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Research article

Contextual fear response is modulated by M-type K+ channels and is associated with subtle structural changes of the axon initial segment in hippocampal GABAergic neurons

  • Background 

    In the fear memory network, the hippocampus modulates contextual aspects of fear learning while mutual connections between the amygdala and the medial prefrontal cortex are widely involved in fear extinction. G-protein-coupled receptors (GPCRs) are involved in the regulation of fear and anxiety, so the regulation of GPCRs in fear signaling pathways can modulate the mechanisms of fear memory acquisition, consolidation and extinction. Various studies suggested a role of M-type K+ channels in modulating fear expression and extinction, although conflicting data prevented drawing of clear conclusions. In the present work, we examined the impact of M-type K+ channel blockade or activation on contextual fear acquisition and extinction. In addition, regarding the pivotal role of the hippocampus in contextual fear conditioning (CFC) and the involvement of the axon initial segment (AIS) in neuronal plasticity, we investigated whether structural alterations of the AIS in hippocampal neurons occurred during contextual fear memory acquisition and short-time extinction in mice in a behaviorally relevant context.

    Results 

    When a single systemic injection of the M-channel blocker XE991 (2 mg/kg, IP) was carried out 15 minutes before the foot shock session, fear expression was significantly reduced. Expression of c-Fos was increased following CFC, mostly in GABAergic neurons at day 1 and day 2 post-fear training in CA1 and dentate gyrus hippocampal regions. A significantly longer AIS segment was observed in GABAergic neurons of the CA1 hippocampal region at day 2.

    Conclusions 

    Our results underscore the role of M-type K + channels in CFC and the importance of hippocampal GABAergic neurons in fear expression.

    Citation: Sara Arciniegas Ruiz, Eliav Tikochinsky, Vardit Rubovitch, Chaim G Pick, Bernard Attali. Contextual fear response is modulated by M-type K+ channels and is associated with subtle structural changes of the axon initial segment in hippocampal GABAergic neurons[J]. AIMS Neuroscience, 2023, 10(1): 33-51. doi: 10.3934/Neuroscience.2023003

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  • Background 

    In the fear memory network, the hippocampus modulates contextual aspects of fear learning while mutual connections between the amygdala and the medial prefrontal cortex are widely involved in fear extinction. G-protein-coupled receptors (GPCRs) are involved in the regulation of fear and anxiety, so the regulation of GPCRs in fear signaling pathways can modulate the mechanisms of fear memory acquisition, consolidation and extinction. Various studies suggested a role of M-type K+ channels in modulating fear expression and extinction, although conflicting data prevented drawing of clear conclusions. In the present work, we examined the impact of M-type K+ channel blockade or activation on contextual fear acquisition and extinction. In addition, regarding the pivotal role of the hippocampus in contextual fear conditioning (CFC) and the involvement of the axon initial segment (AIS) in neuronal plasticity, we investigated whether structural alterations of the AIS in hippocampal neurons occurred during contextual fear memory acquisition and short-time extinction in mice in a behaviorally relevant context.

    Results 

    When a single systemic injection of the M-channel blocker XE991 (2 mg/kg, IP) was carried out 15 minutes before the foot shock session, fear expression was significantly reduced. Expression of c-Fos was increased following CFC, mostly in GABAergic neurons at day 1 and day 2 post-fear training in CA1 and dentate gyrus hippocampal regions. A significantly longer AIS segment was observed in GABAergic neurons of the CA1 hippocampal region at day 2.

    Conclusions 

    Our results underscore the role of M-type K + channels in CFC and the importance of hippocampal GABAergic neurons in fear expression.


    Abbreviations

    AIS:

    Axon initial segment; 

    AnkG:

    Ankyrin G; 

    BLA:

    Basolateral amygdala; 

    CFC:

    Contextual fear conditioning; 

    CRP:

    Context Re-exposure Procedure; 

    CS:

    Conditioned stimulus; 

    DG:

    Dentate gyrus of hippocampus; 

    GPCRs:

    G-protein-coupled receptors; 

    IHC:

    Immunohistochemistry; 

    IL:

    Infralimbic prefrontal cortex; 

    IP:

    Intraperitoneal; 

    MAP2:

    Microtubule-associated protein 2; 

    mPFC:

    Medial prefrontal cortex; 

    RTG:

    Retigabine; 

    US:

    Unconditioned stimulus; 

    vGAT:

    Vesicular GABA transporter; 

    vGLUT:

    Vesicular glutamate transporter

    1. Introduction

    G. Caginalp proposed in [3] and [4] two phase-field system, namely,

    utΔu+f(u)=T, (1.1)
    TtΔT=ut, (1.2)

    called nonconserved system, and

    ut+Δ2uΔf(u)=ΔT, (1.3)
    TtΔT=ut, (1.4)

    called concerved system (in the sense that, when endowed with Neumann boundary conditions, the spacial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as T=˜TTE, where ˜T is the absolute temperature and TE is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F(s)=14(s21)2, hence the usual cubic nonlinear term f(s)=s3s). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., [3,4,5,8,9,10,12,13,14,15,16,18,19,21,22,23,25].

    Both systems are based on the (total Ginzburg-Landau) free energy

    ΨGL=Ω(12|u|2+F(u)uT12T2)dx, (1.5)

    where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R3, with boundary Γ), and the enthalpy

    H=u+T. (1.6)

    As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)

    uu=DΨGLDu, (1.7)

    for the nonconserved model, and

    uu=ΔDΨGLDu, (1.8)

    for the conserved one, where DDu denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation

    Ht=divq, (1.9)

    where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,

    q=T, (1.10)

    we obtain (1.2).

    In (1.5), the term |u|2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones; it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions [11].

    G. Caginalp and Esenturk recently proposed in [6] (see also [20]) higher-order phase-field models in order to account for anisotropic interfaces (see also [7] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these autors proposed the following modified (total) free energy

    ΨHOGL=Ω(12ki=1|β|=iaβ|Dβu|2+F(u)uT12T2)dx,kN, (1.11)

    where, for β=(k1,k2,k3)(N{0})3,

    |β|=k1+k2+k3

    and, for β(0,0,0),

    Dβ=|β|xk11xk22xk33

    (we agree that D(0,0,0)v=v).

    A. Miranville studied in [17] the corresponding nonconserved higher-order phase-field system.

    As far as the conserved case is concerned, the above generalized free energy yields, procceding as above, the following evolution equation for the order parameter u:

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (1.12)

    In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form

    ut+Δ3i=1ai2ux2iΔf(u)=Δ(αtΔαt)

    and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form

    utΔ3i,j=1aij4ux2ix2j+Δ3i=1bi2ux2iΔf(u)=Δ(αtΔαt).

    L. Cherfils A. Miranville and S. Peng have studied in [8] the corresponding higher-order isotropic equation (without the coupling with the temperature), namely, the equation

    utΔP(Δ)uΔf(u)=0,

    where

    P(s)=ki=1aisi,ak>0,k1,

    endowed with the Dirichlet/Navier boundary conditions

    u=Δu=...=Δku=0onΓ.

    Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation

    2αt2Δ2αt2ΔαtΔα=ut. (1.13)

    In particular, we obtain the existence and uniqueness of solutions.


    2. Setting of the problem

    We consider the following initial and boundary value problem, for kN, k2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):

    utΔki=1(1)i|β|=iaβD2βuΔf(u)=Δ(αtΔαt), (2.1)
    2αt2Δ2αt2ΔαtΔα=ut, (2.2)
    Dβu=α=0onΓ,|β|k, (2.3)
    u|t=0=u0,α|t=0=α0,αt|t=0=α1. (2.4)

    We assume that

    aβ>0,|β|=k, (2.5)

    and we introduce the elliptic operator Ak defined by

    Akv,wHk(Ω),Hk0(Ω)=|β|=kaβ((Dβv,Dβw)), (2.6)

    where Hk(Ω) is the topological dual of Hk0(Ω). Furthermore, ((., .)) denotes the usual L2-scalar product, with associated norm .. More generally, we denote by .X the norm on the Banach space X; we also set .1=(Δ)12., where (Δ)1 denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that

    (v,w)Hk0(Ω)2|β|=kaβ((Dβv,Dβw))

    is bilinear, symmetric, continuous and coercive, so that

    Ak:Hk0(Ω)Hk(Ω)

    is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(Ak)=H2k(Ω)Hk0(Ω),

    where, for vD(Ak),

    Akv=(1)k|β|=kaβD2βv.

    We further note that D(A12k)=Hk0(Ω) and, for (v,w)D(A12k)2,

    ((A12kv,A12kw))=|β|=kaβ((Dβv,Dβw)).

    We finally note that (see, e.g., [24]) Ak. (resp., A12k.) is equivalent to the usual H2k-norm (resp., Hk-norm) on D(Ak) (resp., D(A12k)).

    Similarly, we can define the linear operator ¯Ak=ΔAk

    ˉAk:Hk+10(Ω)Hk1(Ω)

    which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(ˉAk)=H2k+2(Ω)Hk+10(Ω),

    where, for vD(ˉAk),

    ˉAkv=(1)k+1Δ|β|=kaβD2βv.

    Furthermore, D(ˉA12k)=Hk+10(Ω) and, for (v,w)D(ˉA12k),

    ((ˉA12kv,ˉA12kw))=|β|=kaβ((Dβv,Dβw)).

    Besides ˉAk. (resp., ˉA12k.) is equivalent to the usual H2k+2-norm (resp., Hk+1-norm) on D(ˉAk) (resp., D(ˉA12k)).

    We finally consider the operator ˜Ak=(Δ)1Ak, where

    ˜Ak:Hk10(Ω)Hk+1(Ω);

    note that, as Δ and Ak commute, then the same holds for (Δ)1 and Ak, so that ˜Ak=Ak(Δ)1.

    We have the (see [17])

    Lemme 2.1. The operator ˜Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

    D(˜Ak)=H2k2(Ω)Hk10(Ω),

    where, for vD(˜Ak)

    ˜Akv=(1)k|β|=kaβD2β(Δ)1v.

    Furthermore, D(˜A12k)=Hk10(Ω) and, for (v,w)D(˜A12k),

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Besides ˜Ak. (resp., ˜A12k.) is equivalent to the usual H2k2-norm (resp., Hk1-norm) on D(˜Ak) (resp., D(˜A12k)).

    Proof. We first note that ˜Ak clearly is linear and unbounded. Then, since (Δ)1 and Ak commute, it easily follows that ˜Ak is selfadjoint.

    Next, the domain of ˜Ak is defined by

    D(˜Ak)={vHk10(Ω),˜AkvL2(Ω)}.

    Noting that ˜Akv=f,fL2(Ω),vD(˜Ak), is equivalent to Akv=Δf, where ΔfH2(Ω), it follows from the elliptic regularity results of [1] and [2] that vH2k2(Ω), so that D(˜Ak)=H2k2(Ω)Hk10(Ω).

    Noting then that ˜A1k maps L2(Ω) onto H2k2(Ω) and recalling that k2, we deduce that ˜Ak has compact inverse.

    We now note that, considering the spectral properties of Δ and Ak (see, e.g., [24]) and recalling that these two operators commute, Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, s1,s2R, (Δ)s1 and As2k commute.

    Having this, we see that ˜A12k=(Δ)12A12k, so that D(˜A12k)=Hk10(Ω), and for (v,w)D(˜A12k)2,

    ((˜A12kv,˜A12kw))=|β|=kaβ((Dβ(Δ)12v,Dβ(Δ)12w)).

    Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm ˜A12k. is equivalent to the norm (Δ)12.Hk(Ω) and, thus, to the norm (Δ)k12..

    Having this, we rewrite (2.1) as

    utΔAkuΔBkuΔf(u)=Δ(αtΔαt), (2.7)

    where

    Bkv=k1i=1(1)i|β|=iaβD2βv.

    As far as the nonlinear term f is concerned, we assume that

    fC2(R),f(0)=0, (2.8)
    fc0,c00, (2.9)
    f(s)sc1F(s)c2c3,c1>0,c2,c30,sR, (2.10)
    F(s)c4s4c5,c4>0,c50,sR, (2.11)

    where F(s)=s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3s satisfies these assumptions.

    Throughout the paper, the same letters c, c' and c" denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.


    3. A priori estimates

    We multiply (2.7) by (Δ)1ut and (2.2) by αtΔαt, sum the two resulting equalities and integrate over Ω and by parts. This gives

    ddt(A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2)+2ut21+2αt2+2Δαt2=0 (3.1)

    (note indeed that αt2+2αt2+Δαt2=αtΔαt2), where

    B12k[u]=k1i=1|β|=iaβDβu2 (3.2)

    (note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality

    B12k[u]=k1i=1|β|=iaβDβu2 (3.3)
    (Δ)i2vc(i)(Δ)m2vimv1im,

    there holds

    vHm(Ω),i{1,...,m1},mN,m2, (3.4)

    This yields, employing (2.11),

    |B12k[u]|12A12ku2+cu2.

    whence

    A12ku2+B12k[u]+2ΩF(u)dx12A12ku2+ΩF(u)dx+cu4L4(Ω)cu2c", (3.5)

    nothing that, owing to Young's inequality,

    A12ku2+B12k[u]+2ΩF(u)dxc(u2Hk(Ω)+ΩF(u)dx)c,c>0, (3.6)

    We then multiply (2.7) by (Δ)1u and have, owing to (2.10) and the interpolation inequality (3.3),

    u2ϵu4L4(Ω)+c(ϵ),ϵ>0.

    hence, proceeding as above and employing, in particular, (2.11)

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(u2+αt2+Δαt2)+c", (3.7)

    Summing (3.1) and δ1 times (3.7), where δ1>0 is small enough, we obtain a differential inegality of the form

    ddtu21+c(u2Hk(Ω)+ΩF(u)dx)c(αt2+Δαt2)+c,c>0. (3.8)

    where

    ddtE1+c(u2Hk(Ω)+ΩF(u)dx+ut21+αt2H2(Ω))c,c>0,

    satisfies, owing to (3.5)

    E1=A12ku2+B12k[u]+2ΩF(u)dx+α2+Δα2+αtΔαt2+δ1u21 (3.9)

    Multiplying (2.2) by Δα, we then obtain

    E1c(u2Hk(Ω)+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0.

    which yields, employing the interpolation inequality

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+Δα2ut2+αt2+Δαt2, (3.10)

    the differential inequality, with 0<ϵ<<1 is small enough

    v2cv1vH1(Ω),vH10(Ω), (3.11)

    We now differentiate (2.7) with respect to time to find, owing to (2.2),

    ddt(Δα22((αt,Δα))+2((Δαt,Δα)))+cα2H2(Ω)c(ut21+ϵut2H1(Ω)+αt2H2(Ω)),c>0. (3.12)

    together with the boundary condition

    tutΔAkutΔBkutΔ(f(u)ut)=Δ(Δαt+Δαut), (3.13)

    We multiply (3.11) by (Δ)1ut and obtain, owing to (2.9) and the interpolation inequality (3.3),

    Dβut=0onΓ,|β|k.

    hence, owing to (3.10), the differential inequality

    ddtut21+cut2Hk(Ω)c(ut2+Δα2+Δαt2),c>0, (3.14)

    Summing finally (3.8), δ2 times (3.11) and δ3 times (3.14), where δ2,δ3>0 are small enough, we find a differential inequality of the form

    ddtut21+cut2Hk(Ω)c(ut21+α2H2(Ω)+αt2H2(Ω)),c>0. (3.15)

    where

    dE2dt+c(E2+ut2Hk(Ω))c,c>0,

    Owing to the continuous embedding H2k+1(Ω)C(ˉΩ), we deduce that

    E2=E1+δ2(Δα22((αt,Δα))+2((Δαt,Δα)))+δ3ut21.

    and since

    |ΩF(u0)dx|Q(u0H2k+1(Ω))

    we see that (Δ)12ut(0)L2(Ω) and

    (Δ)12ut(0)=(Δ)12Aku0(Δ)12Bku0(Δ)12f(u0)+(Δ)12(α1Δα1), (3.16)

    Furthermore E2 satisfies

    ut(0)1Q(u0H2k+1(Ω),α1H3(Ω)). (3.17)

    It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that

    E2c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.18)

    and

    u(t)2Hk(Ω)+ut(t)21+α(t)2H2(Ω)+αt(t)2H2(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c>0,t0, (3.19)

    r>0 given.

    Multiplying next (2.7) by ˜Aku, we find, owing to the interpolation inequality (3.3),

    t+rtut2Hk(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    hence, since f and F are continuous and owing to (3.18),

    ddt˜A12ku2+cu2H2k(Ω)c(u2+f(u)2+αt2+Δαt2),c>0, (3.20)

    Summing (3.15) and (3.22), we have a differential inequality of the form

    ddt˜A12ku2+cu2H2k(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0. (3.21)

    where

    dE3dt+c(E3+u2H2k(Ω)+ut2Hk(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c",c,c>0,t0,

    satisfies

    E3=E2+˜A12ku2 (3.22)

    In particular, it follows from (3.21)-(3.22) that

    E3c(u2Hk(Ω)+ut21+ΩF(u)dx+α2H2(Ω)+αt2H2(Ω))c,c>0. (3.23)

    r>0 given.

    We now multiply (2.7) by u and obtain, employing (2.9) and the interpolation inequality (3.3)

    t+rtu2H2k(Ω)dsectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c(r),c>0,t0,

    whence, proceeding as above,

    ddtu2+cu2Hk+1(Ω)c(u2H1(Ω)+αt2+Δαt2),c>0, (3.24)

    We also multiply (2.7) by ut and find

    ddtu2+cu2Hk+1(Ω)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0.

    where

    ddt(ˉA12ku2+ˉB12k[u])+cut2cΔf(u)22((Δut,αtΔαt)),

    Since f is of class C2, it follows from the continuous embedding H2(Ω)C(ˉΩ) that

    ˉB12k[u]=k1i=1|β|=iaβDβu2.

    hence, owing to (3.18),

    Δf(u)2Q(uH2(Ω)), (3.25)

    Multiply next (2.2) by Δ(αtΔαt), we have

    ddt(ˉA12ku2+ˉB12k[u])+cut2ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))2((Δut,αtΔαt))+c,c,c>0. (3.26)

    (note indeed that αt2+2Δαt2+Δαt2=αtΔαt2).

    Summing (3.25) and (3.26), we obtain

    ddt(Δα2+Δα2+αtΔαt2)+c(Δαt2+Δαt2)2((Δut,αtΔαt)),c>0 (3.27)

    Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form

    ddt(ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2)+c(ut2+Δαt2+Δαt2)ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0. (3.28)

    where

    dE4dt+c(E3+u2Hk+1(Ω)+u2H2k(Ω)+ut2+ut2Hk(Ω)+αt2H3(Ω))ectQ(u0H2k+1(Ω),α0H2(Ω),α1H3(Ω))+c,c,c>0,t0

    satisfies, owing to (2.11) and the interpolation inegality (3.3)

    E4=E3+u2+ˉA12ku2+ˉB12k[u]+Δα2+Δα2+αtΔαt2 (3.29)

    In particular, it follows from (3.28)-(3.29) that

    E4c(u2Hk+1(Ω)+ut21+ΩF(u)dx+α2H3(Ω)+αt2H3(Ω))c,c>0. (3.30)

    and

    u(t)Hk+1(Ω)+α(t)H3(Ω)+αt(t)H3(Ω)ectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0,t0, (3.31)

    r given.

    We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,

    t+rt(ut2+αt2H3(Ω))dsectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c(r),c>0,t0, (3.32)

    Multiplying (3.32) by Aku, we obtain, owing to the interpolation inequality (3.3),

    Aku=(Δ)1utBkuf(u)+αtΔαt,Dβu=0onΓ,|β|k1.

    hence, since f is continuous and owing to (3.18)

    Aku2c(u2+f(u)2+ut21+αt2+Δαt2), (3.33)

    4. Existence and uniqueness of solutions

    We first have the following theorem.

    Theorem 4.1. (i) We assume that (u0,α0,α1)Hk0(Ω)×(H2(Ω)H10(Ω))×(H2(Ω)H10(Ω)), with ΩF(u0)dx<+. Then, (2.1)(2.4) possesses at last one solution (u,α,αt) such that, T>0, u(0)=u0, α(0)=α0, αt(0)=α1,

    u(t)2H2k(Ω)cectQ(u0H2k+1(Ω),α0H3(Ω),α1H3(Ω))+c,c>0t0.
    uL(R+;Hk0(Ω))L2(0,T;H2k(Ω)Hk0(Ω)),
    utL(R+;H1(Ω))L2(0,T;Hk0(Ω)),

    and

    α,αtL(R+;H2(Ω)H10(Ω))
    ddt((Δ)1u,v))+ki=1|β|=iai((Dβu,Dβv))+((f(u),v))=ddt(((u,v))+((u,v))),vCc(Ω),

    in the sense of distributions.

    (ii) If we futher assume that (u0,α0,α1)(Hk+1(Ω)Hk0(Ω))×(H3(Ω)H10(Ω))×(H3(Ω)H10(Ω)), then, T>0,

    ddt(((αt,w))+((αt,w))+((α,w)))+((α,w))=ddt((u,w)),wCc(Ω),
    uL(R+;Hk+1(Ω)Hk0(Ω))L2(R+;Hk+1(Ω)Hk0(Ω))
    utL2(R+;L2(Ω)),

    and

    αL(R+;H3(Ω)H10(Ω))

    The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.

    We then have the following theorem.

    Theorem 4.2. The system (1.1)-(1.4) possesses a unique solution with the above regularity.

    proof. Let (u(1),α(1),α(1)t) and (u(2),α(2),α(2)t) be two solutions to (2.1)-(2.3) with initial data (u(1)0,α(1)0,α(1)1) and (u(2)0,α(2)0,α(2)1), respectively. We set

    αtL(R+;H3(Ω)H10(Ω))L2(0,T;H3(Ω)H10(Ω))

    and

    (u,α,αt)=(u(1),α(1),α(1)t)(u(2),α(2),α(2)t)

    Then, (u,α) satisfies

    (u0,α0,α1)=(u(1)0,α(1)0,α(1)1)(u(2)0,α(2)0,α(2)1). (4.1)
    utΔAkuΔBkuΔ(f(u(1))f(u(2)))=Δ(αtΔαt), (4.2)
    2αt2Δ2αt2ΔαtΔα=ut, (4.3)
    Dβu=α=0 on Γ,|β|k, (4.4)

    Multiplying (4.1) by (Δ)1u and integrating over Ω, we obtain

    u|t=0=u0,α|t=0=α0,αt|t=0=α1.

    We note that

    ddtu21+cu2Hk(Ω)c(u2+αtΔαt2)2((f(u(1))f(u(2),u)).

    with l defined as

    f(u(1))f(u(2))=l(t)u,

    Owing to (2.9), we have

    l(t)=10f(su(1)(t)+(1s)u(2)(t))ds.

    and we obtain owing to the intepolation inequalities (3.3) and (3.10),

    2((f(u(1))f(u(2),u))2c0u2                                      cu2 (4.5)

    Next, multiplying (4.2) by (Δ)1(u+αtΔαt), we find

    ddtu21+cu2Hk(Ω)c(u21+αtΔαt2),c>0. (4.6)

    Summing then δ4 times (4.5) and (4.6), where δ4>0 is small enough, we have, employing once more the interpolation inequality (3.10), a differential inequality of the form

    ddt(α2+α2+u+αtΔαt21)+c(αt2+αt2H1(Ω))c(u2+α2). (4.7)

    where

    dE5dtcE5,

    satisfies

    E5=δ4u21+α2+α2+u+αtΔαt21 (4.8)

    It follows from (4.7)-(4.8) and Gronwall's lemma that

    E5c(u21+α2H1(Ω)+αtΔαt2),c>0. (4.9)

    hence the uniquess, as well as the continuous dependence with respect to the initial data in H1×H1×H1-norm.


    Conflict of Interest

    All authors declare no conflicts of interest in this paper.



    Acknowledgments



    We would like to thank the members of the Animal Facility department for taking care of the animal maintenance and particularly to Dr. Lior Bikoveski, the core facility manager for his technical support in the behavioral equipment during our experiments. Bernard Attali holds the Andy Libach Professorial Chair in clinical pharmacology and toxicology.

    Conflict of interests



    The authors declare that they have no competing interests.

    Funding



    This work was supported by a grant from the Israel Science Foundation (ISF 1365/17) to BA.

    Authors' contributions



    Study concept and design: S.A.R and B.A; Performance of experiments: V.R, S.A.R. and E.T; Analysis and interpretation of data: C.G.P, S.A.R and B.A; Drafting of the manuscript: S.A.R and B.A.

    Availability of data and materials



    All data needed to evaluate the conclusions in the paper are present in the paper. Additional data related to this paper may be requested from the authors.

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