  Associative memory is a fundamental function in the brain. Here, we
generalize the standard associative memory model to include long-range Hebbian
interactions at the learning stage, corresponding to a large synaptic
integration window. In our model, the Hebbian length can be arbitrarily large.
The spectral density of the coupling matrix is derived using the replica
method, which is also shown to be consistent with the results obtained by
applying the free probability method. The maximal eigenvalue is then obtained
by an iterative equation, related to the paramagnetic to spin glass transition
in the model. Altogether, this work establishes the connection between the
associative memory with arbitrary Hebbian length and the asymptotic
eigen-spectrum of the neural-coupling matrix.
