Random Attractor for Stochastic Hindmarsh-Rose Equations with Multiplicative Noise

The longtime and global pullback dynamics of stochastic Hindmarsh-Rose equations with multiplicative noise on a three-dimensional bounded domain in neurodynamics is investigated in this work. The existence of a random attractor for this random dynamical system is proved through the exponential transformation and uniform estimates showing the pullback absorbing property and the pullback asymptotically compactness of this cocycle in the $L^2$ Hilbert space.


Introduction
. The Hindmarsh-Rose equations for neuronal spiking-bursting observed in experiments was initially proposed in [19,20]. This mathematical model originally consists of three coupled nonlinear ordinary differential equations and has been studied through numerical simulations and mathematical analysis in recent years, cf. [19,20,22,24,37,47] and the references therein. It exhibits rich bursting patterns, especially chaotic bursting and dynamics, as well as complex bifurcations.
Very recently in [27], it is shown that there exists a global attractor for the diffusive and partly diffusive Hindmarsh-Rose equations in the deterministic environment.
In this work, we shall study the longtime random dynamics in terms of the existence of a random attractor for the stochastic diffusive Hindmarsh-Rose equations driven by a multiplicative white noise: for t > 0, x ∈ Ω ⊂ R n (n ≤ 3), with the Neumann boundary condition ∂u ∂ν (t, x) = 0, ∂v ∂ν (t, x) = 0, ∂z ∂ν (t, x) = 0, t > 0, x ∈ ∂Ω, (1.4) and an initial condition Here W (t), t ∈ R, is a one-dimensional standard Wiener process or called Brownian motion on the underlying probability space to be specified. The stochastic driving terms with the multiplicative noise indicate that the stochastic PDEs (1.1)-(1.3) are in the Stratonovich sense interpreted by the Stratonovich stochastic integrals and the corresponding differential calculus.
In this system (1.1)-(1.3), the variable u(t, x) refers to the membrane electric potential of a neuronal cell, the variable v(t, x) represents the transport rate of the ions of sodium and potassium through the fast ion channels and is called the spiking variable, while the variables z(t, x) represents the transport rate across the neuronal cell membrane through slow channels of calcium and other ions correlated to the bursting phenomenon and is called the bursting variable.
Assume that all the parameters a, b, α, β, q, r, J and ε in the above equations are positive constants except c (= u R ) ∈ R, which is a reference value of the membrane potential of a neuron cell. In the original model of ODE [47], a set of the typical parameters are (1.6) This neuron model was motivated by the discovery of neuronal cells in the pond snail Lymnaea which generated a burst after being depolarized by a short current pulse. This model characterizes the phenomena of synaptic bursting and especially chaotic bursting in a three-dimensional (u, v, z) space. The chaotic dynamics is mainly reflected by the sensitive dependence of the longtime behavior of solutions on the initial conditions. The presence of the multiplicative noise as well as the diffusion of ions and membrane potential in the neuron model is expected to have large effect on the long-term behavior of the dynamical system in a random environment.
The figure below is an illustration of the chaotic trajectories of the deterministic Hindmarsh-Rose model when the key parameter J of the injected stimulation to the membrane potential varies. 2, (f) the x-z phase portrait when J = 3.1. Source: [25] Neuronal signals are short electrical pulses called spike or action potential. Neurons often exhibit bursts of alternating phases of rapid firing spikes and then quiescence. Bursting constitutes a mechanism to modulate and set the pace for brain functionalities and to communicate signals with the neighbor neurons. Bursting patterns occur in a variety of bio-systems such as pituitary melanotropic gland, thalamic neurons, respiratory pacemaker neurons, and insulin-secreting pancreatic β-cells, cf. [4,7,10,20]. The current mathematical analysis of this neuron model mainly uses bifurcation theory together with numerical simulations, cf. [3,15,23,24,28,37,38,42,47].
Neurons communicate and coordinate actions through synaptic coupling or diffusive coupling (called gap junction) in neuroscience. Synaptic coupling of neurons has to reach certain threshold for release of quantal vesicles and form a synchronization [13,29,34].
The chaotic coupling exhibited in the simulations and analysis of this Hindmarsh-Rose model of ODE shows more rapid synchronization and more effective regularization of neurons due to lower threshold than the synaptic coupling [38,47]. But the dynamics of chaotic bursting is highly complicated.
It is known that Hodgkin-Huxley equations [21] (1952) provided a four-dimensional model for the dynamics of membrane potential taking into account of the sodium, potassium as well as leak ions current. The FitzHugh-Nagumo equations [16] (1961-1962) derived a two-dimensional model for an excitable neuron with the membrane potential and the current variable. This two-dimensional ODE model admits an exquisite phase plane analysis showing spikes excited by supra-threshold input pulses and sustained periodic spiking with refractory period, but due to the 2D nature FitzHugh-Nagumo equations exclude any chaotic solutions and chaotic dynamics so that no chaotic bursting can be generated.
The research on this model (1.6) indicated the possibility to lower down the neuron firing threshold. More observations also indicate that the Hindmarsh-Rose model allows varying interspike-interval when the parameters vary. Therefore, the 3D model (1.6) is a suitable choice for the investigation of both the regular bursting and the chaotic bursting. It is expected that the augmented neuron model of the stochastic Hindmarsh-Rose equations (1.1)-(1.3) studied in this paper will be exposed to a wide range of applications in neuroscience.
The rest of Section 1 is the formulation of the stochastic system (1.1)-(1.3) and provides basic concepts and results in the theory of random dynamics. In Section 2, we convert the stochastic PDEs to a system of random PDEs by the transformation of exponential multiplication. Then the global existence of pullback weak solutions is established. The uniform estimates will show the pullback absorbing property of the Hindmarsh-Rose semiflow in the L 2 space. In Section 3, we shall prove the main result on the existence of a random attractor for the diffusive Hindmarsh-Rose random dynamical system.

Preliminaries and Formulation.
To study the stochastic dynamics in the asymptotically long run, we first recall the preliminary concepts for random dynamical systems, or called cocycles, cf. [1,2,9,11,12,14,17,26,31]. Let (Q, F, P ) be a probability space and let X be a real Banach space. Definition 1.1. (Q, F, P, {θ t } t∈R ) is called a metric dynamical system (MDS), if (Q, F, P ) is a probability space and θ t is a time-shifting mapping with the following conditions satisfied: (i) the mapping θ : (iii) θ t+s = θ t • θ s for all t, s ∈ R, and (iv) θ t is probability invariant, meaning θ t P = P for all t ∈ R. Here B(X) stands for the σ-algebra of Borel sets in a Banach space X and (θ t P )(S) = P (θ t S) for any S ∈ F. Definition 1.2. A continuous random dynamical system (RDS) briefly called a cocycle on X over an MDS (Q, F, P, {θ t } t∈R ) is a mapping which is (B(R + ) ⊗ F ⊗ B(X), B(X))measurable and satisfies the following conditions for every ω in Q: (i) ϕ(0, ω, ·) is the identity operator on X.
A random set S(ω) ⊂ X is called compact (reps. precompact) if for every ω ∈ Q the set S(ω) is a compact (reps. precompact) set in X.
Remark 1. If {B(ω)} ω∈Q is a closed random set of X such that for any fixed x ∈ X the mapping ω → d(x, B(ω)) = inf{ x − y : y ∈ B(ω)} is (F, B(R + )-measurable, then B(ω) is a random set in the sense of Definition 1.3, cf. [5]. Considering that a random set may be neither closed or open, Definition 1.3 is more general.
We shall let D X denote an inclusion-closed family of random sets in X, meaning that if D = {D(ω)} ω∈Q ∈ D X andD = {D(ω)} ω∈Q withD(ω) ⊂ D(ω) for all ω ∈ Q, thenD ∈ D X . Such a family of random sets in X is called a universe. In this work, we define D H to be the universe of all the tempered random sets in the Hilbert space H = L 2 (Ω, R 3 ). Definition 1.5. For a given universe D X of random sets in a Banach space X, a random set K ∈ D X is called a pullback absorbing set with respect to an RDS (cocycle) ϕ over the MDS (Q, F, P, {θ t } t∈R ), if for any bounded random set B ∈ D X and any ω ∈ Q there exists a finite time T B (ω) > 0 such that Definition 1.6. Let a universe D X of random sets in a Banach space X be given, A random dynamical system (cocycle) ϕ is pullback asymptotically compact with respect to D X , if for any ω ∈ Q, the sequence has a convergent subsequence in X, whenever t m → ∞ and x m ∈ B(θ −t ω) for any given B ∈ D X . Definition 1.7. Let a universe D X of tempered random sets in a Banach space X be given. A random set A ∈ D X is called a random attractor for a given random dynamical system (cocycle) ϕ over the metric dynamical system (Q, F, P, {θ t } t∈R ), if the following conditions are satisfied: (i) A is a compact random set in the space X.
(iii) A attracts every B ∈ D X in the pullback sense that where dist X (·, ·) is the Hausdorff semi-distance with respect to the X-norm. Then D X is called the basin of attraction for A.
The existence of random attractors for continuous and discrete random dynamical systems has been studied in the recent three decades by many authors, cf. [1,2,6,9,11,12,18,31,32,36,39,40,41,44,45,46,48]. The following theorem is shown in [12,31]. Theorem 1.8. Given a Banach space X and a universe D X of random sets in X, let ϕ be a continuous random dynamical system on X over the metric dynamical system (Q, F, P, {θ t } t∈R ). If the following two conditions are satisfied: (i) there exists a closed pullback absorbing set K = {K(ω)} ω∈Q ∈ D X for ϕ, (ii) the cocycle ϕ is pullback asymptotically compact with respect to D X , then there exists a unique random attractor A = {A(ω)} ω∈Q ∈ D X for the cocycle ϕ and the random attractor is given by We now formulate the initial-boundary value problem (1.1)-(1.5) of the stochastic Hindmarsh-Rose equations with the multiplicative white noise in the framework of the product Hilbert spaces (1.7) The norm and inner-product of H or L 2 (Ω) will be denoted by · and ·, · , respectively. The norm of space E will be denoted by · E . The norm of L p (Ω) or L p (Ω, R 3 ) will be denoted by · L p for p = 2. W use | · | to denote a vector norm in Euclidean spaces. The nonpositive self-adjoint linear differential operator is the generator of an analytic C 0 -semigroup {e At } t≥0 of contraction on the Hilbert space H. By the Sobolev embedding H 1 (Ω) → L 6 (Ω) for space dimension n ≤ 3, the nonlinear mapping is locally Lipschitz continuous. Thus the initial-boundary value problem (1.1)-(1.5) is formulated into an initial value problem of the following stochastic Hindmarsh-Rose evolutionary equation driven by a multiplicative white noise, (1.10) Here g(t, ω, g 0 ) = col (u(t, ·, ω, g 0 ), v(t, ·, ω, g 0 ), z(t, ·, ω, g 0 )), where dot stands for the hidden spatial variable x.
Assume that {W (t)} t∈R is a one-dimensional, two-sided standard Wiener process in the probability space (Q, F, P ), where the sample space where C(R, R) stands for the metric space of continuous functions on the real line, the σ-algebra F is generated by the compact-open topology endowed in Q, and P is the corresponding Wiener measure [1,9,26] on F. Define the P -preserving time-shift transformations {θ t } t∈R by Then (Q, F, P, {θ t } t∈R ) is a metric dynamical system and the stochastic process , ω ∈ Q} is the canonical Wiener process. Accordingly dW/dt in (1.10) denotes the white noise. The results we shall prove in this paper can be extended to a vector white noise with three different but independent scalar noises in the three component equations.
In the recent paper [27], we have shown the existence of a global attractor for the diffusive deterministic Hindmarsh-Rose equations and for the partly diffusive Hindmarsh-Rose equations in the space H. In this paper, it will be shown that there exists a random attractor in the space H for the random dynamical system generated by the global solutions of the stochastic evolutionary equation (1.10).

Random Hindmarsh-Rose Equations and Pullback Dissipativity
The mathematical treatment of the stochastic PDE such as in the form of (1.1)-(1.3) driven by the multiplicative noise will be facilitated by its conversion to a random PDE with coefficients and initial data being random variables instead. For this purpose, one can exploit the following properties of the Wiener process.
Proposition 2.1. Let the MDS (Q, F, P, {θ t } t∈R ) and the Wiener process W (t) be defined as above. Then the following statements hold.
(1) The Wiener process W (t) has the asymptotically sublinear growth property, (2) For any given positive constant λ, the stochastic process X(t) = e −λW (t) is a solution of the following stochastic differential equation in the Stratonovich sense, is locally Hölder continuous with exponents γ ∈ 0, 1 2 . It means that for any integer n, Proof. By the law of iterated logarithm [26], 2|t| log log |t| = 1, a.s.
Then (2.1) is valid. Next, from Itô's formula [26] we have On the other hand, the transformation formula [26] of the stochastic Itô integral and the Stratonovich integral reads

Hence (2.2) holds. Finally, (2.3) follows from the Kolmogorov Moment Criterion.
We now convert the stochastic PDE (1.1) -(1.3) to a system of random PDE by the exponential multiplication of Q(t, ω) = e −εω(t) : (2.4) According to the second statement in Proposition 2.1, the initial-boundary value problem (1.1)-(1.5) is equivalently converted to the following system of random PDEs: and an initial condition for ω ∈ Q, The equations (2.5)-(2.7) are pathwise nonautonomous random PDEs and (2.5)-(2.9) can be written as the initial value problem of the random evolutionary equation: Here we define the weak solution of the initial value problem (2.10) with the initial state G τ = Q(τ, ω)g 0 , In this section, we first prove the global existence of all the pullback weak solutions of the problem (2.5)-(2.9) and to explore the dissipativity of the generated random dynamical system.
Remark 2. We can certainly merge the above two lemmas into one which gives rise to the bounded estimate (2.27). Here we split the time interval [t 0 , 0] to [t 0 , −1] ∪ [−1, 0] in order to facilitate the argument in the proof of the pullback asymptotic compactness of the associated random dynamical system later in Section 3.

Hindmarsh-Rose Cocycle and Absorbing
Property. Now define a concept of stochastic semiflow, which is related to the concept of cocycle in the theory of random dynamical systems.
Here in the setting of the stochastic evolutionary equation (2.10) formulated from the stochastic Hindmarsh-Rose equations (1.1)-(1.5), we define S(t, τ, ω) : H → H for t ≥ τ ∈ R and ω ∈ Q by and then define a mapping Φ : which is equivalent to The following lemma shows that this mapping Φ is a cocycle on the Hilbert space H over the canonical metric dynamical system (Q, F, P, {θ t } t∈R ) specified in (1.11) and (1.12). Therefore, the following pullback identity is validated: for any t ≥ 0 and ω ∈ Q. We shall call this mapping Φ defined by (2.30) the Hindmarsh-Rose cocycle, which is a random dynamical system on the Hilbert space H. We shall call {Φ(t, θ −t ω, g 0 ) : t ≥ 0} a pullback quasi-trajectory with the initial state g 0 for the Hindmarsh-Rose cocycle.
Remark 3. Here the pullback quasi-trajectory {Φ(t, θ −t ω, g 0 ), t ≥ 0} is not a single trajectory but the set of all the points at time t = 0 of the bunch of trajectories started from the same initial state g 0 but at different pullback initial time −t. Moreover, the one-parameter operators where {g(ω) : ω ∈ Q} can be any H-valued random set on the probability space (Q, F, P ), turns out to be a semigroup of operators on the H-valued random sets.

The Existence of Random Attractor
In this section, we shall prove that this Hindmarsh-Rose cocycle is pullback asymptotically compact on H through the following two lemmas. Then the main result on the existence of a random attractor for the Hindmarsh-Rose cocycle is established.
Consequently, (3.16) is also proved for any random variable ρ(ω) as well, by the remark at the beginning of this proof. It completes the proof.
We complete this paper to present the main result on the existence of a random attractor for the Hindmarsh-Rose random dynamical system Φ in the space H. Theorem 3.3. For any positive parameters d 1 , d 2 , d 3 , a, b, α, β, q, r, J, ε and c ∈ R, there exists a random attractor A(ω) in the space H = L 2 (Ω, R 3 ) with respect to the universe D H for the Hindmarsh-Rose cocycle Φ over the metric dynamical system (Q, F, P, {θ t } t∈R ). Moreover, the random attractor A(ω) is a bounded random set in the space E.
Proof. In Lemma 2.6, we proved that there exists a pullback absorbing set B 0 (ω) in H for the Hindmarsh-Rose cocycle Φ. According to Definition 1.6, Lemma 3.2 and the compact imbedding E → H show that the Hindmarsh-Rose cocycle Φ is pullback asymptotically compact on H with respect to D H . Hence, by Theorem 1.8, there exists a random attractor in H for this random dynamical system Φ, which is given by (3.28) Since A(ω) is an invariant set, Lemma 3.2 implies that the random attractor A(ω) is also a bounded random set in E.