Size of the basin of attraction as a key factor to achieve homeostasis in a mathematical model of synaptic plasticity

Many functions and regimes of the cerebral cortex are characterized by patterns of self-sustained neural activity produced by recurrent excitatory connections. Mathematical models of synaptic plasticity traditionally depend on the pre-existing level of neural activity; therefore, if the initial state during development is silent, modifying the synaptic weights is, in principle, unachievable. To understand the emergence of a stable activity from the mostly silent state in early developmental stages, we addressed this problem using a firing-rate neural model with rectified linear activation functions (ReLU). We performed an exhaustive exploration of the neural model's parameter space (synaptic weight values). Tens of millions of combinations of the four local synaptic weights (self-excitation, self-inhibition, and cross-connections) were simulated. The dynamical analysis focused on quantifying the size of the silent state's basin of attraction in two scenarios: early and late development stages. We found that a significant percentage of combinations (in both the early and late models) meet the condition of having a basin of attraction small enough to align with experimental criteria. Particularly, the basin of attraction in the early developmental model can be smaller than in the late one. This finding shows that, within a certain region of parameter space, the network has an intrinsic ability to "take off" from the silent state via spontaneous fluctuations.