Neuron particles capture network topology and behavior from single units

While networks of neurons, glia and vascular systems enable and support brain functions, to date, mathematical tools to decode network dynamics and structure from very scarce and partially observed neuronal spiking behavior remain underdeveloped. Large neuronal networks contribute to the intrinsic neuron transfer function and observed neuronal spike trains encoding complex causal information processing, yet how this emerging causal fractal memory in the spike trains relates to the network topology is not fully understood. Towards this end, we propose a novel statistical physics inspired neuron particle model that captures the causal information flow and processing features of neuronal spiking activity. Relying on synthetic comprehensive simulations and real-world neuronal spiking activity analysis, the proposed fractional order operators governing the neuronal spiking dynamics provide insights into the memory and scale of the spike trains as well as information about the topological properties of the underlying neuronal networks. Lastly, the proposed model exhibits superior predictions of animal behavior during multiple cognitive tasks.