Abstract
We study the properties of the dynamical phase transition occurring in neural network models in which a competition between associative memory and sequential pattern recognition exists. This competition occurs through a weighted mixture of the symmetric and asymmetric parts of the synaptic matrix. Through a generating functional formalism, we determine the structure of the parameter space at non-zero temperature and near saturation (i.e., when the number of stored patterns scales with the size of the network), identifying the regions of high and weak pattern correlations, the spin-glass solutions, and the order-disorder transition between these regions. This analysis reveals that, when associative memory is dominant, smooth transitions appear between high correlated regions and spurious states. In contrast when sequential pattern recognition is stronger than associative memory, the transitions are always discontinuous. Additionally, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of the same set of patterns, there is a discontinuous transition between associative memory and sequential pattern recognition. In contrast, when the symmetric and asymmetric parts of the synaptic matrix are defined in terms of independent sets of patterns, the network is able to perform both associative memory and sequential pattern recognition for a wide range of parameter values.
Highlights
Neural networks were originally developed to model the behavior of the brain
Using the generating functional formalism developed in Refs. [17], [18], [19], [20,21] we investigate how the network transits from pure Associative Memory (AM) to pure Sequential Pattern Recognition (SPR) for the CS model, first in the case in which the symmetric and asymmetric parts of the synaptic matrix are correlated and when they are independent
Numerical simulations Before introducing the mathematical formalism to analyze the different models presented in the previous section, we illustrate here the coexistence of the two types of dynamics, SPR and AM, with a numerical simulation
Summary
Neural networks were originally developed to model the behavior of the brain. due to the great complexity of the brain’s neural circuitry and of the synaptic interactions, it was necessary to propose simplified models, such as the McCullochPitts model [1] and the Hopfield model [2] which, simple, still capture some important characteristics of the neuronal dynamics. The neural network performs AM when its dynamical attractors are fixed points, each corresponding to one of the patterns that we want to store in the network. This type of dynamic behavior is characterized by a symmetric interaction matrix that contains the connection strength between the neurons. Some examples of this dynamics are the Hopfield model and the Little model [2],[3], [4], [5]. A well known example of this type of dynamics is the asymmetric Hopfield model [2], [6]
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