From spiking neuronal networks to interpretable dynamics: a diffusion-approximation framework

Modeling and interpreting the complex recurrent dynamics of neuronal spiking activity is essential to understanding how networks implement behavior and cognition. Nonlinear Hawkes process models can capture a large range of spiking dynamics, but remain difficult to interpret, due to their discontinuous and stochastic nature. To address this challenge, we introduce a novel framework based on a piecewise deterministic Markov process representation of the nonlinear Hawkes process (NH-PDMP) followed by a diffusion approximation. We analytically derive stability conditions and dynamical properties of the obtained diffusion processes for single-neuron and network models. We established the accuracy of the diffusion approximation framework by comparing it with exact continuous-time simulations of the original neuronal NH-PDMP models. Our framework offers an analytical and geometric account of the neuronal dynamics repertoire captured by nonlinear Hawkes process models, both for the canonical responses of single-neurons and neuronal-network dynamics, such as winner-take-all and traveling wave phenomena. Applied to human and nonhuman primate recordings of neuronal spiking activity during speech processing and motor tasks, respectively, our approach revealed that task features can be retrieved from the dynamical landscape of the fitted models. The combination of NH-PDMP representations and diffusion approximations thus provides a novel dynamical analysis framework to reveal single-neuron and neuronal-population dynamics directly from models fitted to spiking data.