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
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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
A self-mapping F on a convex, closed, and bounded subset K of a Banach space U is known as nonexpansive if ‖Fu−Fv‖ ≤ ‖u−v‖, u,v∈U and need not essentially possess a fixed point. It is widely known that a point u∈U is a fixed point or an invariant point if Fu=u. However, some researchers ensured the survival of a fixed point of nonexpansive mapping in Banach spaces utilizing suitable geometric postulates. Numerous mathematicians have extended and generalized these conclusions to consider several nonlinear mappings. One such special class of mapping is Suzuki generalized nonexpansive mapping (SGNM). Many extensions, improvements and generalizations of nonexpansive mappings are given by eminent researchers (see [8,9,10,13,15,17,19,21,22,25], and so on). On the other hand, Krasnosel'skii [16] investigated a novel iteration of approximating fixed points of nonexpansive mapping. A sequence {ui} utilizing the Krasnosel'skii iteration is defined as: u1=u,ui+1=(1−α)ui+αFui, where α∈(0,1) is a real constant. This iteration is one of the iterative methods which is the extension of the celebrated Picard iteration [24], ui+1=Fui. The convergence rate of the Picard iteration [24] is better than the Krasnosel'skii iteration although the Picard iterative scheme is not essentially convergent for nonexpansive self-mappings. It is interesting to see that the fixed point of a self-mapping F is also a fixed point of the iteration Fn (n∈N), of the self-mapping F but the reverse implication is not feasible. Recently several authors presented extended and generalized results for better approximation of fixed points (see [1,3,11,23,26,27]).
We present convergence and common fixed point conclusions for the associated α-Krasnosel'skii mappings satisfying condition (E) in the current work. Also, we support these with nontrivial illustrative examples to demonstrate that our conclusions improve, generalize and extend comparable conclusions of the literature.
We symbolize F(F), to be the collection of fixed points of a self-mapping F, that is, F(F) = {u∈U:Fu=u}. We begin with the discussion of convex Banach spaces, α-Krasnosel'skii mappings and the condition (E) (see [12,18,20,23]).
Definition 2.1. [14] A Banach space U is uniformly convex if, for ϵ∈(0,2] ∃ δ>0 satisfying, ‖u+v2‖ ≤1−δ so that ‖u−v‖>ϵ and ‖u‖=‖v‖=1, u,v∈U.
Definition 2.2. [14] A Banach space U is strictly convex if, ‖u+v2‖<1 so that u≠v,‖u‖=‖v‖=1, u,v∈U.
Theorem 2.1. [5] Suppose U is a uniformly convex Banach space. Then ∃ a γ>0, satisfying ‖12(u+v)‖≤[1−γϵδ]δ for every ϵ,δ>0 so that ‖u−v‖≥ϵ, ‖u‖≤δ and ‖v‖≤δ, for u,v∈U.
Theorem 2.2. [14] The subsequent postulates are equivalent in a Banach space U:
(i) U is strictly convex.
(ii) u=0 or v=0 or v=cu for c>0, whenever ‖u+v‖ = ‖u‖+‖v‖,u,v∈U.
Definition 2.3. Suppose F is a self-mapping on a non-void subset V of a Banach space U.
(i) Suppose for u∈U, ∃ v∈V so that for all w∈V, ‖v−u‖ ≤‖w−u‖. Then v is a metric projection [6] of U onto V, and is symbolized by PV(.). The mapping PV(u):U→V is the metric projection if PV(x) exists and is determined uniquely for each x∈U.
(ii) F satisfies condition (Eμ) [23] on V if ∃ μ≥1, satisfying ‖u−Fv‖≤μ‖u−Fu‖+‖u−v‖,u,v∈V. Moreover, F satisfies condition (E) on V, if F satisfies (Eμ).
(iii) F satisfies condition (E) [23] and F(F)≠0, then F is quasi-nonexpansive.
(iv) F is a generalized α-Reich-Suzuki nonexpansive [21] if for an α∈[0,1), 12‖u−Fu‖≤‖u−v‖⟹‖Fu−Fv‖≤ max {α‖Fu−u‖+α‖Fv−v‖+(1−2α)‖u−v‖,α‖Fu−v‖+α‖Fv−u‖+(1−2α)‖u−v‖}, ∀u,v∈V.
(v) A self-mapping Fα:V→V is an α-Krasnosel'skii associated with F [2] if, Fαu=(1−α)u+αFu, for α∈(0,1), u∈V.
(vi) F is asymptotically regular [4] if limn→∞‖Fnu−Fn+1u‖=0.
(vii) F is a generalized contraction of Suzuki type [2], if ∃ β∈(0,1) and α1,α2,α3∈[0,1], where α1+2α2+2α3=1, satisfying β‖u−Fu‖≤‖u−v‖ implies
‖Fu−Fv‖≤α1‖u−v‖+α2(‖u−Fu‖+‖v−Fv‖)+α3(‖u−Fv‖+‖v−Fu‖) ,u,v∈U. |
(viii) F is α-nonexpansive [7] if ∃ an α<1 satisfying
‖Fu−Fv‖≤α‖Fu−v‖+α‖Fv−u‖+(1−2α)‖u−v‖,u,v∈U. |
Theorem 2.3. [5] A continuous mapping on a non-void, convex and compact subset V of a Banach space U has a fixed point in V.
Pant et al.[23] derived a proposition that if β=12, then a generalized contraction of Suzuki type is a generalized α-Reich-Suzuki nonexpansive. Moreover, the reverse implication may not necessarily hold.
Lemma 2.1. [2] Let F be a generalized contraction of the Suzuki type on a non-void subset V of a Banach space U. Let β∈[12,1), then
‖u−Fv‖≤(2+α1+α2+3α31−α2−α3)‖u−Fu‖+‖u−v‖. |
Proposition 2.1. [23] Let F be a generalized contraction of the Suzuki type on a non-void subset V of a Banach space U, then F satisfies condition (E).
The converse of this proposition is not true, which can be verified by the following example.
Example 2.1. Suppose U=(R2,‖.‖) with the Euclidean norm and V=[−1,1]×[−1,1] be a subset of U. Let F:V→V be defined as
F(u1,u2)={(u12,u2),if|u1|≤12(−u1,u2),if|u1|>12. |
Case I. Let x=(u1,u2),y=(v1,v2) with |u1|≤12, |v1|≤12. Then,
‖Fx−Fy‖=‖(u12,u2)−(v12,v2)‖=√(u1−v1)24+(u2−v2)2≤√(u1−v1)2+(u2−v2)2=‖x−y‖, |
which implies
‖x−Fy‖≤‖x−Fx‖+‖Fx−Fy‖≤‖x−Fx‖+‖x−y‖. |
Case II. If |u1|≤12, |v1|>12
‖x−Fy‖=√(u1+v1)2+(u2−v2)2‖x−y‖=√(u1−v1)2+(u2−v2)2‖x−Fx‖=|u1|2. |
Consider
‖x−Fy‖=√(u1−v1)2+(u2−v2)2+4u1v1≤√(u1−v1)2+(u2−v2)2+4|u1|≤√(u1−v1)2+(u2−v2)2+4|u1|. |
Hence,
‖x−Fy‖≤8‖x−Fx‖+‖x−y‖. |
Here μ=8 satisfies the inequality.
Case III. If |u1|>12, |v1|≤12
‖x−Fy‖=√(u1−v12)2+(u2−v2)2‖x−y‖=√(u1+v1)2+(u2−v2)2‖x−Fx‖=2|u1|. |
Consider
‖x−Fy‖=√(u1−v12)2+(u2−v2)2≤√(u1−v1)2+(u2−v2)2≤√(u1−v1)2+(u2−v2)2+|u1|≤√(u1−v1)2+(u2−v2)2+2|u1|. |
So,
‖x−Fy‖≤‖x−Fy‖+‖x−y‖. |
Case IV. If |u1|>12 and |v1|>12, then
‖x−Fy‖=√(u1+v1)2+(u2−v2)2‖x−y‖=√(u1−v1)2+(u2−v2)2‖x−Fx‖=2|u1|. |
Since |u1|>12 and |v1|>12, by simple calculation as above, we attain
‖x−Fy‖≤μ‖x−Fx‖+‖x−y‖. |
Thus, F satisfies condition (E) for μ=4.
Now, suppose x=(12,1) and y=(1,1), so
β‖x−Fx‖=β(12−14)=β4≤‖x−y‖=12. |
Clearly, ‖Fx−Fy‖=√(54)2+(1−1)2=54.
Consider
α1‖x−y‖+α2(‖x−Fx‖+‖y−Fy‖)+α3(‖x−Fy‖+‖y−Fx‖)=α1‖(12,1)−(1,1)‖+α2(‖(12,1)−(14,1)‖+‖(1,1)−(−1,1)‖)+α3(‖(12,1)−(−1,1)‖+‖(1,1)−(14,1)‖)=α12+α24+2α2+3α32+3α34=α12+94(α2+α3)=α12+94(1−α12)(by Definition 2.3 (vii))=α12+98−9α18=98−5α18. |
Since α1,α2,α3≥0, therefore
‖Fx−Fy‖>α1‖x−y‖+α2(‖x−Fy‖+‖y−Fy‖)+α3(‖x−Fy‖+‖y−Fx‖), |
which is a contradiction.
Thus, F is not a generalized contraction of the Suzuki type.
Now, we establish results for a pair of α-Krasnosel'skii mappings using condition (E).
Theorem 3.1. Let Fi, for i∈{1,2}, be self-mappings on a non-void convex subset V of a uniformly convex Banach space U and satisfy condition (E) so that F(F1∩F2)≠ϕ. Then the α-Krasnosel'skii mappings Fiα, α∈(0,1) and i∈{1,2} are asymptotically regular.
Proof. Let v0∈V. Define vn+1=Fiαvn for i∈{1,2} and n∈N∪{0}. Thus,
Fiαvn=yn+1=(1−α)vn+αFivnfori∈{1,2}, |
and
Fiαvn−vn=Fiαvn−Fiαvn−1=α(Fivn−vn)fori∈{1,2}. |
It is sufficient to show that limn→∞‖Fivn−vn‖=0 to prove Fiα is asymptotically regular.
By definition, for u0∈F(F1∩F2), we have
‖u0−Fivn‖≤‖u0−vn‖fori∈{1,2} | (3.1) |
and for i∈{1,2},
‖u0−vn+1‖=‖u0−Fiαvn‖=‖u0−(1−α)vn−αFivn‖≤(1−α)‖u0−vn‖+α‖u0−Fivn‖=(1−α)‖u0−vn‖+α‖u0−vn‖=‖u0−vn‖. | (3.2) |
Thus, the sequence {‖u0−vn‖} is bounded by s0=‖u0−v0‖. From inequality (3.2), vn→u0 as n→∞, if vn0=u0, for some n0∈N. So, assume vn≠u0, for n∈N, and
wn=u0−vn‖u0−vn‖anden=u0−Fivn‖u0−vn‖,fori∈{1,2}. | (3.3) |
If α≤12 and using Eq (3.3), we obtain
‖u0−vn+1‖=‖u0−Fiαvn‖,fori∈{1,2}=‖u0−(1−α)vn−αFivn‖,fori∈{1,2}=‖u0−vn+αvn−αFivn−2αu0+2αu0+αvn−αvn‖,fori∈{1,2}=‖(1−2α)u0−(1−2α)vn+(2αu0−αvn−αFivn)‖,fori∈{1,2}≤(1−2α)‖u0−vn‖+α‖2u0−vn−Fivn‖=2α‖u0−vn‖‖wn+en2‖+(1−2α)‖u0−vn‖. | (3.4) |
As the space U is uniformly convex with ‖wn‖≤1, ‖en‖≤1 and ‖wn−en‖=‖vn−Fivn‖‖u0−vn‖≥‖vn−Fivn‖s0=ϵ (say) for i∈{1,2}, we obtain
‖wn+en‖2≤1−δ‖vn−Fivn‖sofori∈{1,2}. | (3.5) |
From inequalities (3.4) and (3.5),
‖u0−vn+1‖≤(2α(1−δ‖vn−Fivn‖so)+(1−2α))‖u0−vn‖=(1−2αδ(‖vn−Fivn‖s0) )‖u0−vn‖. | (3.6) |
By induction, it follows that
‖u0−vn+1‖≤n∏j=1(1−2αδ(‖vn−Fivn‖s0))s0. | (3.7) |
We shall prove that limn→∞‖Fivn−vn‖=0 for i∈{1,2}. On the contrary, consider that {‖Fivn−vn‖} for i∈{1,2} is not converging to zero, and we have a subsequence {vnk}, of {vn}, satisfying ‖Fivnk−vnk‖ converges to ζ>1. As δ∈[0,1] is increasing and α≤12, 1−2αδ‖vk−Fivk‖s0∈[0,1], i∈{1,2}, for all k∈N. Since ‖Fivnk−vnk‖→ζ so, for sufficiently large k,‖Fivnk−vnk‖≥ζ2, from inequality (3.7), we have
‖u0−vnk+1‖≤s0(1−2αδ(ζ2−s0))(nk+1). | (3.8) |
Making k→∞, it follows that vnk→u0. By inequality (3.1), we get Fivnk→u0 and ‖vnk−Fivnk‖→0 as k→∞, which is a contradiction. If α>12, then 1−α<12, because α∈(0,1). Now, for i∈{1,2}
‖u0−vn+1‖=‖u0−(1−α)vn−αFivn‖=‖u0−vn+αvn−αFivn+(2−2α)u0−(2−2α)u0+Fivn−Fivn+αFivn−αFivn‖=‖(2u0−vn−Fivn)−α(2u0−vn−Fivn)+2α(u0−Fivn)−(u0−Fivn)‖≤(1−α)‖2u0−vn−Fivn‖+(2α−1)‖u0−vn‖≤2(1−2α)‖u0−vn‖‖wn+en‖2+(2α−1)‖u0−vn‖. |
By the uniform convexity of U, we attain, for i∈{1,2},
‖x0−yn+1‖≤(2(1−α)−2(1−α)δ‖yn−Fiyn‖so+(1−2α))‖x0−yn‖. | (3.9) |
By induction, we get
‖u0−vn+1‖≤n∏j=1(1−2(1−α)δ(‖vj−Fivj‖s0))s0. |
Similarly, it can be easily proved that ‖Fivn−vn‖→0 as n→∞, which implies that Fiα for i∈{1,2}, is asymptotically regular.
Next, we demonstrate by a numerical experiment that a pair of α-Krasnosel'skii mappings are asymptotically regular for fix α∈(0,1).
Example 3.1. Assume U=(R2,||.||) with Euclidean norm and V={u∈R2:‖u‖≤1}, to be a convex subset of U. Fi for i∈{1,2} be self-mappings on V, satisfying
F1(u1,u2)=(u1,u2)F2(u1,u2)=(u12,0) |
Then, clearly both F1 and F2 satisfy the condition (E) and F(F1∩F2)=(0,0). Now, we will show that the α-Krasnosel'skii mappings Fiα for α∈(0,1) and i∈{1,2} are asymptotically regular.
Since F1 is the identity map, α- Krasnosel'skii mapping F1α is also identity and hence asymptotically regular.
Now, we show F2α is asymptotically regular, let u=(u1,u2)∈V
F2α(u1,u2)=(1−α)(u1,u2)+αF2(u1,u2)=((1−α)u1,(1−α)u2)+α(u12,0)=(u1−αu12,(1−α)u2), |
F22α(u1,u2)=(1−α)(u1−αu12,(1−α)u2)+αF2(u1−αu12,(1−α)u2)=(x1+α2u12−3αu12,(1−α)2u2)+(αu2−α2u14,0)=(u1−αu1+α2u14,(1−α)2x2). |
Continuing in this manner, we get
fn2α(u1,u2)=((u1−α2)n,(1−α)nu2). |
Since (u1,u2)∈V and α∈(0,1), we get that limn→∞(u1−α2)n=0 and limn→∞(1−α)n=0. Now, consider
limn→∞‖Fn2α(u1,u2)−Fn+12α(u1,u2)‖=supu∈Mlimn→∞‖(u1−α2)n−(u1−α2)n+1,((1−α)n−(1−α)n+1)x2‖=0. |
Hence, F2α is also asymptotically regular.
Theorem 3.2. Let Fi be quasi-nonexpansive self-mappings on a non-void and closed subset V of a Banach space U for i∈{1,2}, and satisfy condition (E) so that F(F1∩F2)≠0. Then, F(F1∩F2) is closed in V. Also, if U is strictly convex, then F(F1∩F2) is convex. Furthermore, if U is strictly convex, V is compact, and F is continuous, then for any s0∈V,α∈(0,1), the α-Krasnosel'skii sequence {Fniα(s0)}, converges to s∈F( F1∩F2) .
Proof. (i) We assume {sn}∈F( F1∩F2) so that sn→s∈F(F1∩F2) as n→∞. Hence, Fisn=sn for i∈{1,2}. Next, we show that Fis=s for i∈{1,2}. Since Fi are quasi-nonexpansive, we get
‖sn−Fis‖≤‖sn−s‖fori∈{1,2}, |
that is, Fis=s for i=1,2, hence F(F2∩F2) is closed.
(ii) V is convex since U is strictly convex. Also fix γ∈( 0,1) and u,v∈F(F1∩F2) so that u≠v. Take s=γu+(1−γ)v∈V. Since mapping Fi satisfy condition (E),
‖u−Fis‖≤‖u−Fiu‖+‖u−s‖=‖u−s‖fori∈{1,2}. |
Similarly,
‖v−Fis‖≤‖v−s‖fori∈{1,2}. |
Using strict convexity of U, there is a θ∈[ 0,1] so that Fis=θu+(1−θ)v for i=1,2
(1−θ)‖u−v‖=‖Fiu−Fis‖≤‖u−s‖=(1−γ) ‖u−v‖,fori∈{1,2}, | (3.10) |
and
θ‖u−v‖=‖Fiv−Fis‖≤‖v−s‖=γ‖u−v‖,fori∈{1,2}. | (3.11) |
From inequalities (3.10) and (3.11), we obtain
1−θ≤1−γandθ≤γimplies thatθ=γ. |
Hence, Fis=s for i = 1, 2, implies s∈F(F1∩F2) .
(iii) Let us define {sn} by sn=Fniαs0,s0∈V, where Fiαs0=(1−α)s0+αFis0,α∈( 0,1) . We have a subsequence {snk} of {sn} converging to some s∈V, since V is compact. Using the Schauder theorem and the continuity of Fi, we have F(F1∩F2) ≠ϕ. We shall demonstrate that s∈F(F1∩F2). Let w0∈F(F1∩F2), consider
‖sn−w0‖=‖Fniαs0−w0‖≤‖Fn−1iαs0−w0|=‖sn−1−w0‖. |
Therefore, {‖sn−w0‖} converges as it is a decreasing sequence that is bounded below by 0. Moreover, since Fiα for i=1,2 is continuous, we have
‖w0−s0‖=limk→∞‖snk+1−so‖=limk→∞‖Fiαsnk−s0‖=‖Fiαs−s0‖=‖(1−α)s+αFis−s0‖≤(1−α)‖s−s0‖+α‖Fis−s0‖fori∈{1,2}. | (3.12) |
Since α>0, we get
‖s−s0‖≤‖Fis−s0‖,fori∈{1,2}. | (3.13) |
Since Fi are quasi-nonexpansive maps, we get
‖Fis−s0‖≤‖s−s0‖,fori∈{1,2}, | (3.14) |
and from inequalities (3.13) and (3.14), we get
‖Fis−s0‖=‖s−s0‖,fori∈{1,2}. | (3.15) |
Now, from inequality (3.12), we have
‖s−s0‖≤‖(1−α)s+αFis−s0‖,fori∈{1,2}≤(1−α)‖s−s0‖+α‖Fis−s0‖,fori∈{1,2}=‖s−s0‖, |
which implies that
‖(1−α)s+αFis−s0‖=(1−α)‖s−s0‖+α‖Fis−s0‖,fori∈{1,2}. |
Since U is strictly convex, either Fis−s0=a(s−s0) for some a⪈0 or s=s0. From Eq (15), it follows that a=1, then, Fis=s for i=1,2 and s∈F(F1∩F2). Since limn→∞‖sn−s0‖ exists and {snk} converges strongly to s. Hence, {sn} converges strongly to s∈F(F1∩F2).
The next conclusion for metric projection is slightly more fascinating.
Theorem 3.3. Let Fi be quasi-nonexpansive self-mappings on a non-void, closed, and convex subset V of a uniformly convex Banach space U for i∈{1,2}, and satisfies condition (E) so that F(F1∩F2)≠ϕ. Let P:U→F(F1∩F2) be the metric projection. Then, for every u∈U, the sequence {PFniu} for i={1,2}, converges to s∈F(F1∩F2).
Proof. Let u∈V. For n,m∈N
‖PFniu−Fniu‖≤‖PFmiu−Fniu‖,forn≥m,i∈{1,2}. | (3.16) |
Since u∈F(F1∩F2) , n∈N and Fi are quasi-nonexpansive maps, fori∈{1,2} we have
‖PFmiu−Fniu‖=‖PFmiu−FiFn−1iu‖≤‖PFmiu−Fn−1iu‖. |
Therefore, for n≥m, it follows that
‖PFmiu−Fniu‖≤‖PFmiu−Fmiu‖,fori∈{1,2}. | (3.17) |
From inequalities (3.16) and (3.17), we have
‖PFniu−Fniu‖≤‖PFmiu−Fmiu‖,fori∈{1,2}, |
which implies that limn→∞‖PFniu−Fniu‖ exists. Taking limn→∞‖PFniu−Fniu‖=l.
If l=0, then we have an integer n0( ϵ) for ϵ>0, satisfying
‖PFniu−Fniu‖>ϵ4,fori∈{1,2}, | (3.18) |
for n≥n0. Therefore, if n≥m≥n0 and using inequalities (3.17) and (3.18), we have, for i∈{1,2},
‖PFniu−PFmiu‖≤‖PFniu−PFn0iu‖+‖PFn0iu−Fmiu‖≤‖PFniu−Fniu‖+‖Fniu−PFn0iu‖+‖PFmiu−Fmiu‖+‖Fmiu−PFn0iu‖≤‖PFniu−Fniu‖+‖Fn0iu−PFn0iu‖+‖PFmiu−Fmiu‖+‖Fn0iu−PFn0iu‖≤ϵ4+ϵ4+ϵ4+ϵ4=ϵ. |
That is, {PFniu} for i={1,2} is a Cauchy sequence in F(F1∩F2). Using the completeness of U and the closedness of F(F1∩F2) from the above theorem, {PFnix} for i=1,2, converges in F(F1∩F2). Taking l>0, we claim that the sequence {PFniu} for i=1,2, is a Cauchy sequence in U. Also we have, an ϵ0>0 so that, for each n0∈N, we have some r0,s0≥n0 satisfying
‖PFr0iu−PFs0iu‖≥ϵ0,fori∈{1,2}. |
Now, we choose a θ>0
(l+θ)(1−δϵ0l+θ)<θ. |
Let m0 be as large as possible such that for q≥m0
l≤‖PFqiu−Fqiu‖≤l+θ. |
For this m0, there exist q1,q2 such that q1,q2>m0 and
‖PFq1iu−PFq2iu‖≥ϵ0fori∈{1,2}. |
Thus, for q0≥max{q1,q2}, we attain
‖PFq1ix−Fq0ix‖≥‖PFq1ix−Fq1ix‖<l+θ, |
and
‖PFq2ix−Fq0ix‖≥‖PFq1ix−Fq1ix‖<l+θfori∈{1,2}. |
Now, using the uniform convexity of U, we attain
l≤‖PFq0ix−Fq0ix‖≤‖PFq1ix+PFq2ix2−Fq0ix‖,fori∈{1,2}≤( l+θ) (1−δϵ0l+θ)<θ, |
a contradiction. Hence for every u∈V, the sequence {PFniu} for i=1,2, converges to some s∈F(F1∩F2).
We have proved some properties of common fixed points and also showed that if two mappings have common fixed points, then their α-Krasnosel'skii mappings are asymptotically regular. To show the superiority of our results, we have provided an example. Further, we have proved that the α-Krasnosel'skii sequence and its projection converge to a common fixed whose collection is closed.
Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.
The authors declare no conflict of interest.
[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 |