One of the fundamental problems in neuroscience is characterizing the transfer function that converts noisy synaptic inputs into output firing rates. A common assumption is that the membrane time constant is the dominant factor governing the time course of firing rate responses. However, previous studies have shown that neural response times can be faster than expected from voltage dynamics alone. If the membrane time constant does not solely determine response time, what are the parameters that describe the transformation of inputs into output firing rates?

We investigate this question using integrate-and-fire models (IF). Noisy synaptic inputs are modeled as a white noise process with drift, characterized by a time-varying mean and variance. We use linear perturbation techniques to analyze the response dynamics of several different IF models, for signals encoded in the mean and the variance of the input, and for models operating in two qualitatively different regimes of behavior.

In the perfect integrate-and-fire model (PIF), the sub-threshold membrane dynamics perfectly mimic the integral of the input current. We show that the PIF produces a perfect replica of the time-varying input rate for Poisson distributed input.

Next, we survey the response properties of the leaky integrate-and-fire model (LIF). Our survey covers a wide range of baseline input parameter values and studies perturbations in both the mean or variance of the input. We find that response dynamics are highly dependent on regime, as well as which input parameter encodes the signal.

Additionally we investigate how synaptic dynamics affect LIF response. We find a striking reduction in the overall gain for variance-encoded signals. Mean encoded signals elicit responses with finite high-frequency gain across regimes, and reduced resonances.

Finally, we focus on nonlinear responses, examining the time course of onset and offset responses for two different IF models, the LIF and the more realistic exponential integrate-and-fire model (EIF). The responses of the models differ in that the EIF shows a slight delay before responding to a step increase in input, a delay that is not found for the LIF nor for the response to step decreases in input for either model.