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Associative memory is a fundamental function in the brain. Here, wegeneralize the standard associative memory model to include long-range Hebbianinteractions at the learning stage, corresponding to a large synapticintegration window. In our model, the Hebbian length can be arbitrarily large.The spectral density of the coupling matrix is derived using the replicamethod, which is also shown to be consistent with the results obtained byapplying the free probability method. The maximal eigenvalue is then obtainedby an iterative equation, related to the paramagnetic to spin glass transitionin the model. Altogether, this work establishes the connection between theassociative memory with arbitrary Hebbian length and the asymptoticeigen-spectrum of the neural-coupling matrix.

Eigenvalue spectrum of neural networks with arbitrary Hebbian length