Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?

Tiziana M. Florio; Tiziana M. Florio

doi:10.3934/Neuroscience.2020010

AIMS Neuroscience

2020, Volume 7, Issue 2: 136-152. doi: 10.3934/Neuroscience.2020010

Previous Article Next Article

Review Topical Sections

Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?

Tiziana M. Florio ^,

Department of Life, Health and Environmental Sciences, University of L'Aquila, Italy

Received: 31 January 2020 Accepted: 18 May 2020 Published: 27 May 2020

Comprehensive knowledge about higher executive functions of motor control has been covered in the last decades. Critical goals have been targeted through many different technological approaches. An abundant flow of new results greatly progressed our ability to respond at better-posited answers to look more than ever at the challenging neural system functioning. Behaviour is the observable result of the invisible, as complex cerebral functioning. Many pathological states are approached after symptomatology categorisation of behavioural impairments is achieved. Motor, non-motor and psychiatric signs are greatly shared by many neurological/psychiatric disorders. Together with the cerebral cortex, the basal ganglia contribute to the expression of behaviour promoting the correct action schemas and the selection of appropriate sub-goals based on the evaluation of action outcomes. The present review focus on the basic classification of higher motor control functioning, taking into account the recent advances in basal ganglia structural knowledge and the computational model of basal ganglia functioning. We discuss about the basal ganglia capability in executing ordered motor patterns in which any single movement is linked to each other into an action, and many actions are ordered into each other, giving them a syntactic value to the final behaviour. The stereotypic, automatized and habitual behaviour's constructs and controls are the expression of successive stages of rule internalization and categorisation aimed in producing the perfect spatial-temporal control of motor command.

Keywords:

Citation: Tiziana M. Florio. Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?[J]. AIMS Neuroscience, 2020, 7(2): 136-152. doi: 10.3934/Neuroscience.2020010

Related Papers:

[1]	Mehmet Yavuz, Waled Yavız Ahmed Haydar . A new mathematical modelling and parameter estimation of COVID-19: a case study in Iraq. AIMS Bioengineering, 2022, 9(4): 420-446. doi: 10.3934/bioeng.2022030
[2]	Sarbaz H. A. Khoshnaw, Kawther Y. Abdulrahman, Arkan N. Mustafa . Identifying key critical model parameters in spreading of COVID-19 pandemic. AIMS Bioengineering, 2022, 9(2): 163-177. doi: 10.3934/bioeng.2022012
[3]	Bashdar A. Salam, Sarbaz H. A. Khoshnaw, Abubakr M. Adarbar, Hedayat M. Sharifi, Azhi S. Mohammed . Model predictions and data fitting can effectively work in spreading COVID-19 pandemic. AIMS Bioengineering, 2022, 9(2): 197-212. doi: 10.3934/bioeng.2022014
[4]	Sarbaz H. A. Khoshnaw, Azhi Sabir Mohammed . Computational simulations of the effects of social distancing interventions on the COVID-19 pandemic. AIMS Bioengineering, 2022, 9(3): 239-251. doi: 10.3934/bioeng.2022016
[5]	Ayub Ahmed, Bashdar Salam, Mahmud Mohammad, Ali Akgül, Sarbaz H. A. Khoshnaw . Analysis coronavirus disease (COVID-19) model using numerical approaches and logistic model. AIMS Bioengineering, 2020, 7(3): 130-146. doi: 10.3934/bioeng.2020013
[6]	Honar J. Hamad, Sarbaz H. A. Khoshnaw, Muhammad Shahzad . Model analysis for an HIV infectious disease using elasticity and sensitivity techniques. AIMS Bioengineering, 2024, 11(3): 281-300. doi: 10.3934/bioeng.2024015
[7]	Fırat Evirgen, Fatma Özköse, Mehmet Yavuz, Necati Özdemir . Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 2023, 10(3): 218-239. doi: 10.3934/bioeng.2023015
[8]	Kranthi Kumar Lella, Alphonse PJA . A literature review on COVID-19 disease diagnosis from respiratory sound data. AIMS Bioengineering, 2021, 8(2): 140-153. doi: 10.3934/bioeng.2021013
[9]	Eduardo Federighi Baisi Chagas, Piero Biteli, Bruno Moreira Candeloro, Miguel Angelo Rodrigues, Pedro Henrique Rodrigues . Physical exercise and COVID-19: a summary of the recommendations. AIMS Bioengineering, 2020, 7(4): 236-241. doi: 10.3934/bioeng.2020020
[10]	Cong Toai Truong, Kim Hieu Huynh, Van Tu Duong, Huy Hung Nguyen, Le An Pham, Tan Tien Nguyen . Model-free volume and pressure cycled control of automatic bag valve mask ventilator. AIMS Bioengineering, 2021, 8(3): 192-207. doi: 10.3934/bioeng.2021017

Abstract

1. Introduction

People always love to live healthy. But it was not expected all the time. Over many years, many diseases have been evolved. The root of many diseases are not still known. Some of the natural elements being spoiled by mankind, spoil the mankind in turn. Sometimes, diseases emerge naturally and in sometimes, it may be by a human error. Whatever it is, once a disease has arrived among the mankind, it will spread to maximum number of people irrespective of age, gender, country, etc. Today, mankind is facing one such deadly disease called coronavirus. Coronavirus is not an exception in causing major loss to the society as well as mankind. Its time to quarantine our self to save our precious life. Coronaviruses are actually a group of same kind of viruses. They affect birds and mammals highly. In human beings, these coronaviruses are causing toughness to breath. The respiratory infections caused by COVID-19 may be light, like common cold, or may be severe, like SARS, MERS, and novel corona. The medical world is still struggling to find the medicine that can treat these diseases. The spread of coronaviruses from one human to other is believed to spread among the close communities by means of sneezing or coughing or even by touching. The first human infecting coronaviruses were discovered during 1960. The very first one was found in chickens called as infectious-bronchitis-virus. The other two were found in human beings with common cold. It was named as human coronaviruses 229E and human coronaviruses OC43 later. Also, there are other lists of this coronavirus. They are SARS-CoV which was found in 2003. In the year 2004, HCoV NL63 was found. In the next year, 2005 HKU1 was found. MERS-CoV found in the year of 2012, and SARS-CoV-2 which was well known to be 2019-nCoV, found recently in the year 2019. Still new viruses are persisting in the globe. Most of these viruses have involved very serious problems such as respiratory problems and inner organ failure. Coronaviruses are classified in to four types and they are Alpha-coronaviruses, Beta-coronaviruses, Gamma-coronaviruses, and Delta-coronaviruses. From the observations, the bats, birds and all hot blooded creatures are the only hosts for the alpha and beta coronaviruses, and birds are common to gamma and delta coronaviruses for the evolution and spreading of these viruses. All coronaviruses may change in terms of their risk factor. More than 30% of those infected (such as MERS-CoV) may die and some others such as fever, dry cough and sore throat from swollen adenoids, occurring mostly in the winter and in spring seasons are less harmful. Coronaviruses are able to cause pneumonia and bronchitis. The most threaten human coronavirus was discovered in the year 2003, (SARS-CoV), which causes a severe acute respiratory syndrome called SARS. It has very unique pathogenesis infecting both upper and lower respiratory tracts. During the years of 2002–2004, it was found as 774 people had died of SARS-CoV. In 2012, 2015 and 2018 the MERS-CoV respectively caused over 400, 36 and 41 deaths. Now, during 2019 to 2020, coronavirus pandemic was considered as a global threat with at least 30,935 deaths till March 29 2020.

1.1. Coronavirus disease 2019 (COVID-19)

During December 2019, It was reported in Wuhan, China , a pneumonia type of disease outbreak. On 31 September 2019, that was traced and identified as a novel coronavirus. The World Health Organization (WHO) called it as 2019-nCoV. Later, the International Committee on Taxonomy of viruses called it as SARS-CoV-2. Since, March 29, 2020, there have been found at least 30,935 confirmed deaths and more than 492,571 confirmed cases in the coronavirus pandemic over 199 countries and territories around the world and two international conveyances, one is the Diamond Princess cruise ship of Japan and the other is Holland America's MS Zaandam Cruise ship. The Wuhan novel strain has been identified as a new type of Beta coronavirus and it was from group 2B with approximately 70% genetic similarity with SARS-CoV. As this virus has a 96% of similarity to bat coronaviruses, it was believed that these viruses start from bats. This pandemic has resulted in travel restrictions. Also worldwide, we can see lock-downs in several countries. Since modeling of any pandemic or epidemic disease spread is uncertain, there will be vagueness always. To overcome this vagueness and to study uncertainty, fuzzy sets was first introduced individually by Lofti A. Zadeh in 1965 [1]. Today randomness, vagueness and uncertainty problems were mostly studied everywhere like signal processing, artificial intelligence, population dynamics, optics, oriental medicine, by Engineers, Doctors and Scientists etc. Fuzzy sets, fuzzy differential equations, fuzzy logic are considered to be their important tools to study them. The imprecise nature of medical concepts are always defined with the help of fuzzy logic.

In [1] and [2], L.A. Zadeh, presented fuzzy sets, fuzzy mapping and control. In [3], J.J. Bukley and T. Feuring, in [4], D. Dubois and H. Prade, in [5], Lupulescu, in [6] and in [7], Kaleva did a great research on fuzzy differential equations. In [8], Ma, Friedman and Kandel, and in [9], seikkala, provided a great contribution to fuzzy differential equations, fuzzy differential calculus, fuzzy initial value problem and seikkala dervative is the predominant work of Seikkala in the fields of fuzzy differential equations. In [10], Diamond and Kloedan analyzed metric spaces of fuzzy sets in 1984. In [11], Y. Chalco-Cano and H. Roman-Flores, defined, analyzed and studied new solutions of fuzzy differential equations. In [12] and [13], S. Abbasbandy and T. Allahviranloo, studied numerical solution of fuzzy differential equation by Taylor method and by Runge-Kutta method. In [14], C.F. Curtiss, J.O. Hirschfelder, studied integration of stiff equations. In [15], Gustaff Soderlind, Laurent Jay and Manual Calvo, published a work on Stiffness. In [16], L. Shampine, provided evaluation of a test set for stiff ODE solvers. In [17], D.J. Higham and L.N Trefethen studied stiffness of ODEs. In [18], M.N. Spijker, evaluated stiffness in numerical initial value problems.

In [19], W.O. Kermack and A.G. Mckendrick, gave the first contribution to the mathematical theory of epidemics. In [20], P. Palese and J.F. Young, studied variation of Influenza A, B, and C. In [21], Allen LJS gave an introduction to mathematical biology in 2007. In [22], Sha He, Sanyi Tang and Libinin Rong, framed a discrete stochastic model for COVID-19 outbreak: forecast and control. In [23], Weike zhou, Aili Wang, Fan Xia, Yanni Xiao and Sanyi Tang reported the effects of media on mitigating spread of COVID-19 in the early phase of the outbreak. In [24], Fuliyan Yin, Jiahui Lv, Xiaojian Zhang, Xinyu Xia, and Jianhong Wu, presented COVID-19 information propagation dynamics in the chinese sina-microblog. In [25], Chenxi Dai, Jing Yang and Kaifa Wang, did evaluation of control interventions on the epidemic of coronavirus disease 2019 in Chongqing and Guizhou Provinces. In [26], Xinmiao Rong, Liu Yang, Huidi Chu and Meng Fan, analyzed the effect of delay in diagnosis on spread of coronavirus. In [27], Jingjing Tian, Jiabing wu, Yunting Bao, Xiaoyu Weng, Lei Shi, Binbin Liu, Xinya Yu, did modeling analysis of COVID-19 in Anhui, China. In [28], Chentong Li, Jinhu Xu, Jiawei Liu and Yicang Zhou, particularly studied the within-host viral kinetics COVID-19. In [29], Vitaly Volpert, Malay Banerjee and Sergei Petrovskii, gave their tremendous work on a quarantine model of coronavirus infection along with data analysis. In [30], Biao Tang, Nicola Luigi Bragazzi, Qian Li, Sanyi Tang, Yanni Xiao and Jianhong Wu analysed and provided an updated estimation of the novel coronavirus transmission. Zifeng Yang et al. analysed modified SEIR and AI prediction of the epidemics trend of COVID-19 in China under public health interventions in [31]. ShreshthTuli, Shikhar Tuli et al. predicted the growth and trend of COVID-19 pandemic using machine learning and cloud computing in [32]. Chintamani Pai et al. investigated the dynamics of COVID-19 pandemic in India under lockdown in [33]. Matheus Henrique Dal Molin Ribeiro et al. focused on short-term forecasting COVID-19 cumulative confirmed cases: perspectives for Brazil in [34]. Muhammad Altaf Khan and Abdon Atangana, have modeled the dynamics of novel coronavirus (2019-nCov) with fractional derivative in [35]. In [36], [37], [38] and [39], authors like Muhammad Altaf Khan regularly studied SEIR, dengue and also presented the numerical solution of the competition model among bank data in Caputo-Fabrizio derivative. Authors of [40], [41] and [42] have regularly studied numerical solutions of fuzzy differential equations.

The paper contains preliminaries, which are essential to create some basic knowledge about the study to the readers, in section 2. The remainder of the paper is structured as follows. The formulation of models in terms of ordinary differential equations and in terms of fuzzy differential equations are given in section 3. Stability analysis and test for given system to be stiff differential equations are performed in section 4. Numerical simulation of infected, recovered and death cases are presented in terms of both ordinary differential equations and fuzzy differential equations in section 5.

2. Preliminaries

Before proceeding to some of the prerequisites necessary for the study, we are now presenting the novelty, new contributions and new objectives in this paper which will motivate the readers and researchers. This is the first paper in which the epidemic modeling without considering the susceptible population was framed. This is the first paper in which the stiff differential equations and fuzzy properties are used to frame and study such a model. Many papers are emerging daily to predict the daily cases but this is the first paper in which a new idea called maintaining the 14 day average-infectious, recovered and death populations under threshold populations is given to control the spread and human loss. Also in this paper since susceptible population was not considered, we are introducing the new number called average transmission number-N₀ instead of basic reproduction number R₀. For each country this modeling could be applied as here we are studying globally. In mathematical biology point of view, the advantage of models framed by stiff fuzzy differential equations is used to analyze the model in which the stiffness (stiffness is defined below) arise can be set under fuzzy boundary so that values are predicted at crisp set values and analyzed at the fuzzy set values. That will help us to set the boundary of our analyze to two extreme (minimum and maximum) values and also to the in between values (partial member values). All these are explained in detail below.

2.1. Stiff differential equations

Though stiff differential equations are most important field of research in mathematical modeling, a rigorous, efficient, meaningful and computationally relevant properties or definition of stiffness is still not well defined. Since many years, many different numerical methods and software codes have been designed for the efficient solution of stiff initial value problem. The seminal paper of curtiss and Hirchfelder [14] first opened this field, stiffness. But it was not yet properly defined. According to Shampine [16], “A major difficulty is that stiffness is a complex of related phenomena, so that it is not easy to define what stiffness is”.

Stiff equations are some of the problems in ordinary differential equations for which explicit type of numerical methods does not work. The explicit type of numerical methods may struggle a lot when applied to the IVP's (initial value problems) $\dot{y} = f (y)$ , y(0) = y₀. It is always not possible to solve, when f′(y) is very large. Similarly in non linear equations, fixed point iteration y^m⁺¹ = f(y^m) does not converge for problems whose f′(y) is large. Curtiss and Hirchfelder [14] are the two people who introduced the notation stiffness for which the explicit methods for numerical solutions of ordinary differential equations failed. There is no unique definition of stiffness is found in the literature. But one can call the system of ordinary differential equations is called stiff if Lipschits constant L defined for solution over the interval is L(y₀, y_n)>>1.

The following is the definition of stiff system of fuzzy differential equations.

Definition 2.1 (Stiff system of fuzzy differential equations). The linear system of fuzzy differential equations ${\tilde{y}}^{'} (x) = A \tilde{y} (x), A \in R^{n \times n}$ , is known as fuzzy stiff, if all eigenvalues λ_i posses a negative real part and if

$q = \frac{m a x | (R e (λ_{i})) |, i = 1,2,..., n}{m i n | (R e (λ_{i})) |, i = 1,2,..., n} > > 1$

Remarks:

The stiff differential equations are numerically known to be unstable unless the step size is extremely very very small.
Stiff differential equations are also characterized as one for which exact solution possessed to have a term of the form e^−kt, where k is a large positive constant.
Large derivatives of e^−kt always give error terms and that will always dominate the solutions.

2.2. Fuzzy differential equations

We are interested to study the model in fuzzy differential equations because, it only studies the system completely as non-member, partial members and full member. If y₀ is a fuzzy number found with α-level of intervals ${[y_{0}]}^{α} = [{\underline{y}}^{α}, {\bar{y}}^{α}], 0 \leq α \leq 1$ . The extension principle of Zadeh leads us to the following definition of $\tilde{f} (t, y (t))$ when y is a fuzzy number.

It follows that

$\tilde{f} (t, y (t)) = {\begin{cases} m i n f (t, u) : u ({\underline{y}}_{t}^{α}, {\bar{y}}_{t}^{α}), u \in E \\ m a x f (t, u) : u ({\underline{y}}_{t}^{α}, {\bar{y}}_{t}^{α}), \end{cases}$ (2.1)

for y∈E with α-level sets

${[y]}_{α} = [{\underline{y}}_{t}^{α}, {\bar{y}}_{t}^{α}], 0 \leq α \leq 1$ since the fuzzy derivative y′(t) of a fuzzy process, y:R₊→E is defined by

${[y^{'} (t)]}_{α} = [({\underline{y}}^{α})^{'} (t), ({\bar{y}}^{α})^{'} (t)],0 \leq α \leq 1$ .

We call y:R₊→E a fuzzy solution of (2.1) on the interval I = [0,t] if

${\begin{cases} ({\underline{y}}^{α})^{'} (t) = m i n {f (t, u) : u \in [{\underline{y}}_{t}^{α}, {\bar{y}}_{t}^{α}], \underline{y} (t_{0}) = {\underline{y}}_{0}} \\ ({\bar{y}}^{α})^{'} (t) = m a x {f (t, u) : u \in [{\underline{y}}_{t}^{α}, {\bar{y}}_{t}^{α}], \bar{y (} t_{0}) = {\bar{y}}_{0}} \end{cases}$ (2.2)

for t∈I and 0≤α≤1 Thus for fixed α we have an initial value problem in R². If we can solve it, we have only to verify that intervals

$[y_{t}^{α}, {\bar{y}}_{t}^{α}], 0 \leq α \leq 1,$ define a fuzzy number y(t)∈E.

3. Model formulation

As COVID-19 is pandemic and spreads in several countries of the world very quickly, we are unable to know the exact count of susceptible population. The infected, recovered and dead populations were taken from various online resources like [43]. In order to frame the model, we have taken the data from March 10 to March 23, 2020. All the existing mathematical models deals with growth of the disease day by day and there is no model yet without susceptible population, also they require the susceptible population to calculate the basic reproduction number. Here we take 14 days data of the whole world's COVID-19 cases and take their average as the initial population. The worlds population is very high and the disease is spreading in a very fast manner. so the population of infected, recovered and dead was also increasing day by day. We have to keep in mind that the worlds total population cannot be taken as susceptible. Anyone with common symptoms of the disease but not yet infected is what exactly susceptible population mean. Due to lack of awareness about the symptoms no one can visit the hospital. In such situation its not possible to record the exact population of susceptible cases as recording the exact population of infected, recovered and dead. So we are interested to create a model called IRD-Infected, Recovered and Dead. Since the disease has an average life of 14 days we are also interested to study the average cases of IRD population over 14 days mean. It is quite interesting to study such new way of modeling. In future many researchers may interested to study the model like this. The main advantage of studying this average transmission model is one can easily estimate the average change in population for any years. But the average number should be the life of virus. Before presenting the model, the data from March 10 to March 23, 2020 are presented.

Let β, μ, γ and ζ be the rate of infected I(t), recovered R(t), Death D(t) and the total cases T(t) respectively. We know that the rate of total case is always 100. i.e., ζ = 100.

Table 1. COVID-19, 14 days population and rates [43].

Date(t)	I(t)	β	R(t)	μ	D(t)	γ	T(t)
Mar23	260277	68.70	102069	26.94	16514	4.36	378860
Mar22	224221	66.44	98627	29.22	14641	4.34	337489
Mar21	196523	64.43	95500	31.31	13013	4.26	305036
Mar20	172637	62.64	91573	33.23	11387	4.13	275597
Mar19	146740	59.91	88162	35.99	10031	4.10	244933
Mar18	124538	56.91	85333	39.00	8951	4.09	218822
Mar17	107637	54.30	82623	41.68	7978	4.02	198238
Mar16	95701	52.44	79628	43.64	7161	3.92	182490
Mar15	85622	50.49	77452	45.67	6519	3.84	169593
Mar14	74888	47.81	75932	48.47	5833	3.72	156653
Mar13	67447	46.36	72607	49.91	5429	3.73	145483
Mar12	592212	44.00	70383	52.30	4981	3.70	134576
Mar11	53279	42.21	68307	54.12	4628	3.67	126214
Mar10	48031	40.38	66621	56.01	4296	3.61	118948
Sum	1616753	757.02	1154817	587.49	121362	55.49	2892932
Average	115482.36	54.07	82486.93	41.97	8668.71	3.96	206638

| Show Table

DownLoad: CSV

From the above table it was found that I(t), R(t) and D(t) increases rapidly and also it is clear that I(t) > R(t) > D(t) and obviously β>μ>γ. But without considering from the table we shall calculate β, μ and γ using separate forumalae later. So the COVID-19 model is the exponentially growing model. Since the spread of this pandemic disease is not coming to an end or any proper medicine was not yet found it is not a right way to go for the logistic growth model. If so, what will be the carrying capacity, as total population and susceptible population are not clearly known? Biologist and Doctors says the average life of this COVID-19 is 14 days. At least we can check whether its average population model is growing exponentially or not. If this is not growing exponentially then we can suggest doctors to maintain this average initial populations as threshold numbers. Exceeding this may lead to loss the control of infections and deaths. Through out the paper, Δ represents rate of change in population of Infected, Recovered and Dead cases with respect to time t and D represents death population. With these ideas the following model is made.

Figure 1. Model formulation.

DownLoad: Full-Size Img PowerPoint

$\begin{array}{l} Δ I (t) & = & (β I (t) - μ I (t) R (t) - γ I (t) D (t) \\ Δ R (t) & = & (μ I (t) R (t) - β I (t) R (t) - γ R (t) D (t)) \\ Δ D (t) & = & (γ D (t) - μ R (t) D (t)) \end{array}$ (3.1)

3.1. Description of the model

As already discussed, β, μ and γ be the rate of healthy become infected (I(t)), infected become recovered (R(t)), infected become death (D(t)) respectively. Mathematically death is similar to recovered because both are infection free. The average rate of infectious become death are given by γI(t)D(t). The average rate of infectious become recovered and recovered become infectious are given by μI(t)R(t) and βI(t)R(t). The average rate of recovered become death are represented in two ways as γR(t)D(t) to study death during infection or partial recovery and μR(t)D(t) to study death after being recovered, i.e., natural death.

The exact solutions of (3.1) is found to be

$I (t) = c e^{\int_{t_{0}}^{t_{n}} β - μ R (t) - γ D (t) d t} .$

$R (t) = c e^{\int_{t_{0}}^{t_{n}} μ I (t) - β I (t) - γ D (t) d t} .$

$D (t) = c e^{\int_{t_{0}}^{t_{n}} γ - μ R (t) d t} .$

For any value t₀≤t≤t_n, β−μR(t)−γD(t)<0, μI(t)−βI(t)−γD(t)<0, γ−μR(t)<0 which produces ce^(−ve) values as the solutions of (3.1) Thus we call the model (3.1) as system of stiff differential equations. We shall present the test later to confirm it. Now fuzzy system of stiff differential equation are given in (3.2).

$\begin{array}{l} Δ \tilde{I} (t) & = & (β \tilde{I} (t) - μ \tilde{I} (t) \tilde{R} (t) - γ \tilde{I} (t) \tilde{D} (t)) \\ Δ \tilde{R} (t) & = & (μ \tilde{I} (t) \tilde{R} (t) - β \tilde{I} (t) \tilde{R} (t) - γ \tilde{R} (t) \tilde{D} (t)) \\ Δ \tilde{D} (t) & = & (γ \tilde{D} (t) - μ \tilde{R} (t) \tilde{D} (t)) \end{array}$ (3.2)

where, $\tilde{f} (t) = (0.75 + 0.25 r,1.125 - 0.125 r) f (t)$ is the fuzzy number with r∈[0,1].

The initial conditions are the average populations, we presented in Table 1. Since the population cannot be a fractional number we take

$\begin{matrix} I (t_{0}) = I_{0} = m_{1} = 115482, t = t_{0}, \\ R (t_{0}) = R_{0} = m_{2} = 82487, t = t_{0}, \\ D (t_{0}) = D_{0} = m_{3} = 8669, t = t_{0} . \end{matrix}$

The rate of infection, recovered and death are also found by average of ratios of 14 days. Instead of taking β, μ and γ from the Table 1, we are calculating them using more appropriate formulae given below.

β→the rate of infection=Infected case×ζ/total case

μ→the rate of recovery=Recovered case×ζ/total case

γ→the rate of death=death case×ζ/total case

ζ→the rate of total case.

The total case T(t)=Infected case I(t)+Recovered case R(t)+Death case D(t)=206638 at t=t₀

T′(t)=I′(t)+R′(t)+D′(t)

Now we consider the average of those 14 days rate as the required rate.

We found that β=56, μ=40, γ=4 and ζ=100.

4. Stability analysis and stiff system test

4.1. Equilibrium points

The equilibrium points are found by setting $Δ \tilde{I} (t) = 0$ , $Δ \tilde{R} (t) = 0$ and $Δ \tilde{D} (t) = 0$ . The disease free equilibrium points are (0,0,0) and the disease depending equilibrium points are given by $(\frac{γ - β}{μ - β}, \frac{γ}{μ},1 - \frac{β}{γ})$ thus the values of disease depending equilibrium points are (3.25,0.1,−13).

4.2. Average transmission number: N₀

The average transmission number is the number telling the average spread of disease from one to others. Though it is similar to basic reproduction number R₀, we are not using susceptible population here. That's why we are calling it by a different name “average transmission number N₀”. The calculation of basic reproduction number requires the population of susceptible case. Since we are not aware of susceptible population, we are introducing this new method for calculating this new number called average transmission number. It is a two step method.

From the above average transmission number, we can easily assert that this 14 day average pandemic COVID-19 is spreading from infected one to at least 1 person or to at most 4 persons. This number N₀ found from this new method matches with universally accepted basic reproduction number R₀ of COVID-19.

4.3. Stability analysis using eigen values

Theorem 4.1. When all the eigen values of linearized form of (3.2) are < 0. Then the model considered is asymptotically stable.

Proof. Let us first linearize the model (3.2) in the form of Jacobian matrix and name it as A

$A = (\begin{matrix} (β - μ \tilde{R} (t) - γ \tilde{D} (t)) & - μ \tilde{I} (t) & γ \tilde{I} (t) \\ μ \tilde{R} (t) - β \tilde{R} (t) & (μ \tilde{I} (t) - β \tilde{I} (t) - γ \tilde{D} (t)) & - γ \tilde{R} (t) \\ 0 & - μ \tilde{D} (t) & γ - μ \tilde{R} (t) \end{matrix})$ (4.2)

On substituting the values of all the parameters and solving (4.2), the characteristic polynomial was found to be λ³+8515964λ²+17276845232848λ+422461568851044800=0

The corresponding eigen values are given by

$\begin{matrix} λ_{1} = (- 24753.6 + i (1.5522 \times 10^{- 10})), \\ λ_{2} = (- 5.22464 \times 10^{6}) + i (1.16415 \times 10^{- 10}), \\ λ_{3} = (- 3.26657 \times 10^{6}) - i (4.65661 \times 10^{- 10}) . \end{matrix}$

Since all the eigen values have the negative real parts, the system we considered is asymptotically stable.

4.4. Stiff system of fuzzy differential equation-test

Theorem 4.2. Stiffness test by eigen values

By the definiton 2.1, a linear fuzzy system or a linearized form of a non linear fuzzy system posses negative real parts in all its eigen values and satisfying

$q = \frac{m a x | (R e (λ_{i})) |, i = 1,2,..., n}{m i n | (R e (λ_{i})) |, i = 1,2,..., n} > > 1$

is called stiff system of fuzzy differential equation.

Proof. Let us take (4.2), which is the linerized form of (3.2). We found the eigen values as

All are complex numbers whose real part is negative,

Now, calculating the value of q,

$\begin{matrix} q = \frac{m a x | (R e (λ_{i})) |, i = 1,2,..., n}{m i n | (R e (λ_{i})) |, i = 1,2,..., n} > > 1, \\ q = \frac{24753.6}{3.26657} \approx 7577.8569 > > 1. \end{matrix}$

The (4.2) satisfied the theorem and hence our model (3.2) is the stiff system of fuzzy differential equations which obviously establishes that (3.1) is the stiff system of differential equations.

5. Numerical simulations

Since the model we considered is a stiff system of differential equations, we do not prefer to use Runge-Kutta methods or any other numerical methods. As already discussed the exact solutions of $\tilde{I} (t), \tilde{R} (t), \tilde{D} (t)$ are in the form e^{(−ve) value}. Also those negative values are high requiring very precise small step size which will lead to more number of iterations. So it was found that numerical methods like Runge-Kutta methods are not stable to study any stiff differential equations like (3.1).

So going for software is the better option. Software like Mathematica, Matlab, Maple etc., having build in functions are effective to provide best approximate solutions. We obtain the numerical solutions of both ordinary and fuzzy valued model by using the NDSolve technique of Mathematica software to solve differential equations.

The following plots will be useful to check the average of 14 days pandemic transmission. The plots for March 10 to March 23 are not provided since seeing the table itself we can understand that all the cases like susceptible, infected and recovered are increasing like exponential growth. Without any logistic term to control the growth, the average population itself is giving the stable graphs, which means they are our threshold populations. The graphical representations are given for very minimum time span i.e., 0.00001 days or 0.864 seconds in order to zoom and show how the spread of diseases and each population changes are decreased. Also the main idea is to set a threshold population up to which the spread is under control and possible to completely remove the infection from the globe.

Figure 2. Infected case for every 0.864 seconds (0.00001 days).

DownLoad: Full-Size Img PowerPoint

Figure 3. Recovered case for every 0.864 seconds (0.00001 days).

DownLoad: Full-Size Img PowerPoint

Figure 4. Death case for every 0.864 seconds (0.00001 days).

DownLoad: Full-Size Img PowerPoint

Figure 5. Infected case for every 86400 seconds (1 day).

DownLoad: Full-Size Img PowerPoint

Figure 6. Recovered case for every 86400 seconds (1 day).

DownLoad: Full-Size Img PowerPoint

Figure 7. Dead case for every 86400 seconds (1 day).

DownLoad: Full-Size Img PowerPoint

6. Concluding remarks

Without susceptible population a new model, say IRD-Infection-Recovered and Death model of 14 day average transmission of COVID 19 pandemic system was formulated. It was reformulated as fuzzy system of differential equations. The equilibrium points have been found and stability analysis was done by studying the eigen values. We have given the new method for calculating the new number called average transmission number N₀ which matches with existing result of basic reproduction number R₀. Also we proved that the system considered was stiff system of differential equations as well as stiff system of fuzzy differential equations. From the Table 1, we can see the system is exponentially growing day by day. As a sample, we took the data from March 10, 2020 to March 23, 2020. Since $\tilde{I} (t)$ , $\tilde{R} (t)$ and $\tilde{D} (t)$ are growing exponentially, we took the average of those 14 days as our initial populations, say $\tilde{I} (t_{0})$ , $\tilde{R} (t_{0})$ and $\tilde{D} (t_{0})$ . Also their respective rates are found. From the plots Figures 2–4, we have shown in 0.00001 days (0.864 seconds) in average of 14 days, the cases approaching zero. This means for these initial populations the system produces a stable graph and approaches to zero in every 0.864 seconds per day of 14 day average is clearly noticed in plots. This study makes us to suggest the medical world to fix the threshold population of infectious, recovered and death cases under I(t)=115482, R(t)=82487 and D(t)=8669 and their rates under β=56, μ=40 and γ=4. So that, the whole world will easily overcome this deadly disease COVID-19. If the next 14 day average exceeds this threshold population, then it will make the disease to quickly spread and infect everyone in the globe. The respective fuzzy plots were given in Figures 5–7 for 86400 seconds. i.e., for a day.

In this work, instead of prediction we had founded the threshold numbers maintaining under which will reduce the spread of disease and the number of deaths due to it. But as a future work we are going to predict and compare the growth of new daily cases using commonly used metrics for comparison like mean average precision error, regularized R squared values. This stiff fuzzy model is used here to bound the unbound disease growth by setting 0≤α≤1 depending on which the stiff model will actively work with the boundary framed by 0≤α≤1. This is not the predictive analysis or comparison tool with daily cases data. This stiff fuzzy model is used to analyze the present spread of disease as three cases such as inactive hosts, partially active hosts and completely active hosts. It will be thus useful to confidently propose the threshold number from which medical world and government will get benefited by setting the every 14 day average IRD populations under this threshold number to control the disease and the deaths due to this pandemic disease COVID-19.

Acknowledgments

Author acknowledges Profs. Marcello Alecci for general supervision of the research group and proofreading, Angelo Galante for stimulating discussions and Giuseppe Di Giovanni for encouraging comments and proofreading.

Funding

University of L'Aquila RIA 2019 research funds.

Conflict of interest

Author declares no conflicts of interest in this paper.

References

[1]	Anholt RRC (2020) Evolution of epistatic networks and the genetic basis of innate behaviors. Trends Genet 36: 24-29. doi: 10.1016/j.tig.2019.10.005
[2]	Arber S (2012) Motor circuits in action: specification, connectivity, and function. Neuron 74: 975-989. doi: 10.1016/j.neuron.2012.05.011
[3]	Adolph KE (2002) Babies' steps make giant strides toward a science of development. Infant Behav Dev 25: 86-90. doi: 10.1016/S0163-6383(02)00106-6
[4]	Simion F, Regolin L, Bulf H (2008) A predisposition for biological motion in the newborn baby. PNAS 105: 809-813. doi: 10.1073/pnas.0707021105
[5]	Slater A, Quinn PC (2001) Face recognition in the newborn infant. Inf Child Dev 10: 21-24. doi: 10.1002/icd.241
[6]	Ekman P, Friesen W (1971) Constants across cultures in the face and emotion. J Pers Social Psychol 17: 124-129. doi: 10.1037/h0030377
[7]	Miellet S, Vizioli L, He L, et al. (2013) Mapping face recognition information use across cultures. Front Pshycol 4.
[8]	Grillner S, Robertson B (2016) The basal ganglia over 500 million years. Curr Biol 26: R1088-R1100. doi: 10.1016/j.cub.2016.06.041
[9]	Stephenson-Jones M, Samuelsson E, Ericsson J, et al. (2011) Evolutionary conservation of the basal ganglia as a common vertebrate mechanism for action selection. Curr Biol 21: 1081-1091. doi: 10.1016/j.cub.2011.05.001
[10]	Romo R, de Lafuente V (2013) Conversion of sensory signals into perceptual decisions. Prog Neurobiol 103: 41-75. doi: 10.1016/j.pneurobio.2012.03.007
[11]	Boettiger CA, D'Esposito M (2005) Frontal networks for learning and executing arbitrary Stimulus–Response associations. J Neurosci 25: 2723-2732. doi: 10.1523/JNEUROSCI.3697-04.2005
[12]	Garr E (2019) Contributions of the basal ganglia to action sequence learning and performance. Neurosci Biobehav Rev 107: 279-295. doi: 10.1016/j.neubiorev.2019.09.017
[13]	Wu T, Hallett M, Chana P (2015) Motor automaticity in Parkinson's disease. Neurobiol Dis 82: 226-234. doi: 10.1016/j.nbd.2015.06.014
[14]	Marinelli L, Quartarone A, Hallett M, et al. (2017) The many facets of motor learning and their relevance for Parkinson's disease. Clin Neurophysiol 128: 1127-1141. doi: 10.1016/j.clinph.2017.03.042
[15]	Balleine BW, Liljeholm M, Ostlund SB (2009) The integrative function of the basal ganglia in instrumental conditioning. Behav Brain Res 199: 43-52. doi: 10.1016/j.bbr.2008.10.034
[16]	Kobesova A, Kolar P (2014) Developmental kinesiology: Three levels of motor control in the assessment and treatment of the motor system. J Bodyw Mov Ther 18: 23-33. doi: 10.1016/j.jbmt.2013.04.002
[17]	Alexander GE, Crutcher MD (1990) Functional architecture of basal ganglia circuits: Neural substrates of parallel processing. Trends Neurosci 13: 266-271. doi: 10.1016/0166-2236(90)90107-L
[18]	Yin HH, Knowlton BJ (2006) The role of the basal ganglia in habit formation. Nat Rev Neurosci 7: 464-76. doi: 10.1038/nrn1919
[19]	Hiebert NM, Vo A, Hampshire A, et al. (2014) Striatum in stimulus–response learning via feedback and in decision making. Neuroimage 101: 448-457. doi: 10.1016/j.neuroimage.2014.07.013
[20]	Jenkins IH, Jahanshahi M, Jueptner M, et al. (2000) Self-initiated versus externally triggered movements. II. The effect of movement predictability on regional cerebral blood flow. Brain 123: 1216-1228. doi: 10.1093/brain/123.6.1216
[21]	Gasbarri A, Sulli A, Packard MG (1997) The dopaminergic mesencephalic projections to the hippocampal formation in the rat. Prog Neuropsychopharmacol Biol Psychiatry 21: 1-22. doi: 10.1016/S0278-5846(96)00157-1
[22]	Shu SY, Jiang G, Zeng QY, et al. (2015) The marginal division of the striatum and hippocampus has different role and mechanism in learning and memory. Mol Neurobiol 51: 827-839. doi: 10.1007/s12035-014-8891-6
[23]	Seger CA (2008) How do the basal ganglia contribute to categorization? Their roles in generalization, response selection, and learning via feedback. Neurosci Biobehav Rev 32: 265-278. doi: 10.1016/j.neubiorev.2007.07.010
[24]	Arroyo-García LE, Rodríguez-Moreno A, Flores G (2018) Apomorphine effects on the hippocampus. Neural Regen Res 13: 2064-2066. doi: 10.4103/1673-5374.241443
[25]	Bissonette GB, Roesch MR (2016) Neurophysiology of rule switching in the corticostriatal circuit. Neuroscience 14: 64-76.
[26]	Shu SY, Jiang G, Zeng QY, et al. (2019) A New neural pathway from the ventral striatum to the nucleus basalis of Meynert with functional implication to learning and memory. Mol Neurobiol 56: 7222-7233. doi: 10.1007/s12035-019-1588-0
[27]	Goodman J, Packard MG (2017) Memory Systems of the Basal Ganglia. Handbook of Basal Ganglia Structure and Function Academic Press, 725-740.
[28]	Peall KJ, Lorentzos MS, Heyman I, et al. (2017) A review of psychiatric co-morbidity described in genetic and immune mediated movement disorders. Neurosci Biobehav Rev 80: 23-25. doi: 10.1016/j.neubiorev.2017.05.014
[29]	Crittenden JR, Graybiel AM (2011) Basal ganglia disorders associated with imbalances in the striatal striosome and matrix compartments. Front Neuroanat 5: 59. doi: 10.3389/fnana.2011.00059
[30]	Graybiel AM (1997) The basal ganglia and cognitive pattern generators. Schizophr Bull 23: 459-69. doi: 10.1093/schbul/23.3.459
[31]	Leisman G, Moustafa AA, Shafir T (2016) Thinking, walking, talking: integratory motor and cognitive brain function. Front Public Health 4: 94. doi: 10.3389/fpubh.2016.00094
[32]	Nobili A, Latagliata EC, Viscomi MT, et al. (2017) Dopamine neuronal loss contributes to memory and reward dysfunction in a model of Alzheimer's disease. Nat Commun 8: 14727. doi: 10.1038/ncomms14727
[33]	Intzandt B, Beck EN, Silveira CRA (2018) The effects of exercise on cognition and gait in Parkinson's disease: A scoping review. Neurosci Biobeh Rev 95: 136-169. doi: 10.1016/j.neubiorev.2018.09.018
[34]	Crowley EK, Nolan YM, Sullivan AM (2019) Exercise as a therapeutic intervention for motor and non-motor symptoms in Parkinson's disease: Evidence from rodent models. Prog Neurobiol 172: 2-22. doi: 10.1016/j.pneurobio.2018.11.003
[35]	Feng YS, Yang SD, Tan ZX, et al. (2020) The benefits and mechanisms of exercise training for Parkinson's disease. Life Sci 245: 117345. doi: 10.1016/j.lfs.2020.117345
[36]	Cunnington R, Iansek R, Thickbroom GW, et al. (1996) Effects of magnetic stimulation over supplementary motor area on movement in Parkinson's disease. Brain 119: 815-822. doi: 10.1093/brain/119.3.815
[37]	Grieb B, von Nicolai C, Engler G, et al. (2013) Decomposition of abnormal free locomotor behavior in a rat model of Parkinson's disease. Front Syst Neurosci 7: 95. doi: 10.3389/fnsys.2013.00095
[38]	Bouchekioua Y, Tsutsui-Kimura I, Sanob H, et al. (2018) Striatonigral direct pathway activation is sufficient to induce repetitive behaviors. Neurosci Res 132: 53-57. doi: 10.1016/j.neures.2017.09.007
[39]	Smith KS, Graybiel AM (2016) Habit formation. Dialogues Clin Neurosci 18: 33-43. doi: 10.31887/DCNS.2016.18.1/ksmith
[40]	Jin X, Costa RM (2015) Shaping action sequences in basal ganglia circuits. Curr Opin Neurobiol 33: 188-196. doi: 10.1016/j.conb.2015.06.011
[41]	Graybiel AM (1998) The basal ganglia and chunking of action repertoires. Neurobiol Learn Mem 70: 119-136. doi: 10.1006/nlme.1998.3843
[42]	Thorn CA, Atallah H, Howe M, et al. (2010) Differential Dynamics of activity changes in dorsolateral and dorsomedial striatal loops during learning. Neuron 66: 781-795. doi: 10.1016/j.neuron.2010.04.036
[43]	Smith KS, Graybiel AM (2014) Investigating habits: strategies, technologies and models. Front Behav Neurosci 8: 39.
[44]	Iachini T, Ruggiero G, Ruotolo F, et al. (2014) Motor resources in peripersonal space are intrinsic to spatial encoding: Evidence from motor interference. Acta Psychol 153: 20-27. doi: 10.1016/j.actpsy.2014.09.001
[45]	Wang C, Chen X, Knierim JJ (2020) Egocentric and allocentric representations of space in the rodent brain. Curr Opin Neurobiol 60: 12-20. doi: 10.1016/j.conb.2019.11.005
[46]	Glover S, Bibby E, Tuomi E (2020) Executive functions in motor imagery: support for the motor-cognitive model over the functional equivalence model. Exp Brain Res 238: 931-944. doi: 10.1007/s00221-020-05756-4
[47]	Soriano M, Cavallo A, D'Ausilio A, et al. (2018) Movement kinematics drive chain selection toward intention detection. PNAS 115: 10452-10457. doi: 10.1073/pnas.1809825115
[48]	Del Vecchio M, Caruana F, Sartori I, et al. (2020) Action execution and action observation elicit mirror responses with the same temporal profile in human SII. Commun Biol 3: 80. doi: 10.1038/s42003-020-0793-8
[49]	McBride SD, Parker MO (2015) The disrupted basal ganglia and behavioural control: An integrative cross-domain perspective of spontaneous stereotypy. Behav Brain Res 276: 45-58. doi: 10.1016/j.bbr.2014.05.057
[50]	Guzulaitis R, Alaburda A, Hounsgaard J (2013) Increased activity of pre-motor network does not change the excitability of motoneurons during protracted scratch initiation. J Physiol 120: 2542-2554.
[51]	Grillner S, McClellan A, Perret C (1981) Entrainment of the spinal pattern generators for swimming by mechano-sensitive elements in the lamprey spinal cord in vitro. Brain Res 217: 380-386. doi: 10.1016/0006-8993(81)90015-9
[52]	Grillner S (1991) Recombination of motor pattern generators. Curr Biol 1: 231-3. doi: 10.1016/0960-9822(91)90066-6
[53]	Kalueff AV, Stewart AM, Song C, et al. (2015) Neurobiology of rodent self-grooming and its value for translational neuroscience. Nat Rev Neurosci 17: 45-59. doi: 10.1038/nrn.2015.8
[54]	Ashby FG, Turner BO, Horvitz JC (2010) Cortical and basal ganglia contributions to habit learning and Automaticity. Trends Cogn Sci 14: 208-215. doi: 10.1016/j.tics.2010.02.001
[55]	Redgrave R, Rodriguez M, Smith Y, et al. (2011) Goal-directed and habitual control in the basal ganglia: implications for Parkinson's disease. Nat Rev Neurosci 11: 760-772. doi: 10.1038/nrn2915
[56]	Dezfouli A, Balleine BW (2012) Habits, action sequences and reinforcement learning. Eur J Neurosci 35: 1036-1051. doi: 10.1111/j.1460-9568.2012.08050.x
[57]	Hernández LF, Redgrave P, Obeso JA (2015) Habitual behaviour and dopamine cell vulnerability in Parkinson disease. Front Neuroanat 9: 99.
[58]	Helie S, Chakravarthy S, Moustafa AA (2013) Exploring the cognitive and motor functions of the basal ganglia: an integrative review of computational cognitive neuroscience models. Front Comput Neurosci 7: 174. doi: 10.3389/fncom.2013.00174
[59]	Smith KS, Graybiel AM (2013) A dual operator view of habitual behavior reflecting cortical and striatal dynamics. Neuron 79: 361-374. doi: 10.1016/j.neuron.2013.05.038
[60]	Hart G, Leung BK, Balleine BW (2014) Dorsal and ventral streams: The distinct role of striatal subregions in the acquisition and performance of goal-directed actions. Neurobiol Learn Mem 108: 104-118. doi: 10.1016/j.nlm.2013.11.003
[61]	Hinman JR, Chapman GW, Hasselmo ME (2019) Neural representation of environmental boundaries in egocentric coordinates. Nat Commun 10: 2772. doi: 10.1038/s41467-019-10722-y
[62]	Florio TM (2017) The 6-Hydroxydopamine Hemiparkinsonian Rat Model: Evidence of Early Stage Degeneration of the Nigrostriatal Pathway. Xjenza Online Available from: https://www.xjenza.org/JOURNAL/OLD/5-2-2017/05.pdf.
[63]	Poldrack RA, Packard MG (2003) Competition among multiple memory systems: converging evidence from animal and human brain studies. Neuropsychologia 41: 245-251. doi: 10.1016/S0028-3932(02)00157-4
[64]	Calabresi P, Maj R, Pisani A, et al. (1992) Longterm synaptic depression in the striatum: physiological and pharmacological characterization. J Neurosci 12: 4224-4233. doi: 10.1523/JNEUROSCI.12-11-04224.1992
[65]	Calabresi P, Centonze D, Gubellini P, et al. (2000) Synaptic transmission in the striatum: from plasticity to neurodegeneration. Prog Neurobiol 61: 231-265. doi: 10.1016/S0301-0082(99)00030-1
[66]	Calabresi P, Picconi B, Tozzi A, et al. (2007) Dopamine mediated regulation of corticostriatal synaptic plasticity. Trends Neurosci 30: 211-219. doi: 10.1016/j.tins.2007.03.001
[67]	Pisani A, Centonze D, Bernardi G, et al. (2005) Striatal synaptic plasticity: Implications for motor learning and Parkinson's disease. Movement Disord 20: 395-402. doi: 10.1002/mds.20394
[68]	Centonze D, Picconi B, Gubellini P, et al. (2001) Dopaminergic control of synaptic plasticity in the dorsal striatum. Eur J Neurosci 13: 1071-1077. doi: 10.1046/j.0953-816x.2001.01485.x
[69]	Grahn JA, Parkinson JA, Owen AM (2009) The role of the basal ganglia in learning and memory: Neuropsychological studies. Behav Brain Res 199: 53-60. doi: 10.1016/j.bbr.2008.11.020
[70]	Atallah HE, Frank MJ, O'Reilly RC (2004) Hippocampus, cortex, and basal ganglia: Insights from computational models of complementary learning systems. Neurobiol Learn Mem 82: 253-267. doi: 10.1016/j.nlm.2004.06.004
[71]	Brown JW, Bullock D, Grossberg S (2004) How laminar frontal cortex and basal ganglia circuits interact to control planned and reactive saccades. Neural Networks 17: 471-510. doi: 10.1016/j.neunet.2003.08.006
[72]	Aron AR, Behrens TE, Smith S, et al. (2007) Triangulating a cognitive control network using diffusion-weighted magnetic resonance imaging (MRI) and functional MRI. J Neurosci 27: 3743-52. doi: 10.1523/JNEUROSCI.0519-07.2007
[73]	Baunez C, Nieoullon A, Amalric M (1995) Dopamine and complex sensorimotor integration: further studies in a conditioned motor task in the rat. Neuroscience 65: 375-384. doi: 10.1016/0306-4522(94)00498-T
[74]	Lehéricy S, Benali H, Van de Moortele PF, et al. (2005) Distinct basal ganglia territories are engaged in early and advanced motor sequence learning. Proc Natl Acad Sci 102: 12566-12571. doi: 10.1073/pnas.0502762102
[75]	Alexander GE, Crutcher MD (1990) Functional architecture of basal ganglia circuits: neural substrates of parallel processing. Trends Neurosci 13: 266-271. doi: 10.1016/0166-2236(90)90107-L
[76]	Lawrence AD, Sahakian BJ, Robbins TW (1998) Cognitive functions and corticostriatal circuits: insights from Huntington's disease. Trends Cogn Sci 2: 379-388. doi: 10.1016/S1364-6613(98)01231-5
[77]	Bédard P, Sanes JN (2009) On a basal ganglia role in learning and rehersing visual-MPTPr associations. Neuroimage 47: 1701-1710. doi: 10.1016/j.neuroimage.2009.03.050
[78]	Delgado MR, Miller MM, Inati S, et al. (2005) An fMRI study of reward-related probability learning. Neuroimage 24: 862-873. doi: 10.1016/j.neuroimage.2004.10.002
[79]	Grol MJ, de Lange FP, Verstraten FA, et al. (2006) Cerebral changes during performance of overlearned arbitrary visuomotor associations. J Neurosci 26: 117-125. doi: 10.1523/JNEUROSCI.2786-05.2006
[80]	Seger CA, Cincotta CM (2006) Dynamics of frontal, striatal, and hippocampal systems during rule learning. Cereb Cortex 16: 1546-1555. doi: 10.1093/cercor/bhj092
[81]	Boettiger CA, D'Esposito M (2005) Frontal networks for learning and executing arbitrary stimulus–response associations. J Neurosci 25: 2723-2732. doi: 10.1523/JNEUROSCI.3697-04.2005
[82]	Buch ER, Brasted PJ, Wise SP (2006) Comparison of population activity in the dorsal premotor cortex and putamen during the learning of arbitrary visuomotor mappings. Exp Brain Res 169: 69-84. doi: 10.1007/s00221-005-0130-y
[83]	Charpier S, Mahon S, Deniau JM (1999) In vivo induction of striatal long-term potentiation by low-frequency stimulation of the cerebral cortex. Neuroscience 91: 1209-1222. doi: 10.1016/S0306-4522(98)00719-2
[84]	Bonsi P, Florio T, Capozzo A, et al. (2003) Behavioural learning-induced increase in spontaneous GABAA-dependent synaptic activity in rat striatal cholinergic interneurons. Eur J Neurosci 17: 174-178. doi: 10.1046/j.1460-9568.2003.02410.x
[85]	Frank MJ, Loughry B, O'Reilly RC (2001) Interactions between the frontal cortex and basal ganglia in working memory: A computational model. Cogn Affect Behav Neurosci 1: 137-160. doi: 10.3758/CABN.1.2.137
[86]	Fino E, Glowinski J, Venance L (2005) Bidirectional activity-dependent plasticity at corticostriatal synapses. J Neurosci 25: 11279-11287. doi: 10.1523/JNEUROSCI.4476-05.2005
[87]	Fino E, Deniau JM, Venance L (2008) Cell-specific spike-timing-dependent plasticity in GABAergic and cholinergic interneurons in corticostriatal rat brain slices. J Physiol 586: 265-282. doi: 10.1113/jphysiol.2007.144501
[88]	Fino E, Deniau JM, Venance L (2009) Brief subthreshold events can act as Hebbian signals for long-term plasticity. PloS ONE 4: e6557. doi: 10.1371/journal.pone.0006557
[89]	Bliss TV, Collingridge GL (1993) A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361: 31-39. doi: 10.1038/361031a0
[90]	Martin SJ, Grimwood PD, Morris RG (2000) Synaptic plasticity and memory: an evaluation of the hypothesis. Annu Rev Neurosci 23: 649-711. doi: 10.1146/annurev.neuro.23.1.649
[91]	Martin SJ, Morris RG (2002) New life in an old idea: the synaptic plasticity and memory hypothesis revisited. Hippocampus 12: 609-636. doi: 10.1002/hipo.10107
[92]	Lynch MA (2004) Long-term potentiation and memory. Physiol Rev 84: 87-136. doi: 10.1152/physrev.00014.2003
[93]	Giordano N, Iemolo A, Mancini M, et al. (2018) Motor learning and metaplasticity in striatal neurons: relevance for Parkinson's disease. Brain 141: 505-520. doi: 10.1093/brain/awx351
[94]	Cameron IGM, Coe CB, Watanabe M, et al. (2009) Role of the basal ganglia in switching a planned response. Eur J Neurosci 29: 2413-2425. doi: 10.1111/j.1460-9568.2009.06776.x
[95]	Hikosaka O, Isoda M (2010) Switching from automatic to controlled behavior: cortico-basal ganglia mechanisms. Trends Cogn Sci 14: 154-161. doi: 10.1016/j.tics.2010.01.006
[96]	Obeso JA, Rodriguez-Oroz MC, Benitez-Temino B, et al. (2008) Functional organization of the basal ganglia: Therapeutic implications for Parkinson's Disease. Movement Disord 23: S548-S559. doi: 10.1002/mds.22062
[97]	Mink JW (1996) The basal ganglia: focused selection and inhibition of competing motor programs. Prog Neurobiol 50: 381-425. doi: 10.1016/S0301-0082(96)00042-1
[98]	Reynolds JNJ, Hyland BI, Wickens JR (2019) A cellular mechanism of reward-related learning. Nature 413: 67-70. doi: 10.1038/35092560
[99]	Vandaele Y, Mahajan NR, Ottenheimer DJ, et al. (2019) Distinct recruitment of dorsomedial and dorsolateral striatum erodes with extended training. eLife 8: e49536. doi: 10.7554/eLife.49536
[100]	Klaus A, Martins GJ, Paixao VB, et al. (2017) The spatiotemporal organization of the striatum encodes action space. Neuron 95: 1171-1180. doi: 10.1016/j.neuron.2017.08.015
[101]	Florio TM, Scarnati E, Rosa I, et al. (2018) The basal ganglia: More than just a switching device. CNS Neurosci Ther 24: 677-684. doi: 10.1111/cns.12987
[102]	Florio TM, Confalone G, Sciarra A, et al. (2013) Switching ability of over trained movements in a Parkinson's disease rat model. Behav Brain Res 250: 326-333. doi: 10.1016/j.bbr.2013.05.020
[103]	Patel N, Jankovic J, Hallett M (2014) Sensory aspects of movement disorders. Lancet Neurol 13: 100-112. doi: 10.1016/S1474-4422(13)70213-8
[104]	Williams-Gray CH, Worth PF (2016) Parkinson's disease. Movement Disorders. Medicine (Baltimore) 44: 542-546. doi: 10.1016/j.mpmed.2016.06.001
[105]	Asakawa T, Fang H, Sugiyama K, et al. (2016) Human behavioural assessments in current research of Parkinson's disease. Neurosci Biobehav Rev 68: 741-772. doi: 10.1016/j.neubiorev.2016.06.036
[106]	Obeso JA, Stamelou M, Goetz CG, et al. (2017) Past, present, and future of Parkinson's disease: A special essay on the 200th Anniversary of the Shaking Palsy. Mov Disord 32: 1264-1310. doi: 10.1002/mds.27115
[107]	Berganzo K, Tijero B, González-Eizaguirre A, et al. (2016) Motor and non-motor symptoms of Parkinson's disease and their impact on quality of life and on different clinical subgroups. Neurologia 31: 585-591. doi: 10.1016/j.nrl.2014.10.010
[108]	Cameron IGM, Watanabe M, Pari G, et al. (2010) Executive impairment in Parkinson's disease: Response automaticity and task switching. Neuropsychologia 48: 1948-57. doi: 10.1016/j.neuropsychologia.2010.03.015
[109]	Magrinelli F, Picelli A, Tocco P, et al. (2016) Pathophysiology of Motor Dysfunction in Parkinson's Disease as the Rationale for Drug Treatment and Rehabilitation. Parkinsons Dis 2016: 9832839.
[110]	Mancini M, Carlson-Kuhta P, Zampieri C, et al. (2012) Postural sway as a marker of progression in Parkinson's disease: A pilot longitudinal study. Gait Posture 36: 471-476. doi: 10.1016/j.gaitpost.2012.04.010
[111]	Rosa I, Di Censo D, Ranieri B, et al. (2020) Comparison between Tail Suspension Swing Test and Standard Rotation Test in revealing early motor behavioral changes and neurodegeneration in 6-OHDA hemiparkinsonian rats. Int J Mol Sci 21: 2874. doi: 10.3390/ijms21082874
[112]	Siragy T, Nantel J (2020) Absent arm swing and dual tasking decreases trunk postural control and dynamic balance in people with Parkinson's disease. Front Neurol 11: 213. doi: 10.3389/fneur.2020.00213
[113]	Fox ME, Mikhailova MA, Bass CE, et al. (2016) Cross-hemispheric dopamine projections have functional significance. Proc Natl Acad Sci 113: 6985-6990. doi: 10.1073/pnas.1603629113
[114]	Lisman JE, Grace AA (2005) The Hippocampal-VTA loop: Controlling the entry of information into long-term memory. Neuron 46: 703-713. doi: 10.1016/j.neuron.2005.05.002
[115]	Cordella A, Krashia P, Nobili A, et al. (2018) Dopamine loss alters the hippocampus-nucleus accumbens synaptic transmission in the Tg2576 mouse model of Alzheimer's disease. Neurobiol Dis 116: 142-154. doi: 10.1016/j.nbd.2018.05.006
[116]	Casarrubea M, Di Giovanni G, Crescimanno G, et al. (2019) Effects of Substantia Nigra pars compacta lesion on the behavioural sequencing in the 6-OHDA model of Parkinson's disease. Behav Brain Res 362: 28-35. doi: 10.1016/j.bbr.2019.01.004
[117]	Gilat M, Bell PT, Ehgoetz Martens KA, et al. (2017) Dopamine depletion impairs gait automaticity by altering cortico-striatal and cerebellar processing in Parkinson's disease. Neuroimage 152: 207-220. doi: 10.1016/j.neuroimage.2017.02.073
[118]	Wissner-Gross AD, Freer CE (2013) Causal entropic forces. Phys Rev Lett 110: 168702-1-168702-5.
[119]	Merchant H, Averbeck BB (2017) The computational and neural basis of rhythmic timing in medial premotor cortex. J Neurosci 37: 4552-4564. doi: 10.1523/JNEUROSCI.0367-17.2017
[120]	Athalye VR, Carmena JM, Costa RM (2020) Neural reinforcement: re-entering and refining neural dynamics leading to desirable outcomes. Curr Opin Neurobiol 60: 145-154. doi: 10.1016/j.conb.2019.11.023

This article has been cited by:

1.	Saeed Ahmad, Saud Owyed, Abdel-Haleem Abdel-Aty, Emad E. Mahmoud, Kamal Shah, Hussam Alrabaiah, Mathematical analysis of COVID-19 via new mathematical model, 2021, 143, 09600779, 110585, 10.1016/j.chaos.2020.110585
2.	Abdon Atangana, Seda İğret Araz, Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-03095-w
3.	Isa Abdullahi Baba, Abdullahi Yusuf, Kottakkaran Sooppy Nisar, Abdel-Haleem Abdel-Aty, Taher A. Nofal, Mathematical model to assess the imposition of lockdown during COVID-19 pandemic, 2021, 20, 22113797, 103716, 10.1016/j.rinp.2020.103716
4.	Prasantha Bharathi Dhandapani, Jayakumar Thippan, Dumitru Baleanu, Vinoth Sivakumar, On a novel fuzzy fractional retarded delay epidemic model, 2022, 7, 2473-6988, 10122, 10.3934/math.2022563
5.	Prasantha Bharathi Dhandapani, Dumitru Baleanu, Jayakumar Thippan, Vinoth Sivakumar, New Fuzzy Fractional Epidemic Model Involving Death Population, 2021, 37, 0267-6192, 331, 10.32604/csse.2021.015619
6.	Maheswari Rangasamy, Nazek Alessa, Prasantha Bharathi Dhandapani, Karuppusamy Loganathan, Dynamics of a Novel IVRD Pandemic Model of a Large Population over a Long Time with Efficient Numerical Methods, 2022, 14, 2073-8994, 1919, 10.3390/sym14091919
7.	Kalpana Umapathy, Balaganesan Palanivelu, Renuka Jayaraj, Dumitru Baleanu, Prasantha Bharathi Dhandapani, On the decomposition and analysis of novel simultaneous SEIQR epidemic model, 2022, 8, 2473-6988, 5918, 10.3934/math.2023298
8.	Emmanuel A. Bakare, Snehashish Chakraverty, Radovan Potucek, Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method, 2021, 10, 2075-1680, 114, 10.3390/axioms10020114
9.	Maheswari Rangasamy, Christophe Chesneau, Carlos Martin-Barreiro, Víctor Leiva, On a Novel Dynamics of SEIR Epidemic Models with a Potential Application to COVID-19, 2022, 14, 2073-8994, 1436, 10.3390/sym14071436
10.	Prasantha Bharathi Dhandapani, Víctor Leiva, Carlos Martin-Barreiro, Maheswari Rangasamy, On a Novel Dynamics of a SIVR Model Using a Laplace Adomian Decomposition Based on a Vaccination Strategy, 2023, 7, 2504-3110, 407, 10.3390/fractalfract7050407
11.	Shomaila Mazhar, Zahid Ullah, Syed Inayat Ali Shah, Noor Badshah, Development of a probabilistic model for Covid-19 dynamics with consideration of non-severe and severe infections, 2023, 82, 11100168, 126, 10.1016/j.aej.2023.09.067

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Neuroscience

3.1 4.2

Metrics

Article views(4317) PDF downloads(332) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Neuroscience

Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?

Related Papers:

Abstract

1. Introduction

1.1. Coronavirus disease 2019 (COVID-19)

2. Preliminaries

2.1. Stiff differential equations

2.2. Fuzzy differential equations

3. Model formulation

3.1. Description of the model

4. Stability analysis and stiff system test

4.1. Equilibrium points

4.2. Average transmission number: N₀

4.3. Stability analysis using eigen values

4.4. Stiff system of fuzzy differential equation-test

5. Numerical simulations

6. Concluding remarks

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Neuroscience

Stereotyped, automatized and habitual behaviours: are they similar constructs under the control of the same cerebral areas?

Related Papers:

Abstract

1. Introduction

1.1. Coronavirus disease 2019 (COVID-19)

2. Preliminaries

2.1. Stiff differential equations

2.2. Fuzzy differential equations

3. Model formulation

3.1. Description of the model

4. Stability analysis and stiff system test

4.1. Equilibrium points

4.2. Average transmission number: N0

4.3. Stability analysis using eigen values

4.4. Stiff system of fuzzy differential equation-test

5. Numerical simulations

6. Concluding remarks

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

4.2. Average transmission number: N₀