Number of synapses required to trigger NMDAR spikes in distal dendritic branches

We examined how many synapses onto an individual terminal dendritic branch in the apical tuft were required to trigger a regenerative NMDAR event in the absence of background synaptic input. This condition roughly approximates the in vitro conditions used to study synaptic integration with glutamate uncaging, where background activity is reduced due to long range inputs being severed in acute slices and inhibition is blocked by MNI-glutamate [55]. Since the most likely location for NMDAR spike generation is in the high impendence terminal branches of the tuft [29], [35], [56], we focused on these dendritic compartments (Fig. 1B). To stimulate a particular branch, synapses were randomly distributed along the branch and near-coincident stimulus-evoked synaptic input was mimicked by driving each synapse with a 200 Hz random train for 5 ms. This resulted in each quantal synaptic input (subsequently referred to as ‘synapse’) firing once, on average. When few coincident synapses were stimulated, the EPSP in the stimulated distal branch was small and increased approximately linearly from the resting potential (−58 mV; similar to −57 mV measured in slices [36]) as the number of synapses increased. However, above 15 synapses, the duration of the EPSP increased dramatically and exhibited a flat top, hallmarks of a local regenerative NMDAR spike (Fig. 1C). We identified the presence of NMDAR spikes in a dendritic branch using a −30 mV voltage threshold criterion (Fig. S1). By stochastically varying the spatial and temporal patterns of synaptic input onto a dendritic branch, and measuring the occurrence of NMDAR spikes over 10 trials, it was possible to build an input-output (I-O) relationship for each individual branch. That is, the relationship between the number of stimulated synapses and the probability of generating an NMDAR spike (P(NMDAR spike)). Fig. 1D shows the fits of a sigmoid function (Materials and Methods) to the I-O relationships for all 28 terminal branches in the model (mean length = 72±52 µm). Between 10 and 25 temporally coincident synapses were required to trigger an NMDAR spike with a probability of 0.5 in the terminal branches. On average 18 synapses were required and this increased to 30 coincident synaptic inputs to reliably trigger an NMDAR spike on every trial. These values are in good agreement with experimental estimates from dendritic recordings [31] and glutamate uncaging experiments on L2/3 [35] and L5 neurons [36] in acute slices and previous modelling studies [29], [44]. Given that the average number of spines per terminal branch is approximately 140, assuming a spine density of 2 µm⁻¹ [57], these simulations suggest that nearly coincident activation of a substantial fraction (13–20%) of the synapses on a terminal branch is required to activate a dendritic spike reliably, under quiescent conditions.

Background excitatory synaptic input reduces the threshold for NMDAR spikes

We next examined how asynchronous background excitatory synaptic input affected integration in the tuft. We did not include background inhibition in these initial simulations, in order to study the effects of background excitatory inputs in isolation. We estimated the rate of background excitatory synaptic input in vivo from both anatomical and functional measurements. Tufted L5 pyramidal cells are innervated by ∼15,000 excitatory synaptic contacts (assuming 1 per spine), of which 3500 occur on the apical trunk and dendritic tuft in layers 1–3 [57], [58]. Since many of these contacts arise cortically from other L5 and L2/3 pyramidal cells, which typically fire at ∼1 Hz in vivo [49], [50], a L5 pyramidal cell is expected to experience a background quantal synaptic input rate of ∼1750 Hz on its apical dendritic tree, assuming a release probability of ∼0.5 per synaptic contact [59]. To mimic a modest level of background excitatory synaptic input, we randomly distributed 900–1500 quantal synaptic inputs over the apical tuft (including the Ca²⁺ hot zone; Blue region, Fig. 1A) and drove each synapse with a different 0.85 Hz Poisson train, resulting in a background synaptic input rate of 765–1250 Hz. These modest levels of uncorrelated background excitatory input rarely triggered spontaneous NMDAR spikes (<1% of trials, which were excluded from the analysis whenever possible), but did depolarize the dendritic membrane potential by 2–5 mV, within the range of experimental recordings [36], [47] (Fig. 1E, F) and generated voltage noise at the soma with a standard deviation of 0.86–1.26 mV.

We next examined how the branch I-O relationship was affected by background excitatory synaptic input. These simulations revealed that many fewer coincident synaptic inputs were required to trigger an NMDAR spike than for quiescent control conditions (Fig. 1G). On average, the number of synapses required to trigger an NMDAR spike with 50% probability fell from 18 to 6 in the presence of 1500 background synapses (Fig. 1H). A lower threshold for NMDAR spike generation and an increased in slope (gain) of the branch I-O relationship was also observed with background input when the time window for synaptic input was increased from 5 to 15 ms and the input frequency scaled down to maintain ∼1 quantal conductance per synapse (Fig. 1I). These simulations show that NMDAR spikes can be triggered reliably in terminal branches by much smaller numbers of nearly-coincident synaptic inputs (i.e. activation of 4–7% of the total synapses) in the presence of modest levels of background excitatory synaptic input, than under quiescent conditions.

Background excitatory input decreases the number of stimulated branches required to trigger action potentials

To investigate whether background excitatory synaptic input also affects the spike output of the pyramidal cell model, we determined the number of stimulated branches required to trigger somatic APs. To do this we randomly selected a group of N branches within our set of 28 terminal branches in the tuft and simultaneously stimulated each branch with 30 near-synchronous synaptic inputs (5 ms window) to ensure a high probability of triggering an NMDAR spike in each branch under all conditions (Fig. 1H). We then systematically altered the number of stimulated branches and determined whether the neuron fired APs or not. This rather artificial synaptic input configuration was chosen to quantify the impact of excitatory background input when multiple branches were stimulated. Fig. 2A shows an example where stimulation of 4 apical branches produced only a subthreshold somatic potential during quiescent control conditions. Systematically increasing the number of stimulated branches, while averaging across many combinations of branches to minimize branch specific effects, revealed that in the absence of background input 11 near-synchronous stimulated NMDAR spikes were required to trigger APs with 50% probability, and 15 were required to trigger an AP reliably (Fig. 2B).

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Figure 2. Background excitatory input reduces the number of nearly synchronously stimulated branches required to trigger action potentials.

(A) Somatic voltage response to the activation of 4 apical branches stimulated with 30 nearly synchronous synapses each, in the absence (black) and presence (blue) of 1500 background excitatory synapses distributed on the apical tuft. (B) Probability of triggering action potentials (P(AP)) versus number of stimulated branches, in the absence (black open circles) and presence of different levels of background excitation (blue filled circles). Lines show fits to a sigmoid function. Grey line and circles show results for 15 synaptic inputs per branch and 1500 background excitatory synapses.

https://doi.org/10.1371/journal.pcbi.1003590.g002

By contrast, in the presence of background excitatory synaptic input, the relationship between the probability of triggering APs (P(AP)) and the number of stimulated dendritic branches exhibited a lower activation threshold and was much steeper (Fig. 2A, B). Indeed, the number of stimulated branches required to trigger an AP with 50% reliability was reduced from 11 to <4 for 1500 background excitatory synapses. The enhanced efficacy of NMDAR spikes was not dependent on the strong activation of individual branches, since reducing the number of synapses per stimulated branch from 30 to 15 still produced a leftward shift in the I-O relationship (Fig. 2B, grey). Thus, even the modest level of background excitatory input used here substantially increased the efficacy of NMDAR spikes in triggering APs. These results suggest that, in the absence of inhibition, modest levels of background excitatory synaptic input arising from ongoing network activity could increase the probability of NMDAR spikes occurring in the dendritc tuft and increase AP generation.

Background excitatory input extends the temporal integration window in pyramidal cells

To examine how the L5 pyramidal cell model integrated inputs on longer time scales than the nearly coincident synaptic input examined so far, we desynchronized the stimulation of different dendritic branches located over the apical tuft. To do this we stimulated each branch with a near-synchronous (5 ms) burst of synaptic input, but activated different branches at random times within a time window that ranged from 5 ms to 200 ms and examined how desynchronization of branch activation affected the probability of triggering APs. In the absence of background excitatory input, 4 stimulated branches produced only a subthreshold response (Fig. 3A1). The average number of stimulated branches required to trigger an AP with 50% reliability gradually increased from 11 to 21 as stimulation of the branches was desynchronized from 5 ms to 200 ms (Fig. 3B). This result suggests that under quiescent conditions, L1 synaptic input would need to be both synchronous and strong enough to trigger NMDAR spikes in a substantial fraction of the terminal branches in order to produce an AP.

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Figure 3. Background excitatory input extends the temporal integration of stimulated branches.

(A1) Left panel: example voltage traces in 4 terminal apical branches during near synchronous branch activation (5 ms window, 30 synapses per branch; grey traces) and from the soma (red). Other panels show responses when stimulated branches are desynchronized in progressively larger temporal windows (50 ms, 100 ms, 200 ms; soma traces lighter shades of orange). (A2) Same as A1 but during background input from 1500 excitatory synapses, for near-synchronous branch stimulation (5 ms; dark blue) or desynchronized in progressively larger temporal windows (50 ms, 100 ms, 200 ms; soma traces lighter shades of blue). (B) Probability of action potentials (P(AP)) versus number of stimulated apical branches for different degrees of temporal dispersion during background activity from 1500 background excitatory synapses (blue lines) compared to control condition (red-yellow lines). (C) Number of stimulated branches required to trigger an AP with 50% probability (P(AP) = 0.5), for different levels of temporal dispersion in the stimulated branches (input window), in the absence (red) and presence of background excitatory input (blue).

https://doi.org/10.1371/journal.pcbi.1003590.g003

By contrast, background excitatory synaptic input substantially reduced the number of stimulated branches required to trigger APs (e.g. 4 branches, Fig. 3A2) and asynchronous branch activation within a 100 ms window became just as effective at triggering APs as coincident activation of the dendritic branches (Fig. 3B). The number of stimulated branches required to trigger an AP only slightly increased when branches were stimulated over a 200 ms window (Fig. 3B, C). The longer the time window over which the NMDAR spikes occurred, the larger the absolute reduction in the number of stimulated branches required to trigger APs with background input, compared to control conditions (Fig. 3C). These results suggest that, in the presence of modest levels of background excitatory input, clusters of synchronous synaptic inputs could be integrated over timescales that are considerably longer than both the membrane time constant of L5 pyramidal cells (∼10–20 ms [60]) and the NMDAR-mediated synaptic integration window in individual terminal branches of L2/3 pyramidal cells recorded in acute slices using glutamate uncaging (∼10 ms integration window [35]), where background network activity and inhibition were largely absent.

Background excitatory synaptic input enables integration of spatially and temporally distributed excitatory synaptic input

To investigate how background excitatory synaptic activity affects the integration of spatially distributed input, we examined the I-O relationship of the model when stimulus-evoked synaptic input was randomly distributed across the entire apical tuft (Fig. 4A), rather than being clustered on selected branches. In the absence of background network activity, near synchronous activation (5 ms duration random burst at 200 Hz) of 100 excitatory quantal synapses typically generated subthreshold responses (Fig. 4B1), while 160 synapses triggered an AP with a 50% probability (Fig. 4C). By contrast, 100 near-synchronous synapses triggered numerous NMDAR spikes and somatic APs in the presence of 1500 background excitatory synapses (Fig. 4B2). Indeed, only 60–70 nearly synchronously activated synapses were required to reach P(AP) = 0.5 in the presence of background excitatory input (Fig. 4C). When the temporal coincidence of the stimulus-evoked input was relaxed from 5 ms to 50 ms (and the input frequency scaled down to ensure the same total amount of excitation per synapse), the number of spatially distributed inputs required to reach P(AP) = 0.5 rose to >200 in the absence of background input (Fig. 4C). However, 60–70 spatially and temporally distributed synapses were still sufficient to reach P(AP) = 0.5 in the presence of background excitatory input, only marginally more than for coincident input (Fig. 4C). Temporally dispersing the synaptic input further to 100 ms (Fig. 4B1,B2) or 200 ms required a greater number of synaptic inputs, but the number of synapses required in the presence of background excitatory input remained approximately 3-fold less than in its absence (Fig. 4C,D). These simulations show that modest levels of background excitatory synaptic input can extend both spatial and temporal integration in a L5 pyramidal cell model, when compared to quiescent conditions.

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Figure 4. Background excitatory input enables integration of spatially distributed synaptic input over extended temporal windows.

(A) Apical tuft with an example distribution of 60 synaptic inputs (red dots). (B1) Left: example voltage traces from all terminal apical branches (grey traces, N = 28) and the soma (red) in response to stimulation of 100 spatially distributed synaptic inputs at 200 Hz for 5 ms, during control conditions. Right: example trial for 100 ms stimulation window. (B2) Same as B1 in the presence of background activity from 1500 excitatory synapses. Somatic voltage traces are in blue and are truncated at +10 mV. (C) Probability of triggering action potentials (P(AP)) versus number of stimulated synapses, spatially distributed over the apical tuft. Synapses were driven with random trains with a window of 5 ms, 50 ms, 100 ms and 200 ms, and the frequency was scaled to maintain ∼1 quantal conductance per synapse, in the absence (red-yellow traces) and presence of background excitatory input (blue traces). (D) Number of distributed synapses required to trigger an AP with 50% probability (P(AP) = 0.5), for different levels of temporal dispersion in the stimulus evoked synapses (input window), in the absence (red) and presence of background excitatory input (blue), and during AMPAR-only background synaptic inputs which depolarized the dendrites to comparable levels (green).

https://doi.org/10.1371/journal.pcbi.1003590.g004

Synaptic integration during balanced background excitatory and inhibitory synaptic input

So far we have examined how synaptic integration in the apical tuft of a L5 pyramidal cell model is altered by background excitatory synaptic input. However, under physiological conditions ongoing cortical network activity consists of both excitatory and inhibitory conductances in an approximately balanced configuration [15], [16], [18]. Although our understanding of cortical inhibition is far from complete, our knowledge of the properties of inhibitory synaptic inputs onto pyramidal cells has expanded recently, as a result of a number of in vitro and in vivo studies [48], [61]–[66]. These studies show that Somatostatin (SOM) expressing interneurons (which include Martinotti cells) innervate the dendritic tuft of L5 pyramidal cells with GABA_A receptor mediated synaptic input [63], while neurogliaform (NGF) cells, which have dense axonal plexi, form numerous mixed GABA_A and GABA_B receptor mediated synapses [62]. These dendrite targeting inhibitory interneurons [62], [63], [67] have been shown to powerfully control dendritic gain [68] and neuronal firing during behaviour [48]. Moreover, distributed inhibition has been shown to be particularly effective in shunting excitation in branched structures such as the dendritic tuft [69].

To examine how synaptic integration in L5 pyramidal cells might operate under more physiological conditions, when network activity delivers balanced excitatory and inhibitory background synaptic input, we added 60 SOM-like GABA_A receptor mediated synapses and 13 neurogliaform (NGF) cell-like mixed GABA_A/GABA_B synapses onto the apical dendritc tree. These inputs were driven at firing rates reported for passive touch whisker experiments in awake animals and are likely to reflect the upper end of the rates observed during active touch, which fall to about half this level on average [48]. The GABA_A receptor component had a fast rise and a decay time of 10 ms [66], while the GABA_B receptor-mediated K⁺ conductance had a slow rise and decay time of 50 and 80 ms, respectively, with peak values of 0.5 nS and 0.06 nS and an activation delay of 10 ms [70] (Text S1 Supporting information). This produced somatic IPSPs with properties comparable to those measured experimentally [61], [62] (Fig. S2). Although the inhibitory synapses were fewer in number than the excitatory synapses, their higher firing rate and slower kinetics effectively counterbalanced the AMPAR component of the 1500 excitatory synapses, producing a time averaged GABAR/AMPAR conductance ratio of 1.5, comparable to that estimated from synaptic currents [16], [71]. We refer to this experimentally constrained configuration of asynchronous background excitation and inhibition as “balanced background synaptic input”.

When the balanced background synaptic input included SOM-type and NGF (SOM+NGF) mediated inhibition, the NMDAR spike threshold was reduced compared to quiescent conditions, but not as strongly as during background excitation alone (Fig. 5A). The fact that the NMDAR spike threshold was similar for SOM+NGF and for a SOM-only case where the number of SOM-inputs was doubled (2×SOM), to maintain the same total GABA_A receptor mediated conductance, suggests that the slow low-amplitude GABA_B component and the GABA_A component are equally effective at inhibiting NMDAR spikes (Fig. 5A). Near-synchronous activation of 6 branches triggered an AP burst (P(AP) = 0.5) during SOM+NGF mediated inhibition (Fig. 5B1). However, SOM+NGF mediated inhibition was more effective at counteracting the background excitation-mediated increase in AP probability in response to multiple asynchronously activated branches than the 2×SOM input alone (Fig. 5C1, C2).

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Figure 5. Background inhibition modulates spatio-temporal integration.

(A) Probability of an NMDAR spike (P(NMDAR spike)) occurring in a terminal branch versus number of nearly synchronous stimulus evoked synapses during control (black), in presence of background activity from 1500 excitatory synapses (blue) and during mixed excitatory and inhibitory background (gGABA/gAMPA ratio of 1.5) provided by 130 pure GABA_A receptor (SOM-like) synapses firing random trains at 3 Hz (2×SOM, red) or by a combination of 65 SOM-like synapses and 13 NGF-like synapses providing both GABA_A and GABA_B receptor-mediated inhibition, firing random trains at 14 Hz (SOM+NGF, green). (B1) Example of voltages in 6 apical branches (grey) and soma (green) during nearly synchronous stimulation of 6 branches in the presence of ballanced background input (SOM+NGF). (B2) Example of voltages in all apical branches (grey) and soma (green) during nearly synchronous stimulation of 100 spatially distributed stimulus evokes synaptic inputs in the presence of balanced background input (SOM+NGF). (C1) Probability of triggering APs (P(AP)) versus number of nearly synchronously stimulated branches (5 ms, filled circles, solid lines) or asynchronously stimulated branches during a 100 ms window (open circles, dashed lines) for conditions in (A). (C2) Increase, from the nearly synchronous condition, in the number of stimulated branches required to trigger an AP (P(AP) = 0.5) with different levels of temporal dispersion (input) for conditions in (A) excluding control. (D1) P(AP) versus number of stimulated synapses for spatially distributed input over the apical tuft for conditions in (A). Synapses activated with random trains at 200 Hz for 5 ms (solid markers) or 10 Hz for 100 ms (empty markers). (D2) Number of spatially distributed synapses required to trigger an AP (P(AP) = 0.5) for different levels of temporal dispersion (input window), for conditions in (A).

https://doi.org/10.1371/journal.pcbi.1003590.g005

For spatially distributed synaptic input, the number of nearly synchronous inputs required to trigger APs was only marginally higher for 2×SOM-only balanced background inhibition than background excitation alone (Fig. 5D1, D2). However, for combined SOM+NGF inhibition, the number of synapses required to trigger an AP burst increased, indicating that the presence of the slow inhibitory component was effective at lowering dendritic gain for spatially distributed synaptic inputs (Fig. 5D1, D2). The fact that NMDAR spikes were evident in the voltage recordings from the terminal branches indicates that nonlinear dendritic integration underlies the AP output during distributed input in the presence of SOM+NGF inhibition (Fig. 5B2). Balanced background inhibition mediated by SOM+NGF was also effective at counteracting asynchronous inputs, increasing the number of spatially distributed inputs requited to trigger an AP burst with P(AP) = 0.5, from 100 with background excitation alone to 170 for a temporal window of 100 ms (Fig. 5D1,D2). Nevertheless, this was still far lower than under quiescent condition when 273 synapses were required. These simulations show that GABA_A and GABA_B receptor-mediated inhibitory conductances can counteract the effects of background excitatory input on synaptic integration in the dendritic tuft by raising NMDAR spike threshold and lowering the dendritic gain. Our simulations predict that during periods of lighter dendritic inhibition, for example when both sensory and motor systems are engaged [10], [48], the presence of balanced background synaptic input increases dendritic gain and extends the spatio-temporal integration properties of L5 pyramidal cells.

In the subsequent sections we investigate the mechanisms underlying network activity-dependent spatio-temporal integration in L5 pyramidal cells.

Mechanisms underlying the changes in spatio-temporal integration during background excitatory input

Several dendritic mechanisms could be involved in the network activity-dependent changes in synaptic integration we observe, but two are likely to be particularly important. Background excitatory inputs activate glutamatergic synapses that: 1) depolarize the dendrite (Fig. 1F) and 2) generate glutamate-bound but mostly silent NMDARs over the dendritic tree. Depolarization [33], glutamate spillover [72] and glutamate-bound NMDARs from paired-pulse synaptic activation [73], [74] have been shown to facilitate the generation of local NMDAR spikes in basal dendrites. In contrast to these relatively local effects, background network activity confers a spatially distributed NMDAR conductance that could provide an additional nonlinear resource for regenerative activity across the dendritic tree.

Since the threshold of NMDAR spikes is voltage-dependent [33], we first examined the effect of the depolarization induced by excitatory background activity on NMDAR spike properties. To do this we applied spatially distributed current injections (Fig. S3A1) that depolarized the apical tree to a level comparable to that during background excitatory synaptic input (Fig. S3A3, inset). Current injections accounted for a substantial part of the change in the single-branch I-O relationship observed with background input (Fig. S3A2). However, when spatially distributed AMPAR-only synapses were used to depolarize the apical tree to a comparable level of depolarization obtained with 1500 AMPAR/NMDAR synapses, they were less effective at altering the branch I-O relationship, due to the shunt introduced by these conductances (Fig. S3A2). Similarly, AMPAR-only background activity that generated comparable dendritic depolarization only accounted for a fraction of the changes observed in the neuronal I-O relationship in response to clustered (Fig. S3A3) and spatially distributed inputs (Fig. 4C). Moreover, current injections that produced depolarisations comparable to that obtained during background synaptic input at the Ca²⁺-spike initiation zone and at the soma recovered only a small part of the leftward shift in the neuronal I-O relationship (Fig. S3B, C). These results show that, while depolarization is important in determining NMDAR spike threshold [33], when depolarization is mediated by AMPAR-only background synaptic conductance it does not fully account for the lowering of NMDAR spike threshold or the increased efficacy of apical input in triggering APs during background input. This suggests that the spatially distributed NMDAR conductance arising from background network activity is also playing a key role in synaptic integration.

NMDAR component of background synaptic input extends the duration of NMDAR spikes

Comparison of the properties of NMDAR spikes in apical branches during different levels of background excitatory synaptic input revealed that the duration of NMDAR spikes systematically increased with the amount of background activity (Fig. 6A1–4). The mean duration increased from 45 ms in the absence of background input to 79 ms in the presence of 1500 background synapses (Fig. 6B1–2). Examination of the distribution of spike duration revealed an increased dispersion and some very long events (>150 ms; Fig. 6C1–2). Depolarization with current injections into the apical tuft accounted for part, but not all, of the increase in NMDAR spike duration and the contribution of depolarization became smaller at higher background rates (Fig. 6D). Replacing the current injections with AMPAR synaptic conductances accounted for less than half of the shift in the decay time distribution with 1500 background excitatory synapses (Fig. 6D–E). These results are consistent with the idea that both depolarization and the presence of glutamate-bound NMDARs over the dendritic tree contribute to the lengthening of NMDAR spikes during background input.

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Figure 6. Background excitatory input extends the duration of NMDAR spikes.

(A1–4) Dendritic NMDAR spikes triggered in different terminal branches (30 synapses; 100 trials) in the absence (control) and presence of 900, 1200 and 1500 background excitatory synapses. Single trials (grey) and average (solid colour). (B1) Average NMDAR spikes in (A) overlaid. (B2) Average decay time (37% of peak) of NMDAR spikes in (A). (C1) Cumulative distributions of decay times for control and different levels of background input. (C2) NMDAR spike decay time distribution in the absence (black) and presence (blue) of 1500 background synapses. (D) Fractional increase in average NMDAR spike decay time for excitatory background synapses containing AMPAR/NMDARs (blue) and equivalent dendritic depolarization obtained with background AMPAR-only synapses (green) or current injection (red). (E) Cumulative distributions of NMDAR spike decay times during depolarization mediated by AMPAR-only (green) and mixed AMPAR/NMDAR (blue) background synaptic input.

https://doi.org/10.1371/journal.pcbi.1003590.g006

Propagation and multiplication of dendritic regenerative events during background excitatory synaptic input

The finding that spatially distributed NMDAR conductance arising from background excitatory synaptic input contributes to the lengthening of dendritic spikes, suggests that glutamate-bound NMDARs in the vicinity of stimulated branches are recruited by depolarization. To investigate the extent of this effect, we monitored the voltage in each terminal branch, while triggering an NMDAR spike in only one branch at a time, with and without the background input (Fig. 7A; Movie S1). In the absence of background input, triggering an NMDAR spike in branch 11 with 30 nearly synchronous synaptic inputs produced a large voltage depolarization (Fig. 7B, lower red trace) that spread into the neighbouring branch (10) producing a similar voltage profile (Fig. 7B, black trace). However, the voltage in more distant branches (12 and 13) was markedly attenuated. When the same branch was triggered in the presence of background excitatory synaptic input (upper red trace) both the duration of the NMDAR spikes in branches 10 and 11 were extended and an additional regenerative event was observed in branch 12 and branch 13 (blue traces), suggesting that glutamate-bound NMDARs had become unblocked in neighbouring branches. Since it is difficult to distinguish between regenerative events and passive voltage propagation in the presence of background input (Fig. 7C), we used a more stringent voltage threshold criterion to identify regenerative events in neighbouring branches (peak_background−peak_control≥13 mV). The average number of additional activated branches ranged from 0.46 with 900 background excitatory inputs, to 2.19 with 1500 background excitatory inputs (Fig. 7D). These results show that additional regenerative events can be generated in neighbouring branches during background synaptic activity (Movie S1). This nonlinear dendritic amplification potentially explains why fewer stimulated branches are required to trigger APs in the presence of background excitatory synaptic input (Fig. 4A).

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Figure 7. Stimulation of a dendritic branch triggers regenerative potentials in neighbouring branches during background synaptic input.

(A) Apical tuft with inset showing branch 11 (red) stimulated with 30 synaptic inputs (5 ms window) and branch 12 (blue) that receives no stimulus evoked input. Inset: enlarged tuft region with overlapping branches removed for clarity. (B) Membrane voltage of all 28 terminal apical branches during activation of branch 11 (red trace) in the presence (upper trace) and absence (lower trace) of distributed background synaptic activity from 900 excitatory inputs. Asterisks denote additional regenerative events triggered in branches 12 and 13 in the presence of background activity. (C) Voltage in branch 11 (red) and branch 12 (blue) with (solid lines) and without (dashed lines) background activity. (D) Average number of additional regenerative events triggered in neighbouring branches (identified using a 13 mV increase above level observed in the absence of background input) during different levels of background excitatory input (10 trials per branch, per condition).

https://doi.org/10.1371/journal.pcbi.1003590.g007

Lowered threshold, extended duration and spread of regenerative NMDAR events during background excitatory synaptic input in a passive L5 pyramidal cell model

L5 pyramidal cells contain several nonlinear voltage-dependent dendritic conductances that could interact with synaptic input in complex ways. We therefore examined whether any of the basic changes in synaptic integration we observed in the presence of background excitatory input occurred in a passive L5 model, where all the non-synaptic nonlinear conductances were replaced by a uniform passive leak conductance. Although there were some differences between the resting potential of this passive model compared to the active model (the passive model is more hyperpolarized) asynchronous background synaptic input lowered NMDAR spike threshold and elongated the duration of NMDAR spikes (Fig. 8A, B). These results confirm that synaptic AMPAR and NMDAR conductances are sufficient to change NMDAR spike threshold and duration during asynchronous excitatory background synaptic input.

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Figure 8. Background excitatory input lowers threshold, extends duration and regeneration of NMDAR spikes in a passive model.

(A) Probability of triggering an NMDAR spike (P(NMDAR spike)) in terminal branches versus number of nearly synchronous stimulated synapses in a passive model (solid lines), where all non-synaptic active dendritic conductances have been removed, in the absence (black) and presence of 1500 background excitatory synapses (blue). Original model shown for comparison (dashed lines). (B) NMDAR spike decay time distribution (N = 100, triggered with 30 synapses each) lines as for (A). Excitatory decay times distribution during control and during background activity. Inset, mean voltage (±SD) in randomly selected terminal branches (N = 100) with and without background activity in passive (filled symbols) and original model (open symbols). (C) Membrane voltage traces from all 28 terminal apical branches (as in Fig. 7) in the passive model for a single trial, during nearly synchoronous activation of branch 11 (red traces) in the presence (upper trace; blue) and absence (lower trace; black) of background activity from 1500 excitatory inputs. Linear sum of the depolarization during control and during background input only, shown in green. (D) Peak depolarization induced by an NMDAR spike in branch 11 versus terminal branch number in absence (black) and presence (blue) of background activity. Peak depolarization during background activity alone shown in grey and peak value of linear sum of control during stimulation of branch 11 and background-only shown in green. (E) Voltage traces from branch 11 (red) and branch 7 (blue) in the absence (dashed lines) and presence of 1500 background excitatory inputs (solid lines). Grey and green trace as in (D). (F) Number of additional branches activated during background input when a single branch is stimulated (empty bar). This was estimated by stimulating the terminal branches in turn for 10 trials each and applying the 13 mV criterion, which identified regenerative peaks, although the number of branches exhibiting regenerative events is probably overestimated due to passive spread of voltage. Solid bar shows original model for comparison.

https://doi.org/10.1371/journal.pcbi.1003590.g008

We also examined whether propagation of regenerative events into neighbouring dendritic branches that we observed during background excitatory input could be generated by synaptic AMPARs and NMDARs in the passive model. Fig. 8C shows an example where we stimulated a branch and examined the response in the other distal branches of the tuft. Regenerative events in neighbouring branches were less evident, nevertheless the voltage depolarization in neighbouring dendritic branches during background input was larger than the linear sum of the depolarizations observed with the stimulus alone and the background input alone (Fig. 8C, D). This suggests that, in the presence of background excitatory input, the spread of voltage depolarization from the stimulated branch has a nonlinear regenerative component. Indeed, the duration of the voltage depolarisations in neighbouring branches also increased and some continued to build up well after the peak depolarization in the stimulated branch (Fig. 8E). As for the full active model, regenerative events were observed in neighbouring branches when a single branch was stimulated during background input (Fig. 8F).

These results suggest that synaptic AMPARs and NMDARs activated by background excitatory input can mediate active propagation into neighbouring branches, under certain conditions. However, the presence of dendritic Na⁺ and Ca²⁺ conductances in real cells [75] and in the original model are likely to facilitate this process. The spread of NMDAR-mediated regenerative events across branches and the increased decay time are likely to be interlinked: spread of voltage into neighbouring branches recruits NMDAR conductance activated by background input, potentially triggering a regenerative event. Recruitment of NMDAR conductances will help sustain the depolarization, lengthening the NMDAR spike duration, which in turn propagates more effectively through the dendritic tree. Simulations show that during background activity, a few synchronous distributed inputs can trigger an ‘avalanche’ of activity in multiple branches that feeds and sustains the depolarization for more than one hundred milliseconds. These events, through the recruitment of Ca²⁺ currents in the trunk, can then initiate somatic APs (See Movie S2). Propagation of voltage and activation of spatially distributed NMDAR conductance in neighbouring branches can therefore account for the increased efficacy of NMDAR spikes and the extended spatio-temporal integration we observe in the L5 pyramidal cell model, during background excitatory synaptic input.

Effect of inhibition on NMDAR spike initiation, duration and regeneration in the tuft

Having established that depolarization and spatially distributed NMDAR conductances are principally responsible for extending spatio-temporal integration during background excitatory synaptic input, we next examined how inhibition counteracts these effects. Comparison of individual NMDAR spikes during background excitatory synaptic input and during balanced background synaptic input, showed that the presence of SOM+NGF inhibition had little effect on their amplitude but did shorten their duration [44] (Fig. 9A). Inspection of the NMDAR spike decay time distribution showed that inhibition truncated the longer NMDAR spikes, making the distribution less skewed (Fig. 9B). Moreover, SOM+NGF inhibition was more effective than 2×SOM at truncating NMDAR spikes, explaining why GABA_B receptor inhibition was effective in modulating temporal integration (Fig. 5). However, NMDAR spike duration was still substantially longer in the presence of SOM+NGF based balanced background synaptic input than under quiescent conditions (Kolmogorov–Smirnov test, P<0.05; Fig. 9B).

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Figure 9. Background inhibition modulates NMDAR spike duration and regeneration during background excitatory input.

(A) Single trial of voltage response of apical branch triggered by nearly synchronous activation of 30 synapses (5 ms) during control (black), in presence of background activity from 1500 excitatory synapses (blue) and during mixed excitatory and inhibitory background (gGABA/gAMPA ratio of 1.5) provided by 130 pure GABA_A receptor mediated synapses (2×SOM) firing at 3 Hz (red) or by a mixture of 65 SOM-like synapses and 13 NGF-like synapses providing both GABA_A and GABA_B receptor mediated inhibition firing at 14 Hz (SOM+NGF, green). (B) Cumulative distributions of NMDAR spike decay times for conditions in (A). (C) Additional branches activated during stimulation of a single branch for conditions in (A) (10 trials per branch, per condition).

https://doi.org/10.1371/journal.pcbi.1003590.g009

We next examined the effect of background excitation and inhibition on the spread of NMDAR spikes into neighbouring branches. The average fractional increase in the number of branches exhibiting regenerative potentials was reduced from 2.19 to 1.2 when inhibition was added to background excitation (Fig. 9C). Interestingly, SOM+NGF and 2×SOM inhibition were equally effective, consistent with their comparable effects on NMDAR spike initiation (Fig. 5A). These simulations show that dendritic inhibition counteracts the effects of background excitatory input by increasing the threshold for NMDAR spikes and reducing spike duration and spread in neighbouring branches (see Movie S1 and S2).

Mechanisms underlying dendritic integration in active networks

Our simulations show that spatially distributed synaptic NMDAR conductances arising from ongoing network activity extend spatial and temporal integration in the dendritic tuft of L5 pyramidal cells. This network activity-dependent nonlinear dendritic mechanism is distinct from the well-documented effects of noise and linear synaptic conductances on neuronal I-O relationships [20], [23]–[25], [27]. While AMPAR-mediated conductances do contribute to integration, they summate poorly during asynchronous input due to their rapid decays (τ = 2 ms), producing a mild dendritic depolarization, voltage noise and reduced membrane resistance [19]. By contrast, the slow kinetics of NMDARs (τ = 70 ms) [54], [76] summate effectively, allowing background network activity to set the density of NMDAR conductance on the dendritic tree. The dynamic nature of the spatially distributed NMDAR conductance contrasts with the properties of other spatially distributed voltage-dependent dendritic conductances (e.g. Na⁺, Ca²⁺ and K⁺ channels), which are typically modulated slowly, through plasticity [77]–[79] and homeostatic mechanisms, although the effective number of channels available could change rapidly with voltage due to inactivation.

Several properties of the GluN2A/B containing NMDARs present in L5 pyramidal cells are important for dendritic integration in the presence of background synaptic activity. Their strong block by Mg²⁺ below −60 mV [80], [81] prevents their involvement in signalling except when the dendritic branch on which they are located is depolarized. This, together with their high affinity for glutamate, means that GluN2A/B-containing NMDARs can remain glutamate bound and ‘silent’, yet primed and ready to contribute current if the voltage of the dendritic branch depolarizes [73], [74]. The steep voltage dependence of the NMDAR conductance introduces a highly nonlinear threshold, enabling regenerative depolarization, while the slow kinetics of the conductance extends the time course of local EPSPs beyond the local effective membrane time constant of the fine dendritic branches [31], [56]. Our simulations show that these properties of NMDARs enable them to extend spatial and temporal integration in L5 pyramidal cells during in vivo-like network activity. Moreover, our control simulations indicate this effect is robust across a range of NMDAR kinetics: the slower the decay of the synaptic NMDAR conductance the longer the absolute window for temporal integration in the absence of other counteracting conductances (Fig. S4). However, faster NMDAR kinetics would reduce the impact of background synaptic input unless there was a proportionally higher background input rate or a larger NMDAR/AMPAR peak amplitude ratio to maintain the same time-averaged NMDAR conductance. Another key property of NMDARs that sets them apart from nonlinear dendritic Na⁺ conductances [82] is the fact that they remain active during bursts of backpropagating APs and the sustained depolarisations during up-states [83].

By increasing the gain of dendritic integration, background excitation could trigger an unconstrained positive feedback loop, integrating over longer timescales and resulting in prolonged global activation of the dendritic tuft. In practice, cortical microcircuits operate in a balanced configuration with inhibition closely tracking excitation [16], [18], [48], [71], [84], [85]. Indeed, strong inhibition targeted to the tuft [43], [48] may be required to dampen excessive NMDAR excitation during highly active network states and prevent dendritic ‘chain reactions’ from reaching critical levels. Consistent with this, recent experimental evidence shows that apical inhibition has a stronger effect on the I-O relationship of CA1 pyramidal neurons than somatic inhibition [68]. The apical dendrites of pyramidal cells receive inhibition from SOM expressing interneurons, which include Martinotti cells [48]. SOM-expressing interneurons respond to synaptic inputs from neighbouring pyramidal cells and deliver fast GABA_A receptor mediated inhibition to the tuft region [63]. Interestingly, in vivo recordings show that SOM cell firing is suppressed for about 1 s during active whisker touch behaviour, suggesting that the tuft region is disinhibited during this behaviour [48]. Consistent with this, the G_exc/G_inh ratio is higher during in vivo up-states [13], and engaging the motor and sensory circuits together during whisker touch behaviour results in global tuft activation [10]. Our simulations, which reproduced the excitation/inhibition balance and the firing rates of SOM and NGF cells measured in awake animals [48], show that inhibition reduces NMDAR spike threshold. However, our simulations of asynchronous background inhibition cannot rule out the possibility that fast feed-forward inhibition could veto NMDAR spike generation more effectively if it is present in L1 and L2 and targeted to the same branches as the excitation. Our simulations suggest that slow GABA_B-mediated inhibition arising from NGF cells [61], [62], [65] is particularly effective at truncating NMDAR spikes and reducing dendritic gain. This, together with the fact that the firing rate of NGF cells is elevated during active touch [48], suggests that NGF mediated inhibition may be particularly well placed to control spatio-temporal integration in the tuft (Fig. 5). Such co-variation of background excitatory and inhibitory conductances [85] in recurrent cortical networks [43] is likely to extend the dynamic range over which pyramidal cells can operate by adjusting their integrative properties to match the excitatory drive. These findings extend previous work showing that inhibitory conductances can terminate NMDAR spikes [44] by showing how background inhibition arising from specific interneuron types could counteract the effects of background excitation and shape spatio-temporal integration in L5 pyramidal cells. Our results suggest that during active whisking the lowered level of inhibition [48], could promote NMDAR-mediated distributed signalling, while strong dendritic inhibition could keep it in check at other times, thereby preventing runaway excitation.

Potential interactions between background synaptic input, NMDAR spikes and voltage-gated dendritic conductances

The dendrites of pyramidal cells contain many different voltage-gated conductances [86]. While it was not possible to explore all possible interactions between background input and the 9 conductances in our model, the simulations we carried out do point to some general principles of interaction and highlight specific conductances that could play a key role in shaping the properties of spatiotemporal integration. At the most basic level, some excitatory dendritic conductances in the model such as the Na⁺ and Ca²⁺ conductances clearly aid background induced regenerative activity (Fig. S5). Indeed, we found that Ca²⁺ spikes occur in the tuft during strong tuft stimuli, and help to electrically couple the tuft to the soma, as found experimentally [37]. By contrast, increasing I_h reduced the NMDAR spike threshold in the presence of background input, but had negligible effect on NMDAR spike duration (Fig. S5). Indeed, more complex effects are seen when conductances have activation and/or inactivation properties with significant voltage dependences around −55 mV, because background synaptic input depolarizes the dendrite by several millivolts compared to the quiescent condition. For example, transient A-type K⁺ conductances inactivate strongly over this voltage range making their impact weaker than expected (Fig. S6). By contrast, the sustained slow K⁺ conductance (delayed rectifier) strongly activates during NMDAR spikes and truncates their duration (Fig. S6). Increasing the K⁺ conductances to match levels recently reported experimentally [36], increased the threshold of NMDAR spikes, limited their spread to neighbouring branches and was particularly effective at reducing their decay time (Fig. S7). However, the time course of the resulting dendritic spikes generated under these conditions was markedly different from NMDAR spikes recorded experimentally in L5 pyramidal cells [29], [36] and another recent study estimated much lower densities of K⁺ conductances in the tuft region [87] comparable to that in the original L5 model [29]. Nevertheless, slow K⁺ conductances appear well placed to modulate spatio-temporal integration in L5 pyramidal cells. SK type channels in spines, which are activated by local Ca²⁺ influx though NMDARs [88], may also truncate NMDAR spikes (Fig. S8). Thus the level of expression of the various dendritic conductances and their precise voltage and Ca²⁺ dependencies are likely to tune the spatio-temporal integration. Irrespective of the natural configuration of the dendritic conductances present in vivo, our results using a passive L5 model, show that the spatially distributed NMDAR conductance arising from ongoing network activity will extend spatio-temporal integration unless other dendritic conductances are specifically configured to counteract this basic property. Indeed, spatially distributed glutamate-bound NMDARs arising from ongoing network activity extend the toolbox of dendritic conductances available to pyramidal cells, potentially enabling them to perform a wider range of behaviours.

Enhanced spatial and temporal integration during background network activity

The fine dendrites of pyramidal cells have traditionally been thought to act as coincidence detectors on fast time scales [89], due to their fast local membrane time constant [31], [90]. However, when depolarized, slowly decaying synaptic NMDAR conductances set the dendritic EPSP waveform, extending the integration window to tens of milliseconds [35]. The voltage dependent properties of NMDARs have also been shown to extend the time window for integration, underlying the supra-linear response of the second of a pair of synaptic inputs onto the basal dendrites of L5 pyramidal cells [32]. Moreover, temporal integration can be extended further when pairs of bursts are used [73], [74]. Our simulations extend these findings by showing that a spatially distributed NMDAR conductance arising from in vivo-like background synaptic input can extend both the spatial and temporal integration in L5 pyramidal cells by providing both depolarization [27] and a spatially distributed NMDAR conductance that sustains the NMDAR spike plateau depolarization. Interestingly, these mechanisms act in the opposite direction to passive membrane properties, which reduce the temporal integration window during elevated synaptic input, due to increased membrane shunting by synaptic conductances.

Our simulations show that background synaptic input can lower the number of coincident synaptic inputs required to trigger an NMDAR spike in a distal branch. This suggests that NMDAR spikes are much more likely to occur in vivo during network activity than in vitro when the network is largely quiescent. Another important consequence of spatially distributed NMDAR conductance arising from background network activity is that it enables active propagation of NMDAR-mediated regenerative events into neighbouring branches, resulting in a multi-branch regenerative plateau potential. This prediction is consistent with recent in vivo recordings of NMDAR spikes in L2/3 pyramidal cells, which reported that 83% of events occurred in multiple branches [46]. Spatially distributed NMDAR conductance is therefore well suited for integrating excitatory input that is distributed across the branches of the dendritic tree [39], [40]. Our simulations show that under in vivo-like levels of background input our L5 model could integrate spatially and temporally dispersed synaptic input. This suggests that spatial clustering of synaptic inputs required for triggering NMDAR spikes under quiescent in vitro conditions [35], [36], [64] may not be necessary in vivo. Hence, while spatial clustering of synaptic input onto the dendrites of pyramidal cells has been found [42], [91], our results suggest it is not a prerequisite for activating the tuft of L5 pyramidal cells. Indeed, our finding that network activity extends spatio-temporal integration may explain how NMDAR spikes contribute to orientation selective tuning in L2/3 pyramidal cells in mouse V1 [47], when the synaptic inputs onto these cells are spatially distributed [39].

Widespread NMDAR spikes have been reported in the dendrites of L4 spiny stellate neurons of rat barrel cortex, during whisker deflection [45]. The multiple hot spots of Ca²⁺ influx observed are consistent with a small number of distributed thalamo-cortical inputs being amplified by the dendritic NMDAR conductance arising from local network activity, but alternative scenarios are also possible. In L2/3 pyramidal cells in somatosensory cortex, both spontaneous and hind paw-evoked NMDAR spikes are localized to ∼30 µm regions of the dendritic tree and usually encompass multiple branches. These events are effective at triggering APs but do not appear to involve a dendritic Ca²⁺ spike [46]. More widespread [Ca²⁺] signals have been observed in the tuft of L5 pyramidal cells in awake animals performing active whisker touch, when vibrissal sensory input is combined with L1 input arising from the primary motor cortex [10]. Similar tuft responses could be reproduced in vitro when apical trunk Ca²⁺ spikes were paired with dendritic tuft depolarization or synaptic input. This suggests that in vivo tuft plateau potentials and Ca²⁺ influx are caused by apical dendritic Ca²⁺ spikes spreading into the tuft when it is depolarized by L1 input. Although, Ca²⁺ conductances are clearly important [10], network activity-dependent distributed NMDAR conductance could also contribute to this widespread Ca²⁺ influx. Indeed, recent in vivo recordings from CA1 pyramidal cells show that both NMDARs and Ca²⁺ channels are required to generate the slow, widespread regenerative dendritic events that underlie complex spike bursts [92]. These results suggest that the densities of both glutamate-bound NMDARs and Ca²⁺ channels can set the spatial spread of regenerative dendritic events in pyramidal cells. Our results highlight the role of spatially distributed NMDAR conductances in pyramidal cells under in vivo conditions, by showing how they could enable these cells to integrate both spatially clustered and spatially distributed inputs over longer timescales than previously thought.

Implications for cortical processing

The finding that synaptic input and nonlinear membrane conductances can produce local regenerative dendritic events has lead to the idea that individual dendritic branches can operate as independent thresholding units [33], [73], [93] enabling a pyramidal cell to act as a powerful pattern separator [34]. In this scenario, individual branches could act as feature detectors [77] and neuronal firing could report the coincident occurrence of these features. However, our results showing that NMDAR spikes can spread into neighbouring dendritic branches suggest that they may not operate independently and are more effective at triggering APs than anticipated, consistent with recent experimental findings [46]. This enhancement in spatio-temporal integration during network activity is likely to come at the cost of loss of power to discriminate different spatial patterns of synaptic input. That said, our results do not exclude the possibility that L5 pyramidal cells can still act as coincidence detectors and perform pattern discrimination, since our simulations show that altering the level of dendritic inhibition or the density of voltage-gated channels on the dendrite can tune the spatio-temporal properties of the tuft. Coincidence detection and pattern discrimination could be enhanced by raising the threshold for dendritic spikes, truncating their duration and decoupling dendritic branches. This could be achieved by expressing high levels of K⁺ channels [36], [94], which have been shown to compartmentalize dendritic responses in CA1 pyramidal cells and through experience-dependent adjustment of synaptic weights onto specific branches through NMDAR-dependent Ca²⁺ influx [95]. Moreover, recent work suggests that GIRK channels can also modulate NMDAR spike duration in a branch specific manner [78]. Our results suggest that these slowly modulated dendritic mechanisms are complemented, in vivo, by faster dynamic modulation conferred by the background synaptic excitation and inhibition arising from ongoing network activity. Thus, both the past and the present activity state of the network determines which spatio-temporal patterns of synaptic input a L5 pyramidal cell can respond to.

Network activity-dependent spatio-temporal integration could have important implications for cortico-cortical and thalamo-cortical signalling. While global network activity is maintained at a relatively constant level due to tight inhibitory control [16], [18], [71], [85], the activity of sub-networks of excitatory neurons is likely to vary widely, due to common receptive field properties and preferential local synaptic connectivity [96]. If the elevated activity in L2/3 pyramidal cell ensembles observed during sensory input [96] increases synaptic input onto the apical dendrites of subsets of L5 pyramidal cells [4], [97], the resulting increase in NMDAR conductance will enable those L5 pyramidal cells to integrate both spatio-temporally clustered and dispersed signals. This property suggests that the tuft could combine different modalities, such as motor, sensory and emotional state, which span a wide range of timescales. This effect is predicted to be particularly strong when the apical tuft is disinhibited, as has been observed when motor and sensory systems are engaged together [48]. Although speculative, network-dependent enhancement of spatio-temporal integration could also enable L5 pyramidal cells to communicate selectively across sensory and motor cortical areas, in a context dependent manner. Such reconfigurable functional network connectivity has been reviewed in the context of neuronal gain modulation [98] and could be involved in high-level cortical function including attention [76], [99], [100] and learning [9]. Our results strengthen these concepts by providing a new mechanism that links network activity to the spatio-temporal properties of dendritic integration.

The L5 pyramidal cell model

The version of the L5 pyramidal cell model [29] available from ModelDB (Accession number: 124043) was used as a starting point for our simulations. Synaptic input was added and all simulations were generated in neuroConstruct (neuroConstruct.org) [51] and run in the NEURON 6.2 simulation environment [101]. The default parameters of the model are listed in Text S1 and any changes are noted in the figure legends. Simulations were carried out with a fixed integration time step of 0.025 ms and voltages were recorded from the soma and the proximal segment of terminal dendritic branches.

Synaptic input

The AMPAR conductance (gAMPA) component of each quantal glutamatergic synapse was modelled with an instantaneous rise and a single exponential decay:Where the maximal peak conductance (g_max AMPA) was 0.5 nS and the decay time constant (τ_decay) was 2 ms [56].

The NMDAR conductance (gNMDA) component of each quantal glutamatergic synapse was modelled with an exponential rise and decay, and a voltage-dependence function to mimic Mg²⁺ block:Where the normalization factor A is:g_max NMDA is the maximal peak unblocked conductance (1 nS). Time constants for the rise (τ_rise) and decay (τ_decay) were 3 ms and 70 ms, respectively [76], [97]. The NMDAR model also included Ca²⁺-dependent inactivation (CDI) of the NMDAR current. This was implemented by multiplying the gNMDA by e^−h, where the change in h at each time step (h′) is as follows:and  = 1000 ms.

Quantal GABAergic synapses were modelled with an exponential rise and decay:The fast GABA_A receptor component had a maximal peak conductance (g_maxGABA) of 0.5 nS, a τ_rise = 0.3 ms and a τ_decay = 10 ms [66]. A is the peak normalization factor as defined above and the Cl⁻ reverse potential was −75 mV. Synapse arising from neurogliaform (NGF) cells, also had a slow GABA_B receptor component with a τ_rise = 50 ms and a τ_decay = 80 ms [70] and an onset delay of 10 ms to account for the activation of the K⁺ channels. The maximal peak conductance (g_maxGABA = 0.06 nS) was estimated from [61] who reported that the GABA_B component accounts for ∼1/5 of the peak IPSP for NGF cell to L5 pyramidal cell connections. The conductance was obtained by scaling 0.1 nS by the ratio between the driving forces of the GABA_A and GABA_B components using a K⁺ reverse potential of −87 mV.

Two groups of synaptic input were defined. Those that mimicked a stimulus-evoked event and those that mimicked asynchronous background network activity. For simulations where specific dendritic branches were stimulated, one of the 28 terminal branches was randomly selected and the stimulus evoked quantal synaptic inputs randomly placed along it. For spatially distributed stimulus-evoked inputs, quantal synaptic inputs were randomly distributed over the entire apical tuft (Blue region, Fig. 1A) Each stimulus-evoked input was driven by an independent Poisson train with a frequency that was inversely related to duration of the stimulus, thereby maintaining an average of a single quantal conductance per synapse, independent of stimulus duration. For background excitatory synaptic input, 900–1500 glutamatergic synapses were placed randomly over the apical dendritic region (starting from ∼500 µm from soma). Each of these synapses was driven independently with a Poisson train with a mean rate of 0.85 Hz. Background inhibitory synaptic input was implemented either by 130 GABA_A receptor mediated synapses, each driven with a 3 Hz random train (2×SOM configuration) or by a mix of 65 pure GABA_A receptor-mediated synapses firing random trains at 3 Hz and 13 GABA_A/GABA_B receptor-mediated connections (10 synapses each) firing random trains at 14 Hz (SOM+NGF configuration). In both cases this gave a time averaged GABA_A receptor conductance of 1.5 times the background AMPAR conductance, comparable to that measured experimentally [16], [71].

Analysis

Voltage traces from the simulations were analysed in IGOR Pro using NeuroMatic software (http://www.neuromatic.thinkrandom.com/). NMDAR spikes were identified with a threshold crossing criteria of −30 mV (Fig. S1; unless stated otherwise). Dendritic branch and neuronal I-O functions were fit using a sigmoid function with the form:where n is the rate, P₀ is the offset, P_max is the maximum probability of activation and x₅₀ is the number of synapses or stimulated branches at which P(x) reaches half maximum.

Errors are presented ± standard deviation. Distributions were compared with the Kolmogorov-Smirnov test and considered significant at the p<0.05 level.