Effects of dendritic load on the firing frequency of oscillating neurons

We study the effects of passive dendritic properties on the dynamics of neuronal oscillators. We find that the addition of a passive dendrite can sometimes have counterintuitive effects on firing frequency. Specifically, the addition of a hyperpolarized passive dendritic load can either increase, decrease, or have negligible effects on firing frequency. We use the theory of weak coupling to derive phase equations for "ball-and-stick" model neurons and two-compartment model neurons. We then develop a framework for understanding how the addition of passive dendrites modulates the frequency of neuronal oscillators. We show that the average value of the neuronal oscillator's phase response curves measures the sensitivity of the neuron's firing rate to the dendritic load, including whether the addition of the dendrite causes an increase or decrease in firing frequency. We interpret this finding in terms of to the slope of the neuronal oscillator's frequency-applied current curve. We also show that equivalent results exist for constant and noisy point-source input to the dendrite. We note that the results are not specific to neurons but are applicable to any oscillator subject to a passive load.