Inferring the synaptic plasticity rules that govern learning in the brain is a key challenge in neuroscience. We present a novel computational method to infer these rules from experimental data, applicable to both neural and behavioral data. Our approach approximates plasticity rules using a parameterized function, employing either truncated Taylor series for theoretical interpretability or multilayer perceptrons. These plasticity parameters are optimized via gradient descent over entire trajectories to align closely with observed neural activity or behavioral learning dynamics. This method can uncover complex rules that induce long nonlinear time dependencies, particularly involving factors like postsynaptic activity and current synaptic weights. We validate our approach through simulations, successfully recovering established rules such as Oja's, as well as more intricate plasticity rules with reward-modulated terms. We assess the robustness of our technique to noise and apply it to behavioral data from \textit{Drosophila} in a probabilistic reward-learning experiment. Notably, our findings reveal an active forgetting component in reward learning in flies, improving predictive accuracy over previous models. This modeling framework offers a promising new avenue for elucidating the computational principles of synaptic plasticity and learning in the brain.

Learning in deep neural networks is known to depend critically on the knowledge embedded in the initial network weights. However, few theoretical results have precisely linked prior knowledge to learning dynamics. Here we derive exact solutions to the dynamics of learning with rich prior knowledge in deep linear networks by generalising Fukumizu's matrix Riccati solution \citep{fukumizu1998effect}. We obtain explicit expressions for the evolving network function, hidden representational similarity, and neural tangent kernel over training for a broad class of initialisations and tasks. The expressions reveal a class of task-independent initialisations that radically alter learning dynamics from slow non-linear dynamics to fast exponential trajectories while converging to a global optimum with identical representational similarity, dissociating learning trajectories from the structure of initial internal representations. We characterise how network weights dynamically align with task structure, rigorously justifying why previous solutions successfully described learning from small initial weights without incorporating their fine-scale structure. Finally, we discuss the implications of these findings for continual learning, reversal learning and learning of structured knowledge. Taken together, our results provide a mathematical toolkit for understanding the impact of prior knowledge on deep learning.