Review Special Issues

The Cerebellum and Premenstrual Dysphoric Disorder

  • Received: 30 April 2014 Accepted: 07 July 2014 Published: 30 July 2014
  • The cerebellum constitutes ten percent of brain volume and contains the majority of brain neurons. Although it was historically viewed primarily as processing motoric computations, current evidence supports a more comprehensive role, where cerebro-cerebellar feedback loops also modulate various forms of cognitive and affective processing. Here we present evidence for a role of the cerebellum in premenstrual dysphoric disorder (PMDD), which is characterized by severe negative mood symptoms during the luteal phase of the menstrual cycle. Although a link between menstruation and cyclical dysphoria has long been recognized, neuroscientific investigations of this common disorder have only recently been explored. This article reviews functional and structural brain imaging studies of PMDD and the similar but less well defined condition of premenstrual syndrome (PMS). The most consistent findings are that women with premenstrual dysphoria exhibit greater relative activity than other women in the dorsolateral prefrontal cortex and posterior lobules VI and VII of the neocerebellum. Since both brain areas have been implicated in emotional processing and mood disorders, working memory and executive functions, this greater activity probably represents coactivation within a cerebro-cerebellar feedback loop regulating emotional and cognitive processing. Some of the evidence suggests that increased activity within this circuit may preserve cerebellar structure during aging, and possible mechanisms and implications of this finding are discussed.

    Citation: Andrea J. Rapkin, Steven M. Berman, Edythe D. London. The Cerebellum and Premenstrual Dysphoric Disorder[J]. AIMS Neuroscience, 2014, 1(2): 120-141. doi: 10.3934/Neuroscience.2014.2.120

    Related Papers:

    [1] Muhammad Suhail Aslam, Mohammad Showkat Rahim Chowdhury, Liliana Guran, Isra Manzoor, Thabet Abdeljawad, Dania Santina, Nabil Mlaiki . Complex-valued double controlled metric like spaces with applications to fixed point theorems and Fredholm type integral equations. AIMS Mathematics, 2023, 8(2): 4944-4963. doi: 10.3934/math.2023247
    [2] Afrah Ahmad Noman Abdou . Chatterjea type theorems for complex valued extended $ b $-metric spaces with applications. AIMS Mathematics, 2023, 8(8): 19142-19160. doi: 10.3934/math.2023977
    [3] Asifa Tassaddiq, Jamshaid Ahmad, Abdullah Eqal Al-Mazrooei, Durdana Lateef, Farha Lakhani . On common fixed point results in bicomplex valued metric spaces with application. AIMS Mathematics, 2023, 8(3): 5522-5539. doi: 10.3934/math.2023278
    [4] Jamshaid Ahmad, Abdullah Eqal Al-Mazrooei, Hassen Aydi, Manuel De La Sen . Rational contractions on complex-valued extended $ b $-metric spaces and an application. AIMS Mathematics, 2023, 8(2): 3338-3352. doi: 10.3934/math.2023172
    [5] Afrah Ahmad Noman Abdou . A fixed point approach to predator-prey dynamics via nonlinear mixed Volterra–Fredholm integral equations in complex-valued suprametric spaces. AIMS Mathematics, 2025, 10(3): 6002-6024. doi: 10.3934/math.2025274
    [6] Saif Ur Rehman, Iqra Shamas, Shamoona Jabeen, Hassen Aydi, Manuel De La Sen . A novel approach of multi-valued contraction results on cone metric spaces with an application. AIMS Mathematics, 2023, 8(5): 12540-12558. doi: 10.3934/math.2023630
    [7] Ahmed Alamer, Faizan Ahmad Khan . Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems. AIMS Mathematics, 2024, 9(3): 6266-6280. doi: 10.3934/math.2024305
    [8] Khaled Berrah, Abdelkrim Aliouche, Taki eddine Oussaeif . Applications and theorem on common fixed point in complex valued b-metric space. AIMS Mathematics, 2019, 4(3): 1019-1033. doi: 10.3934/math.2019.3.1019
    [9] Sumaiya Tasneem Zubair, Kalpana Gopalan, Thabet Abdeljawad, Nabil Mlaiki . Novel fixed point technique to coupled system of nonlinear implicit fractional differential equations in complex valued fuzzy rectangular $ b $-metric spaces. AIMS Mathematics, 2022, 7(6): 10867-10891. doi: 10.3934/math.2022608
    [10] Fatima M. Azmi, Nabil Mlaiki, Salma Haque, Wasfi Shatanawi . Complex-valued controlled rectangular metric type spaces and application to linear systems. AIMS Mathematics, 2023, 8(7): 16584-16598. doi: 10.3934/math.2023848
  • The cerebellum constitutes ten percent of brain volume and contains the majority of brain neurons. Although it was historically viewed primarily as processing motoric computations, current evidence supports a more comprehensive role, where cerebro-cerebellar feedback loops also modulate various forms of cognitive and affective processing. Here we present evidence for a role of the cerebellum in premenstrual dysphoric disorder (PMDD), which is characterized by severe negative mood symptoms during the luteal phase of the menstrual cycle. Although a link between menstruation and cyclical dysphoria has long been recognized, neuroscientific investigations of this common disorder have only recently been explored. This article reviews functional and structural brain imaging studies of PMDD and the similar but less well defined condition of premenstrual syndrome (PMS). The most consistent findings are that women with premenstrual dysphoria exhibit greater relative activity than other women in the dorsolateral prefrontal cortex and posterior lobules VI and VII of the neocerebellum. Since both brain areas have been implicated in emotional processing and mood disorders, working memory and executive functions, this greater activity probably represents coactivation within a cerebro-cerebellar feedback loop regulating emotional and cognitive processing. Some of the evidence suggests that increased activity within this circuit may preserve cerebellar structure during aging, and possible mechanisms and implications of this finding are discussed.



    Contemporary theoretical mechanics and engineering technology are mainly concerned with either the isotropic or the anisotropic behavior of elastic materials and structures [1,2]. Specifically, the classical theory of linear elasticity is considered to be an indivisible branch of the more general field of non-linear elasticity [3], wherein the mathematical description of the physical quantities, associated with such media [4,5], incorporates the fundamental characteristics of the corresponding physical problems. Therein, the relationship between the strain and the stress of any solid body, subjected to external forces, is considered to be linear. On the other hand, isotropic elasticity, being set in conjunction with problems of elasticity in the linear regime with [6] or without [7,8] body forces present, is an extensively developed area of continuum mechanics, embodying solid analysis that requires both analytical and numerical attention [9]. Even though the most attractive area for developing new methodologies appears to be the anisotropic linear elasticity [10,11], it is evident that there still exist open problems in the isotropic spectrum. In fact, the continuing interest in elaborating with such kind of aspects is still effective and the necessity in producing purely analytical solutions towards this direction, ready to accept proper numerical handling, stands in the frontline of the current research. Indeed, such a complete and comprehensive survey in linear isotropic elasticity brings insight to new elements of applied mathematics, which lead to the next step of studying the wave propagation [12,13] and, in general, the theory of scattering [14] in elastic materials.

    Linear isotropic behavior of elastic media has an inherent mathematical interest due to the fact that even though the related theory is much simplified, many applications can accept the isotropic character without loss of robustness. In view of this aspect, this work is involved with the production of closed-type analytical formulae for the determination of the fundamental field in solid mechanics, i.e. the displacement. Doing so, we initially utilize the Hook's law, which provides the second order stress tensor in terms of the strain dyadic and the stiffness tetratic, while, in the sequel, we invoke the result into the Newton's law so as to obtain the generalized constitutive equation in elasticity. Thereafter, our purpose is twofold, that is we give special attention to elastostatics, neglecting the temporal derivatives and we exclude any external body forces, since any cause of disturbance can be entered into the boundary conditions of the particular physical problem. Moreover, we assume the specific expression for the stiffness tensor, which incorporates the appropriate components that inherit the isotropic character of the material. That way, we arrive at the known Navier partial differential equation for the displacement field, which accepts an analytical compact solution in the form of a partial differential operator, acting on harmonic functions with vector and scalar character, namely the Papkovich differential representation. This can be derived directly from the Naghdi-Hsu general solution [15] and since the spherical harmonics form a complete system, it also provides complete representations for the elastostatic fields. Hence, Papkovich representation of the solution of Navier equation is complete, that is any solution can be expressed in this form. The utility of such solutions is significant, considering the fact that they provide handy analytical expressions, for instance let us refer to a series of articles [16,17,18], which deal with elastic wave scattering at low frequencies around ellipsoidal solid bodies or cavities, using the Papkovich representation. Otherwise, the representation theory could be applied to inverse elastic scattering [19] or to more complicated mixed-type boundary value problems [20] in linear elasticity. Finally, let us not ignore a major advantage of Papkovich representation, which can offers us a certain degree of freedom, since the potentials needed to describe the field itself are redundant, thus the extra terms can be used conveniently, e.g. to compensate the force term in the case wherein it is included into the Navier equation.

    It is the purpose of this research to provide progress in this interesting mathematical aspect, hence, using the spherical coordinate system [21], we obtain connection formulae, which relate the spherical harmonics that lead, through the Papkovich representation, to the displacement in linear isotropic elasticity. In other words, we calculate the displacement field, generated by the harmonic eigenfunctions [22], through the Papkovich representation, and then we provide the necessary analytical background so as to solve the inverse problem of identifying those harmonic eigenfunctions, which generate the same displacement fields. Henceforth, in the aim of solving sufficiently general interior and exterior problems, we use the internal and external solid spherical harmonic eigenfunctions in their complex form [22]. This way, we demonstrate the importance of attaining such kind of ready-to-use mathematical tools, by showing some interesting example problems inside and outside a sphere. The background of this mathematical procedure comes from the low-frequency scattering in elasticity [23] and shows how one can obtain a solution basis of a finite dimensional subspace. This reduces the calculation of the solution to the calculation of a finite number of scalar coefficients. However, our work leads to building blocks for constructing the solution of any relative problem.

    Notwithstanding the existence of adequate and simultaneously convenient computational codes in solving problems in elasticity, we should not overlook the fact that pure analytical techniques are the backbone of numerical analysis. Hence, the important advantage of the performed mathematical analysis is based upon its ability to understand the physical background and to verify the credibility of numerical methods or other more sophisticated analytical models. On the other hand, the idea of building any solution from ready-to-use eigensolutions goes back to the classic references of Rayleigh [24], Kelvin [25], Maxwell [26], Sommerfeld [27] and Neuber [28]. In the present work, we tried to extend this idea to the theory of elasticity, wherein the ample literature survey of the fundamental classical references [29,30,31,32,33,34,35,36,37,38,39,40] verifies the necessity of such kind of analysis. Finally, by virtue of the representation theory, it is obvious that spherical geometry approximates sufficiently well most basic problems in linear isotropic elasticity. Nevertheless, the extension to spheroidal, ellipsoidal or even more complicated geometries [21,41] provides a challenging area for future investigation.

    Let us introduce an arbitrarily defined smooth, either bounded or unbounded, three-dimensional elastic domain $\Omega \left({{\mathbb{R}^3}} \right)$, which could be designated as interior or interior as the case may be. Then, each field within $\Omega \left({{\mathbb{R}^3}} \right)$ is written in terms of its position vector ${\bf{r}} = {x_1}{\mathit{\boldsymbol{\hat x}}_1} + {x_2}{\mathit{\boldsymbol{\hat x}}_2} + {x_3}{\mathit{\boldsymbol{\hat x}}_3}$, expressed via the Cartesian basis ${\mathit{\boldsymbol{\hat x}}_j}$, $j = 1, 2, 3$ in Cartesian coordinates $\left({{x_1}, {x_2}, {x_3}} \right)$. On the other hand, the current investigation excludes the dependence on time, since we operate according to the steady state status of the particular situation, while the necessary fundamental information, which are adequate for this work can be found collected in [11].

    The physical interpretation of linear isotropic elasticity is involved with the displacement field ${\bf{u}}$, which comprises the measure of deformation of an elastic material. By means of the gradient $ \nabla $ and the Laplacian $ \Delta $ operators, the displacement satisfies the well-known Navier equation in the presence of body forces, i.e.

    $ \mu {\mkern 1mu} \Delta {\bf{u}}\left( {\bf{r}} \right) + \left( {\lambda + \mu } \right)\nabla \left[ {\nabla \cdot {\bf{u}}\left( {\bf{r}} \right)} \right] + {\bf{f}}\left( {\bf{r}} \right) = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (1)

    wherein ${\bf{f}}$ is an external applied force that renders expression (1) non-homogeneous, while $ \lambda, \mu \in \mathbb{R} $ are the elastic parameters of the isotropic theory, being known as the Lamé constants. Papkovich proposed a differential representation of the solution for the homogeneous Navier equation [15] with no body forces, by considering ${\bf{f }}{\rm{ = }}{\bf{0}}$ within (1), which expresses the displacement field in differential form, in terms of two harmonic functions, one vector $ {\bf{A}} $ and one scalar B, that is

    $ {\bf{u}}\left( {\bf{r}} \right) = {\bf{A}}\left( {\bf{r}} \right) - \frac{{\lambda + \mu }}{{2\left( {\lambda + 2\mu } \right)}}\nabla \left[ {{\bf{r}} \cdot {\bf{A}}\left( {\bf{r}} \right) + {\rm{B}}\left( {\bf{r}} \right)} \right], \;{\rm{where}}\;\Delta {\bf{A}}\left( {\bf{r}} \right) = {\bf{0}}, \Delta {\rm{B}}\left( {\bf{r}} \right) = 0\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (2)

    which is known to be complete. It is not hard to prove that (2) satisfies (1), bearing in mind the vector identity $ \Delta \left({{\bf{r}} \cdot {\bf{A}}} \right) = \Delta {\bf{r}} \cdot {\bf{A}} + {\bf{r}} \cdot \Delta {\bf{A}} + 2{\left({\nabla \otimes {\bf{r}}} \right)^{\rm{T}}}:\left({\nabla \otimes {\bf{A}}} \right) = 2{{\bf{\tilde I}}^{\rm{T}}}:\left({\nabla \otimes {\bf{A}}} \right) = 2{\bf{\tilde I}}:\left({\nabla \otimes {\bf{A}}} \right) = 2\nabla \cdot {\bf{A}}$ and the interchange $ \Delta \nabla = \nabla \Delta $, since it readily holds that $\Delta {\bf{r}} = \Delta {\bf{A}} = {\bf{0}}$ and $\Delta {\rm{B}} = 0$, noting that $ {{\bf{\tilde I}}}$ is the unit dyadic, "${\rm T}$" denotes transposition, "$ \otimes $" refers to the tensor product and "$:$" stands for the double inner product. The general differential solution (2) provides a powerful analytical tool for solving the homogeneous and time-independent linearized equation of classical dynamic elasticity.

    At this point we have to mention that even though representation (2) is complete, it is not unique, since we have four potentials (three components of the vector $ {\bf{A}} $ and one component, which is the scalar B) to determine the three-dimensional vector displacement field. Actually, this is more than an advantage rather than a disadvantage, considering the fact that we are equipped with certain degrees of freedom to deploy conveniently the proper manner, which depends on the boundary value problem at hand. On the other end, this flexibility of the representation theory could be used to cancel the difficulty when the source term $ {\bf{f}} $ is not absent (see equation (1) for instance). More precisely, in the same sense, we could use a similar differential solution like (2), in order to obtain a general solution for the non-homogeneous Navier equation (1). In fact, we can keep the same form for the displacement field, potential $ {\bf{A}} $ could be harmonic, but then potential B should satisfy the mixed-type equation $ \Delta \left({\nabla {\rm{B}}} \right) \equiv \nabla \left({\Delta {\rm{B}}} \right) = 2{\bf{f}}/\left({\lambda + \mu } \right)$, in order for (1) to be satisfied. However, obtaining B is quite a difficult task.

    This section includes the main results of our work, which are focused on the direct connection between the homogeneous Navier equation and the harmonic kernels of the implicated potentials $ {\bf{A}} $ and B, using the spherical geometry. Before we proceed to the analysis in spherical coordinates, we take advantage of the already discussed flexibility of the differential general solution (2), thus and without loss of generality, we may suppose that the vector harmonic potential $ {\bf{A}} $ is adequate enough to provide us with a complete solution for the displacement field $ {\bf{u}} $ via (2). Consequently, assuming that $ {\rm{B}} \equiv 0 $, the representation (2) becomes

    ${\bf{u}}\left( {\bf{r}} \right) = {\bf{A}}\left( {\bf{r}} \right) - \frac{{\lambda + \mu }}{{2\left( {\lambda + 2\mu } \right)}}\nabla \left[ {{\bf{r}} \cdot {\bf{A}}\left( {\bf{r}} \right)} \right], {\rm{where}}\;\Delta {\bf{A}}\left( {\bf{r}} \right) = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (3)

    which constitutes a complete solution for the Navier equation (1) if ${\bf{f}}{\rm{ = }}{\bf{0}}$. Taking the curl on both sides of (3) and defining a new arbitrary function $ \Phi $, we have

    $ \nabla \times {\bf{u}}\left( {\bf{r}} \right) = \nabla \times {\bf{A}}\left( {\bf{r}} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Rightarrow {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\bf{u}}\left( {\bf{r}} \right) - {\bf{A}}\left( {\bf{r}} \right) = \nabla \Phi \left( {\bf{r}} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \\ \Rightarrow {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\bf{A}}\left( {\bf{r}} \right) = {\bf{u}}\left( {\bf{r}} \right) - \nabla \Phi \left( {\bf{r}} \right)\;{\rm{for}} \;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right). $ (4)

    Next, we apply the div on (3), which leads us to

    $ u(r)=A(r)λ+μ2(λ+2μ)Δ[rA(r)]=A(r)λ+μλ+2μA(r)=μλ+2μA(r)A(r)=λ+2μμu(r)forrΩ(R3).
    $
    (5)

    Next, we take the divergence of relation (4) and use simultaneously the outcome (5), to arrive at

    $ \nabla \cdot {\bf{A}}\left( {\bf{r}} \right) = \nabla \cdot {\bf{u}}\left( {\bf{r}} \right) - \Delta \Phi \left( {\bf{r}} \right){\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Rightarrow {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \Delta \Phi \left( {\bf{r}} \right) = - \frac{{\lambda + \mu }}{\mu }\nabla \cdot {\bf{u}}\left( {\bf{r}} \right)\;{\rm{for}}\; {\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (6)

    which is a Poisson equation with respect to the unknown function $ \Phi $, whose solution is the result of implying the fundamental solution of the Laplace's operator [20], obtaining

    $ \Phi \left( {\bf{r}} \right) = \frac{{\lambda + \mu }}{{4\pi \mu }}\iiint\limits_{\Omega \left( {{\mathbb{R}^3}} \right)} {\frac{{{\nabla _{{\bf{r}}'}} \cdot {\bf{u}}\left( {{\bf{r}}'} \right)}}{{\left| {{\bf{r}} - {\bf{r}}'} \right|}}{\rm{d}}V\left( {{\bf{r}}'} \right)} \;{\rm{ for}}\; {\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right). $ (7)

    So, if we insert $ \Phi $ from (7) into (4), we recover potential $ A $ as

    ${\bf{A}}\left( {\bf{r}} \right) = {\bf{u}}\left( {\bf{r}} \right) - \frac{{\lambda + \mu }}{{4\pi \mu }}\nabla \iiint\limits_{\Omega \left( {{\mathbb{R}^3}} \right)} \frac{{{\nabla _{{\bf{r'}}}} \cdot {\bf{u}}\left( {{\bf{r'}}} \right)}}{{\left| {{\bf{r}} - {\bf{r'}}} \right|}}{\rm{d}}V\left( {{\bf{r'}}} \right) \;{\rm{ for}}\; {\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right). $ (8)

    Therefore, given the displacement $ {\bf{u}} $, the harmonic potential $ {\bf{A}} $ is given by (8). Recapitulating the above reasoning, we calculate the displacement field $ {\bf{u}} $, generated by the harmonic function $ {\bf{A}} $ through the Papkovich representation (3) and then we face the inverse problem of determining this harmonic function $ {\bf{A}} $, which generates the displacement field $ {\bf{u}} $, via the integro-differential formula (8). This procedure is also invertible, in the sense that we can start with solutions of the form (8) and recover the displacement field via the Papkovich form (3). Both ways lead to the same result, independently of the coordinate system.

    Aiming to approach applications that require an easily amenable expression for the displacement, we focus our attention to the spherical geometry [19], so in terms of the spherical coordinate system

    $ {x_1} = r\sin \theta \cos \varphi , {x_2} = r\sin \theta \sin \varphi \;{\rm{ and}}\; {x_3} = r\cos \theta $ (9)

    for $ 0 \leqslant r < + \infty $, $ 0 \leqslant \theta \leqslant \pi $ and $ 0 \leqslant \varphi < 2\pi $, the differential operators appearing into the Papkovich representation assume the forms

    $ \nabla = {\bf{\hat r}}\frac{\partial }{{\partial r}} + \frac{{{\bf{ \pmb{\mathsf{\hat θ}} }}}}{r}\frac{\partial }{{\partial \theta }} + \frac{{{\bf{ \pmb{\mathsf{\hat φ}} }}}}{{r\sin \theta }}\frac{\partial }{{\partial \varphi }}\;{\rm{ and}}\; \Delta = \frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}\frac{\partial }{{\partial r}}} \right) + \frac{1}{{{r^2}\sin \theta }}\frac{\partial }{{\partial \theta }}\left( {\sin \theta \frac{\partial }{{\partial \theta }}} \right) + \frac{1}{{{r^2}{{\sin }^2}\theta }}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} , $ (10)

    where $ {\bf{\hat r}}$, $ {{\bf{ \pmb{\mathsf{\hat θ}} }}} $, $ {{\bf{ \pmb{\mathsf{\hat φ}} }}}$ (see [19]) denote the coordinate vectors of the spherical system with position vector $ {\bf{r}} = r\, {\bf{\hat r}} $ and unit dyadic $ {\bf{\tilde I}} = {\bf{\hat r}} \otimes {\bf{\hat r}} + {\bf{ \pmb{\mathsf{\hat θ}} }} \otimes {\bf{ \pmb{\mathsf{\hat θ}} }} + {\bf{ \pmb{\mathsf{\hat φ}} }} \otimes {\bf{ \pmb{\mathsf{\hat φ}} }} $. For every value of the nature $ n \geqslant 0 $, there exist $ \left({2n + 1} \right) $ linearly independent complex spherical surface harmonics $ Y_n^m $ of degree $ n $ and of order $ \left| m \right| \leqslant n $ [20], given in the orthonormalized form by

    $ Y_n^m\left( {{\bf{\hat r}}} \right) = \sqrt {\frac{{2n + 1}}{{4\pi }}\frac{{\left( {n - \left| m \right|} \right)!}}{{\left( {n + \left| m \right|} \right)!}}} {{P}}_n^{\left| m \right|}\left( {\cos \theta } \right){e^{i\, m\, \varphi }} \;{\rm{for}}\; {{\bf{\hat r}}} = \left( {\theta , \varphi } \right) \in \left[ {0, \pi } \right] \times \left[ {0, 2\pi } \right) $ (11)

    and with inner product

    $ \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_{{S^2}} Y_n^m\left( {{\bf{\hat r}}} \right)\bar Y_n^m\left( {{\bf{\hat r}}} \right){\rm{d}}s\left( {{\bf{\hat r}}} \right) = 1 \;{\rm{for}}\;{\bf{\hat r}} = \left( {\theta , \varphi } \right) \in \left[ {0, \pi } \right] \times \left[ {0, 2\pi } \right), $ (12)

    wherein $ {{P}}_n^{\left| m \right|} $ are the associated Legendre functions of the first kind [20], $ \bar Y_n^m $ denote the complex conjugate surface spherical harmonics and $ {S^2} $ is the unit sphere in $ {\mathbb{R}^3} $. Thereafter, we introduce the interior $ u_{n, {\rm{in}}}^m $ (regular as $r \to {0^ + }$) and the exterior $ u_{n, {\rm{ex}}}^m $ (regular as $r \to + \infty $), solid spherical harmonic eigenfunctions for every $ n = 0, 1, 2, ... $ and $ m = - n, ..., - 1, 0, 1, ..., n $, which are given by the expression

    $u_{n, {\rm{in}}}^m\left( {\bf{r}} \right) = {r^n}Y_n^m\left( {{\bf{\hat r}}} \right)\;{\rm{and}}\;u_{n, {\rm{ex}}}^m\left( {\bf{r}} \right) = {r^{ - \left( {n + 1} \right)}}Y_n^m\left( {{\bf{\hat r}}} \right)\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (13)

    respectively. Relationships (13) comprise a complete set of eigenfunctions for harmonic functions and belong to the kernel space of the Laplace's operator from (10), i.e. $ \Delta u_{n, {\rm{in}}}^m = 0 $ and $ \Delta u_{n, {\rm{ex}}}^m = 0 $ for $ n \geqslant 0 $ and $ \left| m \right| \leqslant n $, while they are obtained once the classical method of separation of variables [21,22] is applied.

    Adopting the above mathematical analysis, the harmonic function $ A $ in differential representation (3) admits series expansion in terms of functions (13), i.e.

    $ {\bf{A}}\left( {\bf{r}} \right) = \sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {\left[ {{\bf{c}}_{n, {\rm{in}}}^mu_{n, {\rm{in}}}^m\left( r \right) + {\bf{c}}_{n, {\rm{ex}}}^mu_{n, {\rm{ex}}}^m\left( r \right)} \right]} } \;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (14)

    where

    $ cmn,in=cm,1n,inˆx1+cm,2n,inˆx2+cm,3n,inˆx3andcmn,ex=cm,1n,exˆx1+cm,2n,exˆx2+cm,3n,exˆx3withn0and|m|n
    $
    (15)

    are arbitrary constant coefficients. Expansion (14) expresses the completeness of the interior and the exterior solid spherical harmonics. Consequently, substituting the potential (14) into the general solution (3), we obtain

    $ {\bf{u}}\left( {\bf{r}} \right) = \frac{1}{{2\left( {\lambda + 2\mu } \right)}}\sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {\left[ {\left( {\lambda + 3\mu } \right){\bf{c}}_{n, {\rm{in}}}^mu_{n, {\rm{in}}}^m\left( {\bf{r}} \right) - \left( {\lambda + \mu } \right)\left( {{\bf{c}}_{n, {\rm{in}}}^m \cdot {\bf{r}}} \right)\nabla u_{n, {\rm{in}}}^m\left( {\bf{r}} \right)} \right.} } \\+ \left. {\left( {\lambda + 3\mu } \right){\bf{c}}_{n, {\rm{ex}}}^mu_{n, {\rm{ex}}}^m\left( {\bf{r}} \right) - \left( {\lambda + \mu } \right)\left( {{\bf{c}}_{n, {\rm{ex}}}^m \cdot {\bf{r}}} \right)\nabla u_{n, {\rm{ex}}}^m\left( {\bf{r}} \right)} \right]\;{\rm{for}}\; {\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (16)

    where we used the trivial differential identities $ \nabla \left({{\bf{r}} \cdot {\bf{A}}} \right) = {\bf{A}} + \left({\nabla \otimes {\bf{A}}} \right) \cdot {\bf{r}} $ (note that $ \nabla \otimes {\bf{r}} = {\bf{\tilde I}} $) and $ \nabla \left({{\bf{c}}_{n, {\rm{y}}}^mu_{n, {\rm{y}}}^m} \right) = \nabla u_{n, {\rm{y}}}^m \otimes {\bf{c}}_{n, {\rm{y}}}^m $ for $ {\rm{y}} = {\rm{in}}, {\mkern 1mu} {\mkern 1mu} {\rm{ex}}$, where $ n \geqslant 0 $ and $ \left| m \right| \leqslant n $. Nevertheless, expression (16) needs further elaboration in order for its processing to be feasible. To this direction, we analytically work as follows. Since the vector character of the harmonic function $ {\bf{A}} $ is reflected upon the constant coefficients, which are written in Cartesian coordinates, we are obliged to work in the Cartesian system. This is attainable but requires the expression of the displacement field $ {\bf{u}} $ in Cartesian coordinates involving constants and spherical surface harmonics. Therefore, we are able to transfer the connection between $ {\bf{A}} $ and $ {\bf{u}} $ to the corresponding connection via the constant coefficients. In order to do that, it is necessary to express the terms $ \nabla u_{n, {\rm{in}}}^m $ and $ \nabla u_{n, {\rm{ex}}}^m $ for $ n \geqslant 0 $ and $ \left| m \right| \leqslant n $ as a function of spherical surface harmonics in Cartesian coordinates. This is possible, since these terms belong to the subspace that is produced by the spherical surface harmonics, while this task requires certain steps.

    Hence, in the interest of making this work complete and independent, we provide recurrence relations for the associated Legendre functions [20], which, by definition of the conveniently chosen variable $ x = \cos \theta \in \left[{- 1, 1} \right] $, since $ \theta \in \left[{0, \pi } \right] $, they are furnished by the Rodrigues formula

    $ {\rm{P}}_n^{\left| m \right|}\left( x \right) = {\left( {1 - {x^2}} \right)^{\left| m \right|/2}}\frac{1}{{{2^n}n!}}\frac{{{{\rm{d}}^{\left| m \right| + n}}}}{{{\rm{d}}{x^{\left| m \right| + n}}}}{\left( {{x^2} - 1} \right)^n}\;{\rm{for}}\;n \ge 0\;{\rm{and}}\;\left| m \right| \le n, $ (17)

    where obviously $ {\rm{P}}_n^{\left| m \right|}\left(x \right) = 0 $ if $ \left| m \right| > n $. Thus, the associated Legendre functions of the first kind satisfy

    $ \left( {2n + 1} \right)x {\rm{P}}_n^{\left| m \right|}\left( x \right) = \left( {n + \left| m \right|} \right){\rm{P}}_{n - 1}^{\left| m \right|}\left( x \right) + \left( {n - \left| m \right| + 1} \right) {\rm{P}}_{n + 1}^{\left| m \right|} \left( x \right), $ (18)
    $ \left( {2n + 1} \right)\sqrt {1 - {x^2}} {\rm{P}}_n^{\left| m \right|}\left( x \right) = {\rm{P}}_{n + 1}^{\left| m \right| + 1}\left( x \right) - {\rm{P}}_{n - 1}^{\left| m \right| + 1}\left( x \right)\, \\ = \left( {n + \left| m \right|} \right)\left( {n + \left| m \right| - 1} \right){\rm{P}}_{n - 1}^{\left| m \right| - 1}\left( x \right) - \left( {n - \left| m \right| + 1} \right)\left( {n - \left| m \right| + 2} \right){\rm{P}}_{n + 1}^{\left| m \right| - 1}\left( x \right) , $ (19)
    $ \frac{{2mx}}{{\sqrt {1 - {x^2}} }} {\rm{P}}_n^{\left| m \right|}\left( x \right) = {\rm{P}}_n^{\left| m \right| + 1}\left( x \right) + \left( {n + \left| m \right|} \right)\left( {n - \left| m \right| + 1} \right){\rm{P}}_n^{\left| m \right| - 1}\left( x \right) $ (20)

    and the first derivative relation

    $ \frac{{\rm{d}}}{{{\rm{d}}x}} {\rm{P}}_n^{\left| m \right|}\left( x \right) = {\left( {1 - {x^2}} \right)^{ - 1/2}}\left[ { - \left| m \right|x{{\left( {1 - {x^2}} \right)}^{ - 1/2}} {\rm{P}}_n^{\left| m \right|}\left( x \right) + {\rm{P}}_n^{\left| m \right| + 1}\left( x \right)} \right] , $ (21)

    all relationships (18)–(21) being provided for every value of $ n \geqslant 0 $ and $ \left| m \right| \leqslant n $. On the other hand, the classical trigonometric functions imply the trivial identities

    $ \sin \varphi \sin \left| m \right|\varphi = \frac{1}{2}\left[ {\cos \left( {\left| m \right| - 1} \right)\varphi - \cos \left( {\left| m \right| + 1} \right)\varphi } \right] , $ (22)
    $ \cos \varphi \cos \left| m \right|\varphi = \frac{1}{2}\left[ {\cos \left( {\left| m \right| - 1} \right)\varphi + \cos \left( {\left| m \right| + 1} \right)\varphi } \right] , $ (23)
    $ \cos \varphi \sin \left| m \right|\varphi = \frac{1}{2}\left[ {\sin \left( {\left| m \right| + 1} \right)\varphi + \sin \left( {\left| m \right| - 1} \right)\varphi } \right] , $ (24)
    $ \sin \varphi \cos \left| m \right|\varphi = \frac{1}{2}\left[ {\sin \left( {\left| m \right| + 1} \right)\varphi - \sin \left( {\left| m \right| - 1} \right)\varphi } \right] $ (25)

    for any $ \left| m \right| \leqslant n $ ($ n \geqslant 0 $). The derivation process to obtain handy expressions for $ \nabla u_{n, {\rm{in}}}^m $ and $ \nabla u_{n, {\rm{ex}}}^m $ with $ n \geqslant 0 $ and $ \left| m \right| \leqslant n $ involves long and tedious calculations, however it is useful to provide the basic steps that gives us the necessary tools to accomplish this task. Under this aim, the procedure we follow is based on the utilization of the gradient operator within (10) in terms of the decomposition of the orthonormal spherical basis to the Cartesian one via the formulae

    $ {\bf{\hat r}} = \sin \theta \cos \varphi {\mathit{\boldsymbol{\hat x}}_1} + \sin \theta \sin \varphi {\mathit{\boldsymbol{\hat x}}_2} + \cos \theta {\mathit{\boldsymbol{\hat x}}_3}, $ (26)
    $ {\bf{ \pmb{\mathsf{\hat θ}} }} = \cos \theta \cos \varphi {\mathit{\boldsymbol{\hat x}}_1} + \cos \theta \sin \varphi {\mathit{\boldsymbol{\hat x}}_2} - \sin \theta {\mathit{\boldsymbol{\hat x}}_3}, $ (27)
    ${\bf{ \pmb{\mathsf{\hat φ}} }} = - \sin \varphi {\mathit{\boldsymbol{\hat x}}_1} + \cos \varphi {\mathit{\boldsymbol{\hat x}}_2}. $ (28)

    Thus, substituting above (26)–(28) into the spherical gradient operator from (10), we initially invoke the surface spherical harmonics from (11) into the corresponding solid spherical harmonics (13), next we write $ {e^{{\rm{i}}m\varphi }} $ in the form $ {e^{ \pm {\rm{i}}\left| m \right|\varphi }} = \cos \left| m \right|\varphi \pm {\rm{i}}\sin \left| m \right|\varphi $ for every $ \left| m \right| \leqslant n $ ($ n \geqslant 0 $) and, finally, by virtue of $ \cos \theta = x $ and $ \sin \theta = \sqrt {1 - {x^2}} $ with $ x \in \left[{- 1, 1} \right] $, we use recurrence relations (17)–(21) for the associated Legendre functions of the first kind, as well as relationships (22)–(25) for the trigonometric functions, so as to perform our analysis. Consequently, combining properly the relative terms to reproduce surface spherical harmonics, we derive for the interior solid spherical harmonic eigenfunctions $ u_{n, {\rm{in}}}^m $ in $\Omega \left({{\mathbb{R}^3}} \right)$ the expressions

    $\nabla u_{n, {\rm{in}}}^m\left( {\bf{r}} \right) = \frac{1}{2}\kappa _n^m\left[ {\left( {n + \left| m \right|} \right)\left( {n + \left| m \right| - 1} \right)\frac{{Y_{n - 1}^{m - 1}\left( {{\bf{\hat r}}} \right)}}{{\kappa _{n - 1}^{m - 1}}} - \frac{{Y_{n - 1}^{m + 1}\left( {{\bf{\hat r}}} \right)}}{{\kappa _{n - 1}^{m + 1}}}} \right]{r^{n - 1}}{{\bf{\hat x}}_1}\\ \;\;\;+ \frac{{\rm{i}}}{2}\kappa _n^m\left[ {\left( {n + \left| m \right|} \right)\left( {n + \left| m \right| - 1} \right)\frac{{Y_{n - 1}^{m - 1}\left( {{\bf{\hat r}}} \right)}}{{\kappa _{n - 1}^{m - 1}}} + \frac{{Y_{n - 1}^{m + 1}\left( {{\bf{\hat r}}} \right)}}{{\kappa _{n - 1}^{m + 1}}}} \right]{r^{n - 1}}{{\bf{\hat x}}_2} \\ +κmn[(n+|m|)Ymn1(ˆr)κmn1+Ym+1n1(ˆr)κm+1n1]rn1ˆx3forn0andm=n,...,1,1,...,n
    $
    (29)

    and for the case $ m = 0 $

    $u0n,in(r)=12κ0n[Y1n1(ˆr)κ1n1Y1n1(ˆr)κ1n1]rn1ˆx1+i2κ0n[Y1n1(ˆr)κ1n1+Y1n1(ˆr)κ1n1]rn1ˆx2+κ0n[nY0n1(ˆr)κ0n1Y1n1(ˆr)κ1n1]rn1ˆx3forn0,
    $
    (30)

    while for the exterior solid spherical harmonic eigenfunctions $ u_{n, {\rm{ex}}}^m $ in $\Omega \left({{\mathbb{R}^3}} \right)$, we similarly receive the expressions

    $ umn,ex(r)=12κmn[(n|m|+1)(n|m|+2)Ym1n+1(ˆr)κm1n+1Ym+1n+1(ˆr)κm+1n+1]r(n+2)ˆx1+i2κmn[(n|m|+1)(n|m|+2)Ym1n+1(ˆr)κm1n+1+Ym+1n+1(ˆr)κm+1n+1]r(n+2)ˆx2κmn[(n|m|+1)Ymn+1(ˆr)κmn+1+Ym+1n+1(ˆr)κm+1n+1]r(n+2)ˆx3forn0andm=n,...,1,1,...,n
    $
    (31)

    and for the case $ m = 0 $

    $u0n,ex(r)=12κ0n[Y1n+1(ˆr)κ1n+1+Y1n+1(ˆr)κ1n+1]r(n+2)ˆx1+i2κ0n[Y1n+1(ˆr)κ1n+1Y1n+1(ˆr)κ1n+1]r(n+2)ˆx2κ0n[(n+1)Y0n+1(ˆr)κ0n+1]r(n+2)ˆx3forn0,
    $
    (32)

    where

    $ \kappa _n^m = \sqrt {\frac{{2n + 1}}{{4\pi }}\frac{{\left( {n - \left| m \right|} \right)!}}{{\left( {n + \left| m \right|} \right)!}}} \;{\rm{for}}\;n \ge 0\;{\rm{and}}\;\left| m \right| \le n $ (33)

    are the normalizing constants of the spherical surface harmonics. It is obvious, from the definition of the associated Legendre functions, that

    $Y_{ - n}^m\left( {{\bf{\hat r}}} \right) \equiv 0\;{\rm{for}}\;n \ge 0\;{\rm{and}}\;\left| m \right| \le n\;{\rm{with}}\;{\bf{\hat r}} = \left( {\theta , \varphi } \right) \in \left[ {0, \pi } \right] \times \left[ {0, 2\pi } \right), $ (34)

    while

    $ Y_n^m\left( {{\bf{\hat r}}} \right) \equiv 0\;{\rm{for}}\;n \ge 0\;{\rm{and}}\;\left| m \right| > n\;{\rm{with}}\;{\bf{\hat r}} = \left( {\theta , \varphi } \right) \in \left[ {0, \pi } \right] \times \left[ {0, 2\pi } \right), $ (35)

    rendering formulae (29)–(32) with (33) applicable for every value of $ n $ and $ m $.

    In order to validate the correctness and demonstrate the effectiveness of the produced formulae, we provide a simple example of evaluating $ \nabla u_{1, {\rm{in}}}^1 $, utilizing two different approaches, one with direct calculation and the other with the aid of expression (29) for $ n = 1 $ and $ m = 1 $, showing that both the two results coincide. Towards this direction, from (11) and (33) we obtain

    $ Y_1^1\left( {{\bf{\hat r}}} \right) = \kappa _1^1{\rm{P}}_1^1\left( {\cos \theta } \right) {e^{{\rm{i}}\;\varphi }} = \kappa _1^1\sin \theta {e^{{\rm{i}}\;\varphi }} \;{\rm{with}}\;\kappa _1^1 = \sqrt {\frac{3}{{8\pi }}} \;{\rm{for}}\;{\bf{\hat r}} = \left( {\theta , \varphi } \right) \in \left[ {0, \pi } \right] \times \left[ {0, 2\pi } \right), $ (36)

    since $ {\rm{P}}_1^1\left({\cos \theta } \right) = \sin \theta $, therefore (13) yields

    $u_{1, {\rm{in}}}^1\left( {\bf{r}} \right) = {r^1}Y_1^1\left( {{\bf{\hat r}}} \right) = \kappa _1^1r{\rm{P}}_1^1\left( {\cos \theta } \right) {e^{{\rm{i}}\;\varphi }} = \kappa _1^1r\sin \theta {e^{{\rm{i}}\;\varphi }} \;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right). $ (37)

    Thus, by direct action of the gradient operator from (10) on the interior harmonic (37) and in view of equations (26)–(28), it holds

    $ u11,in(r)=[ˆrr+ˆθˆθrθ+ˆφˆφrsinθφ](κ11rsinθeiφ)=κ11ˆrsinθeiφ+κ11ˆθˆθcosθeiφ+iκ11ˆφˆφeiφ
    \\ = \kappa _1^1\left( {\sin \theta \cos \varphi {\mathit{\boldsymbol{\widehat x}}_1} + \sin \theta \sin \varphi {\mathit{\boldsymbol{\widehat x}}_2} + \cos \theta {\mathit{\boldsymbol{\widehat x}}_3}} \right)\sin \theta {e^{{\rm{i}}\; \varphi }} \\+ \kappa _1^1\left( {\cos \theta \cos \varphi {\mathit{\boldsymbol{\widehat x}}_1} + \cos \theta \sin \varphi {\mathit{\boldsymbol{\widehat x}}_2} - \sin \theta {\mathit{\boldsymbol{\widehat x}}_3}} \right)\cos \theta {e^{{\rm{i}}\; \varphi }} \\ + {\rm{i}}\kappa _1^1 \left( { - \sin \varphi {\mathit{\boldsymbol{\widehat x}}_1} + \cos \varphi {\mathit{\boldsymbol{\widehat x}}_2}} \right){e^{{\rm{i}}\; \varphi }}\\ = \kappa _1^1\left( {\cos \varphi - i\sin \varphi } \right){\hat x_1}{e^{{\rm{i}}\; \varphi }} + \kappa _1^1\left( {\sin \varphi + i\cos \varphi } \right){\hat x_2}{e^{{\rm{i}}\; \varphi }} \;{\rm{for}}\; {\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right) $
    (38)

    or

    $\nabla u_{1, {\rm{in}}}^1\left( {\bf{r}} \right) = \kappa _1^1{\mathit{\boldsymbol{\hat x}}_1} + {\rm{i}}\kappa _1^1{\mathit{\boldsymbol{\hat x}}_2} = \kappa _1^1\left( {{{\mathit{\boldsymbol{\hat x}}}_1} + {\rm{i}}{\mkern 1mu} {{\mathit{\boldsymbol{\hat x}}}_2}} \right)\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right) $ (39)

    because $ {e^{{\rm{i}}\; \varphi }} = \cos \varphi + i\sin \varphi $. Otherwise, if we utilize the derived formula (29) for $ n = m = 1 $ and we take profit from the property (35), yielding $ Y_0^1 \equiv 0 $ and $ Y_0^2 \equiv 0 $, we have

    $\nabla u_{1, {\rm{in}}}^1\left( {\bf{r}} \right) = \frac{1}{2}\kappa _1^1\left[ {2\frac{{Y_0^0\left( {{\bf{\hat r}}} \right)}}{{\kappa _0^0}}} \right]{{\bf{\hat x}}_1} + \frac{{\rm{i}}}{2}\kappa _1^1\left[ {2\frac{{Y_0^0\left( {{\bf{\hat r}}} \right)}}{{\kappa _0^0}}} \right]{{\bf{\hat x}}_2}\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right) $ (40)

    or

    $ \nabla u_{1, {\rm{in}}}^1\left( {\bf{r}} \right) = \kappa _1^1{\mathit{\boldsymbol{\hat x}}_1} + {\rm{i}}\kappa _1^1{\mathit{\boldsymbol{\hat x}}_2} = \kappa _1^1\left( {{{\mathit{\boldsymbol{\hat x}}}_1} + {\rm{i}}{\mkern 1mu} {{\mathit{\boldsymbol{\hat x}}}_2}} \right)\;{\rm{for}}\;{\bf{r}} \in \Omega \left( {{\mathbb{R}^3}} \right), $ (41)

    since $ \kappa _0^0 = 1 $ and $ Y_0^0 = 1 $. Obviously, (39) and (41) are identical, not only validating the obtained expression (29), but also confirming that our approach is much more efficient and faster than the direct calculation of the gradient on solid spherical harmonics. It is evident that following a similar way, we can reproduce the subspace of any space of harmonic functions in the Cartesian basis, while a general procedure can be readily established.

    Concluding, the result (16) for the displacement field, via the Papkovich differential representation, can be now rewritten in a handy form, by substituting the outcomes (29)–(32) with (33)–(35) into (16), using the Cartesian expressions for the implicated constant coefficients (15) and the interior, as well as the exterior harmonic eigenfunctions (13). Even though the current theory is applied on spherical boundaries, it can be extended to any surface. For instance, we indicate as an example that in a scattering problem by an arbitrary compact body we can always consider a sphere surrounding the scatterer. In this case, utilizing Green's second identity [22] we can transfer any information from the boundary of the scatterer to the sphere including the scatterer. This way, we reduce the exterior problem to the spherical geometry, where the boundary conditions are globally introduced. After this, it is obvious that the displacement field is now easily accessible to be applied to any kind of boundary value problem that concerns the wide area of linear isotropic elasticity.

    The purpose of this section is to demonstrate the usefulness and the efficiency of the proposed analytical methodology, by invoking some special case problems in spherical geometry, which can be found in simple but quite important physical problems in linear isotropic elasticity. The specific choices of the examples come from specific boundary value problems appearing in the theory of low-frequency scattering [23]. In fact, the assumed elastic fields describe the leading low-frequency approximations of the incident excitation field. Consequently, the forthcoming examples come from real physical problems and they are not artificial.

    We consider a spherical body of radius $ a $, whose center coincides with the center of the Cartesian coordinate system. Therefore, the surrounding boundary $\partial \Omega \equiv S$ corresponds to the spherical variable at $ r = a $ for any $ \theta \in \left[{0, \pi } \right] $ and $ \varphi \in \left[{0, 2\pi } \right) $. This spherical surface separates the domain of interest $\Omega \left({{\mathbb{R}^3}} \right) \equiv \Omega $ into two subdomains, one interior ${\Omega ^ - }$ for every $ r < a $ and one exterior ${\Omega ^ + }$ for every $ r > a $, such as $\Omega = {\Omega ^ - } \cup {\Omega ^ + } \cup S$. In order to facilitate our calculations, we define the ratio of the main phase velocities in an elastic medium [14] via the formula

    $ \tau = \sqrt {\frac{\mu }{{\lambda + 2\mu }}} , $ (42)

    hence, the differential representation of the displacement field (3) becomes

    ${{\bf{u}}^ - }\left( {\bf{r}} \right) = \frac{{{\tau ^2} + 1}}{2}{{\bf{A}}^ - }\left( {\bf{r}} \right) + \frac{{{\tau ^2} - 1}}{2}\left[ {\nabla \otimes {{\bf{A}}^ - }\left( {\bf{r}} \right)} \right] \cdot {\bf{r}}, \;{\rm{where}}\;\Delta {{\bf{A}}^ - }\left( {\bf{r}} \right) = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }$ (43)

    inside the sphere and

    $ {{\bf{u}}^ + }\left( {\bf{r}} \right) = \frac{{{\tau ^2} + 1}}{2}{{\bf{A}}^ + }\left( {\bf{r}} \right) + \frac{{{\tau ^2} - 1}}{2}\left[ {\nabla \otimes {{\bf{A}}^ + }\left( {\bf{r}} \right)} \right] \cdot {\bf{r}}, \;{\rm{where}}\;\Delta {{\bf{A}}^ + }\left( {\bf{r}} \right) = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }$ (44)

    outside the sphere, where we have once more utilized the identity $ \nabla \left({{\bf{r}} \cdot {{\bf{A}}^ \mp }} \right) = {{\bf{A}}^ \mp } + \left({\nabla \otimes {{\bf{A}}^ \mp }} \right) \cdot {\bf{r}}$. The harmonic potentials $ {{\bf{A}}^ - } $ and $ {{\bf{A}}^ + } $ admit expansions similar to (14) with (15), taking profit of their interior and exterior character, respectively, thus, using (13), we have

    $ {{\bf{A}}^ - }\left( {\bf{r}} \right) = \sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {{\bf{c}}_{n, {\rm{in}}}^mu_{n, {\rm{in}}}^m\left( {\bf{r}} \right)} } = \sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {{\bf{c}}_{n, {\rm{in}}}^m{r^n}Y_n^m\left( {{\bf{\hat r}}} \right)\;} } {\rm{for}}\;{\bf{r}} \in {\Omega ^ - }, $ (45)

    in order for the potential $ {{\bf{A}}^ - } $ to be regular at the origin and

    $ {{\bf{A}}^ + }\left( {\bf{r}} \right) = \sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {{\bf{c}}_{n, {\rm{ex}}}^mu_{n, {\rm{ex}}}^m\left( {\bf{r}} \right)} } = \sum\limits_{n = 0}^{ + \infty } {\sum\limits_{m = - n}^n {{\bf{c}}_{n, {\rm{ex}}}^m{r^{ - \left( {n + 1} \right)}}Y_n^m\left( {{\bf{\hat r}}} \right)} } \;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (46)

    in order for the potential $ {{\bf{A}}^ + } $ to remain bounded as we move towards infinity. The unknown vector constant coefficients

    $ {\bf{c}}_{n, {\rm{y}}}^m = c_{n, {\rm{y}}}^{m, 1}{\mathit{\boldsymbol{\hat x}}_1} + c_{n, {\rm{y}}}^{m, 2}{\mathit{\boldsymbol{\hat x}}_2} + c_{n, {\rm{y}}}^{m, 3}{\mathit{\boldsymbol{\hat x}}_3}\;{\rm{for}}\;{\rm{y}} = {\rm{in}}, {\mkern 1mu} {\mkern 1mu} {\rm{ex}}, \;{\rm{where}}\;n \ge 0\;{\rm{and}}\;\left| m \right| \le n$ (47)

    within (45) and (46) are calculated, when a particular set of boundary conditions is applied on the surface boundary $S$ at $ r = a $. In the sequel, we focus ourselves in providing general solutions of particular interest in ${\Omega ^ - }$ and ${\Omega ^ + }$ for specific values of the degree $ n \geqslant 0 $, without getting involved with the boundary value problem itself.

    1st Example: If $ c $ is a constant vector, then it is not difficult to prove, performing direct substitution, that the Papkovich potentials

    $ {{\bf{A}}^ - }\left( {\bf{r}} \right) = {\bf{c}} , r < a \;{\rm{and}}\;{{\bf{A}}^ + }\left( {\bf{r}} \right) = \frac{{\bf{c}}}{r}, r > a $ (48)

    generate the displacement fields

    $ {{\bf{u}}^ - }\left( {\bf{r}} \right) = \frac{{{\tau ^2} + 1}}{2}{\bf{c}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - } $ (49)

    and

    $ {{\bf{u}}^ + }\left( {\bf{r}} \right) = \left[ {\frac{{{\tau ^2} + 1}}{2}{\bf{\tilde I}} - \frac{{{\tau ^2} - 1}}{2}{\bf{\hat r}} \otimes {\bf{\hat r}}} \right] \cdot \frac{{\bf{c}}}{r}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (50)

    respectively.

    2nd Example: If $ \tilde C $ is a constant tensor, let us introduce the Papkovich potentials

    $ {{\bf{A}}^ - }\left( {\bf{r}} \right) = {\bf{\tilde C}} \cdot {\bf{r}}, r < a \;{\rm{and}}\;{{\bf{A}}^ + }\left( {\bf{r}} \right) = {\bf{\tilde C}} \cdot \frac{{\bf{r}}}{{{r^\mathit{3}}}}, r > a , $ (51)

    which, by virtue of the identities

    $\nabla \otimes {{\bf{A}}^ - }\left( {\bf{r}} \right) = \left( {\nabla \otimes {\bf{\tilde C}}} \right) \cdot {\bf{r}} + \left( {\nabla \otimes {\bf{r}}} \right) \cdot {{\bf{\tilde C}}^\intercal} = {{\bf{\tilde C}}^\intercal}$ (52)

    and

    $ \nabla \otimes {{\bf{A}}^ + }\left( {\bf{r}} \right) = \left( {\nabla \otimes {\bf{\tilde C}}} \right) \cdot \frac{{\bf{r}}}{{{r^\mathit{3}}}} + \left( {\nabla \otimes \frac{{\bf{r}}}{{{r^\mathit{3}}}}} \right) \cdot {\tilde C^\intercal} = \frac{{{\bf{\tilde I}} - 3{\bf{\hat r}} \otimes {\bf{\hat r}}}}{{{r^\mathit{3}}}} \cdot {{\bf{\tilde C}}^\intercal}$ (53)

    generate the displacement fields

    $ {{\bf{u}}^ - }\left( {\bf{r}} \right) = \left( {\frac{{{\tau ^2} + 1}}{2}{\bf{\tilde C}} + \frac{{{\tau ^2} - 1}}{2}{{{\bf{\tilde C}}}^\intercal}} \right) \cdot {\bf{r}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }$ (54)

    and

    ${{\bf{u}}^ + }\left( {\bf{r}} \right) = \left[ {\frac{{{\tau ^2} + 1}}{2}{\bf{\tilde C}} + \frac{{{\tau ^2} - 1}}{2}\left( {{\bf{\tilde I}} - 3{\bf{\hat r}} \otimes {\bf{\hat r}}} \right) \cdot {{{\bf{\tilde C}}}^\intercal}} \right] \cdot \frac{{\bf{r}}}{{{r^\mathit{3}}}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (55)

    respectively. Here, we must remark that, if we consider the potentials $ {{\bf{A}}^ - } = {\bf{r}} \cdot {\bf{\tilde C}} = {{\bf{\tilde C}}^\intercal} \cdot {\bf{r}} $ and $ {{\bf{A}}^ + } = \left({{\bf{r}}/{r^\mathit{3}}} \right) \cdot {\bf{\tilde C}} = {{\bf{\tilde C}}^\intercal} \cdot \left({{\bf{r}}/{r^\mathit{3}}} \right) $, we obtain the displacements generated by $ {{\bf{\tilde C}}^ \intercal } $ instead of $ {\bf{\tilde C}} $. On the other hand, if $ {\bf{\tilde C}} $ is symmetric, i.e. if ${\bf{\tilde C}} = {{\bf{\tilde C}}^ \intercal } $, then the interior and exterior displacements (54) and (55) are rewritten as

    $ u(r)=τ2(˜Cr)forrΩandu+(r)=[τ2˜C32(τ21)(˜C:ˆrˆr)˜I]rr3forrΩ+,
    $
    (56)

    respectively. In particular, if ${\bf{\tilde C}} = {\bf{\tilde I}}$, then (56) simplify to

    ${{\bf{u}}^ - }\left( {\bf{r}} \right) = {\tau ^2}{\bf{r}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }\;{\rm{and}}\;{{\bf{u}}^ + }\left( {\bf{r}} \right) = \frac{{3 - {\tau ^2}}}{2}\frac{r}{{{r^\mathit{3}}}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (57)

    respectively. Finally, if $ {\bf{\tilde C}} $ is antisymmetric, i.e. if $ {{\bf{\tilde C}}^ \intercal } = - {\bf{\tilde C}} $, then obviously

    $ {{\bf{\tilde C}}^ \intercal } = \frac{1}{2}\left( {{{{\bf{\tilde C}}}^ \intercal } + {{{\bf{\tilde C}}}^ \intercal }} \right) = \frac{1}{2}\left( {{{{\bf{\tilde C}}}^ \intercal } - {\bf{\tilde C}}} \right) $ (58)

    and since

    ${{\bf{\tilde C}}^\intercal}:{\bf{\hat r}} \otimes {\bf{\hat r}} = {\bf{\tilde C}}:{\bf{\hat r}} \otimes {\bf{\hat r}}, $ (59)

    we obtain

    $ {{\bf{\tilde C}}^ \intercal }:{\bf{\hat r}} \otimes {\bf{\hat r}} = \frac{1}{2}{{\bf{\tilde C}}^\intercal}:{\bf{\hat r}} \otimes {\bf{\hat r}} - \frac{1}{2}{\bf{\tilde C}}:{\bf{\hat r}} \otimes {\bf{\hat r}} = 0.$ (60)

    Taking into account (58)–(60), the interior and exterior fields (54) and (55) are transformed to

    $ {{\bf{u}}^ - }\left( {\bf{r}} \right) = {\bf{\tilde C}} \cdot {\bf{r}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }\;{\rm{and}}\;{{\bf{u}}^ + }\left( {\bf{r}} \right) = {\bf{\tilde C}} \cdot \frac{r}{{{r^\mathit{3}}}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (61)

    respectively, which coincide with the harmonic potentials (51).

    3rd Example: If $ {\bf{\tilde C}} $ is a constant dyadic, then we want to find the displacement field that the exterior harmonic potential

    $ {{\bf{A}}^ + }\left( {\bf{r}} \right) = \nabla \left[ {{\bf{\tilde C}}:\nabla \otimes \nabla \frac{1}{r}} \right], r > a\;{\rm{with}}\;{\bf{\tilde C}} = \sum\limits_{i = 1}^3 {{\mathit{\boldsymbol{a}}_i} \otimes {\mathit{\boldsymbol{b}}_i}} , $ (62)

    produces. To prove it we first show that

    $\nabla \frac{1}{r} = - \frac{{\bf{r}}}{{{r^\mathit{3}}}}\;{\rm{and}}\;\nabla \otimes \nabla \frac{1}{r} = \nabla \otimes \left( { - \frac{r}{{{r^\mathit{3}}}}} \right) = \frac{{3{\bf{\hat r}} \otimes {\bf{\hat r}} - {\bf{\tilde I}}}}{{{r^\mathit{3}}}}. $ (63)

    Then, combining (62) and (63), we have

    $ A+(r)=[˜C:3ˆrˆr˜Ir3]=3i=1[aibi:(3r5rr1r3˜I)]=3i=1[3r5(air)(bir)1r3(aibi)]=3i=1[15r7aibi:rrr+3r5aibir+3r5biair+3r5(aibi)r]=3r53i=1[5aibi:ˆrˆrr+aibir+biair+(aibi)r]
    $
    (64)

    or

    ${{\bf{A}}^ + }\left( {\bf{r}} \right) = \frac{3}{{{r^5}}}\left[ {\left( {{\bf{\tilde C}}:{\bf{\tilde I}}} \right){\bf{\tilde I}} + \left( {{\bf{\tilde C}} + {{{\bf{\tilde C}}}^ \top }} \right) - 5\left( {{\bf{\tilde C}}:{\bf{\hat r}} \otimes {\bf{\hat r}}} \right){\bf{\tilde I}}} \right] \cdot {\bf{r}}, r > a . $ (65)

    Thereafter, the gradient of the Papkovich potential (65), i.e. $ \nabla \otimes {{\bf{A}}^ + } $, is then calculated, using classical identities and algebra as follows,

    $ A+(r)=3[((˜C:˜I)˜I+(˜C+˜C))rr5]153i=1[1r7(air)(bir)r]=3(rr5)[(˜C:˜I)˜I+(˜C+˜C)]153i=1[7r5aibi:ˆrˆrˆr+1r5aibiˆrˆr+1r5biaiˆrˆr+1r5(aibi:ˆrˆr)˜I]=3[5r5ˆrˆr+1r5˜I][(˜C:˜I)˜I+(˜C+˜C)]+15[7r5(˜C:ˆrˆr)ˆrˆr1r5(˜C+˜C)ˆrˆr1r5(˜C:ˆrˆr)˜I],r>a.
    $
    (66)

    Contracting $ \nabla \otimes {{\bf{A}}^ + } $ of (66) with $ {\bf{r}} $ from the right, we obtain

    $ [A+(r)]r=[15r5ˆrˆr+3r5˜I][(˜C:˜I)r+(˜C+˜C)r]+15[7r5(˜C:ˆrˆr)r1r5(˜C+˜C)r1r5(˜C:ˆrˆr)r]=15r5(˜C:˜I)r+3r5(˜C:˜I)r15r5(˜C+˜C):ˆrˆrr+3r5(˜C+˜C)r+90r5(˜C:ˆrˆr)r15r5(˜C+˜C)r=12r5(˜C:˜I)r+60r5(˜C:ˆrˆr)r12r5(˜C+˜C)r=43r5[(˜C:˜I)˜I+(˜C+˜C)5(˜C:ˆrˆr)˜I]r=4A+(r),r>a.
    $
    (67)

    Substituting (67) into the Papkovich representation (44) for the exterior field, we obtain

    $ {{\bf{u}}^ + }\left( {\bf{r}} \right) = \frac{{{\tau ^2} + 1}}{2}{{\bf{A}}^ + }\left( {\bf{r}} \right) + \frac{{{\tau ^2} - 1}}{2}\left[ { - 4{{\bf{A}}^ + }\left( {\bf{r}} \right)} \right] = \frac{{5 - 3{\tau ^2}}}{2}{{\bf{A}}^ + }\left( {\bf{r}} \right)\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }, $ (68)

    which is the requested. Herein, we remark that, since $ {{\bf{A}}^ + } $ is harmonic and $ {{\bf{u}}^ + } $ solves the equation of elastostatics (3), we obtain that $ \nabla \nabla \cdot {{\bf{A}}^ + } = {\bf{0}} $. Indeed,

    $ A+(r)=[(˜C:1r)]=Δ[˜C:1r]=[˜C:(Δ1r)]=0,r>a.
    $
    (69)

    Another interesting remark is that if $ {\bf{\tilde C}} $ is an antisymmetric dyadic, i.e. if $ {{\bf{\tilde C}}^ \intercal } = - {\bf{\tilde C}} $, then it is not hard to prove

    $ {{\bf{A}}^ + }\left( {\bf{r}} \right) = \nabla \left[ {{\bf{\tilde C}}:\nabla \nabla \frac{1}{r}} \right] = {\bf{0}}, r > a\;{\rm{and}}\;{\rm{therefore}}\;{{\bf{u}}^ + }\left( {\bf{r}} \right) = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ + }.$ (70)

    This means that the potential $ {{\bf{A}}^ + } $ is symmetric and it generates a symmetric displacement. In particular, any $ {\bf{\tilde C}} $ generates a symmetric $ {{\bf{A}}^ + } $ and thus, a symmetric $ {{\bf{u}}^ + } $.

    4th Example: In this case, we consider the equation of elastostatics (1) for $ {\bf{f}} = {\bf{0}} $, which in terms of (42) is rewritten as

    ${\tau ^2}{\mkern 1mu} \Delta {{\bf{u}}^ - }\left( {\bf{r}} \right) + \left( {1 - {\tau ^2}} \right)\nabla \left[ {\nabla \cdot {{\bf{u}}^ - }\left( {\bf{r}} \right)} \right] = {\bf{0}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }$ (71)

    and we wish to find the coefficients $ \alpha, \, \beta, \, \gamma $ for which the linear combination of the displacement field

    $ {{\bf{u}}^ - }\left( {\bf{r}} \right) = \alpha {\mkern 1mu} {\bf{\tilde S}}:{\bf{r}} \otimes {\bf{r}} \otimes {\bf{r}} + \beta {r^2}{\bf{\tilde S}} \cdot {\bf{r}} + \gamma S{r^2}{\bf{r}}\;{\rm{for}}\;{\bf{r}} \in {\Omega ^ - }$ (72)

    belongs to the $ {\text{ker}}\left[{{\tau ^2}\, \Delta + \left({1 - {\tau ^2}} \right)\nabla \nabla \cdot } \right] $, where $ {\bf{\tilde S}} = {\bf{a}} \otimes {\bf{b}} = {\bf{b}} \otimes {\bf{a}} = {{\bf{\tilde S}}^ \intercal } $ is a constant symmetric dyadic and $ S = {\bf{a}} \cdot {\bf{b}} $ is its trace. This example makes use of the previous three examples, as well as the linearity of the Navier equation, to expand the physical important interior displacement field in a three-dimensional solution subspace and to calculate the actual solution by calculating the three scalar coefficients. We primarily have to evaluate the following expressions,

    $ Δ(˜S:rrr)=Δ[(ar)(br)r]=Δ[(ar)(br)]r+(ar)(br)Δr+2[(ar)(br)]rr=2(ab)r+2(ab+ba)r=2Sr+4˜Sr,
    $
    (73)
    $ Δ(r2˜Sr)=(Δr2)˜Sr+r2Δ(˜Sr)+2(r2)(˜Sr)=6˜Sr+4r˜S=10˜Sr,
    $
    (74)
    $ \Delta \left( {{r^2}{\bf{r}}} \right) = 6{\bf{r}} + 2 \cdot 2{\bf{r}} \cdot \widetilde {\bf{I}} = 10{\bf{r}}, $ (75)
    $ (˜S:rrr)=[(ar)(br)r]=[(ab+ba):rr+(ar)(br)3]=5(˜S:rr)=5(ab+ba)r=10˜Sr,
    $
    (76)
    $ (r2˜S×r)=[2r˜Sr+r2(˜Sr)]=[2(ar)(br)+r2˜S:˜I]=2(ab+ba)r+2Sr=4˜Sr+2Sr
    $
    (77)

    and

    $ \nabla \nabla \cdot \left( {{r^2}{\bf{r}}} \right) = \nabla \left[ {2{\bf{r}} \cdot {\bf{r}} + 3{r^2}} \right] = 5\nabla {r^2} = 10{\bf{r}}, $ (78)

    wherein we have assumed that $ {\bf{\tilde S}} = \left({{\bf{a}} \otimes {\bf{b}} + {\bf{b}} \otimes {\bf{a}}} \right)/2$, so that ${\bf{\tilde S}}:{\bf{r}} \otimes {\bf{r}} = \left({{\bf{a}} \cdot {\bf{r}}} \right)\left({{\bf{b}} \cdot {\bf{r}}} \right) $ and $S = {\bf{a}} \cdot {\bf{b}}$. In view of the above formulae (73)–(78) and with respect to (71) and (72), we now obtain

    $ [τ2Δ+(1τ2)]u(r)=ατ2[2Sr+4˜Sr]+βτ2[10˜Sr]+γτ2[10r]S+α(1τ2)[10˜Sr]+β(1τ2)[2Sr+4˜Sr]+γ(1τ2)[10r]S=[2ατ2S+10γτ2S+2β(1τ2)S+10γ(1τ2)S]r+[4ατ2S+10βτ2+10α(1τ2)+4β(1τ2)]˜Sr=[2ατ2S+2β(1τ2)S+10γS]r+[α(106τ2)+β(4+6τ2)]˜Sr=0forrΩ,
    $
    (79)

    which implies

    $ {\tau ^2}S\alpha + \left( {1 - {\tau ^2}} \right)S\beta = - 5\gamma S \;{\rm{and}}\; \left( {5 - 3{\tau ^2}} \right)\alpha + \left( {2 + 3{\tau ^2}} \right)\beta = 0 , $ (80)

    leading to the constants

    $ \alpha = 3{\tau ^2} + 2 , \beta = 3{\tau ^2} - 5 \;{\rm{and}}\; \gamma = 1 - 2{\tau ^2} . $ (81)

    Hence, the solution is provided via (72) by

    $ u(r)=(3τ2+2)˜S:rrr+(3τ25)r2˜Sr+(12τ2)Sr2rforrΩ,
    $
    (82)

    which is the interior displacement field of the specific static elasticity problem.

    Recapitulating, we offered the analytical solution for the displacement fields of four interesting problems in linear isotropic elastostatics in the absence of body forces, inside and outside a spherical boundary in the absence of boundary conditions. These solutions are solid and can be directly derived from the theory of differential representation, which we developed and analyzed in the previous section.

    In this study, we presented a mathematical method for recovering the main spherical components of the well-known Navier equation in linear isotropic elastostatics, under the circumstance of no external forces present. To this end, we primarily combined the Hooke's and Newton's law, via the correlation of the displacement field with the strain, the stress and the stiffness tensors, reproducing the linearized equation of dynamic isotropic elasticity. In the sequel, we introduced the Papkovich differential representation, which offered solutions in terms of scalar and vector harmonic functions. Then, connection formulae were obtained, by which we transformed any solution of the Navier system from the Papkovich to the potential-type eigenform and vice versa. In order to enhance this procedure, we implied the commonly used spherical geometry and we calculated the time-independent displacement field, generated by the well-known spherical harmonic eigenfunctions. In order to demonstrate the effectiveness of our method, we presented some important degenerate cases for the evaluation of the interior and the exterior displacement field on either side of a spherical boundary.

    Work under progress involves research directed towards the extension of the current analysis to more complicated geometries, e.g. spheroidal and ellipsoidal, producing ready-to-use functions and their Navier counterparts.

    The authors have declared no conflict of interest.

    [1] Association AP (2013) American Psychiatric Association: Diagnositc and Statistical Manual of Mental Disorders, 5 Eds. American Psychiatric Association: Diagnositc and Statistical Manual of Mental Disorders.
    [2] Epperson CN (2013) Premenstrual dysphoric disorder and the brain. Am J Psychiatry 170:248-252. doi: 10.1176/appi.ajp.2012.12121555
    [3] O'Brien PM, Backstrom T, Brown C, et al. (2011) Towards a consensus on diagnostic criteria, measurement and trial design of the premenstrual disorders: the ISPMD Montreal consensus. Arch Womens Ment Health 14: 13-21. doi: 10.1007/s00737-010-0201-3
    [4] Veith I (1965) The History of a Disease. Chicago: Chicago University Press.
    [5] Herculano-Houzel S (2010) Coordinated scaling of cortical and cerebellar numbers of neurons. Front Neuroanat 4: 12.
    [6] Tien RD, Ashdown BC (1992) Crossed cerebellar diaschisis and crossed cerebellar atrophy: correlation of MR findings, clinical symptoms, and supratentorial diseases in 26 patients. AJR Am J Roentgenol 158: 1155-1159. doi: 10.2214/ajr.158.5.1566683
    [7] Ito M (1993) New concepts in cerebellar function. Rev Neurol (Paris) 149: 596-599.
    [8] Harlow HF, Harlow M (1962) Social deprivation in monkeys. Sci Am 207: 136-146. doi: 10.1038/scientificamerican1162-136
    [9] Prescott JW (1970) Early somatosensory deprivation as ontogenic process in the abnormal development of the brain and behavior. Medical Primatology 1970: 356-375.
    [10] Nashold BS, Jr. , Slaughter DG (1969) Effects of stimulating or destroying the deep cerebellar regions in man. J Neurosurg 31: 172-186. doi: 10.3171/jns.1969.31.2.0172
    [11] Heath RG (1977) Modulation of emotion with a brain pacemaker. Treatment for intractable psychiatric illness. J Nerv Ment Dis 165: 300-317.
    [12] Cooper IS, Amin, L. , Gilman, S. , Waltz, J. M. (1974) The Effect of chronic stimulation of cerebellar cortex on epilepsy in Man. The Cerebellum, Epilepsy and Behavior. New York: Plenum Press. pp. 199-172.
    [13] Heath RG, Franklin DE, Shraberg D (1979) Gross pathology of the cerebellum in patients diagnosed and treated as functional psychiatric disorders. J Nerv Ment Dis 167: 585-592. doi: 10.1097/00005053-197910000-00001
    [14] Heath RG, Llewellyn RC, Rouchell AM (1980) The cerebellar pacemaker for intractable behavioral disorders and epilepsy: follow-up report. Biol Psychiatry 15: 243-256.
    [15] Heath RG, Franklin DE, Walker CF, et al. (1982) Cerebellar vermal atrophy in psychiatric patients. Biol Psychiatry 17: 569-583.
    [16] Schmahmann JD, Sherman JC (1998) The cerebellar cognitive affective syndrome. Brain 121 ( Pt4): 561-579.
    [17] Schmahmann JD, Weilburg JB, Sherman JC (2007) The neuropsychiatry of the cerebellum - insights from the clinic. Cerebellum 6: 254-267. doi: 10.1080/14734220701490995
    [18] Schmahmann JD (1991) An emerging concept. The cerebellar contribution to higher function. Arch Neurol 48: 1178-1187.
    [19] Schmahmann JD (1996) From movement to thought: anatomic substrates of the cerebellar contribution to cognitive processing. Hum Brain Mapp 4: 174-198. doi: 10.1002/(SICI)1097-0193(1996)4:3<174::AID-HBM3>3.0.CO;2-0
    [20] Allen G, Buxton RB, Wong EC, et al. (1997) Attentional activation of the cerebellum independent of motor involvement. Science 275: 1940-1943. doi: 10.1126/science.275.5308.1940
    [21] Stoodley CJ, Schmahmann JD (2009) Functional topography in the human cerebellum: a meta-analysis of neuroimaging studies. Neuroimage 44: 489-501. doi: 10.1016/j.neuroimage.2008.08.039
    [22] Koziol LF, Budding D, Andreasen N, et al. (2014) Consensus paper: the cerebellum's role in movement and cognition. Cerebellum 13: 151-177. doi: 10.1007/s12311-013-0511-x
    [23] Schraa-Tam CK, Rietdijk WJ, Verbeke WJ, et al. (2012) fMRI activities in the emotional cerebellum: a preference for negative stimuli and goal-directed behavior. Cerebellum 11: 233-245. doi: 10.1007/s12311-011-0301-2
    [24] Ferrucci R, Giannicola G, Rosa M, et al. (2012) Cerebellum and processing of negative facial emotions: cerebellar transcranial DC stimulation specifically enhances the emotional recognition of facial anger and sadness. Cogn Emot 26: 786-799. doi: 10.1080/02699931.2011.619520
    [25] Grimaldi G, Argyropoulos GP, Boehringer A, et al. (2014) Non-invasive cerebellar stimulation--a consensus paper. Cerebellum 13: 121-138. doi: 10.1007/s12311-013-0514-7
    [26] West RL (1996) An application of prefrontal cortex function theory to cognitive aging. Psych Bull120: 272-292.
    [27] Hogan MJ (2004) The cerebellum in thought and action: a fronto-cerebellar aging hypothesis. New Ideas in Psychology 22: 97-125. doi: 10.1016/j.newideapsych.2004.09.002
    [28] Eckert MA (2011) Slowing down: age-related neurobiological predictors of processing speed. Front Neurosci 5: 25.
    [29] Woodruff-Pak DS, Vogel RW, Ewers M, et al. (2001) MRI-assessed volume of cerebellum correlates with associative learning. Neurobiology of Learning and Memory 76: 342-357. doi: 10.1006/nlme.2001.4026
    [30] MacLullich AMJ, Edmond CL, Ferguson KJ, et al. (2004) Size of the neocerebellar vermis is associated with cognition in healthy elderly men. Brain and Cognition 56: 344-348. doi: 10.1016/j.bandc.2004.08.001
    [31] Paul R, Grieve SM, Chaudary B, et al. (2009) Relative contributions of the cerebellar vermis and prefrontal lobe volumes on cognitive function across the adult lifespan. Neurobiol Aging 30:457-465. doi: 10.1016/j.neurobiolaging.2007.07.017
    [32] Eckert MA, Keren NI, Roberts DR, et al. (2010) Age-related changes in processing speed: unique contributions of cerebellar and prefrontal cortex. Front Hum Neurosci 4: 10.
    [33] Hogan MJ, Staff RT, Bunting BP, et al. (2011) Cerebellar brain volume accounts for variance in cognitive performance in older adults. Cortex 47: 441-450. doi: 10.1016/j.cortex.2010.01.001
    [34] Rasgon N, Serra M, Biggio G, et al. (2001) Neuroactive steroid-serotonergic interaction: responses to an intravenous L-tryptophan challenge in women with premenstrual syndrome. Eur J Endocrinol 145: 25-33. doi: 10.1530/eje.0.1450025
    [35] Hamakawa H, Kato T, Murashita J, et al. (1998) Quantitative proton magnetic resonance spectroscopy of the basal ganglia in patients with affective disorders. Eur Arch Psychiatry Clin Neurosci 248: 53-58. doi: 10.1007/s004060050017
    [36] Renshaw PF, Levin JM, Kaufman MJ, et al. (1997) Dynamic susceptibility contrast magnetic resonance imaging in neuropsychiatry: present utility and future promise. Eur Radiol 7 Suppl 5:216-221.
    [37] Buchpiguel C, Alavi A, Crawford D, et al. (2000) Changes in cerebral blood flow associated with premenstrual syndrome: a preliminary study. J Psychosom Obstet Gynaecol 21: 157-165. doi: 10.3109/01674820009075623
    [38] Rasgon NL, Thomas MA, Guze BH, et al. (2001) Menstrual cycle-related brain metabolite changes using 1H magnetic resonance spectroscopy in premenopausal women: a pilot study. Psychiatry Res 106: 47-57. doi: 10.1016/S0925-4927(00)00085-8
    [39] Epperson CN, Haga K, Mason GF, et al. (2002) Cortical gamma-aminobutyric acid levels across the menstrual cycle in healthy women and those with premenstrual dysphoric disorder: a proton magnetic resonance spectroscopy study. Arch Gen Psychiatry 59: 851-858. doi: 10.1001/archpsyc.59.9.851
    [40] Jovanovic H, Cerin A, Karlsson P, et al. (2006) A PET study of 5-HT1A receptors at different phases of the menstrual cycle in women with premenstrual dysphoria. Psychiatry Res 148:185-193. doi: 10.1016/j.pscychresns.2006.05.002
    [41] Eriksson O, Wall A, Marteinsdottir I, et al. (2006) Mood changes correlate to changes in brain serotonin precursor trapping in women with premenstrual dysphoria. Psychiatry Res 146: 107-116. doi: 10.1016/j.pscychresns.2005.02.012
    [42] Batra NA, Seres-Mailo J, Hanstock C, et al. (2008) Proton magnetic resonance spectroscopy measurement of brain glutamate levels in premenstrual dysphoric disorder. Biol Psychiatry 63:1178-1184. doi: 10.1016/j.biopsych.2007.10.007
    [43] Protopopescu X, Tuescher O, Pan H, et al. (2008) Toward a functional neuroanatomy of premenstrual dysphoric disorder. J Affect Disord 108: 87-94. doi: 10.1016/j.jad.2007.09.015
    [44] Bannbers E, Gingnell M, Engman J, et al. (2012) The effect of premenstrual dysphoric disorder and menstrual cycle phase on brain activity during response inhibition. J Affect Disord 142:347-350. doi: 10.1016/j.jad.2012.04.006
    [45] Gingnell M, Morell A, Bannbers E, et al. (2012) Menstrual cycle effects on amygdala reactivity to emotional stimulation in premenstrual dysphoric disorder. Horm Behav 62: 400-406. doi: 10.1016/j.yhbeh.2012.07.005
    [46] Gingnell M, Bannbers E, Wikstrom J, et al. (2013) Premenstrual dysphoric disorder and prefrontal reactivity during anticipation of emotional stimuli. Eur Neuropsychopharmacol 23: 1474-1483. doi: 10.1016/j.euroneuro.2013.08.002
    [47] Baller EB, Wei SM, Kohn PD, et al. (2013) Abnormalities of dorsolateral prefrontal function in women with premenstrual dysphoric disorder: a multimodal neuroimaging study. Am J Psychiatry170: 305-314.
    [48] Jeong HG, Ham BJ, Yeo HB, et al. (2012) Gray matter abnormalities in patients with premenstrual dysphoric disorder: an optimized voxel-based morphometry. J Affect Disord 140: 260-267. doi: 10.1016/j.jad.2012.02.010
    [49] Berman SM, London ED, Morgan M, et al. (2013) Elevated gray matter volume of the emotional cerebellum in women with premenstrual dysphoric disorder. J Affect Disord 146: 266-271. doi: 10.1016/j.jad.2012.06.038
    [50] Rapkin A (2003) A review of treatment of premenstrual syndrome and premenstrual dysphoric disorder. Psychoneuroendocrinology 28 Suppl 3: 39-53.
    [51] Halbreich U (2008) Selective serotonin reuptake inhibitors and initial oral contraceptives for the treatment of PMDD: effective but not enough. CNS Spectr 13: 566-572. doi: 10.1017/S1092852900016849
    [52] Nevatte T, O'Brien PM, Backstrom T, et al. (2013) ISPMD consensus on the management of premenstrual disorders. Arch Womens Ment Health 16: 279-291. doi: 10.1007/s00737-013-0346-y
    [53] Raichle M (1987) Circulatory and Metabolic Correlates of brain function in normal humans. Handbook of Physiology-The nervous system Bethesda: American Physiological Society V:643-674.
    [54] Rapkin AJ, Berman SM, Mandelkern MA, et al. (2011) Neuroimaging evidence of cerebellar involvement in premenstrual dysphoric disorder. Biol Psychiatry 69: 374-380. doi: 10.1016/j.biopsych.2010.09.029
    [55] Mackenzie G, Maguire J (2014) The role of ovarian hormone-derived neurosteroids on the regulation of GABA receptors in affective disorders. Psychopharmacology (Berl).
    [56] Schmidt PJ, Nieman LK, Danaceau MA, et al. (1998) Differential behavioral effects of gonadal steroids in women with and in those without premenstrual syndrome. New England Journal of Medicine 338: 209-216. doi: 10.1056/NEJM199801223380401
    [57] Hanstock C, Allen, PS (2000) Segmentation of brain from a PRESS localized single volume using double inversion recovery for simultaneous T1 nulling. 8th Annual Meeting of the International Society for Magnetic Resonance in Medicine. Denver, Colorado.
    [58] Diedrichsen J, Balsters JH, Flavell J, et al. (2009) A probabilistic MR atlas of the human cerebellum. Neuroimage 46: 39-46. doi: 10.1016/j.neuroimage.2009.01.045
    [59] Kalpouzos G, Persson J, Nyberg L (2012) Local brain atrophy accounts for functional activity differences in normal aging. Neurobiol Aging 33: 623 e621-623 e613.
    [60] Diedrichsen J, Verstynen T, Schlerf J, et al. (2010) Advances in functional imaging of the human cerebellum. Curr Opin Neurol 23: 382-387.
    [61] Baldacara L, Nery-Fernandes F, Rocha M, et al. (2011) Is cerebellar volume related to bipolar disorder? J Affect Disord 135: 305-309. doi: 10.1016/j.jad.2011.06.059
    [62] De Bellis MD, Kuchibhatla M (2006) Cerebellar volumes in pediatric maltreatment-related posttraumatic stress disorder. Biol Psychiatry 60: 697-703. doi: 10.1016/j.biopsych.2006.04.035
    [63] Frodl TS, Koutsouleris N, Bottlender R, et al. (2008) Depression-related variation in brain morphology over 3 years: effects of stress? Arch Gen Psychiatry 65: 1156-1165. doi: 10.1001/archpsyc.65.10.1156
    [64] Peng J, Liu J, Nie B, et al. (2011) Cerebral and cerebellar gray matter reduction in first-episode patients with major depressive disorder: a voxel-based morphometry study. Eur J Radiol 80:395-399. doi: 10.1016/j.ejrad.2010.04.006
    [65] Kim D, Cho HB, Dager SR, et al. (2013) Posterior cerebellar vermal deficits in bipolar disorder. J Affect Disord 150: 499-506. doi: 10.1016/j.jad.2013.04.050
    [66] Schutter DJ, Koolschijn PC, Peper JS, et al. (2012) The cerebellum link to neuroticism: a volumetric MRI association study in healthy volunteers. PLoS One 7: e37252. doi: 10.1371/journal.pone.0037252
    [67] Adler CM, DelBello MP, Jarvis K, et al. (2007) Voxel-based study of structural changes in first-episode patients with bipolar disorder. Biol Psychiatry 61: 776-781. doi: 10.1016/j.biopsych.2006.05.042
    [68] Spinelli S, Chefer S, Suomi SJ, et al. (2009) Early-life stress induces long-term morphologic changes in primate brain. Arch Gen Psychiatry 66: 658-665. doi: 10.1001/archgenpsychiatry.2009.52
    [69] Draganski B, Gaser C, Busch V, et al. (2004) Neuroplasticity: changes in grey matter induced by training. Nature 427: 311-312. doi: 10.1038/427311a
    [70] Kwok V, Niu Z, Kay P, et al. (2011) Learning new color names produces rapid increase in gray matter in the intact adult human cortex. Proc Natl Acad Sci U S A 108: 6686-6688. doi: 10.1073/pnas.1103217108
    [71] Oral E, Ozcan H, Kirkan TS, et al. (2013) Luteal serum BDNF and HSP70 levels in women with premenstrual dysphoric disorder. Eur Arch Psychiatry Clin Neurosci 263: 685-693. doi: 10.1007/s00406-013-0398-z
    [72] Anim-Nyame N, Domoney C, Panay N, et al. (2000) Plasma leptin concentrations are increased in women with premenstrual syndrome. Hum Reprod 15: 2329-2332. doi: 10.1093/humrep/15.11.2329
    [73] Oldreive CE, Harvey J, Doherty GH (2008) Neurotrophic effects of leptin on cerebellar Purkinje but not granule neurons in vitro. Neurosci Lett 438: 17-21. doi: 10.1016/j.neulet.2008.04.045
    [74] Riad-Gabriel MG, Jinagouda SD, Sharma A, et al. (1998) Changes in plasma leptin during the menstrual cycle. Eur J Endocrinol 139: 528-531. doi: 10.1530/eje.0.1390528
    [75] Narita K, Kosaka H, Okazawa H, et al. (2009) Relationship between plasma leptin level and brain structure in elderly: a voxel-based morphometric study. Biol Psychiatry 65: 992-994. doi: 10.1016/j.biopsych.2008.10.006
    [76] Matochik JA, London ED, Yildiz BO, et al. (2005) Effect of leptin replacement on brain structure in genetically leptin-deficient adults. J Clin Endocrinol Metab 90: 2851-2854. doi: 10.1210/jc.2004-1979
    [77] London ED, Berman SM, Chakrapani S, et al. (2011) Short-term plasticity of gray matter associated with leptin deficiency and replacement. J Clin Endocrinol Metab 96: E1212-1220. doi: 10.1210/jc.2011-0314
    [78] Tommaselli GA, Di Carlo C, Bifulco G, et al. (2003) Serum leptin levels in patients with premenstrual syndrome treated with GnRH analogues alone and in association with tibolone. Clin Endocrinol (Oxf) 59: 716-722. doi: 10.1046/j.1365-2265.2003.01911.x
    [79] Akturk M, Toruner F, Aslan S, et al. (2013) Circulating insulin and leptin in women with and without premenstrual disphoric disorder in the menstrual cycle. Gynecol Endocrinol 29: 465-469. doi: 10.3109/09513590.2013.769512
    [80] Eikelis N, Esler M, Barton D, et al. (2006) Reduced brain leptin in patients with major depressive disorder and in suicide victims. Mol Psychiatry 11: 800-801. doi: 10.1038/sj.mp.4001862
    [81] Westling S, Ahren B, Traskman-Bendz L, et al. (2004) Low CSF leptin in female suicide attempters with major depression. J Affect Disord 81: 41-48. doi: 10.1016/j.jad.2003.07.002
    [82] Yoshida-Komiya H, Takano K, Fujimori K, et al. (2014) Plasma levels of leptin in reproductive-aged women with mild depressive and anxious states. Psychiatry Clin Neurosci.
    [83] Lawson EA, Miller KK, Blum JI, et al. (2012) Leptin levels are associated with decreased depressive symptoms in women across the weight spectrum, independent of body fat. Clin Endocrinol (Oxf) 76: 520-525. doi: 10.1111/j.1365-2265.2011.04182.x
    [84] Chirinos DA, Goldberg R, Gellman M, et al. (2013) Leptin and its association with somatic depressive symptoms in patients with the metabolic syndrome. Ann Behav Med 46: 31-39. doi: 10.1007/s12160-013-9479-5
    [85] Kloiber S, Ripke S, Kohli MA, et al. (2013) Resistance to antidepressant treatment is associated with polymorphisms in the leptin gene, decreased leptin mRNA expression, and decreased leptin serum levels. Eur Neuropsychopharmacol 23: 653-662. doi: 10.1016/j.euroneuro.2012.08.010
    [86] Johnston JM, Greco SJ, Hamzelou A, et al. (2011) Repositioning leptin as a therapy for Alzheimer's disease. Therapy 8: 481-490. doi: 10.2217/thy.11.57
    [87] Johnston J HW, Fardo D, Greco S, Perry G, Montine T, Trojanowski J, Shaw L, Ashford J, Tezapsidis N (2013) For The Alzheimer's Disease Neuroimaging Initiative. Low Plasma Leptin in Cognitively Impaired ADNI Subjects- Gender Differences and Diagnostic and Therapeutic Potential. Curr Alzheimer Res.
    [88] Rapkin AJ, Morgan M, Goldman L, et al. (1997) Progesterone metabolite allopregnanolone in women with premenstrual syndrome. Obstet Gynecol 90: 709-714. doi: 10.1016/S0029-7844(97)00417-1
    [89] Singh M, Su C (2013) Progesterone and neuroprotection. Horm Behav 63: 284-290. doi: 10.1016/j.yhbeh.2012.06.003
    [90] Azcoitia I, Arevalo MA, De Nicola AF, et al. (2011) Neuroprotective actions of estradiol revisited. Trends Endocrinol Metab 22: 467-473. doi: 10.1016/j.tem.2011.08.002
    [91] Gao Q, Horvath TL (2008) Cross-talk between estrogen and leptin signaling in the hypothalamus. Am J Physiol Endocrinol Metab 294: E817-826. doi: 10.1152/ajpendo.00733.2007
    [92] Hedges VL, Ebner TJ, Meisel RL, et al. (2012) The cerebellum as a target for estrogen action. Front Neuroendocrinol 33: 403-411. doi: 10.1016/j.yfrne.2012.08.005
    [93] Ghidoni R, Boccardi M, Benussi L, et al. (2006) Effects of estrogens on cognition and brain morphology: involvement of the cerebellum. Maturitas 54: 222-228. doi: 10.1016/j.maturitas.2005.11.002
    [94] Boccardi M, Ghidoni R, Govoni S, et al. (2006) Effects of hormone therapy on brain morphology of healthy postmenopausal women: a Voxel-based morphometry study. Menopause 13: 584-591. doi: 10.1097/01.gme.0000196811.88505.10
    [95] Robertson D, Craig M, van Amelsvoort T, et al. (2009) Effects of estrogen therapy on age-related differences in gray matter concentration. Climacteric 12: 301-309. doi: 10.1080/13697130902730742
    [96] Kim SG, Ogawa S (2012) Biophysical and physiological origins of blood oxygenation level-dependent fMRI signals. J Cereb Blood Flow Metab 32: 1188-1206. doi: 10.1038/jcbfm.2012.23
    [97] D'Esposito M, Deouell LY, Gazzaley A. (2003) Alterations in the BOLD fMRI signal with ageing and disease: a challenge for neuroimaging. Nat Rev Neurosci 4: 863-872. doi: 10.1038/nrn1246
    [98] Ances BM, Liang CL, Leontiev O, et al. (2009) Effects of aging on cerebral blood flow, oxygen metabolism, and blood oxygenation level dependent responses to visual stimulation. Hum Brain Mapp 30: 1120-1132. doi: 10.1002/hbm.20574
    [99] Gauthier CJ, Madjar C, Desjardins-Crepeau L, et al. (2013) Age dependence of hemodynamic response characteristics in human functional magnetic resonance imaging. Neurobiol Aging 34:1469-1485. doi: 10.1016/j.neurobiolaging.2012.11.002
    [100] Sui R, Zhang L (2012) Cerebellar dysfunction may play an important role in vascular dementia. Med Hypotheses 78: 162-165. doi: 10.1016/j.mehy.2011.10.017
    [101] arrett DD, Kovacevic N, McIntosh AR, et al. (2010) Blood oxygen level-dependent signal variability si more than just noise. J Neurosci 30: 4914-4921.
    [102] Grady CL, Garrett DD (2013) Understanding variability in the BOLD signal and why it matters for aging. Brain Imaging Behav.
  • This article has been cited by:

    1. D. Labropoulou, P. Vafeas, D. M. Manias, G. Dassios, Generalized Solutions in Isotropic and Anisotropic Elastostatics, 2025, 157, 0374-3535, 10.1007/s10659-025-10123-x
  • Reader Comments
  • © 2014 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(6428) PDF downloads(936) Cited by(0)

Figures and Tables

Tables(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog