We develop mathematical techniques for analyzing detailed Hodgkin–Huxley like models for excitatory–inhibitory neuronal networks. Our strategy for studying a given network is to first reduce it to a discrete-time dynamical system. The discrete model is considerably easier to analyze, both mathematically and computationally, and parameters in the discrete model correspond directly to parameters in the original system of differential equations. While these networks arise in many important applications, a primary focus of this paper is to better understand mechanisms that underlie temporally dynamic responses in early processing of olfactory sensory information. The models presented here exhibit several properties that have been described for olfactory codes in an insect’s Antennal Lobe. These include transient patterns of synchronization and decorrelation of sensory inputs. By reducing the model to a discrete system, we are able to systematically study how properties of the dynamics, including the complex structure of the transients and attractors, depend on factors related to connectivity and the intrinsic and synaptic properties of cells within the network.

我们开发了数学技术来分析兴奋性抑制神经元网络的详细霍奇金赫胥黎样模型。我们研究给定网络的策略是首先将其简化为离散时间动态系统。离散模型在数学和计算上都更容易分析，并且离散模型中的参数直接对应于原始微分方程组中的参数。虽然这些网络出现在许多重要的应用中，但本文的主要重点是更好地理解作为嗅觉信息早期处理中时间动态响应基础的机制。这里介绍的模型展示了几种特性，这些特性已被描述为昆虫触角中的嗅觉代码。这些包括感觉输入的同步和去相关的瞬时模式。通过将模型简化为离散系统，我们能够系统地研究动力学特性，包括瞬态和吸引子的复杂结构，如何取决于与网络内细胞的连通性和内在和突触特性相关的因素。