459SolvDynSystPlaneWithPolynInteract190122

(Submitted to: J. Phys. Conf. Ser., 22.01.2019)

Some Algebraically Solvable Two-Dimensional Dynamical Systems with Polynomial
Interactions

Francesco Calogeroa,b,1 and Farrin Payandeha,c,2

a Physics Department, University of Rome ”La Sapienza”, Rome, Italy

b INFN, Sezione di Roma 1

c Department of Physics, Payame Noor University (PNU), PO BOX 19395-3697
Tehran, Iran

1<EMAIL_ADDRESS><EMAIL_ADDRESS>
2<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract

We tersely review a recently introduced technique to identify systems of two
nonlinearly-coupled Ordinary Differential Equations (ODEs) solvable by
algebraic operations; and we report some specific examples of this kind,
namely systems of $2$ first-order ODEs with polynomial right-hand sides,

$\dot{x}_{n}=P^{\left(n\right)}\left(x_{1},x_{2}\right)~{},~{}~{}~{}n=1,2~{},$

satisfied by the $2$ (possibly complex) dependent variables $x_{n}\equiv
x_{n}\left(t\right)$. Here $P^{\left(n\right)}\left(x_{1},x_{2}\right)$
indicates some specific polynomial. These examples are analogous, but
different, from those previously reported.

##  1 Introduction

A technique to identify dynamical systems characterized by systems of
nonlinearly-coupled Ordinary Differential Equations (ODEs) solvable by
algebraic operations is based on the relations among the $N$ coefficients
$y_{m}\left(t\right)$ and the $N$ zeros $x_{n}\left(t\right)$ of a time-
dependent (monic) polynomial of degree $N$ in the (complex) variable $z$:

$p_{N}\left(z;t\right)=z^{N}+\sum_{m=1}^{N}\left[y_{m}\left(t\right)~{}z^{N-m}\right]=\mathop{\displaystyle\prod}\limits_{n=1}^{N}\left[z-x_{n}\left(t\right)\right]~{}.$
(1)

The basic idea is to consider ”simple” time evolutions of the $N$ coefficients
$y_{m}\left(t\right)$—possibly explicitly solvable evolutions, maybe featuring
remarkable properties such as isochrony (implying that all the $N$
coefficients $y_{m}\left(t\right)$ are periodic in $t$ with the same fixed
period independent of the initial data); and to then consider the
corresponding time evolutions of the $N$ zeros $x_{n}\left(t\right)$, which
are generally characterized by ”more nonlinear” equations of motions, the
solutions of which are then obtainable via the algebraic operation of
computing the $N$ zeros $x_{n}\left(t\right)$ of the polynomial
$p_{N}\left(z;t\right)$ defined in terms of its $N$ coefficients
$y_{m}\left(t\right).$ This generally entails that the time evolution of the
zeros $x_{n}\left(t\right)$ inherits properties (such as isochrony) of the
evolution of the coefficients $y_{m}\left(t\right)$. This idea has a long
history [1], and it has produced many developments, see for instance [2] [3]
[4] [5] [6] and references therein.

An additional recent development has explored the modifications of the
approach outlined above which emerge if, rather than considering generic time-
dependent polynomials, one focusses on special time-dependent polynomials
featuring (for all time) multiple zeros: in particular, as a first step in
this direction, results have been reported for the case of a polynomial
featuring one double zero [7], then some significant progress has been made on
the case of a polynomial featuring one zero of arbitrary multiplicity [8], and
finally the most general case has been treated of a polynomial
$p_{M}\left(z;t\right)$ of degree $M$ featuring an arbitrary number $N$ of
zeros each of them of arbitrary multiplicity $\mu_{n}$ [9]:

$p_{M}\left(z;t\right)=z^{N}+\sum_{m=1}^{M}\left[y_{m}\left(t\right)~{}z^{M-m}\right]=\mathop{\displaystyle\prod}\limits_{n=1}^{N}\left\\{\left[z-x_{n}\left(t\right)\right]^{\mu_{n}}\right\\}~{},$
(2a) of course with the $N$ positive integers $\mu_{n}$ related to the
positive integer parameters $M$ and $N$ by the relation
$M=\sum_{n=1}^{N}\left(\mu_{n}\right)~{}.$ (2b)

A second twist of this development restricted attention to the $N=2$ case—i.
e., polynomials (see (2)) featuring only $2$ zeros [10] [9]. This stringent
limitation has opened the way to the identification of algebraically solvable
two-dimensional dynamical systems with polynomial interactions, namely systems
characterized by the following system of two nonlinearly coupled ODEs:

$\dot{x}_{n}=P^{\left(n\right)}\left(x_{1},x_{2}\right)~{},~{}~{}~{}n=1,2~{}.$
(3)

Notation 1.1. Here and hereafter superimposed dots denote differentiations
with respect to the dependent variable $t$ (”time”), $x_{n}\equiv
x_{n}\left(t\right)$ are the $2$ (generally complex) dependent variables, and
$P^{\left(n\right)}\left(x_{1},x_{2}\right)$ is generally a polynomial (or
”almost a polynomial”: see below) in these $2$ dependent variables
$x_{1},~{}x_{2}$. Parameters such $a,$ $b,$ $c,$ $\alpha,$ $\beta,$ $\gamma$
(possibly equipped with subscripts) are time-independent; they can be
arbitrarily assigned (up to explicitly indicated relations among them). Note
that often, above and hereafter, the $t$-dependence of variables is not
explicitly indicated (when this is unlikely to generate any misunderstanding).
$\square$

A few such systems have been identified and tersely discussed in [10] by
taking as point of departure the case of a polynomial such as (2a) with $M=3$,
and several such cases have been treated in [9] by focussing on the case with
$M=4$. In the present paper—after a terse reminder of the methodology to
obtain these results—we focus again on the $M=3$ case, reporting several
solvable models of type (3) not previously identified. Analogous treatments of
cases with $M>4$ shall be performed by ourselves or by others in the future.

In the following Section 2 we tersely review our notation and some results
which are basic for the following treatment. In Section 3 and its subsections
several specific examples are reported (with some related results confined to
Appendix A). The last Section 4 outlines possible future developments.

##  2 Preliminaries

A basic tool of our treatment are the following definitions of the $3$
variables $y_{m}\equiv y_{m}\left(t\right),$ $m=1,2,3$ in terms of the $2$
variables $x_{n}\equiv x_{n}\left(t\right),$ $n=1,2$:

$y_{1}=-\left(2x_{1}+x_{2}\right)~{},~{}~{}~{}y_{2}=x_{1}\left(x_{1}+2x_{2}\right)~{},~{}~{}\
y_{3}=-\left(x_{1}\right)^{2}x_{2}~{};$ (4a) they of course imply (see (2a))
that $x_{1}$ and $x_{2}$ are the $2$ zeros—with respective multiplicities $2$
and $1$—of the (monic) third-degree polynomial $p_{3}\left(z;t\right)$ with
coefficients $y_{1},$ $y_{2},$ $y_{3}$:
$p_{3}\left(z;t\right)=z^{3}+\sum_{m=1}^{3}\left[y_{m}\left(t\right)~{}z^{3-m}\right]=\left[z-x_{1}\left(t\right)\right]^{2}~{}\left[z-x_{2}\left(t\right)\right]~{}.$
(4b) Note that this entails that $x_{1}$ and $x_{2}$ can be obtained—in terms
of $y_{1}$ and $y_{2}$, or $y_{1}$ and $y_{3}$, or $y_{2}$ and $y_{3}$—by
solving the following algebraic equations:
$3\left(x_{1}\right)^{2}+2y_{1}x_{1}+y_{2}=0~{},~{}~{}~{}x_{2}=-\left(2x_{1}+y_{1}\right)~{},$
(5a) or
$2\left(x_{1}\right)^{3}+y_{1}\left(x_{1}\right)^{2}+y_{3}=0~{},~{}~{}~{}x_{2}=-\left(2x_{1}+y_{1}\right)~{},$
(5b) or
$\left(x_{1}\right)^{3}-y_{2}x_{1}-2y_{3}=0~{},~{}~{}~{}x_{2}=\frac{-\left(x_{1}\right)^{2}+y_{2}}{2x_{1}}~{}.$
(5c)

It can moreover be easily shown—or see [7] or [9] or [10]—that these formulas
imply the following differential relations:

$\dot{x}_{1}=-\frac{2x_{1}\dot{y}_{1}+\dot{y}_{2}}{2\left(x_{1}-x_{2}\right)}~{},~{}~{}~{}\dot{x}_{2}=\frac{\left(x_{1}+x_{2}\right)\dot{y}_{1}+\dot{y}_{2}}{x_{1}-x_{2}}~{},$
(6a)
$\dot{x}_{1}=\frac{-\left(x_{1}\right)^{2}\dot{y}_{1}+\dot{y}_{3}}{2x_{1}\left(x_{1}-x_{2}\right)}~{},~{}~{}~{}\dot{x}_{2}=\frac{x_{1}x_{2}\dot{y}_{1}-\dot{y}_{3}}{x_{1}\left(x_{1}-x_{2}\right)}~{},$
(6b)
$\dot{x}_{1}=\frac{x_{1}\dot{y}_{2}+2\dot{y}_{3}}{2x_{1}\left(x_{1}-x_{2}\right)}~{},~{}~{}~{}\dot{x}_{2}=-\frac{x_{1}x_{2}\dot{y}_{2}+\left(x_{1}+x_{2}\right)\dot{y}_{3}}{\left(x_{1}\right)^{2}\left(x_{1}-x_{2}\right)}~{}.$
(6c)

It is plain from these formulas that if the two variables $y_{1}\equiv
y_{1}\left(t\right)$ and $y_{2}\equiv y_{2}\left(t\right)$, or $y_{1}\equiv
y_{1}\left(t\right)$ and $y_{3}\equiv y_{3}\left(t\right)$, or $y_{2}\equiv
y_{2}\left(t\right)$ and $y_{3}\equiv y_{3}\left(t\right)$ satisfy an
algebraically solvable system of $2$ first-order ODEs, say

$\dot{y}_{m_{1}}=f_{m_{1}}\left(y_{m_{1}},y_{m_{2}}\right)~{},~{}~{}~{}\dot{y}_{m_{2}}=f_{m_{2}}\left(y_{m_{1}},y_{m_{2}}\right)$
(7)

with $m_{1}=1,$ $m_{2}=2$ or $m_{1}=1,$ $m_{2}=3$ or $m_{1}=2,$ $m_{2}=3$,
then the corresponding system of $2$—generally nonlinearly-coupled, first-
order—ODEs satisfied by $x_{1}\equiv x_{1}\left(t\right)$ and $x_{2}\equiv
x_{2}\left(t\right)$ is as well algebraically solvable. Moreover, if the
functions $f_{m_{1}}\left(y_{m_{1}},y_{m_{2}}\right)$ and
$f_{m_{2}}\left(y_{m_{1}},y_{m_{2}}\right)$ are appropriately chosen, then the
right-hand sides of the ODEs (6) become polynomial (or ”almost polynomial”:
see below). In the following Section 3 we report several algebraically
solvable systems satisfied by the $2$ dependent variables $x_{1}\equiv
x_{1}\left(t\right)$ and $x_{2}\equiv x_{2}\left(t\right)$ obtained in this
manner, i. e. by first replacing in the relevant equations (6) the expressions
of the time-derivatives of the relevant $2$ variables $y_{m}$ via (7) and
subsequently replacing the expressions of these $2$ variables $y_{m}$ in terms
of the $2$ variables $x_{n}$ via the relevant equations (4a); of course with
the functions $f_{m_{1}}\left(y_{m_{1}},y_{m_{2}}\right)$ and
$f_{m_{2}}\left(y_{m_{1}},y_{m_{2}}\right)$ appropriately chosen.

##  3 Results

Our first step is to identify $3$ solvable systems of $2$ evolution equations
satisfied by the dependent variables $y_{m_{1}}\equiv y_{m_{1}}\left(t\right)$
and $y_{m_{2}}\equiv y_{m_{2}}\left(t\right)$. Their solvability is discussed
in Appendix A of Ref. [10] and tersely reviewed below in Appendix A.

The first of these $3$ systems—hereafter identified as A1 (see Appendix A)—is
characterized by the following $2$ uncoupled ODEs:

$\dot{y}_{m_{1}}=\sum_{\ell=0}^{L}\left[\alpha_{\ell}\left(y_{m_{1}}\right)^{\ell
m_{2}+1}\right]~{},~{}~{}~{}\dot{y}_{m_{2}}=\sum_{\ell=0}^{L}\left[\beta_{\ell}\left(y_{m_{2}}\right)^{\ell
m_{1}+1}\right]~{}.$ (8)

The second of these $3$ systems—hereafter identified as A2 (see Appendix A)—is
characterized by the following $2$ coupled ODEs:

$\dot{y}_{m_{1}}=\sum_{\ell=0}^{L}\left[\alpha_{\ell}\left(y_{m_{1}}\right)^{\ell}\right]~{},~{}~{}~{}\dot{y}_{m_{2}}=y_{m_{2}}\sum_{\ell=1}^{L}\left[\beta_{\ell}\left(y_{m_{1}}\right)^{\ell-1}\right]+\sum_{\ell=0}^{L}\left[\gamma_{\ell}\left(y_{m_{1}}\right)^{\ell-1+\left(m_{2}/m_{1}\right)}\right]~{}.$
(9)

The third of these $3$ systems—hereafter identified as A3 (see Appendix A)—is
characterized by the following $2$ coupled ODEs:

$\displaystyle\dot{y}_{m_{1}}$ $\displaystyle=$
$\displaystyle\alpha_{0}+\alpha_{1}y_{m_{2}}~{},~{}~{}~{}\dot{y}_{m_{2}}=\beta_{0}\left(y_{m_{1}}\right)^{-1+m}+\beta_{1}\left(y_{m_{1}}\right)^{-1+2m}~{},$
$\displaystyle m$ $\displaystyle=$ $\displaystyle
m_{2}/m_{1}~{},~{}~{}~{}m_{1}=1~{},~{}~{}~{}m_{2}=2,3~{}.$ (10)

Our next step is to list in the following $11$ subsections $11$ algebraically
solvable systems of $2$ nonlinearly-coupled ODEs with polynomial (or ”almost
polynomial”: see below) right-hand sides (see (3)); in each case we identify
the corresponding appropriately chosen algebraically solvable system of
ODEs—see above and Appendix A—satisfied by the corresponding functions
$y_{m_{1}}\left(t\right)$ and $y_{m_{2}}\left(t\right)$. But note that the
algebraically solvable systems thus identified are only $9$, because $2$ pairs
of the systems identified below—although arrived at differently—are in fact
identical (a phenomenon already noted in [9]).

###  3.1 Models of type A1

In the $6$ cases listed in this Subsection the variables
$y_{m_{1}}\left(t\right)$ and $y_{m_{2}}\left(t\right)$ are supposed to
satisfy the system A1 of $2$ uncoupled ODEs (see (8) and Appendix A), with the
indicated assignments of the various parameters.

Model A1.1: $m_{1}=1,~{}m_{2}=2;$ $L=1;$ $\alpha_{0}=a,$
$\alpha_{1}=b;~{}\beta_{0}=2a,~{}\beta_{1}=6b;$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
ax_{1}+bx_{1}\left[5\left(x_{1}\right)^{2}+5x_{1}x_{2}-\left(x_{2}\right)^{2}\right]~{},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
ax_{2}-b\left[2\left(x_{1}\right)^{3}-2\left(x_{1}\right)^{2}x_{2}-8x_{1}\left(x_{2}\right)^{2}-\left(x_{2}\right)^{3}\right]~{}.$
(11)

Model A1.2: $m_{1}=1,~{}m_{2}=3;$ $L=1;$ $\alpha_{0}=a,$
$\alpha_{1}=-2b;~{}\beta_{0}=3a,~{}\beta_{1}=-162b;$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
x_{1}\left\\{a+b\left[16\left(x_{1}\right)^{3}+48\left(x_{1}\right)^{2}x_{2}-9x_{1}\left(x_{2}\right)^{2}-\left(x_{2}\right)^{3}\right]\right\\}~{},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
x_{2}\left\\{a-2b\left[16\left(x_{1}\right)^{3}-33\left(x_{1}\right)^{2}x_{2}-9x_{1}\left(x_{2}\right)^{2}-\left(x_{2}\right)^{3}\right]\right\\}~{}.$
(12)

Model A1.3: $m_{1}=2,~{}m_{2}=3;$ $L=1;$ $\alpha_{0}=2a,$
$\alpha_{1}=2b;~{}\beta_{0}=3a,~{}\beta_{1}=81b;$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
x_{1}\left\\{a+b\left(x_{1}\right)^{3}\left[\left(x_{1}\right)^{3}+9\left(x_{1}\right)^{2}x_{2}+33x_{1}\left(x_{2}\right)^{2}-16\left(x_{2}\right)^{3}\right]\right\\}~{},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
x_{2}\left\\{a-b\left(x_{1}\right)^{3}\left[2\left(x_{1}\right)^{3}+18\left(x_{1}\right)^{2}x_{2}-15x_{1}\left(x_{2}\right)^{2}-32\left(x_{2}\right)^{3}\right]\right\\}~{}.$
(13)

###  3.2 Models of type A2

In the cases listed in this Subsection the variables $y_{m_{1}}\left(t\right)$
and $y_{m_{2}}\left(t\right)$ are supposed to satisfy the system A2 of $2$
coupled ODEs (see (9) and Appendix A), with the indicated assignments of the
various parameters.

Model A2.1: $m_{1}=1,~{}m_{2}=2;$ $L=3;$
$\alpha_{0}=3b_{0},~{}\alpha_{1}=a_{0}-3b_{1},~{}\alpha_{2}=-a_{1}+3b_{2},~{}\alpha_{3}=a_{2}-3b_{3};$
$\beta_{\ell}=\left(-1\right)^{\ell-1}2a_{\ell-1},~{}\ell=1,2,3;~{}\gamma_{\ell}=\left(-1\right)^{\ell}2b_{\ell},~{}\ell=0,1,2,3;$

$\displaystyle\dot{x}_{n}$ $\displaystyle=$ $\displaystyle
x_{n}\left(a_{0}+a_{1}X+a_{2}X^{2}\right)-\left[b_{0}+b_{1}X+b_{2}X^{2}+b_{3}X^{3}\right]~{},$
$\displaystyle n$ $\displaystyle=$ $\displaystyle 1,2;~{}X=2x_{1}+x_{2}~{}.$
(14)

Model A2.2: $m_{1}=1,~{}m_{2}=2;$ $L=3;$
$\alpha_{0}=0,~{}\alpha_{\ell}=c_{\ell};~{}\beta_{\ell}=2c_{\ell};~{}\ell=1,2,3~{};$

$\dot{x}_{n}=x_{n}\left(c_{1}+c_{2}X+c_{3}X^{2}\right)~{},~{}~{}~{}n=1,2,~{}~{}~{}X=x_{1}\left(x_{1}+2x_{2}\right)~{}.$
(15)

Model A2.3: $m_{1}=1,~{}m_{2}=3;$ $L=3;$ $\alpha_{0}=9b_{0},$
$\alpha_{\ell}=\left(-1\right)^{\ell-1}\left(a_{0}-9b_{\ell}\right),~{}\beta_{\ell}=\left(-1\right)^{\ell-1}3a_{\ell-1},$
$\ell=1,2,3;$ $\gamma_{\ell}=\left(-1\right)^{\ell}b_{\ell},~{}\ell=0,1,2,3.$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
x_{1}\left(a_{0}+a_{1}X+a_{2}X^{2}\right)-\left(\frac{5x_{1}+x_{2}}{2x_{1}}\right)\left(b_{0}+b_{1}X+b_{2}X^{2}++b_{3}X^{3}\right)~{},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
x_{2}\left(a_{0}+a_{1}X+a_{2}X^{2}\right)-\left(\frac{4x_{1}-x_{2}}{x_{1}}\right)\left(b_{0}+b_{1}X+b_{2}X^{2}++b_{3}X^{3}\right)~{},$
$\displaystyle X$ $\displaystyle=$ $\displaystyle 2x_{1}+x_{2}~{}.$ (16)

Note that the right hand sides of these ODEs (16) are polynomial only if all
the coefficients $b_{\ell}$ vanish.

Model A2.4: $m_{1}=3,~{}m_{2}=1;$ $L=3;$
$\alpha_{0}=0,~{}\alpha_{\ell}=\left(-1\right)^{\ell-1}3c_{\ell},~{}\beta_{\ell}=\left(-1\right)^{\ell-1}c_{\ell-1},~{}\ell=1,2,3;$
$\gamma_{\ell}=0,~{}\ell=0,1,2,3.$

$\dot{x}_{n}=x_{n}\left(c_{0}+c_{1}X+c_{2}X^{2}\right)~{},~{}~{}~{}n=1,2,~{}~{}~{}X=\left(x_{1}\right)^{2}x_{2}~{}.$
(17)

Model A2.5: $m_{1}=2,~{}m_{2}=3;$ $L=3;$
$\alpha_{0}=0,~{}\alpha_{\ell}=2c_{\ell-1},~{}\beta_{\ell}=3c_{\ell-1},~{}\ell=1,2,3;$
$\gamma_{\ell}=0,~{}\ell=0,1,2,3.$

$\dot{x}_{n}=x_{n}\left(c_{0}+c_{1}X+c_{2}X^{2}\right)~{},~{}~{}~{}n=1,2,~{}~{}~{}X=x_{1}\left(x_{1}+2x_{2}\right)~{}.$
(18)

Model A2.6: $m_{1}=3,~{}m_{2}=2;$ $L=3;$
$\alpha_{0}=0,~{}\alpha_{\ell}=\left(-1\right)^{\ell-1}3c_{\ell-1},~{}\beta_{\ell}=\left(-1\right)^{\ell-1}2c_{\ell-1},~{}\ell=1,2,3;$
$\gamma_{\ell}=0,~{}\ell=0,1,2,3.$

$\dot{x}_{n}=x_{n}\left(c_{0}+c_{1}X+c_{2}X^{2}\right)~{},~{}~{}~{}n=1,2,~{}~{}~{}X=\left(x_{1}\right)^{2}x_{2}~{}.$
(19)

Remark 3.2-1. Note that the systems of ODEs (15) and (18) are identical, and
likewise the systems of ODEs (17) and (19) are identical. $\square$

###  3.3 Models of type A3

In the cases listed in this Subsection the variables $y_{m_{1}}\left(t\right)$
and $y_{m_{2}}\left(t\right)$ are supposed to satisfy the system A3 of $2$
coupled ODEs (see (10) and Appendix A), with the indicated assignments of the
various parameters.

Model A3.1: $m_{1}=1,~{}m_{2}=2;$ $\alpha_{0}=-3a,$
$\alpha_{1}=-9b;~{}\beta_{0}=-2a,~{}\beta_{1}=-2b;$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$ $\displaystyle
a+b\left[\left(x_{1}\right)^{2}+7x_{1}x_{2}+\left(x_{2}\right)^{2}\right]~{},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$ $\displaystyle
a+b\left[7\left(x_{1}\right)^{2}+4x_{1}x_{2}-2\left(x_{2}\right)^{2}\right]~{}.$
(20)

Model A3.2: $m_{1}=1,~{}m_{2}=3;$ $\alpha_{0}=-18a,$
$\alpha_{1}=-486b;~{}\beta_{0}=-2a,~{}\beta_{1}=-2b;$

$\displaystyle\dot{x}_{1}$ $\displaystyle=$
$\displaystyle\left(x_{1}\right)^{-1}\left\\{a\left(5x_{1}+x_{2}\right)\right.$
(21)
$\displaystyle\left.+b\left[32\left(x_{1}\right)^{4}-131\left(x_{1}\right)^{3}x_{2}-51\left(x_{1}x_{2}\right)^{2}-11x_{1}\left(x_{2}\right)^{3}-\left(x_{2}\right)^{4}\right]\right\\},$
$\displaystyle\dot{x}_{2}$ $\displaystyle=$
$\displaystyle\left(x_{1}\right)^{-1}\left\\{2a\left(4x_{1}-x_{2}\right)\right.$
(22)
$\displaystyle\left.-2b\left[32\left(x_{1}\right)^{4}+112\left(x_{1}\right)^{3}x_{2}-51\left(x_{1}x_{2}\right)^{2}-11x_{1}\left(x_{2}\right)^{3}-\left(x_{2}\right)^{4}\right]\right\\}.$

##  4 Outlook

Space limitations do not allow to report here some natural developments of the
results reported above, such as the identification of the more general
solvable systems which obtain from those reported in the preceding Section 3
via linear transformations of the dependent variables $x_{n}\left(t\right)$;
the interested reader is referred to the analogous developments discussed in
Section 5 of Ref. [9].

##  5 Acknowledgements

FP likes to thank the Physics Department of the University of Rome ”La
Sapienza” for the hospitality from April 2018 to January 2019 (during her
sabbatical, when the results reported in this paper were obtained), and the
organizers of DICE2018 (where these results were reported), especially Prof.
Hans Thomas Elze.

##  6 Appendix A

The $2$ ODEs of the system A1 (see (8)) are uncoupled, and each of them is
clearly solvable by quadratures. Note in particular that the initial-value
problem of the ODE

$\dot{y}=ay+by^{m+1}$ (23a) is given by the explicit formula
$y\left(t\right)=y\left(0\right)\exp\left(at\right)\left\\{1+\left(b/a\right)\left[y\left(0\right)\right]^{m}\left[1-\exp\left(mat\right)\right]\right\\}^{-1}~{}.$
(23b)

To solve the system A2 one treats firstly the first of the $2$ ODEs
(9)—essentially as just above. Then one notes—see Appendix A of Ref. [10]—that
the solution of the second ODE of the system (9) reads

$y_{m_{2}}\left(t\right)=F\left(t\right)\left[y_{m_{2}}\left(0\right)+\int_{0}^{t}dt^{\prime}~{}\left[F\left(t^{\prime}\right)\right]^{-1}\sum_{\ell=0}^{L}\left\\{\gamma_{\ell}\left[y_{m_{1}}\left(t^{\prime}\right)\right]^{\ell-1+\left(m_{2}/m_{1}\right)}\right\\}\right]$
(24a) with
$F\left(t\right)=\exp\left\\{\int_{0}^{t}\left[dt^{\prime}\sum_{\ell=1}^{L}\left\\{\beta_{\ell}\left[y_{m_{1}}\left(t^{\prime}\right)\right]^{\ell-1}\right\\}\right]\right\\}~{}.$
(24b)

Explicit solutions can be easily obtained in the following cases:

$m_{1}=1~{},~{}~{}~{}m_{2}=2,3~{},~{}~{}~{}m=m_{2}/m_{1}~{},$ (25a)
$\dot{y}_{m_{1}}=\alpha_{0}+\alpha_{1}y_{m_{1}}+\alpha_{2}\left(y_{m_{1}}\right)^{2}~{},$
(25b)
$\dot{y}_{m_{2}}=y_{m_{2}}\left(\beta_{0}+\beta_{1}y_{m_{1}}\right)+\gamma_{0}\left(y_{m_{1}}\right)^{-1+m}+\gamma_{1}\left(y_{m_{1}}\right)^{m}+\gamma_{2}\left(y_{m_{1}}\right)^{1+m}~{};$
(25c) $\displaystyle y_{{}_{m_{1}}}\left(t\right)$ $\displaystyle=$
$\displaystyle\frac{y_{{}_{m_{1}}}\left(0\right)\left[1+\left(\Delta/\alpha_{1}\right)\tanh\left(\Delta
t\right)\right]-2\left(\alpha_{0}/\alpha_{1}\right)\tanh\left(\Delta
t\right)}{1-\left\\{\left[2\alpha_{2}y_{{}_{m_{1}}}\left(0\right)+\Delta\right]/\alpha_{1}\right\\}\tanh\left(\Delta
t\right)}~{},$ $\displaystyle\Delta^{2}$ $\displaystyle=$
$\displaystyle\left(\alpha_{1}\right)^{2}-4\alpha_{0}\alpha_{2}~{},$ (25d)
$y_{m_{2}}\left(t\right)=f\left(t\right)\left[y_{m_{2}}\left(0\right)+\int_{0}^{t}dt^{\prime}~{}\left[f\left(t^{\prime}\right)\right]^{-1}\sum_{\ell=0}^{L}\left\\{\gamma_{\ell}\left[y_{m_{1}}\left(t^{\prime}\right)\right]^{\ell-1+m}\right\\}\right]~{},$
(25e)
$f\left(t\right)=\exp\left\\{\int_{0}^{t}\left[dt^{\prime}\sum_{\ell=1}^{L}\left\\{\beta_{\ell}\left[y_{\tilde{m}_{1}}\left(t^{\prime}\right)\right]^{\ell-1}\right\\}\right]\right\\}~{}.$
(25f)

Finally, to identify the solution of the system A3 (see (10)) we note that
time-differentiation of the first of its $2$ ODEs entails that
$y_{m_{1}}\left(t\right)$ satisfies the decoupled second-order ODE

$\ddot{y}_{m_{1}}=\alpha_{1}\left[\beta_{0}\left(y_{m_{1}}\right)^{-1+m}+\beta_{1}\left(y_{m_{1}}\right)^{-1+2m}\right]~{},~{}~{}~{}m=m_{2}/m_{1}~{};$
(26)

hence (see (10)) $y_{1}\left(t\right)$ is an elliptic (for $m=2$) or
hyperelliptic (for $m=3$) function; and likewise for $y_{2}\left(t\right)$
(see the first of the $2$ ODEs (10)).

## References

  * [1] F. Calogero, ”Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differential Equations, and Related ”Solvable” Many Body Problems”, Nuovo Cimento 43B, 177-241 (1978).
  * [2] F. Calogero, Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph m66, Springer, Heidelberg, 2001 (749 pages).
  * [3] F. Calogero, Isochronous systems, Oxford University Press, Oxford, U. K., (2008); paperback (2012).
  * [4] F. Calogero, Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge University Press, Cambridge, U. K., 2018 (168 pages).
  * [5] F. Calogero, “Systems of nonlinearly-coupled differential equations solvable by algebraic operations”, in Nonlinear Systems and Their Remarkable Mathematical Structures (edited by N. Euler), CRC Press (Taylor & Francis), 2018, pages 1-14.
  * [6] D. Gómez-Ullate and M. Sommacal, ”Periods of the Goldfish Many-Body Problem”, J. Nonlinear Math. Phys. 12, Suppl. 1, 351-362 (2005).
  * [7] O. Bihun and F. Calogero, “Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations”, Qual. Theory Dyn. Syst. (published online: 26 July 2018). doi.org/10.1007/s12346-018-0282-3; http://arxiv.org/abs/1806.07502.
  * [8] O. Bihun, ”Time-dependent polynomials with one multiple root and new solvable dynamical systems”, arXiv:1808.00512v1 [math-ph] 1 Aug 2018.
  * [9] F. Calogero and F. Payandeh, ”Polynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactions”, J. Math. Phys. (submitted).
  * [10] F. Calogero and F. Payandeh, ”Solvable dynamical systems in the plane with polynomial interactions”, to be published as a chapter in a collective book to celebrate the 65th birthdate of Emma Previato (in press).


# Assessing Viewer’s Mental Health by Detecting Depression in YouTube Videos

Shanya Sharma
SAP Labs
<EMAIL_ADDRESS>Manan Dey
SAP Labs
<EMAIL_ADDRESS>

###### Abstract

Depression is one of the most prevalent mental health issues around the world,
proving to be one of the leading causes of suicide and placing large economic
burdens on families and society. In this paper, we develop and test the
efficacy of machine learning techniques applied to the content of YouTube
videos captured through their transcripts and determine if the videos are
depressive or have a depressing trigger. Our model can detect depressive
videos with an accuracy of 83%. We also introduce a real-life evaluation
technique to validate our classification based on the comments posted on a
video by calculating the CES-D scores of the comments. This work conforms
greatly with UN Sustainable Goal of ensuring Good Health and Well Being with
major conformity with section UN SDG 3.4.

##  1 Introduction

Depression is affecting as high as 300 million people around the world,
according to the World Health Organization. However, the WHO reports that a
clear majority of depressed individuals never seek out treatment, mostly
because of being unaware of what is going on with them [11] .

We propose a solution to aid in early diagnosis of depression by giving a
zoomed-in perspective of a patient’s state of mind and experiences in order to
improve diagnostic accuracy and efficiency. Our solution is to develop a model
with the ability to track an individual’s YouTube watching activity and
predict his/her state of mind by analyzing the patterns in the kind of videos
watched by him/her over a period of time. We realize that a false positive
result from the same can lead to anxiety and other psychological impacts,
therefore we also propose a procedure to validate the results produced by the
model using CES-D scale [15] used for screening of depression by primary
health care and in research. A basic flow describing our solution is given in
Figure 1.

Unlike other work that aims at detecting depression from an individual’s
activity on social media platforms [9], [12], [3], [1], [17], [10], [4]
examining clinical interview videos [7], [2], [14] where posts/interviews by
an individual were self-descriptive, we aim at collecting data passively from
user’s end to get an objective view of his/her mental state.

At a high level, our pipeline is composed of 3 different layers: i) Secured
data acquisition, ii) Feature extraction from individual videos and their
content analysis iii) Analysis of video viewing patterns over time. Examples
of the types of patterns we aim to find are:

  * •

A gradual decline from happy/uplifting videos to depressing/sad videos.

  * •

Videos of same emotional content, since they tend to do repetitive tasks for a
long period of time (rumination).

  * •

High frequency of depressive videos in the watching history

Figure 1: Pipeline of the solution proposed- Detecting depression based on
YouTube watching activity

In this paper, we focus on the second layer, i.e. analyzing the video content
based on the textual features extracted from the transcripts to construct
classifiers that can identify if a video can be termed as depressive or not.
If, on the analysis, it is found that content of most of the videos been
watched by him/her can be classified as depressive (Pattern 3), it gives us a
hint about a decline in his mental state and help us make him self-aware or
direct him to the support portals if needed.

Our major contributions to this paper are:

  * •

We construct a classifier that can help understand the content of a video by
classifying it as Depressive/Non-Depressive.

  * •

To provide a real-life validation of the classification results in above step,
we propose a methodology to evaluate the comments posted for a video and
determine a potential score that would have been obtained on a CES-D scale and
use it as a real life proxy to judge the accuracy of the classification.

##  2 Data and Classification

###  2.1 Data

The data for the analysis of the video content was gathered by extracting
videos using various keywords and extracting the transcript out of it. There
was a total of around 3000transcripts, totalling 1,409,719 words. The videos
were divided into two categories: depressing and not-depressing. Of the data
collected, 1,427( 48%) are considered depressing and these transcripts account
for nearly 754,883 words. The 1,573 (52%) non-depressed transcripts are
responsible for the other 832,117 words. To gather our data, we considered a
video depressing if it was a search result when depressing keywords were used.
While looking for depressing videos, we aimed mostly on videos which had tags
as “self-harm”, “suicidal”, “triggering” etc. For the list of keywords, see
Appendix D. Note that the keywords mentioned in the appendix are manually
validated to contain depressive videos. Similarly, we collected videos around
the categories like educational, funny etc. for non-depressing label.

###  2.2 Classification and Results

With respect to classification, we experimented with three models.

  * •

Naive Bayes with n-grams: We built a Multinomial Naive Bayes (NB) Classifier
using features extracted from Empath model [5]. Empath is a living lexicon
mined from modern text on the web for analyzing text across lexical categories
(similar to LIWC [16]). These features were the normalized score for a set of
lexical categories. Apart from the pre-defined features like positive and
negative emotions, the Empath model also gives the flexibility to create
customized features. Therefore, categories that included seed-terms pertaining
to symptoms diagnosed by the CES-D questionnaire [15] were also created and
included in the set of categories. A list of seed-terms corresponding to
questions is mentioned in Appendix B.

Table 1: Hyperparameter details for Model 3
Hyperparameter | Value(%) (s)
---|---
N-Gram range | 2-3
Maximum Features | 5000

Table 2: Model comparison
Model | Accuracy (%) | Time (s)
---|---|---
Empath + NB | 52 | 5
TF-IDF + Empath + NB | 81.2 | 8
LSTM | 83.4 | 345

  * •

LSTM: While the Naive Bayes Model provided a baseline, we used LSTM to get
better results. The full structure of our LSTM model is shown in Fig.2. For
each comment that’s pre-processed, we make it pass through an Embedding layer
seeded with pre-trained GloVe 300D word embedding weights. We feed this into a
LSTM network with 196 units and tanh activation, applying a recurrent dropout
of 0.2. The outputs of the LSTM are then fed through two dense layers with a
dropout layer of 0.5 in between to output the video classification. We use
binary cross-entropy loss and Adam optimization.

Figure 2: LSTM Model Structure

  * •

Naive Bayes with n-grams: We tried to modify the baseline model by capturing
textual features by extracting TF-IDF weighted combinations of word ngrams
from the transcripts and used it along with the features from previous model
to train the Naive Bayes classifier. The hyper-parameters and their
corresponding values are provided in Table 1.

A comparison of the results obtained is shown in Table 2. From the table it
can be inferred that the combination of the Empath model with TF-IDF features
resulted in almost the same classification accuracy as of LSTM.

##  3 Evaluation

###  3.1 CES-D Scores

For the validation and evaluation of the results produced by the classifier,
we introduce a scoring method, called CES-D score, for each video to analyze
how depressive the video is. The score is the density of the terms, derived
from the various symptoms (Insomnia, Self-hate, Appetite, etc.) considered in
the CES-D scale [15], present in a negative connotation within a given text.
The Center for Epidemiologic Studies Depression Scale is a brief self-report
questionnaire developed to measure depressive symptoms severity in the general
population (See Appendix A). To identify the occurrence of symptoms, we use
Empath Model [5] which combines modern NLP techniques to construct categories
on demand using a few seed terms. For e.g., for the seed term “weight-loss”,
the generated set of words were “weight loss”, “anemia”, ”stress”,
“malnutrition” etc. The various seed-terms used for generation of categories
and examples of the seed-terms identified from the comments’ text are
mentioned in Appendix B.

####  3.1.1 Calculation of CES-D Score

Algorithm 1 CES-D Score

0: data $text$

set ( $categoryTerms$)

$termFreq\leftarrow 0$

while $term$ in $categoryTerms$ do

if $term$ in $text$ then

$termFreq\leftarrow termFreq+1$

end if

end while

$termFreq\leftarrow\frac{termFreq}{len(text)}$

$neg\leftarrow Empath.analyze(text,negative)$

$pos\leftarrow Empath.analyze(text,positive)$

$connotation\leftarrow(neg-pos)$

$CESD\leftarrow termFreq\times connotation$

return $CESD$

Algorithm 1 presents how the score is calculated for each comment in the
video. The categories generated by each seed-term is merged in a set
(categoryTerms). The algorithm iterates through each word of the set and
calculates the termfrequency for it and calculates an aggregate by adding the
term-frequency for each word. Since the words in the set can be used in both
positive and negative connotations, the comment is then sent to the Empath
model for analysis of a score for positive and negative emotion. The aggregate
generated in the previous step is then multiplied by the difference of
positive score form the negative. This is to ensure that only the negative
emotions add up to the score. If for any comment, the positive score exceeds
the negative score, the CES-D score is calculated as 0.
For the calculation of the CES-D score for the video as a whole, in order to
determine the intensity of depression in the comments, the aggregate
calculated in the first step is added up for all comments and a normalized
score is generated by dividing the sum by the total number of comments.

###  3.2 Analyzing Comments

Figure 3: Comparison of CES-D scores in various categories of videos

Figure 4: Confusion Matrix for evaluation of classifications

The comments posted on a video is a great way of understanding how the video
made the viewers feel. They give us a fair idea of the video’s type and
content. So, we examined responses to a video by analyzing the textual
features of the comments considering the idea that the words a person uses
reveal information about their psychological state [6]. We extracted the
comments posted on videos that fall under one of the following categories:

a) Music b) Depression c) Funny d) Self-Help/Motivational

It was determined that a video belonged to a category if it was in the YouTube
search result when the category names were used as keywords. For each keyword
(category), we extracted around 200 videos and analyzed the comments in a
cumulative manner by calculating their CES-D score. On the comparison of
various categories mentioned above, the difference between the average CES-D
scores of depressive vs non-depressive videos was quite evident in Figure 2.
In the plot, the percent of non-depressed comments (CESD = 0) for non-
depressive categories (music, funny) was much higher than that in depressive
category. Also, the range of the CES-D score is the largest in depressive
videos which proves that depressed are induced and attracted by it. As per
different theories, it has been proven that on being sad/depressed, humans
tend to be inclined towards groups where they can find similarities to their
mental states [8]. It can thus be deduced that watching videos of high CES-D
score (depressive videos) on a frequent basis is likely to be an indicator of
an unhealthy mental state.

###  3.3 Evaluation Results

To evaluate the classification results produced by our classifier in a real-
life setting, we tested the results in a less constructed domain. The model
was fed the transcripts of 1500 random videos and the classifications were
compared to the CES-D score obtained by the comments. The method mentioned in
section 3.1 was used to calculate the CES-D score of the video. A CES-D score
greater than 20 was considered depressive. We came to the conclusion of
setting the threshold as 20 after taking the average of CES-D scores for the
depressive videos collected for the training purpose. The accuracy of the
classifier based on this methodology came out to be 84% as can be evaluated
from the confusion matrix shown in Figure 4.

##  4 Conclusion and Future Works

This work demonstrates a method to analyze videos for depressive content. It
also shows that individuals with a depressed state of mind tend to ruminate
and watch depressive videos in order to gain validation or find company. This
behavioral pattern can be useful in determining the mental state of a viewer.
Since an individual should show persistent symptoms for at least two weeks to
be called depressed [13], our tool will extract data over a window size of 24
days with an overlap of 10 days. The kind of videos he has watched will be
analyzed for depressive content and patterns. This can be used to monitor
recommendations and restrict the recommendations of triggering videos based on
the watching activity. In case the patterns reveal the signs on unhealthy
mental state, our tool aims to provide early diagnosis by warning the user
about the change points in his mood and also direct them to the right portals
for help.

The results obtained in this paper are using textual features, therefore our
future work will focus on the usage of audio/visual features, and analyzing
the affective content using continuous dimensions like arousal/valence, we can
infer more and get better results for video content analysis.

##  5 Acknowledgements

We would like to express our gratitude to Kris Sankaran (MILA) for his
continuous support.

## References

[1] Almeida, H., Briand, A., and Meurs, M.-J. Detecting early risk of
depression from social media user-generated content. InCLEF, 2017.

[2] Cohn, J. F., Kruez, T. S., Matthews, I., Yang, Y., Nguyen,M. H., Padilla,
M. T., Zhou, F., and De la Torre, F. Detecting depression from facial actions
and vocal prosody. In 2009 3rd International Conference on Affective Computing
and Intelligent Interaction and Workshops, pp. 1–7.IEEE, 2009.

[3] De Choudhury, M., Counts, S., and Horvitz, E. Predicting postpartum
changes in emotion and behavior via social media. In Proceedings of the SIGCHI
conference on human factors in computing systems, pp. 3267–3276. ACM,2013a

[4] De Choudhury, M., Counts, S., and Horvitz, E. Social media as a
measurement tool of depression in populations. In Proceedings of the 5th
Annual ACM Web Science Conference, pp. 47–56. ACM, 2013.

[5] Fast, E., Chen, B., and Bernstein, M. S. Empath: Understanding topic
signals in large-scale text. In Proceedings of the 2016 CHI Conference on
Human Factors in Computing Systems, CHI ’16, pp. 4647–4657, New York, NY, USA,
2016. ACM. ISBN 978-1-4503-3362-7. URL
http://doi.acm.org/10.1145/2858036.2858535.

[6] Fine, J. Language in psychiatry: A handbook of clinical practice. Equinox
London, 2006.

[7] Giannakakis, G., Pediaditis, M., Manousos, D., Kazantzaki,E., Chiarugi,
F., Simos, P., Marias, K., and Tsik-nakis, M.Stress and anxiety detection
using facial cues from videos. Biomedical Signal Processing and Control, 31:89
– 101, 2017.ISSN 1746-8094. URL
http://www.sciencedirect.com/science/article/pii/S1746809416300805.

[8] Gray, H. M., Ishii, K., and Ambady, N. Misery loves company: When sadness
increases the desire for social connectedness. Personality and Social
Psychology Bulletin, 37(11):1438–1448, 2011.

[9] Guntuku, S. C., Yaden, D. B., Kern, M. L., Ungar, L. H.,and Eichstaedt, J.
C. Detecting depression and mental illness on social media: an integrative
review.Current Opinion in Behavioral Sciences, 18:43–49, 2017.

[10] Jashinsky, J., Burton, S. H., Hanson, C. L., West, J., Giraud-Carrier,
C., Barnes, M. D., and Argyle, T. Tracking suicide risk factors through
twitter in the us. Crisis, 2014.

[11] Levinson, W., Gorawara-Bhat, R., and Lamb, J. A study of patient clues
and physician responses in primary care and surgical settings. Jama,
284(8):1021–1027, 2000.

[12] Moreno, M. A., Jelenchick, L. A., Egan, K. G., Cox, E.,Young, H., Gannon,
K. E., and Becker, T. Feeling bad on facebook: depression disclosures by
college students on a social networking site. Depression and anxiety,
28(6):447–455, 2011.

[13] NIMH, D. Overview. URL
https://www.nimh.nih.gov/health/topics/depression/index.shtml.

[14] Pampouchidou, A., Simantiraki, O., Fazlollahi, A., Pedi-aditis, M.,
Manousos, D., Roniotis, A., Giannakakis, G.,Meriaudeau, F., Simos, P., Marias,
K., et al. Depression assessment by fusing high and low level features from
audio, video, and text. In Proceedings of the 6th International Workshop on
Audio/Visual Emotion Challenge, pp.27–34. ACM, 2016.

[15] Radloff, L. S. The ces-d scale: A self-report depression scale for
research in the general population. Applied Psychological Measurement,
1(3):385–401, 1977.

[16] Tausczik, Y. R. and Pennebaker, J. W. The psychological meaning of words:
Liwc and computerized text analysis methods. Journal of language and social
psychology, 29(1):24–54, 2010.

[17] Yates, A., Cohan, A., and Goharian, N. Depression and self-harm risk
assessment in online forums. arXiv preprint arXiv:1709.01848, 2017.


# Faster Deep Ensemble Averaging for Quantification of DNA Damage from Comet
Assay Images With Uncertainty Estimates

Srikanth Namuduri  Prateek Mehta  Lise Barbe  Stephanie Lam  Zohreh
Faghihmonzavi  Steve Finkbeiner  Shekhar Bhansali  Electrical & Computer
Engineering, Florida International University, Miami, FL 33174  Finkbeiner
Lab, Gladstone Institutes, San Francisco, CA 94158

###### Abstract

Several neurodegenerative diseases involve the accumulation of cellular DNA
damage. Comet assays are a popular way of estimating the extent of DNA damage.
Current literature on the use of deep learning to quantify DNA damage presents
an empirical approach to hyper-parameter optimization and does not include
uncertainty estimates. Deep ensemble averaging is a standard approach to
estimating uncertainty but it requires several iterations of network training,
which makes it time-consuming. Here we present an approach to quantify the
extent of DNA damage that combines deep learning with a rigorous and
comprehensive method to optimize the hyper-parameters with the help of
statistical tests. We also use an architecture that allows for a faster
computation of deep ensemble averaging and performs statistical tests
applicable to networks using transfer learning. We applied our approach to a
comet assay dataset with more than 1300 images and achieved an $R^{2}$ of
0.84, where the output included the confidence interval for each prediction.
The proposed architecture is an improvement over the current approaches since
it speeds up the uncertainty estimation by 30X while being statistically more
rigorous.

###### keywords:

Deep Learning , Uncertainty Estimation , Deep Regression , Comet Assay , DNA
Damage , Neurodegenerative Diseases

##  1 Introduction

Diseases such as Alzheimer’s, Parkinson’s, ALS, frontotemporal body dementia,
and Huntington’s involve a gradual deterioration of the central nervous system
and are called neurodegenerative diseases. Such deterioration is associated
with DNA damage, including breaks to the DNA helix.

DNA damage pathways are altered in neurodegenerative diseases [1, 2, 3, 4, 5,
6, 7], and DNA double-strand breaks (DSBs) increase with the extent of damage
in diseased neurons [8, 9, 10, 11, 12]. This observation suggests a role for
DNA damage in neuronal health and indicates that pathways that control it
could be targeted to reduce neuronal degeneration. Comet assays are used to
assess DNA damage in single cells. In a comet assay, single cells are
immobilized on a slide in an agarose gel, lysed in place, and subjected to an
electric field. Breaks in the DNA allow it to migrate farther in the electric
field than intact DNA can. The migrating DNA forms a trail that looks like the
tail of a comet; the bulk of slow-moving, undamaged DNA forms the comet’s
head. The percentage of DNA material in the tail of the comet is a measure of
the extent of DNA damage. Damaged DNA is a marker for genotoxicity, and hence,
comet assays can be used for screening drugs [13][14].

Calculating the extent of damage from the comet assay images requires an
image-processing tool to estimate the number of pixels and pixel intensity
values in the head and the tail. In the past, this was done using classical
image processing techniques [15], but recently machine learning (ML) has been
leveraged by researchers to automate the scoring process [16].

Among the various ML techniques available, deep learning (DL) approaches,
specifically Convolutional Neural Networks (CNNs), do not require manual
feature engineering and are the most effective when working with images. When
deep learning is applied to regression problems, i.e., the target variable is
continuous, it is called deep regression.

When using ML or DL to assess the extent of damage in human cells, such as in
neurological diseases, accuracy and rigor are both critical. Otherwise, the
results cannot be trusted or relied upon. In the existing literature, the
emphasis is on accuracy and not reliability. There is a need to improve the
rigorousness of the deep learning approaches for better acceptance in fields
such as medicine. The goal of the research presented in this paper was to
identify the deep regression approach that can quantify the percentage of DNA
damage from comet assay images in a rigorous fashion.

Any estimate made using regression techniques involves some uncertainty. In
problems such as assessing the DNA damage from images, it is essential to
quantify the uncertainty because it conveys the extent to which the actual
value may differ from the estimated value. The user then decides whether the
reported uncertainty is acceptable.

One of the challenges with the application of DL in medicine and in
neurodegenerative diseases research is insufficient data. It cannot be
expected that millions of comet assay images from diseased cells will be
available for training. Even with limited data, the other challenge is the
time taken for computation. Transfer learning and an application of multi-task
learning to uncertainty estimation were explored in this research to address
these challenges. The methods presented in this paper can be applied in any
domain where deep regression is being used. The architecture presented in this
paper can be used to improve the reliability of deep regression without
sacrificing the speed of computation.

###  1.1 Deep Learning

Images do not conform to a tabular structure and are considered un-structured
data. They consist of a matrix of pixel values and can be treated as two-
dimensional arrays for computational purposes. Deep Learning (DL) approaches
are suitable for extracting features and making estimations from such data.
While traditional Machine Learning (ML) approaches can also be applied for
unstructured data, DL approaches give superior accuracy. Hence the scope of
this paper is restricted to DL. DL is a subset of ML that uses neural networks
with several layers. The feature learning is hierarchical, with each layer
learning from the previous one.

The layered architecture allows the DL networks to learn complex relationships
between the features and the target variable. In comet assays assessment, the
target variable is the percentage of DNA in the tail.

$y=f(x;\theta)\,$ (1)

$f(x)=f_{3}(f_{2}(f_{1}(x)))\,$ (2)

The layers in the network consist of weights, represented by $\theta$ in
equation 1. The learning process involves updating the weights in the layers
using the backpropagation algorithm [17], and [18]. The learned relationship
can be represented as a sequence of nested functions as shown in equation 2.

When a deep learning architecture is used to perform regression, it is called
deep regression. The focus of this paper is on deep regression.

###  1.2 Convolutional Neural Networks (CNN)

Convolution is a mathematical operation defined on two functions. In the case
of continuous functions, it consists of an integral of the product of one
function and a reversed and shifted version of the second function. The
discrete convolution operation consists of summation is shown in 3. The
discrete version of convolutions is the mathematical basis for the CNNs.

A neural network layer that uses the convolution operation is a convolutional
layer. Figure 1 shows a representative image of a CNN applied to an image. A
CNN consists of a convolutional block, and the fully connected block [17, 16].

Figure 1: A VGG19 network with 2 fully connected layers. The input is a Comet
Assay image and the output is the % DNA damage

The discrete convolution operation, shown in equation 3, is the mathematical
basis for the convolutional layers. $I$ indicates the input, and $K$ denotes a
filter applied to the input. The output of each convolutional layer indicated
by $C$ in Equation 3 is referred to as the feature map [18].

$C(i,j)=\sum_{m}\sum_{n}I(m,n)K(i-m,j-n)$ (3)

###  1.3 Transfer Learning

Transfer learning involves a machine learning model leveraging the learning of
another model. The transfer of knowledge can be from a related learning task
or a completely unrelated task. It leads to faster learning and better
performance on the second task in both cases. Another major advantage is that
the second task can be trained with a smaller dataset. [19, 20, 21, 22, 23,
24]

In the case of DL, transfer of knowledge is achieved by freezing (or not
training) some of the network layers after training on the first task and then
training or fine-tuning the rest of the layers to learn the second task.
Typically the initial layers are kept, and the subsequent layers are fine-
tuned for the second or final task.[24]

The initial training (first task) can be done from scratch by selecting a task
with sufficient data. Alternatively, a pre-trained model can be used. Trained
weights for some of the popular DL architectures like VGG and Resnet, on well-
known datasets such as MNIST and ImageNet, have been open-sourced by the
respective researchers to be used for transfer learning. The study reported
here used such a VGG19 network, pre-trained on ImageNet data.

###  1.4 Multi-Task Learning

The default approach trains a neural network to learn a single task based on a
specific metric. However, by only focusing on a single task, the network may
ignore other information that may be useful. When there are several related
tasks, the respective networks can gain better information from each other’s
learning by sharing representations between the tasks. With comet assays, the
head diameter, the tail length, and the number of pixels in the head are all
related variables. They are all different aspects of the comet but are related
to determining the amount of DNA damage.

Multi-Task Learning (MTL) involves using networks with shared layers to learn
multiple tasks together, enabling the models to generalize better on the
original task. This also leads to several benefits, such as less computational
memory, faster training in some cases, and better generalization of learning
due to regularization. [25].

##  2 Related Work

Researchers have applied Deep Learning (DL), specifically Convolutional Neural
Networks (CNNs), to comet assay images. However, most applications of DL to
comet assays focus on classifying the images. Only a few studies have
attempted to quantify the extent of damage from the images, and none have
attempted to calculate a confidence interval for their estimates. Deep
learning is a stochastic process, and every estimate involves some
uncertainty, and not reporting the uncertainty can mislead the user.

Studies that analyzed comet images via DL have successfully applied transfer
learning [25]. The research presented in this paper also leverages transfer
learning. Multi-task learning (MTL) has been applied to DL by several
researchers. The neural network presented in this paper involves a novel use
of the MTL architecture to calculate the confidence intervals. [26] [27]

##  3 Data Collection and Annotation

The Trevigen comet assay kit (4250-050-ESK, R&D Systems) was used for running
the experiments. Some cells were processed using the alkaline comet assay
protocol, and the others using the neutral comet assay protocol. The slides
used during the experiments were imaged with fluorescent microscopy.
Specifically, a Nikon Ti-E inverted microscope was used with a 10x/0.45
objective.

Images from both neutral and alkaline protocols can train the deep learning
model. A total of 1309 images were collected from the experiments. Out of
these images, 1047 were used for training, and 262 were used to test the
model.

Figure 2: Images of some of the comets in the training data. The comet shape
is not always apparent to the naked eye

###  3.1 Comet Assay Scoring

The ground truth was generated with an algorithm that detects, segments, and
quantifies the comets in the microscopy images. In short, we start with a re-
orientation of the comets as needed to have the tails facing downward of the
heads. Then, binary thresholding is performed to separate meaningful objects
from the background. Noise and comets out of bound were removed, while the
rest of the detected comet objects were cropped into 273 x 143-pixel size
(with 0.6465 µM per pixel). Any crop containing only tail objects, overlapping
comets, or artifacts is filtered using size, circularity, and intensity
criteria. During segmentation, comet heads and tails are identified using
different thresholding. Different measurements are then calculated to quantify
DNA damage, including head diameter, tail length, area, DNA content, and
length. All values are saved into a CSV for any further quantitative analysis.

##  4 Methodology

A 2019 paper [28] has noted that most of the research presented in the current
literature on deep learning lacks a statistical evaluation of the performance.
A lack of confidence interval reporting was also noted in the context of deep
regression. The paper then presents a rigorous approach to optimizing a deep
learning network. This paper applies the same approach to optimizing the
network when applied to the comet assay data, but with more samples per
configuration.

[28] demonstrated that, for Deep Regression, using a standard architecture
such as VGG16 or Resnet and rigorously optimizing the hyper-parameters gives
the same or better performance as compared to more complex architectures. This
paper leverages this finding by using a VGG19 architecture and rigorously
optimizing the parameters.

[29] notes that uncertainty is ubiquitous in natural sciences, but DL methods
in current use do not pay attention to uncertainty estimation. One of the
sections in the paper captures several methods to estimate the uncertainty and
mentions that Deep Ensemble Averaging is the most popular approach.

The approach presented in this paper is a quicker way to perform Deep Ensemble
Averaging. The ensemble of networks is trained in parallel rather than serial
to hasten the process. It is a trade-off between computation time and memory.
However, the use of additional memory is acceptable (in most cases), given the
speed up.

Further, the Multi-Task Learning architecture presented also reduces the
compute time for the statistical tests. Multiple training runs of the network
(to get enough samples to perform a statistical test) are replaced by a single
run (that gives enough samples).

###  4.1 Metrics

The neural network’s output is a continuous variable, namely the percentage
damage. Hence, it is treated as a regression problem. Since the training uses
annotated or labeled samples, it is supervised learning. The cost function
optimized during the learning process is the mean square error.

$MSE=\frac{1}{n}\sum_{i=1}^{n}{(\hat{y}-y_{i})}^{2}\,$ (4)

The coefficient of determination ( R2̂) indicates the amount of variation in
the data explained by the model. R2 is defined as follows:

$R^{2}=1-\frac{SSE}{SST}\,$ (5)

In equation 5, SSE is the Sum of Squares Error of the model, and SST is the
Sum of Squares Total. SST is an estimate of the total variation in the data.

###  4.2 Data Preprocessing

The dataset used for this research included only grayscale images. Any out-of-
sample images were converted to grayscale before being included in our model.
All the images were re-sized to be of the same size for training and testing
purposes. Data augmentation techniques were applied to improve the model
robustness and increase the size of the training dataset. This involved image
rotation, zoom-in/zoom-out (limited magnitude, so as to not cut off the
comets) , flipping, and translation operations. These operations modify the
images without changing their aspect ratios and hence do not impact the ratio
of the tail size to the head size.

###  4.3 Computational Environment

The pre-processing, training, and testing were done on Google Colab
computational environment. The code was written in Python 3, and the packages
that were indispensable for this implementation were NumPy, Pandas, and Keras.
The plotting was done using the Matplotlib library in python. The network
layers table/visualization was generated using the build-in plot-model
facility from Keras.

###  4.4 Network Architecture

Convolutional Neural Networks (CNNs) consists of one or more convolutional
layers and fully connected layers. Deeper networks, which have more layers,
can learn more complex tasks. VGG19 [30] networks use a total of 19 layers
that contain weights, which includes five blocks of convolutional layers
interspersed with pooling layers and fully connected layers.

Prior research from this group found that using a pre-trained VGG19 network
performed better than other models[16]. As an extension, the same pre-trained
convolutional blocks from VGG19 are used, but the fully connected block uses
an MTL type architecture, where 30 replications of the dense layers are
connected in parallel. Since the weights of the dense layers are randomly
initialized, the learned weights at the end of the training process are also
different. Consequently, the estimates made by each of the 30 branches are
slightly different.

Figure 3: A Multi-task learning architecture, with common convolutional layers
and separate dense layers, re-purposed to learn the same task 30 times. Each
of the outputs is an estimate of the percentage DNA damage

The VGG19 network and the weights pre-trained on the Imagenet dataset were
loaded from the Keras library, like in [16]. The convolutional layers were
frozen and unchanged during training, but the dense layers went through
learning. Linear activation is used in the final layer since the output is a
continuous variable.

This architecture was implemented using the Keras Functional API, which allows
the creation of deep learning models with shared layers. The Keras Functional
API developer guide can be found at [31]. The examples in the developer guide
helped develop the code for this research. The output from the convolutional
block was used as input for 30 different fully connected layers, which led to
30 outputs. A simple for-loop was used to generate these 30 layers.

###  4.5 Hyper-parameter Optimization

The optimization approach used in this research is based on the work of [28].
Where [28] used five samples to run statistical tests for each of the
configurations, this research used a parallel architecture to obtain 30
samples for the same.

Optimizing a DL network requires finding the optimal combination of hyper-
parameters, such as the number of nodes in each layer, learning rate, and
batch size. DL models are inherently stochastic, with the weight
initialization, optimizer, and the samples in each batch being some of the
sources of randomness. Due to this, training the network repeatedly with the
same set of parameters results in slightly different estimations of the DNA
damage percentage. So statistical tests are required to compare one
configuration of hyper-parameters with another.

In all the comet assay papers in prior literature and most of the papers on
deep regression, an empirical approach is taken in finding the right
combination of hyper-parameters. In contrast, training each configuration
multiple times, followed by statistical tests for comparisons, the best set of
hyper-parameters can be picked based on statistical significance rather than
the empirical results. [28].

This research uses the Wilcoxon Signed Rank test to compare each combination,
like in [28]. Even though the number of samples used in this research is
higher, we stick to this test and do not use a parametric test since there is
no basis for assuming that the underlying distribution is Gaussian.

First, a list of hyper-parameter combinations is generated, and a baseline
combination is chosen. The rest of the combinations are compared with the
baseline until we find a better performing combination, at which point, the
baseline is updated. After this, the subsequent combinations are compared to
the new baseline until an even better one is found. Comparing and finding a
better baseline is repeated until all the combinations are exhausted. The
median R2̂ of the best combination is considered the coefficient of
determination of the optimized model.

While comparing a configuration with the baseline, the null hypothesis is as
follows:

”$H_{0}$: The $R^{2}$ of the current combination is not significantly better
than the baseline”

Whenever the P-value is less than 0.05, we reject the null hypothesis, which
means the new combination is better than the baseline. This method of
optimizing the model is more scientific and hence more reliable and
trustworthy than an empirical approach.

###  4.6 Uncertainty Estimation

The $R^{2}$ of a model alone does not convey the trustworthiness of the
estimates. Reporting the confidence interval (CI) can take us further in
understanding the estimates. CI indicates the reliability of the process
involved in the estimation. It is a range of values where the actual value can
be found with a certain confidence level. The most common way to estimate CI
in DL methods is the Deep Ensemble Averaging (DEA) [29].

As the name suggests, DEA involves creating an ensemble of predictors to
calculate the predicted value and the CI. In typical implementations of DEA,
the same model is trained several times (typically 30), and then the CI is
calculated as:

$CI=(\mu-\sigma,\mu+\sigma)\,$ (6)

In equation 6, $\mu$ is the mean of the 30 estimates and $\sigma$ is the
standard deviation.

The predicted value is calculated as:

$R^{2}=\frac{\Sigma(R_{i}^{2})}{n}\,$ (7)

In equation 7, $R_{i}$ is the $i^{t}h$ sample and $n$ is the total number of
samples.

Even though DEA is a well-known approach, running a model several times is
very time-consuming. When using large datasets (with millions of images), each
training run can take hours to days, depending on the computing infrastructure
available.

Being able to run several instances of the same network in parallel can speed
up the process of using statistical tests to optimize the network and the
calculation of CI. The network architecture proposed in this paper serves that
purpose. CI can be calculated by replicating the same task in an MTL type
configuration to get 30 different predictions. The predictions are different
because of the underlying stochastic process.

The architecture suggested here is best suited when using transfer learning,
where we freeze the initial convolutional layers and fine-tune the fully
connected layers. This way, doing a single training run with the new
architecture is equivalent to doing 30 runs of the network with a single
output.

##  5 Results

The performance of a VGG19 was compared to that of an MTL-VGG19 network
trained on the same set of comet images. The VGG19 network was trained 30
times, making predictions on the test set each time. Subsequently, CI and the
predicted value were calculated for each test sample. In contrast, the MTL-
VGG19 only needed training once and output 30 predictions for each test
sample. CI and the predicted value were calculated for the MTL-VGG19 network.

The average size of the CI for the test samples using the VGG19 network (and
30 training runs) was 1.89 and for the MTL-VGG19 architecture was 1.36. The
following was the null hypothesis to compare the two groups:

”$H_{0}$: There is no significant difference in the CI calculated using VGG19
network and the CI calculated using MLT-VGG19” The p-value when performing a
Student’s T-test was 0.40, which is not significant. So we could not reject
the null hypothesis. Hence the CI obtained using both approaches can be
concluded to be the same. Hence the MTL-VGG19 architecture can calculate the
CI in place of the existing method, which is more time-consuming.

Once it was established that the results from the new architecture are not
different from repeated training of the original network, it was used to
optimize the network and calculate the median R2̂ and CI. The results (not
exhaustive) from optimizing the network are shown in table 1. The first five
columns of the table show the various hyper-parameters being optimized. The
last two columns show the Test $R^{2}$ and the P-value, respectively. The
P-value in each row results from the statistical test conducted to compare the
configuration with the one in the previous row. If the Test $R^{2}$ over 30
iterations is significantly different from the previous configuration, the
P-value is expected to be less than 0.05. The first row shows the baseline
configuration. Only one of the other configurations was significantly better
than the baseline, as shown in row 4, indicating the final optimized set of
hyper-parameters.

Nodes1 | Nodes2 | Drop Out | LR | Batch Size | Test R2 | P-Value
---|---|---|---|---|---|---
256 | 128 | 0.2 | 0.01 | 32 | 0.69 | NA*
256 | 128 | 0.2 | 0.01 | 64 | 0.73 | 0.99
256 | 128 | 0.2 | 0.001 | 16 | 0.76 | 0.35
256 | 128 | 0.2 | 0.001 | 32 | 0.84 |  $8.67\times 10^{-}7$ **
256 | 128 | 0.2 | 0.001 | 64 | 0.82 | 0.99
256 | 128 | 0.2 | 0.005 | 16 | 0.79 | 0.99
256 | 128 | 0.2 | 0.005 | 32 | 0.81 | 0.99
256 | 128 | 0.4 | 0.01 | 16 | 0.57 | 0.99
256 | 128 | 0.4 | 0.001 | 16 | 0.71 | 0.99
128 | 64 | 0.2 | 0.01 | 16 | 0.70 | 0.99
128 | 64 | 0.2 | 0.01 | 32 | 0.74 | 0.99
128 | 64 | 0.2 | 0.01 | 64 | 0.78 | 0.99
128 | 64 | 0.2 | 0.001 | 16 | 0.83 | 0.99
128 | 64 | 0.2 | 0.001 | 32 | 0.83 | 0.99
128 | 64 | 0.2 | 0.001 | 64 | 0.82 | 0.99
128 | 64 | 0.2 | 0.005 | 32 | 0.79 | 0.99
128 | 64 | 0.4 | 0.01 | 16 | 0.39 | 0.99
128 | 64 | 0.4 | 0.001 | 16 | 0.70 | 0.99
Table 1: Performance of various Hyper-Parameter Combinations. * Baseline
Configuration. ** Significant. LR - Learning Rate

##  6 Conclusion and Future Work

Using the proposed architecture, in conjunction with transfer learning, we
calculated confidence intervals in one shot, i.e., a single training run. This
architecture can be used in most deep regression problems, and it offers a way
to calculate CI and perform statistical tests much faster. The speed could be
increased up to 30-fold with 30 samples, which is typical for calculating CI
and statistical tests.

So far in this research, all the measurements available from the comet images
have not been leveraged. Learning multiple measurements such as the diameter
of the head, length of the tail can improve the overall accuracy and
reliability of the damage assessment. Our research group plans to study the
use of multiple measurements and assess the impact on accuracy to further the
research in the use of Deep Learning for scoring the comets while studying
neurodegenerative diseases.

##  7 Acknowledgements

This work was made possible with support from NIH grants R01 LM013617, R37
NS101996, P01 AG054407 and RF1 AG058476 to S.F. as well as support from the
Koret Foundation Artificial Intelligence Program for Biomedical Research

## References

  * [1] M. A. Bradley-Whitman, M. D. Timmons, T. L. Beckett, M. P. Murphy, B. C. Lynn, M. A. Lovell, Nucleic acid oxidation: an early feature of alzheimer’s disease, Journal of neurochemistry 128 (2) (2014) 294–304.
  * [2] R. Madabhushi, L. Pan, L.-H. Tsai, Dna damage and its links to neurodegeneration, Neuron 83 (2) (2014) 266–282.
  * [3] E. N. Guerrero, H. Wang, J. Mitra, P. M. Hegde, S. E. Stowell, N. F. Liachko, B. C. Kraemer, R. M. Garruto, K. Rao, M. L. Hegde, Tdp-43/fus in motor neuron disease: complexity and challenges, Progress in neurobiology 145 (2016) 78–97.
  * [4] R. Lopez-Gonzalez, Y. Lu, T. F. Gendron, A. Karydas, H. Tran, D. Yang, L. Petrucelli, B. L. Miller, S. Almeida, F.-B. Gao, Poly (gr) in c9orf72-related als/ftd compromises mitochondrial function and increases oxidative stress and dna damage in ipsc-derived motor neurons, Neuron 92 (2) (2016) 383–391.
  * [5] M. A. Farg, A. Konopka, K. Y. Soo, D. Ito, J. D. Atkin, The dna damage response (ddr) is induced by the c9orf72 repeat expansion in amyotrophic lateral sclerosis, Human molecular genetics 26 (15) (2017) 2882–2896.
  * [6] C. Milanese, S. Cerri, A. Ulusoy, S. V. Gornati, A. Plat, S. Gabriels, F. Blandini, D. A. Di Monte, J. H. Hoeijmakers, P. G. Mastroberardino, Activation of the dna damage response in vivo in synucleinopathy models of parkinson’s disease, Cell death & disease 9 (8) (2018) 1–12.
  * [7] N. M. Shanbhag, M. D. Evans, W. Mao, A. L. Nana, W. W. Seeley, A. Adame, R. A. Rissman, E. Masliah, L. Mucke, Early neuronal accumulation of dna double strand breaks in alzheimer’s disease, Acta neuropathologica communications 7 (1) (2019) 1–18.
  * [8] Y. Enokido, T. Tamura, H. Ito, A. Arumughan, A. Komuro, H. Shiwaku, M. Sone, R. Foulle, H. Sawada, H. Ishiguro, et al., Mutant huntingtin impairs ku70-mediated dna repair, Journal of Cell Biology 189 (3) (2010) 425–443.
  * [9] G. S. Jeon, K. Y. Kim, Y. J. Hwang, M.-K. Jung, S. An, M. Ouchi, T. Ouchi, N. Kowall, J. Lee, H. Ryu, Deregulation of brca1 leads to impaired spatiotemporal dynamics of $\gamma$-h2ax and dna damage responses in huntington’s disease, Molecular neurobiology 45 (3) (2012) 550–563.
  * [10] M. L. Ferlazzo, L. Sonzogni, A. Granzotto, L. Bodgi, O. Lartin, C. Devic, G. Vogin, S. Pereira, N. Foray, Mutations of the huntington’s disease protein impact on the atm-dependent signaling and repair pathways of the radiation-induced dna double-strand breaks: Corrective effect of statins and bisphosphonates, Molecular neurobiology 49 (3) (2014) 1200–1211.
  * [11] I. Castaldo, M. De Rosa, A. Romano, C. Zuchegna, F. Squitieri, R. Mechelli, S. Peluso, C. Borrelli, A. Del Mondo, E. Salvatore, et al., Dna damage signatures in peripheral blood cells as biomarkers in prodromal huntington disease, Annals of neurology 85 (2) (2019) 296–301.
  * [12] J. Illuzzi, S. Yerkes, H. Parekh-Olmedo, E. B. Kmiec, Dna breakage and induction of dna damage response proteins precede the appearance of visible mutant huntingtin aggregates, Journal of neuroscience research 87 (3) (2009) 733–747.
  * [13] M. Karbaschi, M. S. Cooke, Novel method for the high-throughput processing of slides for the comet assay, Scientific reports 4 (1) (2014) 1–6.
  * [14] G. Koppen, A. Azqueta, B. Pourrut, G. Brunborg, A. R. Collins, S. A. Langie, The next three decades of the comet assay: a report of the 11th international comet assay workshop, Mutagenesis 32 (3) (2017) 397–408.
  * [15] T. Lee, S. Lee, W. Y. Sim, Y. M. Jung, S. Han, J.-H. Won, H. Min, S. Yoon, Hicomet: a high-throughput comet analysis tool for large-scale dna damage assessment, BMC bioinformatics 19 (1) (2018) 49–61.
  * [16] S. Namuduri, B. N. Narayanan, M. Karbaschi, M. Cooke, S. Bhansali, Automated quantification of dna damage via deep transfer learning based analysis of comet assay images, in: Applications of Machine Learning, Vol. 11139, International Society for Optics and Photonics, 2019, p. 111390Y.
  * [17] Y. Lecun, Y. Bengio, G. Hinton, Deep learning (may 2015).  doi:10.1038/nature14539.
  * [18] I. Goodfellow, Y. Bengio, A. Courville, Deep learning, MIT press, 2016.
  * [19] K. Weiss, T. M. Khoshgoftaar, D. Wang, A survey of transfer learning, Journal of Big data 3 (1) (2016) 1–40.
  * [20] Y. Bengio, Deep learning of representations for unsupervised and transfer learning, in: Proceedings of ICML workshop on unsupervised and transfer learning, JMLR Workshop and Conference Proceedings, 2012, pp. 17–36.
  * [21] D. Cook, K. D. Feuz, N. C. Krishnan, Transfer learning for activity recognition: A survey, Knowledge and information systems 36 (3) (2013) 537–556.
  * [22] B. Tan, Y. Zhang, S. Pan, Q. Yang, Distant domain transfer learning, in: Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 31, 2017.
  * [23] S. J. Pan, Q. Yang, A survey on transfer learning, IEEE Transactions on knowledge and data engineering 22 (10) (2009) 1345–1359.
  * [24] N. Tajbakhsh, J. Y. Shin, S. R. Gurudu, R. T. Hurst, C. B. Kendall, M. B. Gotway, J. Liang, Convolutional neural networks for medical image analysis: Full training or fine tuning?, IEEE transactions on medical imaging 35 (5) (2016) 1299–1312.
  * [25] Y. Zhang, Q. Yang, A survey on multi-task learning, IEEE Transactions on Knowledge and Data Engineering (2021).
  * [26] E. Anarossi, R. D. Yanuaryska, F. U. Nuha, S. Mulyana, et al., Comet assay classification for buccal mucosa’s dna damage measurement with super tiny dataset using transfer learning, in: Asian Conference on Intelligent Information and Database Systems, Springer, 2019, pp. 279–289.
  * [27] R. Rosati, L. Romeo, S. Silvestri, F. Marcheggiani, L. Tiano, E. Frontoni, Faster r-cnn approach for detection and quantification of dna damage in comet assay images, Computers in Biology and Medicine 123 (2020) 103912.
  * [28] S. Lathuilière, P. Mesejo, X. Alameda-Pineda, R. Horaud, A comprehensive analysis of deep regression, IEEE transactions on pattern analysis and machine intelligence 42 (9) (2019) 2065–2081.
  * [29] J. Jiménez-Luna, F. Grisoni, G. Schneider, Drug discovery with explainable artificial intelligence, Nature Machine Intelligence 2 (10) (2020) 573–584.
  * [30] K. Simonyan, A. Zisserman, Very deep convolutional networks for large-scale image recognition, arXiv preprint arXiv:1409.1556 (2014).
  * [31] F. Chollet, Keras functional api, last accessed December 6th 2021 (2019).
URL https://keras.io/guides/functional_api/