A key element of our model is the topology of network connections, which differs from all other topologies of biological networks in the literature. The network has n neurons and n_b bundles, and each neuron has two dendrites in different bundles [4],[14],[15]. The network can be described by a bipartite graph G = {N∪B, E}, where N is the set of neurons, B the set of bundles, and E the set of connections from neurons to bundles such that, for a neuron a ∈ N and a bundle b ∈ B, (a ,b) ∈ E if a has a dendrite in b. The network topology is thus specified by the adjacency matrix O = {o_ij}, i = 1,…,n, j = 1,…,n_b, where o_ij = 1 if neuron i has a dendrite in bundle j, and o_ij = 0 otherwise. If dendro-dendritic connections are formed at random, then O is a random binary matrix whose rows satisfy . Figure 1 shows such a matrix for a network of 48 neurons and 12 bundles. We considered two procedures in order to assign dendrites to bundles. In both cases, for a network of n neurons, and a given integer d>0, we start with an empty adjacency matrix of n rows and columns. Then, for each neuron we select two bundles as follows. The index of the first bundle i₁ is selected uniformly at random in the set {1,2,…,n_b}, the second index is selected uniformly at random in the set {1,2,…,n_b}/{i₁}, ensuring that no neuron has two dendrites in the same bundle. For the first procedure, this selection is repeated for all neurons, leading to a completely random allocation of dendrites into bundles. There is a finite probability that some bundles are never selected, and these are removed from the network. In the second procedure, we keep track of the number of dendrites in each bundle and, once one bundle contains d dendrites, this is excluded from further selection. This we refer to as a “homogeneous arrangement of the connections” as each of the n_b bundles contains the same number of dendrites.

As observed in vivo [5] we found no fixed ‘leader’ or ‘follower’ cells, and the order in which neurons start to burst varies randomly with each burst (Fig. 4A). Thus bursting in the model is an emergent activity due to the interplay between the single neuron dynamics and network dynamics. The lack of a marked leader/follower character of the model neurons might have been accentuated by the homogeneous arrangement of the connections in the network used for simulations, as all bundles contained the same number of dendrites (d_i = d̅, i = 1,…,n_b). Therefore, we also considered a network with the same number of cells and bundles (and the same mean connectivity d̅) but where the number of dendrites varied in each bundle. For each cell, the leader/follower character was measured by its mean ‘advantage’(12)where denotes the time of the onset of the kth burst in cell i, and p is the total number of bursts. A_i is strongly correlated with the number of dendritic connections (r = 0.8796, P = 10⁻¹⁶; Fig. 4B). Thus bursts are more likely to start in regions of the network where dendritic bundling is more pronounced.

With no suckling input, the firing of oxytocin cells in the model is uncorrelated (as in vivo), as each receives a wholly independent synaptic input. Between bursts, spiking activity in the network is characterised by small but increasing cross-correlation of firing rates (Fig. 4C), a consequence of the strengthening of the interactions between cells. The background spike activity becomes progressively more irregular approaching a burst, as indicated by an increasing index of dispersion of the firing rate (Fig. 4D). Both results are in agreement with experimental findings in vivo [34]–[36]. In the model the increased variability arises because, towards a burst, activity produces dendritic oxytocin release, with excitatory consequences, but also endocannabinoid production, with inhibitory consequences but with different timescales; if endocannabinoid release is eliminated (by setting α = 0) then there is no increase in variability.

We observed bursting in networks with varying number of neurons and/or bundles. In a network of 1000 neurons with limited bundling (d̅ = 2), bursts occur rarely, propagate slowly, and involve only some cells (Fig. 5A). Increasing the degree of bundling, i.e. decreasing n_b, leads to faster propagation and better synchronization (Fig. 5B). Figure 5C shows the propagation of a burst by plotting the temporal course of the number of cells recruited into a burst. The burst ‘wavefront’ grows exponentially with time, implying that even large networks can be rapidly synchronized. An example is given in Fig. 5D where we show a synchronized burst occurring in a network of 3000 neurons (d̅ = 8).

The bursts are followed by long silent periods (up to 20 s). In vivo [7] the post-burst inhibition is the most variable component of the burst, both in duration (7–56 s) and intensity, indicating that it is not simply the deterministic consequence of an activity-dependent AHP. In the model, the post burst silence is mainly a consequence of the prolonged suppression of afferent input, following the increase in endocannabinoid concentration after a burst. In vivo, some otherwise typical oxytocin cells have been observed occasionally which show no bursts at milk ejection but instead fall silent (Fig. 6A). A similar phenomenon can be replicated in the model by assuming that some neurons do not express oxytocin receptors (i.e. by setting k_OT  = 0 for these neurons, Fig. 6B).

In the model, the dendritic stores of readily-releasable vesicles are continuously incremented by the suckling-related ‘priming’ input. Their level, averaged over the entire network, increases relatively steadily between bursts despite activity-dependent depletion (Fig. 7A), and bursts tend to occur when the stores are relatively large. The mean level at the time of bursts correlates strongly with the logarithm of the inter-burst interval (r = 0.99; P<10⁻⁹; Fig. 7B). Fig. 7C plots the rate of change of the stores against the store level (both averaged over the network). The decrease in slope at high levels reflects a reduction of the average release rate, and is a consequence of the suppression of afferent input as a result of endocannabinoid release. This stops the release from becoming regenerative, and allows the stores to increase further. In this phase, the network activity becomes more irregular because of the opposing feedback mechanisms: local activity-dependent excitation through the effects of dendritic oxytocin release, and inhibition due to suppression of afferent input. When the stores are large, spatially coordinated fluctuations of release can have a large impact on the dynamics. If just a few neighbouring cells show coincidentally increased activity due to stochastic variation in their input rates, and have large enough stores, then enough oxytocin can be released to trigger positive feedback and start a burst.

Increased spike activity between the bursts enhances depletion of the stores and so can delay or even suppress bursting (Fig. 8A); conversely, an increase in inhibitory input can promote the reflex in a system which fails to express bursting because of an insufficient priming (Fig. 8B). Such “paradoxical” behaviours have been extensively described in vivo; for example, injections of the inhibitory neurotransmitter GABA into the supraoptic nucleus of a suckled, lactating rat can trigger milk-ejection bursts [37] (Fig. 8B right); conversely, many stimuli that activate oxytocin cells, including the systemic administration of cholecystokinin, relaxin, or hypertonic saline, all suppress the reflex [e.g. 37] (Fig. 8A right). Very occasionally a single burst can occur shortly after removing the suckling stimulus (Fig. 8C). This feature is also shared (very occasionally) by the reflex in vivo, and indicates that suckling itself is not a strictly necessary trigger.

Figure 9 shows the response of a model neuron in absence of suckling input; in this case, cells increase their mean firing rate more strongly in response to an increase of the excitatory input. In the model, during suckling, neurons that are strongly excited produce endocannabinoids that reduce the overall input level. This negative feedback defends the system from over-excitation, and maintains the network activity in an optimal range for bursting. This is an important feature, because bursting in the model is possible only within a range of values of excitatory input (Fig. 9C). The exact range depends on the strength of the coupling between spike activity and dendritic secretion (as measured by the frequency threshold for release f_rel = 1/τ_rel). At a low level of excitation, an increase in the excitatory rate favours bursting by increasing the frequency of release episodes which can trigger a burst. However, beyond a critical level, release events may be so frequent that stores are not replenished fast enough to reach the critical level required to trigger a burst. Under such conditions, bursts become rarer and less predictable, until eventually over-excitation disrupts the reflex.

As illustrated in Fig. 9E, the inhibitory effect of endocannabinoids reduces the likelihood of a burst being triggered at low synaptic rates, but also reduces the rate of depletion of dendritic oxytocin, thus increasing the probability of bursting at high synaptic input rates. The overall effect is to increase the range of synaptic input rates compatible with bursting, and to make the mean rate at which bursts occur relatively independent of synaptic input rate within this range.

Spatial inhomogeneity in the stochastic input can also degrade the reflex (Fig. 9D). With increasing spatial inhomogeneity, for a given average firing rate, there are more faster firing cells, and also more slowly firing cells. The faster firing cells will generate more short intervals – potential burst triggering events - but those events will be less potent because of greater depletion of their stores. For these events to trigger bursts, they must recruit responses from other cells to which they are connected – but the slower firing cells are less excitable (although they have higher store levels). The net result is that bursts are triggered less often. Thus the system performs optimally when the activity is relatively homogeneous between oxytocin cells, a conclusion previously drawn from experimental studies [33].

Whereas neurons exchange information mostly via spikes, endocrine cells rely on hormonal pulses to signal to their target tissues. For many neurons, clustered spike activity can be optimally effective in inducing the required changes on the targets, but for endocrine cells to generate a signal large enough to be read at a distance, their secretory activity must not only be optimal for each cell, their activity must also be co-ordinated; hence peptide hormone signals are generally pulsatile [41]. Gonadotrophin-releasing hormone (GnRH) neurons also display synchronised bursts, possibly as a result of direct positive feedback from GnRH release [42]. Neuroendocrine cells are perhaps a special case in generating a classical hormone signal by co-ordinated electrical activity. However, many populations of neurons in the brain produce a peptide product as well as a conventional transmitter, and many of these peptides have effects on organismal behaviour that are hormone-like [17],[43], in that they act at dispersed and often distant targets to produce prolonged organisational changes. For a hormone-like, pulsatile signal to be produced reliably, the activity of a population of peptide-secreting neurons must be co-ordinated in a physiologically plastic manner. Such co-ordinated signals, coming from the individual nodes of an interactive network, must emerge from the dynamics at the lower level of organization (for the neuron case, from the dynamics of stochastic ionic channels coupled via the membrane potential). In the present model, network interactions are solely mediated by spikes with interspike intervals less than τ_rel; similar spike doublets are thought to play a critical role in the synchronization of network activity in many neural systems [44]–[46].

The present model clearly produces a close match to electrophysiological data at the level of spike output, and its main strength is the simplicity of the representation of a single neuron; this makes it feasible to use the model to explore how properties of the network (connectivity and dynamics of intercommunication) affect the system behaviour. We believe that the simplifications are unlikely to have had any major influence, with two possible exceptions. First, we have not included intracellular [Ca²⁺] as a variable, although mobilisation of intracellular Ca²⁺ can trigger dendritic oxytocin release, and therefore probably contributes to the central oxytocin release during milk-ejection. Implicitly we have assumed that this overlaps with activity-induced oxytocin release and can be neglected, but it is possible that in some circumstances oxytocin release triggered by Ca²⁺ release from intracellular stores might precipitate a burst. Second, we modelled dendritic release as a relatively common deterministic event – small packets released fairly frequently. Dendritic release probably involves the relatively rare exocytosis of large vesicles that each contains a very large amount of oxytocin – and release is likely to be highly stochastic, with interval length governing the probability of release rather than determining it. Whether this will affect the model behaviour substantially remains to be tested.