Bilinearity in Spatiotemporal Integration of Synaptic Inputs

Neurons process information via integration of synaptic inputs from dendrites. Many experimental results demonstrate dendritic integration could be highly nonlinear, yet few theoretical analyses have been performed to obtain a precise quantitative characterization analytically. Based on asymptotic analysis of a two-compartment passive cable model, given a pair of time-dependent synaptic conductance inputs, we derive a bilinear spatiotemporal dendritic integration rule. The summed somatic potential can be well approximated by the linear summation of the two postsynaptic potentials elicited separately, plus a third additional bilinear term proportional to their product with a proportionality coefficient . The rule is valid for a pair of synaptic inputs of all types, including excitation-inhibition, excitation-excitation, and inhibition-inhibition. In addition, the rule is valid during the whole dendritic integration process for a pair of synaptic inputs with arbitrary input time differences and input locations. The coefficient is demonstrated to be nearly independent of the input strengths but is dependent on input times and input locations. This rule is then verified through simulation of a realistic pyramidal neuron model and in electrophysiological experiments of rat hippocampal CA1 neurons. The rule is further generalized to describe the spatiotemporal dendritic integration of multiple excitatory and inhibitory synaptic inputs. The integration of multiple inputs can be decomposed into the sum of all possible pairwise integration, where each paired integration obeys the bilinear rule. This decomposition leads to a graph representation of dendritic integration, which can be viewed as functionally sparse.Author Summary: A neuron, as a fundamental unit of brain computation, exhibits extraordinary computational power in processing input signals from neighboring neurons. It usually integrates thousands of synaptic inputs from its dendrites to achieve information processing. This process is known as dendritic integration. To elucidate information coding, it is important to investigate quantitative spatiotemporal dendritic integration rules. However, there has yet to be extensive experimental investigations to quantitatively describe dendritic integration. Meanwhile, most theoretical neuron models considering time-dependent synaptic inputs are difficult to solve analytically, thus impossible to be used to quantify dendritic integration. In this work, we develop a mathematical method to analytically solve a two-compartment neuron model with time-dependent synaptic inputs. Using these solutions, we derive a quantitative rule to capture the dendritic integration of all types, including excitation-inhibition, excitation-excitation, inhibition-inhibition, and multiple excitatory and inhibitory inputs. We then validate our dendritic integration rule through both realistic neuron modeling and electrophysiological experiments. We conclude that the general spatiotemporal dendritic integration structure can be well characterized by our dendritic integration rule. We finally demonstrate that the rule leads to a graph representation of dendritic integration that exhibits functionally sparse properties.