Partial annealing and pattern decorrelation in associative neural networks

Using the Hopfield model as a benchmark case, the present work focuses on the investigation of partially annealed associative neural networks, wherein neural dynamics is coupled to slowly evolving patterns within the two-temperature-two-timescale framework. This setting inherently introduces a real parameter n , reminiscent of the number of replicas in the celebrated replica trick, that tunes the separation of timescales and the effective interaction between fast (i.e. the neurons) and slow (i.e. the synapses) degrees of freedom. By adapting Guerra’s interpolation to the case, we derive the Replica Symmetric pseudo-replicated free energy while treating n directly as a real physical parameter, without relying on analytic continuation. The resulting thermodynamics reproduces the previous replica result of Dotsenko, Franz and Mézard in 1994, including the enhancement of the retrieval region and the limiting storage capacity α c = 1 for n < 0 at zero temperature. This induces a progressive decorrelation of the stored patterns, thereby effectively reducing interference, promoting orthogonal configurations and ultimately conferring to the network the aforementioned maximal storage. Numerical simulations based on a mean-field Monte Carlo dynamics confirm this scenario and show that partial annealing restores retrieval in challenging regimes, such as in the presence of biased patterns. Importantly, this gain in retrieval does not generally preserve the microscopic identity of the initial patterns: the slow dynamics reorganizes the memory set into evolved internal representations that may retain only weak overlap with the original ones. We therefore interpret partial annealing as an adaptive reorganization of the internal memory code, trading representational fidelity for reduced interference and improved associative retrieval. We further show, however, that an explicit hetero-associative memory can reliably map the evolved representations back onto their corresponding original targets, thereby decoupling the optimization of internal representations from the preservation of the original information.