Artificial intelligence (AI) is abundant but features a critical flaw -- the lack of explainability. Namely, its operation is opaque, the way it makes decisions is unclear, which undermines our trust in its decision-making. Modern AI is mostly based on artificial neural networks (ANNs) [1], which work as ''black boxes'' and possess unwanted features, such as false memories, catastrophic forgetting of old memories, and inability to learn on the go like humans [2-5].

These cannot eradicated because nobody understands how they arise.

Within the current project, ANNs will be considered as a special case of dynamical systems (DSs) of a novel type, those with a plastic self-organising vector field [6-9]. Plastic DSs will be used to better explain how the existing ANNs operate, and to design new intelligent (cognitive) systems with prescribed features and without undesirable traits. Such systems will be fully explaianble, and would form a mathematical basis for the development of safe AI in the future.