# On a Unique Solution of a T-Maze Model Arising in the Psychology and Theory of Learning

Authors: Ali Turab; Wajahat Ali; Juan J. Nieto
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In the form of a T, a T-maze is an experimental design in which each trial consists of decisions between two or more options. It contains choices with particular kinds of symmetries that have gained considerable attention in psychology and learning theories. One of the simplest mazes utilized by rats is the T-maze since it requires just a single point of preference. At a T-maze base, the mouse chooses to turn right or left to get food. This paper aims at analyzing the rat’s behavior in such circumstances and proposing a suitable mathematical model for it. The existence and uniqueness of a solution to the proposed T-maze model are investigated by using the appropriate fixed point method.


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## Body

## 1. Introduction

Mathematical psychology is an approach to psychological study focused on mathematical modeling of perceptual, thinking, cognitive, and motor processes. The mathematical methods are used to develop more reliable theories and thus produce more rigorous empirical validations. The biggest issue with today’s application of mathematics to psychological problems and most likely for some time to come is modeling these problems.In an animal or human being, the learning phase may often be viewed as a series of choices between multiple possible reactions. Even in basic repetitive experiments under strictly regulated conditions, preference sequences are typically volatile, suggesting that the probability governs the choice of responses. It is also helpful to identify structural adjustments in the series of alternatives that reflect changes in trial-to-trial outcome probabilities. From this perspective, most of the learning analysis explains the probability of a trial-to-test occurrence that describes a stochastic mechanism.In modern mathematical learning experiments, the researchers concluded that a basic learning experiment was compatible with any stochastic process. It is not a new idea (see [1] for a summary of its history). After 1950, two critical features emerged mainly in the research initiated by Bush, Estes, and Mosteller. In the first instance, the learning method egalitarian essence was a core feature of the developed model. Second, these frameworks were studied and applied in areas that did not conceal their quantitative aspects.Several studies (Estes and Straughan [2], Grant et al. [3], Humphreys [4], and Jarvik [5]) on human actions in probability–learning scenarios have produced results aligned with the so–called event-matching hypothesis that the allocation of incentives would mirror the asymptotic distribution of answers in a two-choice setting. Conflicting findings have been reported in other studies. For example, if subjects choose the correct option in most trials, then, it would accelerate the probability close to 1 (for the detail, see [1, 6]).Turab and Sintunavarat [7, 8] presented a functional equation to analyze Bush and Wilson’s experimental study on a paradise fish [9], in which they offered the fish two options for swimming. As the starting gate was raised, swimmers had two options: swim on the right-hand side or the left-hand side of the tank’s far end.Recently, in [10], the authors discussed a particular type of traumatic avoidance learning experiment of normal dogs proposed by Solomon and Wynne [11]. They examined 30 mongrel dogs weighing between 9 and 13 kg and observed a particular form of emotional resistance performed in a tiny box with a steel grid floor. Turab and Sintunavarat [10] analyzed the dogs’ behavior in such situations and proposed a mathematical model and also presented the existence of solutions of such model by using the fixed point technique.On the other hand, the genesis of the fixed point theory was primarily for the use of successive approximations to prove the existence and uniqueness of solutions, primarily of differential and integral equations, in the second half of the nineteenth century. It is indeed a beautiful blend of pure and applied analysis, topology, and geometry. Picard’s work demonstrates the fundamental concepts of a fixed point theoretic perspective. However, it is attributed to the Polish mathematician Banach for abstracting the fundamental ideas into a framework applicable to a wide variety of applications beyond ordinary differential and integral equations (see [12]). It has been generalized and extended in various directions (for the detail, see [13–16]). For more details about the fixed point theory and its applications in different spaces, we refer the reader to [17–22].In this paper, we present a specific type of psychological learning theory experiment related to the T-maze model proposed by Brunswik and Stanley in [23, 24], and suggest a mathematical model that is appropriate for it. The existence and uniqueness of the proposed model’s solution are investigated by using the suitable fixed point theorem. Later on, to check the proposed model’s validity, we shall highlight some particular aspects of the T-maze model under the experimenter-subject controlled events. In the end, we raise an open problem for the interested readers.


## 2. A T–Maze Experiment Proposed by Brunswik and Stanley

A T-maze [23, 24] is a unique design that has gotten much attention in the past few years. It is a classic maze for rats since it has only one choice point. While experimental design modifications and generalizations have been used with mice and other subjects, we shall concentrate on the primary form of the open maze used with rats.In Figure1, a schematic of the apparatus can be seen. At the starting position, s, a rat is put and it runs to the point of preference, w. After that, the rat moves to one of the two aim bins, A and B, where it may get food.Figure 1
The behavior of a rat in a T-maze experiment.Here, the experimental trials constitute the series of behavior. With the same rat, the process is usually performed several times. A very elaborate course of action is the overall activity of a rat during an experimental study. The rat is in a particular stimulation position when it crosses the labyrinth and is in two potential stimulus conditions after reaching the preference stage. Of course, this overall activity on a test may be divided up and appropriate measurements or indexes used by each part can be used. For example, we could ask about the starting location latency,s, or the running momentum between s and the w-point of preference. However, it seems to us that the part of the rat’s conduct that is unique to this experiment is the behavior at the option stage, w.In the study that follows, we only consider a rat’s choice of the path on a trial instead of any other actions it may exhibit. The rat arrives at the decision point at a complete experimental study where the stimulation factor population is kept unchanged from trial to trial. Corresponding to the target box achieved,A or B, two groups of responses are listed in Table 1. One and only one of these response groups take part in each study trial. Then, an experimental study compares to a trial as described in [12]—a chance to select between alternatives that are mutually exclusive and exhaustive.Table 1
Alternative definitions of experimental events in a T-maze experiment depending on the placement of the food and chosen side.
ResponsesOutcomesA: left turnO1: food side (reward)B: left turnO1: food side (reward)A: right turnO2: nonfood side (no reward)B: right turnO2: nonfood side (no reward)The condition of the organism on a specific test, according to the model, is fully specified by a probabilityk that the rat will go to goal box A and a probability 1−k that it will go to goal box B. We have complete information about the learning process model when these probabilities are recognized for each trial. These probabilities can be estimated from the proportion of turns made by a single rat on several trials to goal box A or from the proportion of a population of rats that go on a specific trial to goal box B.


## 3. Mathematical Modeling of the Proposed T-Maze Experiment

In the above experiment, the significant interest lies in the behavior of a rat; turn left or right, “A” or “B,” and get the food which is dependent on where the food is placed and the movement of a rat towards that compartment. In our view, if a rat chooses the food side, there would be an occurrence of alternative O1, and if a rat made a move to the other side, then, there will be an occurrence of alternative O2. According to the mathematical point of view, there would be four possibilities of events, depending on the movement of the rat and the placement of the food. These events are listed in Table 2.Table 2
The movement of a rat and its corresponding events.
ResponseOutcomesEventsAO1E1BO1E2AO2E3BO2E4The probability of the responsesA and B are x and 1−x, respectively, where x∈0,1. The experimental pattern asks for the outcomes of the responses (whether the rat gets the food or not), trials’ fixed proportion of ς∈0,1. Therefore, we get the event probabilities stated below (see Table 3).Table 3
Corresponding probabilities of the four events.
EventProbability of occurrenceE1ςxE21−xςE31−ςxE41−ς1−xLet us assume thatϑ1,ϑ2∈0,1 are the learning–rate parameters and their values measure the ineffectiveness of the events E1−E4 in altering the response probability. Also, λk∈0,1, where k=1,2 is the constant of the corresponding event E1−E4. If ςx is the probability of response A with outcome O1 on some trial and A is fulfilled, the new probability of A with outcome O1 is ϑ1x+1−ϑ1λ1, and if A is achieved with outcome O2, then, the new probability will be ϑ1x+1−ϑ11−λ1 with the event probability 1−ςx. Similarly, if B is performed with outcomes O1 and O2, then, the new probabilities of B are ϑ2x+1−ϑ2λ2 and ϑ2x+1−ϑ21−λ2, having event probabilities of occurrence 1−xς and 1−ς1−x, respectively. For the four events E1−E4, we can define the transition operators P1,P2,P3,P4:0,1⟶0,1 as
(1)P1x=ϑ1x+1−ϑ1λ1,P2x=ϑ2x+1−ϑ2λ2,P3x=ϑ1x+1−ϑ11−λ1,P4x=ϑ2x+1−ϑ21−λ2,for all x∈0,1 and 0<ϑ1,ϑ2<1.It can be observed that a rat following such description, in the long run, will stop giving feedback to one of the responses and react only with the other (with probability one). Now, givingx,ϑ1,ϑ2, λ1,λ2, what is the probability that the rat stops providing B’s, that is, consumed by A? We define such probability by Px,ϑ1,ϑ2,λ1,λ2 as a function of x, and it depends on the path as well as the responses and outcomes. After one trial, the rat has a new probability shown in (1) depending on the events E1−E4 with the respective probabilities of occurrence. Thus, if its first trial is A with outcomes O1 and O2, its new probability of consumption by A will be Pϑ1x+1−ϑ1λ1,ϑ1,λ1 and Pϑ1x+1−ϑ11−λ1,ϑ1,λ1, respectively. But, if the first trial is B with outcomes O1 and O2, then, the new probability of absorption by B will be Pϑ2x+1−ϑ2λ2,ϑ2,λ2 and Pϑ2x+1−ϑ21−λ2,ϑ2,λ2, respectively.By considering the above transition operators with their corresponding probabilities and events given in Table3, we have the following functional equation
(2)Px,ϑ1,ϑ2,λ1,λ2=ςxPϑ1x+1−ϑ1λ1,ϑ1,λ1+1−xςPϑ2x+1−ϑ2λ2,ϑ2,λ2+1−ςxPϑ1x+1−ϑ11−λ1,ϑ1,λ1+1−ς1−xPϑ2x+1−ϑ21−λ2,ϑ2,λ2.In the progression, the following noted result will be needed.(3)dMℓ,Mm≤ϖdℓ,m,for some ϖ<1 and for all ℓ,m∈S. Then, M has precisely one fixed point. Moreover, the Picard iteration ℓn in S which is defined by ℓn=Mℓn−1 for all n∈ℕ, where ℓ0∈S, converges to the unique fixed point of M.Theorem 1.
(Banach fixed point theorem [12]). Let S,d be a complete metric space and M:S⟶S be a mapping defined by


## 4. Main Results

LetA=0,1. Throughout this article, we denote by B the class of all continuous real-valued functions W:A⟶ℝ such that W0=0 and
(4)supx≠yWx−Wyx−y<∞.It is easy to see thatB,· is a Banach space, where · is defined by
(5)W=supx≠yWx−Wyx−y,for all W∈B.For the computational convenience, we define an operatorW:A⟶ℝ and write functional equation (2) as
(6)Wx=ςxWϑ1x+1−ϑ1λ1+1−xςWϑ2x+1−ϑ2λ2+1−ςxWϑ1x+1−ϑ11−λ1+1−ς1−xWϑ2x+1−ϑ21−λ2,where 0<ϑ1,ϑ2<1 and λ1,λ2∈A. Our objective is to investigate the existence and uniqueness of a solution to functional equation (6) by using the fixed point technique. We begin with the following outcome.Theorem 2.
Let0<ϑ1,ϑ2<1 and λ1,λ2,ς∈A such that ϖ<1, where
(7)ϖ≔2ς−11−ϑ1λ1+1−ϑ2λ2+1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(8)ZWx=ςxWϑ1x+1−ϑ1λ1+ς1−xWϑ2x+1−ϑ2λ2+1−ςxWϑ1x+1−ϑ11−λ1+1−ς1−xWϑ2x+1−ϑ21−λ2,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Proof.
LetW1,W2∈Λ. For each distinct points x,y∈A, we obtain
(9)ZW1−ZW2x−ZW1−ZW2yx−y=1x−yςxW1−W2ϑ1x+1−ϑ1λ1+ς1−xW1−W2ϑ2x+1−ϑ2λ2+1−ςxW1−W2ϑ1x+1−ϑ11−λ1+1−ς1−xW1−W2ϑ2x+1−ϑ21−λ2−ςyW1−W2ϑ1y+1−ϑ1λ1−ς1−yW1−W2ϑ2y+1−ϑ2λ2−1−ςyW1−W2ϑ1y+1−ϑ11−λ1−1−ς1−yW1−W2ϑ2y+1−ϑ21−λ2=1x−yςxW1−W2ϑ1x+1−ϑ1λ1−ςxW1−W2ϑ1y+1−ϑ1λ1+ς1−xW1−W2ϑ2x+1−ϑ2λ2−ς1−xW1−W2ϑ2y+1−ϑ2λ2+1−ςxW1−W2ϑ1x+1−ϑ11−λ1−1−ςxW1−W2ϑ1y+1−ϑ11−λ1+1−ς1−xW1−W2ϑ2x+1−ϑ21−λ2−1−ς1−xW1−W2ϑ2y+1−ϑ21−λ2+ςxW1−W2ϑ1y+1−ϑ1λ1−ςyW1−W2ϑ1y+1−ϑ1λ1+ς1−xW1−W2ϑ2y+1−ϑ2λ2−ς1−yW1−W2ϑ2y+1−ϑ2λ2+1−ςxW1−W2ϑ1y+1−ϑ11−λ1−1−ςyW1−W2ϑ1y+1−ϑ11−λ1+1−ς1−xW1−W2ϑ2y+1−ϑ21−λ2−1−ς1−yW1−W2ϑ2y+1−ϑ21−λ2=1x−yςxW1−W2ϑ1x+1−ϑ1λ1−ςxW1−W2ϑ1y+1−ϑ1λ1+1x−yς1−xW1−W2ϑ2x+1−ϑ2λ2−ς1−xW1−W2ϑ2y+1−ϑ2λ2+1x−y1−ςxW1−W2ϑ1x+1−ϑ11−λ1−1−ςxW1−W2ϑ1y+1−ϑ11−λ1+1x−y1−ς1−xW1−W2ϑ2x+1−ϑ21−λ2−1−ς1−xW1−W2ϑ2y+1−ϑ21−λ2+1x−yςxW1−W2ϑ1y+1−ϑ1λ1−ςyW1−W2ϑ1y+1−ϑ1λ1+1x−yς1−xW1−W2ϑ2y+1−ϑ2λ2−ς1−yW1−W2ϑ2y+1−ϑ2λ2+1x−y1−ςxW1−W2ϑ1y+1−ϑ11−λ1−1−ςyW1−W2ϑ1y+1−ϑ11−λ1+1x−y1−ς1−xW1−W2ϑ2y+1−ϑ21−λ2−1−ς1−yW1−W2ϑ2y+1−ϑ21−λ2≤ϑ1ςxW1−W2+ϑ2ς1−xW1−W2+ϑ11−ςxW1−W2+ϑ21−ς1−xW1−W2+ςW1−W2ϑ1y+1−ϑ1λ1−ςW1−W20+ςW1−W2ϑ2y+1−ϑ2λ2−ςW1−W20+1−ςW1−W2ϑ1y+1−ϑ11−λ1−1−ςW1−W20+1−ςW1−W2ϑ2y+1−ϑ21−λ2−1−ςW1−W20=ϑ1ςxW1−W2+ϑ2ς1−xW1−W2+ϑ11−ςxW1−W2+ϑ21−ς1−xW1−W2+ςϑ1y+1−ϑ1λ1W1−W2+ςϑ2y+1−ϑ2λ2W1−W2+1−ςϑ1y+1−ϑ11−λ1W1−W2+1−ςϑ2y+1−ϑ21−λ2W1−W2≤ϖW1−W2,where ϖ is defined in (7). This gives that
(10)dZW1,ZW2=ZW1−ZW2≤ϖW1−W2=ϖdW1,W2.
It follows from0<ϖ<1 that Z is a Banach contraction mapping with the metric d induced by ·.☐We get the following conclusion fromTheorem2 about the uniqueness of a solution to functional equation (6).Theorem 3.
Functional equation (6) has a unique solution provided that ϖ<1, where ϖ is defined in (7), and there exists a closed subset Λ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(11)ZWx=ςxWϑ1x+1−ϑ1λ1+ς1−xWϑ2x+1−ϑ2λ2+1−ςxWϑ1x+1−ϑ11−λ1+1−ς1−xWϑ2x+1−ϑ21−λ2,for all x∈A. Moreover, the iteration Wn in Λ is defined by
(12)Wnx=ςxWn−1ϑ1x+1−ϑ1λ1+ς1−xWn−1ϑ2x+1−ϑ2λ2+1−ςxWn−1ϑ1x+1−ϑ11−λ1+1−ς1−xWn−1ϑ2x+1−ϑ21−λ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (6) in the sense of the metric d induced by ·.Proof.
We derive the result of this theorem by combining the Banach fixed point theorem with Theorem3.☐


## 5. A Certain Case of a T-Maze Experiment with Experimenter-Subject-Controlled Events

It has been highlighted that the examination of any experiment is based on suppositions, which are assembled about the subject. Experiments are classified as contingent and noncontingent, based on the occurrences of the result.In the previous models on imitation problems such as T-maze experiments with fish, dogs, and humans (see [7, 10, 25]), it was already mentioned that such experiments required a contingent approach; the result of the trials was entirely dependent on the subject’s choice. Such types of models required experimenter-subject-controlled events. The two responses A and B along with outcomes O1 and O2 are choosing the right or left side or pushing the right or left button, which coincides with rewarding and nonrewarding or choosing the correct and incorrect side, respectively. Now, we define the probabilities ς1 and ς2 which indicate the conditional probability of outcomes O1 and O2 of the given alternatives A and B, respectively, such that
(13)ς1+ς2=1.With such conditions, we have Table4.Table 4
Four events under conditional probabilities of occurrence.
EventsOutcomesTransition operatorsProbability of occurrenceAO1P1x=ϑ1x+1−ϑ1λ1ς1xBO1P2x=ϑ2x+1−ϑ2λ2ς21−xAO2P3x=ϑ1x+1−ϑ11−λ11−ς1xBO2P4x=ϑ2x+1−ϑ21−λ21−ς21−xWe have the following functional equation from the data given above(14)Wx=ς1xWϑ1x+1−ϑ1λ1+ς21−xWϑ2x+1−ϑ2λ2+1−ς1xWϑ1x+1−ϑ11−λ1+1−ς21−xWϑ2x+1−ϑ21−λ2,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and λ1,λ2,ς1,ς2∈A with ς1+ς2=1. We shall begin with the following outcome.Theorem 4.
Let0<ϑ1,ϑ2<1 and λ1,λ2,ς1,ς2∈A such that ϖå<1, where
(15)ϖ⋆≔2λ1−1ς11−ϑ1+2λ2−1ς21−ϑ2+1−ϑ11−λ1+1−ϑ21−λ2+2ϑ1+ϑ2.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(16)ZWx=ς1xWϑ1x+1−ϑ1λ1+ς21−xWϑ2x+1−ϑ2λ2+1−ς1xWϑ1x+1−ϑ11−λ1+1−ς21−xWϑ2x+1−ϑ21−λ2,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Proof.
LetW1,W2∈Λ. For each distinct points x,y∈A, we obtain
(17)ZW1−ZW2x−ZW1−ZW2yx−y=1x−yς1xW1−W2ϑ1x+1−ϑ1λ1+ς21−xW1−W2ϑ2x+1−ϑ2λ2+1−ς1xW1−W2ϑ1x+1−ϑ11−λ1+1−ς21−xW1−W2ϑ2x+1−ϑ21−λ2−ς1yW1−W2ϑ1y+1−ϑ1λ1−ς21−yW1−W2ϑ2y+1−ϑ2λ2−1−ς1yW1−W2ϑ1y+1−ϑ11−λ1−1−ς21−yW1−W2ϑ2y+1−ϑ21−λ2=1x−yς1xW1−W2ϑ1x+1−ϑ1λ1−ς1xW1−W2ϑ1y+1−ϑ1λ1+ς21−xW1−W2ϑ2x+1−ϑ2λ2−ς21−xW1−W2ϑ2y+1−ϑ2λ2+1−ς1xW1−W2ϑ1x+1−ϑ11−λ1−1−ς1xW1−W2ϑ1y+1−ϑ11−λ1+1−ς21−xW1−W2ϑ2x+1−ϑ21−λ2−1−ς21−xW1−W2ϑ2y+1−ϑ21−λ2+ς1xW1−W2ϑ1y+1−ϑ1λ1−ς1yW1−W2ϑ1y+1−ϑ1λ1+ς21−xW1−W2ϑ2y+1−ϑ2λ2−ς21−yW1−W2ϑ2y+1−ϑ2λ2+1−ς1xW1−W2ϑ1y+1−ϑ11−λ1−1−ς1yW1−W2ϑ1y+1−ϑ11−λ1+1−ς21−xW1−W2ϑ1y+1−ϑ11−λ1−1−ς21−yW1−W2ϑ1y+1−ϑ11−λ1=1x−yς1xW1−W2ϑ1x+1−ϑ1λ1−ς1xW1−W2ϑ1y+1−ϑ1λ1+1x−yς21−xW1−W2ϑ2x+1−ϑ2λ2−ς21−xW1−W2ϑ2y+1−ϑ2λ2+1x−y1−ς1xW1−W2ϑ1x+1−ϑ11−λ1−1−ς1xW1−W2ϑ1y+1−ϑ11−λ1+1x−y1−ς21−xW1−W2ϑ2x+1−ϑ21−λ2−1−ς21−xW1−W2ϑ2y+1−ϑ21−λ2+1x−yς1xW1−W2ϑ1y+1−ϑ1λ1−ς1yW1−W2ϑ1y+1−ϑ1λ1+1x−yς21−xW1−W2ϑ2y+1−ϑ2λ2−ς21−yW1−W2ϑ2y+1−ϑ2λ2+1x−y1−ς1xW1−W2ϑ1y+1−ϑ11−λ1−1−ς1yW1−W2ϑ1y+1−ϑ11−λ1+1x−y1−ς21−xW1−W2ϑ2y+1−ϑ21−λ2−1−ς21−yW1−W2ϑ2y+1−ϑ21−λ2≤ϑ1ς1xW1−W2+ϑ2ς21−xW1−W2+ϑ11−ς1xW1−W2+ϑ21−ς21−xW1−W2+ς1W1−W2ϑ1y+1−ϑ1λ1−ς1W1−W20+ς2W1−W2ϑ2y+1−ϑ2λ2−ς2W1−W20+1−ς1W1−W2ϑ1y+1−ϑ11−λ1−1−ς1W1−W20+1−ς2W1−W2ϑ2y+1−ϑ21−λ2−1−ς2W1−W20=ϑ1ς1xW1−W2+ϑ2ς21−xW1−W2+ϑ11−ς1xW1−W2+ϑ21−ς21−xW1−W2+ς1ϑ1y+1−ϑ1λ1W1−W2+ς2ϑ2y+1−ϑ2λ2W1−W2+1−ς1ϑ1y+1−ϑ11−λ1W1−W2+1−ς2ϑ2y+1−ϑ21−λ2W1−W2≤ϖ⋆W1−W2,
whereϖ⋆ is defined in (15). This gives that
(18)dZW1,ZW2=ZW1−ZW2≤ϖ⋆W1−W2=ϖ⋆dW1,W2.
It follows from0<ϖ⋆<1 that Z is a Banach contraction mapping with the metric d induced by ·.☐We get the following conclusion from Theorem4 about the uniqueness of a solution to functional equation (14).Theorem 5.
Functional equation (14) has a unique solution provided that ϖ⋆<1, where ϖ⋆ is given in (15), and there exists a closed subset Λ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(19)ZWx=ς1xWϑ1x+1−ϑ1λ1+ς21−xWϑ2x+1−ϑ2λ2+1−ς1xWϑ1x+1−ϑ11−λ1+1−ς21−xWϑ2x+1−ϑ21−λ2,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(20)Wnx=ς1xWn−1ϑ1x+1−ϑ1λ1+ς21−xWn−1ϑ2x+1−ϑ2λ2+1−ς1xWn−1ϑ1x+1−ϑ11−λ1+1−ς21−xWn−1ϑ2x+1−ϑ21−λ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (14) in the sense of the metric d induced by ·.Proof.
We derive the result of this theorem by combining the Banach fixed point theorem with Theorem4.☐


## 6. Some Particular Aspects of the Proposed T-Maze Model

In this section, we have discussed some particular cases of the proposed T-maze model.


### 6.1. Events with Equal Lambda Conditions

This condition (sometimes called commutative condition) says that the transition operatorsP1−P4 (none of them is an identity operator) have the same lambda conditions λ1=λ=λ2. These conditions reduce our transition operators (1) to
(21)P1x=ϑ1x+1−ϑ1λ,P2x=ϑ2x+1−ϑ2λ,P3x=ϑ1x+1−ϑ11−λ,P4x=ϑ2x+1−ϑ21−λ.Now, we can write our functional equation (6) as
(22)Wx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1 and λ,ς∈A. The following conclusions are drawn as a result of Theorem 3.Corollary 6.
Let0<ϑ1,ϑ2<1 and λ,ς∈A with
(23)1−ς1−λ+λς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(24)ZWx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 7.
Functional equation (22) has a unique solution provided that
(25)1−ς1−λ+λς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(26)ZWx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(27)Wnx=ςxWn−1ϑ1x+1−ϑ1λ+ς1−xWn−1ϑ2x+1−ϑ2λ+1−ςxWn−1ϑ1x+1−ϑ11−λ+1−ς1−xWn−1ϑ2x+1−ϑ21−λ,for all n∈ℕ, where W0∈Λ, converges to the unique solution the functional equation (22) in the sense of the metric d induced by ·.


### 6.2. Extinction of an Operant Response

In some cases, nonfood side responses (turning right or left) frequently by the mouse decrease the probability of that event towards an asymptote to zero. In this situation, we haveλ1=0=λ2. These conditions reduce our operators (1) to
(28)P1x=ϑ1x,P2x=ϑ2x,P3x=ϑ1x+1−ϑ1,P4x=ϑ2x+1−ϑ2.Now, we can write our functional equation (6) as
(29)Wx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,where W:A⟶ℝ is an unknown function such that, 0<ϑ1,ϑ2<1, and ς∈A. We have the following corollaries of Theorem 3.Corollary 8.
For0<ϑ1,ϑ2<1 and ς∈A with
(30)1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(31)ZWx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 9.
Functional equation (29) has a unique solution provided that
(32)1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(33)ZWx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(34)Wnx=ςxWnϑ1x+ς1−xWnϑ2x+1−ςxWnϑ1x+1−ϑ1+1−ς1−xWnϑ2x+1−ϑ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (29) in the sense of the metric d induced by ·.
Similarly, if the mouse chooses the food side repeatedly, then, the probability of that specific event will increase. Thus, we haveλ1=1=λ2. In this situation, our four operators (1) will be
(35)P1x=ϑ1x+1−ϑ1,P2x=ϑ2x+1−ϑ2,P3x=ϑ1x,P4x=ϑ2x.
Now, we can write our functional equation (6) as
(36)Wx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and ς∈A. Now, we have the following corollaries of Theorem 3.Corollary 10.
For0<ϑ1,ϑ2<1 and ς∈A with
(37)ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(38)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 11.
Functional equation (36) has a unique solution provided that
(39)ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(40)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(41)Wnx=ςxWnϑ1x+1−ϑ1+ς1−xWnϑ2x+1−ϑ2+1−ςxWnϑ1x+1−ς1−xWnϑ2x,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (36) in the sense of the metric d induced by ·.


### 6.3. Attraction towards the Specific Choice

In some specific cases, it is possible that the mouse always follows theO1 outcome and never choose O2. For such a case, we choose λ1=1. Similarly, if the mouse chooses O2 again and again, then, the probability of that event should turn towards zero. It means that λ2=0. These conditions reduce our four operators (1) to
(42)P1x=ϑ1x+1−ϑ1,P2x=ϑ2x,P3x=ϑ1x,P4x=ϑ2x+1−ϑ2.Now, we can write our functional equation (6) as
(43)Wx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and ς∈A. We have the following results of Theorem 3.Corollary 12.
For0<ϑ1,ϑ2<1 and ς∈A with
(44)2ς−11−ϑ1+1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(45)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,for all x∈A, then Z is a Banach contraction mapping with the metric d induced by ·.Corollary 13.
The functional equation (43) has a unique solution provided that
(46)2ς−11−ϑ1+1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1,
and there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(47)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(48)Wnx=ςxWnϑ1x+1−ϑ1+ς1−xWnϑ2x+1−ςxWnϑ1x+1−ς1−xWnϑ2x+1−ϑ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (36) in the sense of the metric d induced by ·.


## 6.1. Events with Equal Lambda Conditions

This condition (sometimes called commutative condition) says that the transition operatorsP1−P4 (none of them is an identity operator) have the same lambda conditions λ1=λ=λ2. These conditions reduce our transition operators (1) to
(21)P1x=ϑ1x+1−ϑ1λ,P2x=ϑ2x+1−ϑ2λ,P3x=ϑ1x+1−ϑ11−λ,P4x=ϑ2x+1−ϑ21−λ.Now, we can write our functional equation (6) as
(22)Wx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1 and λ,ς∈A. The following conclusions are drawn as a result of Theorem 3.Corollary 6.
Let0<ϑ1,ϑ2<1 and λ,ς∈A with
(23)1−ς1−λ+λς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(24)ZWx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 7.
Functional equation (22) has a unique solution provided that
(25)1−ς1−λ+λς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(26)ZWx=ςxWϑ1x+1−ϑ1λ+ς1−xWϑ2x+1−ϑ2λ+1−ςxWϑ1x+1−ϑ11−λ+1−ς1−xWϑ2x+1−ϑ21−λ,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(27)Wnx=ςxWn−1ϑ1x+1−ϑ1λ+ς1−xWn−1ϑ2x+1−ϑ2λ+1−ςxWn−1ϑ1x+1−ϑ11−λ+1−ς1−xWn−1ϑ2x+1−ϑ21−λ,for all n∈ℕ, where W0∈Λ, converges to the unique solution the functional equation (22) in the sense of the metric d induced by ·.


## 6.2. Extinction of an Operant Response

In some cases, nonfood side responses (turning right or left) frequently by the mouse decrease the probability of that event towards an asymptote to zero. In this situation, we haveλ1=0=λ2. These conditions reduce our operators (1) to
(28)P1x=ϑ1x,P2x=ϑ2x,P3x=ϑ1x+1−ϑ1,P4x=ϑ2x+1−ϑ2.Now, we can write our functional equation (6) as
(29)Wx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,where W:A⟶ℝ is an unknown function such that, 0<ϑ1,ϑ2<1, and ς∈A. We have the following corollaries of Theorem 3.Corollary 8.
For0<ϑ1,ϑ2<1 and ς∈A with
(30)1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(31)ZWx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 9.
Functional equation (29) has a unique solution provided that
(32)1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(33)ZWx=ςxWϑ1x+ς1−xWϑ2x+1−ςxWϑ1x+1−ϑ1+1−ς1−xWϑ2x+1−ϑ2,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(34)Wnx=ςxWnϑ1x+ς1−xWnϑ2x+1−ςxWnϑ1x+1−ϑ1+1−ς1−xWnϑ2x+1−ϑ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (29) in the sense of the metric d induced by ·.
Similarly, if the mouse chooses the food side repeatedly, then, the probability of that specific event will increase. Thus, we haveλ1=1=λ2. In this situation, our four operators (1) will be
(35)P1x=ϑ1x+1−ϑ1,P2x=ϑ2x+1−ϑ2,P3x=ϑ1x,P4x=ϑ2x.
Now, we can write our functional equation (6) as
(36)Wx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and ς∈A. Now, we have the following corollaries of Theorem 3.Corollary 10.
For0<ϑ1,ϑ2<1 and ς∈A with
(37)ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(38)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,for all x∈A, then, Z is a Banach contraction mapping with the metric d induced by ·.Corollary 11.
Functional equation (36) has a unique solution provided that
(39)ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
Also, there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(40)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ϑ2+1−ςxWϑ1x+1−ς1−xWϑ2x,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(41)Wnx=ςxWnϑ1x+1−ϑ1+ς1−xWnϑ2x+1−ϑ2+1−ςxWnϑ1x+1−ς1−xWnϑ2x,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (36) in the sense of the metric d induced by ·.


## 6.3. Attraction towards the Specific Choice

In some specific cases, it is possible that the mouse always follows theO1 outcome and never choose O2. For such a case, we choose λ1=1. Similarly, if the mouse chooses O2 again and again, then, the probability of that event should turn towards zero. It means that λ2=0. These conditions reduce our four operators (1) to
(42)P1x=ϑ1x+1−ϑ1,P2x=ϑ2x,P3x=ϑ1x,P4x=ϑ2x+1−ϑ2.Now, we can write our functional equation (6) as
(43)Wx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and ς∈A. We have the following results of Theorem 3.Corollary 12.
For0<ϑ1,ϑ2<1 and ς∈A with
(44)2ς−11−ϑ1+1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1.
If there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(45)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,for all x∈A, then Z is a Banach contraction mapping with the metric d induced by ·.Corollary 13.
The functional equation (43) has a unique solution provided that
(46)2ς−11−ϑ1+1−ς1−ϑ1+1−ϑ2+2ϑ1+ϑ2<1,
and there exists a closed subsetΛ of B such that Λ is Z invariant, that is, ZΛ⊆Λ, where Z is the operator from Λ defined for each W∈Λ by
(47)ZWx=ςxWϑ1x+1−ϑ1+ς1−xWϑ2x+1−ςxWϑ1x+1−ς1−xWϑ2x+1−ϑ2,for all x∈A. Moreover, the iteration Wn in Λ which is defined by
(48)Wnx=ςxWnϑ1x+1−ϑ1+ς1−xWnϑ2x+1−ςxWnϑ1x+1−ς1−xWnϑ2x+1−ϑ2,for all n∈ℕ, where W0∈Λ, converges to the unique solution of functional equation (36) in the sense of the metric d induced by ·.


## 7. Conclusion

In an animal or a human being, the learning phase may also be analyzed through a sequence of choices between multiple possible answers. The choice sequence is usually unpredictable, even in basic experiments conducted under highly regulated conditions, indicating that probabilities govern the selection of responses. Thus, it is helpful to think of the sequential changes in a sequence of choices in response probabilities from trial to trial. In this paper, we have discussed a particular type of stochastic process related to the T-maze experiment [23, 24], which plays a vital role in observing the behavior of the mouse in a two-choice situation. We analyzed the rat’s behavior in such situations and proposed a mathematical model for it. The existence and uniqueness of a solution to the proposed model have been investigated by using the Banach fixed point theorem. To observe the flexibility of the T-maze model, we examined it under the experimenter-subject-controlled events. Furthermore, the proposed model depends only on the contingent reinforcement behavior of rats in which a rat gets the reward for choosing the food side. However, in general, a natural question arises, which we present here to make this interaction more interesting.


### 7.1. Question

What happens if a mouse does not move to any side (left or right) on a specific trialk and remains sticking to its starting position?Moreover, one of the critical issues in functional equations is to find out its stability regarding Hyers-Ulam- and Hyers-Ulam-Rassias-type stability (see for the detail, [26–30]). We leave the stability question to the following functional equation as an open problem:
(49)Wx=ςxWϑ1x+1−ϑ1λ1+ς1−xWϑ2x+1−ϑ2λ2+1−ςxWϑ1x+1−ϑ11−λ1+1−ς1−xWϑ2x+1−ϑ21−λ2,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and λ1,λ2,ς∈A.


## 7.1. Question

What happens if a mouse does not move to any side (left or right) on a specific trialk and remains sticking to its starting position?Moreover, one of the critical issues in functional equations is to find out its stability regarding Hyers-Ulam- and Hyers-Ulam-Rassias-type stability (see for the detail, [26–30]). We leave the stability question to the following functional equation as an open problem:
(49)Wx=ςxWϑ1x+1−ϑ1λ1+ς1−xWϑ2x+1−ϑ2λ2+1−ςxWϑ1x+1−ϑ11−λ1+1−ς1−xWϑ2x+1−ϑ21−λ2,where W:A⟶ℝ is an unknown function, 0<ϑ1,ϑ2<1, and λ1,λ2,ς∈A.



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*Source: 6081250-2022-01-18.xml*