The stochastic activity of neurons is caused by various sources of correlated fluctuations and can be described in terms of simplified, yet biophysically grounded, integrate-and-fire models. One paradigmatic model is the quadratic integrate-and-fire model and its equivalent phase description by the theta neuron. Here we study the theta neuron model driven by a correlated Ornstein-Uhlenbeck noise and by periodic stimuli. We apply the matrix-continued-fraction method to the associated Fokker-Planck equation to develop an efficient numerical scheme to determine the stationary firing rate as well as the stimulus-induced modulation of the instantaneous firing rate. For the stationary case, we identify the conditions under which the firing rate decreases or increases by the effect of the colored noise and compare our results to existing analytical approximations for limit cases. For an additional periodic signal we demonstrate how the linear and nonlinear response terms can be computed and report resonant behavior for some of them. We extend the method to the case of two periodic signals, generally with incommensurable frequencies, and present a particular case for which a strong mixed response to both signals is observed, i.e. where the response to the sum of signals differs significantly from the sum of responses to the single signals. We provide Python code for our computational method: https://github.com/jannikfranzen/theta_neuron.

神经元的随机活动是由各种相关波动源引起的，可以用简化的、但具有生物物理基础的积分和激发模型来描述。一种范例模型是二次积分激发模型及其由 θ 神经元进行的等效相位描述。在这里，我们研究由相关 Ornstein-Uhlenbeck 噪声和周期性刺激驱动的 theta 神经元模型。我们将矩阵连分数法应用于相关的福克-普朗克方程，以开发一种有效的数值方案来确定稳态放电率以及刺激引起的瞬时放电率调制。对于静止情况，我们确定了发射率因有色噪声的影响而降低或增加的条件，并将我们的结果与极限情况的现有分析近似值进行比较。对于额外的周期信号，我们演示了如何计算线性和非线性响应项并报告其中一些的谐振行为。我们将该方法扩展到两个周期信号的情况，通常具有不可通约的频率，并提出了一种特殊情况，观察到对两个信号的强烈混合响应，即对信号总和的响应与响应总和显着不同的情况到单个信号。我们为我们的计算方法提供Python代码：https://github.com/jannikfranzen/theta_neuron。