Matignon-Based Stability and Weight Synchronization of a Fractional Time Delay Neural Network Model

Artificial neural networks (ANNs) are powerful models inspired by the structure and function of the human brain. They are widely used for tasks such as classification, prediction, and model recognition. This study examines the stability of fractional-order neural networks with neuronal conditions, dynamic behavior, synchronization, and delays of time "σ" . Synchronization and stability for delayed neural network models are two important aspects of dynamic behavior. For a calculated fractional-order, the state of the state variable w<sub>i</sub>(t) are synchronized with each other. Weight synchronization of w<sub>i</sub> (i=1, 2, 3,..., 6) provides coherent updates during training, helping neural networks to study stable models. The incommensurate fractional-orders are linked to a system where each dynamic component develops with a different value, i.e. q<sub>i</sub> ≠ q<sub>j</sub> (i≠j) is inconsistent. These fractional-orders are calculated for the system’s eigenvalues and their singular points within the stability region defined by the Matignon-based stability. As the time delay decreases, more activation functions are induced, and the variable state of w<sub>4</sub>(t) requires longer relaxation times to be more stable than the variable state of w<sub>3</sub>(t). The Grunwald-Letnikov method is used to solve a fractional neural network system numerically and effectively handle fractional derivatives. This approach helps to more accurately simulate memory in neural networks.