Glassy and retrieval properties of Potts neural networks with constrained mean activity

Neural network models are powerful tools in understanding complex systems such as the brain. Network models that are mathematically analyzable as well as biologically plausible are particularly popular in making sense of empirical data on brain functions. Here, we study one such a model, the Potts neural network, focusing on its thermodynamic properties. We add an extra term in the dynamics of Potts attractor neural networks, which have been proposed as a model of global cortical dynamics. The new term forces the network to hover around trajectories where the mean activity of the network takes a predefined value. We discuss the effectiveness of the term and phase transitions that occur under the influence of this constraint. The new term lowers the critical temperature for the onset of the spin glass phase, while increasing the maximum storage capacity at the cost of a reduced basin of attraction. Transient properties are studied by means of the dynamical mean-field formalism, which is in good agreement with computer simulations. Our results contribute to the growing body of theoretical works on complex neural networks that can potentially lead to the understanding of cognitive functions.