11institutetext: Center for Nonlinear Science, University of North Texas, P.O.
Box 311427, Denton, Texas 76203-1427, USA

Dynamics of evolution Social and economic systems Decision theory and game
theory

# Imitation-Induced Criticality: Network Reciprocity and Psychological Reward

Korosh Mahmoodi  Paolo Grigolini

###### Abstract

The nodes of a regular two-dimensional lattice play a game based on the joint
action of two distinct levels. At the first step of the game, using a random
prescription half players are assigned the cooperation and half the defection
state. At the bottom level the strategy choice is done on the mere basis of
imitation according to the _homo imitans_ principle, generating a form of
collective intelligence that makes the system sensitive to the criteria
determining the strategy choice adopted at the top level. The units of the top
level, in fact, play the prisoner’s dilemma game and are allowed to update
their strategy either by selecting the strategy of the most successful nearest
neighbor, _success model_ , or merely on the basis of the criterion of the
best financial benefit, _selfishness model_. The intelligence emerging from
imitation-induced criticality leads in the former case to the extinction of
defection and in the latter case to the extinction of cooperation. The former
case is interpreted as a form of network reciprocity enhanced by the
imitation-induced criticality and contributing to the evolution towards
cooperation. We perturb the selfishness model with a form of morality
pressure, exerted by a psychological reward $\lambda$ for cooperation, to
establish the sensitivity of collective intelligence to morality. We find that
when $\lambda$ gets a crucial value $\lambda_{c}$, exceeding the temptation to
cheat, the system makes a transition from the supercritical defection state to
the critical regime, with the warning that an excess of morality and religion
pressure may annihilate the criticality-induced resilience of the system.

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##  1 Introduction

The unification of behavioral sciences is an attractive and challenging
enterprise that would be impossible without using game theory [1]. The recent
book by Gintis [1] aims at the unification of Behavioral Sciences, namely, at
the ambitious purpose of unifying biology, psychology, economics, anthropology
and political science, stressing however that game theory alone is not enough
to realize this important goal. Game theory is a theoretical attempt at
explaining why the selfish action of single individuals is compatible with the
emergence of altruism and cooperation. Nowak and May [4] have attracted the
attention of an increasing number of researchers on the network reciprocity, a
special condition where spatial structure favors the emergence of altruism,
see also [2], in spite of its nodes playing the prisoner’s dilemma game [3],
which is expected to favor defection. Actually, since the patch owner, either
a cooperator $C$ or a defector $D$ is replaced by the neighbor with the
largest payoff [4], this is equivalent to updating the strategy of each player
adopting that of the most successful neighbor, and a cluster of cooperators
has the effect of protecting the units of the cluster from the exploitation of
the defectors.

As illustrated in the recent review paper of Wang _et al_ [5], the research
work in this field is now focusing on the topology of networks and especially
on the emerging field of multilayers networks to explain the pattern formation
with production of clusters protecting cooperators from the exploitation of
the defectors so as to favor their survival of cooperators. This Letter adopts
a multilayers perspective, by using however _dynamical_ rather than merely
topological arguments, in such a way as to be as close as possible to the
project of unification of behavioral sciences recently proposed by Grigolini
_et al_ [6, 7], as an attempt at addressing the challenge of Gintis [1].
Furthermore the multiple layers do not necessarily correspond to different
nodes, but in this Letter refer to different levels of human behavior, the
social, the financial and the spiritual. The bottom layer of this Letter is
based on the observation that the individuals of a network playing the
prisoner’s dilemma game are the units of a human society and are expected to
be strongly influenced by imitation [8, 9, 10]. The individuals of this
network make a choice between the cooperation state $C$ and the defection
state $D$, without any form of cognition. This choice is not determined by the
wish of maximizing the personal benefit through imitation of the most
successful neighbor [2] or by the greedy choice of an immediate payoff, but
imitation is as blind as the bird tendency to select their flying direction on
the basis of the flying directions of the neighbor birds [11], with no
consideration of the personal benefit, either direct or indirect. The
imitation strength is a control parameter hereby denoted by the symbol
$K_{1}$. A critical value of $K_{1}$, called $K_{1c}$, exists making it
possible for the swarm to fly as a whole. Although the action of the single
individuals of the network does not require any form of cognition, as stressed
by the authors of Ref. [12, 13], criticality generates a form of collective
intelligence. This collective intelligence is characterized not only by the
criticality-induced long-range correlation but also, and especially, by
temporal complexity [14, 15, 16, 17], a condition making the complex network
flexible and resilient. It is important to stress that the supercritical
condition is characterized by fluctuations around a non vanishing mean field
as random as the fluctuations around the vanishing mean field of the
subcritical regime, thereby lacking the flexibility and resilience of
criticality.

At the top layer the units play the prisoner’s dilemma game [3] and exert an
influence on the bottom level choosing their strategy according to either the
success or selfishness criterion.

The _success model_ is realized as follows. For most of time steps the choice
of strategy is determined by the bottom level, and from time to time the units
are allowed to select their strategy adopting that of their most successful
nearest neighbor, as suggested by the pioneer work of Ref. [4]. The success
model yields the impressive effect of annihilating the emergence of the branch
with the majority of defectors when the system adopting this choice is made
intelligent by imitation-induced criticality. We are convinced that this
effect affords a solid explanation of the evolution towards cooperation, this
being a very important result of this Letter.

We compare the _success model_ to the _selfishness model_ , where the player
does not adopt the strategy of the most successful nearest neighbor, but she
makes her choice only on the basis of her personal benefit. She evaluates the
financial benefit derived from the defection choice and the financial benefit
that she would get from the cooperation choice, giving larger weight to the
maximal profit. The benefit of a given choice is done assuming that the unit
in action can play with equal probability with all her neighbors. The
selection of the convenient strategy is weighted with a second control
parameter $K_{2}$, with the ratio $\rho_{K}=K_{2}/K_{1}$ establishing if the
link with the top layer is stronger, $\rho_{K}>1$, or weaker, $\rho_{K}<1$,
than the link with the bottom layer. Switching on the interaction with the top
layer has the effect of leading to the extinction of cooperation with a big
loss for society, even if, as we shall see hereby, $K_{1}>0$ yields imitation-
induced clusters of cooperators with financial benefit for society, this being
the reason why the _success model_ , for $K\gg K_{1c}$ leads to the extinction
of defectors. The strategy choice determined by the criterion of maximal
personal benefit, rather than by the choice of the strategy of the most
successful nearest neighbor [4], on the contrary, yields the extinction of
cooperation, with an even very small value of $\rho_{K}$, as an effect of
imitation-induced intelligence.

To complete the illustration of the role of criticality-induced intelligence
we study the influence of morality on the dynamics of the _selfishness model_
showing that as an effect of imitation-induced intelligence the system becomes
so sensitive to morality as to make a psychological reward moderately
exceeding the temptation to cheat, robust enough as to prevent the collapse of
the social system into the sub-criticality disorder. In principle the
influence of morality on the system should be established by the interaction
of the network with an additional layer. For simplicity’s sake, we modify the
conventional prisoner’s dilemma game [1, 18] through the introduction of the
psychological reward for the choice of cooperation. The strength of
psychological, called $\lambda$, affords a simplified way to describe the
influence that an additional layer, concerning morality and religion, may have
on the selfishness model. We find that, when the bottom layer operates at
criticality, a crucial value $\lambda_{c}$ exists with the effect of
preventing the extinction of cooperators and of recovering the criticality-
induced temporal complexity that is essential for the healthy behavior of the
social system. We call this _Asbiyyah_ effect [19], this arabic world meaning
_group feeling_ , namely the natural tendency of human beings to cooperate.
This natural disposition is enhanced by religion and it has the eventual
effect of increasing the social prosperity, but, in accordance with the
observation of Ahmed [20], we find that $\lambda>\lambda_{c}$, the super-
asbiyyan condition, may be as bad as the lack of social cohesion.

##  2 Game theory

As well known [3], game theory rests on the crucial inequality

$T>R>P>S,$ (1)

where $R$ denotes the reward that a cooperator gets when playing with another
cooperator. The parameter $T>R$ is the payoff of a defector: it is larger than
$R$ thereby $t=T-R$ is a measure of the temptation to cheat. $S$ is the payoff
of a cooperator playing with a defector, and $P<R$ is the payoff of a defector
playing with another defector. The condition

$2R>T+S$ (2)

indicates that the community gets a larger benefit from the play between two
cooperators than from the play between a cooperator and a defector. Of course,
also the play between two defectors with $2R>2P$ is less beneficial to the
community than the play between two cooperators. We adopt the choice made by
Gintis [1, 18] and we set $R=1$, $P=0$, $T=1+t$ and $S=-s$. To study the
influence of morality on the selfishness model we introduce the psychological
reward $\lambda$, setting

$R=1+\lambda,$ (3) $T=1+t,$ (4) $P=-\lambda$ (5)

and

$S=-s.$ (6)

In this Letter we adopt always but in Fig. 4 the choice

$s=3$ (7)

and

$t=2.$ (8)

Note that we denote by $N$ the total number of players. They are the nodes of
a two-dimensional regular network of size $32$. Therefore throughout the whole
Letter we adopt $N=32X32$. We use the symbol $L$ to denote the number of time
steps, ranging from $L=10^{4}$ to $L=10^{6}$.

##  3 Decision-Making, Success, Selfishness and Influence of Morality Model

To establish an interaction between the bottom and the top level we adopt a
natural extension of the _Decision Making Model_ (DMM) of Ref. [8]. The
transition rate from the cooperator to the defector state, $g_{12}$, is given
by

$g_{12}=g_{0}exp\left[-K_{1}\left(\frac{M_{1}-M_{2}}{M}\right)-K_{2}\left(\frac{\Pi_{C}-\Pi_{D}}{|\Pi_{C}|+|\Pi_{D}|}\right)\right]$
(9)

and the transition rate from the defector to the cooperator state, $g_{21}$,
is given by

$g_{21}=g_{0}exp\left[K_{1}\left(\frac{M_{1}-M_{2}}{M}\right)+K_{2}\left(\frac{\Pi_{C}-\Pi_{D}}{|\Pi_{C}|+|\Pi_{D}|}\right)\right].$
(10)

The meaning of this prescription is as follows. The parameter $1/g_{0}$
defines the time scale and we set $g_{0}=0.1$ throughout. We consider $N$
units. Each unit has $M$ neighbors (four in the case of the regular two-
dimensional lattice used in this Letter). The cooperation state corresponds to
the state $|1>$ and the defector state to the state $|2>$. In the case
$K_{2}=0$, this is the ordinary DMM of Ref. [8]. If the unit is in the
cooperator state, $|1>$, and the majority of its neighbors are in the same
state, then the transition rate becomes smaller and the units sojourns in the
cooperation state for a longer time. If the majority of its neighbors is in
the defector state, then the unit that has to make a decision sojourns in the
cooperator state for a shorter time. Analog prescription is used if the unit
is in the defector state. Note that at $K=K_{1c}$ the units move from a
dynamical condition where they are virtually independent the ones from the
others to a condition where global order emerges. In the case of two-
dimensional regular lattice of this article, ${K_{1}}_{c}\approx 1.65$.

[width= 0.4 ]Fig1.jpg

Figure 1: Mean field of the bottom level when $K_{2}=0$. We make 10
realizations.

Fig. 1 illustrates the second-order phase transition generated by the DMM,
namely, the model of this Letter when $K_{2}=0$. This corresponds to the
condition where no bias exists for the choice of either cooperation or
defection. In the initial condition the $N$ units of the network are randomly
assigned to either the cooperation or the defection state with the same
probability, making zero the mean field of the network

$\left<\xi\right>\equiv\frac{\sum_{i}^{N}\xi_{i}}{N},$ (11)

where either $\xi_{=}1$ or $\xi=-1$, according to whether the $i-th$ unit is
in the cooperation or in the defection state. Of course, due to the fact that,
as earlier stated, we are using a finite number of units, $N=1024$, the mean
field fluctuates around the vanishing mean value. As we increase $K_{1}$ the
intensity of fluctuations tends to increase. This is a finite-size effect
discussed in detail in the applications of the DMM model [14, 15]. In the case
where each node is coupled to all the other nodes, the DMM yields analytical
results for the mean field that can be interpreted as the space coordinate of
a particle in a non-linear over-damped potential under the action of a random
fluctuation generated by the finite size effect. At criticality, the potential
is quartic, exerts a weaker containment on diffusion and makes larger the
width of its equilibrium distribution. The mean field fluctuation around the
origin has an inverse power law with index $\mu=1.5$: a form of temporal
complexity ensuring the maximal efficiency in the transport of information
from one network to another with the same complexity[16]. This important
effect has been proved also in the case of a neural network model [17],
thereby suggesting that criticality-induced temporal complexity may be a
condition of great importance for the sensitivity of a complex system to its
environment. Temporal complexity must not be confused with critical slowing
down [15]. Both properties are manifestations of criticality and both
properties are characterized by the emergence of an inverse power law, which
makes the survival probability not integrable. However, critical slowing down
is a property of the thermodynamic limit, implying that the number of units
$N$ is virtually infinite, whereas temporal criticality is a finite size
effect [21]. Temporal complexity, in the ideal case where the inverse power
law is not truncated has to be thought of as a form of perennial out of
equilibrium condition, which should force physicists to extend the linear
response theories to the non-ergodic condition [7].

Note that the cooperator and the defector payoffs are determined by the states
of the nearest neighbors. Thus, we have

$\Pi_{C}=(1+\lambda)\frac{M_{C}}{M}-s\frac{M_{D}}{M}$ (12)

and

$\Pi_{D}=(1+t)\frac{M_{C}}{M}-\lambda\frac{M_{D}}{M},$ (13)

where $M_{C}$ is the numbers of neighbors in the cooperative state and $M_{D}$
is the number of neighbors in the defector state. We remind the readers that
in this Letter $M=4$. It is important to state that we evaluate also the
financial benefit for the community by making an average over all possible
pairs of interacting units, according to the prescription:

$B_{ij}=2,$ (14)

if both units of the pair $(i,j)$ are cooperators,

$B_{i,j}=1+t-s,$ (15)

if one unit of the pair $(i,j)$ is a cooperator and the other is a defector,

$B_{i,j}=0,$ (16)

if both units of the pair $(i,j)$ are defectors.

Notice that the choice between the cooperation and defection state is done
with $\lambda\geq 0$, while the financial benefits for the society are
evaluated, as shown by Eqs. (14), (15) and (16), by setting $\lambda=0$. This
is so because $\lambda$, the psychological reward, is an incentive to
cooperate that does not directly increase the social payoff, even if it has
the eventual effect of increasing the society wealth by stimulating
cooperation. In conclusion, the societal benefit $\Pi$ is given by

$\Pi=\sum_{(i,j)}B_{ij},$ (17)

denoting the sum over all possible pairs $(i,j)$.

To define the _selfishness model_ we set $\lambda=0$ and we establish the
interaction between the bottom and the top level by assigning a positive value
to the coupling coefficient $K_{2}$. In this case, a cooperator is encouraged
to adopt the cooperator state for a longer time if the financial benefit
associated the choice of the cooperator condition, $\Pi_{C}$, is larger than
the financial benefit $\Pi_{D}$, corresponding to selecting the defector
state. The _success model_ is established by setting both $\lambda=0$ and
$K_{2}=0$. The influence of the top on the bottom level is established by
randomly selecting a fraction $r$ of the total number $L$ to allow each unit
to adopt the strategy of the most successful nearest neighbor. We note that
small values of $r$ play the same role as small values of $\rho_{K}$ in the
selfishness model. The influence of morality on the selfishness model is
studied by setting $\lambda>0$, while keeping $\lambda=0$ for the evaluation
of the social benefit, as earlier stated.

[width= 0.4 ]Fig2.jpg

Figure 2: Social benefit as a function of $K_{1}$ when $K_{2}=0$. We make ten
realizations. We notice that the social benefit increases upon increase of
$K_{1}$ in the whole subcritical region.

[width= 0.4 ]Fig3.jpg

Figure 3:  The lines of the triangles represents the social benefit. The line
of black squares denotes the number of units with four neighbors in the same
state.

##  4 Criticality-induced network reciprocity

In Fig. 2 we show the social benefit as a function of $K_{1}$ when $K_{2}=0$,
namely, the strategy choice is only determined by imitation. The social
benefit in the supercritical regime is obviously maximal when all the units
select the cooperation strategy and it vanishes when all the units select
defection. Much more interesting is the social benefit for $K<K_{1c}$. We see
that there is an increase of social benefit with increasing $K_{1}$. This is a
consequence of the fact that imitation generates clusters of cooperators and
clusters of defectors and the increasing social benefit is due to the
increasing size of the clusters of cooperators.

The results of Fig. 3 support our claim. In fact, we see that the social
benefit increases in a way that is qualitatively very similar to the increase
of the number of units that have four neighbors in the same state, either
cooperation or defection state. A cluster of units belonging to the same state
increases as a function of $K_{1}$ with a prescription qualitatively similar
to the increase of the number of units with four neighbors in the same state.

[width= 0.4 ]Fig4.jpg

Figure 4: The mean field of the imitation model as a function of $K_{1}$
randomly selecting $0.1\%$ of $L$ steps to update the strategy of each unit by
adopting the strategy of the most successful nearest neighbor.

We now adopt the _success model_ , namely, we perturb the imitation-induced
strategy choice making the units pay some attention to the success of their
nearest neighbors. We run the model for $L$ time steps, and we randomly select
$1\%$ of them to update their strategy adopting the one of their most
successful nearest neighbor. As earlier stated, we use a two-dimensional
regular network, where each unit has $4$ neighbors. The results illustrated by
Fig. 4 are impressive. The adoption of the strategy of the most successful
nearest neighbor has a very modest effect in the subcritical region. At
criticality, on the contrary, the effects of this choice become macroscopic
and the imitation-induced phase transition, when the two branches, one with a
majority of cooperator and one with a majority of defectors, are generated
with equal probability, the success model selects the branch with a majority
of cooperators. At very large values of $K_{1}$ this leads to the extinction
of defectors. On the basis of the results done by our group on the DMM
dynamics we make the very plausible conjecture that this effect is independent
of the topology of the adopted network. In fact, moving from one topology to
another has only the effect of reducing the intensity of the effort necessary
to get consensus, the condition $K_{1}=1$ representing the ideal topology
requiring the weakest effort to get consensus [8].

##  5 Morality stimulus on the selfishness model at criticality

As earlier mentioned the parameter $\lambda$ is only a psychological benefit
implying no direct financial benefit for society. We interpret $\lambda$ as
the strength of the influence that the morality network has on the selfishness
model.

[width= 0.4 ]Fig5.jpg

Figure 5:  The mean field of the selfishness model as a function of $K_{2}$
for different values of $K_{1}$. We make ten realizations.

Here we find the imitation-induced intelligence has a twofold effect. The
collective intelligence is proved to lead to the abrupt extinction of
cooperators, if $\lambda=0$, and to the abrupt extinction of defectors when
$\lambda=\lambda_{c}$. With the values of the parameters adopted in this
Letter, $\lambda_{c}=2.5$. Fig. 5, illustrating the effect of a strategy
choice when $\lambda=0$, shows that in the sub-critical regime a very large
value of $\rho_{K}$ is required for the extinction of cooperation. As we
approach criticality, namely as the system becomes more and more intelligent,
an even very weak interest for the personal financial benefit leads to the
extinction of cooperators, which, in fact, is shown to occur for
$\rho_{K}\approx 0.06$ when $K_{1}=1.6$.

[width= 0.4 ]Fig6.jpg

Figure 6:  The personal benefit of the selfishness model as a function of
$\lambda$. We make an average on ten realizations.

The swarm intelligence of the system makes the model very sensitive to the
influence of morality, as it is clearly shown by Fig. 6. We see, in fact, that
as consequence of the imitation-induced criticality, the average social
benefit undergoes a kind of first-order transition at $\lambda_{c}=2.5$, with
a jump from the lack of benefit to a very large value, when $K_{1}=1.65$ and
$\rho_{K}\approx 0.12$. This indicates that the criticality-induced
intelligence wisely turns the psychological reward for the choice of
cooperation into a significant social benefit.

However, a value of $\lambda$ too large may have the same bad effects as a
value of $\lambda$ too small. The corresponding cumulative probability is
illustrated by Fig. 7, which shows that the distribution density of the time
distances between two consecutive renewal events is an inverse power law with
the power index $\mu=1.5$ when $\lambda=\lambda_{c}=2.5$ and it is an
exponential function for both $\lambda<\lambda_{c}$ and $\lambda>\lambda_{c}$.
This is a clear sign that the collective intelligence generated by criticality
[14, 15, 16, 17] is lost if the moral incentive to altruism $\lambda$ is
either weak or excessive.

[width= 0.4 ]Fig7.jpg

Figure 7:  The cumulative probability of the time distances between two
consecutive regressions to the origin.

##  6 Concluding Remarks

In conclusion we have proved that imitation-induced criticality has the effect
of enhancing the phenomenon of network reciprocity. The adoption of the
strategy of the most successful nearest neighbor not only protects the
cooperator from extinction, as in the pioneer work of Nowak and May [4], but,
at criticality, it annihilates the branch with the majority of defectors. If
we adopt the selfishness model for the choice of strategy, the imitation-
induced criticality has the effect of favoring the extinction of cooperators.
Under the influence of morality stimulus, however, imitation-induced
criticality has the opposite effect of leading to the extinction of defectors.
However, the temporal complexity of the system is lost for both
$\lambda<\lambda_{c}$ and $\lambda>\lambda_{c}$, indicating, in accordance
with Ahmed [20], that a condition of super-asabiyya is detrimental for human
society as the lack of asabiyya.

###### Acknowledgements.

The authors warmly thank ARO for support through Grants No. W911NF-11-1-0478.

## References

  * [1] H. Gintis, _The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences_ , Princeton University Press, revised version (2014).
  * [2] M. A. Nowak, _Five Rules for the Evolution of Cooperation_ , Science, 314, 1560 (2006).
  * [3] R. Axelrod, _The Evolution of Cooperation_ , revised edition, Basic Books (2006).
  * [4] M. A. Nowak, R. M. May, _Evolutionary games and spatial chaos_ , Nature 359, 826-829 (1992).
  * [5] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Eur. Phys. J. B 88, 124 (2015).
  * [6] P. Grigolini, N. Piccinini, A. Svenkeson, P. Pramukkul, D. Lambert, B. J. West, Front. Bioeng. Biotechnol. Published on 16 Jun 2015, doi:10.3389/fbioe.2015.00078
  * [7] P. Grigolini, Chaos, Solitons and Fractals, in press (2015).
  * [8] B. J. West, M. Turalska, P. Grigolini, _Networks of Echoes, Imitation, Innovation and Invisible Leaders_ , Computational Social Sciences, Springer, New York (2014).
  * [9] L. Herrero, _Homo Imitans_ , meetingminds, UK (2010).
  * [10] A. Pentland, _Social Physics: How Good Ideas Spread-The Lesson from a New Science_ , The Penguin Press, New York (2014).
  * [11] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
  * [12] F. Vanni, M. Luković, P. Grigolini, Phys. Rev. Lett. 107, 078103 (2011).
  * [13] S. Rosenfeld, Cancer informatics, 12, 221-229 (2013); Gene Regulation and System Biology, 7, 23-39 (2013).
  * [14] A. Svenkeson, M. Bologna, P. Grigolini, Phys. Rev. E, 86, 041145 (2012).
  * [15] M. T. Beig, A. Svenkeson, M. Bologna, B. J. West, and P. Grigolini, Physical Review E 91, 012907 (2015).
  * [16] M. Lukovic, F. Vanni, A. Svenkeson, P. Grigolini, Physica A, 416, 430-438 (2014).
  * [17] J. U. Mafahim, D. Lambert, M. Zare, P. Grigolini, New Journal of Physics, 17, 015003, 1-17 (2015).
  * [18] H. Gintis, _Game Theory Evolving_ , second edition, Princeton University Press, Princeton and Oxford (2009).
  * [19] Ibn Khaldun, The Muqaddimah, An Introduction to History, Tr. Franz Rosenthal, Bollingen Series XLIII. Princeton University Press, Princeton, 1967 3 Vols.
  * [20] A. S. Ahmed, _Islam under siege: Living dangerously in a post-honor world_ , Polity Press, Cambridge, UK (2002).
  * [21] P. Pramukkul, A. Svenkeson, B. J. West, P. Grigolini, Europhys. Lett. 111, 58003 (2015).


# Alzheimer’s Dementia Detection from Audio and Text Modalities

###### Abstract

Automatic detection of Alzheimer’s dementia by speech processing is enhanced
when features of both the acoustic waveform and the content are extracted.
Audio and text transcription have been widely used in health-related tasks, as
spectral and prosodic speech features, as well as semantic and linguistic
content, convey information about various diseases. Hence, this paper
describes the joint work of the GTM-UVIGO research group and acceXible startup
to the ADDReSS challenge at INTERSPEECH 2020. The submitted systems aim to
detect patterns of Alzheimer’s disease from both the patient’s voice and
message transcription. Six different systems have been built and compared:
four of them are speech-based and the other two systems are text-based. The
x-vector, i-vector, and statistical speech-based functionals features are
evaluated. As a lower speaking fluency is a common pattern in patients with
Alzheimer’s disease, rhythmic features are also proposed. For transcription
analysis, two systems are proposed: one uses GloVe word embedding features and
the other uses several features extracted by language modelling. Several
intra-modality and inter-modality score fusion strategies are investigated.
The performance of single modality and multimodal systems are presented. The
achieved results are promising, outperforming the results achieved by the
ADDReSS’s baseline systems.

Index Terms: Alzheimer’s disease detection, i-vector, x-vector, speech
fluency, word embedding, recurrent neural networks, score level fusion,
ADDReSS challenge

##  1 Introduction

The Alzheimer’s disease (AD) is a neurodegenerative illness that represents
the most common cause of dementia in the world. It is provoked by the damage
of neurons involved in thinking, learning and memory. This disease has three
main stages. In the first one, called preclinical, patients do not present
clear symptoms because the brain initially compensate for them, enabling
individuals to continue to function normally. The second one is defined as
Mild Cognitive Impairment (MCI). In this stage, patients show greater
cognitive decline then expected for their age, having problems to express and
connect ideas. However, these changes may be only noticeable to family members
and friends. The critical phase is the last one, which is called as dementia.
It is characterized by noticeable memory, thinking and behavioural symptoms
that impair a person’s ability to function in daily life [1][2].

Common signs of AD are related to problems with uttering words; consequently,
people with AD may have trouble following or joining a conversation, they may
stop in the middle of a sentence and have no idea how to continue. As a
result, analysis of speech and its transcription may represent a suitable
mechanism for detecting the AD during the second or third stage of the disease
[1] [3]. According to the literature[4][5][6], using information from the
patient’s voice, as well as from its transcription would ease the early AD
detection task.

In this work, four speech-based systems and two text-based systems are
proposed for automatic distinction between patient’s with and without AD.
Those based on the speech signal are compound by four approaches to represent
their spectral and prosodic content, namely: i-vector and x-vector embeddings,
rhythmic features, and statistical functionals. The first three use super
vector machine (SVM) as classifiers and the last one uses a linear
discriminant analysis (LDA) classifier. As for systems that use speech
transcriptions, one relies on GloVe word embedding features [7] using a
recurrent neural network (RNN) as classifier, and the other one is based on
features extracted by language modelling, using a SVM as classifier. Finally,
an intra-modality and inter-modality score fusion strategy was done to improve
the final results.

All the individual systems, and its fusion, are evaluated on the AD
recognition task within the framework of the ADReSS Challenge [8]. This
challenge targets the AD detection using spontaneous speech. The data used in
the Challenge consists of speech recordings, and their transcripts,
corresponding to the description of the Cookie Theft picture. Specifically, it
is a selection of Alzheimer and control patients from of the DementiaBank’s
Pitt corpus 111http://dementia.talkbank.org/.

The rest of the paper is organized as follows. Sections 2 and 3 describe the
speech-based and text-based systems, respectively. Section 4 outlines the
experimental framework. The experimental results are exposed and discussed in
Section 5. Finally, Section 6 draws some conclusions and future work.

##  2 Speech-based systems

In this section, a description of speech feature extraction strategies and
classification methods is done, with special attention to different
statistical features extracted from the patient’s voice.

###  2.1 Speech embedding features

Speech embedding features are considered as state-of-art speech representation
for speaker recognition application. These speech representations can be
applied for AD detection, as long as they preserve those spectral patterns in
the speaker’s voice that allow the distinction between patient with and
without AD. In this paper, two strategies were analyzed, the first one uses
the i-vector paradigm [9], and the second one uses an x-vector [10] based
representation. The main characteristics of these approaches are briefly
described below.

  * A.

_i-vectors_

To extract the i-vectors an universal background model (UBM) and a total
variability matrix $\mathbf{T}$ model must be trained. As speech
parameterization these models use 13 perceptual linear prediction (PLP)
cepstral coefficients, combined with two pitch-related features (F0 and
voicing probability) [11]. This features are augmented with their delta and
acceleration coefficients, leading to vectors of dimension 45. These features
were chosen since that combination achieves a representation of speech that
includes spectral information and prosodic features such as rhythm or
intonation that are embedded in the fundamental frequency. The UBM was a
diagonal covariance, 512-component gaussian mixture model (GMM), and was
trained with data from outside this task; and $\mathbf{T}$ was trained using
the task training data. The dimension of the i-vectors was set to 125 and they
were length normalized.

  * B.

_x-vectors_

A pretrained time delay deep neural network (TDNN) 222http://kaldi-
asr.org/models.html, with 5 time delay layers and two dense layers, and
trained to discriminate between speakers, was used. The network was
implemented using the nnet3 neural network library in the Kaldi Speech
Recognition Toolkit [12] and trained on augmented VoxCeleb 1 and VoxCeleb 2
datasets [13]. The input to the TDNN are 30 mel-frequency cepstral
coefficients (MFCC), and the embeddings are extracted from the first dense
layer with a dimensionality of 512. The output of this layer (x-vector) is
first projected using latent discriminant analysis (LDA) into a 200
dimensional space and then length normalized.

Instead of extracting an i-vector or x-vector (embeddings) to represent the
entire audio signal, a set of these vectors is obtained applying a sliding
window. In this way, each audio file is represented by a certain number of
embeddings, which are then used for classification.

Both systems use as classifier an SVM with a linear kernel. Since a number of
embeddings are extracted from each audio, there will also be a set of
classification results (one for each embedding), which must be combined to
obtain a patient’s classification in AD or non-AD. In this work, the mean of
the classifications was used as score for the final decision.

###  2.2 Statistical functionals

This system represents the audio waveform with a set of vectors called
functionals. These are mainly statistical and regression features extracted
from a wide variety of low-level acoustic attributes using the openSMILE [14]
software. Specifically, the set of functionals proposed in the AVEC 2013
evaluation (Audiovisual Emotion Recognition Challenge and Depression) [15] was
used, which consist of 2,268 descriptors extracted from 32 low-level features
of the audio signal related to energy, the spectrum, volume and tone.

As the number of descriptors is very high in relation to the amount of data
available, a selection of those most relevant was done. For this, a
correlation based feature subset selection (CFS) algorithm [16] was used,
resulting in 57 functionals related with the low level descriptors shown in
Table 1. The functionals were extracted using a sliding window of 3 s and an
overlap of 1 s. As with the i-vectors and x-vectors this system uses an SVM
with a linear kernel as classifier.

Table 1: Low level descriptors in the set of 57 functionals Energy and
spectral features
---
Loudness, MFCC and delta-MFCC,
energy in bands 250–650Hz and 1 kHz–4 kHz,
25% spectral roll-of points,
spectral flux, entropy, skewness,
psychoacousitc sharpness, harmonicity, flatness
Voicing related
$F_{0}$,voicing probability, jitter

###  2.3 Speech fluency

A common pattern in patient with AD is the lack of speaking fluency, being the
rhythm a viable clue to detect that behavior. However, prosodic information
(e.g., mean energy) was also used because the correct pronunciation of words
does not only depends on the rhythm but intonation, tone and stress. For this
system, the selected parameters were based on [4] [17], and they are as
follows:

  * •

Number of syllables

  * •

Rate of speech (syllables / original duration)

  * •

Speaking duration

  * •

Average fundamental frequency

  * •

Median of fundamental frequency

  * •

Minimum fundamental frequency

  * •

Pronunciation posterior probability

  * •

Average voice interval duration

  * •

Average duration of pairs 333Pairs: consecutive voiced and unvoiced segments

  * •

Mean energy

  * •

The ratio between the energy mean and its standard-deviation

The extraction process was done using the Python library My-Voice Analysis
444https://github.com/Shahabks/my-voice-analysis, the voice activity detection
of the SIDEKIT software [18], and our own algorithms.

The classification process was done by the LDA algorithm, projecting the
rhythmic feature vector into an one-dimensional space where every projected
point represents a classification score.

##  3 Text-based systems

Two different text-based approaches were analysed. The first one uses the
manual transcripts to extract linguistic information for creating the input
features of a classifier. The second one is a sequential model based on deep
learning, which classifies directly from the sequence of GloVe word
embeddings.

###  3.1 Text preprocessing

The transcripts contained in the dataset are in CHAT format [19], which
facilitates speech annotation and analysis. They include the transcript of
both the subject and the investigator in charge of the test, as well as
additional non-speech annotations: times, researcher interventions, errors,
morphological or syntactic analysis.

Transcripts are divided into several interventions, i.e. sentences or parts of
complete sentences with meaning, and this granularity has been maintained in
the preprocessing. Metadata, researcher interventions and linguistic analysis
included in the files have been removed. We keep only subject’s words as input
for the classifiers, as it better reflects the real-world task of detecting
Alzheimer’s based only on speech analysis.

###  3.2 Sequential model

The first approach consists of training a RNN at intervention level, by
classifying if a given sentence belongs to an AD subject. Then, the
probability that the subject had AD given all his/her interventions is
computed as the mean of the probabilities of each intervention. By using the
interventions as independent samples the training dataset is increased to a
size of 1492 records from the 108 subjects. In the training process, words are
tokenized, and interventions are padded up to 20 tokens.

The RNN is composed of 3 layers: an Embedding layer with pre-trained weights
of GloVe 50-dimensional representation of words, a long short term memory
(LSTM) layer with 4 units and an output layer with 1 unit and sigmoid
activation. Additionally, a Dropout at 10% rate is performed in the Embedding
and LSTM layers for regularization and to prevent overfitting. The model is
trained for 10 epochs, using a batch size of 16 and Adam optimization.

###  3.3 Model based on linguistic features

In this approach, several linguistic features and indicators are built from
subject’s interventions and, using this feature vector as input, an SVM is
trained. Unlike the previous method, the classification here is performed at
subject level, by considering the full transcript of each participant.

Previous works [20] have shown that certain linguistic features are useful for
detecting AD using Cookie Theft test. Here 13 features have been built,
grouped into 4 categories:

  * •

Extension information such as the number of interventions, number of words per
intervention and mean word length.

  * •

Vocabulary richness, by measuring the number of unique words used by the
subject.

  * •

Presence of key informational concepts: kitchen, mother, stool, boy and girl.

  * •

Frequency of verbs, nouns, adjectives and pronouns from POS-tagging.

Each feature is then rescaled with min-max normalization in range [0, 1] and
an SVM with radial basis function (RBF) kernel and C = 1.0 is trained, whose
output is the probability that the subject had AD given the 13-dimensional
feature vector.

##  4 Experimental framework

The training dataset [8] consists of the recordings and manual transcripts of
108 subjects performing the test known as Cookie Theft, whose objective is to
describe an image. Out of the 108 participants, 54 are patients diagnosed with
Alzheimer’s. Table 2 shows the training data distribution in detail.

Table 2: ADReSS training dataset

| AD | non-AD
---|---|---
Age interval | Male | Female | Male | Female
$[50,55)$ | 1 | 0 | 1 | 0
$[55,60)$ | 5 | 4 | 5 | 4
$[60,65)$ | 3 | 6 | 3 | 6
$[65,70)$ | 6 | 10 | 6 | 10
$[70,75)$ | 6 | 8 | 6 | 8
$[75,80)$ | 3 | 2 | 3 | 2
Total | 24 | 30 | 24 | 30

The evaluation metrics for the AD classification task are:
$Accuracy=\frac{TN+TP}{N}$, Precision $\pi=\frac{TP}{TP+FP}$, Recall
$\rho=\frac{TP}{TP+FN}$ and $F_{1}=2\frac{{\pi}*{\rho}}{{\pi}+{\rho}}$.

where N is the number of patients, TP, FP and FN are the number of true
positives, false positives and false negatives, respectively.

All the systems have been trained using leave-one-subject-out (LOSO) cross-
validation strategy for measuring the generalization error. Therefore, models
use 107 subjects as training data and are validated on the held-out subject.

##  5 Results

This section presents the results in the AD classification for both the leave-
one-subject-out (LOSO) and test settings of the ADReSS challenge.

###  5.1 Results for LOSO setting

Table 3 illustrates the individual results achieved by the four speech-based
systems. These results show that the x-vectors, functionals, and fluency based
systems have similar performance regarding the accuracy, as well as Area Under
Curve (AUC). The i-vector was the weakest model, being the only one with an
accuracy under 70%, although it is still above the challenge baseline results
[8].

Table 3: Results of the proposed speech-based systems

| class | Precision | Recall | F1 Score | Accuracy | AUC
---|---|---|---|---|---|---
i-vector | non-AD | 0.7058 | 0.6666 | 0.6857 | 0.6944 | 0.6806
AD | 0.6842 | 0.7222 | 0.7027
x-vector | non-AD | 0.7400 | 0.6851 | 0.7115 | 0.7222 | 0.7615
AD | 0.7068 | 0.7592 | 0.7321
functionals | non-AD | 0.7454 | 0.7592 | 0.7522 | 0.7500 | 0.7435
AD | 0.7547 | 0.7407 | 0.7476
fluency | non-AD | 0.7450 | 0.7037 | 0.7238 | 0.7314 | 0.7613
AD | 0.7192 | 0.7592 | 0.7387

The results for both text-based systems are shown in the table 4. Concerning
the RNN model, the AUC in held-out set is 0.8563. For a selected threshold
probability, we also obtain an accuracy of 0.7407, recall of 0.8704 and
F1-score of 0.7705. The false negative rate is 13%. For the linguistic model,
the AUC in held-out set is 0.7510. For a selected threshold probability the
accuracy is 0.6852, recall of 0.7037 and F1-score of 0.6909. The false
negative rate is 30%.

Table 4: Results of the proposed text-based systems

| class | Precision | Recall | F1 Score | Accuracy | AUC
---|---|---|---|---|---|---
RNN model | non-AD | 0.7424 | 0.9074 | 0.8166 | 0.7962 | 0.8563
AD | 0.8809 | 0.6851 | 0.7708
Linguistic model | non-AD | 0.7017 | 0.7407 | 0.7207 | 0.7129 | 0.7510
AD | 0.7254 | 0.6851 | 0.7047

Moreover, two different score fusion strategies were carried out. In the first
one (referred as Fusion I), the scores of every system was normalized by
z-norm, and merged by the average. In the second one (referred as Fusion II),
the same normalization strategy was used and the average was computed on the
speech-based and text-based system scores separately. Finally, the new scores
were again merged by the average. The figure 2 shows the ROC curves of the
described fusion systems. The Fusion I model had an accuracy of 0.8056, a
F1-score of 0.8073, and an AUC of 0.8930. On the other hand, the Fusion II
model presented an accuracy of 0.8241, a F1-score of 0.8190 and an AUC of
0.9023. The results show that the fusion of both text-based and speech-based
modality improves the detection of AD.

For further insight, figures 1 and 2 compares the ROC curves of all individual
systems and their fusion, respectively.

Figure 1: ROC curve of all systems Figure 2: ROC curve of fusion

###  5.2 Results for test setting

Based on the previous results, five systems were submitted to the challenge.
The table 5 shows the performance achieved by them on the testing dataset,
which consist of recordings of 48 subjects.

Table 5: Results on the testing dataset

| class | Precision | Recall | F1 Score | Accuracy
---|---|---|---|---|---
Fusion II | non-AD | 0.8000 | 0.6667 | 0.7273 | 0.7500
AD | 0.7143 | 0.8333 | 0.7692
Fusion I | non-AD | 0.8235 | 0.5833 | 0.6829 | 0.7292
AD | 0.6774 | 0.8750 | 0.7636
RNN model | non-AD | 0.6667 | 1.0000 | 0.8000 | 0.7500
AD | 1.0000 | 0.5000 | 0.6667
fluency | non-AD | 0.6087 | 0.5833 | 0.5957 | 0.6042
AD | 0.6000 | 0.6250 | 0.6122
x-vector | non-AD | 0.5553 | 0.4167 | 0.4762 | 0.5417
AD | 0.5333 | 0.6667 | 0.5926

The results achieved by the RNN model were really similar in training and
testing, which is a good sign of the lack of overfitting on the training stage
as a result of a Dropout at 10% rate performed in the Embedding and LSTM (Long
Short Term Memory) layers. However, the submitted speech-based systems had a
significant decrease in performance. This decline would be a consequence of a
poor generalization capacity. As a result, the performance of the fusion
systems also declines.

##  6 Conclusions and Further work

Two lines of work have been developed on the ADReSS Challenge dataset: one
based on speech processing and the other based on text processing.

It is important to highlight the following points when assessing both
solutions:

  * •

Performance: x-vector, functionals, and fluency speech features have shown a
competitive performance in the LOSO cross-validation. However, their
performance decreases on the test setting. This would be a result of a low
generalization level achieved at the training stage. On the other hand, the
text-based RNN model has obtained superior results both in the LOSO and
testing settings, but with unbalanced values of recall and precision in the
last one. Nevertheless, the Fusion II strategy was a solution to that problem,
presenting the same accuracy that the RNN model but with more balanced
measures. As a result, multimodal systems demonstrate to be a better strategy
for the detection of AD than individual sytems.

  * •

Complexity: The x-vector and RNN systems, as deep-learning-based models, need
a large amount of training data to be able to generalize well.

  * •

Explainability: deep learning models are black-box models, being very
difficult to interpret from a human perspective. On the contrary, the
linguistic, fluency, functionals and i-vector features created here are
explainable and one could determine the weight of them in the classification.

Several future research lines have been identified for further work. Firstly,
investigation on improved classification based on deep neural networks and
novel acoustical feature extraction algorithms. Secondly, addition of new
linguistic features and non-verbal information (breaks, silence duration, word
mistakes, etc.) in text-based systems. Thirdly, analysis of strategies for
increasing the generalization capacity of the proposed speech-based systems;
for example: to find new rhythmic parameters more discriminatory between
patients with and without AD. It is also important to conduct experiments in
other experimental settings, for example using other questions of the mini-
mental state examination test, to validate the results obtained and, above
all, to increase the size of the data settings.

##  7 Acknowledgements

This work has received financial support from the Spanish “Ministerio de
Economia y Competitividad” through the project Speech&Sign
RTI2018-101372-B-100, and also from Xunta de Galicia (AtlanTTic and ED431B
2018/60 grants) and European Regional Development FundERDF.

## References

  * [1] A. Association, “2019 alzheimer’s disease facts and figures,” _Alzheimer’s & Dementia_, vol. 15, no. 3, pp. 321–387, 2019.
  * [2] P. J. Nestor, P. Scheltens, and J. R. Hodges, “Advances in the early detection of alzheimer’s disease,” _Nature medicine_ , vol. 10, no. 7, pp. S34–S41, 2004.
  * [3] S. Ahmed, A.-M. F. Haigh, C. A. de Jager, and P. Garrard, “Connected speech as a marker of disease progression in autopsy-proven alzheimer’s disease,” _Brain_ , vol. 136, no. 12, pp. 3727–3737, 2013.
  * [4] J. J. G. Meilán, F. Martínez-Sánchez, J. Carro, D. E. López, L. Millian-Morell, and J. M. Arana, “Speech in alzheimer’s disease: Can temporal and acoustic parameters discriminate dementia?” _Dementia and Geriatric Cognitive Disorders_ , vol. 37, no. 5-6, pp. 327–334, 2014.
  * [5] K. Lopez-de Ipiña, J. B. Alonso, J. Solé-Casals, N. Barroso, P. Henriquez, M. Faundez-Zanuy, C. M. Travieso, M. Ecay-Torres, P. Martinez-Lage, and H. Eguiraun, “On automatic diagnosis of alzheimer’s disease based on spontaneous speech analysis and emotional temperature,” _Cognitive Computation_ , vol. 7, no. 1, pp. 44–55, 2015.
  * [6] K. C. Fraser, J. A. Meltzer, and F. Rudzicz, “Linguistic features identify alzheimer’s disease in narrative speech,” _Journal of Alzheimer’s Disease_ , vol. 49, no. 2, pp. 407–422, 2016.
  * [7] J. Pennington, R. Socher, and C. D. Manning, “Glove: Global vectors for word representation.” in _In Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , vol. 14, 2014, pp. 1532–1543.
  * [8] S. Luz, F. Haider, S. de la Fuente, D. Fromm, and B. MacWhinney, “Alzheimer’s dementia recognition through spontaneous speech: The ADReSS Challenge,” in _Proceedings of INTERSPEECH 2020_ , Shanghai, China, 2020\. [Online]. Available: https://arxiv.org/abs/2004.06833
  * [9] N. Dehak, P. J. Kenny, R. Dehak, P. Dumouchel, and P. Ouellet, “Front end factor analysis for speaker verification,” _IEEE Transactions on Audio, Speech and Language Processing_ , 2010.
  * [10] D. Snyder, D. Garcia-Romero, G. Sell, D. Povey, and S. Khudanpur, “X-vectors: Robust dnn embeddings for speaker recognition,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , 2018, pp. 5329–5333.
  * [11] P. Ghahremani, B. BabaAli, D. Povey, K. Riedhammer, J. Trmal, and S. Khudanpur, “A pitch extraction algorithm tuned for automatic speech recognition,” in _2015 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP_ , 2014, pp. 2494–2498.
  * [12] D. Povey, A. Ghoshal, G. Boulianne, L. Burget, O. Glembek, N. Goel, M. Hannemann, P. Motlicek, Y. Qian, P. Schwarz, J. Silovsky, G. Stemmer, and K. Vesely, “The kaldi speech recognition toolkit,” 2011.
  * [13] A. Nagrani, J. S. Chung, W. Xie, and A. Zisserman, “Voxceleb: Large-scale speaker verification in the wild,” _Computer Speech and Language_ , vol. 60, 2020.
  * [14] F. Eyben, M. Wöllmer, and B. W. Schuller, “Opensmile: the munich versatile and fast open-source audio feature extractor.” in _ACM Multimedia_. ACM, 2010, pp. 1459–1462.
  * [15] M. Valstar, B. Schuller, K. Smith, F. Eyben, B. Jiang, S. Bilakhia, S. Schnieder, R. Cowie, and M. Pantic, “AVEC 2013 - the continuous audio/visual emotion and depression recognition challenge,” in _Proceedings of the 3rd International Worskhop on Audio/Visual Emotion Challenge, Proceedings of AVEC’13_ , 2013.
  * [16] M. A. Hall, “Correlation-based feature subset selection for machine learning,” Ph.D. dissertation, University of Waikato, Hamilton, New Zealand, 1998\.
  * [17] M. Ajili, S. Rossato, D. Zhang, and J.-F. Bonastre, “Impact of rhythm on forensic voice comparison reliability,” 2018.
  * [18] A. Larcher, K. A. Lee, and S. Meignier, “An extensible speaker identification sidekit in python,” in _2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2016, pp. 5095–5099.
  * [19] B. Macwhinney, “The childes project: tools for analyzing talk,” _Child Language Teaching and Therapy_ , vol. 8, 01 2000.
  * [20] K. C. Fraser, J. A. Meltzer, and F. Rudzicz, “Linguistic features identify alzheimer’s disease in narrative speech,” _Journal of Alzheimer’s disease : JAD_ , vol. 49, no. 2, pp. 407–422, 2016.