#  preprint SIN-PR-80-008
Post-Newtonian Generation of Gravitational Waves in a Theory of Gravity with
Torsion

M. Schweizer, N. Straumann, and A. Wipf 111present address: Theoretisch-
Physikalisches-Institut, Friedrich-Schiller-Universität Jena, 07743 Jena,
Germany; email<EMAIL_ADDRESS>
_Institute for Theoretical Physics of the University of Zürich,_
_Schönberggasse 9, 8001 Zürich, Switzerland_

We adapt the post-Newtonian gravitational-radiation methods developed within
general relativity by Epstein and Wagoner to the gravitation theory with
torsion, recently proposed by Hehl et al., and show that the two theories
predict in this approximation the same gravitational radiation losses. Since
they agree also on the first post-Newtonian level, they are at the present
time – observationally – indistinguishable.

Published version: General Relativity and Gravitation, Vol. 12 (1980) 951-961
doi: 10.1007/BF00757366
arXiv-ed February 1980 ; LaTeX-ed May 1, 2023

Uploaded to arXiv to help researchers with limited library facilities

###### Contents

  1. 1 Introduction
  2. 2 The Poincaré Gauge Theory of Hehl et al.
  3. 3 The First Post-Newtonian Approximation
  4. 4 Conservation Laws
  5. 5 Post-Newtonian Generation of Gravitational Radiation
  6. 6 Summary

##  1 Introduction

The binary pulsar data obtained by Taylor and coworkers (Taylor [1]) have
already ruled out many alternative gravitation theories. Will [2] has adopted
the post-Newtonian gravitational-radiation methods developed within general
relativity by Epstein and Wagoner [3] and by Wagoner and Will [4] to
alternative metric theories of gravitation and thus has shown that in most (if
not all) of these dipole gravitational radiation can also exist. The dipole
radiation of a two-body system, say, is generated by the varying dipole moment
of the gravitational binding energy and is typically much too large for the
binary pulsar PSR 1913+ 16. There are examples (e.g. the Rosen bimetric
theory) where the dipole gravitational radiation causes the system even to
gain energy at a relatively high rate and strong dissipative mechanisms would
have to be invented to account for the observed decrease. In an earlier paper
[5] we have shown that the "Poincaré gauge theory of gravity" proposed by Hehl
et al. [6] agrees with general relativity on the first post-Newtonian level.
The purpose of the present work is to demonstrate that the "dipole
catastrophe" mentioned above does not occur in this theory and that – within
the approximation scheme of [3] – the gravitational radiation loss is the same
as in general relativity. From the observational point of view it is,
therefore, practically impossible to distinguish the theory of Hehi et al.
from general relativity. In Section 2, the theory of Hehl et al. (slightly
generalized) is briefly summarized in the language of differential forms. In
Section 3, we recast the field equations into a form from which one can deduce
easily that the first post-Newtonian approximation of the theory agrees with
that of general relativity. In Section 4, we determine the energy-momentum
forms of the gravitational field, and finally we demonstrate in Section 5 that
the post-Newtonian generation of gravitational radiation agrees also with
general relativity. The results are summarized in Section 6

##  2 The Poincaré Gauge Theory of Hehl et al.

In this section, we give a short description of the gravitation theory which
was proposed by Hehl et al. [6]. In contrast to Reference [6] we use Cartan’s
calculus of differential forms and follow the conventions of Trautman [7].
(See also [5, 8, 9] .) Let $\theta^{\alpha}$ denote an orthonormal tetrad
field of 1-forms, and ${\omega^{\alpha}}_{\beta}$ the connection forms of a
metric connection with torsion. The exterior covariant derivatives of
$\theta^{\alpha}$ and ${\omega^{\alpha}}_{\beta}$ are the torsion forms
$\Theta^{\alpha}$ and the curvature forms ${\Omega^{\alpha}}_{\beta}(c=1,8\pi
G=1)$. The 1-forms $(\theta^{\alpha};{\omega^{\alpha}}_{\beta})$ can be
considered as gauge potentials of the Poincaré group because they determine a
connection in the Poincaré bundle (which is a subbundle of the affine bundle).
In this interpretation, the 2-forms
$(\Theta^{\alpha},{\Omega^{\alpha}}_{\beta})$ are the corresponding field
strengths. A slight generalization of the gravitational Lagrangian which was
chosen in [6] reads as follows

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\mathrm{transl}}+\mathcal{L}_{\mathrm{rot}}\,,$
(2.1) $\displaystyle\mathcal{L}_{\mathrm{transl}}$
$\displaystyle=-\frac{1}{2l^{2}}\Big{[}(\Theta^{\alpha}\wedge\theta^{\beta})\wedge{{}^{*}}(\Theta_{\beta}\wedge\theta_{\alpha})-\frac{\lambda}{2}(\Theta^{\alpha}\wedge\theta_{\alpha})\wedge{{}^{*}}(\Theta^{\beta}\wedge\theta_{\beta})\Big{]}\,,$
(2.2) $\displaystyle\mathcal{L}_{\mathrm{rot}}$
$\displaystyle=-\frac{1}{2k}\Omega^{\alpha\beta}\wedge{{}^{*}}\Omega_{\alpha\beta}\,.$
(2.3)

Here $l$ is the Planck length and $k$ is a dimensionless coupling constant.
Hehl et al. use $\lambda=0$ in (2.2). To the Lagrangian (2.1) we have to add a
matter term, $L_{m}$. Independent variations of $L_{g}+L_{m}$ with respect to
$\theta^{\alpha}$ and ${\omega^{\alpha}}_{\beta}$ give the field equations.
The variation of $L_{m}$ with respect to ${\omega^{\alpha}}_{\beta}$
determines the spin density which is for macroscopic matter very small
compared to the energy-momentum density. For this reason Hehl et al. choose
for astronomical systems the "translational gauge limit" in which the
curvature vanishes

${\Omega^{\alpha}}_{\beta}=0\,.$ (2.4)

Consequently the rotational (Yang-Mills type) part (2.3) in the Lagrangian
(2.1) has to be dropped. The resulting theory has an obvious geometric
interpretation in a Weitzenböck space. Relative to a teleparallel tetrad the
connection forms ${\omega^{\alpha}}_{\beta}$ also vanish and the exterior
covariant derivative reduces the ordinary exterior derivative. The torsion
$\Theta^{\alpha}$ is then equal to $\mathrm{d}\theta^{\alpha}$. Thus, we use
the following total Lagrangian relative to an arbitrary (orthonormal) frame:

$\mathcal{L}=-{\textstyle\frac{1}{2}}(\Theta^{\alpha}\wedge\theta^{\beta})\wedge{{}^{*}}(\Theta_{\beta}\wedge\theta_{\alpha})+{\textstyle\frac{\lambda}{4}}(\Theta^{\alpha}\wedge\theta_{\alpha})\wedge{{}^{*}}(\Theta^{\beta}\wedge\theta_{\beta})+\mathcal{L}_{m}\,,$
(2.5)

which reduces to

$\mathcal{L}=-{\textstyle\frac{1}{2}}(\mathrm{d}\theta^{\alpha}\wedge\theta^{\beta})\wedge{{}^{*}}(\mathrm{d}\theta_{\beta}\wedge\theta_{\alpha})+{\textstyle\frac{\lambda}{4}}(\mathrm{d}\theta^{\alpha}\wedge\theta_{\alpha})\wedge{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})+\mathcal{L}_{m}$
(2.6)

for a teleparallel frame.

Now we note that the Hilbert-Einstein Lagrangian can be written in the
following way (up to an exact differential):

$\mathcal{L}=-{\textstyle\frac{1}{2}}(\mathrm{d}\theta^{\alpha}\wedge\theta^{\beta})\wedge{{}^{*}}(\mathrm{d}\theta_{\beta}\wedge\theta_{\alpha})+{\textstyle\frac{1}{4}}(\mathrm{d}\theta^{\alpha}\wedge\theta_{\alpha})\wedge{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})+\mathcal{L}_{m}\,.$
(2.7)

This can easily be shown by putting
$\mathrm{d}\theta^{\alpha}={\textstyle\frac{1}{2}}C^{\alpha}_{\beta\gamma}\theta^{\beta}\wedge\theta^{\gamma}$
and expressing both sides of (2.7) in terms of $C^{\alpha}_{\beta\gamma}$.
Thus the gravitational part of (2.6) reduces to the Hilbert-Einstein
Lagrangian for $\lambda=1$. This remark is crucial for all that follows and
was also used in [5]. It is not surprising that the gravitational Lagrangian
is invariant under local Lorentz transformations

$\theta^{\alpha}(x)\longrightarrow\Lambda^{\alpha}_{\;\,\beta}(x)\theta^{\beta}(x),\qquad\Lambda(x)\in\mathcal{L}^{\uparrow}_{+}\,,$
(2.8)

for $\lambda=1$ only.

Variation of the tetrad fields leads to the field equations. We introduce the
notations:

$\displaystyle\delta\mathcal{L}_{E}$
$\displaystyle=\delta\theta^{\alpha}\wedge\epsilon^{E}_{\alpha}\,,$ (2.9)
$\displaystyle\delta\mathcal{L}_{m}$
$\displaystyle=\delta\theta^{\alpha}\wedge t_{\alpha}\,,$ (2.10)
$\displaystyle\delta(\Delta\mathcal{L})$
$\displaystyle=\delta\theta^{\alpha}\wedge\Delta\epsilon_{\alpha}\,,$ (2.11)

where $(\lambda-1)\Delta\mathcal{L}$ is the difference between the
gravitational part of (2.6) and (2.7). The explicit expressions of
$\epsilon^{E}_{\alpha}$ and $\Delta\epsilon_{\alpha}$ are

$\displaystyle\epsilon^{\alpha}_{E}\;$
$\displaystyle=-\mathrm{d}\big{[}\theta^{\beta}\wedge{{}^{*}}(\mathrm{d}\theta_{\beta}\wedge\theta^{\alpha})\big{]}-\mathrm{d}\theta^{\beta}\wedge{{}^{*}}(\mathrm{d}\theta^{\alpha}\wedge\theta_{\beta})+{\textstyle\frac{1}{2}}\mathrm{d}\big{\\{}\theta^{\alpha}\wedge{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})\big{\\}}$
$\displaystyle\hskip
213.39566pt+{\textstyle\frac{1}{2}}\mathrm{d}\theta^{\alpha}\wedge{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})\,,$
(2.12) $\displaystyle\Delta\epsilon^{\alpha}$
$\displaystyle=\mathrm{d}\theta^{\alpha}\wedge{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})-{\textstyle\frac{1}{2}}\theta^{\alpha}\wedge
d\,{{}^{*}}(\mathrm{d}\theta^{\beta}\wedge\theta_{\beta})\,.$ (2.13)

and the field equations read as follows:

$\epsilon^{\alpha}_{E}+(\lambda-1)\Delta\epsilon^{\alpha}=-t^{\alpha}\,.$
(2.14)

$t^{\alpha}$ are the energy-momentum 3-forms of matter. The components of the
energy-momentum tensor, $T^{\alpha\beta}$, relative to $\theta^{\alpha}$ are
given by

${{}^{*}}t^{\alpha}=T_{\beta}^{\;\,\alpha}\theta^{\beta}\,.$ (2.15)

The field equations (2.14) and the Einstein equations have a large family of
common solutions. For any metric of the diagonal form

$\mathrm{d}s^{2}=a_{0}^{2}(\mathrm{d}x^{0})^{2}-a_{1}^{2}(\mathrm{d}x^{1})^{2}-a_{2}^{2}(\mathrm{d}x^{2})^{2}-a_{3}^{2}(\mathrm{d}x^{3})^{2}$

with arbitrary functions $a_{i}$ the forms $\Delta\epsilon^{\alpha}$ vanish
for the orthonormal basis

$\theta^{0}=a_{0}\mathrm{d}x^{0},\quad\theta^{1}=a_{1}\mathrm{d}x^{1},\quad\theta^{2}=a_{2}\mathrm{d}x^{2},\quad\theta^{3}=a_{3}\mathrm{d}x^{3}$

because $\mathrm{d}\theta^{\alpha}\wedge\theta_{\alpha}=0$. This proves in
particular that all spherically symmetric solutions of the Einstein equations
are also solutions of (2.14) and thus a large body of astrophysical
applications remains unchanged. The Kerr solution is, however, no vacuum
solution of (2.14) for $\lambda\neq 1$. It is an open problem to generalize
the stationary black hole solution to this case.

##  3 The First Post-Newtonian Approximation

For all the further discussions, it will turn out to be useful to split the
field equations (2.14) into symmetrized and antisymmetrized parts. In contrast
to $\epsilon^{\alpha}_{E}$ and $t^{\alpha}$ the forms
$\Delta\epsilon^{\alpha}$ are not symmetric:

$\Delta\epsilon^{\alpha}\wedge\theta^{\beta}\neq\Delta\epsilon^{\beta}\wedge\theta^{\alpha}$

because $\Delta\mathcal{L}$ is not locally Lorentz invariant. The symmetric
and antisymmetric parts of $\Delta\epsilon^{\alpha}$ are

$\displaystyle\Delta\epsilon^{\alpha}_{s}$
$\displaystyle=-{\textstyle\frac{1}{2}}\,{{}^{*}}\big{[}(\mathrm{d}\theta^{\alpha}\wedge\theta^{\beta}+\mathrm{d}\theta^{\beta}\wedge\theta^{\alpha})\wedge{{}^{*}}(\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma})\big{]}\eta_{\beta}\,,$
(3.1) $\displaystyle\Delta\epsilon^{\alpha}_{a}$
$\displaystyle=-{\textstyle\frac{1}{2}}\,{{}^{*}}\big{\\{}\mathrm{d}[\theta^{\alpha}\wedge\theta^{\beta}\wedge{{}^{*}}(\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma})]\big{\\}}\eta_{\beta}\,,$
(3.2)

where $\eta^{\beta}={{}^{*}}\theta^{\beta}$. Hence the field equations (2.14)
are equivalent to the pair of equations

$\displaystyle\epsilon^{\alpha}_{E}-{\textstyle\frac{1}{2}}(\lambda-1)$
$\displaystyle\,{{}^{*}}\big{[}(\mathrm{d}\theta^{\alpha}\wedge\theta^{\beta})+\mathrm{d}\theta^{\beta}\wedge\theta^{\alpha})\wedge{{}^{*}}(\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma})\big{]}\eta_{\beta}=-t^{\alpha}\,,$
(3.3)
$\displaystyle(\lambda-1)\mathrm{d}\big{[}\theta^{\alpha}\wedge\theta^{\beta}\wedge{{}^{*}}(\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma})\big{]}=0\,.$
(3.4)

Now we expand the teleparallel frames $\theta^{\alpha}$ in terms of a
coordinate basis

$\theta^{\alpha}=\mathrm{d}x^{\alpha}+\Phi^{\alpha}_{\;\;\beta}\mathrm{d}x^{\beta}$
(3.5)

and decompose $\Phi_{\alpha\beta}$ into its symmetric and antisymmetric pieces

$\displaystyle\Phi_{\alpha\beta}=\phi_{\alpha\beta}+a_{\alpha\beta}\,,$ (3.6)
$\displaystyle\phi_{\alpha\beta}$ $\displaystyle=\Phi_{(\alpha\beta)},\qquad
a_{\alpha\beta}=\Phi_{[\alpha\beta]}\,.$ (3.7)

Let us first consider the linearized approximations of (3.3) and (3.4). The
second term on the left-hand side of (3.3) contains obviously no linear terms.
Furthermore, the antisymmetric field $a_{\alpha\beta}$ drops out identically
in the linearized part of $\epsilon^{\alpha}_{E}$ (as a consequence of the
local Lorentz invariance of (2.7)). Thus equation (3.3) is reduced to the
linearized Einstein equation for $\phi_{\alpha\beta}$. For $\lambda=1$ (i.e.,
general relativity) equation (3.4) is empty; $a_{\alpha\beta}$ is just a gauge
degree of freedom. For $\lambda\neq 1$ equation (3.4) leads in the linearized
approximation to the decoupled source free equation

$\Box\,a^{\alpha\beta}+a^{\lambda\alpha,\beta}_{\hskip
17.07164pt\lambda}-a^{\lambda\beta,\alpha}_{\hskip 17.07164pt\lambda}=0$ (3.8)

for $a^{\alpha\beta}$ which is invariant under the gauge transformation

$a_{\alpha\beta}\longrightarrow
a_{\alpha\beta}+\xi_{\alpha,\beta}-\xi_{\beta,\alpha}\,.$ (3.9)

Imposing the gauge condition ${a^{\alpha\beta}}_{,\beta}=0$, we are left with
$\Box\,a^{\alpha\beta}=0$. In the Newtonian approximation $a_{\alpha\beta}$
vanishes in this gauge and the theory gives the correct Newtonian limit. Next
we show that the first post-Newtonian approximation of (3.3) and (3.4) agrees
with that of general relativity. (This was shown already in [5], but the
following discussion is simpler.) Since the term proportional to $(\lambda-1)$
in (3.3) is of higher order in $\Phi_{\alpha\beta}$, it contains in the first
post-Newtonian approximation at most quadratic expressions of the Newtonian
approximation of $\phi_{\alpha\beta}$. But these vanish identically because
$\mathrm{d}\theta^{\alpha}\wedge\theta_{\alpha}$ is already quadratic in
$\phi_{\alpha\beta}$. (The linear term vanishes identically.) Hence (3.3)
reduces to the first post-Newtonian approximation of general relativity. For
the discussion of equation (3.4), we note first that the Newtonian
approximation, $\phi^{\text{(N)}}_{\alpha\beta}$, of $\phi_{\alpha\beta}$ is
diagonal in a suitable coordinate system:
$\phi^{\text{(N)}}_{\alpha\beta}=\delta_{\alpha\beta}\Phi$, $\Phi$ = Newtonian
potential. From this, one concludes easily that the quadratic terms in
$\phi^{\text{(N)}}_{\alpha\beta}$ in (3.4) vanishes identically and thus the
first post-Newtonian approximation of (3.4) reduces to equation (3.8) for the
post-Newtonian approximation, $a^{\text{PN}}_{\alpha\beta}$, od
$a_{\alpha\beta}$. This implies that $a^{\text{PN}}_{\alpha\beta}$ vanishes
also in a suitable gauge and hence our claim is proven. Hehl and Nitsch [10]
have shown that the post-post-Newtonian approximation no longer agrees with
that of general relativity. The deviations are, however, unmeasurably small.

##  4 Conservation Laws

We start by writing the symmetric field equation (3.3) in arbitrary (not
necessarily teleparallel or orthonormal) frames:

$\epsilon^{\alpha}_{E}-{\textstyle\frac{1}{2}}(\lambda-1)\,{{}^{*}}\big{[}(\Theta^{\alpha}\wedge\theta^{\beta}+\Theta^{\beta}\wedge\theta^{\alpha})\wedge{{}^{*}}(\Theta^{\gamma}\wedge\theta_{\gamma})\big{]}\eta_{\beta}=-t^{\alpha}\,.$
(4.1)

For $\epsilon^{\alpha}_{E}$ we use the "Landau decomposition" derived in [11]

$\epsilon^{\alpha}_{E}=\big{[}2(-g)^{1/2}\big{]}^{-1}\cdot\mathrm{d}\big{[}(-g)^{1/2}\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}\\!_{\beta}^{\;\;\gamma}\\!\wedge\,\eta^{\alpha\beta}_{\hskip
11.38109pt\gamma}\big{]}+t^{\alpha}_{\mathrm{LL}}\,.$ (4.2)

Here ${\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}}\,\\!_{\beta}^{\;\;\gamma}$ are the Levi-Civith connection
forms and $t^{\alpha}_{\mathrm{LL}}$ are the Landau-Lifschitz energy-momentum
forms of the metric field given explicitly by

$t^{\alpha}_{\mathrm{LL}}=-{\textstyle\frac{1}{2}}\eta^{\alpha\beta\gamma\delta}\big{(}\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\sigma\beta}\wedge\,{\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}}\,\\!^{\sigma}_{\;\;\gamma}\wedge\theta_{\delta}-\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\beta\gamma}\wedge\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\sigma\delta}\wedge\theta^{\sigma}\big{)}\,.$ (4.3)

Relative to a coordinate basis $\theta^{\alpha}=\mathrm{d}x^{\alpha}$ the
3-forms $t^{\alpha}_{\mathrm{LL}}$ are symmetric. Inserting (4.2) into (4.1)
the symmetric field equation takes the form

$-{\textstyle\frac{1}{2}}\mathrm{d}\big{[}(-g)^{1/2}\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\beta\gamma}\wedge\eta^{\alpha\beta\gamma}\big{]}=(-g)^{1/2}\tau^{\alpha}\,,$
(4.4)

where

$\displaystyle\tau^{\alpha}$
$\displaystyle=t^{\alpha}+t^{\alpha}_{\mathrm{LL}}-(\lambda-1)\Delta
t^{\alpha}\,,$ (4.5) $\displaystyle\Delta t^{\alpha}$
$\displaystyle={\textstyle\frac{1}{2}}{{}^{*}}\big{[}(\Theta^{\alpha}\wedge\theta^{\beta}+\Theta^{\beta}\wedge\theta^{\alpha})\wedge{{}^{*}}(\Theta^{\gamma}\wedge\theta_{\gamma})\big{]}\eta_{\beta}\,.$
(4.6)

It may be useful to note that the left-hand side of (4.4) relative to a
coordinate basis can be expressed in terms of the Landau-Lifschitz
superpotential as follows:

$\mathrm{d}\big{[}(-g)^{1/2}\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\beta\gamma}\wedge\eta^{\mu\beta\gamma}\big{]}=(-g)^{1/2}{H^{\mu\alpha\nu\beta}}_{,\alpha}\,\eta_{\nu}\,,$

where

$H^{\mu\alpha\nu\beta}={\hat{g}}^{\mu\nu}{\hat{g}}^{\alpha\beta}-{\hat{g}}^{\mu\beta}{\hat{g}}^{\nu\alpha}$

with

${\hat{g}}^{\mu\nu}=(-g)^{1/2}g^{\mu\nu}\,.$

As in general relativity, the $t^{\alpha}$ are interpreted as the total
energy-momentum forms. By construction, they are symmetric relative to a
coordinate basis:

$\tau^{\alpha}\wedge\mathrm{d}x^{\beta}=\tau^{\beta}\wedge\mathrm{d}x^{\alpha}\,.$
(4.7)

From (4.4) we conclude that the field equations imply the conservation laws

$\mathrm{d}\big{[}(-g)^{1/2}\tau^{\alpha}\big{]}=0\,.$ (4.8)

The last two equations imply

$\mathrm{d}\big{[}(-g)^{1/2}\mathrm{M}^{\alpha\beta}\big{]}=0\,,$ (4.9)

where

$\mathrm{M}^{\alpha\beta}=x^{\alpha}\tau^{\beta}-x^{\beta}\tau^{\alpha}$
(4.10)

is the total angular momentum density. For isolated systems the total momentum

$\mathrm{P}^{\alpha}=\int_{\Sigma}(-g)^{1/2}\tau^{\alpha}$ (4.11)

and the total angular momentum

$\mathrm{J}^{\alpha\beta}=\int_{\Sigma}(-g)^{1/2}\mathrm{M}^{\alpha\beta}$
(4.12)

($\Sigma$: spacelike surface) can be expressed with the help of the field
equations (4.4) in terms of flux integrals at infinity

$\displaystyle\mathrm{P}^{\alpha}$
$\displaystyle=-{\textstyle\frac{1}{2}}\oint(-g)^{1/2}\,\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}_{\beta\gamma}\wedge\eta^{\alpha\beta\gamma}\,,$ (4.13)
$\displaystyle\mathrm{J}^{\alpha\beta}$
$\displaystyle={\textstyle\frac{1}{2}}\oint(-g)^{1/2}\,\big{[}(x^{\alpha}\eta^{\beta}_{\;\;\sigma\gamma}-x^{\beta}\eta^{\alpha}_{\;\;\sigma\gamma})\wedge\stackrel{{\scriptstyle\text{\tiny
LC}}}{{\omega}}\\!^{\sigma\gamma}+\eta^{\alpha\beta}\big{]}\,.$ (4.14)

As usual, the coordinates have to be asymptotically Lorentzian.
$\mathrm{P}^{\alpha}$ and $\mathrm{J}^{\alpha\beta}$ transform like Lorentz
tensors under coordinate transformations which preserve this property.
Clearly, the flux integrals (4.13) and (4.14) are the same as in general
relativity. Since we have not seen equation (4.14) in the literature, we
derive it in the Appendix.

##  5 Post-Newtonian Generation of Gravitational Radiation

In this section we adapt the method of Epstein and Wagoner [3] and show that
the post-Newtonian generation of gravitational radiation is the same as in
general relativity.

First we have to bring the field equations (3.3) and (3.6) into a convenient
form by separating explicitly the linear terms in the fields
$\phi_{\alpha\beta}$ and $a_{\alpha\beta}$.

We have already noted in Section 3 that the term proportional to $(\lambda-1)$
in (3.3) contains no linear terms and that the field $a_{\alpha\beta}$ does
not appear in the linearized part of $\epsilon^{\alpha}_{E}$. Hence we only
have to split off the linear part in the Einstein form

$\epsilon^{\alpha}_{E}=\epsilon^{\alpha}_{L}+\epsilon^{\alpha}_{Q}\,.$ (5.1)

The linear part, $\epsilon^{\alpha}_{L}$, is given by

$\epsilon^{\alpha}_{L}=-G^{\alpha\beta}_{L}\eta_{\beta}\,,$ (5.2)

where $G^{\alpha\beta}_{L}$ is the linearized Einstein tensor

$-G_{L\beta}^{\alpha}=\Box(\phi^{\alpha}_{\;\,\beta}-\delta^{\alpha}_{\;\,\beta}\phi^{\gamma}_{\;\,\gamma})+\delta^{\alpha}_{\;\,\beta}{\phi^{\lambda\sigma}}_{,\lambda\sigma}+{\phi^{\lambda}_{\;\,\lambda}}^{,\alpha}_{\;\;\,\beta}-({\phi^{\alpha\lambda}}_{,\beta}+{\phi^{\lambda}_{\;\,\beta}}^{,\alpha})_{,\lambda}\,.$
(5.3)

The quadratic and higher-order terms are easily obtained from (2.12):

$\epsilon^{\alpha}_{Q}=\Delta\epsilon^{\alpha}_{S}+\\{A^{\beta\alpha}+B^{\beta\alpha}+C^{\beta\alpha}\\}\eta_{\beta}\,,$
(5.4a) where $\displaystyle A^{\nu\mu}$
$\displaystyle=2{{}^{*}}\big{[}\theta^{\nu}\wedge\mathrm{d}\theta^{\alpha}\wedge{{}^{*}}(\mathrm{d}\theta_{\alpha}\wedge\theta^{\mu})\big{]}\,,$
(5.4b) $\displaystyle B^{\nu\mu}$
$\displaystyle={\textstyle\frac{1}{2}}{{}^{*}}\big{[}\theta^{\nu}\wedge\mathrm{d}\theta^{\alpha}\wedge{{}^{*}}\mathrm{d}(\theta^{\mu}\wedge\theta_{\alpha})+\theta^{\mu}\wedge\mathrm{d}\theta^{\alpha}\wedge{{}^{*}}\mathrm{d}(\theta^{\nu}\wedge\theta_{\alpha})\big{]}\,,$
(5.4c) $\displaystyle C^{\nu\mu}$
$\displaystyle=-{\textstyle\frac{1}{2}}{{}^{*}}\big{[}\theta^{\nu}\wedge\theta^{\alpha}\wedge\mathrm{d}{{}^{*}}(\mathrm{d}\theta_{\alpha}\wedge\Phi^{\mu}_{\;\,\lambda}\mathrm{d}x^{\lambda})+(\mathrm{d}x^{\nu}\wedge\Phi^{\alpha}_{\;\,\sigma}\mathrm{d}x^{\sigma}+\Phi^{\nu}_{\;\,\lambda}\mathrm{d}x^{\lambda}\wedge\mathrm{d}x^{\alpha}$
$\displaystyle\hskip
14.22636pt+\Phi^{\nu}_{\;\,\lambda}\Phi^{\alpha}_{\;\,\sigma}\mathrm{d}x^{\lambda}\wedge\mathrm{d}x^{\sigma})\wedge\mathrm{d}{{}^{*}}(\mathrm{d}\theta_{\alpha}\wedge\mathrm{d}x^{\mu})$
$\displaystyle\hskip
14.22636pt+\theta^{\mu}\wedge\theta^{\alpha}\wedge\mathrm{d}{{}^{*}}(\mathrm{d}\theta_{\alpha}\wedge\Phi^{\nu}_{\;\,\lambda}\mathrm{d}x^{\lambda})+(\mathrm{d}x^{\mu}\wedge\Phi^{\alpha}_{\;\,\lambda}\mathrm{d}x^{\lambda}+\Phi^{\mu}_{\;\,\sigma}\mathrm{d}x^{\sigma}\wedge\mathrm{d}x^{\alpha}$
$\displaystyle\hskip
14.22636pt+\Phi^{\mu}_{\;\,\sigma}\Phi^{\alpha}_{\;\,\lambda}\mathrm{d}x^{\sigma}\wedge\mathrm{d}x^{\lambda})\wedge\mathrm{d}{{}^{*}}(\mathrm{d}\theta_{\alpha}\wedge\mathrm{d}x^{\nu})\big{]}\,.$
(5.4d)

The decomposition (5.4) is equivalent to equation (31) of Reference [3] and is
quite useful for practical (post-Newtonian) calculations. Inserting (5.1) into
(3.3) gives

$G^{\alpha\beta}_{L}=T^{\alpha\beta}_{\mathrm{eff}}$ (5.5)

with

$T^{\alpha\beta}_{\mathrm{eff}}\eta_{\beta}=t^{\alpha}+\epsilon^{\alpha}_{Q}+(\lambda-1)\Delta\epsilon^{\alpha}_{S}\,,$
(5.6)

where $\Delta\epsilon^{\alpha}_{S}$ is given by (3.1) (and contains no linear
terms). The last term in (5.6) is absent in general relativity. We also
rewrite (3.4) (for $\lambda\neq 1$) in a similar way:

$\Box\,a^{\alpha\beta}+{a^{\lambda\alpha,\beta}}_{\lambda}-{a^{\lambda\beta,\alpha}}_{\lambda}=A^{\alpha\beta}_{\mathrm{ef}}\,,$
(5.7)

where $A^{\alpha\beta}_{\mathrm{ef}}$ collects the quadratic and higher-order
terms of

${\textstyle\frac{1}{2}}\mathrm{d}\big{[}\theta^{\alpha}\wedge\theta^{\beta}\wedge{{}^{*}}(\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma})\big{]}\,.$

For outgoing-wave boundary conditions, we obtain from (5.5) and (5.7) the
integral equations

$\displaystyle\phi_{\alpha\beta}(t,\boldmathe{x})$
$\displaystyle=-\frac{1}{4\pi}\int\frac{\hat{T}_{\alpha\beta}(t-|\boldmathe{x}-\boldmathe{x}^{\prime}|,\boldmathe{x}^{\prime})}{|\boldmathe{x}-\boldmathe{x}^{\prime}|}\mathrm{d}^{3}x^{\prime}+\xi_{\alpha,\beta}+\xi_{\beta,\alpha}\,,$
(5.8) $\displaystyle(\hat{T}^{\alpha}_{\;\,\beta}$
$\displaystyle={T_{\mathrm{eff}}}^{\alpha}_{\;\,\beta}-{\textstyle\frac{1}{2}}\delta^{\alpha}_{\;\,\beta}{T_{\mathrm{eff}}}^{\lambda}_{\;\,\lambda})$
$\displaystyle a_{\alpha\beta}(t,\boldmathe{x})$
$\displaystyle=\frac{1}{4\pi}\int\frac{\hat{A}_{\alpha\beta}^{\mathrm{eff}}(t-|\boldmathe{x}-\boldmathe{x}^{\prime}|,\boldmathe{x}^{\prime})}{|\boldmathe{x}-\boldmathe{x}^{\prime}|}\mathrm{d}^{3}x^{\prime}+\eta_{\alpha,\beta}-\eta_{\beta,\alpha}\,.$
(5.9)

The gauge terms added on the right-hand sides of (5.8) and (5.9) are, of
course, not determined.

Let us now consider the fields $\phi_{\alpha\beta}$ and $a_{\alpha\beta}$ far
from the source ($r=|\boldmathe{x}|\gg R=$"size" of source,
$|\phi_{\alpha\beta}|\ll 1,|a_{\alpha\beta}|\ll 1$) where the radiation is
detected. In the approximation scheme of Reference [3], one first expands the
$1/r$ terms with respect to the retardation within the source (slow motion).
In the next step, one constructs a post-Newtonian expansion of the sources
$T^{\alpha\beta}_{\mathrm{eff}}$ and $A^{\alpha\beta}_{\mathrm{eff}}$. The
necessary orders in the post-Newtonian expansion parameter which are needed
for the various pieces are described in [3] and will not be repeated. In this
approximation, the effective sources are contained within the near zone. We
may then employ (5.8) and (5.9) for field points within this region as well,
again in terms of a post-Newtonian expansion. But the near-zone post-Newtonian
expansion was already studied in Section 3. To the necessary orders
$a_{\alpha\beta}$ vanishes and $\phi_{\alpha\beta}$ agrees with general
relativity if we impose suitable gauge conditions.
[$A^{\alpha\beta}_{\mathrm{eff}}$ and the last term in (5.6) for
$T^{\alpha\beta}_{\mathrm{eff}}$ both vanish in the near-zone post-Newtonian
approximation.] Hence, we conclude from (5.8) and (5.9) that far away from the
source $a_{\alpha\beta}$ is just equal to a gauge term and that
$\phi_{\alpha\beta}$ agrees also there with general relativity. But in this
weak field region, the equation for the a field decouples and has a separate
gauge invariance. Thus the a field can be gauged away.

Finally we note that the modified energy-momentum
$t^{\alpha}_{\mathrm{LL}}-(\lambda-1)\Delta t^{\alpha}$ [see equation (4.5)]
does not change the gravitational energy loss because $\Delta t^{\alpha}$ is
cubic in $\phi_{\alpha\beta}$. This follows from (4.6) and from the fact that
the factor $\mathrm{d}\theta^{\gamma}\wedge\theta_{\gamma}$ in (4.6) is
already quadratic in $\phi_{\alpha\beta}$. (The linear term vanishes
identically.) Taken all together, this proves our claim at the beginning of
this section.

##  6 Summary

The gravitational theory with torsion corresponding to the one-parameter
family of Lagrangians (2.6) has many exact solutions in common with general
relativity. We have shown that they agree also with general relativity on the
first post-Newtonian level (but not in higher orders.). The main new result of
this paper is contained in Section 5, where we demonstrate that even the post-
Newtonian generation of gravitational waves (developed within general
relativity by Epstein and Wagoner [3] ) is the same as in general relativity.
In particular, the "dipole catastrophe," described in the Introduction, which
occurs in many alternative metric theories of gravitation, is absent.
Therefore, the results obtained in [3] and [4] also hold in the theory by Hehl
et al. [6]. At the present time, this theory can thus not be distinguished
observationally from general relativity.

This is, of course, only true if the approximation scheme of Reference [3] is
numerically reliable. Various authors (see, e.g., [12] and references therein)
have criticized the presently existing approximation methods for treating the
radiation problem. We are aware of the critical questions that have been
raised, but we think that the method used here is physically plausible.

Apart from aesthetic arguments, we see no way to favor the Lagrangian with
$\lambda=1$ in (2.6), i.e., general relativity.

## Appendix

In this Appendix, we derive the expression (4.14) for the angular momentum.
From the definition (4.10) and the field equations in the form (4.4) we
conclude that

$\displaystyle(-g)^{1/2}\mathrm{M}^{\sigma\alpha}$
$\displaystyle={\textstyle\frac{1}{2}}(x^{\sigma}\mathrm{d}h^{\alpha}-x^{\alpha}\mathrm{d}h^{\sigma})$
$\displaystyle={\textstyle\frac{1}{2}}\mathrm{d}(x^{\sigma}h^{\alpha}-x^{\alpha}h^{\sigma})-{\textstyle\frac{1}{2}}(\mathrm{d}x^{\sigma}\wedge
h^{\alpha}-\mathrm{d}x^{\alpha}\wedge h^{\sigma})\,,$ (A.1)

where

$h^{\alpha}=-(-g)^{1/2}(\omega^{\beta\gamma}\wedge{\eta^{\alpha}}_{\beta\gamma})\,.$
(A.2)

In this Appendix ${\omega^{\alpha}}_{\beta}$ denote always the Levi-Cività
connection forms. Now we write also the last term in (A.1) as an exact
differential. We have

$\displaystyle\mathrm{d}x^{\sigma}\wedge h^{\alpha}-\mathrm{d}x^{\alpha}\wedge
h^{\sigma}$
$\displaystyle=(-g)^{1/2}\omega^{\beta\gamma}\wedge\mathrm{d}x^{\sigma}\wedge{\eta^{\alpha}}_{\beta\gamma}-(\alpha\longleftrightarrow\sigma)$
$\displaystyle=(-g)^{1/2}\big{(}{\omega_{\beta}}^{\sigma}\wedge\eta^{\alpha\beta}+{\omega^{\sigma}}_{\beta}\wedge\eta^{\beta\alpha}$
$\displaystyle\hskip
51.21495pt-{\omega_{\beta}}^{\alpha}\wedge\eta^{\sigma\beta}-{\omega^{\alpha}}_{\beta}\wedge\eta^{\beta\sigma}\big{)}\,.$

Here we use

$\mathrm{d}\eta^{\sigma\alpha}+{\omega^{\sigma}}_{\beta}\wedge\eta^{\beta\alpha}+{\omega^{\alpha}}_{\beta}\wedge\eta^{\sigma\beta}=0$

and obtain

$\mathrm{d}x^{\sigma}\wedge h^{\alpha}-\mathrm{d}x^{\alpha}\wedge
h^{\sigma}=(-g)^{1/2}\big{[}{\omega_{\beta}}^{\sigma}\wedge\eta^{\alpha\beta}-(\sigma\longleftrightarrow\alpha)-\mathrm{d}\eta^{\sigma\alpha}\big{]}\,.$

But

${\omega_{\beta}}^{\sigma}\wedge\eta^{\alpha\beta}=\Gamma_{\beta\mu}^{\;\,\sigma}\,\mathrm{d}x^{\mu}\wedge\eta^{\alpha\beta}={\Gamma_{\beta}}^{\beta\sigma}\eta^{\alpha}-{\Gamma_{\beta}}^{\alpha\sigma}\eta^{\beta}$

and hence

$\mathrm{d}x^{\sigma}\wedge h^{\alpha}-\mathrm{d}x^{\alpha}\wedge
h^{\sigma}=(-g)^{1/2}\big{(}\Gamma^{\beta\sigma}_{\beta}\eta^{\alpha}-\Gamma^{\beta\alpha}_{\beta}\eta^{\sigma}-\mathrm{d}\eta^{\sigma\alpha}\big{)}\,.$

If we use the expression

$\Gamma^{\beta\sigma}_{\beta}=(-g)^{1/2}\cdot
g^{\mu\sigma}\partial_{\mu}(-g)^{1/2}$

then we find easily

$\mathrm{d}x^{\sigma}\wedge h^{\alpha}-\mathrm{d}x^{\alpha}\wedge
h^{\sigma}=-\mathrm{d}\big{[}(-g)^{1/2}\cdot\eta^{\sigma\alpha}\big{]}\,.$
(A.3)

With this result and (A.2) equation (A.1) becomes

$(-g)^{1/2}\mathrm{M}^{\sigma\alpha}={\textstyle\frac{1}{2}}\mathrm{d}\big{\\{}(-g)^{1/2}[\eta^{\sigma\alpha}+(x^{\sigma}{\eta^{\alpha}}_{\beta\gamma}-x^{\alpha}{\eta^{\sigma}}_{\beta\gamma})\wedge\omega^{\beta\gamma}]\big{\\}}\,.$
(A.4)

Inserting this into (4.14) and using Stokes’ theorem finally gives equation
(4.14)

## Acknowledgement

We thank Dr. M. Camenzind for many useful discussions

## References

  * [1] Taylor, J. H., Hulse, R. A., Fowler, L. A., Gullahorn, G. E., and Rankin, J. M. (1976). Astrophys. J., 206, L53; Taylor, J. H. (1978). Talk given at the Ninth Texas Symposium on Relativistic Astrophysics (München, 1978); Taylor, J. H., Fowler, L. A., and McCullough, P. M. (1979). Nature, 277,437.
  * [2] Will, C. M. (1977). Astrophys. J., 214, 826-839; (1978). Talk given at the Ninth Texas Symposium on Relativistic Astrophysics (München, 1978).
  * [3] Epstein, R., and Wagoner, R. V. (1975). Astrophys. J., 197,717.
  * [4] Wagoner, R. V., and Will, C. M. (1976). Astrophys. J., 210, 764.
  * [5] Schweizer, M., and Straumann, N. (1979). Phys. Lett., 71A, 493.
  * [6] Hehl, F. W., Ne’eman, Y., Nitsch, J., and Von der Heyde, P. (1978). Phys. Lett., 788, 102; Hehl, F. W., Nitsch, J., and Von der Heyde, P. (1980). Einstein Commemorative Volume, ed. Held, A. Plenum, New York (to appear); Hehl, F. W. (1980). Four Lectures on Poincaré Gauge Field Theory, Proceedings of the International School of Cosmology and Gravitation, Erice, May 1979, eds. Bergmann, P. G. and de Sabbata, V. Plenum, New York (to appear).
  * [7] Trautman, A. (1973). Symposia Mathematica, Vol. 12, Academic, New York, p. 139.
  * [8] Rumpf, H. (1978). Z. Naturforsch., 33a, 1224.
  * [9] Wipf, A. (1979). Diplomarbeit, "Analyse einer Poincaré Eichfeldtheorie der Gravitation und Emission yon Gravitationsstrahlung in derselben," Institute for Theoretical Physics, Schönberggasse 9, University of Zürich, CH-8001 Zürich.
  * [10] Hehl, F. W., and Nitsch, J. (1979). University of Cologne, preprint; Nitsch, J. (1980). Seminar Talk, Proceedings of the International School of Cosmology and Gravitation, Erice, May 1979, eds. Bergmann, P. G., and de Sabbata, V. Plenum, New York (to appear).
  * [11] Thirring, W. (1978). Lehrbuch der Mathematischen Physik, Vol. 2, Springer-Verlag, Wien.
  * [12] Ehlers, J. (1978). "Isolated Systems in General Relativity," Talk given at the Ninth Texas Symposium on Relativistic Astrophysics, (München, 1978).


# Accurate and Energy-Efficient Classification with Spiking Random Neural
Network:
Corrected and Expanded Version

Khaled F. Hussain 1, Mohamed Yousef Bassyouni 1 and Erol Gelenbe 2
1 Computer Science Department, Faculty of Computers and Information, Assiut
University, Assiut 71515, Egypt<EMAIL_ADDRESS><EMAIL_ADDRESS>
2 Institute of Theoretical & Applied Informatics, Polish Academy of Sciences,
ul. Baltycka 5, 44100 Gliwice, Poland, & Imperial College London, SW7 2BT
United Kingdom<EMAIL_ADDRESS>
###### Abstract

Artificial Neural Network (ANN) based techniques have dominated state-of-the-
art results in most problems related to computer vision, audio recognition,
and natural language processing in the past few years, resulting in strong
industrial adoption from all leading technology companies worldwide. One of
the major obstacles that have historically delayed large scale adoption of
ANNs is the huge computational and power costs associated with training and
testing (deploying) them. In the mean-time, Neuromorphic Computing platforms
have recently achieved remarkable performance running more bio-realistic
Spiking Neural Networks at high throughput and very low power consumption
making them a natural alternative to ANNs. Here, we propose using the Random
Neural Network (RNN), a spiking neural network with both theoretical and
practical appealing properties, as a general purpose classifier that can match
the classification power of ANNs on a number of tasks while enjoying all the
features of a spiking neural network. This is demonstrated on a number of
real-world classification datasets.

###### Index Terms:

Random Neural Network; Spiking Neural Networks; Artificial Neural Network;
Neuromorphic Computing

##  I Introduction

Despite being first proposed about 60 years ago [1], only in the past few
years have artificial neural networks (ANNs) become the de facto standard
machine learning model [2] achieving accurate state-of-the-art results for a
wide range of problems ranging from image classification [3, 4, 5], object
detection [6, 7], semantic segmentation [8, 9], face recognition [10, 11], and
text recognition [12, 13], to speech recognition [14, 15, 16], natural
language processing problems such as machine translation [17, 18], language
modeling [19], and question answering [20]. This has resulted in a huge
industry-wide adoption from leading technology companies such as Google,
Facebook, Microsoft, IBM, Yahoo!, Twitter, Adobe, and a quickly growing number
of start-ups.

One of the prominent reasons for this recent revival is that in order for ANNs
to achieve such performance they need very large labeled datasets and huge
computational power at a scale that only recently came into the hands of
individual researchers in the form of GPUs [21], which kick-started the deep
learning revolution in 2012 [3]. Since then, the trend for demanding more
computation and more power consumption for such applications has largely
increased.

Despite being initially bio-inspired architectures, ANNs have significant
differences from actual biological neurons in how computations are performed
by neurons, their structure (connection patterns and topologies of neurons),
learning (how neurons adapt themselves to new observations), and communication
(how inter-neuron data is encoded and passed).

One of the main differences of ANNs compared to biological neurons, is how
communication is done. While biological neurons use asynchronous trains of
spikes in an event-based, data-driven manner that adapts locally to its
external stimulation pattern to communicate and encode data (though the
specific encoding mechanism used by neurons is not totally understood), ANNs
communicate in dense, continuous valued activations, which means that all ANN
neurons are working at the same time, thus using lots of computation and
energy to operate.

Spiking neural networks leverage the benefit from biological neurons to
communicate asynchronously in trains of spikes. Thus, spiking neural networks
incorporate the concept of time, and instead of all neurons firing at the same
time as the case with ANNs, in spiking neural networks neurons fire only when
thier intrinsic potential (i.e. membrane voltage) reaches a specific threshold
[22, 23].

Neuroscientists have historically suggested several models for simulating how
biological neurons communicate, and one of the simplest that is widely used is
the integrate-and-fire (IF) model [24], in which the change in the membrane
voltage $v_{mem}$ is given by the equation:

$\frac{dv_{mem}(t)}{dt}=\sum_{i}\sum_{s\in S_{i}}w_{i}\delta(t-s)$ (1)

where $w_{i}$ is the weight of the $i$th incoming synapse, $\delta(.)$ is the
Dirac delta function, and $S_{i}={t^{0}_{i},t^{1}_{i},...}$ contains the spike
times of the $i$th presynaptic neuron. If the membrane voltage crosses the
spiking threshold $v_{thr}$, a spike is generated and the membrane voltage is
reset to a reset potential $v_{res}$ [25]. Several other models have also been
proposed, such as te spike response model (SRM) [26], and the Izhikevich model
[27].

One of the prominent differences between spiking neural networks and ANNs is
how they learn and adapt to new signals. ANNs have been predominantly trained
in the literature using Backpropagation [28], and with variants of stochastic
gradient descent (SGD), which can summarised as moving the vector of network
parameters or weights $\theta$ in the direction of the negative gradient of
some loss function that characterizes the deviation network’s current output
from the ground truth labels of input data.

Training spiking neural networks on the other hand is still an open research
issue with many proposed solutions and no consensus [29]. One of the most
popular and biologically plausible learning methods in spiking neural networks
is unsupervised learning using Spike Timing Dependent Plasticity (STDP) [30,
31], in which the synaptic weight is adjusted in accordance with the relative
spike times of the presynaptic and postsynaptic neurons.

An important problem that has always faced using the popular gradient-based
optimization algorithms in spiking neural networks is that both spike trains
and the underlying membrane voltage are not differentiable at the time of
spikes,. Thus researchers tried different approaches to alleviate this
problem, and one of the most successful has been the workaround of first
training an ANN and then converting it to a corresponding spiking neural
network [32, 25, 33].

Though von Neumann architectures [34] work very well to run ANNs, it was
suggested as early as the 1980s that they were not adequate for running the
more realistic spiking neural networks models efficiently, and a new
architecture was needed to realize thier power and computational efficiency
[35]. The recent success of ANNs has pushed this trend much faster.

The main idea behind Neuromorphic Computing was to design Integrated Circuits
(ICs) that are arranged and behave like living neurons (i.e. to mimic how the
brain performs computation) [36], and spiking neural network models of
bioglogical neurons have historically been used as a guiding design in this
process. After years of attempts, the past few years saw the demonstration of
Neuromorphic Computing platforms with millions of neurons while requiring only
milliWatts of power for thier operation such as TrueNorth [37], SpiNNaker
[38], and Loihi [39]. A number of pattern classification applications were
demonstrated to run efficiently and accurately on such chips while being
orders of magnitude more efficient in terms of power consumption than an ANN
on a von Neumann CPU or GPU running a similar task.

The main source of this power saving is the asynchronous working and firing of
spiking neural networks described earlier, so neurons fire and the chip
consumes power only when needed, which is completely different than what
happens in an ANN when all neurons are obliged to fire synchronously leading
to unnecessary computation and energy consumption. This efficiency can be even
increased when using input from neuromorphic sensors such as silicon retinas
[40] or cochleas [41], which create sparse, frame-free, and precisely timed
trains of signals, with substantially reduced latencies compared to
traditional frame-based approaches which produce large volumes of redundant
data and therefore consume much power. This line of neuromorphic sensor design
has been applied to vision sensors, auditory sensors, and olfactory sensors
[42].

This paper presents results that show how a spiking model known as the Random
Neural Network (RNN) [43] can be used effectively and efficiently for Machine
Learning applications, resulting in a level of performance comparable to the
best ANN results.

The rest of the paper is structured as follows: The Random Neural Network is
described and reviewed in Section II; our experimental setup is described and
results presented in Section III; the conclusions and future work are drawn in
Section IV

##  II The Random Neural Network (RNN)

Figure 1: Schematic representation of a Random Neural Network (RNN).

The Random Neural Network (RNN) [43, 44, 45] has a powerful property of
approximating continuous and bounded real-valued functions [46]. This property
serves as the foundation for RNN based learning algorithms, both for recurrent
(containing feedback) and feedforward networks [47], and for Deep Learning
[48, 49].

The RNN has been used for modelling natural neuronal networks [50], and for
protein alignment [51]. It has been used in several image processing
applications including the accurate evaluation of tumors from brain MRI scans
[52] and the compression of still and moving images [53, 54, 55]. It was
recently introduced as a tool for predicting the toxicity of chemical
compounds [56].

In the field of computer networks, the RNN has been used to build distributed
controllers for quality of service routing in packet networks [57, 58, 59] and
in the design of Software Defined Network controllers for the Internet [60,
61]. Real-time optimised task allocation algorithms in Cloud systems [62] have
also been built and tested. Recent applications has addressed the use of the
RNN to detect network attacks [63] and attacks on Internet of Things (IoT)
gateways [64].

On the other hand, G-Networks [65] are a family of queueing networks that
generalize the RNN model, and like the RNN they have a convenient and
computationally efficient “product form ”mathematical solution. The
computation of the state of a G-Network is obtained via a simple fixed-point
iteration, and the existence and uniqueness of the solution to the key
G-Network state equation is shown in [66]. G-Networks incorporate useful
primitives, such as the transfer of jobs between servers or the removal of
batches of jobs from excessively busy servers, that were developed in a series
of successive papers [67, 68, 69, 70]. They have a wealth of diverse
applications as a tool to analyse and optimise the effects of dynamic load
balancing in large scale networks and distributed computer systems [71]. They
are also used to model Gene Regulatory Networks [72, 73]. A recent application
of G-Networks is to the modelling of systems which operate with intermittent
sources of energy, known as Energy Packet Networks [74, 75, 76, 77, 78].

###  II-A The RNN as a Mathematical Model

Figure 1 gives a schematic diagram of a RNN for a system with $L$ neurons. The
state of neuron $i$ at time $t$ is represented by a non-negative integer
called its potential, denoted by $k_{i}(t)\geq 0$. Network state at time $t$
is a vector $k(t)=(k_{1}(t),...,k_{i}(t),\dotsc,k_{L}(t))$. When an excitation
spike arrives to neuron $l$, the state of neuron $i$ is changed from
$k_{i}(t)$ to $k_{i}(t)+1$ . When an inhibition spike arrives to neuron $i$,
its state from $k_{i}(t)$ to $k_{i}(t)-1$ if $k_{i}(t)>0$, and does not change
if $k_{i}(t)=0$.

Each neuron $i$ receives exogenous or external excitation spikes in the form
of two independent Poisson processes of rates $\Lambda_{i}\geq 0$ and
$\lambda_{i}\geq 0$, respectively. These represent the external inputs to the
network.

Neuron $i$ can emit a spike if $k_{i}(t)$ is positive (i.e. it is excited);
this occurs with probability $r_{i}\Delta(t)+o(\Delta(t)$, where $r_{i}\geq 0$
is the “firing rate” of neuron $i$. Ehen this happens, neuron $i$’s state
changes from $k_{i}(t)$ to $k_{i}(t)-1$. The spikes are then sent .

Spikes are sent from the outside world to neuron $i$ as a positive signal
according to Poisson processes of rate $\Lambda_{l}$ or as a negative signal
according also to Poisson processes of rate $\lambda_{l}$

Each spike is sent out from neuron $i$ to neuron $j$ as a positive or
excitatory spike with probability $p^{+}_{ij}$, or as a negative or inhibitory
spike spike with probability $p^{-}_{ij}$ , or it departs from the network
with probability $d_{i}$ which represents the probability that spikes
emanating from a neuron may be “lost” for a variety of reasons, and does not
reach any target neuron $j$. The sum of these probabilities must be one:

$d_{i}+\sum_{j=1}^{L}[p^{+}_{ij}+p^{-}_{ij}]=1,\forall i~{}.$ (2)

The spikes are sent out from neuron $i$ to neuron $j$ at rates:

$w^{+}_{ij}=r_{i}p^{+}_{ij}\geq 0,~{}w^{-}_{ij}=r_{l}i\>p^{-}_{ij}\geq 0,$ (3)

and $w^{+}_{ij}$ and $w^{-}_{ij}$ are called the excitatory and inhibitory
weights, respectively.

Combining equations (2) and (3) we get:

$r_{i}=\frac{\sum_{j=1}^{L}[w^{+}_{ij}+w^{-}_{ij}]}{1-d_{i}}~{}.$ (4)

Let $q_{i}=\lim_{t\to\infty}Prob(k_{l}it)>0)$ denote the stationary excitation
probability of neuron $i$. The total arrival rates of positive signals
$\Omega_{i}$ and negative signals $\Omega^{-}_{i}$ , for $i=1,\dotsc,L$, can
be calculated from the following nonlinear system of equations:

$\Omega^{+}_{i}=\Lambda_{i}+\sum_{j=1}^{L}q_{i}w^{+}_{ij},~{}\Omega^{-}_{i}=\lambda_{i}+\sum_{j=1}^{L}q_{i}w^{-}_{ij}$
(5)

It has been proven that the $q_{i}$ can be directly calculated by the
following system of equations:

$q_{i}=\min\left\\{1,\frac{\Omega^{+}_{i}}{r_{i}+\Omega^{-}_{i}}\right\\},~{}1\leq
i\leq L,$ (6)

and furthermore that if all the resulting $q_{i}$ are strictly less than one,
then:

$Prob[k_{1}(t)=k_{1},~{}...~{},k_{L}(t)=k_{L}]=\Pi_{i=1}^{L}q_{i}^{k_{i}}(1-q_{i}),$
(7)

which is known as a “Product Form Solution”.

The existence of a solution to the system of $L$ non-linear equations (6) and
the uniqueness of the solution has been proven [47]. Therefore, the states of
the RNN can be obtained by solving (6), for instance using a fixed-point
iteration.

##  III Experimental Results using the RNN

We know that spiking neural network-based RNNs are empirically at least as
powerful as ANNs in this category of classification problems. It was already
shown that RNNs are universal function approximators for continuous and
bounded functions [79, 80], and hence as computationally capable as multi-
layer perceptrons in this respect [81]. However, RNNs have been shown to have
an unique solution even in the recurrent case [47, 82, 55], and furthermore in
the recurrent case the RNN’s learning algorithm is still of low polynomial
time and space complexity [83, 84, 85].

Here we will provide experimental results on a number of benchmark real world
classification datasets that are widely used in the literature. These
experimental results can be reproduced using the software and and data stored
at the web site www.github.com/ASDen/Random_Neural_Network.

Another web site that specializes in Deep Learning with the Random Neural
Network can be found at https://github.com/yinyongh/DenseRandomNet.

###  III-A Evaluation Setup

_Dataset_ | _# Attributes_ | _# Features_ | _# Output Classes_
---|---|---|---
Iris | 4 | 150 | 3
Breast Cancer Wisconsin | 9 | 699 | 2
Glass | 9 | 214 | 7
Ovarian cancer | 100 | 216 | 2
TABLE I: Names and statistics of the Datasets that are used.

Table VI shows statistics about the used datasets. We use the Iris, Breast
Cancer, and Glass datasets from the UCI machine learning repository [86] and
the Ovarian cancer dataset [87]. We train the RNN using the procedure
described in [88] based on [89]; the related software and data sets are at
www.github.com/ASDen/Random_Neural_Network. The algorithm can be summarized as
follows:

  1. 1.

Assume the given dataset has $K$ pairs of input training patterns for an
$L-vector$ $x_{k}=(x_{1l},~{}...~{},x_{Lk})$ associated with the output value
$L-vector$ $y_{k}=(y_{1l},~{}...~{},y_{Lk})$.

  2. 2.

Initialize the weights $w^{+}_{ij}$ and $w^{-}_{ij}$ , $\forall(i,~{}j)$, to
random values between zero and one.

  3. 3.

Set the external inhibitory arrival rates to zero.

  4. 4.

Set the external excitatory input rates for the input neurons,
$\Lambda_{k}=x_{k}$, where $x_{k}$ is the $k$th input training pattern.

  5. 5.

Solve the nonlinear system of equations (6) to obtain each neuron’s stationary
excitation probability $q_{i},\forall i$ .

  6. 6.

For all the given $(input,~{}output)$ values $(x_{k},~{}y_{k})$, iterate
through the RNN recurrent network learning algorithm [89, 90] till
convergence, updating at each step the weights $w^{+}_{ij}$ and $w^{-}_{ij}$ ,
$\forall(i,~{}j)$, so as to minimize the following error function:

$E=\sum_{k=1}^{K}\sum_{i=1}^{L}[q_{ik}-y_{ik}]^{2}$ (8)

###  III-B Dataset Description

Here we give a brief description of the four datasets used for evaluation:

  1. 1.

Iris dataset [86]: Each instance is described by four plants attributes (sepal
length and width, and petal length and width) all are real numbers and the
task is to recognize which class of Iris plants (Iris Setosa, Iris
Versicolour, or Iris Virginica) a given test instance belongs to.

  2. 2.

Breast Cancer Wisconsin dataset [86]: Each instance is described by 9
numerical attributes, that range from 1 to 10. The attributes include the
clump thickness, uniformity of cell size, uniformity of cell shape, marginal
adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal
nucleoli and mitoses. This breast cancer databases was obtained from the
University of Wisconsin Hospitals, Madison from Dr. William H. Wolberg. The
task is to recognize the class of the beast cancer (benign or malignant).

  3. 3.

Glass dataset [86]: Each instance has 9 continuous attributes, including the
refractive index and the unit measurements of sodium, magnesium, aluminum,
silicon, potassium, calcium, barium and iron. There are in total 7 types of
glass, while there are instances of only 6 types of glass in the dataset. The
task is to using the 9 attributes to recognize which type of glass this
instance belongs to (whether it is windows glass or non-window glass).

  4. 4.

Ovarian Cancer dataset [87]: From the FDA-NCI Clinical Proteomics Program
Databank, the dataset comprises 216 patients, out of which 121 are ovarian
cancer patients and 95 are normal patients. Each instance has 100 attributes,
each of which represents the ion intensity level at a specific mass-charge
value . The task is to recognize the class of the ovarian cancer (benign or
malignant).

###  III-C Results

Tables II, III, IV, and V display confusion matrices of RNN on the four
datasets. Table VI compares the accuracy of RNNs against ANNs some UCI
datasets. For the ANN results, we use results obtained on UCI datasets from
the comprehensive study in [91]. We can clearly see in VI that RNNs are at
least as powerful as ANNs in these datasets and can deliver excellent
classification accuracy.

_Class_ | _Setosa_ | _Versicolour_ | _Virginica_
---|---|---|---
_Setosa_ | 1.0 | 0.0 | 0.0
_Versicolour_ | 0.0 | 1.0 | 0.0
_Virginica_ | 0.0 | 0.0 | 1.0
TABLE II: Confusion matrix for Iris dataset. _Class_ | _Positive_ | _Negative_
---|---|---
_Postivie_ | 0.984 | 0.016
_Negative_ | 0.067 | 0.933
TABLE III: Confusion matrix for Breast Cancer dataset. _Class_ | _Positive_ | _Negative_
---|---|---
_Postivie_ | 1.0 | 0.0
_Negative_ | 0.127 | 0.8139
TABLE IV: Confusion matrix for Glass dataset. _Class_ | _Postive_ | _Negative_
---|---|---
_Postive_ | 1.0 | 0.0
_Negative_ | 0.1064 | 0.8936
TABLE V: Confusion matrix for Ovarian Cancer dataset. _Dataset_ | _RNN_ | _ANN_
---|---|---
Iris | 1.0 | 0.959
Breast Cancer Wisconsin | 0.964 | 0.963
TABLE VI: Accuracy comparison between RNN, and ANN. Best result in each
dataset is in bold.

##  IV Conclusions

In this paper, we have motivated the need for power and computation efficiency
in neuromorphic models that are used in Machine Learning, and discussed some
of the shortcomings of current ANN models in this respect. We have briefly
presented and reviewed some spiking neural networkmodels used for neuromorphic
computing as a possible alternative due to their low power usage and
computational efficiency, but we have also indicated their limitations with
regard to being harder to train and their generalization performance.

We have then presented a specific spiking neural network model, the Random
Neural Network (RNN) that was first introduced in [92]. We have indicated that
the RNN’s special analytical properties makes it much easier to train, and we
have shown empirically that it provides a generalization performance that is
at least as powerful as conventional ANNs for a number of real world
classification datasets, while achieving the efficiency associated with
spiking neural networks.

## References

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