Stability of travelling waves in stochastic Nagumo equations

Stability of travelling waves for the Nagumo equation on the whole line is proven using a new approach via functional inequalities and an implicitely defined phase adaption. The approach can be carried over to obtain pathwise stability of travelling wave solutions in the case of the stochastic Nagumo equation as well. The noise term considered is of multiplicative type with variance proportional to the distance of the solution to the orbit of the travelling wave solutions.


Introduction
The purpose of this paper is to introduce a new approach to the study of (local) stability of travelling waves and pulses in excitable media that is in particular well-suited for stochastic perturbations. We are interested in the classical equations modelling the propagation of the action potential travelling along the axon of a neuron. As a starting point in this paper we consider the Nagumo equation on the real line (cf. [11]) perturbed by stochastic forcing terms. We make particular use of the explicit knowledge of the travelling waves in this case. However, our approach will be robust w.r.t. small perturbations in the coefficients. Since the spectral considerations, employed in the classical stability analysis of nerve axon equations (cf. [4,6,7] and the recent monograph [3]) are not easy to carry over to the stochastic case, we look for a pathwise stability analysis in the sense of the classical Lyapunov approach to the stability of dynamical systems. A first novelty of the paper is the introduction of an additional dynamics of gradient type that adapts a given solution of the stochastic Nagumo equation to the correct phase of the travelling wave. This explicitely given phase adaption, which is in addition easy to implement numerically, is the analogue of the phase conditions introduced as algebraic constraints in the classical stability analysis (see in particular [6]). As a second novelty in this paper, we replace the usual spectral considerations, applied to the Schrödinger operator, obtained as linearization of the underlying dynamics along a given travelling wave, by functional inequalities of Poincare type. Our hope is that the latter method will be generalizable also to general systems of reaction diffusion type because it only uses partial information of the travelling wave solutions. Certainly, it is well suited for stochastic perturbations as demonstrated in this paper. An additional advantage is that, in contrast to the usual spectral considerations, our approach allows explicit quantitative estimates, both, in the deterministic and in the stochastic case and sensitivity considerations w.r.t. the coefficients. The paper is organized as follows: In Section 2 we first present our new approach in the case of the deterministic Nagumo equation, to demonstrate the main arguments in a somewhat easier setting. The analogue to the usual spectral considerations of the Schrödinger operator associated with the linearization along a travelling wave is contained in Theorem 2.3. Our result obtained on the spectral gap is optimal (see Proposition 4.3). Theorem 2.6 then contains our main result on the local stability of travelling wave solutions. In Section 3 we consider the Nagumo equation perturbed with multiplicative noise. Combining our stability analysis of Section 2 with a careful analysis of the stochastic perturbation, we obtain in Theorem 3.1 the stochastic analogue of our local stability result in the deterministic case. Our identification of the implicitely defined phase allows to rigorously set up a stochastic differential equation for the speed of the wave front and thus gives rise to the correct decomposition of the stochastic dynamics into the travelling wave and random fluctuations. Work is in progress to generalize the approach to the stochastic neural fields equations considered in [1]. In addition to the new approach to the stability analysis via functional inequalities we also would like to mention that the type of stochastic Nagumo equations considered in this paper are also new in comparison with the models of spatially extended neurons subject to noise studied numerically and analytically by Tuckwell and Jost in [13,14] and also by Lord and Thümmler in [10]. In order to ensure existence and uniqueness of a solution to our stochastic partial differential equation we use the variational approach to stochastic evolution equations as presented in monograph [12] with recent extensions presented in [9]. In particular, we make use of the Ito formula, that can be obtained for the Hilbert space norm of the variational solution. The implied semimartingale decomposition can then be used to apply the one-dimensional (timedependent) Ito formula to any smooth transformation of the Hilbert norm.

The deterministic case
Consider the Nagumo equation The equation is obtained from the well-known Fitz-Hugh Nagumo system by letting ε ↓ 0, i.e., setting the recovery variable w constant, and further equal to the input current I. It is well-known that for parameters in the exitable region, the Fitz-Hugh Nagumo system admits a travelling pulse solution modelling signal propagation along the axon of a single neuron. The analogue for the Nagumo equation is a travelling [2]). We are interested in the local stability of this wave front in the function space H = L 2 (R).
Before we can state a precise definition of stability, we need to introduce first our concept of a solution that we are working with. To simplify notations in the following we write v T W (t) = v T W (·+ct). Next (formally) decompose the function v(t, ·) = u(t, ·) + v T W (t) w.r.t. the travelling wave. The resulting equation for u is then given by For the precise definition of the Laplacian ∂ 2 xx we need to introduce the Sobolev space V = H 1,2 (R) of order 1, equipped with the usual norm u 2 V := (∂ x u) 2 dx + u 2 H . Clearly, V ֒→ H densely and continuously. Identifying H with its dual H ′ we obtain the embeddings V ֒→ H ≡ H ′ ֒→ V ′ . Recall that w.r.t. this embedding the dualization i.e. the scalar product in H in the case where f ∈ H. The Laplacian ∂ 2 xx then induces a linear continuous mapping ∆ : The nonlinear term in equation (4) can be realized as a continuous mapping that is Lipschitz w.r.t. the second variable on bounded subsets of V with Lipschitz constant independent of t. Indeed, due to the elementary estimate u ∞ ≤ u V , the Taylor representation and uniform bounds on for finite constants c 1 and c 2 depending on f |[0,1] only.
Note also that the sum ν∆u + bG(t, u) satisfies the (global) monotonicity condition , since (f (s) − f (t))(s − t) ≤ η(s − t) 2 for all s, t ∈ R, and the coercivity condition It is now standard (see, e.g. Theorem 1.1 in [9]) to deduce for all u 0 ∈ H and all finite times T existence and uniqueness of a variational solution associated with (4). Clearly, we may consider this solution u as a solution on the whole time axes t ≥ 0.

The integral
t 0 ν∆u(s, ·)+bf (u(s)+v T W (s))−bf (v T W (s)) ds appearing in the integral equation (10) is well-defined as a Bochner integral in and therefore also locally Lipschitz.
The stability of travelling wave fronts for the Nagumo equation has been studied in many papers as a prototype example for metastability. The mathematical analysis of stability properties of v T W faces two major difficulties. The first one is the obvious fact that the reaction term f (u) in the equation (1) is not strictly dissipative in the sense that for some κ * > 0, or equivalently, the associated potential is not uniformly strictly convex, but a double-well potential. This remains true if we fix v 2 to be equal to the travelling wave v T W or any of its spatial translates v T W (· − y). A first naive calculation, exploiting the coercivity condition (9), only yields the following a priori estimate.
Proof. The coercivity condition (9) implies that d dt Integrating up the last inequality w.r.t. t yields the desired inequality.
However, restricting u to the orthogonal component of the derivative ∂ x v T W of the travelling wave solution, i.e. u∂ x v T W dx = 0, we have the following local dissipativity estimate according to the following a)) and C * = 6(ν + b) . Then The proof of the Theorem is postponed to Section 4.
The second difficulty in the mathematical analysis of the stability properties of v T W is to identify the correct phase-shift of v T W (t + t 0 ) to which to compare the given solution v of (1). To this end we introduce an auxiliary ordinary differential equation of gradient descent type associated with the minimization of the distance between v and the set N = v T W (· + C) | C ∈ R of all phase-shifted travelling waves. More precisely, given a solution v to the Nagumo equation (1) with initial condition v 0 satisfying v 0 − v T W ∈ H 1,2 (R), and given any relaxation rate m > 0 (that will be specified later) we consider the ordinary differential equation The next proposition states that the ordinary differential equation is well-posed.
Proof. Using the representation the continuity of B follows from the continuity of (t, C) → ∂ x v T W (· + C + ct) as a mapping with values in V , the continuity of (t, For the proof of the Lipschitz property w.r.t. C note that We next assume w.l.o.g. C 1 ≤ C 2 . According to the explicit represen- so that we can further estimate (13) Similarly, using |I| ≤ Inserting (13) and (14) into (12) yields the desired assertion.
According to the last Proposition the function C defined by (11) is well-defined. As already indicated, C will adapt to the correct phase of the v if we choose m ≥ C * (cf. Theorem 2.3) and our aim is to prove in the following that the difference converges to zero as t → ∞ if the initial condition u 0 = v 0 − v T W is sufficiently small in the H-norm. In the next Proposition we first identify the resulting evolution equation forũ.
be a solution of (4) and letũ be defined by (15).
In particular, The proof of the Proposition is an immediate consequence of the properties of v T W and the equations (4) and (11).
As usual we will now consider the linearization of the mappingG(t, u) around zero. To simplify notations, letṽ T W (t) := v T W (· + C(t) + ct). Then we can write Similar to the classical stability analysis of the Nagumo equation we now use the information on the spectrum of the Schrödinger operator ν∆u + bf ′ (v T W )u contained in Theorem 2.3 with the above localization to obtain the first local stability result.

2.1.
Main result in the deterministic case.
Theorem 2.6. Recall the definition of κ * and C * in Theorem 2.3. If the initial condition v 0 = u 0 + v T W is close to v T W in the sense that Proof. Letũ(t) := v(t) −ṽ T W (t) be as in (15). Then Proposition 2.5 and (18) imply that (20) Using translation invariance of ν∆ and (∂ x u) 2 dx, Theorem 2.3 yields the estimate (21)

Inserting (21) into (20) yields that
In the next step we define the stopping time with the usual convention inf ∅ = ∞. Continuity of t → ũ(t) H implies that T > 0 since u 0 H < δ κ * b(4+a) . For t < T note that 1 2 for t < T . Suppose now that T < ∞. Then continuity of t → ũ(t) H implies on the one hand that ũ(T ) H = δ κ * b(4+a) and on the other hand, using the last inequality, which is a contradiction. Consequently, T = ∞ and thus which implies the assertion.
Denote with Lip σ its Lipschitz constant. As the covariance operator Q is of trace class, positive semi-definite and symmetric, it has a positive semi-definite square root √ Q of Hilbert-Schmidt type. If we denote the representing integral kernel with k √ Q (x, y) ∈ L 2 (R 2 ) we assume that The theory of Wiener processes on Hilbert spaces and associated stochastic evolution equations can be found in the monograph [12].
As in the deterministic case we will give the equation a rigorous meaning by decomposing v(t) = u(t) + v T W (t) w.r.t. the (deterministic) travelling wave (3). The stochastic evolution equation for u is then given by where the nonlinear term G is as in (5), H . Indeed, note that the assumption on √ Q implies for any complete orthonormal system (e n ) n≥1 of H that hence the Lipschitz continuity of Σ in the Hilbert-Schmidt norm (27) follows. Similarly, using the pointwise inequality We now consider the equation (25) w.r.t. the same triple V ֒→ H ≡ H ′ ֒→ V ′ as in Section 2. Due to the properties (6), (7), (8) and (9), we can deduce from Theorem 1.1. in [9] for all finite T and all (deterministic) initial conditions u 0 ∈ H the existence and uniqueness of a solution (u(t)) t∈[0,T ] of (25) satisfying the moment estimate As a consequence we can apply Proposition 2.4 to a typical trajectory u(·)(ω) to obtain a unique solution C(·)(ω) of equation (11). It is also clear that the resulting stochastic process (C(t)) t≥0 is (F t ) t≥0 -adapted, since (u(t)) t≥0 is. We will assume as in the deterministic case that the relaxation rate m is sufficiently large, i.e., m > C * .
Similar to the deterministic case we now define the stochastic process which is (F t ) t≥0 adapted too and satisfies the stochastic evolution equation and the moment estimates Due to [12], Theorem 4.2.5, we have the Ito-formula It follows from the above representation that ũ(t) 2 H is a (scalarvalued) continuous local semimartingale, in particular we have also the (one-dimensional) time-dependent Ito-formula for any ϕ ∈ C 1,2 ([0, T ] × R + ). Here,Σ * (s, u) denotes the adjoint operator ofΣ(s, u).
Remark 3.2. The theorem establishes a global bound on the error between the solution v of the stochastic Nagumo equation (22) and the phase-shifted travelling wave v T W (· + ct + C(t)) on the set T = ∞.
The probability that T is infinite depends on two parameters, one is the initial error ũ(0) = v − v T W and the other component depends on the covariance operator of the noise term. In particular, the smaller the noise amplitude in the sense that Lip σ and/or M √ Q are small, the smaller the probability for T being finite. In this sense the stochastic process C(t) + ct gives the correct speed of the wave front and we will use the associated random ordinary differential equation in future work to study rigorously its statistical properties.

Proof of Theorem 2.3
The proof of Theorem 2.3 requires a number of preliminary results. To simplify notations in the following we simply write v instead of v T W in the whole section. Let w( Proposition 4.1. Let u ∈ C 1 c (R) and write u = hw. Then Proof. First note that We will prove in Lemma 4.2 below that This proves the assertion. Then The proof of the Lemma requires additional information on functional inequalities satisfied by the gradient form h 2 x w 2 dx, which will be provided in Proposition 4.3 and in Lemma 4.5 first: Proposition 4.3. The following inequality Here, the constant 4 3k 2 is the best possible. Proof. We will first show that To this end we will split up the estimate w.r.t. x ≥ 0 (resp. x ≤ 0) and show that Indeed note that for x ≥ 0, using Integrating the last inequality against w 2 dx we obtain that which gives (35).
For the proof of (36) note that w 2 (−x) = w 2 (x), so that (35) implies (36). Clearly, combining (35) and (36) implies (34). For the final step of the proof of inequality (33) let us consider the probability measure which implies the desired inequality.
To see that the constant 4 3k 2 is the best possible one, consider the and due to (37) h 0, Combining all these equalities yields Hence, if κ denotes the minimal constant for which the inequality holds for any h ∈ C 1 b (R) with hw 2 dx = 0, it follows by approximation of h 0 and its derivative in L 2 (w 2 dx) with functions in C 1 b (R) that the same inequality also holds for h 0 which implies κ ≥ k 4 3k 2 .
The Poincaré inequality proven above will be only sufficient to control the lower order term c h 2 w x w dx if the wave speed c is sufficiently small which means that a is sufficiently close to 1 2 . For small a however, we will need an additional information provided by inequalities contained in the following two lemmas: Proof. Let v * = 1 2 + 1 2 √ 3 be the unique solution v * ∈ ( 1 2 , 1) of 6v * (1 − v * ) = 1 and let x * := v −1 (v * ) > 0. We will now first show that For the proof of both inequalities note that It follows for x ≥ x * that Integrating the last inequality against w 2 dx yields and integrating the last inequality against w 2 dx again yields Combining (39) and (40) we obtain the inequality For the final step let us consider the probability measure µ(dx) := Z −1 w 2 dx on R + , where is a normalizing constant. Then (41) implies Proof. For the proof of the inequality note that using Rh 2 w x w dx = Rĥ 2 w x w dx = 0. The previous Lemma 4.5 and