The steady state and response to a periodic stimulation of the firing rate for a theta neuron with correlated noise

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.