In this portion, certain situations of the Wyckoff stochastic model have been examined.

Situation with identical lambda constraints

This condition, also known as the commutative condition, states that our transition operators \(P_\) through \(P_\) (none of which are identity operators) have identical lambda conditions \((\lambda _=\lambda _=\lambda _=\lambda _=\lambda _=\lambda _=\lambda _=\lambda _=\lambda )\). Due to these constraints, our transition operators (4.6) are reduced to

$$\begin \left\ P_x&= \vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda , \\ P_x&=\vartheta _x+(1-\vartheta _)\lambda . \ \end\right. \end$$

(6.1)

By ensuring that all transition operators share the same lambda value, this condition maintains the model’s stability across different maze structures and configurations. This uniformity allows the model to generalize beyond a specific experimental setting, making it adaptable to alternative maze layouts without requiring extensive parameter adjustments. Now, our functional Eq. (5.1) can be expressed as

$$\begin \mathcal (x)= & \fracu_x\mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\fracu_x\mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal (\vartheta _x+(1-\vartheta _)\lambda ), \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal (\vartheta _x+(1-\vartheta _)\lambda ), \end$$

(6.2)

where \(0<\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _, \vartheta _, \vartheta _<1\), \(\lambda ,u_,u_\in \mathcal \) and \(\mathcal :\mathcal \rightarrow \mathbb \) is an unknown function. The Theorem 5.1 has the following findings as a result.

Corollary 6.1

Let \(0<\vartheta _,\vartheta _,\dots ,\vartheta _<1\) and \(\lambda ,u_,u_\in \mathcal \) with \(\mathfrak _^ <1\), where \(\mathfrak _^\) is defined in (4.3). If an operator Z from \(\Lambda\) associated for every \(\mathcal \) \(\in\) \(\Lambda\) and for all \(x\in \mathcal \) by

$$\begin (Z\mathcal )(x)= & \fracu_x\mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\fracu_x\mathcal (\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & + \frac(1-u_)x\mathcal (\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal (\vartheta _x+(1-\vartheta _)\lambda ), \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal (\vartheta _x+(1-\vartheta _)\lambda ), \end$$

(6.3)

and satisfies property (\(\mathfrak ^\)), then Z is a BCM.

Corollary 6.2

The Eq. (6.2) has a unique solution having that \(\mathfrak _^ <1\), where \(\mathfrak _^\) is given in (4.3), and there exists an operator Z defined in (6.3) satisfies property (\(\mathfrak ^\)). Furthermore, the iteration \(\_\}\) (\(\forall n\in \mathbb \)) in \(\Lambda\), where \(\mathcal _\in \Lambda\), is given by

$$\begin (\mathcal _)(x)= & \fracu_x\mathcal _(\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & +\frac(1-u_)x\mathcal _(\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\fracu_x\mathcal _(\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & +\frac(1-u_)x\mathcal _(\vartheta _x+(1-\vartheta _)\lambda ) \nonumber \\ & +\frac(1-x)u_ \mathcal _(\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal _(\vartheta _x+(1-\vartheta _)\lambda ),\nonumber \\ & +\frac(1-x)u_ \mathcal _(\vartheta _x+(1-\vartheta _)\lambda )\nonumber \\ & + \frac(1-x)(1 - u_) \mathcal _(\vartheta _x+(1-\vartheta _)\lambda ), \end$$

(6.4)

converges to the unique solution of (6.2).

Elimination of a behavioral reflex

It’s possible that the mouse’s frequent right- or left-turning side reactions might reduce an event’s probability to the point of asymptote to zero. Such a situation necessitates the assumption that \(\lambda _=\lambda _=\dots =\lambda _ =0\). These constraints decrease our four operators (4.6) to

$$\begin \left\ P_x&=\vartheta _x, \\ P_x&=\vartheta _x, \\ P_x&=\vartheta _x, \\ P_x&=\vartheta _x, \\ P_x&=\vartheta _x, \\ P_x&=\vartheta _x \\ P_x&=\vartheta _x \\ P_x&=\vartheta _x \ \end\right. \end$$

(6.5)

By setting lambda values to zero in cases where habitual side preferences emerge, this condition prevents the model from being influenced by fixed behavioral reflexes. As a result, decision-making remains dynamically responsive to environmental cues rather than being constrained by preconditioned biases, thereby improving the model’s generalizability across different testing conditions. So, we can write (5.1) as

$$\begin \mathcal (x)= & \fracu_x\mathcal (\vartheta _x) +\frac(1-u_)x\mathcal (\vartheta _x)\nonumber \\ & +\fracu_x\mathcal (\vartheta _x) \nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x) +\frac(1-x)u_ \mathcal (\vartheta _x)\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal (\vartheta _x), \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x) +\frac(1-x)(1 - u_) \mathcal (\vartheta _x), \end$$

(6.6)

where \(0<\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _, \vartheta _, \vartheta _<1\), \(u_,u_\in \mathcal \) and \(\mathcal :\mathcal \rightarrow \mathbb \) is an unknown function. The following are the Theorem 5.1’s corollaries.

Corollary 6.3

Let \(0<\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _,\vartheta _, \vartheta _, \vartheta _<1\) and \(u_,u_\in \mathcal ,\) with \(\mathfrak _^ <1\), where \(\mathfrak _^\) is defined in (4.4). If an operator Z from \(\Lambda\) associated for every \(\mathcal \) \(\in\) \(\Lambda\) and for all \(x\in \mathcal \) by

$$\begin (Z\mathcal )(x)= & \fracu_x\mathcal (\vartheta _x) +\frac(1-u_)x\mathcal (\vartheta _x)\nonumber \\ & +\fracu_x\mathcal (\vartheta _x) \nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x) +\frac(1-x)u_ \mathcal (\vartheta _x)\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal (\vartheta _x) \nonumber \\ & +\frac(1-x)u_\mathcal (\vartheta _x) +\frac(1-x)(1 - u_) \mathcal (\vartheta _x), \end$$

(6.7)

and satisfies property (\(\mathfrak ^\)), then Z is a BCM.

Corollary 6.4

The stochastic Eq. (6.6) has a unique solution with \(\mathfrak _^ <1\), where \(\mathfrak _^\) is given in (4.4), and there exists an operator Z defined in (6.7) satisfies property (\(\mathfrak ^\)). Furthermore, the iteration \(\_\}\) (\(\forall n\in \mathbb \)) in \(\Lambda\), where \(\mathcal _\in \Lambda\), is given by

$$\begin (\mathcal _)(x)= & \fracu_x\mathcal _(\vartheta _x) +\frac(1-u_)x\mathcal _(\vartheta _x)\nonumber \\ & +\fracu_x\mathcal _(\vartheta _x) +\frac(1-u_)x\mathcal _(\vartheta _x) \nonumber \\ & +\frac(1-x)u_ \mathcal _(\vartheta _x) \nonumber \\ & +\frac(1-x)(1 - u_)\mathcal _(\vartheta _x) \nonumber \\ & +\frac(1-x)u_\mathcal _(\vartheta _x) \nonumber \\ & +\frac(1-x)(1 - u_)\mathcal _(\vartheta _x), \end$$

(6.8)

converges to the unique solution of (6.6).

In the same way, if the rat frequently selects the food side, the chance of that event occurring increases. Therefore, we have

$$\begin \lambda _ = \lambda _ = \dots = \lambda _ =1. \end$$

In this situation, our four operators will be

$$\begin \left\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _). \\ P_x&=\vartheta _x+(1-\vartheta _), \\ P_x&=\vartheta _x+(1-\vartheta _). \ \end\right. \end$$

(6.9)

Now, our functional Eq. (5.1) may be expressed as

$$\begin \mathcal (x)= & \fracu_x\mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x + (1-\vartheta _)) \nonumber \\ & + \fracu_x\mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & + \frac(1-u_)x\mathcal (\vartheta _x + (1-\vartheta _)) \nonumber \\ & + \frac(1-x)u_ \mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal (\vartheta _x + (1-\vartheta _)),\nonumber \\ & + \frac(1-x)u_ \mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_)\mathcal (\vartheta _x + (1-\vartheta _)), \end$$

(6.10)

where \(0<\vartheta _, \vartheta _, \vartheta _, \vartheta _,\vartheta _, \vartheta _, \vartheta _, \vartheta _ <1\), \(u_, u_ \in \mathcal \) and \(\mathcal :\mathcal \rightarrow \mathbb \) is an unknown function. The following outcomes are the findings of Theorem 5.1.

Corollary 6.5

Let \(0<\vartheta _, \vartheta _, \vartheta _, \vartheta _, \vartheta _, \vartheta _, \vartheta _, \vartheta _<1\) and \(u_, u_ \in \mathcal \) with \(\mathfrak _^ <1\), where \(\mathfrak _^\) is defined in (4.5). If an operator Z from \(\Lambda\) associated for every \(\mathcal \) \(\in\) \(\Lambda\) and for all \(x\in \mathcal \) by

$$\begin (Z\mathcal )(x)= & \fracu_x\mathcal (\vartheta _x +(1-\vartheta _))\nonumber \\ & +\frac(1-u_)x\mathcal (\vartheta _x +(1-\vartheta _)) \nonumber \\ & + \fracu_x\mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & + \frac(1-u_)x\mathcal (\vartheta _x + (1-\vartheta _)) \nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal (\vartheta _x + (1-\vartheta _)),\nonumber \\ & +\frac(1-x)u_ \mathcal (\vartheta _x + (1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal (\vartheta _x + (1-\vartheta _)), \end$$

(6.11)

and satisfies property (\(\mathfrak ^\)), then Z is a BCM.

Corollary 6.6

The Eq. (6.10) has a unique solution claiming that \(\mathfrak _^ <1\), where \(\mathfrak _^\) is defined in (4.5), and there exists an operator Z defined in (6.11) satisfies property (\(\mathfrak ^\)). Furthermore, the iteration \(\_\}\) (\(\forall n\in \mathbb \)) in \(\Lambda\), where \(\mathcal _\in \Lambda\), is given by

$$\begin (\mathcal _)(x)= & \fracu_x\mathcal _(\vartheta _x+(1-\vartheta _))\nonumber \\ & +\frac(1-u_)x\mathcal _(\vartheta _x+(1-\vartheta _))\nonumber \\ & +\fracu_x\mathcal _(\vartheta _x+(1-\vartheta _))\nonumber \\ & +\frac(1-u_)x\mathcal _(\vartheta _x+(1-\vartheta _)) \nonumber \\ & +\frac(1-x)u_ \mathcal _(\vartheta _x+(1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal _(\vartheta _x+(1-\vartheta _)), \nonumber \\ & +\frac(1-x)u_ \mathcal _(\vartheta _x+(1-\vartheta _))\nonumber \\ & +\frac(1-x)(1 - u_) \mathcal _(\vartheta _x+(1-\vartheta _)), \end$$

(6.12)

converges to the unique solution of (6.10).