Dynamical mean-field theory for a highly heterogeneous neural population

Abstract: Large-scale systems with inherent heterogeneity often exhibit complex dynamics that are crucial for their functional properties. However, understanding how such heterogeneity shapes these dynamics remains a significant challenge, particularly in systems with widely varying time scales. To address this, we extend Dynamical Mean Field Theory$\unicode{x2014}$a powerful framework for analyzing large-scale population dynamics$\unicode{x2014}$to systems with heterogeneous temporal properties. Using the population dynamics of a biological neural network as an example, we develop a theoretical framework that determines how inherent heterogeneity influences the critical transition point of the network. By introducing a model that incorporates graded-persistent activity$\unicode{x2014}$a property where certain neurons sustain activity over extended periods without external inputs$\unicode{x2014}$we show that neurons with extremely long timescales shift the system's transition point and expand its dynamical regime, enhancing its suitability for temporal information processing. Furthermore, we validate our framework by applying it to a system with heterogeneous adaptation, demonstrating that such heterogeneity can reduce the dynamical regime, contrary to previous simplified approximations. These findings establish a theoretical foundation for understanding the functional advantages of diversity in complex systems and offer insights applicable to a wide range of heterogeneous networks beyond neural populations.