Zeki Mustafa, Dag Tamer
College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait.
Front Comput Neurosci. 2025 Jun 13;19:1565552. doi: 10.3389/fncom.2025.1565552. eCollection 2025.
Mathematical analysis of biological neural networks, specifically inhibitory networks with all-to-all connections, is challenging due to their complexity and non-linearity. In examining the dynamics of individual neurons, many fast currents are involved solely in spike generation, while slower currents play a significant role in shaping a neuron's behavior. We propose a discrete map approach to analyze the behavior of inhibitory neurons that exhibit bursting modulated by slow calcium currents, leveraging the time-scale differences among neural currents. This discrete map tracks the number of spikes per burst for individual neurons. We compared the map's predictions for the number of spikes per burst and the long-term system behavior to data obtained from the continuous system. Our findings demonstrate that the discrete map can accurately predict the canonical behavioral signatures of bursting performance observed in the continuous system. Specifically, we show that the proposed map a) accounts for the dependence of the number of spikes per burst on initial calcium levels, b) explains the roles of individual currents in shaping the system's behavior, and c) can be explicitly analyzed to determine fixed points and assess their stability.
对生物神经网络，特别是具有全对全连接的抑制性网络进行数学分析具有挑战性，因为它们具有复杂性和非线性。在研究单个神经元的动力学时，许多快速电流仅参与动作电位的产生，而较慢的电流在塑造神经元行为方面起着重要作用。我们提出一种离散映射方法来分析由缓慢钙电流调制而呈现爆发行为的抑制性神经元的行为，利用神经电流之间的时间尺度差异。这种离散映射跟踪单个神经元每次爆发的动作电位数量。我们将该映射对每次爆发动作电位数量的预测以及长期系统行为与从连续系统获得的数据进行了比较。我们的研究结果表明，离散映射可以准确预测在连续系统中观察到的爆发行为的典型行为特征。具体而言，我们表明所提出的映射a）说明了每次爆发动作电位数量对初始钙水平的依赖性，b）解释了各个电流在塑造系统行为中的作用，并且c）可以进行明确分析以确定不动点并评估其稳定性。
