In the nervous system, neurons are the fundamental component used to transmit information, conveyed along axons by a series of action potentials. However, it is unclear how information is encoded in these trains of action potentials. To study the computational function of neurons, we focused on the firing rate outputs of single neurons as the statistics of their inputs varied. Our goal was to uncover principles governing the dependence of single neuron firing rates on input statistics. This could be thought of as exploring further what is meant by “hyper-excitable” and “hypo-excitable” neuronal firing rates. We studied responses from single-compartment neuron models, in vitro recordings from rat neocortical brain slices in response to injected current, and in vivo single unit recordings from anesthetized rats in response to whisker movements.

Using phase planes from dynamical systems analysis, we found that the firing rates of neurons can be categorized into three fundamental types, which describe how the mean firing rates depend on input fluctuations at steady state. We found that changes in biophysical parameters can alter a model neuron’s type. For example, to what extent a neuron’s firing rate is sensitive to quick input fluctuations was found to depend on two factors: a ratio of inward to outward currents and the separation of time scale between the fast and slow currents. Thus, decreasing a neuron’s sodium conductance caused its firing rate to be more sensitive to input fluctuations in a nonlinear fashion.

We further studied firing rate adaptation of single neurons, such as how firing rates change over time in response to input fluctuations. Although rate adaptation is commonly observed, simple yet precise descriptions of this adaptation remain illusive. Using in vitro and in vivo data from the rat cortex and thalamus, we found that rate adaptation is similar to the mathematical operation of fractional differentiation. This discovery suggests that adaptation is fundamentally a multiple time scale process with a frequency-independent phase advance, and that rate adaptation can be approximated as a simple linear filter specified by the order of differentiation. Thus, using this non-integer order derivative, we can predict neuronal firing rates to time-varying stimuli. Further, we find that in the rat the order of differentiation increases along the sensory pathway from the thalamus to the cortex, suggesting an emphasis on high-pass filtering in the cortex.

In summary, we found that the sensitivity of single neuron firing rates to input fluctuations depends nonlinearly on biophysical parameters such as sodium conductance. Processes that alter channel conductances can fundamentally alter how the firing rate responds to input fluctuations. Further, we show that rate adaptation in cortical and thalamic rat neurons is consistent with fractional differentiation, which has properties that are potentially useful for coding and motor control. Together, these results suggest computations more complex than a simple one-to-one relation between stimulus intensity and single neuron firing rate.