Weakly nonlinear response of noisy neurons

We calculate the instantaneous firing rate of a stochastic integrate-and-fire neuron driven by an arbitrary time-dependent signal up to second order in the signal amplitude. For cosine signals, this weakly nonlinear theory reveals: (i) a frequency-dependent change of the time-averaged firing rate reminiscent of frequency locking in deterministic oscillators; (ii) higher harmonics in the rate that may exceed the linear response; (iii) a strong nonlinear response to two cosines even when the response to a single cosine is linear. We also measure the second-order response numerically for a neuron model with excitable voltage dynamics and channel noise, and find a strong similarity to the second-order response that we obtain analytically for the leaky integrate-and-fire model. Finally, we illustrate how the transition of neural dynamics from the linear rate response regime to a mode-locking regime is captured by the second-order response. Our results highlight the importance and robustness of the weakly nonlinear regime in neural dynamics.