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.
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.
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
[1] | Miao Fu, Yuqin Zhang . Results on monochromatic vertex disconnection of graphs. AIMS Mathematics, 2023, 8(6): 13219-13240. doi: 10.3934/math.2023668 |
[2] | Fan Wu, Xinhui An, Baoyindureng Wu . Sombor indices of cacti. AIMS Mathematics, 2023, 8(1): 1550-1565. doi: 10.3934/math.2023078 |
[3] | Yinkui Li, Jiaqing Wu, Xiaoxiao Qin, Liqun Wei . Characterization of Q graph by the burning number. AIMS Mathematics, 2024, 9(2): 4281-4293. doi: 10.3934/math.2024211 |
[4] | A. El-Mesady, Y. S. Hamed, M. S. Mohamed, H. Shabana . Partially balanced network designs and graph codes generation. AIMS Mathematics, 2022, 7(2): 2393-2412. doi: 10.3934/math.2022135 |
[5] | Yixin Zhang, Yanbo Zhang, Hexuan Zhi . A proof of a conjecture on matching-path connected size Ramsey number. AIMS Mathematics, 2023, 8(4): 8027-8033. doi: 10.3934/math.2023406 |
[6] | Haicheng Ma, Xiaojie You, Shuli Li . The singularity of two kinds of tricyclic graphs. AIMS Mathematics, 2023, 8(4): 8949-8963. doi: 10.3934/math.2023448 |
[7] | Yuan Zhang, Haiying Wang . Some new results on sum index and difference index. AIMS Mathematics, 2023, 8(11): 26444-26458. doi: 10.3934/math.20231350 |
[8] | Syafrizal Sy, Rinovia Simanjuntak, Tamaro Nadeak, Kiki Ariyanti Sugeng, Tulus Tulus . Distance antimagic labeling of circulant graphs. AIMS Mathematics, 2024, 9(8): 21177-21188. doi: 10.3934/math.20241028 |
[9] | Muhammad Kamran Jamil, Muhammad Imran, Aisha Javed, Roslan Hasni . On the first general Zagreb eccentricity index. AIMS Mathematics, 2021, 6(1): 532-542. doi: 10.3934/math.2021032 |
[10] | Sakander Hayat, Bagus Imanda, Asad Khan, Mohammed J. F. Alenazi . Three infinite families of Hamilton-connected convex polytopes and their detour index. AIMS Mathematics, 2025, 10(5): 12343-12387. doi: 10.3934/math.2025559 |
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.
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.
Our results underscore the role of M-type K + channels in CFC and the importance of hippocampal GABAergic neurons in fear expression.
Axon initial segment;
Ankyrin G;
Basolateral amygdala;
Contextual fear conditioning;
Context Re-exposure Procedure;
Conditioned stimulus;
Dentate gyrus of hippocampus;
G-protein-coupled receptors;
Immunohistochemistry;
Infralimbic prefrontal cortex;
Intraperitoneal;
Microtubule-associated protein 2;
Medial prefrontal cortex;
Retigabine;
Unconditioned stimulus;
Vesicular GABA transporter;
Vesicular glutamate transporter
G. Caginalp proposed in [3] and [4] two phase-field system, namely,
∂u∂t−Δu+f(u)=T, | (1.1) |
∂T∂t−ΔT=−∂u∂t, | (1.2) |
called nonconserved system, and
∂u∂t+Δ2u−Δf(u)=−ΔT, | (1.3) |
∂T∂t−ΔT=−∂u∂t, | (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=˜T−TE, 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(s2−1)2, hence the usual cubic nonlinear term f(s)=s3−s). 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)−uT−12T2)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)
∂u∂u=−DΨGLDu, | (1.7) |
for the nonconserved model, and
∂u∂u=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
∂H∂t=−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(12∑ki=1∑|β|=iaβ|Dβu|2+F(u)−uT−12T2)dx,k∈N, | (1.11) |
where, for β=(k1,k2,k3)∈(N∪{0})3,
|β|=k1+k2+k3 |
and, for β≠(0,0,0),
Dβ=∂|β|∂xk11∂xk22∂xk33 |
(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:
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (1.12) |
In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form
∂u∂t+Δ∑3i=1ai∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂t) |
and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form
∂u∂t−Δ∑3i,j=1aij∂4u∂x2i∂x2j+Δ∑3i=1bi∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂t). |
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
∂u∂t−ΔP(−Δ)u−Δf(u)=0, |
where
P(s)=∑ki=1aisi,ak>0,k⩾1, |
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
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t. | (1.13) |
In particular, we obtain the existence and uniqueness of solutions.
We consider the following initial and boundary value problem, for k∈N, k⩾2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.1) |
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t, | (2.2) |
Dβu=α=0onΓ,|β|⩽k, | (2.3) |
u|t=0=u0,α|t=0=α0,∂α∂t|t=0=α1. | (2.4) |
We assume that
aβ>0,|β|=k, | (2.5) |
and we introduce the elliptic operator Ak defined by
⟨Akv,w⟩H−k(Ω),Hk0(Ω)=∑|β|=kaβ((Dβv,Dβw)), | (2.6) |
where H−k(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)→H−k(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 v∈D(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)→H−k−1(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 v∈D(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:Hk−10(Ω)→H−k+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)=H2k−2(Ω)∩Hk−10(Ω), |
where, for v∈D(˜Ak)
˜Akv=(−1)k∑|β|=kaβD2β(−Δ)−1v. |
Furthermore, D(˜A12k)=Hk−10(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 H2k−2-norm (resp., Hk−1-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)={v∈Hk−10(Ω),˜Akv∈L2(Ω)}. |
Noting that ˜Akv=f,f∈L2(Ω),v∈D(˜Ak), is equivalent to Akv=−Δf, where −Δf∈H2(Ω)′, it follows from the elliptic regularity results of [1] and [2] that v∈H2k−2(Ω), so that D(˜Ak)=H2k−2(Ω)∩Hk−10(Ω).
Noting then that ˜A−1k maps L2(Ω) onto H2k−2(Ω) and recalling that k⩾2, 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,s2∈R, (−Δ)s1 and As2k commute.
Having this, we see that ˜A12k=(−Δ)−12A12k, so that D(˜A12k)=Hk−10(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)k−12.‖.
Having this, we rewrite (2.1) as
∂u∂t−ΔAku−ΔBku−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.7) |
where
Bkv=∑k−1i=1(−1)i∑|β|=iaβD2βv. |
As far as the nonlinear term f is concerned, we assume that
f∈C2(R),f(0)=0, | (2.8) |
f′⩾−c0,c0⩾0, | (2.9) |
f(s)s⩾c1F(s)−c2⩾−c3,c1>0,c2,c3⩾0,s∈R, | (2.10) |
F(s)⩾c4s4−c5,c4>0,c5⩾0,s∈R, | (2.11) |
where F(s)=∫s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3−s 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.
We multiply (2.7) by (−Δ)−1∂u∂t and (2.2) by ∂α∂t−Δ∂α∂t, sum the two resulting equalities and integrate over Ω and by parts. This gives
ddt(‖A12ku‖2+B12k[u]+2∫ΩF(u)dx+‖∇α‖2+‖Δα‖2+‖∂α∂t−Δ∂α∂t‖2)+2‖∂u∂t‖2−1+2‖∇∂α∂t‖2+2‖Δ∂α∂t‖2=0 | (3.1) |
(note indeed that ‖∂α∂t‖2+2‖∇∂α∂t‖2+‖Δ∂α∂t‖2=‖∂α∂t−Δ∂α∂t‖2), where
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.2) |
(note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.3) |
‖(−Δ)i2v‖⩽c(i)‖(−Δ)m2v‖im‖v‖1−im, |
there holds
v∈Hm(Ω),i∈{1,...,m−1},m∈N,m⩾2, | (3.4) |
This yields, employing (2.11),
|B12k[u]|⩽12‖A12ku‖2+c‖u‖2. |
whence
‖A12ku‖2+B12k[u]+2∫ΩF(u)dx⩾12‖A12ku‖2+∫ΩF(u)dx+c‖u‖4L4(Ω)−c′‖u‖2−c", | (3.5) |
nothing that, owing to Young's inequality,
‖A12ku‖2+B12k[u]+2∫ΩF(u)dx⩾c(‖u‖2Hk(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),
‖u‖2⩽ϵ‖u‖4L4(Ω)+c(ϵ),∀ϵ>0. |
hence, proceeding as above and employing, in particular, (2.11)
ddt‖u‖2−1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx)⩽c′(‖u‖2+‖∂α∂t‖2+‖Δ∂α∂t‖2)+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
ddt‖u‖2−1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx)⩽c′(‖∂α∂t‖2+‖Δ∂α∂t‖2)+c″,c>0. | (3.8) |
where
ddtE1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx+‖∂u∂t‖2−1+‖∂α∂t‖2H2(Ω))⩽c′,c>0, |
satisfies, owing to (3.5)
E1=‖A12ku‖2+B12k[u]+2∫ΩF(u)dx+‖∇α‖2+‖Δα‖2+‖∂α∂t−Δ∂α∂t‖2+δ1‖u‖2−1 | (3.9) |
Multiplying (2.2) by −Δα, we then obtain
E1⩾c(‖u‖2Hk(Ω)+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω))−c′,c>0. |
which yields, employing the interpolation inequality
ddt(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+‖Δα‖2⩽‖∂u∂t‖2+‖∇∂α∂t‖2+‖Δ∂α∂t‖2, | (3.10) |
the differential inequality, with 0<ϵ<<1 is small enough
‖v‖2⩽c‖v‖−1‖v‖H1(Ω),v∈H10(Ω), | (3.11) |
We now differentiate (2.7) with respect to time to find, owing to (2.2),
ddt(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+c‖α‖2H2(Ω)≤c′(‖∂u∂t‖2−1+ϵ‖∂u∂t‖2H1(Ω)+‖∂α∂t‖2H2(Ω)),c>0. | (3.12) |
together with the boundary condition
∂∂t∂u∂t−ΔAk∂u∂t−ΔBk∂u∂t−Δ(f′(u)∂u∂t)=−Δ(Δ∂α∂t+Δα−∂u∂t), | (3.13) |
We multiply (3.11) by (−Δ)−1∂u∂t and obtain, owing to (2.9) and the interpolation inequality (3.3),
Dβ∂u∂t=0onΓ,|β|⩽k. |
hence, owing to (3.10), the differential inequality
ddt‖∂u∂t‖2−1+c‖∂u∂t‖2Hk(Ω)⩽c′(‖∂u∂t‖2+‖Δα‖2+‖Δ∂α∂t‖2),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
ddt‖∂u∂t‖2−1+c‖∂u∂t‖2Hk(Ω)⩽c′(‖∂u∂t‖2−1+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω)),c>0. | (3.15) |
where
dE2dt+c(E2+‖∂u∂t‖2Hk(Ω))⩽c′,c>0, |
Owing to the continuous embedding H2k+1(Ω)⊂C(ˉΩ), we deduce that
E2=E1+δ2(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+δ3‖∂u∂t‖2−1. |
and since
|∫ΩF(u0)dx|⩽Q(‖u0‖H2k+1(Ω)) |
we see that (−Δ)−12∂u∂t(0)∈L2(Ω) and
(−Δ)−12∂u∂t(0)=−(−Δ)12Aku0−(−Δ)12Bku0−(−Δ)12f(u0)+(−Δ)12(α1−Δα1), | (3.16) |
Furthermore E2 satisfies
‖∂u∂t(0)‖−1⩽Q(‖u0‖H2k+1(Ω),‖α1‖H3(Ω)). | (3.17) |
It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that
E2⩾c(‖u‖2Hk(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω))−c′,c>0. | (3.18) |
and
‖u(t)‖2Hk(Ω)+‖∂u∂t(t)‖2−1+‖α(t)‖2H2(Ω)+‖∂α∂t(t)‖2H2(Ω)≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(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+rt‖∂u∂t‖2Hk(Ω)ds≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c′(r),c>0,t≥0, |
hence, since f and F are continuous and owing to (3.18),
ddt‖˜A12ku‖2+c‖u‖2H2k(Ω)⩽c′(‖u‖2+‖f(u)‖2+‖∂α∂t‖2+‖Δ∂α∂t‖2),c>0, | (3.20) |
Summing (3.15) and (3.22), we have a differential inequality of the form
ddt‖˜A12ku‖2+c‖u‖2H2k(Ω)≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c",c,c′>0,t≥0. | (3.21) |
where
dE3dt+c(E3+‖u‖2H2k(Ω)+‖∂u∂t‖2Hk(Ω))≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c",c,c′>0,t≥0, |
satisfies
E3=E2+‖˜A12ku‖2 | (3.22) |
In particular, it follows from (3.21)-(3.22) that
E3⩾c(‖u‖2Hk(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(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+rt‖u‖2H2k(Ω)ds⩽e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c′(r),c>0,t⩾0, |
whence, proceeding as above,
ddt‖u‖2+c‖u‖2Hk+1(Ω)⩽c′(‖u‖2H1(Ω)+‖∂α∂t‖2+‖Δ∂α∂t‖2),c>0, | (3.24) |
We also multiply (2.7) by ∂u∂t and find
ddt‖u‖2+c‖u‖2Hk+1(Ω)⩽e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0. |
where
ddt(‖ˉA12ku‖2+ˉB12k[u])+c‖∂u∂t‖2⩽c′‖Δf(u)‖2−2((Δ∂u∂t,∂α∂t−Δ∂α∂t)), |
Since f is of class C2, it follows from the continuous embedding H2(Ω)⊂C(ˉΩ) that
ˉB12k[u]=∑k−1i=1∑|β|=iaβ‖∇Dβu‖2. |
hence, owing to (3.18),
‖Δf(u)‖2⩽Q(‖u‖H2(Ω)), | (3.25) |
Multiply next (2.2) by −Δ(∂α∂t−Δ∂α∂t), we have
ddt(‖ˉA12ku‖2+ˉB12k[u])+c‖∂u∂t‖2≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))−2((Δ∂u∂t,∂α∂t−Δ∂α∂t))+c″,c,c′>0. | (3.26) |
(note indeed that ‖∇∂α∂t‖2+2‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2=‖∇∂α∂t−∇Δ∂α∂t‖2).
Summing (3.25) and (3.26), we obtain
ddt(‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2)+c(‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2)≤2((Δ∂u∂t,∂α∂t−Δ∂α∂t)),c>0 | (3.27) |
Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form
ddt(‖ˉA12ku‖2+ˉB12k[u]+‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2)+c(‖∂u∂t‖2+‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2)≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0. | (3.28) |
where
dE4dt+c(E3+‖u‖2Hk+1(Ω)+‖u‖2H2k(Ω)+‖∂u∂t‖2+‖∂u∂t‖2Hk(Ω)+‖∂α∂t‖2H3(Ω))≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0,t≥0 |
satisfies, owing to (2.11) and the interpolation inegality (3.3)
E4=E3+‖u‖2+‖ˉA12ku‖2+ˉB12k[u]+‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2 | (3.29) |
In particular, it follows from (3.28)-(3.29) that
E4⩾c(‖u‖2Hk+1(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H3(Ω)+‖∂α∂t‖2H3(Ω))−c′,c>0. | (3.30) |
and
‖u(t)‖Hk+1(Ω)+‖α(t)‖H3(Ω)+‖∂α∂t(t)‖H3(Ω)≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c′,c>0,t≥0, | (3.31) |
r given.
We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,
∫t+rt(‖∂u∂t‖2+‖∂α∂t‖2H3(Ω))ds≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c′(r),c>0,t≥0, | (3.32) |
Multiplying (3.32) by Aku, we obtain, owing to the interpolation inequality (3.3),
Aku=−(−Δ)−1∂u∂t−Bku−f(u)+∂α∂t−Δ∂α∂t,Dβu=0onΓ,|β|⩽k−1. |
hence, since f is continuous and owing to (3.18)
‖Aku‖2⩽c(‖u‖2+‖f(u)‖2+‖∂u∂t‖2−1+‖∂α∂t‖2+‖Δ∂α∂t‖2), | (3.33) |
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,∂α∂t) such that, ∀T>0, u(0)=u0, α(0)=α0, ∂α∂t(0)=α1,
‖u(t)‖2H2k(Ω)⩽ce−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c″,c′>0t⩾0. |
u∈L∞(R+;Hk0(Ω))∩L2(0,T;H2k(Ω)∩Hk0(Ω)), |
∂u∂t∈L∞(R+;H−1(Ω))∩L2(0,T;Hk0(Ω)), |
and
α,∂α∂t∈L∞(R+;H2(Ω)∩H10(Ω)) |
ddt((−Δ)−1u,v))+∑ki=1∑|β|=iai((Dβu,Dβv))+((f(u),v))=ddt(((u,v))+((∇u,∇v))),∀v∈C∞c(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(((∂α∂t,w))+((∇∂α∂t,∇w))+((∇α,∇w)))+((∇α,∇w))=−ddt((u,w)),∀w∈C∞c(Ω), |
u∈L∞(R+;Hk+1(Ω)∩Hk0(Ω))∩L2(R+;Hk+1(Ω)∩Hk0(Ω)) |
∂u∂t∈L2(R+;L2(Ω)), |
and
α∈L∞(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
∂α∂t∈L∞(R+;H3(Ω)∩H10(Ω))∩L2(0,T;H3(Ω)∩H10(Ω)) |
and
(u,α,∂α∂t)=(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) |
∂u∂t−ΔAku−ΔBku−Δ(f(u(1))−f(u(2)))=−Δ(∂α∂t−Δ∂α∂t), | (4.2) |
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t, | (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,∂α∂t|t=0=α1. |
We note that
ddt‖u‖2−1+c‖u‖2Hk(Ω)⩽c′(‖u‖2+‖∂α∂t−Δ∂α∂t‖2)−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)+(1−s)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))≤2c0‖u‖2 ≤c‖u‖2 | (4.5) |
Next, multiplying (4.2) by (−Δ)−1(u+∂α∂t−Δ∂α∂t), we find
ddt‖u‖2−1+c‖u‖2Hk(Ω)⩽c′(‖u‖2−1+‖∂α∂t−Δ∂α∂t‖2),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(‖α‖2+‖∇α‖2+‖u+∂α∂t−Δ∂α∂t‖2−1)+c(‖∂α∂t‖2+‖∂α∂t‖2H1(Ω))≤c′(‖u‖2+‖α‖2). | (4.7) |
where
dE5dt⩽cE5, |
satisfies
E5=δ4‖u‖2−1+‖α‖2+‖∇α‖2+‖u+∂α∂t−Δ∂α∂t‖2−1 | (4.8) |
It follows from (4.7)-(4.8) and Gronwall's lemma that
E5⩾c(‖u‖2−1+‖α‖2H1(Ω)+‖∂α∂t−Δ∂α∂t‖2),c>0. | (4.9) |
hence the uniquess, as well as the continuous dependence with respect to the initial data in H−1×H1×H1-norm.
All authors declare no conflicts of interest in this paper.
[1] | Zelikowsky M, Hersman S, Chawla MK, et al. (2014) Neuronal ensembles in amygdala, hippocampus, and prefrontal cortex track differential components of contextual fear. J Neurosci 34: 8462-8466. https://doi.org/10.1523/JNEUROSCI.3624-13.2014 |
[2] | Orsini CA, Yan C, Maren S (2013) Ensemble coding of context-dependent fear memory in the amygdala. Front Behav Neurosci 7: 199-199. https://doi.org/10.3389/fnbeh.2013.00199 |
[3] | Poulos AM, Ponnusamy R, Dong H-W, et al. (2010) Compensation in the neural circuitry of fear conditioning awakens learning circuits in the bed nuclei of the stria terminalis. PNAS 107: 14881-14886. https://doi.org/10.1073/pnas.1005754107 |
[4] | Helmstetter FJ, Bellgowan PS (1994) Effects of muscimol applied to the basolateral amygdala on acquisition and expression of contextual fear conditioning in rats. Behav Neurosci 108: 1005-1009. https://doi.org/10.1037/0735-7044.108.5.1005 |
[5] | Muller J, Corodimas KP, Fridel Z, et al. (1997) Functional inactivation of the lateral and basal nuclei of the amygdala by muscimol infusion prevents fear conditioning to an explicit conditioned stimulus and to contextual stimuli. Behav Neurosci 111: 683-691. https://doi.org/10.1037/0735-7044.111.4.683 |
[6] | Corcoran KA, Quirk GJ (2007) Activity in prelimbic cortex is necessary for the expression of learned, but not innate, fears. J Neurosci 27: 840-844. https://doi.org/10.1523/JNEUROSCI.5327-06.2007 |
[7] | Goshen I, Brodsky M, Prakash R, et al. (2011) Dynamics of retrieval strategies for remote memories. Cell 147: 678-689. https://doi.org/10.1016/j.cell.2011.09.033 |
[8] | Zhu LJ, Liu MY, Li H, et al. (2014) The different roles of glucocorticoids in the hippocampus and hypothalamus in chronic stress-induced HPA axis hyperactivity. PLoS One 9: e97689. https://doi.org/10.1371/journal.pone.0097689 |
[9] | Chaaya N, Battle AR, Johnson LR (2018) An update on contextual fear memory mechanisms: Transition between Amygdala and Hippocampus. Neurosci Biobehav Rev 92: 43-54. https://doi.org/10.1016/j.neubiorev.2018.05.013 |
[10] | Nees F, Pohlack ST, Grimm O, et al. (2019) White matter correlates of contextual pavlovian fear extinction and the role of anxiety in healthy humans. Cortex 121: 179-188. https://doi.org/10.1016/j.cortex.2019.08.020 |
[11] | Silva BA, Burns AM, Graff J (2019) A cFos activation map of remote fear memory attenuation. Psychopharmacology (Berl) 236: 369-381. https://doi.org/10.1007/s00213-018-5000-y |
[12] | Tronson NC, Schrick C, Guzman YF, et al. (2009) Segregated populations of hippocampal principal CA1 neurons mediating conditioning and extinction of contextual fear. J Neurosci 29: 3387-3394. https://doi.org/10.1523/JNEUROSCI.5619-08.2009 |
[13] | Lebois LAM, Seligowski AV, Wolff JD, et al. (2019) Augmentation of Extinction and Inhibitory Learning in Anxiety and Trauma-Related Disorders. Annu Rev Clin Psychol 15: 257-284. https://doi.org/10.1146/annurev-clinpsy-050718-095634 |
[14] | Craske MG, Treanor M, Conway CC, et al. (2014) Maximizing exposure therapy: an inhibitory learning approach. Behav Res Ther 58: 10-23. https://doi.org/10.1016/j.brat.2014.04.006 |
[15] | Myers KM, Ressler KJ, Davis M (2006) Different mechanisms of fear extinction dependent on length of time since fear acquisition. Learn Memory (Cold Spring Harbor, NY) 13: 216-223. https://doi.org/10.1101/lm.119806 |
[16] | Westbrook RF, Iordanova M, McNally G, et al. (2002) Reinstatement of fear to an extinguished conditioned stimulus: two roles for context. J Exp Psychol Anim Behav Process 28: 97-110. https://doi.org/10.1037/0097-7403.28.1.97 |
[17] | Maes JHR, Vossen JMH (1994) The effect of separate reinforced and nonreinforced exposures to a context participating in a Pavlovian discrimination procedure. Elsevier Science, : 231-246. https://doi.org/10.1016/0376-6357(94)90009-4 |
[18] | Maren S (2001) Neurobiology of Pavlovian fear conditioning. Annu Rev Neurosci 24: 897-931. https://doi.org/10.1146/annurev.neuro.24.1.897 |
[19] | Sotres-Bayon F, Quirk GJ (2010) Prefrontal control of fear: more than just extinction. Curr Opin Neurobiol 20: 231-235. https://doi.org/10.1016/j.conb.2010.02.005 |
[20] | Milad MR, Orr SP, Lasko NB, et al. (2008) Presence and acquired origin of reduced recall for fear extinction in PTSD: results of a twin study. J Psychiatr Res 42: 515-520. https://doi.org/10.1016/j.jpsychires.2008.01.017 |
[21] | Gowrishankar R, Bruchas MR (2019) Defining circuit-specific roles for G protein-coupled receptors in aversive learning. Curr Opin Behav Sci 26: 146-156. https://doi.org/10.1016/j.cobeha.2019.01.002 |
[22] | Quirk GJ, Garcia R, Gonzalez-Lima F (2006) Prefrontal mechanisms in extinction of conditioned fear. Biol Psychiatry 60: 337-343. https://doi.org/10.1016/j.biopsych.2006.03.010 |
[23] | Quirk GJ, Russo GK, Barron JL, et al. (2000) The role of ventromedial prefrontal cortex in the recovery of extinguished fear. J Neurosci 20: 6225-6231. https://doi.org/10.1523/JNEUROSCI.20-16-06225.2000 |
[24] | Lebron K, Milad MR, Quirk GJ (2004) Delayed recall of fear extinction in rats with lesions of ventral medial prefrontal cortex. Learn Mem 11: 544-548. https://doi.org/10.1101/lm.78604 |
[25] | Sierra-Mercado D, Corcoran KA, Lebron-Milad K, et al. (2006) Inactivation of the ventromedial prefrontal cortex reduces expression of conditioned fear and impairs subsequent recall of extinction. Eur J Neurosci 24: 1751-1758. https://doi.org/10.1111/j.1460-9568.2006.05014.x |
[26] | Santini E, Quirk GJ, Porter JT (2008) Fear conditioning and extinction differentially modify the intrinsic excitability of infralimbic neurons. J Neurosci 28: 4028-4036. https://doi.org/10.1523/JNEUROSCI.2623-07.2008 |
[27] | Santini E, Porter JT (2010) M-type potassium channels modulate the intrinsic excitability of infralimbic neurons and regulate fear expression and extinction. J Neurosci 30: 12379-12386. https://doi.org/10.1523/JNEUROSCI.1295-10.2010 |
[28] | Young MB, Thomas SA (2014) M1-muscarinic receptors promote fear memory consolidation via phospholipase C and the M-current. J Neurosci 34: 1570-1578. https://doi.org/10.1523/JNEUROSCI.1040-13.2014 |
[29] | Fanselow MS (1980) Conditional and Unconditional Components of Post-Shock Freezing. Pavlovian J Biol Sci 15: 177-182. https://doi.org/10.1007/BF03001163 |
[30] | Anagnostaras SG, Josselyn SA, Frankland PW, et al. (2000) Computer-assisted behavioral assessment of Pavlovian fear conditioning in mice. Learn Mem 7: 58-72. https://doi.org/10.1101/lm.7.1.58 |
[31] | Cima G (2013) AVMA Guidelines for the Euthanasia of Animal: 2013 Edition. Javma-J Am Vet Med A 242: 715-716. |
[32] | Alshammari MA, Alshammari TK, Laezza F (2016) Improved Methods for Fluorescence Microscopy Detection of Macromolecules at the Axon Initial Segment. Front Cell Neurosci 10. https://doi.org/10.3389/fncel.2016.00005 |
[33] | de los Santos-Arteaga M, Sierra-Dominguez SA, Fontanella GH, et al. (2003) Analgesia induced by dietary restriction is mediated by the kappa-opioid system. J Neurosci 23: 11120-11126. https://doi.org/10.1523/JNEUROSCI.23-35-11120.2003 |
[34] | Lezmy J, Lipinsky M, Khrapunsky Y, et al. (2017) M-current inhibition rapidly induces a unique CK2-dependent plasticity of the axon initial segment. Proc Natl Acad Sci U S A 114: E10234-e10243. https://doi.org/10.1073/pnas.1708700114 |
[35] | Navarro D, Alvarado M, Navarrete F, et al. (2015) Gestational and early postnatal hypothyroidism alters VGluT1 and VGAT bouton distribution in the neocortex and hippocampus, and behavior in rats. Front Neuroanat 9: 9. https://doi.org/10.3389/fnana.2015.00009 |
[36] | Kim WB, Cho J-H (2020) Encoding of contextual fear memory in hippocampal–amygdala circuit. Nat Commun 11: 1382. https://doi.org/10.1038/s41467-020-15121-2 |
[37] | Anagnostaras SG, Gale GD, Fanselow MS (2001) Hippocampus and contextual fear conditioning: Recent controversies and advances. Hippocampus 11: 8-17. https://doi.org/10.1002/1098-1063(2001)11:1<8::AID-HIPO1015>3.0.CO;2-7 |
[38] | Lara Aparicio SY, Laureani Fierro ÁdJ, Aranda Abreu GE, et al. (2022) Current Opinion on the Use of c-Fos in Neuroscience. NeuroSci 3: 687-702. https://doi.org/10.3390/neurosci3040050 |
[39] | Milad MR, Quirk GJ (2012) Fear extinction as a model for translational neuroscience: ten years of progress. Ann Rev Psychol 63: 129-151. https://doi.org/10.1146/annurev.psych.121208.131631 |
[40] | Orsini CA, Maren S (2012) Neural and cellular mechanisms of fear and extinction memory formation. Neurosci Biobehav R 36: 1773-1802. https://doi.org/10.1016/j.neubiorev.2011.12.014 |
[41] | Santini E, Sepulveda-Orengo M, Porter JT (2012) Muscarinic receptors modulate the intrinsic excitability of infralimbic neurons and consolidation of fear extinction. Neuropsychopharmacology 37: 2047-2056. https://doi.org/10.1038/npp.2012.52 |
[42] | Slomko AM, Naseer Z, Ali SS, et al. (2014) Retigabine calms seizure-induced behavior following status epilepticus. Epilepsy Behav 37: 123-132. https://doi.org/10.1016/j.yebeh.2014.06.010 |
[43] | Criado-Marrero M, Santini E, Porter JT (2014) Modulating fear extinction memory by manipulating SK potassium channels in the infralimbic cortex. Front Behav Neurosci 8: 96. https://doi.org/10.3389/fnbeh.2014.00096 |
[44] | Victoria NC, Marron Fernandez de Velasco E, Ostrovskaya O, et al. (2016) G Protein-Gated K(+) Channel Ablation in Forebrain Pyramidal Neurons Selectively Impairs Fear Learning. Biol Psychiatry 80: 796-806. https://doi.org/10.1016/j.biopsych.2015.10.004 |
[45] | Tischmeyer W, Grimm R (1999) Activation of immediate early genes and memory formation. Cell Mol Life Sci 55: 564-574. https://doi.org/10.1007/s000180050315 |
[46] | Dragunow M (1996) A role for immediate-early transcription factors in learning and memory. Behav Genet 26: 293-299. https://doi.org/10.1007/BF02359385 |
[47] | Peng Z, Houser CR (2005) Temporal patterns of fos expression in the dentate gyrus after spontaneous seizures in a mouse model of temporal lobe epilepsy. J Neurosci 25: 7210-7220. https://doi.org/10.1523/JNEUROSCI.0838-05.2005 |
[48] | Kasugai Y, Vogel E, Hörtnagl H, et al. (2019) Structural and Functional Remodeling of Amygdala GABAergic Synapses in Associative Fear Learning. Neuron 104: 781-794.e784. https://doi.org/10.1016/j.neuron.2019.08.013 |
[49] | Temel Y, Blokland A, Lim LW (2012) Deactivation of the parvalbumin-positive interneurons in the hippocampus after fear-like behaviour following electrical stimulation of the dorsolateral periaqueductal gray of rats. Behav Brain Res 233: 322-325. https://doi.org/10.1016/j.bbr.2012.05.029 |
[50] | Caliskan G, Muller I, Semtner M, et al. (2016) Identification of Parvalbumin Interneurons as Cellular Substrate of Fear Memory Persistence. Cereb Cortex 26: 2325-2340. https://doi.org/10.1093/cercor/bhw001 |
[51] | Whissell PD, Bang JY, Khan I, et al. (2019) Selective Activation of Cholecystokinin-Expressing GABA (CCK-GABA) Neurons Enhances Memory and Cognition. eNeuro 6. https://doi.org/10.1523/ENEURO.0360-18.2019 |
[52] | Grubb MS, Burrone J (2010) Activity-dependent relocation of the axon initial segment fine-tunes neuronal excitability. Nature 465: 1070-1074. https://doi.org/10.1038/nature09160 |
[53] | Kuba H, Oichi Y, Ohmori H (2010) Presynaptic activity regulates Na+ channel distribution at the axon initial segment. Nature 465: 1075-1078. https://doi.org/10.1038/nature09087 |
[54] | Grubb MS, Shu Y, Kuba H, et al. (2011) Short- and long-term plasticity at the axon initial segment. J Neurosci 31: 16049-16055. https://doi.org/10.1523/JNEUROSCI.4064-11.2011 |
[55] | Adachi R, Yamada R, Kuba H (2015) Plasticity of the axonal trigger zone. Neuroscientist 21: 255-265. https://doi.org/10.1177/1073858414535986 |
[56] | Yamada R, Kuba H (2016) Structural and Functional Plasticity at the Axon Initial Segment. Front Cell Neurosci 10: 250. https://doi.org/10.3389/fncel.2016.00250 |
[57] | Petersen AV, Cotel F, Perrier JF (2017) Plasticity of the Axon Initial Segment: Fast and Slow Processes with Multiple Functional Roles. Neuroscientist 23: 364-373. https://doi.org/10.1177/1073858416648311 |
[58] | Leterrier C (2016) Chapter Six - The Axon Initial Segment, 50Years Later: A Nexus for Neuronal Organization and Function. Current Topics in Membranes . Academic Press pp. 185-233. https://doi.org/10.1016/bs.ctm.2015.10.005 |
[59] | Kole Maarten HP, Stuart Greg J (2012) Signal Processing in the Axon Initial Segment. Neuron 73: 235-247. https://doi.org/10.1016/j.neuron.2012.01.007 |
[60] | Holmes A, Chen A (2015) GABA receptors in a state of fear. Nat Neurosci 18: 1194-1196. https://doi.org/10.1038/nn.4098 |
[61] | Rovira-Esteban L, Gunduz-Cinar O, Bukalo O, et al. (2019) Excitation of Diverse Classes of Cholecystokinin Interneurons in the Basal Amygdala Facilitates Fear Extinction. eNeuro 6: ENEURO.0220-0219.2019. https://doi.org/10.1523/ENEURO.0220-19.2019 |
[62] | Saha R, Knapp S, Chakraborty D, et al. (2017) GABAergic Synapses at the Axon Initial Segment of Basolateral Amygdala Projection Neurons Modulate Fear Extinction. Neuropsychopharmacology 42: 473-484. https://doi.org/10.1038/npp.2016.205 |
1. | Talat Nazir, Mujahid Abbas, Safeer Hussain Khan, Fixed Point Results of Fuzzy Multivalued Graphic Contractive Mappings in Generalized Parametric Metric Spaces, 2024, 13, 2075-1680, 690, 10.3390/axioms13100690 | |
2. | Joginder Paul, Mohammad Sajid, Naveen Chandra, Umesh Chandra Gairola, Some common fixed point theorems in bipolar metric spaces and applications, 2023, 8, 2473-6988, 19004, 10.3934/math.2023969 | |
3. | Charu Batra, Renu Chugh, Mohammad Sajid, Nishu Gupta, Rajeev Kumar, Generalized viscosity approximation method for solving split generalized mixed equilibrium problem with application to compressed sensing, 2023, 9, 2473-6988, 1718, 10.3934/math.2024084 |