends to the full interval [0, t,], it suffices to show that if w solves (1.8) locally, then llAw\l,, < C’. (For C’ large, the other conditions in (1.10) then follow.) Since t, was arbitrary, showing that solution exists in 10,c, ] implies that it exists for [0, co). To obtain the a priori estimate that any solution of (1.8) in (0, 51E [O,t, J satisfies llAwllv < C’, we use a Lyopunov functional. The method is essentially the same as that used in 111and [ 16). Suppose that w = (u,, u2, y,, y2) satisfies (1.8) in [0, t2j C_10,t, 1. Consider the functional E,(t)= j” f[U:,t u:, t u;, t u& t c; + u: t y; t y;l dx. 0 We can compute E’,(t) by substituting for ulrr and utll the expressions obtained by subtracting I out of (1.2) and differentiating, then using integration by parts. If we assumethat the coefficients off, satisfy 0 < a, t bi < 3, 0 < a, < bi for i=l,2, (1.11) then we obtain from Cauchy’s inequality the estimate E’,(t) < c,E,(t) $ c2 for some constants, c, and c,. Thus, for any E > 0, E,(t) <K,(t,) + K,(t,) for all t E (0, 1,) such that w(t) exists and satisfies (1.8) on [O, t]. But Sobolev’s inequality implies that for some c > 0, i = 1, 2, 308 BELL AND COSNER Also, Ml: <E,(t), and since yi, is a linear combination of ci, JJ~, and components of Y for i = 1,2, ]IWI]: < K,E,(I) for some constant K,. Hence, if we choose E > 0 so that w exists in 10,t,J with E < t,, then for all tE [s,t,J for which w continues to exist, (]wJJY,(]w!(~, and ]zI~~,i= 1, 2, are bounded uniformly in terms of E,(E) and t,. Thus, Ijg(t, w)l], is similarly bounded. Hence, as long as w exists, llAwl!v= II-w, + s(c w>ll,< KdE,(&). (1.12) This is exactly the estimate needed in (1.10). Fix t, > 0. We obtain the existence of a unique solution of (1.2), (1.4) in (0, t”) for some t* > 0 by a direct application of Theorem 1 to (1.8). Choose c E (0, t*); then E,(c) will be some fixed number. It then follows from (1.2) that I(Aw](, is uniformly bounded throughout any subinterval of (c, r,] for which w exists. Hence, (1.10) is satisfied so Theorem 1 can be applied repeatedly in [e, E + t** 1, [a f f**, E t 2P*], and so on, for some t** > 0, until (0, t, ] is exhausted. Hence there exists a solution w of (1.8) on (0, f,]. A solution v of (1.2), (1.4) is obtained by taking v = w + Y. Since f, > 0 was arbitrary, the solution must exist for all time. We thus have proved the following: THEOREM 2. Suppose thaf Hypothesis S is satisfied and (1.11) holds. Therz fhe problem (1.2), (1.4) has a unique solution in (0, 03) X (0, a~). The same sort of argument can be applied with more general nonlinearities, and in more space dimensions. Such results are obtained in ] 161 for systems with a slightly different structure than (1.2). 2. STABILITY OF THE REST STATE We again consider system (1.2). In the derivation of the model from electrical circuit considerations of the two parallel membranes [2], it turns out that p, =p2 =p. We can rescale the x variable so that system (1.2) can be rewritten as Y,r=~IUI-YlYl~ u21 +f?(uJ PUIXX, + Y, = uzxxY2t =~2u2 - (2.1) Y2Y23 where all parameters are nonnegative and p* < /i, that is, the diffusion matrix is positive definite. Thus if B(p, q) = Ap* - 2ppq + q*, then there exists a positive c, such that (2.2) A MODEL OF PARALLEL 309 NERVE FIBERS The only constant solution to (2.1) is the rest state (u,, yr, u,, yz) = (0, O?0,O). We expect the origin to be.only locally stable, which is the case for the single nerve fiber model. Yamaguchi 1231and Yoshizawa and Kitada 1241 showed for the single nerve case (system (1.1) with 7 = 0) that under certain conditions on the nonlinearity, the solution tends to zero uniformly in x as t --t co. We will derive a similar result. Since the solution of (2.1) satisfies (1.4), where the pi’s, j = 1, 2, are assumed to have compact support, there is a f,, > 0 such that for t > t,, ,uu,(f)=p2(0) = 0. Let FJ(u)q"fi(s)ds=u2 Caj $u2- ‘0 + 3 bj) u + ujbj j= 1,2, y-' I I and consider the following Lyopunov functional for t > t,: E(t) - I.= jk$u:,- kpu,,uzX +;z& + kF,(u,) “0 B +kF,(u,)+;u:+++k 4 + Y: (Ul + Y,12 u:+y: i-K -++- 2 + k 042 2 2 dx I + Y2J2 2 . (2.3) We also define E,(t) = jr {u:, + u:, + uf + u: + yt + y:] dx. By Sobolev’s lemma, there is a constant c > 0 such that SUP o<x<w t)l< c{E~(~)}“2, l"j(x, j= 1,2. (2.4) In effect, (2.2) and (2.4) define c and co for the rest of this section. We define the following quantities: I,=(K+k)a,-ky,, aJ =.‘aa,+b. 2 r/2+k &=(L+k)o,-ky,, (uJ - bj)2 I j= 1,2, 4 $(aj+6j)‘-ajb,+ l+Yj---j 1 * 310 BELL AND COSNER We now choose any k that satisfies k > 2/c,. Define b and /I by the expressions b=(a,-l)K+k(a,-y,-l), B=(a2-l)L+k(o,-y,-l), and let K be defined by (2.7) We then let L be suffkiently large such that the following holds: L >max(2,y;‘-kk,p2,q2,r2/~,J, (2.8) K > max(2, Y; ’ - k A, ql, r,la, 1. Let 0 < a ,< mini, ,.2 aj; then THEOREM 3. Zf (u, , y, , u2, yz) is a solution )uj(x, t,,)] < a, j = 1,2, and (E(f,)) “’ < a/c, then ‘,s,“,p, Proof: lu,(x, t)l, o,s,u,p, lu*(x, a+ (O,O) to (2.1) us t++co. such that By subtracting E, from E, it follows that the integrand becomes L + -j- (u: + y:> - u:, - u:, - u: - 24:- y: - y: = +?(u 1x9u*x) 1 - (4, + &I a, + b, + ku; -u:----u, 4 + ku: 1 -us--u*+ a2 + b2 4 3 + 3 +;(uI +y,)* + 5 (242 + ~2)’ a,& 2+ +; bfK-2 2k [(K - 2)~: + (L - 2)~: 1 7 A MODEL OF PARALLEL 311 NERVE FIBERS which is positive if the u:, u:, y:, y: terms are positive and (k/2) B(u,x, u,,) > (u:, + nix), where B was defined previously. The latter is true becauseof the choice of k and the yf , y: terms are positive becauseK and L are greater than 2. The U: and U: terms are positive if the discriminants associated with the quadratic expressions are negative, which is the case if K + b > 2 + k[(2/9)(a, + b,)‘- a,b,], (2.9) L + P > 2 + k[(2/9>(a, + b,)* - a,b,]. But (2.8) implies a,L > r2 and a,K > r, and becauseof (2.6), the inequality (2.9) holds; hence E(t) > E,(t). Now taking the derivative of E(r) and substituting in the differential equations (2.1), E’(t) = fin (-k&f, + u:,) - B(u,,, ~4~) + y,u,(A, - (b + k -I-K)) '0 + ~2~21~2 - (P + k + L)] - y,(k WY: + - +L)Y: Yz(k +k(a,u:+o,u:)-(b+k+K)u,f,(u,) - (P + k + L) ~2.W2) 1 k where B(ulx, uzx) = A(b + k + K) u;, - ,@ + P + 2k + K + L) u,,u~~ + (p + k + L) u;,. Since (2.6) implies A,=b+k+K, A,=j3+k+L, and (2.7) implies 1, = A,, then they, u, and y2u2 terms are zero and we can rewrite (2.10) as E’(f) = rp ‘0 I -W:t + u:t) - 12W,x9 uzx) - y,(k + WY: -yz(k+L)+h fw-,(.,)-$41 1 kff, -12 dx M*(u*)-~u* 2 [ = -E,(t) 2 + O” -k(u:, f I -71 (K+“+$+-l, -4 [u,hw II + u:t> - lJ,B(u,,> (L+k-;)Y: (,,,Y l )u:] uzx)- (u:, + &>I 312 BELL AND COSNER But (2.8) guarantees each of the square-bracketedquantities is positive. Thus we have (2.11) E’(t) < --E,(t) < 0, provided that uj < ai, j = 1, 2. By hypothesis E’(t,) < 0 and we claim E’(t) < 0 for all t > to. Supposethare exists a t* > I, such that u,(x, t*) > a, for some x > 0 and let T, = inf(t > t,: u,(x, t) > a1 for some x > 0). If for some x > 0 and some t** > to, z+(x, t*,) > a2, then we can define an analogous T2. Let T = min{ T, , T,}, then for t & T, u, < a, and u, < a2 for all x > 0 so that E’(t) < 0 for all t < T. Hence E(T) < E(t,), but by Sobolev’s lemma, SUP,,<~<~1uj(x, t)l < c( E(T)} li2 < a, j = 1, 2, which is a contradiction. Therefore E(t) is a nonincreasing, bounded, positive function for t > to and so lim,, E(t) exists; call it e. Note also this implies that u, and u2 remain bounded for all t > to so that by Cauchy’s inequality E(I) < ME,(t) for some positive constant M. Now if e = 0, then lim,, E,(t) = 0 and by Sobolev’s lemma, the conclusion of the theorem follows. Thus suppose that e > 0. Then there exists a 7 > to such that for t > r, fe < E(I) ( $e. In particular, je < ME,(t) so that by (2.1 l), E’(t) < -e/2M < 0. But this contradicts the fact that for all t > t,, 0 < E,(t) < E(t). Remarks. Using another Lyopunov functional, we can give a condition on the solution to (2.1) which does not tend uniformly to zero as t + +co. That is we can find an E,(t) such that if, for some t, > 0, E,(t,) < 0, we can force, for all t > t,, uj > aj for somej. Coupled together with Theorem 3, this indicates a threshold behavior for the system. For purposes of the following computations, we assume Yi’>Q (2.12) j= 1,2. Consider E,(t)qpx -pu,,u,,+~u:,+F,(u,)+F*(u*)+pU: 0 -bu,y,+py:+~u:--u,y,+~y: = * 1 p@b J0 I + B,(u,, I %x) + F,(u,) + Y,) + B204*9 Y2) d-G I F2&2) dx A MODEL OF PARALLEL NERVE FIBERS 313 where the bilinear form B was defined earlier and B&y)=+‘-buy+;y’, B,(u, y) = + u2 -Buy + + y’. Then since -. ( U2r A --242-t2 B+l 2 ’ y2 ) = .-uzt2 y;+Au,u,,-(B+l)y,u,,++I)u,y, (and similarly the form for -(uI1 - (a/2) u, + ((b-t- 1)/2)y,)‘) E#)=jo* I- [u,t-~u,+~Y,]2-[u2t-qu,+~Y*]2 + (f&b+:+ $ we can write E;(t) in ($kIo,) u: by,++f(b+l) (b+ l)* 4 The condition (2.12) comes in in the following a2/4=ba,, (b + 1)‘/4=dy,r way. Set A2/4 = Ba2, (B + 1)2/4 = Dy,, (2.13) and define a = 2{y, t- d5}, A = 2(7, + d5,. (2.14) Then a2 = 4[ay, - O, 1 (respectively A ’ = 4[ay, - u2 J) and b = ay,/a, - i (respectively B ==Ap/02 -. 1). It then follows from (2.13)-(2.14) that 314 BELL AND COSNER 67, + da, - (a/2)@ + 1) = 0 and By, + Da, - (A/2)(B + 1) = 0. Also B,(u, 9YJ = (a/2) u: - (a2/40,) u, Y, + (a2~,/40:)~: > 0 and similarly, B,(u,, yZ) > 0. Thus (2.13)-(2.14) imply that b+l 2 Ulf -“,*t2 + [u2,-++- B+l 2 Y2 Yl \* II2 dx<O Therefore, if for some to < 0, E,(t,) < 0, then E,(t) < 0 for all t > to. Then, from (2.2), (2.13) and (2.14) 0 > em(+B(u1x3u2.r) + F,(u,) J0 a 1% ‘0 IFI + F2@2)1 + F2k2) + B,(u,, Y,) f B,(u,, u2)l dx (2.15) fix. But (2.14) implies that for all t > to, at least one of the uj is such that u, > uj, j = 1, 2, becauseof the nature of the nonlinearity. We want to emphasize that the single nerve case, system (1.1) has analogous behavior about the origin because similar Lyopunov functionals can be constructed. A large invariant region can also be constructed for system (1.1) (see [ 171). This appears not to be the case for the parallel fiber case becauseof the nature of the coupling, and hence we do not seemto have that tool at our disposal. REFERENCES 1. R. AROMAAND Y. HASAGAWA,On the global solutions for mixed problem of a semilinear differential equation, froc. Japan Acad. 29 (1963), 721-725. 2. J. BELL, Modeling parallel, unmyelinated axons: pulse trapping and ephaptic transmission, SIAM. J. Appl. Moth., to appear. 3. J. 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