Mathematical Modeling and Analysis of Spatial Neuron Dynamics: Dendritic Integration and Beyond

Neurons compute by integrating spatiotemporal excitatory (E) and inhibitory (I) synaptic inputs received from the dendrites. The investigation of dendritic integration is crucial for understanding neuronal information processing. Yet quantitative rules of dendritic integration and their mathematical modeling remain to be fully elucidated. Here neuronal dendritic integration is investigated by using theoretical and computational approaches. Based on the passive cable theory, a PDE-based cable neuron model with spatially branched dendritic structure is introduced to describe the neuronal subthreshold membrane potential dynamics, and the analytical solutions in response to conductance-based synaptic inputs are derived. Using the analytical solutions, a bilinear dendritic integration rule is identified, and it characterizes the change of somatic membrane potential when receiving multiple spatiotemporal synaptic inputs from the dendrites. In addition, the PDE-based cable neuron model is reduced to an ODE-based point-neuron model with the feature of bilinear dendritic integration inherited, thus providing an efficient computational framework of neuronal simulation incorporating certain important dendritic functions. The above results are further extended to active dendrites by numerical verification in realistic neuron simulations. Our work provides a comprehensive and systematic theoretical and computational framework for the study of spatial neuron dynamics.