Mathematical Modeling of Human Emotions using Neural Network - A Qualitative Approach G. Shirisha* 1Department of Mathematics, Stanley College of Engineering and Technology for Women, Abids, Hyderabad-500001, Telangana, India. deepasiri82@gmail.com Abstract In this article, a model is proposed to study the impact of concepts stored in the memory from earlier experiences on the output of emotions. The model with constant and time-varying inputs has been considered. To show that the systems are well behaved, sufficient conditions for global asymptotic stability are derived for the system with constant inputs, and sufficient conditions for asymptotic nearness and boundedness are derived for the system with time-varying inputs. Numerical examples with simulations are illustrated to support the theory. Keywords: Cooperative and supportive networks, Time delays, Equilibria, Global stabil- ity, Emotions, Memory. Mathematics Subject Classification (2010) :34D23, 34K20, 91E40, 92B20. 1 Introduction Emotions are a class of feelings which act as a source of non-verbal information to better understand what is being communicated. A human being is exposed to different kinds of emotions with different degrees of intensity by various experiences in life. Emotions have a huge impact on the behavior of a person as they get influenced through motivation and aggression. Emotions steer the decision-making process by creating certain feelings. Dif- ferent emotions affect decisions in different ways, like anger can lead to impatience, rash decision-making, and if the person is afraid, the decisions may be uncertain, and it might take them longer to choose. Emotions and social life are intimately connected; as to how a person judges or understands or forms a relationship with another person depends on his/her emotions. Thus emotions have a vital role in day to day life of a human being. The role of emotions from their perspective is being studied by several researchers from a variety of disciplines, including the social sciences, biological sciences, mathematical sci- ences, engineering sciences, and many more. The psychological aspects of the causes and Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133x 2346 effects of these emotions have been the attention of psychologists. They have described a number of emotional theories to anticipate or identify an individual’s emotion. The im- pact of brain activity on emotions and their reactions has been investigated by biologists. Computer scientists have attempted to use artificial neural networks to identify or predict emotions. Various methods have been used by mathematicians to attempt to formulate a mathematical model for emotions. In psychology, according to the classical view of emotion, emotions can be assessed objec- tively and accurately through facial expressions. But in recent literature it is said that the facial expressions or heart rate variability or body movements don’t give fingerprints to a particular emotion, i.e., the particular pattern of body movement or facial expression may not represent a particular emotion i.e., a smile on a face or furrow on your brow may not always represent happiness or anger [2, 9, 10]. When a person is angry, the way he gives response varies from person to person, they may express it with a scowl or with glowering looks or by shouting or they may become quiet. Thus, there is no particular fingerprint for any emotion, it varies from person to person and time to time. But how does the brain guide us to take a particular action for different emotions based on the situation? This is a key point on which many researchers are working. It is said in [2] that these actions for emotions are generated by the brain by using the concepts stored in memory from earlier experiences. We will try to model this idea of how the past experiences will contribute to the present outcome of emotion. As we will be dealing with mathematical models and neural networks, we will see some of the works of mathematicians and computer scientists on emotions. Mathematical models to study emotions are proposed in [3,7,11]. Various neural networks for studying emotions were introduced and studied in [4- 6, 8, 16-17, 21, 23]. Most of the research was to predict or recognize the emotions, which has a wide application in the fields of marketing, media and communication, economic theory, banking, hospitality industry, etc. We will focus on the study of emotions based on the previous experiences, which will help us to predict the outcome of it. In this paper, we have tried to understand how the concepts stored in the memory of past experiences contribute to the output of an emotion using a Cooperative and Supportive Neural Network (CSNN) model. CSNN concentrates on the contribution of collective ca- pabilities and distributive operations of neurons. CSNN is a more reliable model that can be used for classification and clustering problems, data mining, financial and economic systems, and industrial information systems. CSNN model was propounded in [19] and various modifications of the system were studied in [12-14]. This network was considered for estimation of key parameters in infectious disease models [17] and in a recent study, it was used to understand the interaction between the focal and non-focal parts of the human brain [15]. The paper is organised as follows. In Section 2, the generalised CSNN model with constant inputs has been described in terms of our preposition, and its behaviour has 2 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2347 been studied by deriving sufficient conditions for global asymptotic stability. In Section 3, the generalised CSNN model with time-varying inputs has been described, and its be- haviour has been studied by deriving conditions for asymptotic nearness and boundedness of solutions. Followed by a discussion in Section 4. 2 Description of the Model with Constant Inputs-Behavior of the Solutions A CSNN model comprises of two neuronal fields Fx and Fy. The neurons in Fx are denoted by xi’s and the neurons in Fy are denoted by yik ’s, for i = 1, 2, 3....n, k = 1, 2, 3.....ri, 1 ≤ ri ≤ n. Each neuron xi in Fx is connected to the other neurons xj for i ̸= j in the same neuronal field and they are also connected to ri number of neurons (yi1 , yi2 ....yiri ) in the neuronal field Fy. The pictorial representation of this network can be seen in the Figure 1. Figure 1 The dynamics of the model will be given by the following system of equations x ′ i = −aixi + n∑ j=1 bijfj(xj(t)) + ri∑ k=1 ciikgik(xi, yik(t)) + Ii, y ′ ik = −cikyik + ri∑ l=1 dilhil(yil(t)) + Jik , i = 1, 2, 3....n, k = 1, 2, 3.....ri, 1 ≤ ri ≤ .n (1) Here xi’s be a state of particular emotion like happiness, anger, sadness, fear, etc., and yik ’s be the neurons corresponding to different memories stored from the previous experiences for a particular emotion xi. ai be the resting potential of an emotion xi(i.e., 3 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2348 the rate at which the neuron of a particular emotion is not active). cik be the resting potential of yik (i.e., the rate at which the neuron corresponding to a particular memory is not active). bij be the synaptic connection strength among emotions xi and xj . dil be the synaptic connection strength among the memories of a particular emotion xi. ciik be the rate at which particular memory is influencing the emotion. The functions fj be the functional relation among the emotions xj ’s (for example, emotions like joy and happiness will inhibit the occurrence of anger or sadness, whereas emotion like enjoyment will enhance the occurrence of happiness and satisfaction [22]), gik be the functional relation which shows how yik ’s are influencing xi’s, hil be the response function of yil towards yik (i.e., how the memories of particular emotions are interrelated). Ii, Jik be the external inputs which may be information from the outside world or information from inside the body (or sensory inputs). They could as well be parts of the input that provoked and invoked particular emotion and its past memory. It is noted that a person may not react to an emotion immediately, it may be because of the concurrency of many emotions at the same time, so a processing delay may arise among xi’s. Also, it’s quite natural that the brain may take some time to recollect previously stored memory, which results in the processing delay among yik ’s. In certain situations, it may take some time for the concepts of previous memories to show their influence on emotions, as the brain may be engaged with another activity, which leads to transmission delays. To make the model more realistic, we will incorporate these delays in (1). Hence, we get x ′ i = −aixi + n∑ j=1 bijfj(xj(t− τj)) + ri∑ k=1 ciikgik(xi, yik(t− ϑik)) + Ii, y ′ ik = −cikyik + ri∑ l=1 dilhil(yil(t− ζil)) + Jik , i = 1, 2, 3....n, k = 1, 2, 3.....ri, 1 ≤ ri ≤ n. (2) Where in τi’s and ζik ’s are the processing delays among xi’s and yik ’s respectively. ϑik is the transmission delay from yik ’s to xi’s. The occurrence of delay will depend on the requirement of the problem under study, so the presence of all the three delays may not require always. Hence, we can deduce different models by considering one or two delays to exist. Some of these deduced models are the open problems II, III, and IV, which are proposed in [19] and one of them is the model that has been studied in [12]. The results that are going to be derived for (2) will apply to all the deduced models and also to the basic CSNN model (1). We assume the following Lipschitz conditions on the response functions, as they are needed for the solutions to exist. ∥gik(xi, yik)− gik(xi, yik)∥ ≤ M1ik |yik − yik |+M2ik |xi − xi|, 4 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2349 |fj(xj)− fj(xj)| ≤ pj |xj − xj |, |hik(yik)− hik(yik)| ≤ qik |yik − yik |, (3) for some positive constants M1ik ,M2ik , pj and qik . By the theory of delay differential equations, we know that the local Lipschitz conditions on response functions guarantee the existence of solutions. Thus, the system (2) possesses unique solutions that are continuous in their maximal interval of existence with suitable initial conditions [12]. Now we know the solutions of the system (2) exist (i.e., concepts stored in the previous memory are provoking a person to respond with an emotion). But how do they behave? As we are considering system (2) to study the outcome of emotions based on the concepts stored in memory, solutions should be controllable, i.e., the solutions should be either bounded or converge to a particular solution (i.e., the outcome of emotion should not be too wild). As a system (2) is an autonomous system, it may have an equilibrium solution. We can consider that equilibrium is the optimal way of responding to emotion, showing one’s emotional stability. So first, we will check for what conditions does the equilibrium of the system exist. Since the existence of equilibria is not affected by the presence of time delays, as in [12, 13, 19], we may establish that Theorem 2.1. If the output functions satisfy conditions (3) and the parameters satisfy conditions n∑ j=1 |bji|pj + ri∑ k=1 |ciik |M2ik < ai, 1 ai ri∑ k=1 |ciik |M1ik + 1 cik ri∑ k=1 |dik |qik < 1. (4) Then the system (2) has a unique equilibrium solution. Thus, under conditions (4), system (2) possesses a unique equilibrium, and they may typically be represented by (x∗i , y ∗ ik ). As we know that equilibria are stationary solutions of the system, we can write (xi − x∗i ) ′ = −ai(xi − x∗i ) + n∑ j=1 bij(fj(xj(t− τj))− fj(x ∗ j )) + ri∑ k=1 ciik(gik(xi, yik(t− ϑik))− gik(x ∗ i , y ∗ ik )), (yik − y∗ik) ′ = −cik(yik − y∗ik) + ri∑ l=1 dil(hil(yil(t− ζil))− hil(y ∗ il )), (5) where i = 1, 2, 3....n, k = 1, 2, 3.....ri and 1 ≤ ri ≤ n. We will be using equation (5) whenever required in our results. 5 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2350 Remark 2.2. From here on, we assume that the equilibrium always exists in our system. We notice that conditions (4) are sufficient and simpler conditions may exist that ensure existence. However, we may recall Theorem 2.1 whenever necessary. Now the question arises, under what conditions do the solutions of (2) converge to the equilibrium point. So next we try to derive sufficient conditions for the solutions to converge to equilibria, which are nothing but the conditions for global asymptotic stability of the system. Two types of conditions are derived here using the Lyapunov functional technique. First one is delay independent stability where the parametric conditions are derived without restricting the delays and the second one is delay-dependent stability where the conditions are derived by restricting the delay parameters to a particular region. Let us first derive the delay- independent conditions. 2.1 Delay Independent Here, we have obtained parametric conditions for asymptotic stability without restricting delay parameters. These conditions are for those situations where emotional stability is not affected by time delays. In this regard, we state the following theorem Theorem 2.3. Assume that the output function fi, gik and hil satisfy the conditions (3) then the equilibrium (x∗i , y ∗ ik ) of (2) is globally asymptotically stable, provided the parameters of the system satisfy the inequalities ai > n∑ j=1 |bji|pi + ri∑ k=1 |ciik |M2ik , cik > ri∑ k=1 |dik |qik . (6) Proof. Consider the functional V1 = |yik − y∗ik |+ ri∑ l=1 |dil |qil ∫ t t−ζil |yil(z)− y∗il |dz 6 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2351 Then the upper dini derivatives along the solutions of (2) is given by D+V1 ⩽ −cik |yik − y∗ik |+ ri∑ l=1 |dil ||hil(yil(t− ζil))− hil(y ∗ il )| + ri∑ l=1 |dil |qil [ |yil − y∗il | − |yil(t− ζil)− y∗il | ] ⩽ − [ cik − ri∑ k=1 |dik |qik ] |yik − y∗ik | ⩽ −AV1 Where A =min { cik − ∑ri k=1 |dik |qik , i = 1, 2...n, 1 ≤ k ≤ ri } By hypothesis A > 0, therefore we get D+V1 < −AV1 < 0 ⇒ D+V1 +AV1 < 0. By comparision theorem using V1 ≥ 0, we have V1 −→ 0 as t −→ ∞. Thus for sufficiently large t, say t > t∗, we have yik −→ y∗ik Now in order to prove xi −→ x∗i , we consider V2 = n∑ i=1 [ |xi − x∗i |+ n∑ j=1 |bij |pj ∫ t t−τj |xj(z)− x∗j |dz + ri∑ k=1 |ciik |M1ik ∫ t t−ϑik |yik(z)− y∗ik |dz ] for t > T = t∗ +Max{τj} for all j and proceeding as above we get D+V2 ≤ − n∑ i=1 [ ai − ri∑ k=1 |ciik |M2ik − n∑ j=1 |bji|pi ] |xi − x∗i | = − n∑ i=1 B|xi − x∗i | (7) where B = min { ai − ∑ri k=1 |ciik |M2ik − ∑n j=1 |bji|pi } By hypothesis B > 0, therefore D+V2 < 0. If we integrate (7) from 0 to t with respect to t we get V2(t) + ∫ t 0 n∑ i=1 B|xi − x∗i |dt ≤ V2(0) < ∞ Therefore we can say that V2(t) & xi and there derivatives are bounded on [0,∞). Thus xi’s are uniformly continuous. Applying Barbalat’s Lemma we get |xi − x∗i | −→ 0 as t −→ ∞. Thus xi(t) −→ x∗i as t −→ ∞. Hence the equilibrium (x∗i , y ∗ ik ) is globally asymptotically stable. 7 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2352 Remark 2.4. Theorem 2.3 gives sufficient conditions for the asymptotic stability of the system (2). These conditions depend on the Lyapunov functions that have been used. So, as the Lyapunov function varies, these conditions also vary. In the next section, we derive the delay-dependent stability conditions. 2.2 Delay Dependent Emotions are interrelated to each other, for instance, if a person is having the guilt of hurting his friend, as long as he overcomes it by expressing or giving an apology to his friend, he will not be happy and joyful (processing delays among xi’s). Thus, the delay in overcoming the feeling of guilt will disturb him mentally. It is quite natural for a human being to forget things, like remembering birthdays. Many of us remember the month of the birthday or anniversary of our friends, but don’t remember the exact date (processing delays among yik ’s). Many times, humans tend to neglect certain things when they are indulging in many activities, or maybe because of priorities. For instance, a person may know that he will feel happy by talking to his friends or family, but he may not do it because of his work schedule (transmission delays). Which may lead him to feel lonely and depressed. Thus, delay affects the emotions in many ways. In this section, we have obtained conditions on parameters for global asymptotic stability of the system with suit- able restrictions on delay parameters. These types of conditions were not discussed in the previous study of this model. We propose the following change of variables, for simplicity We let zi(t) = xi − x∗i , wik = yik − y∗ik , Hik(wik) = hik(yik)− hik(y ∗ ik ) Fi(zi) = fi(xi)− fi(x ∗ i ), and Gik(zi, wik) = gik(xi, yik)− gik(x ∗ i , y ∗ ik ) Then, using (4,) system (2) can be written as z ′ i = −aiz ′ i + n∑ j=1 bijFj(zj(t− τj)) + ri∑ k=1 ciikGik(zi, wik(t− ϑik)) w ′ ik = −cikwik + ri∑ l=1 dilHil(wil(t− ζil)) (8) Further conditions (3) reduce to |Fj(zj(t− τj))| ≤ pj |zj(t− τj)| |Gik(zi, wik(t− ϑik))| ≤ M1ik |wik(t− ϑik)|+M2ik |zi| |Hil(wil(t− ζil))| ≤ qil |wil(t− ζil)| (9) Theorem 2.5. Suppose the output functions of the system (2) satisfy conditions(3) and 8 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2353 the parameters satisfy. A = Min [ ai − n∑ j=1 |bji|pi − ri∑ k=1 |ciik |M2ik ] > 0 B = Min [ cik − |ciik |M1ik − ri∑ k=1 |dik |qik ] > 0 C = Min [ n∑ j=1 |bji|pi ( ai + n∑ j=1 |bji|pi + ri∑ k=1 |ciik |M2ik )] D = Min [ n∑ j=1 |bji|pi|ciik |M1ik ] E = Min [ |ciik |M1ik ( cik + ri∑ k=1 |dik |qik )] F = Min [ ri∑ k=1 |dik |qik ( cik + ri∑ k=1 |dik |qik )] Let τ∗ = Max { τi, 1 ≤ i ≤ n } , ϑ∗ = Max { ϑik , 1 ≤ i ≤ n, 1 ≤ k ≤ ri } and ζ∗ = Max { ζik , 1 ≤ i ≤ n, 1 ≤ k ≤ ri } . Then the equilibrium (x∗i , y ∗ ik ) of (2) is globally asymp- totically stable for 0 < τ∗ < r, 0 < ϑ∗ < s and 0 < ζ∗ < p where r = Min { A C , B D } , s = B E and p = B F . Proof. Let V1 = ∑n i=1 |xi(t)− x∗i | = ∑n i=1 |zi(t)| 9 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2354 The upper dini derivative of V1 along the solutions of (7) is given by D+V1(t) ≤ n∑ i=1 [ − ai|zi(t)|+ n∑ j=1 |bji|pi|zi(t− τi)|+ ri∑ k=1 |ciik | [ M1ik |wik(t− ϑik)|+M2ik |zi(t)| ]] (10) zi(t− τi) = zi(t)− ∫ t t−τi z ′ i(s)ds = zi(t)− ∫ t t−τi [ − aizi(s) + n∑ j=1 bijFj(zj(s− τj)) + ri∑ k=1 ciikGik(zi(s), wik(s− ϑik)) ] ds (11) wik(t− ϑik) = wik(t)− ∫ t t−ϑik w ′ ik (s)ds = wik(t)− ∫ t t−ϑik [ − cikwik(s) + ri∑ l=1 dilHil(wil(s− ζil)) ] ds (12) Substituting (10) and (11) in (9) D+V1(t) ≤ n∑ i=1 [ − ai|zi(t)|+ n∑ j=1 |bji|pi [ |zi(t)|+ ∫ t t−τi ai|zi(s)|ds+ n∑ j=1 |bij | ∫ t t−τi |Fj(zj(s− τj))|ds + ri∑ k=1 |ciik | ∫ t t−τi |Gik(zi(s), wik(s− ϑik)|ds ] + ri∑ k=1 |ciik |M1ik [ |wik(t)|+ ∫ t t−ϑik cik |wik(s)|ds+ ri∑ l=1 |dil | ∫ t t−ϑik |Hil(wil(s− ζil))|ds ] + ri∑ k=1 |ciik |M2ik |zi(t)| ] Using conditions (8), we get D+V1(t) ≤ n∑ i=1 [ − ai|zi(t)|+ n∑ j=1 |bji|pi [ |zi(t)|+ ∫ t t−τi ai|zi(s)|ds+ n∑ j=1 |bji| ∫ t t−τi pi|(zi(s− τi))|ds + ri∑ k=1 |ciik | ∫ t t−τi (M2ik |zi(s)|+Mik |wik(s− ϑik)|)ds ] 10 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2355 + ri∑ k=1 |ciik |M1ik [ |wik(t)|+ ∫ t t−ϑik cik |wik(s)|ds+ ri∑ l=1 |dil | ∫ t t−ϑik qil |(wil(s− ζil))|ds ] + ri∑ k=1 |ciik |M2ik |zi(t)| ] (13) Let V2(t) = n∑ i=1 [ n∑ j=1 |bji|pi [ ai ∫ t t−τi ds ∫ t s zi(u)du+ n∑ j=1 |bji|pi ∫ t t−τi ds ∫ t s |(zi(u− τi))|du + ri∑ k=1 |ciik |M2ik ∫ t t−τi ds ∫ t s |(zi(u))|du+ ri∑ k=1 |ciik |M1ik ∫ t t−τi ds ∫ t s |wik(u− ϑik)|du + τi n∑ j=1 |bji|pi ∫ t t−τi |(zi(s))|ds+ τi ri∑ k=1 |ciik |M1ik ∫ t t−ϑik |wik(s)|ds ] + ri∑ k=1 |ciik |M1ik [ cik ∫ t t−ϑik ds ∫ t s |wik(u)|du+ ri∑ l=1 |dil |qil ∫ t t−ϑik ds ∫ t s |(wil(u− ζil))|du + ϑik ri∑ l=1 |dil |qil ∫ t t−ζil |wil(s)|ds ]] D+V2(t) ≤ n∑ i=1 [ n∑ j=1 |bji|pi [ τi ( ai|zi(t)|+ n∑ j=1 |bji|pi|(zi(t))|+ ri∑ k=1 |ciik |(M2ik |zi(t)|+M1ik |wik(t)|) ) − ai ∫ t t−τi |zi(s)|ds− n∑ j=1 |bji| ∫ t t−τi pi|(zi(s− τi))|ds− ri∑ k=1 |ciik | ∫ t t−τi M2ik |zi(s)|ds − ri∑ k=1 |ciik | ∫ t t−τi M1ik |wik(s− ϑik)|ds ] + ri∑ k=1 |ciik |M1ik [ ϑik ( cikwik(t) + ri∑ l=1 |dil |qilwil(t) ) − cik ∫ t t−ϑik |wik(s)|ds− ri∑ l=1 |dil |qil ∫ t t−ϑik |(wil(s− ζil))|ds ] (14) Let V3(t) = ∑n i=1 ∑ri k=1 |wik(t)|. Then the dini derivative along the solutions of (7) is given by D+V3(t) ≤ n∑ i=1 ri∑ k=1 [ − cik |wik(t)|+ ri∑ l=1 |dil |qil |wil(t− ζil)| ] (15) 11 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2356 wil(t− ζil) = wil(t)− ∫ t t−ζil w ′ il (s)ds = wil(t)− ∫ t t−ζil [ − cilwil(s) + ri∑ l=1 dilHil(wil(s− ζil)) ] ds (16) Substituting (15) in (14) D+V3(t) ≤ n∑ i=1 ri∑ k=1 [ − cik |wik(t)|+ ri∑ l=1 |dil |qil ( |wil(t)|+ ∫ t t−ζil ( cil |wil(s)|+ ri∑ l=1 |dil |qil |wil(s− ζil)| ) ds ) | ] (17) V4(t) = n∑ i=1 ri∑ k=1 [ ri∑ l=1 |dil |qil (∫ t t−ζil ds ∫ t s cil |wil(u)|du+ ri∑ l=1 |dil |qil ∫ t t−ζil ds ∫ t s |(wil(u− ζil))|du + ζil ri∑ l=1 |dil |qil ∫ t t−ζil |wil(s)|ds )] D+V4(t) ≤ n∑ i=1 ri∑ k=1 [ ri∑ l=1 |dil |qil ( ζil ( cil |wil(t)|+ ri∑ l=1 |dil |qil |wil(t)| ) − ∫ t t−ζil cil |wil(s)|ds − ri∑ l=1 |dil |qil ∫ t t−ζil |(wil(s− ζil))|ds )] (18) Let V (t) = V1(t) + V2(t) + V3(t) + V4(t) Using (12),(13),(16) and (17) we get D+V (t) ≤ n∑ i=1 [ − ai|zi(t)|+ n∑ j=1 |bji|pi|zi(t)|+ ri∑ k=1 |ciik |M1ik |wik(t)|+ ri∑ k=1 |ciik |M2ik |zi(t)| ] + n∑ i=1 [ n∑ j=1 |bji|piτi ( ai|zi(t)|+ n∑ j=1 |bij |pj |(zj(t))|+ ri∑ k=1 |ciik |(M2ik |zi(t)|+M1ik |wik(t)|) ) + ri∑ k=1 |ciik |M1ikϑik ( cikwik(t) + ri∑ l=1 |dil |qilwil(t) )] + n∑ i=1 ri∑ k=1 [ − cik |wik(t)| + ri∑ l=1 |dil |qil |wil(t)| ] + n∑ i=1 ri∑ k=1 [ ri∑ l=1 |dil |qilζil ( cilwil(t) + ri∑ l=1 |dil |qil |wil(t)| )] 12 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2357 D+V (t) ≤ − n∑ i=1 [(( ai − n∑ j=1 |bji|pi − ri∑ k=1 |ciik |M2ik ) − τi n∑ j=1 |bji|pi ( ai + n∑ j=1 |bij |pj + ri∑ k=1 |ciik |M2ik )) |zi(t)| + ri∑ k=1 (( cik − |ciik |M1ik − ri∑ k=1 |dik |qik ) − τi n∑ j=1 |bji|pi|ciik |M1ik − ϑik |ciik |M1ik ( cik + ri∑ k=1 |dik |qik ) − ζik( ri∑ k=1 |dik |qik ( cik + ri∑ k=1 |dik |qik )) |wik(t)| ] ≤ − n∑ i=1 [( A− Cτ∗ ) |zi(t)|+ ri∑ k=1 ( B −Dτ∗ − Eϑ∗ − Fζ∗ ) |wik(t)| ] (19) Let r = Min { A C , B D } , s = B E and p = B F . We take τ∗ = Max { τi, 1 ≤ i ≤ n } , ϑ∗ = Max { ϑik , 1 ≤ i ≤ n, 1 ≤ k ≤ ri } and ζ∗ = Max { ζik , 1 ≤ i ≤ n, 1 ≤ k ≤ ri } Clearly from hypothesis and (18), we have 0 < τ∗ < r, 0 < ϑ∗ < s, 0 < ζ∗ < p. Thus we have D+V (t) < 0. Therefore, the equilibrium (x∗i , y ∗ ik ) is globally asymptotically stable (See [19, 13]). Remark 2.6. From the above results, we have obtained two types of conditions for global asymptotic stability of the system (2). One is by restricting the delays, and the other is by not restricting delays but putting some limitations on the parameters of the system. Thus, the solutions of the system (2) will converge to its equilibrium point under given conditions. So we can say our system (2) with constant inputs is well-behaved and controllable under certain conditions. We illustrate the above results by using the numerical example, Example 2.7. Consider the system of equations with the 2 main components and two sub-components attached to the main components x ′ 1 = −1.9x1 + 0.2f1(x1(t− τ1)) + 0.32f2(x2(t− τ2)) + 0.23g11(x1, y11(t− ϑ11)) + 0.44g12(x1, y12(t− ϑ12)) + 1 x ′ 2 = −2.2x2 + 0.6f1(x1(t− τ1)) + 0.25f2(x2(t− τ2)) + 0.31g21(x2, y21(t− ϑ21)) + 0.26g22(x2, y22(t− ϑ22)) + 1 y ′ 11 = −1.5y11 + 0.22h11(y11(t− ζ11)) + 0.12h12(y12(t− ζ12)) + 1 y ′ 12 = −2y12 + 0.31h11(y11(t− ζ11)) + 0.25h12(y12(t− ζ12)) + 1 y ′ 21 = −1.8y21 + 0.5h21(y21(t− ζ21)) + 0.2h22(y22(t− ζ22)) + 1 y ′ 22 = −1.6y21 + 0.3h21(y21(t− ζ21)) + 0.1h22(y22(t− ζ22)) + 1 13 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2358 Choose the response function as fi(xi) = tanh(xi), hil(yil) = tanh(yil) and gik(xi, yik) = xi + yik . For this choice of the functions pj = qil = M1ik = M2ik = 1 for i = 1, 2, 3, k = 1, 2, 3. The equilibrium point of this system is (1.5853, 1.3682, 0.8123, 0.6777, 0.8157, 0.7924). Substituting the parameters in the above Theorem 2.2.1, we get A = 0.43, B = 0.79, C = 1.9038, D = 0.1482, E = 0.4232, F = 0.6256 and r = Min{A C , B D} = 0.229, s = B E = 1.8667, p = B F = 1.2628 0 < τ∗ < r, 0 < ϑ∗ < s, 0 < ζ∗ < p. The following (Figure 2) is the simulation when the delays lie within and outside the region derived in Theorem 2.5 (a) (b) Figure 2 Remark 2.8. This example satisfies the conditions of both Theorem 2.1 and Theorem 2.3. As it satisfies the conditions of Theorem 2.3, in spite of the presence of delays the system will be globally asymptotically stable. We can also observe that when the delays are outside the range of our results in Theorem 2.5, there is a disturbance in the solutions to reach the equilibria when compared to that of when they are within the range. Thus, our system (2) is in a controllable state under specific conditions. Hence, it is suitable for our proposition. We further see that model (2) under consideration has constant exogenous inputs, but in some situations, the inputs may vary according to time. So, to suit such situations, we consider the model to have varying input in the next section. 3 Model with Time Varying Inputs-Behavior of the Solu- tions The external inputs, like sensory inputs, information from the outside world, etc., may not always be constant. It may vary according to time. In order to study the impact of such time-varying inputs on the output of emotions, we modify the model (2) by changing the exogenous inputs from constant functions to a function of time t, i.e., we take the exogenous inputs Ii and Jik as a function of t then the system will take the form 14 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2359 x ′ i = −aixi + n∑ j=1 bijfj(xj(t− τj)) + ri∑ k=1 ciikgik(xi, yik(t− ϑik)) + Ii(t) y ′ ik = −cikyik + ri∑ l=1 dilhil(yil(t− ζil)) + Jik(t), (20) where i = 1, 2, 3....n, k = 1, 2, 3.....ri and 1 ≤ ri ≤ n. Under the conditions (3) on response functions and assuming the inputs Ii(t) and Jik(t) to be bounded and continuous on [0,∞), we can say that (19) poses unique solutions in there maximal intervals of existence [14, 20]. When we take τi = 0 in (19) it will be deduce to the model which was studied in [14]. And if we take τi = 0, ϑik = 0 and ζik = 0 for i = 1, 2, ...n and k = 1, 2, 3.....ri in (19) it will deduce to the modified model that was mentioned as the open problem (V) in [19]. As (19) is a non-autonomous system, it may not possess equilibrium solutions. So we study the behaviour of solutions of the system based on the asymptotic nearness and boundedness of solutions. Asymptotic nearness or closeness shows that the solutions have similar or predictable behaviour with respect to one another. In other words, if one solution is controllable, so are the remaining. On the other hand, if one of the solutions is wild, the system as a whole may be regarded as wild. First, we start with the asymptotic nearness of solutions. Theorem 3.1. For any pair of solutions (xi, yik) and (xi, yik) of (19), we have limt→∞ |(xi, yik)− (xi, yik)| = 0 provided the response functions satisfy (3) and the param- eters satisfy A = min{A,B} > 0, where A = min { ai − n∑ j=1 |bji|pi − ri∑ k=1 ( |ciik |M2ik )} B = min { cik − ri∑ l=1 |dil |qil − |ciik |M1ik } , for i = 1, 2, ..., n and k = 1, 2, ..., ri. (21) Proof. By considering the functional V (t) = n∑ i=1 [ |xi − xi|+ n∑ j=1 |bij |pj ∫ t t−τj |xj(z)− xj(z)|dz + ri∑ k=1 |ciik |M1ik ∫ t t−ϑik |yik(z)− yik(z)|dz + ri∑ k=1 [ |yik − yik |+ ri∑ l=1 |dil |qil ∫ t t−ζil |yil(z)− yil(z)|dz ]] . and proceeding as in Theorem 2.1 of [14] we can prove the result. 15 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2360 Now we will obtain conditions for the solutions of the system to be bounded under the conditions of Theorem 4.1, so that all the solutions stay near a bounded solution and hence, we may predict the behaviour of the system. Theorem 3.2. Assume that the parameters satisfy the condition (19) and let the re- sponse functions, besides (3), satisfy fi(0) = 0, gik(0, 0) = 0, and hil(0) = 0 for i = 1, 2, 3...., n, k = l = 1, 2, 3...., ri where 1 ≤ ri. Further if the inputs satisfy ∫∞ 0 ∑n i=1 |Ii(s)+∑ri k=1 Jik(s)|ds < ∞, then all the solutions of (19) are bounded. Proof. We employ the functional V (t) = n∑ i=1 [ |xi(t)|+ n∑ j=1 |bij | ∫ t t−τj |fj(xj(z))|dz + ri∑ k=1 |ciik | ∫ t t−ϑik |gik(yik(z))|dz + ri∑ k=1 [ |yik(t)|+ ri∑ l=1 |dil | ∫ t t−ζil hil |yil(z)|dz ]] Doing the upper Dini derivative of V along the solutions of (19) and rearranging the terms after using conditions(3), we get D+V (t) ≤ − n∑ i=1 [[ ai − n∑ j=1 |bji|pi − ri∑ k=1 ( |ciik |M2ik )] |xi| + ri∑ k=1 [ cik − ri∑ l=1 |dil |qil − |ciik |M1ik ] |yil | ] ≤ −A n∑ i=1 [ |xi(t)|+ ri∑ k=1 |yik(t)| ] + n∑ i=1 [ |Ii(t) + ri∑ k=1 Jik(t)| ] . Integrating on both sides from 0 to t, we get V (t) +A ∫ t 0 n∑ i=1 [ |xi(s)|+ ri∑ k=1 |yik(s)| ] ≤ V (0) + ∫ t 0 n∑ i=1 |Ii(s) + ri∑ k=1 Jik(s)|ds. From our assumptions on the inputs and parameters, it is easy to see that V (t), xi’s and yik ’s are bounded (See [20], for argument). Thus, the solutions of (19) are bounded. Remark 3.3. From Theorem 3.1 and Theorem 3.2, it is clear that the solutions of the system (19) are near to each other and bounded, so the system is well behaved and under control (i.e, as the input varies the response to emotion will not differ drastically). The effectiveness of the above result can be shown through the following example 16 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2361 Example 3.4. x ′ 1 = −1.65x1 + 0.21f1(x1(t− τ1)) + 0.29f2(x2(t− τ2)) + 0.25g11(x1, y11(t− ϑ11)) + 0.4g12(x1, y12(t− ϑ12)) + I1(t) x ′ 2 = −2.38x2 + 0.5f1(x1(t− τ1)) + 0.2f2(x2(t− τ2)) + 0.3g21(x2, y21(t− ϑ21)) + 0.24g22(x2, y22(t− ϑ22)) + I2(t) y ′ 11 = −2y11 + 0.19h11(y11(t− ζ11)) + 0.3h12(y12(t− ζ12)) + J11(t) y ′ 12 = −2.4y12 + 0.27h11(y11(t− ζ11)) + 0.3h12(y12(t− ζ12)) + J12(t) y ′ 21 = −2.8y21 + 1.5h21(y21(t− ζ21))) + 0.2h22(y22(t− ζ22)) + J21(t) y ′ 22 = −1.8y22 + 0.5h21(y21(t− ζ21)) + 0.12h22(y22(t− ζ22)) + J22(t) (22) Choose the response function as fi(xi) = tanh(xi), hil(yil) = tanh(yil) and gik(xi, yik) = xi + yik . For this choice of the functions, we have pj = qil = M1ik = M2ik = 1, & we choose the delays as τi = ϑik = ζik = 1 for i = 1, 2, k = 1, 2. Clearly all the parametric conditions of Theorem 3.1 and Theorem 3.2 are satisfied and we choose exogenous varying inputs Ii(t) and Jik in such a way that they are bounded, then the simulations of the system (22) with different choices of input functions is as follows Figure 3 (a) shows the simulations when we take Ii(t) = Ii− 1 1 + t2 and Jik(xi) = Jik − e−t (increasing functions), where Ii = 1 and Jik = 1. And Figure 3 (b) shows the simulations when we take Ii(t) = Ii + 1 1 + t2 and Jik(xi) = Jik + e−t (decreasing functions), where Ii = 1 and Jik = 1 (a) (b) Figure 3 Remark 3.5. We can observe that both the solutions of Figure 3 (a) and Figure 3 (b) are converging. But we have noticed an interesting point here, that there is a slight variation 17 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2362 in Figure 3 (b), i.e., when we are taking the inputs as decreasing function there is a dis- turbance in the solutions before it converge when compared to that of taking an increasing function. If we take the increasing function as positive inputs and the decreasing function as negative inputs, then we can say from our observation that the emotional stability of a person may not deviate if he is getting positive inputs. But if he is getting negative inputs, then he needs to struggle to maintain it. 4 Discussion In this article, the CSNN model is used to understand how the concepts stored in the memory contribute to the outcome of emotions. The dynamical system of the model with constant inputs and time-varying inputs has been considered. For a system with constant input, conditions for the existence and uniqueness of equilibria, delay-independent and delay-dependent conditions for global asymptotic stability have been derived. For a system with time-varying input, conditions for asymptotic nearness and boundedness have been derived. So, both in the case of constant input and time varying input, our systems are well behaved and under control, hence they can be used for proposition. Here, a point is noted that when the input function is an increasing function, the stability will not be disturbed compared to when the input is a decreasing function. Numerical examples are illustrated to support our results. References [1] Alberto Prieto, Beatriz Prieto, Eva Martinez Ortigosa, Eduardo Ros, Francisco Pelayo, Julio Ortega, Ignacio Rojas, Neural networks: An overview of early research, current frameworks and new challenges, Neurocomputing, Volume 214, 2016, Pages 242-268, ISSN 0925-2312, https://doi.org/10.1016/j.neucom.2016.06.014. [2] Barrett, L. F. (2017), How emotions are made: The secret life of the brain. Houghton Mifflin Harcourt. [3] Hartmann, Kim & Siegert, Ingo & Glüge, Stefan & Wendemuth, Andreas & Kotzyba, Michael & Deml, Barbara. (2012). Describing Human Emotions Through Mathematical Modelling. IFAC Proceedings Volumes, Volume 45, Issue 2, 2012, Pages 463-468, ISSN 1474-6670, ISBN 9783902823236, https://doi.org/10.3182/20120215-3-AT-3016.00081. [4] Levine, Daniel. (2007). Neural Network Modeling of Emotion. Physics of Life Reviews. 4. 37-63. 10.1016/j.plrev.2006.10.001. [5] Jayapradha Soumya Sharma, J. and Yash Dugar, Detection and Recognition of Human Emo- tion using Neural Network, International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 8 (2018) pp. 6472-6477, http://www.ripublication.com 18 Communications on Applied Nonlinear Analysis Vol 32 No. 10s(2025) https://internationalpubls.com ISSN:1074-133X 2363 [6] Lee, Chung & Yoo, S.K. & Park, Yoonj & Kim, Namhyun & Jeong, Keesam & Lee, Byungchae. (2005). Using Neural Network to Recognize Human Emotions from Heart Rate Variability and Skin Resistance. Conference proceedings : Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Conference. 5. 5523-5. 10.1109/IEMBS.2005.1615734. [7] M. Islam, M. Ahmad, M. S. U. Yusuf and T. Ahmed, ”Mathematical modeling of human emotions using sub-band coefficients of wavelet analysis,” 2015 International Conference on Electrical Engineering and Information Communication Technology (ICEEICT), Dhaka, 2015, pp. 1-6. [8] Minaee, Shervin & Abdolrashidi, Amirali. (2019). Deep-Emotion: Facial Expression Recog- nition Using Attentional Convolutional Network. [9] Patrick Zimmerman, How emotions are made, Behavioral Research Blog, 11 Jun. 2019, https://www.noldus.com/blog/how-emotions-are-made. [10] Patrick Zimmerman, How to measure emotions - I, Behavioral Research Blog, 30 Jul. 2019, https://www.noldus.com/blog/how-to-measure-emotions. [11] Prisnyakov, V.F., Prisnyakova, L.M. Mathematical modeling of emotions. Cybern Syst Anal 30, 142–149 (1994). https://doi.org/10.1007/BF02366374. [12] Raja Sekhara Rao P, Venkata Ratnam K and Lalitha P, Delay Independent Stability of Cooperative and Supportive Neural Network, Nonlinear Dynamics and System Theory, 2015, Volume 15(2): 184-197, http://e-ndst.kiev.ua184. [13] Raja Sekhara Rao P, Venkata Ratnam K, Lalitha P and Dipak Kumar Satpathi, Global Dynamics of a Cooperative and Supportive Network with Subnetwork Deactivation, Nonlinear Dynamics and System Theory, 2017, Volume 17(2): 205-216, http://e-ndst.kiev.ua. [14] Raja Sekhara Rao P, Venkata Ratnam K, and Lalitha P, Estimation of Inputs for Desired Output of a Cooperative and Supportive Neural Network, IJETCAS 14-536, 2014, Issue 9 Volume 1, ISSN (Online): 2279-0055. [15] Raja Sekhara Rao P, Venkata Ratnam K, and Shirisha G, A Study on Interactions Between Focal and Non Focal Parts of a Human Brain using a Cooperative and Supportive Neural Network, Communicated. [16] Rahul Mahadeo Shahane, Ramakrishna Sharma.K, Md.Seemab Siddeeq, Emotion Recogni- tion using Feed Forward Neural Network & Näıve Bayes, International Journal of Innovative Technology and Exploring Engineering (IJITEE) ISSN: 2278-3075, Volume-9 Issue-2, Decem- ber 2019. [17] R. Santhoshkumar, M. Kalaiselvi Geetha, Deep Learning Approach for Emotion Recog- nition from Human Body Movements with Feedforward Deep Convolution Neural Net- works, Procedia Computer Science, Volume 152, 2019, Pages 158-165, ISSN 1877-0509, https://doi.org/10.1016/j.procs.2019.05.038. [18] Sree Hari Rao, V. and Naresh Kumar, M. Estimation of the parameters of an infectious disease model using neural networks. Nonlinear Analysis: Real World Applications. 2010; 11(3): 1810-1818. ISSN 1468-1218. https://doi.org/10.1016/j.nonrwa. 2009.04.006. 19 Communications on Applied Nonlinear Analysis Vol 32 No. 10s(2025) https://internationalpubls.com ISSN:1074-133X 2364 [19] Sree Hari Rao V, Raja Sekhara Rao P, Cooperative and Supportive Neural Network, 2007, Physics Letters A 371(1):101-110, https://doi.org/10.1016/j.physleta. 2007.06.049. [20] Sree Hari Rao V and Raja Sekhara Rao P, (2016). Time Varying Stimulations in Simple Neural Networks and Convergence to Desired Outputs, Differential Equations and Dynamical Systems, 26(6), doi:10.1007/s12591-016-0312-z. [21] Thenius, Ronald. (2013). EMANN - a model of emotions in an artificial neural network. [22] Trampe D, Quoidbach J, Taquet M (2015) Emotions in Everyday Life. PLOS ONE 10(12): e0145450. https://doi.org/10.1371/journal.pone.0145450 [23] Unluturk, Mehmet & Oguz, Kaya & Atay, Coskun. (2009). Emotion recognition using neural networks. 20 Vol 32 No. 10s(2025) https://internationalpubls.com Communications on Applied Nonlinear Analysis ISSN:1074-133X 2365