Maltba Tyler E, Zhao Hongli, Tartakovsky Daniel M
Department of Statistics, UC Berkeley, Berkeley, CA 94720 USA.
Department of Mathematics, UC Berkeley, Berkeley, CA 94720 USA.
Cogn Neurodyn. 2022 Jun;16(3):683-705. doi: 10.1007/s11571-021-09731-9. Epub 2021 Nov 3.
Neuronal dynamics is driven by externally imposed or internally generated random excitations/noise, and is often described by systems of random or stochastic ordinary differential equations. Such systems admit a distribution of solutions, which is (partially) characterized by the single-time joint probability density function (PDF) of system states. It can be used to calculate such information-theoretic quantities as the mutual information between the stochastic stimulus and various internal states of the neuron (e.g., membrane potential), as well as various spiking statistics. When random excitations are modeled as Gaussian white noise, the joint PDF of neuron states satisfies exactly a Fokker-Planck equation. However, most biologically plausible noise sources are correlated (colored). In this case, the resulting PDF equations require a closure approximation. We propose two methods for closing such equations: a modified nonlocal large-eddy-diffusivity closure and a data-driven closure relying on sparse regression to learn relevant features. The closures are tested for the stochastic non-spiking leaky integrate-and-fire and FitzHugh-Nagumo (FHN) neurons driven by sine-Wiener noise. Mutual information and total correlation between the random stimulus and the internal states of the neuron are calculated for the FHN neuron.
神经元动力学由外部施加或内部产生的随机激发/噪声驱动，通常由随机或随机常微分方程系统来描述。这样的系统允许解的分布，其（部分）由系统状态的单时联合概率密度函数（PDF）来表征。它可用于计算诸如随机刺激与神经元的各种内部状态（例如膜电位）之间的互信息以及各种放电统计量等信息论量。当随机激发被建模为高斯白噪声时，神经元状态的联合PDF恰好满足福克 - 普朗克方程。然而，大多数生物学上合理的噪声源是相关的（有色的）。在这种情况下，所得的PDF方程需要一个封闭近似。我们提出了两种封闭此类方程的方法：一种改进的非局部大涡扩散率封闭和一种基于稀疏回归来学习相关特征的数据驱动封闭。对由正弦 - 维纳噪声驱动的随机非放电泄漏积分发放神经元和菲茨休 - 纳古莫（FHN）神经元测试了这些封闭方法。计算了FHN神经元的随机刺激与内部状态之间的互信息和总相关性。
