Magnocellular vasopressin neurons, together with their oxytocin-secreting neighbors, have been important “model systems” in neuroscience because of circumstances that make them exceptionally open to experimental study. The supraoptic nucleus contains only oxytocin and vasopressin neurons, and this greatly facilitates identifying these neurons to make recordings of their electrical activity. Most importantly, however, these neurons all project to the pituitary gland, so their primary output can be measured directly, in their respective plasma oxytocin and vasopressin concentrations. Thus, these systems are a “window on the brain”, they allow us to look directly at the relationship between spiking activity and physiological function.

Most modeling papers present the finished product. This is fine if you want to know what it contributes to the physiology, or to use it as a tool, but to learn how to model, or to assess how good a model is, it helps to understand exactly how the model ended up in its final form. The final model would have been preceded by many earlier versions, even whole previous models which never made it to publication. These hidden parts can be useful to other modelers, and can also help give more confidence in the final model to the physiologists. Here, we present this story for the vasopressin neuron.

The rat hypothalamus contains about 9000 magnocellular vasopressin neurons, mostly within the supraoptic and the paraventricular nuclei. The supraoptic nucleus contains only magnocellular neurons, about 70% of which make vasopressin and 30% oxytocin. Because all magnocellular neurons project to the posterior pituitary gland, and because only magnocellular neurons project there, it is possible to identify these neurons antidromically: if a stimulating electrode is placed on the neural stalk through which all axons en route to the pituitary pass, then stimuli will trigger spikes in those axons; these spikes are propagated antidromically to invade the cell bodies, with a latency that reflects the axonal length and the propagation speed.

Our focus is on modeling the activity of vasopressin neurons in vivo (in anaesthetized rats), but our model must be reconciled with data obtained both in vivo and in vitro. However, there are differences between the behavior of vasopressin cells in vivo and in vitro (and differences between in vitro hypothalamic slice preparations, hypothalamic explants, isolated cell preparations, and organotypic cultures). Many of these differences relate to the sparseness of synaptic input in vitro. The few intracellular recordings that have been made in vivo confirm that supraoptic neurons in vivo receive intense synaptic input, whereas vasopressin neurons in vitro typically receive very little. In slice preparations, magnocellular neurons have truncated dendrites, few of their sources of input remain within the slice, those that are may not still be connected to the vasopressin neurons, and those that do remain are themselves deafferented. The neurons still generate spikes in response to direct depolarization, but show little spontaneous spiking unless the bathing medium is adjusted by, for instance, an increase in K⁺ concentration or by the addition of glutamate.

This is a general issue with in vitro recordings; an intact neuron in the brain may receive 10000 synaptic inputs, and if most come from active neurons, then they receive a massive barrage of excitatory and inhibitory synaptic potentials (EPSPs and IPSPs). Because most classical neurotransmitters open ligand-gated ion channels, the overall conductance state of the neuron at rest is different in vivo and in vitro; a high rate of input results in extensive opening of ion channels, reducing the cell’s input resistance, with consequences for many membrane properties. The effect of the voltage-gated opening of ion channels depends on the cell’s input resistance; if this is high, then a given change in conductance will produce a larger change in membrane voltage. Accordingly, activity-dependent mechanisms are exaggerated in in vitro preparations; these preparations give good access to the components of the spiking mechanism, but properties such as the magnitude and time course of voltage-dependent conductances may be altered.

Over and above this, the membrane noise that synaptic input provides can change how neurons process information. We often assume that noise is undesirable, but in complex nonlinear systems “stochastic resonance” can occur. Stochastic resonance is when an increase in levels of unpredictable fluctuations improves the quality of signal transmission or detection performance, rather than degrading it, raising the possibility that the brain has evolved to utilize random noise as part of the “neural code.” Stochastic resonance is an essential part of osmoresponsiveness in vasopressin neurons. Vasopressin neurons possess stretch-sensitive ion channels that support depolarization in response to the cell shrinkage that accompanies a rise in extracellular osmolarity. In the physiological range of osmotic stimuli, this depolarization is too small to result in spike activity in vasopressin cells that are isolated from a synaptic input. However, the presence of fluctuations in membrane potential changes this – even a small depolarization will increase the probability that fluctuations will exceed spike threshold, allowing small sustained depolarizations to translate into increased spiking activity.

Thus, synaptic “noise” is not merely something that adds imprecision to the behavior of a neuron; it can alter how the neuron processes information. To model vasopressin cells as they operate physiologically, we must include a realistic simulation of synaptic noise. We can incorporate intrinsic mechanisms as established by in vitro experiments, but must be aware that the magnitude and dynamics of conductance changes will be different in vivo from those observed in vitro.

Vasopressin cells project to the posterior pituitary gland, where spike activity triggers exocytosis of hormone-containing vesicles. The peptide secreted from the nerve terminals of the 9000 vasopressin cells forms a relatively smooth summed signal, which passes into the bloodstream and on to targets across the body. This is the important output, the population signal, and to understand the function of the phasic firing pattern, we must recognize that the pattern of secretion from any single cell is lost in the output of a large and asynchronous population.

In many neuronal network models, it is assumed that the output of neurons is synonymous with their spike activity. But the biologically relevant output of any neuron is not its spike activity, but the chemicals that it secretes. This secretion is regulated by spike activity, but the relationship is not linear, is state dependent, and often is subject to retrograde modulation by factors secreted from the post-synaptic target. Thus, the presumption that spike activity is an adequate representation of the information transmitted from a neuron is always unsafe. However, for vasopressin cells, we know the properties of stimulus–secretion coupling, and so we can relate our understanding of spiking behavior to the real output of these neurons.

Vasopressin cells in vivo are not spontaneously active; without synaptic input, they remain silent. With low rates of synaptic input, they exhibit very slow (1 Hz) or irregular spiking. However, given enough synaptic input, they fire phasically. This is not because the input comes in a phasic pattern. Phasic firing involves a combination of activity-dependent excitation and a delayed activity-dependent inhibition. Activity-dependent excitation of bursts can be demonstrated by perturbing a vasopressin cell during its silent period with a stimulus strong enough to trigger just a few spikes. Such stimuli will trigger the onset of a burst depending (a) on the intensity and (b) on the time that has elapsed since the end of the previous burst: it becomes progressively easier to trigger a burst with increasing time. Activity-dependent inhibition can be observed similarly, by delivering stimuli during a burst. Stimuli will stop a burst depending (a) on the intensity and (b) on the time that has elapsed since the beginning of the burst: it becomes progressively easier to stop a burst with increasing time. Thus, the same stimulus can stop or start bursts depending on exactly when in the burst cycle it is delivered. This identifies the vasopressin cell as a bistable oscillator – it has two relatively stable states, activity and silence, and even small perturbations can “flip” the cell from either state into the other.

Knowing that vasopressin cells fire asynchronously, we can see that a brief stimulus may activate some cells, inhibit others, and have no effect on many – and which cells it activates and which it inhibits depend purely on the state of the different cells at the time of application. Phasic activity in vasopressin cells, therefore, does not depend critically on the patterning of their inputs and must be generated by an intrinsic mechanism. However, excitatory inputs are essential for any spiking in vasopressin cells in vivo, and these inputs can be modulated by several retrograde signals that are released from the dendrites of vasopressin cells. So in some circumstances, the input to vasopressin cells is correlated with the phasic activity, but it is the phasic activity that determines the input pattern, not the input pattern that determines the phasic pattern.

The inputs to vasopressin cells are diverse, reflecting the multiple functions of vasopressin. Some inputs come from osmoresponsive neurons of the forebrain, including from two “circumventricular organs” – the subfornical organ (SFO) and the organum vasculosum of the lamina terminalis (OVLT). These inputs involve both excitatory inputs mediated by the neurotransmitter glutamate and inhibitory inputs mediated by gamma-aminobutyric acid (GABA).

Synaptic input consists of many small events. We expect a typical neuron to receive inputs from many different neurons, via as many as perhaps 10000 synapses. Assuming that these inputs are independent on a short timescale (tens of milliseconds), the timing of each synaptic event is essentially random. This gives us a simple way to simulate the spontaneous inputs for a model neuron, by using a random number generator to generate inputs, using a defined mean rate. Each input event generates a postsynaptic potential (PSP) which rises rapidly over 1–2 ms, to a magnitude of 1–5 mV and then decays more slowly, over 10–20 ms. These PSPs can be either excitatory (depolarizing) EPSPs or inhibitory (hyperpolarizing) IPSPs. Because the rise is rapid, we can simplify the PSP to an instantaneous step and model the decline using a single exponential decay. PSP magnitude varies between synapses and events, and also depends on the difference between the membrane potential and the reversal potential for the ion that carries the PSP (usually mainly Na⁺ for EPSPs and Cl⁻ for IPSPs). Just after a spike, IPSPs are smaller because the vasopressin cell is hyperpolarized, but because of this hyperpolarization, the vasopressin cell is not spiking then, so the difference in IPSP size has no consequences for spike activity. Otherwise, changes in PSP magnitude are small, and if we simplify by using a fixed magnitude for PSPs, we do not affect anything noticeably. It is always worth checking conclusions using a detailed model with variable PSP magnitudes and half-lives, and with realistic reversal potentials, but in our models of vasopressin cells, qualitative behavior is robust to such changes. Hence, in our working model, synaptic input is defined by just three parameters for EPSPs and three for IPSPs: event rate, magnitude, and decay rate. If we make these the same for EPSPs and IPSPs, we can study the neuron under conditions where the input is, on average, an exact balance between inhibition and excitation – the state at which we expect neurons generally to be, in resting conditions in vivo, given that we know that inhibitory synapses and excitatory synapses are approximately equally abundant on magnocellular neurons.

Hodgkin–Huxley-type models model a cell’s membrane potential in detail, using representations of the dynamics driving individual ionic conductances. In contrast, in the integrate-and-fire (or spike-response) model, we only model influences on the neuron’s ability to fire a spike (its “excitability”). The first of these “influences” are the PSPs. Each of these perturbs the membrane potential, which starts at a fixed resting value (the resting potential). If the membrane potential reaches a spike threshold, we record a spike. We do not model the spike’s rapid changes in the membrane potential, only record its time of occurrence, but we do model the slower changes in the membrane potential which follow.

The other “influences” consist of potentials generated by the cell after a spike. Vasopressin neurons express several types of voltage-gated Ca²⁺ channels, and every spike is accompanied by Ca²⁺ entry. Accordingly, repetitive spiking results in a large increase in intracellular (cytosolic) Ca²⁺ concentration ([Ca²⁺]_i). This is cleared quite slowly, and it functions as a short-term memory of spike activation, influencing membrane properties via Ca²⁺-sensitive channels. There are several types of Ca²⁺-sensitive channels, and their time course and magnitude of activation differ, depending on channel properties, spatial distribution, proximity to the Ca²⁺ entry channels, and variable Ca²⁺ dynamics within the cell.

We fit the model by applying the same analysis techniques that we use to examine the experimental data. Our analysis of experimental data begins with the series of spike times from cells recorded in vivo. It is possible to record spike activity from a single vasopressin cell for several hours, and, because extracellular recording does not involve damaging the cells membrane integrity, activity patterns are very stable. For a cell firing at a mean rate of 5 spikes/s (about average for vasopressin cells recorded in vivo), 1 h of recording yields a time series of about 18000 spike events.

In vivo (unlike in vitro), spiking in vasopressin cells is never fully regenerative. The DAP never brings the membrane potential of a vasopressin cell above spike threshold, but merely facilitates the depolarizing effects of membrane fluctuations. We can add a DAP to our spike-response model using the same form of decaying exponential as that for the HAP (but with opposite sign), and the parameters for magnitude and decay can be fitted to match the peak in the hazard function. This match predicts a DAP with a half-life of 150 ms. However, models with this DAP show no bursting: the half-life is too short. In vivo, the ISI histogram shows that ISIs of 300 ms are common in vasopressin cells – and a modestly sized DAP (that does not lead to regenerative spiking) decays considerably in this time. We can simulate the effects of dynorphin by adding another, slower variable to suppress the DAP, but this is of no use if the DAP cannot sustain a burst in the first place. Accordingly, it seems that this type of DAP cannot explain the burst mechanism, at least not within a spike-response model.

Bursting and silence are two distinct activity states, both of which can be maintained for tens of seconds. When the neuron shifts between these states, the change is rapid, occurring over just a few seconds. Such bistability can be represented most simply by a differential equation for the membrane potential (), , where (as used in Brown et al. (1994) to model LHRH neurons). This defines a system with two stable states ( and ) separated by an unstable equilibrium (), and gives a simple way of producing phasic bursting. This was used in a bursting model, with the DAP still included to match the short-term patterning detected in the hazard, but not part of the bursting mechanism. However, while this could produce bursting, it was too simple a model to reproduce the detailed features of bursts observed in vivo. The next attempt (Clayton et al., 2010) retained this explicit bistability, but made it more dependent on spiking activity, using a more sophisticated form informed by dynamic systems theory. This created a model with 14 parameters, and fitting the parameters by hand was replaced with fitting using a genetic algorithm along with fit measures designed to detect the match to the histogram, hazard, and measures of the bursting, including shape, mean burst duration and mean silence duration. This model produced extremely close qualitative and quantitative fits to the in vivo data, matching the spiking to the point of being difficult to distinguish between the model and the cell. It could also fit a range of cells with highly varied spike rates and forms to their bursting.

The focus thus far had been on producing a model that could match the form of bursts in vivo. The assumption was that if you could match this in detail, then the important dynamics of how the model cell responds to input would follow. Vasopressin cells only fire phasically under intense osmotic challenge when they receive a high rate of synaptic input. At low input rates, they show slow firing, and as input increases, they progress to an irregular pattern, before transitioning into full phasic firing. As input continues to increase, the bursts intensify and may lengthen, until the cells eventually shift to continuous firing. These transitions are functionally important, because phasic firing is much more effective at triggering hormone secretion than continuous spiking. The secretion mechanism is very nonlinear, and in particular, subject to fatigue: it can only maintain a high secretion rate in response to sustained spiking for about 20 s before dropping to a low rate of response, and it requires a period of silence to recover. This makes the phasic pattern optimal at triggering secretion. The Clayton et al. (2010) model was good at reproducing the shift from slow firing through irregular spiking into a phasic pattern and came close to matching the shift into continuous spiking but would still tend to drop into silent or slow firing periods. This could be fixed, but only by adjusting the sensitivity of the bistable mechanism. This was problematical; the model would reproduce the response of a vasopressin cell to increasing input only if we assumed that parameters of the bistable mechanism were themselves activity dependent.

The problem with the Clayton model is the explicit bistability. Instead of using representations of the cell’s mechanisms to reproduce bursting behavior, it abstracts directly at the required behavioral level: the resulting model thus reproduces the outward appearance of bursting, but not the underlying mechanism, and matches the in vivo spiking response only within a limited input range. Developing a model which reproduces the emergent bistability of the neuronal mechanisms, rather than using an explicit mechanism, is more challenging. The original DAP-based bursting model had attempted this and failed.