Associative memory model with arbitrary Hebbian length*

Our brains build connections within events, i.e. associative memory. Specifically, the brain can establish correlation between temporal sequential events by conversion the correlation into the spatial-structured synapses, which is related to the function of learning and the feeling of time passing. However, The correlation conversion can happens in a wide integration window, whose influence on the correlation conversion remains elusive.

Here, we propose a generalized associative memory model with arbitrary width of integration window and other parameters. This model can be analytically solved by replica method and predict how model parameters affect the conversion. A series of previous models can be degenerated from the models. We highlight that a small increment in integration window width can significantly enhance the correlation conversion. Moreover, the anti-Hebbian component is able to reshape the energy landscape of memories, akin to the function of sleep.  Besides, we apply random matrix theory to show how the maximum eigenvalue is related to the transition from paramagnetic to spin glass in the model. 

Altogether, our work gives a full view of a series models and establishes the connection between associative memory, Hebbian length, and correlation conversion in the brain.