Kinetic modeling approach for a heterogeneous neuronal network activity using adjacency matrices

The heterogeneity of neuronal networks plays a crucial role in shaping emergent dynamics. In this work, we introduced a kinetic modeling approach to describe the activity of heterogeneous neuronal networks through transition probabilities and adjacency matrices. The model explicitly accounts for both structural and functional heterogeneity by considering two interacting neuronal populations—excitatory pyramidal neurons and inhibitory interneurons—distributed across network slices. The transition probabilities encode the binary stochastic interactions between neurons, capturing both the neuronal types involved (excitatory or inhibitory) and the connectivity structure within and between slices. Complementarily, adjacency matrices define the weighted connections among neurons, specifying the structural organization of each slice and the interactions across slices. Together, these two components characterize the functional and the structural heterogeneity of the system. From this framework, we derived a system of nonlinear ordinary differential equations describing the mesoscopic dynamics of the network. First, for the one-slice model, we provided analytical results on the existence and stability of equilibrium states. Then, we presented numerical simulations for two- and four-slice networks to investigate the role of functional and structural heterogeneity. In particular, after defining the excitatory-, inhibitory-, and balanced count regimes and introducing an a priori criterion for their identification, we demonstrated how heterogeneity influences both the short- and long-term dynamics of the network. Our findings revealed that increasing heterogeneity not only alters the proportion of active neurons but also induces more complex dynamical behaviors, potentially driving shifts between excitatory-count- and inhibitory-count-dominated regimes.