Research Article
Classical Conditioning Formulas: Classical Mathematics of Conditioning: Extinction, Spontaneous Recovery, and Advanced Processes
Sanele Inathi Ndaba*
Issue:
Volume 12, Issue 2, June 2026
Pages:
18-28
Received:
26 March 2026
Accepted:
13 April 2026
Published:
27 July 2026
Abstract: This paper presents a formal mathematical theory of classical conditioning that reinterprets the fundamental principles of behavior, including acquisition, extinction, spontaneous recovery, stimulus generalization, stimulus discrimination, and higher-order conditioning, as a coherent system of symbolic equations. The aim is to bridge empirical psychology and quantitative modeling by expressing learning mechanisms in precise mathematical terms. Within this framework, unconditioned and conditioned stimuli are represented as variables whose interactions over time determine associative strength, learning rates, and the probability of conditioned responses. To demonstrate the structure and application of the model, a Pavlovian conditioning example is presented in which a neutral sound stimulus (bell) becomes associated with an unconditioned stimulus (food), resulting in the development of a conditioned response. The proposed equations describe the progressive accumulation of associative strength during acquisition, its reduction through extinction, and its partial restoration through spontaneous recovery, thereby capturing key experimental patterns observed in classical conditioning research. Furthermore, the model enables detailed examination of influential parameters such as stimulus intensity, reinforcement frequency, and temporal relationships between stimuli. Beyond behavioral psychology, the mathematical conditioning framework has broad interdisciplinary applicability. In neuroscience, it can contribute to modeling synaptic plasticity and reward prediction processes; in artificial intelligence, it can inform reinforcement learning algorithms and adaptive agent design; in education, it offers quantitative insights into habit formation and feedback-based learning; and in behavioral economics, it provides a basis for simulating consumer learning, preference development, and decision-making under uncertainty. By translating established behavioral principles into a scalable, predictive, and computationally rigorous framework, this work advances the integration of psychology with computational and systems sciences and contributes toward the development of unified theories of learning that are theoretically coherent, empirically grounded, and computationally efficient.
Abstract: This paper presents a formal mathematical theory of classical conditioning that reinterprets the fundamental principles of behavior, including acquisition, extinction, spontaneous recovery, stimulus generalization, stimulus discrimination, and higher-order conditioning, as a coherent system of symbolic equations. The aim is to bridge empirical psyc...
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Research Article
Applications of Mamadu–Njoseh Polynomials in the Analysis of Dynamical System Fluctuations: Nonlinear Duffing Oscillator
Ebimene James Mamadu*,
Marcus Ifeanyi Ossaiugbo,
Jude Chukwuyem Nwankwo,
Ebikonbo-owei Anthony Mamadu,
Ignatius Nkonyeasua Njoseh,
Henrietta Ify Jarikre
Issue:
Volume 12, Issue 2, June 2026
Pages:
29-41
Received:
10 October 2026
Accepted:
21 October 2026
Published:
6 November 2026
DOI:
10.11648/j.ml.20261202.12
Downloads:
Views:
Abstract: Duffing oscillator, being a fundamental model for investigating nonlinear vibrations, resonance, and complex dynamics, is a key component of nonlinear dynamical systems. However, the challenge remains in obtaining accurate numerical solutions to its governing equation, especially in capturing the transient response and understanding long-lasting oscillatory motions. In this study, a Mamadu–Njoseh Polynomial (MNP) based spectral collocation method is formulated and investigated for the numerical solution and dynamical analysis of the nonlinear Duffing oscillator. The main goal of this study is to evaluate the accuracy, convergence, numerical stability, and computational efficiency of the proposed method. Specifically, the proposed method utilizes Mamadu–Njoseh polynomials as a basis to approximate the solution to the governing nonlinear differential equation through a spectral collocation framework. The accuracy and convergence properties of the proposed method are investigated by comparing the numerical approximations with the exact solution (when available). The method is also applied to capture the system’s transient response and study resonance characteristics, bifurcation behavior, and long-lasting oscillatory motions to further assess its accuracy and convergence properties. The numerical results demonstrate that the proposed method offers accurate approximations with fast convergence to the exact solution for the cases considered. In addition, the method is found to exhibit improved numerical stability and computational efficiency as a small number of basis functions are sufficient to achieve a reasonable level of accuracy. Thus, the MNP based spectral collocation method appears to be a computationally efficient and accurate method that can effectively capture various aspects of the dynamics of nonlinear Duffing oscillators. The method’s accuracy and efficiency can potentially benefit the study of nonlinear vibration dynamics and complex motions of other nonlinear dynamical systems, though a proper validation of the proposed method is essential for each specific case.
Abstract: Duffing oscillator, being a fundamental model for investigating nonlinear vibrations, resonance, and complex dynamics, is a key component of nonlinear dynamical systems. However, the challenge remains in obtaining accurate numerical solutions to its governing equation, especially in capturing the transient response and understanding long-lasting os...
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