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4.5.2: Oscillation of a Membrane

4.5.2: Oscillation of a Membrane


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Let (Omegasubsetmathbb{R}^2) be a bounded domain. We consider the initial-boundary value problem

egin{eqnarray}
label{mem1} ag{4.5.2.1}
u_{tt}(x,t)&=& riangle_xu mbox{in} Omega imesmathbb{R}^1,
label{mem2} ag{4.5.2.2}
u(x,0)&=&f(x), xinoverline{Omega},
label{mem3} ag{4.5.2.3}
u_t(x,0)&=&g(x), xinoverline{Omega},
label{mem4} ag{4.5.2.4}
u(x,t)&=&0 mbox{on} partialOmega imesmathbb{R}^1.
end{eqnarray}

As in the previous subsection for the string, we make the ansatz (separation of variables)

$$
u(x,t)=w(t)v(x)
]

which leads to the eigenvalue problem

egin{eqnarray}
label{evpmem1} ag{4.5.2.5}
- riangle v&=&lambda v mbox{in} Omega,
label{evpmem2} ag{4.5.2.6}
v&=&0 mbox{on} partialOmega.
end{eqnarray}

Let (lambda_n) are the eigenvalues of (( ef{evpmem1})), (( ef{evpmem2})) and (v_n) a complete associated orthonormal system of eigenfunctions. We assume (Omega) is sufficiently regular such that the eigenvalues are countable, which is satisfied in the following examples. Then the formal solution of the above initial-boundary value problem is

$$
u(x,t)=sum_{n=1}^inftyleft(alpha_ncos(sqrt{lambda_n}t)+eta_nsin(sqrt{lambda_n}t) ight)v_n(x),
]

where

egin{eqnarray*}
alpha_n&=&int_Omega f(x)v_n(x) dx
eta_n&=&frac{1}{sqrt{lambda_n}}int_Omega g(x)v_n(x) dx.
end{eqnarray*}

Note

In general, eigenvalues of ( ef{evpmem1}), ( ef{evpmem1}) are not known explicitly. There are numerical methods to calculate these values. In some special cases, see next examples, these values are known.

Examples

Example 4.5.2.1: Rectangle membrane

Let
$$
Omega=(0,a) imes (0,b).
$$
Using the method of separation of variables, we find all eigenvalues of ( ef{evpmem1}), ( ef{evpmem2}) which are given by
$$
lambda_{kl}=sqrt{frac{k^2}{a^2}+frac{l^2}{b^2}}, k,l=1,2,ldots
$$ and associated eigenfunctions, not normalized, are
$$
u_{kl}(x)=sinleft(frac{pi k}{a}x_1 ight)sinleft(frac{pi l}{b}x_2 ight).
]

Example 4.5.2.2: Disk membrane

Set
$$
Omega={xinmathbb{R}^2: x_1^2+x_2^2$$
In polar coordinates, the eigenvalue problem ( ef{evpmem1}), ( ef{evpmem2}) is given by
egin{eqnarray}
label{evppol1} ag{4.5.2.6}
-frac{1}{r}left((ru_r)_r+frac{1}{r}u_{ heta heta} ight)&=&lambda u
label{evppol2} ag{4.5.2.7}
u(R, heta)&=&0,
end{eqnarray}
here is (u=u(r, heta):=v(rcos heta,rsin heta)). We will find eigenvalues and eigenfunctions by separation of variables
$$
u(r, heta)=v(r)q( heta),
$$
where (v(R)=0) and (q( heta)) is periodic with period (2pi) since (u(r, heta)) is single valued.
This leads to
$$
-frac{1}{r}left((rv')'q+frac{1}{r}vq'' ight)=lambda v q.
$$
Dividing by (vq), provided (vq ot=0), we obtain
egin{equation}
label{disk1} ag{4.5.2.8}
-frac{1}{r}left(frac{(rv'(r))'}{v(r)}+frac{1}{r}frac{q''( heta)}{q( heta)} ight)=lambda,
end{equation}
which implies
$$
frac{q''( heta)}{q( heta)}=const.=:-mu.
$$
Thus, we arrive at the eigenvalue problem
egin{eqnarray*}
-q''( heta)&=&mu q( heta)
q( heta)&=&q( heta+2pi).
end{eqnarray*}
It follows that eigenvalues (mu) are real and nonnegative. All solutions of the differential equation are given by
$$
q( heta)=Asin(sqrt{mu} heta)+Bcos(sqrt{mu} heta),
$$
where (A), (B) are arbitrary real constants. From the periodicity requirement
$$
Asin(sqrt{mu} heta)+Bcos(sqrt{mu} heta)=Asin(sqrt{mu}( heta+2pi))+Bcos(sqrt{mu}( heta+2pi))
$$
it followsegin{eqnarray*}
sin x-sin y&=&2cosfrac{x+y}{2}sinfrac{x-y}{2}
cos x-cos y&=&-2sinfrac{x+y}{2}sinfrac{x-y}{2}end{eqnarray*}
$$
sin(sqrt{mu}pi)left(Acos(sqrt{mu} heta+sqrt{mu}pi)-Bsin(sqrt{mu} heta+sqrt{mu}pi) ight)=0,
$$
which implies, since (A), (B) are not zero simultaneously, because we are looking for (q) not identically zero,
$$
sin(sqrt{mu}pi)sin(sqrt{mu} heta+delta)=0
$$
for all ( heta) and a (delta=delta(A,B,mu)). Consequently the eigenvalues are
$$
mu_n=n^2, n=0,1,ldots .
$$
Inserting (q''( heta)/q( heta)=-n^2) into ( ef{disk1}), we obtain the boundary value problem
egin{eqnarray}
label{disk2} ag{4.5.2.9}
r^2v''(r)+rv'(r)+(lambda r^2-n^2)v&=&0 mbox{on} (0,R)
label{disk3} ag{4.5.2.10}
v(R)&=&0
label{disk4} ag{4.5.2.11}
sup_{rin(0,R)}|v(r)|&<&infty.
end{eqnarray}
Set (z=sqrt{lambda}r) and (v(r)=v(z/sqrt{lambda})=:y(z)), then, see ( ef{disk2}),
$$
z^2y''(z)+zy'(z)+(z^2-n^2)y(z)=0,
$$
where (z>0). Solutions of this differential equations which are bounded at zero are Bessel functions of first kind and (n)-th order (J_n(z)). The eigenvalues follows from boundary condition ( ef{disk3}), i. e., from (J_n(sqrt{lambda}R)=0). Denote by ( au_{nk}) the zeros of (J_n(z)), then the eigenvalues of ( ef{evppol1})-( ef{evppol1}) are
$$
lambda_{nk}=left(frac{ au_{nk}}{R} ight)^2
$$
and the associated eigenfunctions are
egin{eqnarray*}
J_n(sqrt{lambda_{nk}}r)sin(n heta), && n=1,2,ldots
J_n(sqrt{lambda_{nk}}r)cos(n heta), && n=0,1,2,ldots.
end{eqnarray*}
Thus the eigenvalues (lambda_{0k}) are simple and (lambda_{nk}, nge1), are double eigenvalues.

Remark. For tables with zeros of (J_n(x)) and for much more properties of Bessel functions see cite{Watson}. One has, in particular, the asymptotic formula
$$
J_n(x)=left(frac{2}{pi x} ight)^{1/2}left(cos(x-npi/2-pi/5)+Oleft(frac{1}{x} ight) ight)
$$
as (x oinfty). It follows from this formula that there are infinitely many zeros of (J_n(x)).


Coexistence of Multiple Stable States and Bursting Oscillations in a 4D Hopfield Neural Network

Neurons are regarded as basic, structural and functional units of the central nervous system. They play an active role in the collection, storing and transferring of the information during signal processing in the brain. In this paper, we investigate the dynamics of a model of a 4D autonomous Hopfield neural network (HNN). Our analyses highlight complex phenomena such as chaotic oscillations, periodic windows, hysteretic dynamics, the coexistence of bifurcations and bursting oscillations. More importantly, it has been found several sets of synaptic weight for which the proposed HNN displays multiple coexisting stable states including three disconnected attractors. Besides the phenomenon of coexistence of attractors, the bursting phenomenon characterized by homoclinic/Hopf cycle–cycle bursting via homoclinic/fold hysteresis loop is observed. This contribution represents the first case where the later phenomenon (bursting oscillations) occurs in an autonomous HNN. Also, PSpice simulations are used to support the results of the previous analyses.


Introduction

Talin was discovered nearly three decades ago as a highly abundant cytosolic protein important for cytoskeleton organization and cell-extracellular matrix (ECM) adhesion 1 . Extensive genetic and cell biological studies have established that talin is crucial for regulating a wide variety of integrin-mediated cell adhesion-dependent processes, such as cell-shape change, growth, differentiation, and migration 2,3,4 . Talin is large in size, with 2541 amino acids, and can be divided into two major segments, an N-terminal head (1-433, talin-H, 50 kDa) that contains a FERM (four-point-one-protein/ezrin/radixin/moesin) domain (86-400, talin-FERM) and a C-terminal rod (482-2541, talin-R, 220 kDa) that contains a series of consecutive helical bundles followed by an actin-binding motif 2,3,4 . Talin-FERM, which engages with heterodimeric (α/β) integrin adhesion receptors, can be further divided into F1, F2, and F3 subdomains, with F3 specifically interacting with integrin β cytoplasmic tails (CTs) 2,3,4 . Because of its capacity to bind both integrin and actin, talin has long been recognized as a mechanical linker between the ECM and actin cytoskeleton to regulate cell adhesion and morphology 5 . However, about a decade ago, it was found that the binding of talin-FERM to integrin β CTs can also promote the conversion of integrins from a low-affinity into a high-affinity ligand-binding state, a dynamic process termed integrin inside-out signaling or integrin activation 6,7,8,9 . This finding suggests that the talin-FERM/integrin interaction plays a multifunctional role in promoting integrin activation as well as integrin-actin coupling during dynamic cell adhesion processes 10,11,12 . Structural and biochemical analyses have shown that the talin-FERM binding to integrin β CT triggers the separation of a key integrin α/β CT clasp, thus facilitating the global conformational change and activation of the receptor 7,8,13 . Stretches of positively charged surfaces on F1, F2, and F3 were also found to be crucial for this process by targeting talin-FERM to the membrane and enhancing the talin-FERM/integrin β CT interaction 13,14,15,16,17 . However, how the membrane-targeting and integrin binding of talin-FERM are spatiotemporally regulated to control the dynamics of integrin adhesion is not understood. This issue is important, since uncontrolled talin activity may lead to dysfunction of integrins, which is linked to many human disorders, including thrombosis, stroke, bleeding, infections, and cancer metastasis 18 . Talin can adopt inactive and active states during dynamic cell adhesion events, but the underlying molecular basis for this transition is elusive 2 . Previous studies suggested that the inactive talin has an autoinhibited conformation, where a key integrin-binding site on talin-F3 is self-masked by a talin-R segment (talin-RS) 19,20 . Membrane lipid phosphotidylinositol-4,5-bisphosphate (PIP2) 19,21 , which is locally enriched by talin-recruited PIPKIγ kinase 22,23 , was shown to activate talin 19,21 and promote integrin-mediated cell adhesion 15,24,25,26,27,28 , but the detailed structural basis of this is not clear. The prevailing hypothesis is that PIP2 may sterically induce the conformational change of talin 2,15,16,19,21 , a mechanism widely employed in the known PIP2-mediated activation of proteins, including FERM domain proteins 29,30 .

Here we describe the first high-resolution crystal structure of the principal autoinhibitory unit of talin. The structure reveals an unexpected dual inhibitory topology in which talin-RS not only sterically masks the integrin-binding site on talin-FERM via one large interface, but also electrostatically hinders the membrane-targeting of talin-FERM via another extensively negatively charged surface. Such a topology is consistent with the biochemical and functional data, but strikingly different from a previous model 20 . The topology further suggests a “pull-push” mechanism for talin activation by membrane enriched with PIP2, which differs distinctly from the classic steric-clash mechanism for the conventional PIP2-mediated activation of FERM proteins activation. We verify this “pull-push” mechanism by a combination of nuclear magnetic resonance (NMR), biochemical, and functional experiments. Our data lay down a new foundation for understanding talin inhibition and activation through novel regulatory mechanisms. They also provide significant insight into how the changes of the membrane surface promote an on/off switch of a cytosolic protein to control the receptor transmembrane signaling – an emerging area in signal transduction that is poorly explored 30 .


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