SPONGE FUNCTIONS: A MATHEMATICAL MODEL OF LEARNING AND PLASTICITY AT THE CELLULAR AND MOLECULAR LEVELS

Source abstract

Recent studies show that some aspects of learning at the brain level are replicated at the cellular level, and this learning influences the behavior of later generations. There are no systematic mathematical models of the mechanisms that support such cellular-level learning. As a starting point to develop a mechanistic account of cellular-level learning, we present a novel sponge function model, integrating fractal geometry, dynamical systems, and memory fields. To model information processing at the micro-granular level, we assume that the disordered states of cells function as a condensed network. We then propose a sedimentary space model, where the Menger sponge serves as an initial geometric structure. Operations traverse this fractal medium, and memory fields capture the residual effects of information flow. This proof-of-concept model establishes a complete dynamical system, characterized by bidirectional coupling between path evolution and memory field updates. Error backpropagation mechanisms are incorporated to enable learning, while inference states allow for recall and prediction. This work provides an initial mathematical foundation for systematically modeling how information can be processed, stored, and retrieved within geometrically complex disordered structures at the cellular and molecular levels. We outline some limitations of the specific sponge model we propose, and possible future work that could address these issues. We conclude with a discussion of the wider research possibilities opened up by the sponge function model.