Our computational study of E–I networks reveals that network dynamics depend jointly on the network connectivity structure, namely the relative strength of inter- and intra-connectivity between and within the excitatory and inhibitory subnetworks, and the neuromodulation of excitatory and inhibitory cells. We consider ACh’s effect on both excitatory and inhibitory cells that switches neuronal response properties, as measured by the PRC and I–F curves, from Type II to Type I, thus affecting the cellular propensity of synchronization. Considering the possibility of nonuniform cholinergic release or that excitatory and inhibitory cells exhibit Type I or Type II properties without the presence of an M-current, we investigate all four combinations of excitatory and inhibitory cells, namely excitatory or inhibitory cells with high (Type I) or low (Type II) modulatory tone. To vary network connectivity structure, we focus on the effects of the strengths of inter-connections between excitatory and inhibitory cells (E–I synapses and I–E synapses), and the strengths of intra-connections among excitatory cells (E–E synapses) and among inhibitory cells (I–I synapses). Figures 3 and 4 show measures of network dynamics as E–E and I–I intra-connectivity strengths are varied together (horizontal axes) and E–I and I–E inter-connectivity strengths are varied together (vertical axes) for networks with Type I excitatory cells (Fig. 3) and Type II excitatory cells (Fig. 4), and inhibitory cells exhibiting either Type I or Type II properties. Network dynamics roughly divide into three parameter regions in which synchronous bursting of the excitatory cells is differentially affected by a combination of the modulation of cellular properties and network connectivity structure.
High E–I and I–E Inter-connectivity Promotes Synchronous Excitatory Bursting Regardless of Cellular Properties
When E–I and I–E inter-connectivity strength dominates over E–E and I–I intra-connectivity strength (upper-left corners of heatmaps in Figs. 3, 4), all networks exhibit synchronous excitatory bursting regardless of the cellular propensity for synchrony modulated by ACh. Values of the Synchrony Measure (panels a and e) are high in this parameter regime for all modulatory conditions, with networks with Type II excitatory cells (Fig. 4) reporting higher values due to more coincident spike firing predicted by their cellular properties. Synchronous activity is robust with approximately all cells participating in the activity bursts (panels b and f), and the frequency of bursts is similar in all networks (panels c and g). Furthermore, the widths of both the excitatory and inhibitory bursts remain narrow in this regime, with narrower excitatory bursts exhibited by networks with Type II excitatory cells due to the additional synchrony promoted by excitatory intra-connectivity and the properties of Type II PRCs (Figs. 3, 4d, h). We note that the detection of excitatory bursts was robust to repeated simulations of these networks, with only one set of network connectivities (whose position in the heatmap is identified by the bolded outline in Fig. 3c, d for which bursts are detected in some, but not all, of the simulations run. As shown in the raster plots in Fig. 5, whether excitatory cells are Type I (Fig. 5a, b) or Type II (Fig. 5c, d) they fire in synchronous bursts. Thus, for this network connectivity, neuromodulation of cellular propensity for synchronization has little effect on the generation of excitatory bursting.
The synchronous excitatory bursting in these networks with high inter-connectivity is predicted and governed largely by the PING mechanism. In the PING mechanism, the inhibitory cells serve to “silence” the excitatory cells following an inhibitory burst, which causes all of the excitatory cells to return to the same point of their oscillatory firing cycle and subsequently fire synchronously when released from inhibition (Traub et al. 1997; Kopell et al. 2010; Whittington et al. 2000; Ermentrout and Kopell 1998). Evidence of the PING mechanism at work lies in the occurrence of inhibitory cell synchronous bursts near the end of excitatory cell synchronous bursts or immediately following these bursts, and in the effective silencing of excitatory activity by the inhibitory burst.
For additional verification, we simulated the same networks, but with all synaptic connections from the inhibitory cells to the excitatory cells (I–E synapses) removed (Fig. 6). In the parameter region discussed above, no synchronous activity is obtained when excitatory cells are Type I, confirming that synchronous inhibitory signaling is necessary to induce excitatory synchronous bursting in these networks. When excitatory cells are Type II, synchronous activity emerges as intra-connectivity strength increases in this parameter regime, as expected from the propensity for Type II neurons to synchronize in response to excitatory connectivity. However, obtaining synchrony for the weakest intra-connectivity strength values, as obtained in the networks containing I–E synapses, depends critically on inhibitory signaling.
In these networks, cellular properties influence the patterning of spike firing within the PING-driven synchrony. The patterning of inhibitory cell activity in response to an excitatory burst relies heavily on the inhibitory cell type, which in turn can cause subtle changes in excitatory cell dynamics. When the excitatory cells are Type I, this effect is primarily seen through the burst frequencies of the excitatory and inhibitory subpopulations (Fig. 3c, g). Comparing the example raster plot with Type I excitatory cells and Type I inhibitory cells (Fig. 5a) to that with Type I excitatory cells and Type II inhibitory cells (Fig. 5b), we see that the slower excitatory burst frequency of the former network is due to the multiple bursts of inhibitory activity in response to a burst of excitatory activity, which provides a longer lasting inhibitory synaptic signal to the excitatory cells. In contrast, when the inhibitory cells are Type II, only one instance of inhibitory activity follows excitatory activity, allowing the excitatory cells quicker release from this inhibitory signal.
The role of inhibitory cell patterning when the excitatory cells are Type II is seen primarily via differences in the Synchrony Measure and is shown by comparing example networks with Type I inhibitory cells (Fig. 5c) to those with Type II inhibitory cells (Fig. 5d). In the latter case, each excitatory burst elicits a single inhibitory burst of activity including all inhibitory cells, which ensures each excitatory cell receives a near-identical profile of inhibitory synaptic current. This allows the excitatory cells to organize based upon their external driving current, with cells with the highest external current firing earliest in the burst and those with the lowest firing latest. This organization causes the burst to occur over a longer time interval, slightly lowering the Synchrony Measure. However, when the inhibitory cells are Type I, we again see them respond to an instance of excitatory network bursting with multiple instances of activity; importantly, in this case the profile of these bursts varies in response to different instances of excitatory activity due to the randomness in the connectivity of the network as well as the heterogeneity in external drive to the cells. Variations in the inhibitory activity cause a disparity in the inhibitory signal felt by each excitatory cell, which disrupts organization in the excitatory bursts. However, this disorder also allows the burst to occur over a shorter timescale, increasing the Synchrony Measure. Thus, the inhibitory cell type plays a key role in explaining the slight difference in the Synchrony Measure seen in these networks when comparing Fig. 4a, e, while a negligible effect is seen in the frequency of the bursts (Fig. 4c, g).
Cellular Properties Dictate Synchronous Excitatory Bursting When E–E and I–I Intra-connectivity is High
When E–E and I–I intra-connectivity dominates over E–I and I–E inter-connectivity, obtaining synchronous excitatory bursting depends on the cellular propensity for synchrony. For the ranges of synaptic strengths considered here, this parameter regime begins when intra-connectivity strength is slightly higher than inter-connectivity strength (near 7a, b labels in Figs. 3, 7c, d labels in Fig. 4). When intra-connectivity is much higher than inter-connectivity (lower-right corners of heatmaps in Figs. 3, 4), the high I–I synaptic strength acts to slow firing of the inhibitory cells to the point that they cannot fire synchronously, thus minimizing their influence on excitatory subnetwork dynamics. In this regime, networks with Type I excitatory cells have Synchrony Measure values close to zero for excitatory cells (Fig. 3a, e left panels) reflecting asynchronous firing (as shown in the examples in Fig. 7a, b), as predicted by their cellular properties. While the majority of excitatory cells are firing (Fig. 3b, f left panels), no synchronous excitatory bursts were detected, as reflected by the lack of a burst frequency value (Fig. 3c, g left panels). Networks with Type II excitatory cells, on the other hand, display synchronous excitatory bursting (as shown in the examples in Fig. 7c, d), as predicted by cellular properties, with high Synchrony Measure (Fig. 4a, d), full cell participation in synchronous bursts (Fig. 4b, e) and similar burst frequencies (Fig. 4c, g). Thus, in this network structure, ACh governs the generation of synchronous excitatory activity.
The PING mechanism is not involved in generating synchronous excitatory bursts in this parameter regime as evidenced by the long gap between the excitatory cell burst and inhibitory cell firing. Inhibitory cells fire in bursts, with higher Synchrony Measure when they are Type I (Fig. 7c, d), in response to the oscillatory excitatory signal, but inhibition is not responsible for silencing the excitatory cells after the burst since they stop firing well in advance of the inhibitory bursts. Additionally, the burst frequency is not affected by the different profiles of inhibitory firing when the inhibitory cells are Type I or Type II, reflecting their lack of influence on excitatory bursting. This is confirmed by fully removing the I–E synapses and continuing to see synchronous excitatory cell firing in this regime (Fig. 6c, d).
In this regime of dominant intra-connectivity, inhibitory cell type can play a role in the dynamics of the inhibitory cell population without significantly influencing the patterning of the excitatory subpopulation. When excitatory cells are Type I, their asynchronous firing provides a weak, nearly tonic drive to the inhibitory cells. Type I inhibitory cells with weak I–I connectivity can form synchronous patterns in response to such a drive, as shown by the example raster plot in Fig. 7a and discussed in detail in our previous work (Rich et al. 2017). In contrast, Type II inhibitory cells are less excitable and more susceptible to suppression by inhibitory signaling via the dominant intra-connectivity, preventing them from forming clearly synchronous dynamics, as shown in the example raster plot in Fig. 7b.
Meanwhile, Type II excitatory cells can synchronize driven by E–E connectivity and not network inter-connectivity as discussed above, allowing for the synchronous excitatory subpopulation dynamics shown in Fig. 7c, d. Here again, though, the type of inhibitory cell dictates the dynamics of the inhibitory subpopulation in response to this weak, but synchronous, drive to the inhibitory cells. Given that Type I inhibitory cells are more excitable, the weak burst of excitation is sufficient to prompt all of the inhibitory cells to fire in a closely clustered fashion, an example of which is shown by the raster in Fig. 7c. In contrast, properties of our Type II neuron model lead these inhibitory cells to respond with a more sparse burst to a nearly identical excitatory synaptic drive, an example of which is shown by the raster in Fig. 7d.
We note that the interaction between Type II PRC properties and strong E–E connectivity can cause some complex burst patterns in the excitatory population. This is evidenced in Fig. 4d, h by the significantly wider excitatory bursts seen in the middle of our range of network intra-connectivities. However, this behavior does not disrupt the overall oscillatory behavior of the network driven by the network intra-connectivity.
Cellular Properties Influence Ability of Inter-connectivity to Generate Synchronous Excitatory Bursting When Inter- and Intra-connectivity are Balanced
When E–I, I–E inter-connectivity and E–E, I–I intra-connectivity are both strong in our parameter space, corresponding to the upper-right corner of the heatmaps in Figs. 3 and 4, both cellular properties and network connectivity contribute to the network’s tendency to exhibit synchronous excitatory bursting. When excitatory cells are Type I, while their cellular properties resist synchronization, loose synchronous bursting is obtained when inhibitory cells are Type I (example raster in Fig. 8a), but not Type II (example raster in Fig. 8b). This example represents one of the few instances in this parameter regime in which excitatory bursting activity was detected in some, but not all, of the simulations run (as represented by the grid squares with a bolded outline in Fig. 3c, d). As reflected in their Synchrony Measures (Fig. 3a, e), Type I inhibitory cells synchronize tightly with high I–I intra-connectivity (A, right panel), while Type II inhibitory cells do not (E, right panel). The strong inhibitory signal provided to the excitatory cells from Type I inhibitory cells silences their activity and produces a weak synchronous excitatory burst by the PING mechanism. When inhibitory cells are Type II, however, their more sparse firing has little effect on the excitatory subnetwork, and asynchronous excitatory firing persists.
When excitatory cells are Type II, their propensity for synchronization strengthens the influence of high inter-connectivity to generate robust excitatory synchronous bursting for both types of inhibitory cells (Fig. 8c, d). Indeed, Synchrony Measures for both excitatory and inhibitory subnetworks are the highest in this parameter regime (Fig. 4a, e) with full network participation in the bursts (Fig. 4b, f). Additionally, values of all measures are the same for Type I and Type II inhibitory cells. There is conflicting evidence as to whether the PING mechanism or excitatory network intra-connectivity drives this synchrony: while the inhibitory network bursts do closely follow the excitatory network bursts (Fig. 8c, d), as predicted by the PING mechanism, the removal of I–E synapses does not eliminate excitatory synchrony (Fig. 6c, d), meaning that inhibition may not serve a causal role in synchronizing excitatory cells. It is likely that some combination of these two mechanisms is what results in the strong synchrony of the excitatory network seen here.
Thus, in this network structure, cholinergic modulation acts in conjunction with high inter-connectivity to generate synchronous excitatory bursting. Synchronous excitatory bursting fails to exist only when both excitatory and inhibitory populations have a low propensity for synchronization.
In this regime of balanced inter- and intra-connectivity, cell properties contribute to the characteristics of excitatory synchronous bursting, as shown by the example raster plots in Fig. 8. For Type II excitatory cells, spikes in the synchronous bursts are highly coincident (Fig. 8c, d) due to the strong E–E intra-connectivity overpowering the heterogeneity in firing frequency that created more variation in spike timing in the low E–E intra-connectivity regime (examples in Fig. 5c, d) This induces a highly coincident burst of inhibitory cells immediately following the excitatory burst. The strong inhibitory signal to all excitatory cells coupled with their previous coincident firing leads to a long silent period between excitatory bursts and low burst frequency (Fig. 4c, g). For Type I excitatory and inhibitory cells, burst frequencies are the highest since the high E–E intra-connectivity drives the excitatory cells to recover quickly from the inhibitory signal and initiate the next excitatory burst (Fig. 3c).
The inter- and intra-connectivity strengths can also be balanced at a weak level in our parameter space, which corresponds with the lower-left corner of the heatmaps in Figs. 3 and 4. In this regime, networks with Type I excitatory cells exhibit completely asynchronous firing due to an inability to achieve PING rhythmicity, while inhibitory cells may be able to synchronize themselves due to the near-tonic drive provided by the asynchronous excitatory cells (Fig. 3a, e). Meanwhile, in networks with Type II excitatory neurons, excitatory synchrony can be achieved in networks with all, but the weakest connectivity strengths (Fig. 4a, e) driven by the ability for Type II excitatory neurons to synchronize themselves even with weak E–E synapses.
Thus, when the inter- and intra-connectivity strengths are balanced, but at weak levels, the tendency to achieve synchronous excitatory cell dynamics is controlled by ACh. In this regime, synchronous excitatory bursting occurs only due to the tendency of Type II neurons to synchronize due to E–E synapses, even when these synapses may be weak.
Dynamics of Inhibitory Subnetworks
Across our parameter space of network connectivity structures, there are regimes where activity in the inhibitory subnetwork does not correlate with activity in the excitatory subnetwork. These instances are largely robust to repetition, as there are few instances where inhibitory bursts are detected in some, but not all, of the repetitions for a given network (cases in which this occurs are represented the grid squares with bolded outlines in Figs. 3, 4c, d, g, h). For example, as discussed above, Type I inhibitory cells can form oscillatory synchronous bursts independent of synchronous activity in the excitatory subnetwork. This behavior is seen in our networks with Type I excitatory cells when intra-connectivity is larger than inter-connectivity (diagonal band of high Synchrony Measure in Fig. 3a, right panel). In this regime, the excitatory cells are asynchronous and provide a tonic excitatory drive to the inhibitory cells, inducing repetitive firing. When coupled with inhibitory synapses, repetitively firing Type I cells receiving strong tonic drive have a high propensity for synchronization (Rich et al. 2016). As intra-connectivity increases, maintaining synchronous bursting requires increased excitatory input to counteract the increased inhibitory signaling within the inhibitory subnetwork (resulting in the diagonal band). In fact, inhibitory activity becomes sparse and asynchronous when intra-connectivity is much higher than inter-connectivity (Fig. 3a–c right panels, lower-right corners).
In other parameter regimes, firing in the inhibitory subnetwork is almost completely suppressed. This occurs in three parameter regimes that are most easily identified in the right panels of Figs. 3 and 4b, f, which show the number of active inhibitory cells: a small region of high inter-connectivity strength for networks of Type I excitatory and inhibitory neurons (Fig. 3b), a small region with low inter-connectivity strength for networks of Type II excitatory and inhibitory neurons (Fig. 4f), and a relatively large parameter regime for networks of Type I excitatory neurons and Type II inhibitory neurons (Fig. 3f). The first two cases are easily explained. The first is a case of classic depolarization block of the Type I inhibitory neurons: the moderate intra-connectivity strength is enough to force excitatory cells to fire very quickly due to the E–E connectivity, and the high inter-connectivity strength leads these fast firing excitatory cells to provide excessive excitation to the inhibitory cell population, driving those cells into depolarization block. The second case is the opposite situation, as weak inter-connectivity combined with low excitability of Type II inhibitory cells result in insufficient excitatory synaptic signal to the inhibitory cells to induce firing.
The final case seen in networks of Type I excitatory neurons and Type II inhibitory neurons is more complex, involving intricacies of the dynamics of the M-type potassium current. The activity of this ionic current not only serves to shift the properties of the neuron PRC from Type I to Type II as discussed previously, but also imbues the neurons with spike frequency adaptation. Given the slow timescale of the z gating variable that governs this current when compared to the extremely fast timescale of the m, n and h variables governing the currents directly inciting action potential firing, the M-current acts to slow down the firing of the neuron following repetitive action potentials. Thus, the neuron “adapts” its firing frequency given the recent past, firing slower if a quick burst of action potentials occurred previously. This adaptation is reflected by a rise in the value of z as action potential firings occur. As the potassium current is a hyperpolarizing current, larger values of z that arise from action potential firing invoke a larger amplitude of the M-type potassium current which in turn slows down cell firing.
When Type II neurons are provided a tonic excitatory drive that induces repetitive firing, the adaptation current eventually settles into a stable periodic pattern that allows repetitive action potential firing at a constant frequency. However, in our E–I networks, the drive to the inhibitory population is provided by the synaptic drive from the excitatory population, which has a distinctly nontonic profile. In particular, properties of the reversal potential in the synaptic current term in Eq. 10 speed up action potential firing when compared to a tonic current with a similar maximum amplitude. This increase in firing frequency prevents z from settling into a stable oscillation, instead causing it to steadily increase. When this gating variable rises too high, the hyperpolarizing current from the M-type potassium channel exceeds the depolarizing excitatory input current from the excitatory cell population, causing a net hyperpolarizing current and in turn quiescence. In short, in certain E–I networks where the excitatory cells do not synchronize, the adaptation current prevents the inhibitory cells from exhibiting repetitive firing due to the form of the excitatory synaptic current.