# Why does neutron transfer play different roles in sub-barrier fusion
reactions 32S+94,96Zr and 40Ca+94,96Zr?

V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, W. Scheid3, and H.Q.Zhang4
1Joint Institute for Nuclear Research, 141980 Dubna, Russia
2International Center for Advanced Studies, Yerevan State University, M.
Manougian 1, 0025, Yerevan, Armenia
3Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392
Giessen, Germany
4China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China

###### Abstract

The sub-barrier capture (fusion) reactions 32S+90,94,96Zr, 36S+90,96Zr,
40Ca+90,94,96Zr, and 48Ca+90,96Zr with positive and negative $Q$-values for
neutron transfer are studied within the quantum diffusion approach and the
universal fusion function representation. For these systems, the $s$-wave
capture probabilities are extracted from the experimental excitation functions
and are also analyzed. Different effects of the positive $Q_{xn}$-value
neutron transfer in the fusion enhancement are revealed in the relatively
close reactions 32S+94,96Zr and 40Ca+94,96Zr.

###### pacs:

25.70.Jj, 24.10.-i, 24.60.-k
Key words: sub-barrier capture (fusion), nucleon transfer, quantum diffusion
approach

##  I Introduction

The nuclear deformation effects are identified as playing a major role in the
magnitude of the sub-barrier fusion (capture) cross sections Gomes ; Back .
There are a several experimental evidences which confirm the straightforward
influence of nuclear deformation on the fusion. If the target nucleus is
prolate in the ground state, the Coulomb field on its tips is lower than on
its sides. Thus, the capture or fusion probability increases at energies below
the barrier corresponding to the spherical nuclei.

The dynamics of neutron transfer-mediated sub-barrier capture and fusion is
not yet revealed Back . The cross section enhancement in the sub-barrier
fusion of 58Ni+64Ni, with respect to 58Ni+58Ni, Beckerman  is interpreted in
Ref. Broglia  as a kinematic effect due to the positive $Q_{2n}$-value of the
ground-state-to-ground-state two-neutron transfer ($2n$-transfer) channel. A
correlation is observed between considerable sub-barrier fusion enhancement
and positive $Q_{xn}$-values for neutron transfer in the reactions
40Ca+94,96Zr TimmersCa40Zr96 ; StefaniniCa40Zn94 ; StefMont40ca96zr , and
40Ca+116,124,132Sn Stefanini40ca116124sn ; Kol40Ca124132Sn .

The importance of neutron transfer with positive $Q_{xn}$-values in nuclear
fusion (capture) originates from the fact that neutrons are insensitive to the
Coulomb barrier and their transfer starts at quite larger separations, before
the projectile is captured by target-nucleus. It is generally thought that the
sub-barrier capture (fusion) cross section increases because of the neutron
transfer TimmersCa40Zr96 ; StefaniniCa40Zn94 ; StefMont40ca96zr ;
Stefanini40ca116124sn ; Kol40Ca124132Sn ; Pengo ; Henning ; Stelson ; Roberts
; Stefanini3236s110pd ; Sonzogni ; Tripathi ; Jiang40ca48ca ; ZhangS32Zr9096 ;
Kola40ca134Te ; Jia32s94zr . However, the reduced excitation functions for the
reactions 16,17,18O+ASn ($A$=112,116-120,122,124) AOASn , scaled to remove the
effects of smoothly varying barrier parameters, do not show any strong
dependence on the mass number of target or projectile. The relative changes
are within a factor two and are not correlated with the positive
$Q_{xn}$-values of neutron-transfer channels in these reactions. As shown in
Ref. Scarlassara , the neutron transfer channels with positive $Q_{xn}$-value
weakly influence the capture (fusion) cross section in the 60Ni+100Mo reaction
at sub-barrier energies. In the reactions 40Ca+116,124Sn ($Q_{2n}>0$) and
132Sn,130Te+58,64Ni ($Q_{2n}>0$) at energies above and a few MeV below the
Coulomb barrier, the effect of transfer channels on the capture (fusion) is
demonstrated to be very weak with no significant differences observed in the
reduced excitation functions Stefanini40ca116124sn ; Liang . In comparison
with the 16O+76Ge reaction Jia18O74Ge , the fusion enhancement due to the
positive $Q_{2n}$-value is not revealed in the 18O+74Ge reaction.

It is presently not clear why the neutron transfers with positive
$Q_{xn}$-values play a decisive role in the fusion reactions 40Ca+48Ca,
58Ni+64Ni, 40Ca+94,96Zr, 40Ca+116,124,132Sn and weakly influence the fusion
reactions 58,64Ni+132Sn, 58,64Ni+130Te, 60Ni+100Mo, 18O+74Ge, 18O+ASn Back ;
Jiang2014 . Although the enhancement appears to be related to the existence of
large positive $Q_{xn}$-values for neutron transfer, it is not proportional to
the magnitudes of those $Q_{xn}$-values, which are larger for 40Ca+96Zr
[40Ca+132Sn or 40Ca+124Sn] than for 40Ca+94Zr [40Ca+124Sn or 40Ca+116Sn]. The
sub-barrier enhancements are similar in these reactions. So, the influence of
neutron transfer on the capture process is not trivial to be easily explained.

The quantum diffusion approach EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD  was
applied to study the role of the neutron transfer with positive $Q_{xn}$-value
in the capture (fusion) reactions at sub-, near- and above-barrier energies. A
good agreement of the theoretical calculations with the experimental data was
demonstrated . As found, the change of the capture cross section after the
neutron transfer occurs due to the change of the deformations of nuclei EPJSub
; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD . Thus, the effect of the neutron transfer
is an indirect influence of the quadrupole deformation. As demonstrated in
Ref. EPJSub1 , the neutron transfer can weakly influence or even suppress the
capture (fusion) cross section in some reactions.

Applying the quantum diffusion approach EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21
; SD  (Sect. IV), the universal fusion function representation Gomes1 ; Gomes2
(Sect. II), and capture probabilities extracted from the experimental
excitation functions (Sect. III), we try to answer the question how the
neutron transfer influence the sub-barrier capture cross section in the
reactions 32S+90,94,96Zr, 36S+90,96Zr, 40Ca+90,94,96Zr, and 48Ca+90,96Zr at
near and sub-barrier energies. We will show why the influence of positive
$Q_{xn}$-value neutron transfer is completely different in the relatively
close reactions 32S+94,96Zr and 40Ca+94,96Zr.

##  II Experimental reduced capture cross sections

To analyze the capture cross sections in the reactions with different Coulomb
barrier heights $V_{b}$ and radius $R_{b}$ calculated in the case of spherical
nuclei, it is useful to compare not the excitation functions, but the
dependence of the dimensionless quantities $2E_{\rm c.m.}\sigma_{cap}(E_{\rm
c.m.})/(\hbar\omega_{b}R_{b}^{2})$ versus $(E_{\rm
c.m.}-V_{b})/(\hbar\omega_{b})$ or $(E_{\rm c.m.}-V_{b})$ Gomes1 ; Gomes2 .
Here, $\omega_{b}$ and $\mu$ are the frequency of an inverted oscillator
approximated the barrier and the reduced mass of the system, respectively. In
the reactions, where the capture and fusion cross sections coincide, the
comparison of experimental data with the universal fusion function Gomes1 ;
Gomes2  allows us to conclude about the role of static deformations of the
colliding nuclei and the nucleon transfer between them in the capture cross
section. Indeed, the universal function disregards these effects.

For the reactions 40Ca+90Zr, 48Ca+90,96Zr, and 36S+90,96Zr, with almost
spherical nuclei and without neutron transfer [the negative $Q_{xn}$-values],
the experimental cross sections are rather close and fall with the same rate
like the universal fusion function (Fig. 1). For the reactions 40Ca+94,96Zr
with the neutron transfer [the positive $Q_{xn}$-values], one can see that the
reduced cross sections strongly deviate from the universal function in
contrast to the reactions 40Ca+90Zr, 36S+90,96Zr, 48Ca+90,96Zr, where the
neutron transfer is suppressed.

In the reactions 32S+90,94,96Zr with strongly deformed projectile 32S (Fig.1),
the deviations from the universal function are mainly caused by the static
deformation effects. In spite of the $Q_{xn}$-values for the neutron transfer
range from the negative [32S+90Zr] to large and positive values [32S+94,96Zr],
the reduced capture (fusion) cross sections appear to be almost the same. So,
we observe the strong and weak influence of neutron transfer on the capture
cross sections in the reactions 40Ca+94,96Zr and 32S+94,96Zr, respectively.

##  III Capture probabilities extracted from experimental capture excitation
functions

Shifting the energy by the rotational energy
$E_{R}(J)=\frac{\hbar^{2}J(J+1)}{2\mu R_{b}^{2}}$ Bala , one can approximate
the angular momentum $J$ dependence of the transmission (capture) probability
$P_{cap}(E_{\mathrm{c.m.}},J)$, at a given $E_{\mathrm{c.m.}}$:

$P_{cap}(E_{\mathrm{c.m.}},J)\approx P_{cap}(E_{\mathrm{c.m.}}-E_{R}(J),J=0).$
(1)

If we use the formula for the capture cross section, convert the sum over the
partial waves $J$ into an integral, and express $J$ by the variable
$E=E_{\mathrm{c.m.}}-E_{R}(J)$, we obtain the following simple expression:

$\sigma_{cap}(E_{\mathrm{c.m.}})=\frac{\pi
R_{b}^{2}}{E_{\mathrm{c.m.}}}\int_{0}^{E_{\mathrm{c.m.}}}dEP_{cap}(E,J=0).$
(2)

Multiplying this equation by $E_{\mathrm{c.m.}}/(\pi R_{b}^{2})$ and
differentiating over $E_{\mathrm{c.m.}}$, one obtains Bala :

$P_{cap}(E_{\mathrm{c.m.}},J=0)=\frac{1}{\pi
R_{b}^{2}}\frac{d[E_{\mathrm{c.m.}}\sigma_{cap}(E_{\mathrm{c.m.}})]}{dE_{\mathrm{c.m.}}}.$
(3)

One can see that
$\frac{d[E_{\mathrm{c.m.}}\sigma_{cap}(E_{\mathrm{c.m.}})]}{dE_{\mathrm{c.m.}}}$
has a meaning of the $s$-wave transmission in the entrance channel. Therefore,
the $s$-wave capture probability can be extracted with a satisfactory accuracy
from the experimental capture cross sections $\sigma_{cap}(E_{\mathrm{c.m.}})$
at energies near and below the Coulomb barrier. Note that at energies
considered the dependence of the Coulomb barrier radius on the angular
momentum is very weak.

The extraction method just described requires some procedure to smooth the
experimental data since the values of
$E_{\mathrm{c.m.}}\sigma_{cap}(E_{\mathrm{c.m.}})$ have error bars. We spline
the experimental points of $E_{\mathrm{c.m.}}\sigma_{cap}(E_{\mathrm{c.m.}})$
by the Bézier parametric curve Bezier .

In Figs. 2 and 3, the extracted capture probabilities $P_{cap}(E,J=0)$
demonstrate the influence of nucleon transfer on the capture (fusion)
excitation function. In the reactions 40Ca+90Zr and 48Ca+90,96Zr with the
negative $Q_{xn}$-values for nucleon transfer, the capture probability
exhibits a steep falloff of the probability at low energies. However, the
first derivatives of $P_{cap}(E,J=0)$ are almost the same. Conversely, the
reactions 40Ca + 94,96Zr have positive $Q_{xn}$-values for neutron transfer.
This leads to the smaller slope of probability functions at sub-barrier
energies. The capture probabilities in the reactions 40Ca+94,96Zr are close to
each other.

Since the nucleus 36S is spherical, the slopes of functions $P_{cap}(E,J=0)$
for the reactions 36S+90,96Zr are larger than those for the reactions
32S+94,96Zr and 32S+90Zr with the strongly deformed 32S. The slopes of
functions $P_{cap}(E,J=0)$ are rather similar (Fig. 3) in the 32S+90Zr
reaction with the negative $Q_{xn}$-values for neutron transfer and in the
reactions 32S+94,96Zr with the positive $Q_{xn}$-values for neutron transfer.
Thus, the enhancement of capture probability in these reactions has the same
origin. It arises due to the large static deformations of nuclei 32,34S and
the neutron transfer is not responsible for the capture (fusion) enhancement.

As follows from the extracted capture probabilities, the experimental
normalizations of the cross sections are different in the reactions
32S+90,94,96Zr and 36S+90,96Zr. One should think about the experimental
reasons for such deviations.

##  IV Calculations within the quantum diffusion approach

In the quantum diffusion approach EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD ;
PRCPOP  the collisions of nuclei are described with a single relevant
collective variable: the relative distance between the colliding nuclei. This
approach takes into consideration the fluctuation and dissipation effects in
collisions of heavy ions which model the coupling with various channels (for
example, the non-collective single-particle excitations, low-lying collective
dynamical modes of the target and projectile). We have to mention that many
quantum-mechanical and non-Markovian effects accompanying the passage through
the potential barrier are taken into consideration in our formalism EPJSub ;
EPJSub1 ; EPJSub3 ; EPJSub21 ; SD ; PRCPOP . The nuclear deformation effects
are taken into account through the dependence of the nucleus-nucleus potential
on the deformations and mutual orientations of the colliding nuclei. To
calculate the nucleus-nucleus interaction potential $V(R)$, we use the
procedure presented in Refs. EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD ;
PRCPOP . For the nuclear part of the nucleus-nucleus potential, the double-
folding formalism with the Skyrme-type density-dependent effective nucleon-
nucleon interaction is used. With this approach many heavy-ion capture
reactions at energies above and well below the Coulomb barrier have been
successfully described EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD ; PRCPOP .

Following the hypothesis of Ref. Broglia , we assume that the sub-barrier
capture in the reactions under consideration mainly depends on the two-neutron
transfer with the positive $Q_{2n}$-value. Our assumption is that, just before
the projectile is captured by the target-nucleus (just before the crossing of
the Coulomb barrier) which is a slow process, the $2n$-transfer ($Q_{2n}>0$)
transfer occurs and leads to the population of the first excited collective
state in the recipient nucleus SSzilner  (the donor nucleus remains in the
ground state). The absolute values of the quadrupole deformation parameters
$\beta_{2}$ in 2+ state of even-even deformed nuclei are taken from Ref. Ram .
For the nuclei deformed in the ground state, the $\beta_{2}$ in the first
excited collective state is similar to that in the ground state. For the
double magic and semi-magic nuclei, we take $\beta_{2}=0$ in the ground state.

The motion to $N/Z$ equilibrium starts in the system before the capture occurs
because it is energetically favorable in the dinuclear system in the vicinity
of the Coulomb barrier. For the reactions under consideration, the average
change of mass asymmetry is related to the two-neutron transfer. In these
reactions, $Q_{2n}>Q_{1n}$ and during the capture the $2n$-transfer is more
probable than $1n$-transfer. After the $2n$-transfer the mass numbers, the
deformation parameters of the interacting nuclei, and, correspondingly, the
height $V_{b}$ and shape of the Coulomb barrier change. Then one can expect an
enhancement or suppression of the capture. If after the neutron transfer the
deformations of interacting nuclei increase (decrease), the capture
probability increases (decreases). If after the transfer the deformations of
interacting nuclei do not change, there is no effect of the neutron transfer
on the capture. This scenario was verified in the description of many
reactions EPJSub ; EPJSub1 ; EPJSub3 ; EPJSub21 ; SD ; PRCPOP .

In Fig. 4 one can see a good agreement between the calculated and the
experimental capture cross sections in the reactions 40Ca+94,96Zr with the
positive $Q$ values for neutron transfer and in the reactions 40Ca+90Zr,
48Ca+90,96Zr with negative $Q$-values for neutron transfer. The theoretical
calculations describe the strong deviation of the slopes of excitation
functions in the reactions 40Ca+94,96Zr with positive $Q$-values for neutron
transfer from those in the reactions 40Ca+90Zr, 48Ca+90,96Zr, where the
neutron transfers are suppressed because of negative $Q$-values. This means
that the observed capture enhancements in the reactions 40Ca+94,96Zr at sub-
barrier energies are related to the two-neutron transfer effect. After
$2n$-transfer in the reactions
40Ca($\beta_{2}=0$)+94Zr($\beta_{2}=0.09$)$\to^{42}$Ca($\beta_{2}=0.25$)+92Zr($\beta_{2}=0.1$)
[$Q_{2n}$=4.9 MeV]
and
40Ca($\beta_{2}=0$)+96Zr($\beta_{2}=0.08$)$\to^{42}$Ca($\beta_{2}=0.25$)+94Zr($\beta_{2}=0.09$)
[$Q_{2n}$=5.5 MeV],
the deformation of the light nucleus strongly increases and, thus, the height
of the Coulomb barrier decreases and the capture cross section becomes larger
(Fig. 4). So, because of the neutron-transfer effect the reactions
40Ca+94,96Zr show large sub-barrier enhancements with respect to the reactions
48Ca+90,96Zr and 40Ca+90Zr. One can see in Fig.5 that with decreasing the sub-
barrier energy the cross sections with and without two-neutron transfer
strongly deviate. The slopes of the excitation functions in the reactions
40Ca+94,96Zr are almost the same because in both cases after the neutron
transfer the nuclei have similar deformations. The relative enhancement of the
sub-barrier fusion cross sections in the reactions 40Ca+94,96Zr with respect
to those in the reactions 48Ca+90,96Zr and 40Ca+90Zr is mainly related to the
deformation of 42Ca in the $2^{+}$ state. Thus, the observed capture
enhancement at sub-barrier energies in the reactions 40Ca+94,96Zr is purely
related to the transfer effects.

Since the sub-barrier enhancements are surprisingly similar for the two
reactions 40Ca+94,96Zr with different positive $Q$-values for neutron
transfer, one can assume that the absolute value of the positive $Q$-value is
rather unimportant for the capture following transfer. If the transfer is
energetically favorable it occurs during the capture process. In this case the
transfer influences the capture (fusion) [or the height and width of the
Coulomb barrier] through the change of the isotopic composition of interacting
nuclei and, correspondingly, through the change of their deformations.

Figure 6 shows the capture (fusion) excitation functions for the reactions
32S+90,94,96Zr and 36S+90,96Zr. The $Q_{2n}$-values for the $2n$-transfer
processes are positive (negative) for the reactions 32S+94,96Zr (32S+90Zr,
36S+90,96Zr). After the $2n$-transfer (before the capture) in the reactions
(Figs. 4 and 6)
32S($\beta_{2}=0.31$)+94Zr($\beta_{2}=0.09$)$\to^{34}$S($\beta_{2}=0.25$)+92Zr($\beta_{2}=0.1$)
[$Q_{2n}$=5.1 MeV]
and
32S($\beta_{2}=0.31$)+96Zr($\beta_{2}=0.08$)$\to^{34}$S($\beta_{2}=0.25$)+94Zr($\beta_{2}=0.09$)
[$Q_{2n}$=5.7 MeV],
the deformation of S slightly decreases and the values of the corresponding
Coulomb barriers slightly increase. As a result, the transfer weakly
suppresses the capture process at the sub-barrier energies. This suppression
becomes stronger with decreasing energy. One can see in Fig. 5 that at
energies above, near, and below the Coulomb barrier the cross sections with
and without two-neutron transfer are almost similar in the reactions
32S+94,96Zr. The relative enhancement of the sub-barrier fusion cross sections
in the reactions 32S+94,96Zr with respect to that in the reactions 36S+90,96Zr
is mainly related to the deformation of 34S in the $2^{+}$ state. With respect
to the reactions 36S+94,96Zr the enhancements of cross sections in the
reactions 32S+94,96Zr and 32S+90Zr are similar because of the close
deformations of interacting nuclei after neutron transfer. So, the observed
capture enhancement at sub-barrier energies in the reactions 32S+94,96Zr and
32S+90Zr is not related to the transfer effects but to the direct static
deformation effects.

##  V Summary

The quantum diffusion approach, the universal fusion function representation,
the extracted capture probabilities from the experimental excitation functions
are applied to study the role of the neutron transfer with positive
$Q_{xn}$-values in the capture (fusion) reactions 40Ca+94,96Zr and
32S+94,96Zr. We found that the change of the capture (fusion) cross section
after the two-neutron transfer occurs due to the change of the deformations of
nuclei. When after the neutron transfer the deformations of nuclei strongly
(weakly) change, the neutron transfer strongly (weakly) influences the fusion
cross section. We clearly showed that the neutron transfer effects on the
excitation functions in the reactions 40Ca+94,96Zr and 32S+94,96Zr are
completely different. The calculations pointed a strong increase of the fusion
enhancement due to the neutron transfer for the systems with the spherical
acceptor-nuclei as in the case of the reactions 40Ca+94,96Zr. In the reactions
32S+94,96Zr with the well deformed acceptor-nucleus 32S, the strong fusion
enhancement arises due to the static deformation effects.

Combining all our calculations within the quantum diffusion approach, one can
come to the following conclusions about the role of neutron transfer in the
capture (fusion) reactions with positive $Q_{xn}$-values for the neutron
transfer.
(a) If the acceptor-nucleus is spherical or slightly deformed (relatively
strongly bound) nucleus and the donor-nucleus is the spherical or deformed
nucleus, the neutron transfer may lead to the strong capture (fusion)
enhancement [for example, 40Ca+94,96Zr,116,124,132Sn,154Sm, 58Ni+64Ni, and
40Ca+48Ca] or to the weak influence or even to the weak suppression [for
example, 18O+92Mo,112,118,124Sn].
(b) If the acceptor-nucleus is strongly deformed (relatively weakly bound)
nucleus and the donor-nucleus is the spherical or deformed nucleus, the
neutron transfer may lead to the weak influence or even to the weak
suppression [for example, 18O+74Ge, 60Ni+100Mo, 28Si+142Ce,154Sm, and
32S+94,96Zr,96-100Mo,100-104Ru,110Pd,154Sm,208Pb]. In these reactions with
strongly deformed nuclei 28Si, 32S, and 74Ge the fusion enhancement is caused
by the static deformation effects.

Thus, the point of view that the sub-barrier capture (fusion) cross section
can be weakly influenced or even suppressed because of the neutron transfer
with positive $Q_{xn}$-values has to be carefully studied. We predict the weak
neutron transfer effects in the fusion reactions 60,62Ni+150Nd,
18O+64Ni,114,116,120,122,126Sn,204,206Pb, and 28Si,32S+116-134Sn,148-152Sm. As
shown with the quantum diffusion approach, the capture cross sections almost
match in the reactions 16O+52Cr and 18O+50Cr, 16O+94Mo and 18O+ 92Mo,
16O+114,120,124,126Sn and 18O+112,118,122,124Sn, respectively. The same
reduced fusion cross sections for the reactions 58,60,62Ni+150Nd with positive
$Q_{2n}$-values [$Q_{2n}$=8.2, 6.0, 4.1 MeV, respectively] are predicted.

G.G.A. and N.V.A. acknowledge the partial support from the Alexander von
Humboldt-Stiftung (Bonn). This work was supported by RFBR and NSFC. The
IN2P3(France)-JINR(Dubna) Cooperation Programme is gratefully acknowledged.

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Figure 1: (Color online) Comparison of the universal fusion function (solid
line) with the experimental values TimmersCa40Zr96 ; StefaniniCa40Zn94 ;
StefMont40ca96zr ; ZhangS32Zr9096 ; Jia32s94zr ; StefaniniCa48Zn9096 ;
StefaniniS36Zn9096 ; Stefan40ca40ca  (symbols) of $\dfrac{2E_{\rm
c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma_{cap}(E_{\rm c.m.})$ vs $\dfrac{E_{\rm
c.m.}-V_{b}}{\hbar\omega_{b}}$ for the reactions indicated.  Figure 2:  The
extracted $s$-wave capture probabilities for the reactions indicated by
employing Eq. (3) [symbols connected by lines]. The used experimental capture
(fusion) excitation functions are from Refs. TimmersCa40Zr96 ;
StefaniniCa40Zn94 ; StefMont40ca96zr ; ZhangS32Zr9096 ; Jia32s94zr .  Figure
3:  The extracted $s$-wave capture probabilities for the reactions indicated
by employing Eq. (3) [symbols connected by lines]. The used experimental
capture (fusion) excitation functions are from Refs. TimmersCa40Zr96 ;
ZhangS32Zr9096 ; StefaniniCa48Zn9096 ; StefaniniS36Zn9096 .  Figure 4: (Color
online) The calculated capture cross sections vs $E_{\rm c.m.}$ for the
reactions (a) 40Ca+96Zr (solid line), 40Ca+94Zr (dotted line), 40Ca+90Zr
(dashed line) and (b) 48Ca+96Zr (solid line), 48Ca+90Zr (dashed line). The
experimental data (symbols) are from Refs. TimmersCa40Zr96 ; StefaniniCa40Zn94
; StefMont40ca96zr ; StefaniniCa48Zn9096 .  Figure 5: (Color online) The
calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions (b)
40Ca+96Zr (solid line), 40Ca+94Zr (dotted line), 40Ca+90Zr (dashed line) and
(a) 32S+96Zr (solid line), 32S+94Zr (dotted line), 32S+90Zr (dashed line). For
the reactions 32S,40Ca+96Zr and 32S,40Ca+94Zr, the capture cross sections
calculated without taking into consideration the neutron transfer are shown by
dash-dot-dotted and dash-dotted [it is matched with solid line in the part
(a)] lines, respectively.  Figure 6: (Color online) The calculated capture
cross sections vs $E_{\rm c.m.}$ for the reactions (a) 32S+96Zr (solid line),
32S+94Zr (dotted line), 32S+90Zr (dashed line) and (b) 36S+96Zr (solid line),
36S+90Zr (dashed line). The experimental data (symbols) are from Refs.
ZhangS32Zr9096 ; Jia32s94zr ; StefaniniS36Zn9096 .


# Quantifying Spatial Domain Explanations in BCI using Earth Mover’s Distance

Param Rajpura  Human-AI Interaction (HAIx) Lab
IIT Gandhinagar, India
<EMAIL_ADDRESS>Hubert Cecotti  Department of Computer Science
California State University,Fresno, USA
<EMAIL_ADDRESS>Yogesh Kumar Meena  Human-AI Interaction (HAIx) Lab
IIT Gandhinagar, India
<EMAIL_ADDRESS>
###### Abstract

Brain-computer interface (BCI) systems facilitate unique communication between
humans and computers, benefiting severely disabled individuals. Despite
decades of research, BCIs are not fully integrated into clinical and
commercial settings. It’s crucial to assess and explain BCI performance,
offering clear explanations for potential users to avoid frustration when it
doesn’t work as expected. This work investigates the efficacy of different
deep learning and Riemannian geometry-based classification models in the
context of motor imagery (MI) based BCI using electroencephalography (EEG). We
then propose an optimal transport theory-based approach using earth mover’s
distance (EMD) to quantify the comparison of the feature relevance map with
the domain knowledge of neuroscience. For this, we utilized explainable AI
(XAI) techniques for generating feature relevance in the spatial domain to
identify important channels for model outcomes. Three state-of-the-art models
are implemented - 1) Riemannian geometry-based classifier, 2) EEGNet, and 3)
EEG Conformer, and the observed trend in the model’s accuracy across different
architectures on the dataset correlates with the proposed feature relevance
metrics. The models with diverse architectures perform significantly better
when trained on channels relevant to motor imagery than data-driven channel
selection. This work focuses attention on the necessity for interpretability
and incorporating metrics beyond accuracy, underscores the value of combining
domain knowledge and quantifying model interpretations with data-driven
approaches in creating reliable and robust Brain-Computer Interfaces (BCIs).

###### Index Terms:

Explainable AI, Brain-Computer Interface, Motor Imagery, Optimal Transport
Theory

##  I Introduction

A Brain-Computer Interface (BCI), also known as a Brain-Machine Interface
(BMI) or Neural Interface, is a technology that establishes a direct
communication pathway between the brain and an external device, such as a
computer or prosthetic limb [34]. The primary goal of a BCI is to enable the
exchange of information between the brain and an external system, bypassing
traditional sensory or motor pathways like muscles or nerves.

BCIs can be designed for various purposes. BCIs can assist individuals with
disabilities by allowing them to control devices or communicate through direct
brain signals. This is particularly relevant for people with paralysis or
severe motor impairments [5]. BCIs can be used to provide individuals with
real-time feedback on their brain activity. This can be employed in areas like
Neurotherapy, where individuals can learn to regulate their brain activity for
therapeutic purposes [8]. BCIs can also enhance virtual or augmented reality
experiences by allowing users to interact with digital environments using
their thoughts. BCIs are valuable tools in neuroscience research. They can be
used to study brain function, cognitive processes, and neurological disorders
[28]. In the medical field, BCIs may be employed for diagnostic purposes or as
part of rehabilitation strategies [47]. In such a case, it is necessary to
have results that can be fully explainable.

Selecting the right sensors for a BCI is important for achieving the desired
performance, user acceptance, and practicality in various applications[1]. In
BCIs designed for practical applications, user experience is critical. The
sensors should be non-intrusive, easy to wear, and comfortable for the user.
This is particularly important for applications that require long-term use,
such as assistive technologies or rehabilitation devices. The spatial
resolution of sensors determines how finely the brain activity can be
localized. Some sensors may have better spatial resolution, allowing for more
precise identification of the brain regions involved in a particular task or
thought[23]. This is especially important for applications that require
detailed mapping of neural activity, such as neuroscientific research or
precise control in neuroprosthetics.

The recent approaches towards adopting deep learning (DL) based Motor Imagery
(MI) models in BCI research are driven by the advantages associated with
learning complex tasks in an end-to-end fashion [38, 24]. This paradigm shift
accompanies a challenge in offering interpretable and robust solutions[43, 13,
44, 35]. Explainable AI(XAI) [17] has been applied to BCIs to improve
performance [40, 36, 24] or for optimal channel selection [29]. Though XAI
approaches are utilized to justify the model performance, the explanations
from a specific model architecture are subjectively interpreted in isolation.
Evaluating model explanations with quantified metrics is a stepping stone
toward interpretable models. Ravindran and Contreras-Vidal [42] used a
simulation framework to evaluate the robustness and sensitivity of twelve
back-propagation-based visualization methods by comparing them to ground truth
features of the simulated EEG data. Cui et al. [12] quantitatively evaluate
seven different deep interpretation techniques across different models and
datasets for EEG-based BCI. Both works evaluate the robustness of XAI
approaches by perturbing the data. Complementing these insights, we compare
and ground these insights with the domain knowledge in neuroscience.

The key contributions of the paper are: 1) a novel approach using Earth
Mover’s distance metric based on optimal transport theory is proposed to
analyze and compare the spatial domain explanations for predicted outcomes of
the models used in EEG-based BCI with existing domain knowledge; 2) a detailed
benchmark of state-of-the-art models is provided on a dataset with 109
participants to evaluate their performance and explanations. The rest of the
paper is organized as follows. The dataset and the different classification
algorithms are presented in Section II. The classification performance is
given in Section III. The impact of the results on eXplainable AI for BCI is
discussed in Section IV. The key findings are summarised in Section V.

##  II Methods

###  II-A Dataset Description

The EEGMMID-Physionet dataset is used in this work[16, 37]. It is one of the
largest datasets on motor imagery (MI) and movement, with 64-channel EEG
recordings collected from 109 participants. We considered the two tasks: 1)
open and close left or right fist and 2) imagine opening and closing left or
right fist. It is a binary classification problem, i.e., with two classes.

We merged both tasks to augment the dataset. Each subject, on average, did 93
trials, each 4 seconds long. We divided each 4-second trial into four 1-second
length runs, resulting in 160 sample recordings in each sequence and an
average of 370 sequences for each subject. A bandpass filter of 8-30 Hz was
applied to remove any noise in the EEG signals, including the mu (8-12 Hz) and
beta (12-30 Hz) bands that contain event synchronization/desynchronization
(ERS/ERD) information [27, 30, 10]. Each sequence was labeled based on the
activity for left and right fist movement/imagery respectively.

###  II-B Model Architecture and Training

Three diverse model architectures are explored and utilised to evaluate and
compare explanations from different families of classification methods. 1)
Riemannian geometry-based classifier (density-based classifier using the
distance to the mean)[4], 2) Convolutional feedforward neural network
architecture: EEGNet (discriminant approach) [24], and 3) Transformer and
self-attention-inspired architecture: EEG Conformer (discriminant approach)
[40].

####  II-B1 Minimum distance to mean (MDM) using Riemannian distance

The proposed methodology by Barachant et al. [4] involves utilizing covariance
matrices derived from raw EEG signals. Each value in these matrices represents
the variance across two EEG channels. Since these matrices are symmetric
positive definite (SPD), the underlying data structure is non-euclidean.
Leveraging Riemannian geometry approaches facilitates directly classifying EEG
signals without spatial filtering. It is widely successful in classifying
relevant signals for motor imagery using EEG [46, 11].

To derive an explanation using feature relevance, we extract the most relevant
channels for predictions, as proposed by Barachant and Bonnet [3] and
implemented in PyRiemannian library [2], for each class, a centroid is
estimated, i.e., the average across examples of a given class, and the channel
selection is based on the maximization of the distance between centroids. The
principle is related to the k-nearest neighbor but instead of estimating the
distance to examples, the distance is estimated to the centroids. Backward
elimination removes the electrode that carries the least distance from the
subset at each iteration. The process is repeated until the required number of
electrode channels remains in the subset.

####  II-B2 EEGNet

EEGNet is a convolutional neural network consisting of two convolutional steps
in subsequent layers [24]. The first layer with temporal convolution learns
frequency filters, followed by depthwise convolution to learn spatial filters
for specific frequencies. The architecture then uses a depthwise convolution
and pointwise convolution to classify the features using a softmax layer. We
follow the experimental setup based on its Torch library [32] implementation
in the TorchEEG package.

####  II-B3 EEG Conformer

EEG Conformer has three stages: the convolutional stage consisting of two one-
dimensional convolutions to learn temporal and spatial dependencies, followed
by the self-attention module to learn global dependencies, and finally, the
classifier stage, two fully connected layers along with the softmax function
to predict the activity in hands or limb movement/imagery. We follow the
experimental setup and implementation of Song et al. [40] for this work. The
model is trained separately for each participant with the same hyperparameters
except for the second convolutional stage, where the kernel size is adjusted
as per the number of EEG channels used in the dataset.

###  II-C Explainable AI (XAI) technique: GradCAM

For the visualization and explanation interface, leveraging the statistical
data-driven observations, we use Gradient-weighted Class Activation Mapping
(Grad-CAM) [39] to generate the feature relevance maps (montage images
highlighting significant EEG channels based on feature relevance scores from
GradCAM).

GradCAM is a class-specific approach combining the gradients and the input
features to interpret the results. Such a class discriminative approach
enables the comparison between the features learned by the model for the two
tasks, respectively.

For EEGNet architecture, we use the gradient information from the first
convolution layer that represents the feature relevance for each channel at
different frequencies or temporal spans. For EEG Conformer, we use the
gradients from the self-attention module as implemented in the original
work[40, 39]. For both models, to generate the explanations using the
relevance scores, the required number of the most relevant channels is
extracted.

###  II-D Experimental Setup

For our first experiment, three models, MDM, EEGConformer, and EEGNet, are
trained using all 64-channel data for each subject. The trained models are
evaluated on the randomly shuffled epochs selected for the test set with a
fixed seed value. Considering the MDM model as a baseline, we choose 14
participants among the 109 in the dataset with task accuracy at least 10%
higher than chance level accuracy. This step ensures that the subsequent
analysis is unbiased and robust. Further, to evaluate the feature relevance
among the 64 channels, we use Backward elimination for the MDM classifier and
GradCAM technique for EEGConformer and EEGNet. The most relevant 21 channels,
as per feature relevance scores for left and right-hand movement/imagery, are
used to train the models. Finally, to compare feature relevance with domain
knowledge, the models are trained considering the 21 channels positioned near
the motor cortical regions, especially central, frontal-central, and parietal
regions [26]. These combinations result in three configurations: $1)$ Using
all 64 EEG channels, $2)$ Using 21 feature-relevant channels, and $3)$ Using
21 motor imagery (MI) relevant channels for each classification model
respectively, totalling nine unique configurations of classification models.

ID |  | Chance level
---
Accuracy
Accuracy using all EEG channels |  | Accuracy using 21
---
MI relevant channels
| Accuracy using top 21
---
feature relevant channels
|  |  | Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
7 | 59.86 | 83.87 | 83.93 | 83.78 | 75.27 | 71.43 | 81.08 | 74.19 | 75.00 | 72.97
12 | 57.61 | 73.12 | 77.78 | 66.67 | 69.89 | 74.07 | 64.10 | 73.12 | 74.07 | 71.79
22 | 58.78 | 73.12 | 80.00 | 63.16 | 61.29 | 63.64 | 57.89 | 63.44 | 65.45 | 60.53
42 | 56.88 | 78.49 | 83.02 | 72.50 | 75.27 | 79.25 | 70.00 | 77.42 | 75.47 | 80.00
43 | 57.61 | 68.82 | 64.81 | 74.36 | 64.52 | 59.26 | 71.79 | 61.29 | 57.41 | 66.67
48 | 56.52 | 77.42 | 81.13 | 72.50 | 75.27 | 79.25 | 70.00 | 72.04 | 75.47 | 67.50
49 | 58.70 | 72.04 | 81.48 | 58.97 | 73.12 | 74.07 | 71.79 | 65.59 | 81.48 | 43.59
53 | 57.25 | 74.19 | 77.36 | 70.00 | 62.37 | 58.49 | 67.50 | 73.12 | 69.81 | 77.50
70 | 59.06 | 69.89 | 72.73 | 65.79 | 60.22 | 52.73 | 71.05 | 67.74 | 65.45 | 71.05
80 | 59.06 | 70.97 | 78.18 | 60.53 | 60.22 | 65.45 | 52.63 | 64.52 | 63.64 | 65.79
82 | 57.97 | 70.97 | 85.19 | 51.28 | 66.67 | 77.78 | 51.28 | 67.74 | 79.63 | 51.28
85 | 58.70 | 70.97 | 72.22 | 69.23 | 74.19 | 75.93 | 71.79 | 65.59 | 62.96 | 69.23
94 | 58.33 | 77.42 | 79.63 | 74.36 | 84.95 | 90.74 | 76.92 | 75.27 | 81.48 | 66.67
102 | 57.61 | 69.57 | 71.70 | 66.67 | 71.74 | 73.58 | 69.23 | 58.70 | 56.60 | 61.54
Mean$\pm$SD | 58.14$\pm$0.94 | 73.63$\pm$4.26 | 77.80$\pm$5.65 | 67.84$\pm$7.96 | 69.64$\pm$7.35 | 71.12$\pm$10.13 | 67.65$\pm$8.52 | 68.56$\pm$5.69 | 70.28$\pm$8.44 | 66.15$\pm$9.68

TABLE I: Chance level accuracy compared to the performance of MDM classifier
using Riemannian distance on covariance matrix using 1) all 64 EEG channels,
2) using 21 Motor Imagery and movement-related EEG channels leveraging the
domain knowledge and 3) using 21 most relevant EEG channels identified by
relevance scores

###  II-E Performance evaluation

Due to the unbalanced prior probability of the classes, we define the accuracy
for each class as the number of correctly detected examples for a given class
by the total number of examples of this class. The overall accuracy is
weighted based on the prior class probabilities, calculated using the average
of recall obtained on each class to avoid any bias in an imbalanced dataset
[7]. All the results are given as percentages.

###  II-F Problem Formulation for Quantifying explanations

To quantify the comparison of relevant channels when the three models are
trained on all 64 EEG channels, the montage (as visualized in Figure 1) as per
the international 10-10 system [37] is projected onto a matrix of order
$N$x$N$ where $N$ is the maximum number of electrodes in any of the two axes
on the projected space. For this case, $N=11$ corresponds to eleven electrodes
corresponding to the central and temporal regions. Hence, each element in the
matrix represents a spatial location that may correspond to the position of an
electrode in the projected space. For simplicity, the channels considered in
the top 21 features extracted using the XAI technique are marked with 1, while
the rest of the locations are marked as 0, denoting the feature relevance as a
binary value. Such a matrix (as visualized in Figure 2) can now be used to
compare the explanations for a model outcome that are derived in terms of the
feature relevance of an EEG channel.

###  II-G Proposed Approach for Quantifying explanations

We compare such matrices by calculating the Earth Mover’s distance [6]
implemented using Python Optimal Transport library [14]. This method involves
assessing the dissimilarity between two matrices, measuring the minimum work
required to transform one into the other. Earth Mover’s distance(EMD), also
known as Wasserstein distance, evaluates the optimal mass transportation plan,
considering both values and spatial arrangements. This approach provides a
robust metric for matrix dissimilarity, capturing both local and global
differences.

Posing the dissimilarity as an optimal transport problem, let $P$ and $Q$
represent two spatial maps, where each map is a set of spatial elements, i.e.,
the relevance of each EEG channel location, denoted as $p_{i}$ and $q_{j}$.
Here, $n$ and $m$ are the number of channels in each map.

The ground distance matrix $C$ signifies the cost of moving mass from one
channel location in $P$ to another in $Q$. $C_{ij}$ represents the cost of
transporting one unit of mass from $p_{i}$ to $q_{j}$, capturing the pairwise
distances between channel locations.

The transportation matrix $T$ outlines the optimal plan for moving mass from
$P$ to $Q$. Each $T_{ij}$ element indicates the amount of mass transported
from $p_{i}$ to $q_{j}$. The objective is to find $T$ that minimizes the total
transportation cost.

The EMD is then calculated as the sum of the element-wise product of $C$ and
$T$:

$\text{EMD}(P,Q)=\sum_{i=1}^{n}\sum_{j=1}^{m}C_{ij}\cdot T_{ij}$

Subject to the constraints:

1\. The sum of mass transported from any $p_{i}$ to all $q_{j}$ must be equal
to $p_{i}$: $\sum_{j=1}^{m}T_{ij}=p_{i}$ for all $i$.

2\. The sum of mass transported to any $q_{j}$ from all $p_{i}$ must be equal
to $q_{j}$: $\sum_{i=1}^{n}T_{ij}=q_{j}$ for all $j$.

ID |  | Chance level
---
Accuracy
Accuracy using all EEG channels |  | Accuracy using 21
---
MI relevant channels
| Accuracy using top 21 relevant
---
channels from GradCAM
|  |  | Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
7 | 59.86 | 73.12 | 69.64 | 78.38 | 80.65 | 78.57 | 83.78 | 60.22 | 76.79 | 35.14
12 | 57.61 | 74.19 | 75.93 | 71.79 | 67.74 | 70.37 | 64.10 | 75.27 | 81.48 | 66.67
22 | 58.78 | 68.82 | 67.27 | 71.05 | 60.22 | 65.45 | 52.63 | 70.97 | 85.45 | 50.00
42 | 56.88 | 78.49 | 83.02 | 72.50 | 82.80 | 84.91 | 80.00 | 67.74 | 66.04 | 70.00
43 | 57.61 | 62.37 | 72.22 | 48.72 | 62.37 | 72.22 | 48.72 | 63.44 | 87.04 | 30.77
48 | 56.52 | 74.19 | 81.13 | 65.00 | 73.12 | 69.81 | 77.50 | 67.74 | 75.47 | 57.50
49 | 58.70 | 69.89 | 81.48 | 53.85 | 74.19 | 77.78 | 69.23 | 73.12 | 75.93 | 69.23
53 | 57.25 | 69.89 | 77.36 | 60.00 | 60.22 | 67.92 | 50.00 | 75.27 | 84.91 | 62.50
70 | 59.06 | 61.29 | 85.45 | 26.32 | 56.99 | 58.18 | 55.26 | 59.14 | 78.18 | 31.58
80 | 59.06 | 60.22 | 70.91 | 44.74 | 59.14 | 61.82 | 55.26 | 56.99 | 69.09 | 39.47
82 | 57.97 | 53.76 | 62.96 | 41.03 | 59.14 | 64.81 | 51.28 | 60.22 | 57.41 | 64.10
85 | 58.70 | 68.82 | 70.37 | 66.67 | 58.06 | 57.41 | 58.97 | 55.91 | 61.11 | 48.72
94 | 58.33 | 79.57 | 92.59 | 61.54 | 77.42 | 85.19 | 66.67 | 68.82 | 79.63 | 53.85
102 | 57.61 | 66.30 | 58.49 | 76.92 | 56.52 | 77.36 | 28.21 | 58.70 | 54.72 | 64.10
Mean$\pm$SD | 58.14$\pm$0.94 | 68.64$\pm$7.30 | 74.92$\pm$9.29 | 59.89$\pm$15.21 | 66.33$\pm$9.42 | 70.84$\pm$9.00 | 60.12$\pm$14.76 | 65.25$\pm$6.85 | 73.80$\pm$10.52 | 53.12$\pm$14.09

TABLE II: Chance level accuracy compared to the performance of EEG Conformer
model using 1) all 64 EEG channels, 2) using 21 Motor Imagery and movement-
related EEG channels leveraging the domain knowledge, and 3) using 21 most
relevant EEG channels identified by GradCAM.

##  III Results

A comprehensive overview of model performance under different setups, using
all 64 EEG channels, 21 feature-relevant channels, and 21 motor imagery (MI)
relevant channels, respectively, for the three model architectures is
represented in Table I, Table II, and Table III, respectively. MDM classifier
using Riemannian distance achieves 73.63% accuracy using all channels. When
trained on 21 MI-relevant channels, the performance reduces by 3.99% to 69.64%
(p=0.0279 using Wilcoxon signed-rank test). Using the backward elimination
method for channel selection, accuracy is 68.56%, decreasing by 5.07%
(p=0.0014 using Wilcoxon signed-rank test). A similar trend is observed for
EEGConformer and EEGNet across the configurations; however, comparing the
three model’s performance MDM classifier outperforms EEGConformer by 4.99%
(p=0.0029 using Wilcoxon signed-rank test) and EEGNet by 6.61% (p=0.0028 using
Wilcoxon signed-rank test) using all channels.

ID |  | Chance level
---
Accuracy
Accuracy using all EEG channels |  | Accuracy using 21
---
MI relevant channels
| Accuracy using top 21 relevant
---
channels from GradCAM
|  |  | Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
| Overall
---
| Left fist
---
| Right fist
---
7 | 59.86 | 78.49 | 83.93 | 70.27 | 81.72 | 82.14 | 81.08 | 66.67 | 75.00 | 54.05
12 | 57.61 | 65.59 | 61.11 | 71.79 | 58.06 | 53.70 | 64.10 | 62.37 | 79.63 | 38.46
22 | 58.78 | 59.14 | 63.64 | 52.63 | 61.29 | 58.18 | 65.79 | 56.99 | 65.45 | 44.74
42 | 56.88 | 68.82 | 69.81 | 67.50 | 78.49 | 79.25 | 77.50 | 56.99 | 66.04 | 45.00
43 | 57.61 | 63.44 | 64.81 | 61.54 | 58.06 | 53.70 | 64.10 | 59.14 | 62.96 | 53.85
48 | 56.52 | 70.97 | 75.47 | 65.00 | 63.44 | 62.26 | 65.00 | 64.52 | 73.58 | 52.50
49 | 58.70 | 75.27 | 77.78 | 71.79 | 73.12 | 66.67 | 82.05 | 73.12 | 81.48 | 61.54
53 | 57.25 | 68.82 | 71.70 | 65.00 | 64.52 | 66.04 | 62.50 | 65.59 | 73.58 | 55.00
70 | 59.06 | 59.14 | 56.36 | 63.16 | 58.06 | 63.64 | 50.00 | 61.29 | 63.64 | 57.89
80 | 59.06 | 60.22 | 63.64 | 55.26 | 59.14 | 60.00 | 57.89 | 51.61 | 54.55 | 47.37
82 | 57.97 | 62.37 | 72.22 | 48.72 | 54.84 | 57.41 | 51.28 | 63.44 | 85.19 | 33.33
85 | 58.70 | 70.97 | 70.37 | 71.79 | 65.59 | 62.96 | 69.23 | 63.44 | 74.07 | 48.72
94 | 58.33 | 77.42 | 87.04 | 64.10 | 78.49 | 85.19 | 69.23 | 62.37 | 72.22 | 48.72
102 | 57.61 | 57.61 | 56.60 | 58.97 | 60.87 | 66.04 | 53.85 | 59.78 | 71.70 | 43.59
Mean$\pm$SD | 58.14$\pm$0.94 | 67.02$\pm$7.02 | 69.61$\pm$9.33 | 63.40$\pm$7.35 | 65.41$\pm$8.87 | 65.51$\pm$10.01 | 65.26$\pm$10.14 | 61.95$\pm$5.12 | 71.36$\pm$8.18 | 48.91$\pm$7.63

TABLE III: Chance level accuracy compared to the performance of EEGNet model
using 1) all 64 EEG channels, 2) using 21 Motor Imagery and movement-related
EEG channels leveraging the domain knowledge, and 3) using 21 most relevant
EEG channels identified by GradCAM.

(a)

(b)

(c)

Figure 1: Montage images highlighting the significant channels among the 64
channels, identified by a) using feature relevance scores based on Riemannian
distance on covariance matrix, b) using GradCAM with EEGConformer model, and
c) using GradCAM with EEGNet model.

(a) Baseline

(b) MDM-Riemannian
Earth mover’s distance: 7.1612

(c) EEG Conformer
Earth mover’s distance: 8.9289

(d) EEGNet
Earth mover’s distance: 9.2948

Figure 2: 2D spatial representation of the significant channels among the 64
channels used for the comparison based on EMD. a) Baseline representation
using 21 channels proximal to motor cortical regions, b) Relevance from MDM
classifier using Riemannian distance is closest to baseline, b) followed by
EEGConformer, and c) EEGNet model, when trained on top 14 participants.

(a) Baseline

(b) MDM-Riemannian
Earth mover’s distance: 7.0931

(c) EEG Conformer
Earth mover’s distance: 9.1471

(d) EEGNet
Earth mover’s distance: 9.7759

Figure 3: 2D spatial representation of the significant channels with weighted
relevance among the 64 channels used for the comparison based on EMD. a)
Baseline representation with equal importance to 21 channels proximal to motor
cortical regions, b) Relevance from MDM classifier using Riemannian distance
is closest to baseline, b) followed by EEGConformer, and c) EEGNet model, when
trained on top 14 participants.

Comparisons of global-level model explanations are facilitated through montage
images in Figure 1, allowing visualization of the distribution differences in
feature importance across configurations. While channel importance for the MDM
classifier is class agnostic, GradCAM provides channel importance for right
and left-hand movement classification separately.

The spatial maps represented in Figure 2 offer a concise representation of
feature relevance and channel positioning, aiding in quantifying model
explanations in the spatial domain. Following a similar trend in accuracy, the
EMD for the MDM classifier is 7.1612, closest to the baseline condition of MI-
relevant electrodes. Explanations from EEGConformer and EEGNet are at 8.9289
and 9.2948 distance, respectively, from the baseline. Considering the
frequency of each channel in the top relevant features across participants,
the weighted spatial map is represented in Figure 3 for each model with
distance from the baseline. Comparing the distances to unweighted spatial
maps, EMD for the MDM classifier decreases to 7.0931 as it has greater
weightage to channels in the subset of the baseline, while EEGConformer and
EEGNet distances increase to 9.1471 and 9.7759, respectively.

##  IV Discussion

The obtained results reveal intriguing insights into the performance of
classification models and the feature relevance-driven explanations derived
from channel selection or XAI techniques in the context of MI tasks.

The observed trend indicating Riemannian geometry outperforming EEGConformer
and EEGNet showcases the effectiveness of Riemannian approaches in capturing
the intrinsic geometry of symmetric positive definite (SPD) matrices, common
in EEG covariance matrices. Features related to motor imagery are based on
rhythms, not in the time domain, which the variance can also capture as it is
proposed in the common spatial pattern (CSP)-based classifiers. However, since
we chose the baseline model to be the MDM classifier, the results possibly
ignored the participant’s data where EEGConformer and EEGNet outperformed the
Riemannian geometry-based approach.

The marginal performance advantage of EEGConformer over EEGNet is consistent
with the nuanced differences in the neural network architecture. EEGConformer
uses multi-head attention layers that help the network identify features
across long sequences compared to EEGNet. The results motivate us to
investigate feature relevance in other domains besides spatial information.
Possible approaches could include comparisons in spectral and temporal domains
by dividing each trial into shorter epochs and comparing feature relevance in
those epochs.

The consistently better model performance of MI-relevant channels over
feature-relevant channels, as demonstrated by GradCAM and backward
elimination, adds depth to the understanding of spatial information. It
underscores the significance of domain-specific knowledge in training data-
driven models and carefully selecting channels that align with motor cortical
regions.

The consistency of this trend in both accuracy and the calculated Wasserstein
distance or EMD reinforces the reliability of the findings. The parallel
outcomes across different metrics enhance the robustness of the conclusions.
An interesting direction to explore is to understand neuroscience experts’
opinions on reliability and their choice of model given the explanations.

An interpretation of relevant channels that were significant in data-driven
approaches may be linked to artefacts or activities that correlated with the
task but were not causally involved. The fact that there are sensors that are
needed and do not correspond to where the expected brain responses are
projected on the scalp, holds for sensor-level analysis. However, an
investigation to understand further the temporal and spectral domain
explanations holds an interesting prospect to support such interpretations.

While the selection of MI-relevant channels based on the domain knowledge of
specific regions associated with motor functions is a known practice, data-
driven model outcomes need to be verified in the context where the signal-to-
noise ratio is typically high and raw EEG data is used to train an end-to-end
learning model. This insight has practical implications, especially in
optimizing Brain-Computer Interface (BCI) systems. There is an increased
interest in exploring neural network architectures optimising on Riemannian
manifolds[18, 21, 31, 41]. Therefore, an approach towards quantifying feature
relevance and evaluation with respect to facts grounded in neuroscience is
even more relevant.

Explanations are used as tools for model improvement in BCIs[33] and across
domains [45]. In this context, the proposed metric for comparing spatial maps
can be employed in model regularisation, gradient, and feature augmentation.
Concurrent efforts to understand the needs and the nuances of explanations for
relevant stakeholders involved in the domain applications of BCIs are also
required[19]. Kim et al. [20] propose a theoretical framework for XAI
interfaces that captures the requirements of a domain expert in BCI. Combining
quantitative explanations with user-centric interfaces appears to be a
promising approach towards reliable and trustworthy BCIs.

In the context of BCIs, where the interpretability of models is crucial for
understanding and trusting the technology, the susceptibility to adversarial
attacks could pose significant concerns. Galli et al. [15] highlight the
susceptibility of DNNs and XAI outcomes to Adversarial Perturbations (APs),
which are procedures designed to mislead a target model using imperceptible
noise. Ravindran and Contreras-Vidal [42] also conclude that XAI techniques
also show reliability issues with simulated EEG data in the context of BCIs
after randomizing either model weights or labels. Such works indicate the
necessity to study the reliability of XAI outcomes vigorously.

Acknowledging potential limitations, such as dataset-specific characteristics,
is essential for this study. The study’s generalizability will benefit from
exploring diverse datasets to validate the observed trends across different
experimental setups. Future work will include the analysis of other BCI
paradigms, such as the event-related paradigm [22, 9, 25]. It is indeed
important to determine if a BCI is truly a BCI and not capturing muscle
artefacts that are synchronized with the presentation of target stimuli.

##  V Conclusion

In this paper, we have analyzed and compared different state-of-the-art
machine-learning algorithms for motor imagery classification. The results
demonstrate that efficacy cannot be determined by using performance metrics
like accuracy on limited datasets. The work proposed quantifying the
explanations or feature relevance extracted using different techniques for
three model architectures. The results confirm the performance of the
Riemannian geometry-based classifier and suggest substantial discrepancies
between the best sensors and where they are expected. The findings indicate
that end-to-end data-driven approaches need to be validated in the context of
BCIs, and the model interpretations should be compared against what can be
expected from the literature by adding the XAI framework to the data-driven
pipeline.

##  VI Acknowledgment

Our source code is publicly available on https://github.com/HAIx-
Lab/SpatialExplanations4BCI. This work was supported by Indian Institute of
Technology Gandhinagar startup grant IP/IITGN/CSE/YM/2324/05.

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