Theoretical Modeling And Analysis Of Neuronal Dendritic Integration

A neuron, as a fundamental unit of brain computation, exhibits great computation-al power in processing input signals from neighboring neurons. It receives thousands of spatially distributed synaptic inputs from its dendrites and then integrates them at the soma, leading to the neuronal information processing. This procedure is called den-dritic integration. Dendritic integration rules are under active investigation in order to elucidate information coding in the brain.In the thesis, we present our work on theoretical modeling and analysis of dendritic integration in the following six chapters. To be specific, we introduce the background knowledge in Chapter 1 and Chapter 2, and introduce our own research work from Chapter 3 to Chapter 6.In Chapter 1, we introduce basic neurophysiology for mathematicians who are not familiar with neuroscience. We also review the current progress in the experimental and theoretical investigation on dendritic integration. We finally point out the scientific contribution and novelty of our work.In Chapter 2, we introduce two types of neuron models to characterize neuronal electrophysiological properties described in Chapter 1. A neuron can be modeled as an idealized point with electrical circuit structure, such as the HH model and the IF model. On the other hand, a neuron can also be modeled as a spatially extended tree with conductive cable structure, such as the two-compartment and multi-compartment cable models. All these models can describe neuronal behavior effectively in different aspects, therefore, they will be used in our following theoretical study of dendritic integration.In Chapter 3, we reveal theoretically the underlying mechanism of a dendritic in-tegration rule for a pair of excitatory(E) and inhibitory(I) synaptic inputs discovered in a recent experiment. Starting with the two-compartment neuron model, we construct its Green’s function and carry out an asymptotic analysis to obtain its solutions. Using these asymptotic solutions, in the presence of E-I inputs, we can fully explain all the experimental observations. We then extend our analysis with multi-compartment neuron model to characterize the E-I dendritic integration on dendritic branches. The novel characterization is con?rmed by a numerical simulation of a biologically realistic neuron as well as published experimental results.In Chapter 4, we theoretically generalize the dendritic integration rule in Chapter3 to describe the spatiotemporal dendritic integration for all types of inputs, including a pair of E-I, E-E, I-I inputs and multiple inputs with mixed types. In addition, the general dendritic integration rule is valid at any time during the dendritic integration process for inputs with arbitrary arrival time difference. The general rule is derived analytically from the two-compartment neuron model. However, we also verify it in a simulation of the realistic neuron and in experiments. The general rule ?nally leads us to a novel graph representation of the dendritic integration process, which is demonstrated to be functionally sparse.In Chapter 5, we address the theoretical issue of how much the dendritic integration rule discovered in the experiment can be accounted for using the somatic membrane potential dynamics described by the point neuron model. We demonstrate both analytically and numerically that the IF model can explain partial of the experimental results. Inspired by a two-port analysis, we then modify the IF model to the DIF model to characterize all the experimental observations. Meanwhile, the DIF model provides experimental testable predictions.In Chapter 6, we systematically investigate the performance of the point neuron models in characterizing the spatiotemporal dendritic integration effect. We demonstrate numerically that, compared with the standard IF model and HH model, our DIF model and DHH model can accurately capture the membrane potential produced by the two-compartment neuron model with a passive or an active soma, respectively. In particular, our DHH model can accurately predict the spike time of the two-compartment neuron model, whereas the prediction error made by the HH model is signi?cantly large. In addition, the HH model occasionally predicts a fake spike.The scienti?c contribution and the novelty of our work can be summarized as follows. First, the nonlinearity in the cable equation with time-dependent synaptic inputs makes its analytical solution di?cult to obtain. Here we analytically solve the cable equation via the asymptotic analysis, and apply the asymptotic solutions to reveal the underlying mechanism of the dendritic integration rule discovered experimentally.In addition, the previous research work on dendritic integration are mainly qualitative and speci?c. Here we propose a general dendritic integration rule to quantitatively describe dendritic integration for all types of synaptic inputs. The general rule is further con?rmed in the realistic simulations and real experiments. Moreover, point neuron models are considered only to describe the somatic membrane potential in previous works. Here we incorporate the dendritic integration effect into point neuron models successfully. Contrast to the cable model, our effective point neuron model can be potentially used in a large scale simulation of a network of neurons with dendrites to reduce the computational cost.