Single-cell learning in Stentor coeruleus is governed by a fractional-order low-pass filter

Single cells display a range of complex behaviors normally associated with a nervous system, including basic forms of learning like habituation. The giant ciliate Stentor coeruleus habituates to mechanical stimuli and shows many of the hallmark features characteristic of habituation in animal cells. When Stentor cells are mechanically stimulated by a predator or other stimuli, an action potential fires and leads to calcium-dependent contraction. When the same cell is repeatedly stimulated, it becomes less likely to respond, thus showing habituation. While the molecular basis of habituation in Stentor is not yet known, it has been shown to involve CaMKII, which also plays a key role in learning in neurons. The presence of an action potential, the role of calcium in the response, and the involvement of CaMKII in habituation, all suggest a potential deep conservation of learning mechanisms between single-celled protists and the neurons of animals. A number of different models have been proposed to explain habituation in a single cell but existing data in Stentor are unable to clearly rule out any of these models or favor others. Here we report a frequency domain analysis of habituation in which we measure the response probability of Stentor cells to pulsatile stimuli delivered at a range of frequencies. We find that the Bode plot of the frequency response resembles a classic low pass filter, with a flat passband at low frequencies, a clear corner frequency, and a linear roll-off. However, unlike standard low pass filter, the roll-off occurs with a slope of -30dB/decade, thus showing a fractional-order behavior. None of the existing models for habituation in Stentor, at least in their current form, predict this form of the frequency response, leading us to look for other explanations. We tested, and ruled out, a model based on a refractory period associated with the re-extension of cells following contraction. Inspired by methods used in analog circuit design to approximate fractional order systems using conventional lumped devices, we developed a model in which a series of distinct molecular species, such as different multimeric complexes of CaMKII, acting in parallel to inhibit the response, produce a fractional-order effect. The fractional order behavior of habituation in Stentor resembles the fractional-order behavior of adaptation in neurons, further supporting the idea that neurons may employ similar mechanisms for learning as were already present in unicellular eukaryotes prior to the evolution of metazoa.