We have constructed mathematical models of the electrical activity of two hypothalamic supraoptic neuro-secretory cell-types, and we support our models with new calcium imaging and in vitro electrophysiological data. These cells are neurones that project to the pituitary gland and secrete either of two hormones, oxytocin or vasopressin, into the blood from their axonal terminals. Oxytocin-secreting and vasopressin-secreting cells are closely related and physically they differ only subtly, however when physiologically stressed their discharge patterns are dramatically distinct. We first show how each potassium current contributes to the action-potentials and after-potentials observed in these cells, and we show how these after-potentials are correlated to intra-cellular calcium elevations. We then show how these currents regulate the excitability of these cells and consequently shape their discharge pattern.

Vasopressin secreting neurons of the rat hypothalamus discharge lengthy, repeating bursts of action potentials in response to physiological stress. Although many electrical currents and calcium-dependent processes have been isolated and analyzed in these cells, their interactions are less well fathomed. In particular, the mechanism of how each burst is triggered, sustained, and terminated is poorly understood. We present a mathematical model for the bursting mechanism, and we support our model with new simultaneous electrical recording and calcium imaging data. We show that bursts can be initiated by spike-dependent calcium influx, and we propose that the resulting elevation of bulk calcium inhibits a persistent potassium current. This inhibition depolarizes the cell above threshold and so triggers regenerative spiking and further calcium influx. We present imaging data to show that bulk calcium reaches a plateau within the first few seconds of the burst, and our model indicates that this plateau occurs when calcium influx is balanced by efflux and uptake into stores. We conjecture that the burst is terminated by a slow, progressive desensitization to calcium of the potassium leak current. Finally, we propose that the opioid dynorphin, which is known to be secreted from the somatodendritic region and has been shown previously to regulate burst length and phasic activity in these cells, is the autocrine messenger for this desensitization.

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We present a tractable stochastic phase model of the temperature sensitivity of a mammalian cold receptor. Using simple linear dependencies of the amplitude, frequency, and bias on temperature, the model reproduces the experimentally observed transitions between bursting, beating, and stochastically phase-locked firing patterns. We analyze the model in the deterministic limit and predict, using a Strutt map, the number of spikes per burst for a given temperature. The inclusion of noise produces a variable number of spikes per burst and also extends the dynamic range of the neuron, both of which are analyzed in terms of the Strutt map. Our analysis can be readily applied to other receptors that display various bursting patterns following temperature changes.