Jorge I Facio

Jernej Mravlje

Leonid Pourovskii

Pablo S Cornaglia

V Vildosola

Spin-orbit and anisotropic strain effects on the electronic correlations of Sr 2 RuO 4

We present an implementation of the rotationally invariant slave boson technique as an impurity solver for density functional theory plus dynamical mean field theory (DFT+DMFT). Our approach provides explicit relations between quantities in the local correlated subspace treated with DMFT and the Bloch basis used to solve the DFT equations. In particular, we present an expression for the mass enhancement of the quasiparticle states in reciprocal space. We apply the method to the study of the electronic correlations in Sr2RuO4 under anisotropic strain. We find that the spin-orbit coupling plays a crucial role in the mass enhancement differentiation between the quasione-dimensional α and β bands, and on its momentum dependence over the Fermi surface. The mass enhancement, however, is only weakly affected by either uniaxial or biaxial strain, even accross the Lifshitz transition induced by the strain.

I. INTRODUCTION

The electronic properties of transition metal oxides continue to be a central problem in condensed matter physics. Part of the challenge is due to the sensitivity of the low energy physics of these systems to the complex interplay between the crystal structure, the amount of hybridization between oxygens and the transition metal [1], the Coulomb interaction, including the correct differentiation between intraorbital e interorbital interactions [2,3], and the spin-orbit coupling (SOC) [4,5]. This interplay seems to be the key to understanding the normal state, magnetic, and superconducting properties of some of these compounds [6]. In particular, the role of the SOC and of the proximity to a Lifshitz transition have recently attracted much interest.

It has been reported that the SOC is crucial to explaining the insulating character of Sr 2 IrO 4 [7], and the magnetic properties of Ca 2 RuO 4 [8] and Sr 3 Ru 2 O 7 [9]. It may also be relevant to determine the nature of the superconducting state in Sr 2 RuO 4 [10,11]. At the local density approximation (LDA) level, its inclusion is necessary to improve the Fermi surface shape of Sr 2 RuO 4 and Sr 2 RhO 4 as compared to ARPES data [4,5,[12][13][14].

Recently, the possibility of inducing a Lifshitz transition in Sr 2 RuO 4 by applying external strains has been addressed experimentally [15,16]. Single crystals under a uniaxial strain applied in the [100] direction present a peak in the superconducting critical temperature as a function of the stress. The peak position seems to coincide with the value of the stress at which the Lifshitz transition takes place and the associated Van Hove singularity (VHS) crosses the Fermi level [15,17]. This transition has been observed in ARPES measurements of epitaxial thin films of Sr 2 RuO 4 and Ba 2 RuO 4 grown over different substrates [16]. Different values of the lattice parameter a are obtained adjusting the lattice mis-match and the results can be interpreted as the behavior of Sr 2 RuO 4 under a biaxial strain in the [100] and [010] directions.

These experiments have triggered theoretical investigations focused on understanding how the Lifshitz transition affects the pairing properties in the superconducting phase [18,19] or the spin susceptibility in the normal phase [20]. An interesting open question is to what extent this transition affects the electronic correlations in the normal phase. Indeed, the proximity of the VHS to the Fermi level has been found to be important to understand the anisotropic mass enhancement of quasiparticles. The Fermi surface of this material is composed by the sheets α, β and γ of mainly Ru t 2g character, whose respective renormalized masses, m * /m LDA , have been reported to be 3, 3.5, and 5, respectively [21]. Density functional theory suplemented by dynamical mean field theory (DFT+DMFT) calculations, have shown that the larger renormalization of the γ sheet can be associated with the proximity of the VHS to the Fermi level, and Hund's rule coupling effects [2].

The purpose of this article is twofold. First, to introduce an implementation of the rotationally invariant slave-boson (RISB) method [22][23][24][25][26] as an impurity solver for DFT+DMFT. The RISB method is a low weight numerical method geared to describe Fermi liquid behavior and has been successfully used, supplemented by DFT calculations, to describe the low energy correlations of materials [9,[27][28][29][30][31]. Our approach provides an explicit relation between the mass enhancement calculated in the quantum impurity problem and the mass enhancement that different quasiparticle states acquire after embedding the impurity self-energy back in the lattice problem. This relation can also be applied when using other techniques to solve the quantum impurity problem. Second, we apply this methodology to Sr 2 RuO 4 . For the unstressed compound, using the above mentioned rela-tion, we explain why Bloch states on the α and β sheets of the Fermi surface, having mostly xz and yz character (which correspond in the quantum impurity problem to degenerated cubic Wannier orbitals), acquire a different renormalization, as experimentally observed. To finalize, motivated by the experiments of Ref. [15,16], we analyze how the electronic correlations, as measured by the mass enhancement, evolve under biaxial or uniaxial stress, and to what extent they are affected by the Lifshitz transition.

The rest of this paper is organized as follows. In Section II we introduce our implementation of RISB as an impurity solver for DFT+DMFT. In Section III we analyze how the method describes the correlated metal Sr 2 RuO 4 . In Section IV we present results for the material under anisotropic stress. Finally, in Section V we present our concluding remarks.

II. METHOD: DFT+DMFT USING RISB AS IMPURITY SOLVER

In this section we first outline the DFT+DMFT scheme as implemented in Refs. [32,33] in order to define the notation. We next describe the implementation of the RISB method as a multiorbital impurity solver and its use for DMFT calculations. Finally, we derive a relation between quantities in the correlated subspace and the Bloch space which allows to determine the mass renormalization of the Bloch states.

A. DFT+DMFT scheme.

The first step is the solution of the DFT Kohn-Sham equations, which yield the Kohn-Sham energies, ε kν , and the corresponding states, |ψ k,ν , classified by the crystal momentum k and a band index ν. The second step is the treatment of the strong local correlations using DMFT. To that aim, a set of Wannier orbitals |χ R m is constructed, where R labels a lattice site and m denotes the orbital and spin degrees of freedom [34]. We define C as the space spanned by the set of correlated Wannier orbitals at a given site, and we omit the site label in the following. In the so-called projective method, only Bloch bands whose energy lies within a predefined energy window W are used in the construction of the Wannier orbitals. We define K as the space spanned by all the Bloch states whose energy lies within W, and K( k) as a subspace of K formed by states with a definite crystal momentum k. We also define P ( k) as the transformation operator from K( k) to C whose matrix elements are P m,ν ( k) = χ m |ψ k,ν . Denoting by 1 C and 1 K( k) the identity matrices in C and K( k) respectively, these transformations satisfy: P ( k)P † ( k) = 1 C , but the converse (P † ( k)P ( k) = 1 K( k) ) only is fulfilled if the number of bands at k and within W is equal to the number of Wannier orbitals in C.

The lattice Green's function in Matsubara representation reads:

G -1 ( k, iω n ) = iω n + µ 1 K( k) -ε( k) -Σ( k, iω n ), (1)
where ε( k) is a diagonal matrix formed by the Kohn-Sham eigenvalues of all the bands within W at a given k point, and Σ( k, iω n ) is constructed by embedding in K a local self-energy, Σ imp (iω n ), calculated through an auxiliary quantum impurity problem introduced by DMFT:

Σ( k, iω n ) = P † ( k) Σ imp (iω n ) -Σ dc P ( k), (2)
where Σ dc is a correction included to reduce the double counting of interactions in the DFT+DMFT method.

The quantum impurity problem consists of a local term, H loc , which includes one-body energies and interaction terms in C; and an hybridization term, which describes the coupling of these Wannier orbitals to an effective non-interacting fermionic bath, which is determined selfconsistently. The local Hamiltonian reads

H loc = mm ε 0 mm d † m d m + H int , (3)
where H int describes the interactions, and ε 0 are the one-body energies, which we compute as

ε 0 = 1 N k P ( k)ε( k)P † ( k) -Σ dc . ( 4
)
where N is the number of k-points. The effective bath is described by an hybridization function, ∆(iω n ), which is determined at each step of the DMFT cycle.

From the lattice Green's function [see Eq. ( 1)] we define a local Green's function in C

G loc (iω n ) = k P ( k)G( k, iω n )P † ( k), (5)
while the impurity Green's function of the DMFT auxiliary problem reads

G imp (iω n ) = [(iω n +µ)1 C -ε 0 -∆(iω n )-Σ imp (iω n )] -1 , (6)
where Σ imp (iω n ) is determined solving the auxiliary quantum impurity problem for a given ∆(iω n ). The DMFT self-consistency is fulfilled for a ∆(iω n ) such that G imp = G loc .

B. RISB as impurity solver

In the RISB formalism [22][23][24][25]35] the physical fermionic operators d m , which destroy an electron in the Wannier state |χ m , are represented as a linear combination of an equal number of auxiliary fermionic operators f m :

d m = m R Cmm [{φ AB }]f m . (7)
Here, the matrix R C is a function of a set of auxiliary boson fields {φ AB }, where the indices A and B refer to the local multiplets, and describes the different processes by which an electron can be destroyed. Its form ensures that the matrix elements of the d m operators remain the same in the new representation [23].

In the enlarged Hilbert space, spanned by the auxiliary fermion and boson fields, H loc is represented as a quadratic form in the auxiliary boson operators:

H loc = AB A|H loc |B n φ † An φ Bn . (8)
A one-to-one mapping with the original local Hilbert space is obtained with the introduction of timeindependent Lagrange multipliers λ 0 and Λ Cmm that enforce the following constraints:

A,B,C φ † CA φ CB B| Ô|A = Ô, (9)
where Ô = {1, f † m f m }. In the saddle-point approximation, the boson fields are replaced by classical numbers, and the self-energy acquires a simple form

Σ imp (iω n ) = iω n Σ imp 1 + Σ imp 0 , (10)
where

Σ imp 1 = 1 C -[R C R † C ] -1 , (11)
effectively renormalizes the hybridization with the noninteracting bath, and

Σ imp 0 = 1 C -[R C R † C ] -1 µ + R †-1 C Λ C R -1 C -ε 0 , (12)
renormalizes the level positions.

Replacing the self-energy of Eq. ( 10) in Eq. ( 6) the impurity Green's function reads

G imp = R C iω n + µ 1 C -Λ C -R † C ∆(iω n )R C -1 R † C . (13)
The auxiliary fermionic fields f m can be interpreted as quasiparticle degrees of freedom with a quasiparticle weight

Z C = R C R † C .
Their associated Green's function is [using Eq. ( 7)]

G imp qp = iω n + µ 1 C -Λ C -R † C ∆(iω n )R C -1 . (14)
C. RISB method in the DFT+DMFT scheme

When the RISB technique is used to solve DMFT's impurity problem, the relations between physical and quasiparticle quantities (e.g. operators and correlators) introduced in the subspace C are expected to have analogues in K. In particular, the physical fermionic operator c † kν which creates an electron in the Kohn-Sham state |ψ k,ν can be related to quasiparticle operators ckν through transformation matrices R Kνν ( k):

c kν = ν R Kνν ( k)c kν . ( 15
)
and accordingly, the lattice Green's function [see Eq. ( 1)] be written in terms of the quasiparticle lattice Green's function, G qp ( k, iω n ):

G( k, iω n ) = R K ( k)G qp ( k, iω n )R † K ( k). (16)
In the RISB saddle-point approximation the lattice self-energy reads:

Σ( k, iω n ) = iω n P † ( k)Σ imp 1 P ( k) + P † ( k) Σ imp 0 -Σ dc P ( k), (17)
which leads to

P † ( k)Σ imp 1 P ( k) = 1 K( k) -Z -1 K ( k),
and using Eq. ( 11) to

Z -1 K ( k) = 1 K( k) -P † ( k)P ( k) + P † ( k)Z -1 C P ( k), (18)
that accounts for the mass renormalization of the Bloch states due to the electronic correlations [36]. Equation (18) gives the quasiparticle weight in momentum space in terms of the quasiparticle weights obtained from DMFT's auxiliary quantum impurity problem. Note that this relation was obtained using general assumptions for the low energy behavior of the selfenergy in the Bloch and Wannier basis and can therefore be used independently of the quantum impurity solver employed to calculate Z C . Within this theory, the k dependence of the mass enhancement stems from the different amplitudes the Bloch states have in the Wannier states of the correlated subspace C.

The quasiparticle lattice Green's function can be calculated from Eqs. ( 1) and ( 16) (see Appendix B). We have:

G -1 qp ( k, iω n ) = iω n + µ 1 K -Λ K ( k) -R † K ( k)ε( k)R K ( k), ( 19
) where Λ K ( k) = P † ( k)Λ C P ( k) and ε( k) = ε( k) - P † ( k) ε 0 + Σ dc P ( k).
From the definition of ε 0 [see Eq. ( 4)] it follows that the energies ε( k) satisfy k P ( k)ε( k)P † ( k) = 0, and the level energies are therefore controlled by Λ K ( k) while the transformation matrix R K ( k) renormalizes the bandwidth. In Appendix A we show that R C and R K can be related by an equation analogous to Eq. (18).

The quasiparticle wave-functions |ψ QP k,ν can be obtained as the eigenvectors of G -1 qp ( k, iω n → 0), and are linear combinations of the Kohn-Sham basis provided by the DFT calculation, |ψ k,ν .

|ψ QP k,ν = ν U † ν,ν ( k)|ψ k,ν . (20)
The mass renormalization of the quasiparticle states is obtained applying the unitary transformation U to the low energy expansion of the lattice self-energy function [given by Eq. ( 2)], which leads to

Z QP K ( k) = U( k)Z K ( k)U † ( k). (21)
A point k F belongs to the renormalized Fermi surface S if there is a state |ψ QP k F ,ν F which is a zero energy eigenvector of the function G -1 qp ( k F , iω n → 0). In the general case, the matrix Z QP K ( k) is non-diagonal. We may, however estimate the quasiparticle weight associated with a Fermi sheet ν F by projecting Z K ( k) onto the corresponding quasiparticle states at the Fermi level:

Z QP ν F ( k F ) = ψ QP k F ,ν F |Z K ( k F )|ψ QP k F ,ν F . (22)
The quasiparticle weight Z QP K ( k) provides information about the mass renormalization obtained at particular line cuts of the Brilloin zone, which can be obtained performing Angle Resolved Photoemission (ARPES) or dHvA oscillations experiments.

As a benchmark of the method, in Appendix C we present a comparison of results obtained by solving the quantum impurity problem with RISB or with CTQMC for the cubic perovskite SrVO 3 .

III. APPLICATION TO Sr2RuO4

The ruthenate Sr 2 RuO 4 was analyzed in Ref. [2] using LDA+DMFT(CTQMC) in the absence of SOC and more recently in Ref. [13] including the SOC at the DFT level. These studies show that the Hund's rule coupling plays a central role in explaining the magnitude of the quasiparticle weight and its orbital differentiation. The magnitude of the latter being also affected by the presence of a Van Hove's singularity near the Fermi level. These results present a remarkable agreement with the experimental effective masses and with ARPES and NMR data. Although including the SOC reduces the degeneracy of the local multiplet structure it does not change the Hund's correlated metal nature of the compound.

We apply below the DFT+DMFT(RISB) method detailed in the previous section to analyze the electronic structure of Sr 2 RuO 4 . We take the experimental crystal structure extracted from Ref. [37] with lattice parameters a = 3.862 Å, c = 12.722 Å. We use the wien2k code [START_REF] Blaha | An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties[END_REF] with the triqs interface for DFT+DMFT [32,33,[START_REF] Aichhorn | [END_REF] and consider the Local Density Approximation (LDA) for the exchange and correlation potential at the DFT level using a dense k mesh of 39 × 39 × 39 points [40]. To construct the Wannier orbitals we take an energy window, W s = [-3 eV, 1. The Coulomb interaction within the t 2g manifold is described with the rotationally invariant Kanamori Hamiltonian:

H int = U m n m↑ n m↓ + (U -2J) m =m n m↑ n m ↓ + + (U -3J) m>m ,σ n mσ n m σ + (23)
+ J m =m (d † m↑ d † m↓ d m ↓ d m ↑ -d † m↑ d m↓ d † m ↓ d m ↑ ).
Here, n mσ = d † mσ d mσ , U is the intraorbital interaction and J is the Hund's rule coupling. The values of U and J have been estimated in Ref. [2] using the constrained random phase approximation (U =2.3eV and J=0.4eV, with J/U =0.17) and in Ref. [41] using the constrained local density approximation (U =3.1eV and J=0.7eV, J/U =0.23). If the SOC interaction is not included in the calculations, the local multiplet structure is formed by a doublet |xyσ and a quartet {|xzσ , |yzσ }, the former having a lower energy. Upon inclusion of the SOC in the DFT calculation, the degeneracy of the quartet is broken leading to three doublets, |0 , |1 and |2 . These were calculated by diagonalization of Eq. 4, and can be approximately associated with the states |J = 3/2, m J = ±1/2 , |J = 1/2, m J = ±1/2 , and |J = 3/2, m J = ±3/2 , respectively. The results for the one body energies and the quasiparticle weights are presented in Table I for calculations using U = 3.1eV and J/U = 0.2 and T = 5K [42].

Although the DFT-DMFT(RISB) results capture the Hund's metal behavior of the system, showing a strong suppression of the charge fluctuations to states with spin other than the maximum, it underestimates the enhancement of quasiparticle renormalization compared to the one obtained by other numerically exact impurity solvers as CTQMC [43]. This has been established before [9] and is a feature shared with other related techniques like the Gutzwiller approximation [44][45][46], or the slavespins formalism [47]. One finds moderate enhancement of the quasiparticle mass ∼ 2 with a slightly smaller enhancement in the xy orbital. This is different to what is obtained using CTQMC where larger mass enhancements ∼ 4 are found, even at smaller values of the interaction parameters. Additionally, the fact that the mass enhancement in the xy orbital is smaller may suggest that the slave-bosons are more sensitive to the overall bandwidth than to the low-energy fine-structure in the DOS. Later in the text we will see that in spite of this discrepancies the material trends, as the epitaxial strain is changed, are consistent with what is found using CTQMC.

As it was mentioned in Section I, the renormalized masses (with respect to LDA) of the α and β sheets of the Fermi surface have been reported to be ∼ 3 and ∼ 3.5, respectively [21]. Note that this experimental finding suggests that, altough possible smaller, there must be additional sources of momentum differentiation of the mass enhacement. Indeed, the inequality m * m LDA | α < m * m LDA | β , or rather Z α > Z β , is not evident since the Bloch states that conform these sheets have mainly xz and yz symmetry and, as a result, their weight in the correlated subspace lies on degenerate cubic Wannier orbitals. In the following, we show that the spin-orbit coupling enhances the k dependence of Z QP K ( k). When the SOC is turned on, the doublet |2 is lower in energy than the degenerate orbitals xz and yz in the absence of SOC, as can be observed in Table I. Consequently, while |2 increases its occupancy, which drives the orbital away from half-filling and, in turn, gives place to a larger quasiparticle weight, the opposite happens for |1 . To better compare with the calculation done without SOC, it is naturally convenient to analyze the changes in the basis of cubic Wannier functions. We find that both the orbital polarization n xy -n xz and the quasiparticle weights Z C xy and Z C xz remain essentially constant upon inclusion of the SOC [48]. This agrees with the CTQMC results of Ref. [13] and supports the conclusion that the SOC does not affect the coherence scale of Sr 2 RuO 4 .

The SOC does, however, modify the way the quasiparticle weight varies in K space. Figures 2(a) and 2(b) present the projected quasiparticle weight Z QP ν F ( k F ) (for ν F = α, β, γ) [see Eq. ( 22)] associated with the Fermi surface Bloch states calculated without or with the SOC turned on in the DFT calculations, respectively. In the absence of SOC the only dependence on k comes from the difference between Z C xy and Z C xz . Along each Fermi surface sheet the quasiparticle weight is essentially constant. The inclusion of the SOC leads to a richer momentum differentiation, which is larger at k x = k y , consistently with the stronger spin-orbital entanglement experimentally observed by spin-resolved ARPES at that line cut of the Brillouin zone [10,14]. In particular, the relation Z α > Z β can only be accounted for in our calculations if the SOC is included.

The effect of the SOC on Z QP ν F ( k F ) can be understood by analyzing the change in the local multiplets and the subsequent induced charge redistribution. Figs. 2(c-e) present the projections of the Bloch states of each sheet onto the different SO multiplets |0 , |1 and |2 along the corresponding parametrized angles θ α and θ βγ , indicated in Fig. 2(b). The γ band has most of its amplitude on the state |0 along almost the whole range of θ βγ . The α sheet has a larger amplitude on the doublet |2 , while the β one has it on |1 , specially at θ α = π/4. The charge redistribution produced by the SOC, which acts to decrease the quasiparticle weight of |1 while incresing that of |2 contributes, therefore, to obtaining Z α > Z β .

IV. EFFECTS OF ANISOTROPIC STRAINS

Motivated by the recent experimental results reported in Ref. [15] and [16], in this section we analyze how the electronic correlations evolve in the presence of anisotropic strains. An interesting question is to what extent the Lifshitz transition, in which the Van Hove singularity (VHS) crosses the Fermi level, affects the electronic correlations. We study two situations that have been experimentally addressed and can induce this transition: a biaxial tension that leads to an increase of the lattice parameters a and b, and also the response of the system to an uniaxial compression along the [100] direction. In all cases, we use the experimental lattice parameters a, b and c [START_REF]A given value of uniaxial (biaxial) deformation fixes a[END_REF] and relax within LDA the internal positions of the apical oxygen and of the transition metal atom.

Figure 3(a) shows the evolution of h/d as a function of a, where h is the calculated apical oxygen height and d=a/2 the distance between the Ru and its next oxygen neighbors. For both uniaxial and biaxial strains, h/d evolves towards a more regular octahedron, from 1.07 to 1.03. In the biaxial case, consequently, the splitting be- tween the t 2g bands diminishes. That is, the on-site xy energy decreases while the {xz, yz} increases, contributing to transfer charge from the {xz, yz} orbitals to the xy one, as confirmed by the evolution of the corresponding occupancies [see Fig. In the presence of a VHS, it is convenient to analyze the strength of the correlations in terms of an effective bandwidth W ef f calculated through the second moment of the density of states, as suggested in Ref. [START_REF] Žitko | [END_REF]. The calculated W ef f for the t 2g bands are shown in Figs. 3(c) and 3(d). It can be observed that, even for the xy orbital, the W ef f decreases smoothly with a. The relative change is only slightly larger in the {xz, yz} bands than in the xy one.

| χi|ψ kν | 2 θ α 0 π 4 (c) | 2|α | | 1|α | | 0|α | θ βγ 0 π 4 (d) | 1|β | | 2|β | | 0|β | θ βγ 0 π 4 (e) | 0|γ | | 1|γ | | 2|γ |
In the following, we present how the electronic correlations, as measured by the quasiparticle weight, evolve under the two kinds of deformation mentioned above.

A. Biaxial strain

Here we present results for strained Sr 2 RuO 4 obtained with the formalism introduced in Sec. II and DMFT calculations using CTQMC as impurity solver.

Figures 4(a tion of the lattice parameter a. The RISB results depend very weakly on the temperature for T 30K. The inclusion of the SOC introduces small changes in the quasiparticle weight. In the range of lattice parameters considered, the observed reduction upon including the SOC is less than 1%. Both Z xy and Z xz,yz decrease monotonically as a increases from the unstrained case, in line with the monotonic reduction of W ef f shown in Figs. 3(c) and 3(d). The smaller variation rate of Z xy can be associated with two effects. First, the bandwidth reduction of the xy band is percentually smaller than that of the {xz,yz} bands. Second, there is a compensating effect in the correlations generated by the increase in the occupancy of the xy orbital with a which drives it further away from half-filling.

We also performed CTQMC calculations for Sr 2 RuO 4 under biaxial strain. For these calculations we used the parameters U =2.3eV and J=0.4eV. The obtained quasiparticle weights, presented in Fig. 4, indicate, similarly to the RISB results, that the biaxial stress has a stronger effect on the xz, yz orbitals. The mass enhancement is computed using a 4th order polynominal fit to the lowest six Matsubara points of the self-energy shown in Fig. 4(cd). Typically, the values of mass renormalization fluctuate by the order of 10% from DMFT iteration to iteration, hence we present the data with such errorbar. At T = 29 K, the effective mass of xz-yz orbitals presents an increment relative to the unstressed compound of ∼ 35% (from 3.4 to 4.6), while that of the xy orbital is of the order of the statistical error, ∼ 10%.

It is important to point out that the range of values of the lattice parameter a studied, includes the Lifshitz transition of this compound. Figure 5 shows the Fermi surfaces calculated with RISB including the SOC for three different values of a= 3.862, 3.890, 3.929 Å. It can be observed that while the electron-like β and the hole-like α sheets shrink with increasing a, the γ one (having mainly xy symmetry) undergoes a Lifshitz transition, changing its character from electron-like to holelike. This theoretical result agrees with the experimental data reported in Ref. [16], obtained for thin films grown on top of different substrates.

The Lifshitz transition occurs already at the LDA level [18]. The effect of an improved treatment of the correlations through RISB is to slightly decrease the value of a at which the transition takes place (a c ). More precisely, LDA gives a c =3.98 Å, RISB a c =3.91 Å and RISB with SOCa c =3.89 Å. The value of a c naturally depends on U because this parameter directly affects the occupancy of the xy orbital.

The main conclusion to point out from these results is that there is no significant effect on the electronic correlations, as measured by Z, associated with the occurrence of the Lifshitz transition (see also Ref. [3]). This observation is in qualitative agreement with the experimental results of Ref. [16].

B. Uniaxial strain

In this section, we study the evolution of the correlation strength as a function of the uniaxial compressive stress xx . As mentioned before, we use the experimental lattice parameters a, b and c [15] and relax the internal positions. The results are expected to be symmetric with respect to tensile stress.

Overall, the effect of uniaxial stress on Z is much weaker than in the biaxial case, basically because the induced [100] relative distortion is smaller (up to 0.8 %). On average, the quasiparticle weight of the t 2g states slightly increases with uniaxial pressure. Moreover, as the case of biaxial distortion, the correlation strength evolves monotonously through the Lifshitz transition. Figures 6(a) y 6(b) show the evolution of the quasiparticle weigth corresponding to the xz, yz and xy states, respectively. We present only the RISB results without SOC, since the effect of SOC is negligible for these small values of compression.

It can be observed that, again, the variation rate of Z xy is much smaller than that of Z xz,yz . This is consistent with the almost constant behaviour of the corresponding effective bandwidth and occupancy under uniaxial strain in the former case [see Fig. 3 (d)].

V. CONCLUSIONS

In this article we presented an implementation of the multiorbital slave-boson rotationally invariant (RISB) quantum impurity solver within the LDA+DMFT approach to ab initio calculations of strongly correlated systems. The main disadvantage of the RISB solver is that in general it can only be trusted at a qualitative level and that it fails to describe non-Fermi liquid behaviors besides trivial insulating phases. Its main advantages, compared to other numerically exact solvers as Quantum Monte Carlo, are its much lower computational cost at low temperatures which allows to rapidly explore a wide range of parameters and materials, and the abilty of easily handling off-diagonal hybridizations at arbitrary low temperatures in problems that in general give place to a sign problem in CTQMC. It can therefore be used as a tool to explore and identify correlated regimes worth of a more detailed study using other techniques as CTQMC or numerical renormalization group approaches. An important result of our implementation is that we obtained transformation relations between physical quantities, as the quasiparticle mass renormalization, in the local correlated space and their counterparts in Bloch space. This transformation allows to obtain the momentum dependence of the quasiparticle mass renormalization independently of the quantum impurity solver used to treat the DMFT equations. We applied the RISB approach to study the electronic correlations, as measured by the quasiparticle mass enhancement, of Sr 2 RuO 4 . We found that it is necessary to include the spin-orbit coupling in the DFT calculations to explain the experimentally observed mass enhancement differentiation between the α and β sheets of the Fermi surface. We also find that the SOC strongly enhances the momentum dependence of the quasiparticle weight on both sheets while the γ sheet is largely unaffected. A biaxial stretching of the compound on the a-b plane leads to a monotonic increase of the mass enhancement that is larger for the xz and yz orbitals, which determine the mass enhancement of the α and β Fermi sheets, than for the xy orbital, which determines the mass enhancement of the γ Fermi sheet. While the sample suffers a Lifshitz transition for a = b ∼ 3.9Å, we did not find any significant change of behavior of the mass enhancement across it. We also performed calculations using the numerically exact CTQMC solver which confirmed the absence of any dramatic effect of the electronic correlations at the Lifshitz transition.

Finally, we analyzed the behavior of the mass enhancement when the sample is under uniaxial strain. As in the biaxial case, we found a monotonic behavior with no significant features across the Lifshitz transition.

FIG. 1 .

 1 FIG.1. LDA density of states of Sr2RuO4 projected on the Ru-t2g orbitals and within the energy window Ws considered in this work to construct the Wannier orbitals.

FIG. 2 .

 2 FIG. 2. Quasiparticle weight on the Fermi surface calculated without including the spin-orbit interaction (a) or including it (b). (c) Projection of the Bloch states of the α sheet of the Fermi surface on atomic multiplets, as a function of the angle θα defined in (b). (d) and (e) present as a function of the angle θ βγ the projection on atomic multiplets of the Bloch states from the β and γ sheets, respectively.

  3(b)] [50]. The uniaxial compression mainly transfers charge from xz to yz, the reason being the increased level energy of the xz orbital due to the shorter distances along x.

FIG. 3

 3 FIG. 3. a) h/d as a function of a, where h is the height of oxygen octahedron and d = a/2 the distance betwen the rutenium atom and its first neighbor oxygen. Empty circles and red filled squares correspond to uniaxial and biaxial stress, respectively. b) Occupancy of Wannier orbitals according to LDA. Effective bandwidth of c) xz, yz and d) xy orbitals, as defined in the text.

FIG. 4 . 1 FIG. 5 .

 415 FIG. 4. Quasiparticle weight of the {xz, yz} (a) and xy orbitals (b) as a function of lattice parameter a calculated using RISB at T = 5K. Empty red and filled blue symbol datapoints are obtained without and with SOC, respectively. Triangle symbols are CTQMC results without SOC. (c-d) Imaginary part of the self-energy calculated with CTQMC at different temperatures. The interaction parameters are U = 2.3 eV and J = 0.4 eV.

FIG. 6 .

 6 FIG. 6. (a) Quasiparticle weight of the orbital {xz, yz} as function of the stress xx at T = 5 K. (b) Quasiparticle weight of the orbital xy as function of the stress xx. c-e) Evolution of the renormalized Fermi surfaces for the percentage uniaxial stress xx= 0, -0.2 y -0.5% (from left to right) showing that the system undergoes a Lifshitz transition close to xx=0.2% around the ( π 2a , 0) point.

TABLE I

 I

	. Diagonal one-body energies (ε 0 ), quasiparticle
	weights (ZC), and occupancies (n) corresponding to Wan-
	nier orbitals constructed using the energy window Ws = [-3 eV, 1.3 eV], both with and without including the SOC.
	Occupancies and quasiparticle weights correspond to the
	LDA+RISB calculation with parameters U = 3.1 eV and
	J/U = 0.2.

ACKNOWLEDGMENTS

This work was partly supported by program ECOS-MINCyT France-Argentina under project A13E04 (J.F, L.P., V.V, P.C). J.F and P.C. acknowledge support from SeCTyP UNCuyo Grant 06/C489 and PICT-201-0204 ANPCyT. V.V. thanks also financial support from AN-PCyT (PICT 20150869, PICTE 2014 134), CONICET (PIP 00273/12 GI). This work was also supported by the European Research Council grants ERC-319286-QMAC (J.M., L.P.). J.M. acknowledges support by the Slovenian Research Agency (ARRS) under Program P1-0044.

Appendix A: Inverse relations

The inverse of Eq. ( 18) can be obtained by applying P ( k) and P † ( k) at left and right, respectively, and using P ( k)P † ( k) = 1 C , it reads:

The asymmetry between Eqs. (18) and (A1) comes from the fact that the transformations P ( k) are not unitary in the general case. Based on Eq. ( 18), we define:

and in analogy with Eq. (A1) we obtain:

From Eq. (A2) it can be shown by direct calculation that the quasiparticle mass renormalization in K space satisfies

The relation between transformation matrices given by Eq. (A2) is consistent with the relations between quasiparticle and physical Green's functions in both K and C spaces. To show this we transform Eq. ( 16) to C space. We first derive the following relation between the transformation matrices in C and K.

by applying P ( k) to the left of Eq. (A2). Similarly, we also obtain:

From Eq. ( 16) it follows that:

We project this equation to C multiplying at left by P ( k), at right by P † ( k) and summing over k:

Using Eqs. (A5) and Eq. (A6), we have:

and finally

which is the expected relation between the physical and the quasiparticle Green's functions in C.

Appendix B: Derivation of Eq. ( 19).

The quasiparticle lattice Green's function can be computed using the DFT+DMFT equations and the selfenergy obtained in the RISB saddle-point approximation. To this aim, we substitute in Eq. ( 1) the self-energy given by Eq. (10). The terms linear in frequency and in the chemical potential read (in the following we omit the dependence in k of P and R K ):

where we have used Eq. ( 18). The term proportional to Λ C reads:

where the equality follows using Eqs. (A5) and (A6).

The other terms are:

The summation of these terms and use of Eq. ( 16) leads to Eq. ( 19).

Appendix C: Application to SrVO3

Here we present as a benchmark a comparison between results obtained with LDA+RISB and LDA+DMFT (CTQMC) for the compound SrVO 3 . We use values of U = 4.5 eV and J = 0.6 eV, and construct Wannier functions for the t 2g bands within the energy window [-1, 2] eV. Fig. 7 presents the spectral density as function of the energy for each of the three-fold degenerated t 2g Wannier orbitals. It can be observed that the RISB method provides a reasonably good approximation at low energies in spite of subestimating the bandwith renormalization. The values of Z obtained are ∼ 0.5 and ∼ 0.65 for CTQMC and RISB, respectively.


Tamar R Makin

Elena Amoruso

Maite Arribas

Sarah Buehler

Lucy Dowdall

Paulina Kieliba

Benjamin Kop

Mathew Kollamkulam

Maria Kromm

Roni Maimon Mor

Maria Molina Sanchez

Yagmur Dollyane Muret

Idil Ozdemir

Vicky Root

Phantom limbs and brain plasticity in amputees (pre-review version)

Keywords: Assistive technology, motor, phantom limb pain, prosthetic limbs, rehabilitation, remapping, reorganisation, sensations, sensorimotor, somatosensory

Phantom sensations are experienced by almost every person who has lost their hand in adulthood. This mysterious phenomenon spans the full range of bodily sensations, including the sense of touch, temperature, movement, and even the sense of wetness. For a majority of upper limb amputees, these sensations will also be at times unpleasant, painful, and for some even excruciating to the point of debilitating, causing a serious clinical problem, termed phantom limb pain (PLP). Considering the sensory organs (the receptors in the skin, muscle or tendon) are physically missing, to understand the origins of phantom sensations and pain we must turn to the nervous system, and the brain in particular. This raises the question -what happens to a fully developed part of the brain that becomes functionally redundant (e.g. the sensorimotor hand area after arm amputation)? Relatedly, what happens to the brain representation of a body part that becomes over-used (e.g. the intact hand, on which most amputees heavily rely for completing all daily tasks). Classical studies in animals show that the brain territory in primary somatosensory cortex (S1) that was "freed up" due to input loss (hereafter deprivation) becomes activated by other body part representations, neighbouring the deprived cortex. This process is termed reorganisation, or more accurately, remapping.

If neural resources in the hand area get redistributed to facilitate the representation of other body parts following amputation, how does this process relate to persistent phantom sensation arising from the amputated hand? Using arm amputation as a model, subsequent work in humans, mostly with non-invasive neuroimaging and brain stimulation techniques, have expanded on the initial observations of cortical remapping in two important ways. First, research with humans allows us to study the perceptual consequence of remapping, particularly with regards to phantom sensations and pain. Second, by considering the various compensatory strategies amputees adopt in order to account for their disability, including over-usage of their intact hand and learning to use an artificial limb, use-dependent plasticity can also be studied in amputees, and its relationship to deprivation-triggered plasticity. Both of these topics are of great clinical value, as these could inform clinicians how to treat phantom limb pain, and how to facilitate rehabilitation and prosthesis usage in particular. Here we review these studies in humans, with emphasis on the presumably paradoxical relationship between brain remapping in sensorimotor cortex, phantom sensations and emerging behaviours and technologies for substituting the missing hand's functionality. We consider the role of remapping and persistent representation in facilitating (or hindering) the realisation of these technologies, with special emphasis on the role of embodiment. Together, this chapter is aimed at providing a comprehensive outlook at the functional consequences of cortical remapping in amputees' primary sensorimotor cortex.

Cortical remapping in monkeys 1.

Cortical remapping in monkeys

What happens to the brain areas controlling the hand, once they become functionally redundant following arm amputation? Our original understanding of what happens to the cortical territories of the hand in the primary somatosensory cortex (S1) comes from studies in animals, and non-human primates in particular, using direct brain recordings (for review see [START_REF] Kaas | Cortical and subcortical plasticity in the brains of humans, primates, and rats after damage to sensory afferents in the dorsal columns of the spinal cord[END_REF]. In primates, S1 is organised in a somatotopic fashion, where inputs of different body parts are represented separately, with neighbouring body parts often represented adjacently, with some exceptionsnotably the hand representation neighbours that of the face ( [START_REF] Penfield | SOMATIC MOTOR AND SENSORY REPRESENTATION IN THE CEREBRAL CORTEX OF MAN AS STUDIED BY ELECTRICAL STIMULATION[END_REF]; Figure 1A). The hand area itself also contains a detailed somatotopy of the hand, where neighbouring digits are represented separately and adjacently ( [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF]; Figure 1B). This hand map makes it easy to notice any changes in the spatial layout of the map. What would happen to the highly organised hand area in S1 following amputation? A series of pioneering studies show that the brain territory that was "freed up" due to input loss (hereafter deprivation) becomes activated by other body part representations. If, for example, digit 3 of the hand gets amputated, over the course of weeks and months the digit 3 (now deprived) part of the hand map becomes activated by digits 2 and 4 ( [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF]; Figure 1C). If input from the entire hand and arm is deprived, due to deafferentation (cutting of the peripheral nerve), the hand area becomes activated by inputs from the lips and jaw [START_REF] Pons | Massive cortical reorganization after sensory deafferentation in adult macaques[END_REF]. This process has been termed cortical reorganisation, because the resulting boundaries of the body-part map are now changed. But perhaps a better term for describing this process is 'cortical remapping'. This is because, as we will explain below, the fact that the spatial layout of the map has been altered does not necessarily entail that the native functional organisation (i.e. the canonical digit maps) have been changed (see [START_REF] Makin | Stability of Sensory Topographies in Adult Cortex[END_REF] and [START_REF] Makin | Reorganisation in adult primate sensorimotor cortex: Does it really happen?[END_REF] for further reading). Cortical remapping has been ascribed to multiple plasticity mechanisms, such as unmasking of normally inhibited connections between representational areas [START_REF] Li | Rapid homeostasis by disinhibition during whisker map plasticity[END_REF], changes in subcortical projections to cortex [START_REF] Jain | Large-scale reorganization in the somatosensory cortex and thalamus after sensory loss in macaque monkeys[END_REF] and even axonal sprouting [START_REF] Florence | Large-scale sprouting of cortical connections after peripheral injury in adult macaque monkeys[END_REF][START_REF] Jones | Thalamic and brainstem contributions to large-scale plasticity of primate somatosensory cortex[END_REF] (though see [START_REF] Chand | Intracortical and Thalamocortical Connections of the Hand and Face Representations in Somatosensory Area 3b of Macaque Monkeys and Effects of Chronic Spinal Cord Injuries[END_REF] on changes after spinal cord injury). Since in both cases, the body-part representation that takes over the deprived territory is a cortical neighbour, it has been subsequently established that the opportunities for remapping in S1 are determined by the topographic neighbourhood relationships [START_REF] Kaas | Cortical and subcortical plasticity in the brains of humans, primates, and rats after damage to sensory afferents in the dorsal columns of the spinal cord[END_REF] (though see [START_REF] Hahamy | Remapping in Cerebral and Cerebellar Cortices Is Not Restricted by Somatotopy[END_REF]).

Subsequent research in humans has led to multiple useful models for studying S1 remapping, for example, temporary deafferentation (Dempsey- Jones, et al., 2019), sensory manipulation for altered synchronisation of tactile inputs, such as syndactyly, [START_REF] Kolasinski | Perceptually relevant remapping of human somatotopy in 24 hours[END_REF] or tactile training (e.g. perceptual learning, [START_REF] Harrar | Topographic generalization of tactile perceptual learning[END_REF], repetitive stimulation [START_REF] Muret | Touch improvement at the hand transfers to the face[END_REF] or tactile acuity in blind individuals [START_REF] Grant | Tactile perception in blind Braille readers: a psychophysical study of acuity and hyperacuity using gratings and dot patterns[END_REF]). Here we review recent research in humans, aimed at studying brain remapping following arm amputation, and how this relates to the mysterious phenomenon of phantom limb sensations and pain in particular. Many of the studies reviewed here are unable to reliably delineate changes occurring in S1 from primary motor cortex (M1) due to methodological limitations. In that case, we refer to the evidence as relating to sensorimotor cortex. [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF]J. M. Zook, 1984, Journal of Comparative Neurology, 224 (4), p. 594. Copyright 1984 by John Wiley and Sons.

2.

Phantom sensations 2.1 Phantom sensations phenomenology and prevalence

Phantom sensations (both painful and non-painful) are highly common following amputation, with the majority of amputees reporting some form of sensations from their missing (phantom) hand (see Table 1 in section 2.2). Non-painful phantom sensations are typically described as tingling, itching or numbness -like the limb is falling asleep [START_REF] Ehde | Chronic phantom sensations, phantom pain, residual limb pain, and other regional pain after lower limb amputation[END_REF][START_REF] Henderson | PHANTOM LIMBS[END_REF][START_REF] Van Der Schans | Phantom pain and health-related quality of life in lower limb amputees[END_REF], and are not necessarily bothersome. Interestingly, many amputees also report being able to voluntarily move their phantom hand, even being able to accurately report the range of movement for individual fingers [START_REF] Henderson | PHANTOM LIMBS[END_REF][START_REF] Shukla | Phantom limb: a phenomenological study[END_REF]. Whilst many amputees report some level of phantom mobility, there are a plethora of cases where involuntary movement or immobility of the phantom hand are reported [START_REF] Henderson | PHANTOM LIMBS[END_REF]. Similar to these varying reports in phantom mobility, the position of the phantom hand is also variable, with many amputees reporting the phantom fingers to be partially curled or even clenched into a fist or the phantom arm to be slightly bent at the elbow [START_REF] Henderson | PHANTOM LIMBS[END_REF]. For the majority of amputees, the phantom limb can also be the source of incredibly bothersome and even highly painful sensations, commonly referred to as phantom limb pain (PLP). These sensations are most commonly described as feeling sharp, shooting, stabbing, throbbing, burning, aching, and/or cramping [START_REF] Ehde | Chronic phantom sensations, phantom pain, residual limb pain, and other regional pain after lower limb amputation[END_REF][START_REF] Schley | Painful and nonpainful phantom and stump sensations in acute traumatic amputees[END_REF], but are not necessarily persistent, with PLP episodes varying in intensity, duration and frequency [START_REF] Whyte | Variation in phantom limb pain: results of a diary study[END_REF]. We surveyed amputee studies that included a sample size of at least 100 amputees (Table 1). We found that non-painful phantom sensations were typically more prevalent than painful phantom sensations in limbless populations (see Table 1 below). A recent meta-analysis found that PLP incidence rates are, typically, between 50% and 85% (mean 63%; 95% CI: 58.23-67.05; [START_REF] Limakatso | The prevalence and risk factors for phantom limb pain in people with amputations: a systematic review and meta-analysis[END_REF]. More research is necessary to characterise the onset and development of phantom sensations following amputation, and the significance of the body part that had been amputated (e.g. finger, arm, leg).

Telescoping of the phantom

Right after amputation, the phantom limb is typically perceived to be in the same spatial location as that previously occupied by the missing limb. With time however, some amputees report that the phantom hand starts to retract, towards the body, moving closer to the residual arm. This phenomenon is called 'telescoping'. Between 30% to 66% of amputees report experiencing some form of telescoping, the numbers vary across studies [START_REF] Carlen | Phantom limbs and related phenomena in recent traumatic amputations[END_REF][START_REF] Katz | Psychophysical correlates of phantom limb experience[END_REF][START_REF] Shukla | Phantom limb: a phenomenological study[END_REF][START_REF] Weiss | Extended and telescoped phantom limbs in unilateral amputees[END_REF]. The time course of telescoping is highly variable, and may take several years. The phantom limb decreases gradually with the proximal portion disappearing first, while the fingers usually retain their size [START_REF] Henderson | PHANTOM LIMBS[END_REF]. Once the phantom limb is 'fully telescoped', amputees describe the phantom fingers to be on or inside the residual limb (Figure 2).

Despite the high prevalence of amputees experiencing telescoping phenomenon, little is known about its cause. One theory is that telescoping might be related to cortical remapping and PLP. Early studies, however, did not find a relationship between cortical remapping and telescoping [START_REF] Flor | Phantom-limb pain as a perceptual correlate of cortical reorganization following arm amputation[END_REF][START_REF] Knecht | Cortical reorganization in human amputees and mislocalization of painful stimuli to the phantom limb[END_REF][START_REF] Knecht | Reorganizational and perceptional changes after amputation[END_REF]. A more recent unpublished observation [START_REF] Flor | Phantom limb pain: a case of maladaptive CNS plasticity[END_REF] indicated that movements of telescoped phantoms showed activity in the cortical shoulder/arm regions whereas non-telescoped phantoms showed activity in the hand region. While some authors observed a positive relationship between PLP and telescoping [START_REF] Grüsser | The relationship of perceptual phenomena and cortical reorganization in upper extremity amputees[END_REF], other studies found telescoping to be negatively associated with PLP [START_REF] Cronholm | Phantom limbs in amputees; a study of changes in the integration of centripetal impulses with special reference to referred sensations[END_REF][START_REF] Katz | Psychophysical correlates of phantom limb experience[END_REF][START_REF] Weiss | Extended and telescoped phantom limbs in unilateral amputees[END_REF] or did not find a relationship between PLP and telescoping [START_REF] Lacoux | Treatment of neuropathic pain in Sierra Leone[END_REF][START_REF] Montoya | The relationship of phantom limb pain to other phantom limb phenomena in upper extremity amputees[END_REF]. To date, the nature of the cortical mechanisms underlying telescoping and its relation to PLP is still not well understood.

Referred Sensations onto the phantom

For some individuals, phantom sensations can be triggered through tactile stimulation of the residual limb [START_REF] Grüsser | The relationship of perceptual phenomena and cortical reorganization in upper extremity amputees[END_REF], which may simply be a consequence of peripheral reinnervation. A severed sensory axon typically regrows and spontaneously reinnervates intact skin, for example, on the residual arm [START_REF] Devor | Phantom Pain as an Expression of Referred and Neuropathic Pain[END_REF]. As a result, it is thought that touch applied to the reinnervated skin can produce signals that are mislabelled by the central nervous system as arising from the missing hand and result in a sensation projected to the missing hand.

A more curious phenomenon is when phantom sensations are triggered through stimulation of other body parts, such as the face. In a famous series of case studies, three upper-limb amputees reported feeling referred phantom sensations on the missing hand following touch to the lower part of the face [START_REF] Ramachandran | Behavioral and magnetoencephalographic correlates of plasticity in the adult human brain[END_REF][START_REF] Ramachandran | Review of Perceptual correlates of massive cortical reorganization[END_REF]. Importantly, neighbouring sites on the face elicited sensations on neighbouring fingers, suggesting a topographical organisation of referred sensations. These findings were interpreted as the perceptual correlate for the remapping of the face onto the cortical area of the missing hand. But other studies have shown that touches applied on many different body parts elicit reported referred phantom sensations, including those which were not known to remap into the missing hand cortex (e.g., feet, trunk, and neck, or even body parts contralateral to the missing limb [START_REF] Grüsser | The relationship of perceptual phenomena and cortical reorganization in upper extremity amputees[END_REF][START_REF] Knecht | Reorganizational and perceptional changes after amputation[END_REF]. Such findings weaken the, unsupported, hypothesis that referred sensations result from S1 remapping.

Maladaptive plasticity and phantom limb pain

As highlighted in section 2.1, many amputees experience PLP. However PLP is notoriously difficult to treat [START_REF] Knotkova | Current and future options for the management of phantom-limb pain[END_REF]. To date, there is a lack of consensus regarding the neural basis of this chronic condition [START_REF] Makin | Brain (re)organisation following amputation: Implications for phantom limb pain[END_REF]. Early theories proposed PLP to be the result of ectopic discharge originating from stump neuromas [START_REF] Mitchell | Injuries of Nerves and Their Consequences[END_REF][START_REF] Wall | Properties of afferent nerve impulses originating from a neuroma[END_REF] citing peripheral, nociceptive input as a primary cause. Nevertheless, clinical approaches using local anaesthetic to the nerve endings produced only partial pain relief [START_REF] Borghi | The use of prolonged peripheral neural blockade after lower extremity amputation: the effect on symptoms associated with phantom limb syndrome[END_REF][START_REF] Nyström | Microelectrode recordings from transected nerves in amputees with phantom limb pain[END_REF], suggesting central mechanisms also play a role in PLP development.

Seminal work addressing central drivers of PLP ascribe to a maladaptive plasticity model, whereby pain arises as a perceptual consequence of cortical reorganisation following limb loss [START_REF] Flor | Phantom-limb pain as a perceptual correlate of cortical reorganization following arm amputation[END_REF]. Specifically, it is based on the idea that S1 sensory deprivation after an amputation leads to degradation of the missing hand representation, which is consequently 'invaded' by neighbouring representations, such as the lips [START_REF] Ramachandran | The perception of phantom limbs. The DO Hebb lecture[END_REF]. As a result, the deprived hand area responds to input intended for adjacent cortical territories, and the mismatch between the body part representations generates an 'error' signal, which is subsequently interpreted as pain arising from the phantom hand. The first evidence supporting this model demonstrated a correlation between the shift in the lip representation towards the missing hand region and the intensity of PLP reported by amputees [START_REF] Flor | Phantom-limb pain as a perceptual correlate of cortical reorganization following arm amputation[END_REF]. This finding has been further replicated and confirmed in other studies using different paradigms and participant groups (see [START_REF] Flor | Phantom limb pain: a case of maladaptive CNS plasticity[END_REF] for a review).

However, recent studies have contested the fundamental assumptions of the maladaptive theory [START_REF] Jutzeler | Relationship between chronic pain and brain reorganization after deafferentation: A systematic review of functional MRI findings[END_REF][START_REF] Mezue | Immutable body representations: lessons from phantoms in amputees[END_REF]. Amputees' deprived hand area is not remapped by the lower face relative to controls [START_REF] Kikkert | Reaffirming the link between chronic phantom limb pain and maintained missing hand representation[END_REF]Makin, Scholz, et al., 2013;[START_REF] Philip | Reorganization of primary somatosensory cortex after upper limb amputation may lack functional significance[END_REF]. Shifts in the lip representations in arm amputees are smaller than previously reported, and seem to be limited to the face region itself rather than 'invading' the missing hand cortex [START_REF] Makin | Reassessing cortical reorganization in the primary sensorimotor cortex following arm amputation[END_REF][START_REF] Raffin | Primary motor cortex changes after amputation correlate with phantom limb pain and the ability to move the phantom limb[END_REF]. In fact, Kambi and colleagues [START_REF] Kambi | Largescale reorganization of the somatosensory cortex following spinal cord injuries is due to brainstem plasticity[END_REF] show that the expansion of the lower face area into the deafferented hand region of S1 in monkeys depends on subcortical rather than cortical 'reorganisation' (Figure 3). These findings highlight the need for continued research into the central and peripheral drivers of PLP.

A key implication of the maladaptive theoretical framework is that reversing cortical remapping by re-establishing the missing hand representation in its initial cortical position, would alleviate PLP. However, numerous treatments have attempted this with mixed effectiveness [START_REF] Barbin | The effects of mirror therapy on pain and motor control of phantom limb in amputees: A systematic review[END_REF][START_REF] Thieme | The Efficacy of Movement Representation Techniques for Treatment of Limb Pain-A Systematic Review and Meta-Analysis[END_REF], including the famous mirror-box treatment (see [START_REF] Herrador Colmenero | Effectiveness of mirror therapy, motor imagery, and virtual feedback on phantom limb pain following amputation: A systematic review[END_REF] for an overview), virtual reality paradigms [START_REF] Ortiz-Catalan | Phantom motor execution facilitated by machine learning and augmented reality as treatment for phantom limb pain: a single group, clinical trial in patients with chronic intractable phantom limb pain[END_REF] and graded motor imagery [START_REF] Moseley | Targeting cortical representations in the treatment of chronic pain: a review[END_REF]. Crucially, the quality of the evidence supporting these treatments has been criticised [START_REF] Aternali | Recent advances in understanding and managing phantom limb pain[END_REF]. As such, despite the popularity of these treatments, as well as others including conventional analgesic drugs, PLP still remains difficult to treat, suggesting the disease mechanism is still undetermined. [START_REF] Kambi | Largescale reorganization of the somatosensory cortex following spinal cord injuries is due to brainstem plasticity[END_REF] 4.

Persistent representation following amputation 4.1 Maintained representation of the missing hand

An increasingly popular alternative theory to the maladaptive plasticity model proposes that canonical cortical representations of the missing hand area persist after amputation [START_REF] Makin | Stability of Sensory Topographies in Adult Cortex[END_REF]. By taking advantage of amputees' ability to voluntarily move their phantom hand (see section 2.1), phantom limb movements have been shown to elicit both central and peripheral signals that are different from those found during movement imagery (Raffin, Giraux, & Reilly, 2012;[START_REF] Raffin | Disentangling motor execution from motor imagery with the phantom limb[END_REF][START_REF] Reilly | Persistent hand motor commands in the amputees' brain[END_REF]. Activity elicited by phantom hand movements in the sensorimotor missing hand area was found to be similar in amplitude to controls [START_REF] Kikkert | Reaffirming the link between chronic phantom limb pain and maintained missing hand representation[END_REF]Makin, Scholz et al., 2013). Crucially, the organising principles underlying hand representation were found to be consistent in amputees' S1 missing hand cortex, in comparison to controls. For example, the canonical finger map (illustrated in Figure 1B) was uncovered in amputees with vivid phantom kinesthesia sensations, even 3 decades after amputation ( [START_REF] Kikkert | Revealing the neural fingerprints of a missing hand[END_REF]; Figure 4). This was also found in a participant who underwent an arm amputation following a brachial plexus avulsion injury, suggesting that hand representations are preserved even in the absence of peripheral input. Other research using brain decoding techniques has demonstrated the stability of hand representation. One study showed that the unique configuration of hand representation (termed here representational typicality, Figure 5A) is preserved in a large sample of amputees even after an average of 18 years since amputation, regardless of the vividness of amputees' phantom sensations or PLP (Wesselink et al., 2019). Furthermore, missing hand representation was not observed in congenital one-handers, indicating that cortical hand representation is predominantly an experience-driven process (Figure 5B). A further study successfully decoded a range of different hand gestures performed with the phantom hand, with similar success to controls in the higher sub-divisions of S1 [START_REF] Bruurmijn | Preservation of hand movement representation in the sensorimotor areas of amputees[END_REF].

Figure 5. Similar inter-finger configuration in primary somatosensory cortex (S1) for amputees' phantom hand and controls' non-dominant hand, but not for congenital one-handers' missing hand. (A)

Two-dimensional projection (using multidimensional scaling) of the representational structure across the three groups. Individual fingers are shown with different colours (see colour key; right side) and ellipses reflect the betweensubject standard error after Procrustes alignment; The greater the distances between two fingers on the plot, the more distinct or dissimilar the representation of the fingers is. As found in controls and amputees, neighbouring fingers on the hand are represented more similar to each other, resulting in a typical hand representation. Please note the different scale for congenital one-handers compared to amputees and controls. (B) Average typicality of the representational structure of contralateral S1 activity for the three groups. Typicality was measured as the correlation of the representational dissimilarity matrix (RDM) with a dataset of hand RDMs in two-handed controls drawn from a different study. In the amputee group, the typicality was high (rho = 0.75) and was not significantly different from controls, whereas the congenital one-handers' missing hand representation did not correlate with a normal hand pattern. Adapted from "Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness" by D. B. Wesselink, F. MZ van der Heiligenberg, N. Ejaz, H. Dempsey-Jones, L. Cardinali, A. Tarall-Jozwiak, J. Diedrichsen, T. R. Makin, 2019, eLife, 8, p. 37227. Copyright 2019 by eLife Sciences Publications Ltd. [START_REF] Wesselink | Obtaining and maintaining cortical hand representation as evidenced from acquired and congenital handlessness[END_REF] Further support for preserved phantom hand representations is provided by studies using cortical stimulation, where sensations can be directly produced. Flesher and colleagues [START_REF] Flesher | Intracortical microstimulation of human somatosensory cortex[END_REF][START_REF] Flesher | Restored tactile sensation improves neuroprosthetic arm control[END_REF] reported that intracortical microstimulation within the S1 hand area of a tetraplegic patient elicited tactile sensations localised to the insensate hand, and not any other body part. Stimulating the missing hand M1 area using transcranial magnetic stimulation (TMS) in arm amputees elicits reports of phantom hand movement sensations [START_REF] Bestmann | Cortical correlates of TMS-induced phantom hand movements revealed with concurrent TMS-fMRI[END_REF][START_REF] Cohen | Motor reorganization after upper limb amputation in man. A study with focal magnetic stimulation[END_REF][START_REF] Hess | Magnetic stimulation of the human brain: facilitation of motor responses by voluntary contraction of ipsilateral and contralateral muscles with additional observations on an amputee[END_REF][START_REF] Pascual-Leone | Reorganization of human cortical motor output maps following traumatic forearm amputation[END_REF][START_REF] Reilly | The motor cortex and its role in phantom limb phenomena[END_REF], even when the same phantom movements could not be made voluntarily [START_REF] Mercier | Mapping phantom movement representations in the motor cortex of amputees[END_REF]. Importantly, painful percepts were not evoked during cortical stimulation of the preserved sensorimotor hand representation in the above studies, again suggesting that other neural mechanisms may underlie PLP. Phantom sensations evoked during microstimulation of the thalamus indicate that representations of the missing limb at the level of the thalamus also remain functional following amputation [START_REF] Davis | Phantom sensations generated by thalamic microstimulation[END_REF]. Finally, electrical stimulation of peripheral residual (injured) nerves have been reported to evoke phantom sensations [START_REF] Anani | Discrimination of phantom hand sensations elicited by afferent electrical nerve stimulation in below-elbow amputees[END_REF][START_REF] D'anna | A somatotopic bidirectional hand prosthesis with transcutaneous electrical nerve stimulation based sensory feedback[END_REF][START_REF] Dhillon | Residual function in peripheral nerve stumps of amputees: implications for neural control of artificial limbs[END_REF][START_REF] Granata | Phantom somatosensory evoked potentials following selective intraneural electrical stimulation in two amputees[END_REF][START_REF] Tan | A neural interface provides long-term stable natural touch perception[END_REF]. This indicates that the entire somatosensory pathways associated with the missing hand is functionally preserved despite amputation.

Persistent hand representation and phantom limb pain

With the key assumptions underlying the maladaptive plasticity model of PLP coming into question, and the unclarity of its relationship to persistence representation, alternative accounts as to what drives PLP must be addressed. Recent work has revisited the importance of the periphery as a main driver. Vaso and colleagues [START_REF] Vaso | Peripheral nervous system origin of phantom limb pain[END_REF] pharmacologically blocked the dorsal root ganglion of lower limb amputees, resulting in rapid, and often complete, attenuation of PLP, as well as nonpainful phantom limb sensations. Thus, findings of persistent activity in the sensorimotor missing hand region of amputees positively correlating to PLP [START_REF] Kikkert | Reaffirming the link between chronic phantom limb pain and maintained missing hand representation[END_REF]Makin, Scholz et al., 2013) could be viewed as a consequence of persistent, nociceptive firing from the periphery. This theory is further evidenced by reports of brain activity in typical pain processing regions, such as the second somatosensory cortex and the posterior insula, during phantom hand movements (Makin, Scholz et al., 2013). Additionally, a reduction in missing hand activity and PLP, via the use of non-invasive brain stimulation during phantom hand movements, has been linked to proceeding upregulation from these same pain processing areas [START_REF] Kikkert | Neural basis of induced phantom limb pain relief[END_REF].

However, the experience of pain is not merely a reflection of peripheral input, and arises as a result of prior knowledge and current sensory information [START_REF] Wiech | Deconstructing the sensation of pain: The influence of cognitive processes on pain perception[END_REF], suggesting that ascribing PLP solely to aberrant peripheral firing may be too simplistic. Studies have demonstrated that predictions relating to expectant pain can modulate the resulting subjective rating of a painful stimulus [START_REF] Grahl | The periaqueductal gray and Bayesian integration in placebo analgesia[END_REF], as well as prediction errors produced when the sensory stimulus is in conflict with the current prediction [START_REF] Hird | Boundary effects of expectation in human pain perception[END_REF]. Neuroimaging and computational work have further shown that these predictions and prediction errors reside in pain processing regions, such as the insula [START_REF] Geuter | Functional dissociation of stimulus intensity encoding and predictive coding of pain in the insula[END_REF]. The chronicity of PLP, therefore, may manifest from similar mechanisms attributable to other neuropathic chronic pain conditions, and stem from an imbalance, or biasing, of predictions and/or prediction errors associated with painful inputs. In support of this, work has shown that individuals are more likely to develop PLP if they experience pain before an amputation [START_REF] Larbig | Pre-and postoperative predictors of phantom limb pain[END_REF], with one study reporting that pain catastrophizing (a cognitive, psychological trait, or 'prediction') accounts for 35% of variance in PLP reports [START_REF] Vase | Cognitive-emotional sensitization contributes to wind-uplike pain in phantom limb pain patients[END_REF]. Most likely, a complex interplay between on-going peripheral input, preserved sensorimotor representation of the missing limb and activity within pain-processing networks lead to the generation and maintenance of PLP.

Adaptive plasticity and compensatory behaviour

As discussed above, the seminal work in amputees was focused at uncovering maladaptive consequences of brain remapping, primarily in the form of PLP. However, as highlighted in the seminal monkey studies (e.g. [START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF]), plasticity in the sensorimotor cortex, and particularly the acquisition of additional cortical territory to support the representation of spared body parts, could also lead to enhanced perceptual and motor consequences, termed adaptive plasticity. For example, S1 remapping following finger amputation results in increased representation of the neighbouring fingers (Figure 1C), which could in turn lead to increased acuity for these fingers. While conceptually compelling, this prediction has been empirically disputed [START_REF] Vega-Bermudez | Spatial acuity after digit amputation[END_REF]. Therefore, it appears that adaptive plasticity may not occur spontaneously. Instead, a growing body of evidence has shown that in some cases sensorimotor remapping in amputees can be associated with the behavioural strategies that they employ to compensate for their disability, most commonly overusage of the intact hand. Recent work highlights how such changed behaviour in amputees may drive usage-dependent patterns of cortical reorganisation within the deprived sensorimotor hand area [START_REF] Makin | Deprivation-related and use-dependent plasticity go hand in hand[END_REF][START_REF] Philip | Compensatory changes accompanying chronic forced use of the nondominant hand by unilateral amputees[END_REF][START_REF] Valyear | Interhemispheric transfer of post-amputation cortical plasticity within the human somatosensory cortex[END_REF][START_REF] Van Den Heiligenberg | Artificial limb representation in amputees[END_REF]. However, it should be noted that the evidence so far is correlational, and a causal relationship between cortical remapping and functional benefits has not been demonstrated. Therefore, any conclusions suggesting a direct association between remapping following amputation and adaptive behaviour must be taken with caution.

Research of adaptive plasticity has been conducted more extensively with people born without one hand (hereafter, termed one-handers) or both hands (hereafter, termed no-handers), due to developmental malformation. Since these individuals had never developed a hand representation (Figure 5A), opportunities for reorganisation during the critical period might be more pervasive [START_REF] Hensch | Critical period plasticity in local cortical circuits[END_REF]. Moreover, due to their congenital disability, these individuals exhibit rich compensatory behavioural strategies, which they develop from an early age, thus providing behavioural context for adaptive reorganisation. Original studies in no-handers have demonstrated, using fMRI, that the sensorimotor missing hand areas become activated by inputs/movements of the feet [START_REF] Stoeckel | Congenitally altered motor experience alters somatotopic organization of human primary motor cortex[END_REF][START_REF] Yu | Somatotopic reorganization of hand representation in bilateral arm amputees with or without special foot movement skill[END_REF][START_REF] Yu | The activation of the cortical hand area by toe tapping in two bilateral upper-extremities amputees with extraordinary foot movement skill[END_REF]) see also [START_REF] Striem-Amit | Limitations of compensatory plasticity: the organization of the primary sensorimotor cortex in foot-using bilateral upper limb dysplasics[END_REF]). Importantly, single-pulse TMS to the missing hand area was shown to elicit motor evoked potentials on the feet, demonstrating that the missing hand area engages motor control of the feet. This remapping was also found to be behaviourally relevant since disruption of this area using TMS interfered with foot reaction times [START_REF] Stoeckel | Congenitally altered motor experience alters somatotopic organization of human primary motor cortex[END_REF]. Later work demonstrated that remapping in no-handers is not restricted to the feet, in fact the missing hand areas are activated even more strongly by body parts that are adjacent to the deprived cortex, i.e. the shoulders and abdomen [START_REF] Striem-Amit | Limitations of compensatory plasticity: the organization of the primary sensorimotor cortex in foot-using bilateral upper limb dysplasics[END_REF]. Changes to the S1 somatotopy were also found more locally within the foot area, where no-handers were shown to possess a detailed map of their toes, structured in a hand-like representational pattern, not found in typically-developed twohanded controls ((Dempsey-Jones, Wesselink, Friedman, & Makin, 2019); Figure 6A).

Similar to no-handers, research in one-handers also show that movements with multiple body parts elicits activity in the sensorimotor missing-hand area. As with nohanders, remapping was found to not be limited by somatotopic proximity, with the missing-hand area displaying increased activity relative to controls during movements of the residual arm, lips and feet ( [START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF]; Figure 6B). Similar patterns of remapping were observed in other sensorimotor areas with different somatotopic layouts (i.e., in the cerebellum and putamen [START_REF] Hahamy | Remapping in Cerebral and Cerebellar Cortices Is Not Restricted by Somatotopy[END_REF]). Thus, remapping does not appear to be entirely restricted by the canonical brain organisation. If remapping is not driven or limited by somatotopy, one of the potential drivers of remapping that have been proposed is compensatory behavior, which is the utilisation of other body parts to substitute for the original function of the missing limb. For example, daily behavioral strategies of one-handers with their residual arm, mouth and feet were reflected in brain activity and connectivity patterns, and were found to be distinct from those of controls, or amputees who use their residual limb less often [START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF][START_REF] Hahamy | Normalisation of brain connectivity through compensatory behaviour, despite congenital hand absence[END_REF][START_REF] Makin | Deprivation-related and use-dependent plasticity go hand in hand[END_REF]. Based on this evidence, it has been proposed that remapping may be facilitated or even limited by the functional role of the deprived brain region [START_REF] Hahamy | Representation of Multiple Body Parts in the Missing-Hand Territory of Congenital One-Handers[END_REF], such that the deprived region would only be activated by body parts that engage in the same functional role of the missing limb. However, this theory has been contested [START_REF] Striem-Amit | Limitations of compensatory plasticity: the organization of the primary sensorimotor cortex in foot-using bilateral upper limb dysplasics[END_REF][START_REF] Yu | Somatotopic reorganization of hand representation in bilateral arm amputees with or without special foot movement skill[END_REF] and awaits more direct empirical confirmation. showing toe selectivity (measured as increased activity for each toe relative to all others), revealing a toe map with similar features as found for fingers in the hand area (Figure 4). The image on the right shows representational similarity of the five toes (projection of multidimensional scaling), with similar features as found for the fingers in the hand area (Figure 5A). (B) Increased activity in the missing-hand sensorimotor area in one-handers (blue) relative to two-handed controls (red), elicited by movement of body parts involved in their compensatory behavior. Adapted from "Neural basis of induced phantom limb pain relief" by S. Kikkert, M. M. MD, J. O'Shea, D. H. S. MD, H. Johansen-Berg, I. Tracey, T. R. Makin, 2019, Annals of Neurology, 85(1), p. 59-73. Copyright 2019by John Wiley & Sons. (Kikkert et al., 2019) 6.

Artificial limbs usage and representation

A further form of adaptive behaviour is the usage of artificial limbs (hereafter prosthetic limbs or prostheses) -devices that are specifically designed to substitute for the lost functionality and/or appearances of the missing limb. The function of the upper extremity is inherently complex and diverse. Replacing a missing arm with a prosthetic device is thus not a trivial task. Despite tremendous resources dedicated to the development of sophisticated devices, studies show only 45% of amputees choose to use their prostheses throughout their day [START_REF] Jang | A survey on activities of daily living and occupations of upper extremity amputees[END_REF]. The most commonly mentioned causes of upper limb prostheses abandonment are level of amputation, prosthesis weight, socket discomfort, ease of control, phantom pain, lack of sensory feedback and unsatisfactory functionality [START_REF] Mcfarland | Unilateral upper-limb loss: satisfaction and prosthetic-device use in veterans and servicemembers from Vietnam and OIF/OEF conflicts[END_REF][START_REF] Østlie | Prosthesis rejection in acquired major upper-limb amputees: a population-based survey[END_REF]. A prominent current hypothesis links users' positivity towards using their prosthesis with embodiment [START_REF] Murray | Embodiment and Prosthetics[END_REF]. Put simply, embodiment can be thought of as the process enabling our brain to share resources, originally devoted to controlling our own body, to represent and operate external objects, such as artificial limbs. More specifically from a neural perspective, embodiment occurs when brain resources that previously represented real body parts are reallocated to representing an artificial limb (de Vignemont, 2011). The intuitive, though untested, view is that embodiment of an artificial limb will allow amputees' brains to seamlessly control it. Successful technological implementation of embodiment could lay the groundwork for users to perceive through and react with their prostheses on a neural, cognitive and phenomenological levels as may otherwise not be possible [START_REF] Makin | Neurocognitive barriers to the embodiment of technology[END_REF]. Research using the rubber hand illusion has shown that an artificial limb can feel like a real body part [START_REF] Kammers | The rubber hand illusion in action[END_REF], but such lab-based manipulations do not necessarily provide a realistic model for everyday prosthesis usage. Despite embodiment now being relatively well recognised as an important potential facilitator of prosthesis usage, it is not yet well studied, with first evidence for a relationship between embodiment and daily usage only beginning to emerge (Maimon-Mor et al., 2020). A first challenge in studying embodiment is coming up with experimental approaches to objectively and reliably quantify it. One appealing approach for studying embodiment of artificial limbs is using neuroimaging, and in particular brain decoding. Investigating prosthesis-related activity within the sensorimotor hand area using fMRI is complex, given methodological restrictions on metal objects and movements in the scanner. One study looking at the impact of prosthesis usage on sensorimotor organisation has studied it indirectly by looking at mouth remapping [START_REF] Lotze | Does use of a myoelectric prosthesis prevent cortical reorganization and phantom limb pain[END_REF]. However, the human brain contains multiple hand-selective areas, even beyond the sensorimotor system, providing more direct opportunities for testing the relationship between prosthesis and hand representation. For example, studies have demonstrated the existence of sub-regions within the occipitotemporal cortex which selectively process visual information relating to the hand (Bracci, Letswaart, Peelen, & Cavina-Pratesi, 2010). Individuals with congenital or acquired limb loss who use their prosthesis more in daily-life show greater activity in this visual hand-selective area, as well as greater functional coupling between the visual and sensorimotor hand areas (van den [START_REF] Van Den Heiligenberg | Artificial limb representation in amputees[END_REF]. However, despite benefiting from hand-selective cortical resources, the prosthesis is not necessarily 'embodied' into people's visual hand representation (Figure 7B). Instead, prosthesis users represented their own prosthesis more dissimilarly to hands, compared to controls, challenging current views of prosthesis embodiment (Maimon-Mor & Makin, 2020).

Brain plasticity and future technologies

Brain-machine interfaces (BMIs; also known as brain computer interfaces) are communication pathways that allow users to control their prosthesis using a direct link to the brain. BMIs work by extracting brain signals from pools of neurons and translating (decoding) those into commands for communication or motor control. Decoders use mathematical algorithms to process the recorded brain data and interpret the underlying neural activity, based on explicit mapping defined by the experimenter. As such, BMIs rely on an implicit feedback loop, where it is up to the brain to learn this predefined mapping through trial and error. Extracting brain signals can be done non-invasively (EEG), but provides the most impressive results with invasive neural recordings (ECoG, microelectrodes arrays). With respect to deprivation-driven plasticity, invasive BMI research has traditionally been focused on intact non-human primates. While those studies improved our understanding of the sensorimotor cortex, they did not account for brain changes that may occur following a major sensory input loss (i.e. limb amputation). However, with BMI research growing rapidly, more and more invasive studies are conducted in human patients, including clinical trials with chronically implanted intracortical electrodes [START_REF] Ethier | Restoration of grasp following paralysis through brain-controlled stimulation of muscles[END_REF][START_REF] Hochberg | Reach and grasp by people with tetraplegia using a neurally controlled robotic arm[END_REF]. Due to sensor limitations, current BMIs usually rely on small populations of neurons, requiring some level of arbitrary assignment of the readout signals. This becomes especially visible when the recorded neurons are not behaviourally relevant to the task at hand (Carmena et al. 2003, see Bensmaia andMiller, 2014 for current effort on selecting the most task relevant neuronsbiomimetic BMIs). In such cases, the brain needs to associate a specific, though arbitrary, brain pattern with a novel behaviour. As such BMIs introduce new challenges and opportunities for persistent representation and brain plasticity: can we harness existing neural infrastructure to control a BMI? And under which conditions can brain activity be remapped to control a machine?

The new frontiers for interfacing man and machine achieved with BMI have inspired other emerging augmentative technologies which aim not only to improve the natural abilities of the human body (e.g. lifting more weight or running faster with an exoskeleton (de Looze, Bosch, Krause, Stadler, & O'Sullivan, 2016), but also to give us new senses and body parts (e.g. radar vision or a third arm). Supernumerary robotic limbs [START_REF] Prattichizzo | Supernumerary Robotic Fingers to Compensate and Augment Human Manipulation Abilities[END_REF][START_REF] Sasaki | MetaLimbs: Multiple Arms Interaction Metamorphism[END_REF][START_REF] Wu | Supernumerary Robotic Fingers: An Alternative Upper-Limb Prosthesis[END_REF] provide users with wearable body parts designed to supplement our own. By widening our working space, supernumerary limbs allow us to carry out tasks normally considered impossible to perform (e.g. tri-manual coordination with a supernumerary robotic arm).

Augmentative devices are meant to change the way we interact with the environment, which often entails changes to how we move and operate our biological body. But can our brain accommodate those changes? Can it represent novel body parts and senses that have never been there before? Is it possible to merge technology with the human body by modulating our own, innate body representation? And if yes, what would be the consequences to sensorimotor control of our biological bodies? Given that we only have limited resources devoted to controlling our body, hijacking those could potentially disorganise our pre-existing neural representations (Makin et al., 2017). Successful implementation of augmentative devices could profoundly change the way we perceive and interact with the world, and have a significant healthcare, industrial and societal impact. But since the realisation of this vision relies on our brain's ability to adapt, in order to work, the next generation of augmentative technologies needs to integrate the advancements of modern robotics with the understanding of cognitive neurosciences.

Figure 1 :

 1 Figure 1: Somatotopic representation in primary somatosensory cortex in primates. (A) The primary somatosensory cortex (S1; orange) and primary motor cortex (M1; blue) are located on both sides of the central sulcus (CS; grey arrow). Right: An illustration of the human somatosensory homunculus. Photo credit from 'The Homunculus Mapper'; Max Planck Florida Institute of Neuroscience. (B-C) S1 hand map, showing somatotopic digit representation in an intact monkey (B) and 62 days post middle finger amputation (C). Following digit amputation, the former middle finger area became responsive to inputs from the neighbouring fingers (index and ring).; index (yellow); middle (green); ring (blue) and little finger (purple). Adapted from "Somatosensory cortical map changes following digit amputation in adult monkeys" by M. M.[START_REF] Merzenich | Somatosensory cortical map changes following digit amputation in adult monkeys[END_REF] J. M. Zook, 1984, Journal of Comparative Neurology, 224 (4), p. 594. Copyright 1984 by John Wiley and Sons.

Figure 2 .

 2 Figure 2. Illustration of telescoping. The grey color represents the phantom hand. From left to right: non-telescoped phantom; partially telescoped phantom; fully telescoped phantom. Adapted from "Chapter 2 -Phantom Limb Pain" by L. Nikolajsen and K. F. Christensen, in Nerves and Nerve Injuries (Vol 2, p. 25), 2015, Academic Press. Copyright 2015 by Elsevier Ltd.[START_REF] Nikolajsen | Chapter 2 -Phantom Limb Pain[END_REF]

Figure 4 .

 4 Figure 4. The somatotopic hand representation is preserved in the primary somatosensory cortex (S1), despite years of deprivation of hand inputs. The grey area in the body drawing shows the body part from which sensory inputs were lost. Black arrows indicate the representational selectivity for fingers 1-5: thumb (red); index (yellow); middle (green); ring (blue) and little finger (purple) in an above-elbow amputee (31 years since amputation) and a representative two-handed control. Participants performed single digit flexion and extension movements with their phantom (amputees) or non-dominant hand hand (controls). A univariate travelling-wave analysis technique was used to identify selectivity gradients for single fingers. Qualitatively similar digit topographies were found in amputees and controls. White arrows indicate the central sulcus. A = anterior; P = posterior. Adapted from "Revealing the neural fingerprints of a missing hand" by S. Kikkert, J. Kolasinski, S. Jbabdi, I. Tracey, C. F. Beckmann, H. Johansen-Berg and T. R.Makin, 2016, eLife, 5, p. 15292. Copyright 2016 by eLife Sciences Publications Ltd.

Figure 6 .

 6 Figure 6. Remapping in individuals with congenital hand malformation. (A) Toe representation in the foot area of a no-hander who possesses extreme functionality with his right foot (a foot painter). The image on the left shows sub-regions within the right foot area,showing toe selectivity (measured as increased activity for each toe relative to all others), revealing a toe map with similar features as found for fingers in the hand area (Figure4). The image on the right shows representational similarity of the five toes (projection of multidimensional scaling), with similar features as found for the fingers in the hand area (Figure5A). (B) Increased activity in the missing-hand sensorimotor area in one-handers (blue) relative to two-handed controls (red), elicited by movement of body parts involved in

Figure 7

 7 Figure 7 Effects of prosthesis use on brain representation. (A) Hand-selective voxels were identified bilaterally in lateral occipitotemporal cortex (green) and in the sensorimotor cortex (white). The putative sensorimotor missing-hand area was estimated (hatched white) by mirror projecting the intact hand region of interest across hemispheres. Functional connectivity between the bilateral visual hand areas and the sensorimotor missing hand area was measured for each individual during a resting-state scan, and was correlated with everyday prosthesis usage in prosthesis users with congenital and acquired hand loss. CS = central sulcus; R/L = right/left hemispheres; STS = superior temporal sulcus. Visuomotor connectivity with the intact hand sensorimotor region of interest was regressed out of the missing hand visuo-motor measure. Adapted from "Artificial limb representation in amputees" by F. M. Z. van den Heiligenberg, T. Orlov, S. N. Macdonald, E. P. Duff, D. H. Slater, C. F. Beckmann, H. Johansen-Berg, J. C. Culham and T. R. Makin, 2018, Brain, 141(55), p. 1427. Copyright 2018 by Oxford University Press. (B) Representational structure in occipitotemporal cortex in twohanded controls and prosthesis users. Pairs of stimuli that have more similar patterns are clustered together under the same branch. Longer connections across the vertical axis indicate greater relative distances. In controls, cosmetic prostheses are clustered with hands and active prostheses with tools, reflecting their visual similarities. In prosthesis users, however, the two prosthesis types (cosmetic and active) are clustered together, indicating the emergence of a dissociated prosthesis categorisation, tools and hands represented dissimilarly from both prostheses. Greater daily-life prosthesis usage correlated with greater prosthesis categorisation (convergence of prosthesis representation away from the natural categories and onto other prosthesis representation). Adapted from "Is an artificial limb

Table of Contents 1 .

 of1

Table 1 . Incidence of phantom limb sensations (PLS) and phantom limb pain (PLP) in selected studies with at least 100 amputees. -not specified.

 1

Figure 3. Remapping in the primary somatosensory cortex is due to brainstem plasticity

. A) In intact non-human primates, inputs from the chin (orange) ascend to S1 via the trigeminal nucleus of the brainstem and the ventroposterior medial nucleus (VPM) of the thalamus. Inputs from the hand (blue) ascend to S1 via cuneate nucleus and the ventroposterior lateral nucleus (VPL). Representation of other parts of the face is shown in red. Dots mark locations of the recording sites where receptive fields of the neurons were mapped. B) After a dorsal column lesion, the deprived hand area became responsive to touch on the chin. The grey arrow marks the location of the hand-face border. Crosses illustrate recording sites for which no receptive field could be mapped. C) Inactivation of the reorganised (hand) cuneate nucleus silenced the expanded chin representation in the deprived hand area. This demonstrated that S1 reorganisation is due to upstream effects of axonal sprouting at the brainstem level or spinal cord. Adapted from "Large-scale reorganization of the somatosensory cortex following spinal cord injuries is due to brainstem plasticity" by N.

Kambi, P. Halder, R. Rajan, V. Arora, P. Chand, M. Arora and N. Jain, 2014, Nature Communications, 5, p. 3602.

Copyright 2014 by Springer Nature. Reprinted with permission from Macmillan Publishers Limited.