Burst control: Synaptic conditions for burst generation in cortical layer 5 pyramidal neurons

The output of neocortical layer 5 pyramidal cells (L5PCs) is expressed by a train of single spikes with intermittent bursts of multiple spikes at high frequencies. The bursts are the result of nonlinear dendritic properties, including Na+, Ca2+, and NMDA spikes, that interact with the ~10,000 synapses impinging on the neuron’s dendrites. Output spike bursts are thought to implement key dendritic computations, such as coincidence detection of bottom-up inputs (arriving mostly at the basal tree) and top-down inputs (arriving mostly at the apical tree). In this study we used a detailed nonlinear model of L5PC receiving excitatory and inhibitory synaptic inputs to explore the conditions for generating bursts and for modulating their properties. We established the excitatory input conditions on the basal versus the apical tree that favor burst and show that there are two distinct types of bursts. Bursts consisting of 3 or more spikes firing at < 200 Hz, which are generated by stronger excitatory input to the basal versus the apical tree, and bursts of ~2-spikes at ~250 Hz, generated by prominent apical tuft excitation. Localized and well-timed dendritic inhibition on the apical tree differentially modulates Na+, Ca2+, and NMDA spikes and, consequently, finely controls the burst output. Finally, we explored the implications of different burst classes and respective dendritic inhibition for regulating synaptic plasticity.

Introduction threshold for a Ca 2+ spike in the apical tree and, consequently, may lead to the generation of a somatic spike burst. They termed this phenomenon backpropagation activated Ca 2+ spike firing (BAC firing). Many replicated and expanded their findings, including in some elaborate neuron models [33,34]. Without a bAP, dendritic input required for Ca 2+ spike generation is much stronger and produces less somatic spikes [30]. The lowest threshold for burst generation was found when tuft activation followed the somatic action potential by~5 ms.
There are several outstanding questions regarding the conditions for the initiation of bursts and the criteria for manipulating their characteristics. Details about the varied effects of dendritic inhibition on bursts are not yet clear. The necessity of the bAP for promoting Ca 2+ spike firing and consequently bursting [30], is still questionable [35]. Furthermore, it is uncertain whether functional clusters of adjacent and temporally correlated inputs (whose existence was recently debated) [36,37] are sufficient for generating bursts?
The present work aims primarily to test various timing and location conditions of (basal and apical) excitatory synapses and of dendritic inhibition, for their effect on the initiation and control of somatic bursts. Towards this end, we employed the model built by Hay et al. [33], that utilized an automated feature-based parameter tuning to faithfully replicate both dendritic Ca 2+ and perisomatic Na + electrogenesis, and the interaction between these two spiking regions (i.e., BAC-firing). Shai et al. [38] employed this model to show how basal and tuft synapse numbers control high frequency bursting. This coincidence detection mechanism depends on voltage gated Ca 2+ channels (VGCC), is approximated by a composite sigmoidal function, and can create orientation tuning. Our experiments introduce additional critical parameters that control this bursting mechanism, including dendritic inhibition, and suggest the involvement of bursts in Ca 2+ -dependent long-term synaptic plasticity.

Temporal characteristics of excitatory dendritic inputs for burst generation
Spiking output of a cell depends critically on the number of activated synapses and their temporal correlation. To examine timing constraints on synaptic inputs for burst generation, we simulate a L5 PC model ( Fig 1A; [33] and see Methods), and look at the response as we vary activation times of a fixed suprathreshold number of synaptic activations ( Fig 1B). The synapses, which incorporated AMPA-and NMDA-dependent conductances (see Methods) were uniformly distributed on the entire basal dendrite (blue region in Fig 1A), and on 25% of the apical tuft (750 μm continuous stretch, red region in Fig 1A). Changing the percentage of the apical dendrite on which we distribute synapses promoted fewer bursts while the same manipulation in the basal tree did not alter it (S1 Fig, and Methods). We first examined the minimal number of excitatory synapses over these dendritic subtrees that, when activated synchronously, elicit a burst of somatic Na + spikes. We found a threshold for synchronous activation of all synapses with a peak AMPA and NMDA conductance of g max = 0.4 nS per synapse, at 50 ± 20 basal and 30 ± 10 apical tuft synapses.
Next, we explored the effect of the activation time of the synaptic input on burst generation. We randomly selected activation times for basal and apical synapses from two normal distributions of identical standard deviation σ (Fig 1B). Typical voltage traces for different σ's and activation of synapses at either or both apical and basal dendrites are shown in Fig 1C. Increasing σ values from 0 (instantaneous) to 3 ms did not change the output burst substantially (data supplied in repository). The somatic burst comprised two Na + spikes that were associated with a prominent Ca 2+ spike (blue and orange traces respectively, in Fig 1C top left).
The voltage measured in the distal tuft compartment (green in Fig 1C) follows closely the voltage measured in the nexus (orange traces) except when Ca 2+ spiking is partially abolished.
With apical activation only (second column from the left) a Ca 2+ spike is still generated, followed by a burst. With basal activation alone (third column) only one somatic spike is generated. With activation of both dendrites but without a Ca 2+ hotspot (rightmost column), again only one somatic spike is fired. When increasing σ further to 6 ms and 9 ms (middle and bottom left traces respectively, in Fig 1C), the burst first consisted of 3 spikes (due to decreased voltage saturation) and then only two, with the Ca 2+ spike essentially intact. With apical excitation only, higher σ resulted in Ca 2+ spikes with lower maximum voltage and no burst (center and bottom rows in Fig 1C). With basal only excitation or no VGCC it finally resulted with no somatic spike at all. We conclude that initiation of bursts occurs by either coincidental basal +tuft excitatory inputs, or by substantially strong excitatory input to the tuft alone.
In Fig 2 we increase the average delay Δt between the apical and basal synapses to 20 ms (right column in Fig 2C). With σ = 3 ms or 6 ms it resulted in a burst of only two late spikes, and the dendritic Ca 2+ spike remained intact. With further increase in σ to 9 ms, the Ca 2+ spike was abolished and thus the somatic firing also terminated (lower trace in right column, Fig 2C).
The summary graph (Fig 2D) of the mean number of spikes per burst for a range of Δt and σ values, shows that more spikes/burst are fired for small values of σ and Δt. The range of Δt with > 2 spikes per trial on average (above the dashed line in Fig 2D) defines the conditions for burst generation. Specifically, for σ = 3 ms (red line in Fig 2D), the range of Δt for burst generation spans 40 ms from Δt = -15 ms (basal synapses preceding tuft synapses) to Δt = +25 ms (tuft synapses then basal synapses). Interestingly, this window is slightly biased towards Δt > 0 (tuft preceding basal activation), as evident from the center value marked by a small upwards arrow in Fig 2D. Increasing σ decreases the number of spikes per burst and the range  of Δt giving rise to a burst (Fig 2C and 2D). For σ = 9 ms, the Δt window for burst generation narrows linearly to 13 ms (from Δt = -8 ms to +5 ms; Fig 2F). At this high σ, Δt values for burst generation are centered near -5 ms (basal input before tuft), which was found experimentally to have the minimum threshold of current injection for burst generation (Fig 2D and 2G as in [30]). At even higher σ = 12 ms, the mean number of spikes per burst dropped to 0.5 and bursting vanished (dark blue in Fig 2D). For control we ran this experiment again without voltage-gated Ca 2+ channels at the Ca 2+ hotspot. This condition results with 1-spike maximum and no bursting, as there is no Ca 2+ spike to boost the tuft input ( Fig 2E, compare to Fig 2D).
Stretching synaptic activation times using σ allows a proxy for testing lower input intensities and their requirements of tuft-to-basal delay for burst initiation. Increasing σ, monotonically decreased both the Δt range for burst generation and the Δt-window center (small upwards pointing arrows in Fig 2D). Increasing σ shifted the center from positive delays at low values (tuft input preceding the basal input) to small negative delays at larger values ( Fig 2G). We conclude that broadening the timing of synaptic activation narrows the tuft-to-basal delay window for burst generation (Fig 2D and 2F) and shifts the preferred order of activation from tuft-then-basal to basal-then-tuft (Fig 2D and 2G).
Ca 2+ spikes are considered to be all-or-none signals, yet their duration may be modified, for example by the excitatory synaptic input that can prolong them or by inhibitory synapses that might cut them short. However, our traces show no substantial difference in Ca 2+ spike duration for the activation of excitatory synapses at the range of σ = 3 and 9 ms (Figs 1B and 2B), nor for σ = 0 (data supplied in repository). A more systematic study is required to examine the correlation between the duration of the Ca 2+ spike and the number and frequency of somatic Na + spikes in a burst; this is beyond the scope of the present study.
To verify the mechanisms for burst generation, we compared our model with the established backpropagation-activated Ca 2+ spike firing paradigm (BAC) [30] in Fig 1. In this paradigm, a backpropagating somatic AP lowers the threshold for an apical Ca 2+ spike and a burst. A somatic depolarizing current injection results in a single somatic AP. This AP backpropagates up to the apical tuft, where, if it meets a subthreshold current injection, they may together generate a dendritic Ca 2+ spike and consequently a burst consisting of 3 APs at the soma. The somatic and dendritic ("nexus" main apical bifurcation) voltage traces in Fig 1C show responses to synaptic activations with variable σ at either or both basal and tuft dendritic trees. Our model reproduces BAC-firing using conductance-based synaptic activation alone on the dendritic trees (Fig 1C middle-row), instead of the original experimental and more artificial current injection (see [33]). For both the original clamp experiment and our simulation, stronger tuft input without perisomatic activation (achieved here by decreasing σ; Fig 1C bottom) generates a Ca 2+ spike (orange) and a burst of only two APs (blue; Fig 1C top-center).
To check for burst dependence on voltage-gated Ca 2+ channels (VGCC), we set their conductance to zero at the Ca 2+ hotspot and noticed that all spikes disappeared, except for a single Temporal conditions for generating somatic spike bursts. a. Morphology of the modelled L5PC. Fifty excitatory synapses were distributed on the basal tree (blue synapses) and thirty on a subtree of the apical tuft (red synapses). Dashed line encircles the Ca 2+ hotspot in the apical nexus. Blue and orange electrodes measure voltage traces shown in c. b. Two normal distributions of synaptic activation times for the basal and apical synapses are shown. The distributions have identical standard deviations, σ, and their means are shifted by Δt from each other. c. Voltage traces from the somatic and main apical bifurcation electrodes in a for a combination of σ and Δt values (Na + spikes in blue and Ca 2+ spike in orange). d. Mean number of somatic spikes per burst for a range of Δt values. Colored lines correspond to different σ values. The extent of Δt for generating a burst is reduced with increase in σ. Dashed line denotes curve width at 2 spikes per burst; arrows denote their centers. e. Same as d but without voltage-gated calcium channels, resulting in the absence of bursts. f, g. Width (f) and center (g) of the curves at two mean spikes per burst shown in d. Both width and mean reduce with increasing σ. Note in g, that the window's center is positive (basal input following apical input) for smaller σ, and negative for larger σ.
https://doi.org/10.1371/journal.pcbi.1009558.g002 somatic one (Figs 1C and 2E right). Thus, we confirmed that low σ tuft input, combined with a somatic AP, creates a burst only via VGCC. Preserving VGCC and increasing σ to 9 ms caused tuft synapses to initiate a subthreshold voltage plateau, and prevented perisomatic synapses from generating an AP. Coincidental activation of the synapses on both trees may still elicit a full-blown Ca 2+ spike and a somatic burst (Fig 1C bottom).

Spatial input conditions and temporal burst characteristics
We next explored the effects of the number of activated excitatory synapses that impinge on various parts of the dendritic tree, on burst generation (Fig 3). To do this we fixed the temporal parameters σ = 10 ms and Δt = 0, and separately manipulated the number of activated synapses on the basal tree and on a distal apical subtree (red subtree in Fig 1A). We used σ = 10 ms for a more realistic (dispersed) activation, and for better comparison with previous models that found some different results. Fig 3A depicts three example voltage traces. When 100 basal and 100 tuft excitatory synapses were activated, a dendritic Ca 2+ spike was initiated, accompanied by a burst of 2 somatic spikes (orange and blue respectively, Fig 3A, left). When the number of activated tuft synapses was reduced to 50, the burst disappeared and so did the dendritic Ca 2+ spike (Fig 3A, center). Keeping the number of activated tuft synapses at 50 but increasing the activated basal excitatory synapses to 150 resulted in a burst of 3 spikes (with lower FR, see below) accompanied by a dendritic Ca 2+ spike (Fig 3A, right).
Next, we measured the mean number of spikes per burst while independently varying the number of activated synapses impinging on both the basal dendritic tree and the apical tuft. Activating up to 50 apical and 140 basal synapses (with Δt = 0, σ = 10 ms) is mostly insufficient for generating a burst, and produces up to one spike (blue regions, Fig 3B; " � " indicates the example trace 3a center). 60 ± 10 apical synapses without basal synapses are sufficient for the generation of a burst of two spikes (top green region, Fig 3B left; "+" denotes the example Bursts with 3-4 spikes appear only for strong basal input (within the green dashed lines). Apical tuft input above threshold results in shorter bursts of 2 spikes and higher rates (within the purple dashed lines). c. Scatter plot for each combination of synapses (x-y coordinates) in b, sorted by the mean # of spikes per burst (left frame) and the intraburst firing rate (right frame). Each value in right-hand heat map is plotted with its corresponding value in the lefthand heat map. The color of each point in c represents the density of data points with the same values, equivalent to their probability. The purple and green histograms along the x & y-axes correspond to the data in the 2 clusters encircled in purple and green, accordingly. Temporal input parameters were fixed at Δt = 0, σ = 10 ms.
https://doi.org/10.1371/journal.pcbi.1009558.g003 trace, 3a left). A burst of two spikes is observed when 150-200 basal synapses (without apical synapses) are activated; increasing the number of basal synapses to 200-300 results in bursts of 3 or 4 spikes (orange and red regions, respectively, Fig 3B left; "#" denotes the example trace 3a, right). This finding is counterintuitive as tuft synapses are thought to be less potent in generating somatic spikes, due to the attenuation of their effect along the~1 mm distance of the tuft dendrites from the soma. However, tuft synapses compensate more for this attenuation by NMDA spike generation, allowing burst-firing by fewer synapses (see Discussion).
The need to activate more synapses for burst generation in Fig 3 compared to Fig 2 is caused by the higher σ used in Fig 3; synaptic activation dispersed in time keeps local voltage from building up (by decaying towards the resting membrane potential), and specifically from promoting supralinear input summation by voltage-dependent NMDA receptors and Ca 2+ channels (and thus their appropriate dendritic spikes too). What we found different than Shai et al. [38] was the addition of the basal-independent burst that we attribute to the smaller region we distributed synapses on the apical tuft, allowing stronger cooperation and dendritic spikes.
A supplementary analysis is shown in S2 Fig 3C). It is characterized by a high FR of~250 Hz and a mean of two spikes per burst. The second cluster appears as a rectangle to the right and is characterized by bursts of three or four spikes at a lower FR of~200 Hz (green dashed line, Fig 3C). It is associated with coincidence detection of input to the basal dendrites and to the apical tuft, resembling BAC-firing as suggested by Larkum et al. [30]. The third cluster is located near the origin and reflects the case where no spikes were initiated, corresponding to the subthreshold blue areas in the two frames of Fig 3B. A further quantification is attained by superimposed histograms of both burst classes ( Fig  3C): in purple the high FR and two spikes class, and in green the lower FR and 3-4 spikes class (see Methods). These reveal clear separation in spike number and partial discrimination in intra-burst FR (histograms on x and y axes, respectively, Fig 3C). The scatterplot transforms the input-aligned picture of the heatmaps into a map of output burst characteristics showing each mean spikes-per-burst value with its corresponding FRs. Accompanied by the associated histograms, the scatterplot further elucidates the clear separation between the two burst classes.

Editing bursts with inhibition
A local modulation of pyramidal cell activity is provided by a spectrum of inhibitory interneuron types that are distributed over its dendritic tree [1]. We tested the sensitivity of burst generation to different locations of inhibitory synapses along the apical tree (Fig 4). Fig 4 shows a burst of 3-spikes, and the various edits that targeted inhibition may perform on it for different inhibition locations. We first selected a single continuous apical branch of length 600 μm ( Fig  4A; compared to a random branching 750 μm in Fig 3) to receive excitatory synapses. The mean distance both between any two synapses, and from any synapse to the hotspot, is decreased by~20% compared to Fig 3. The proximity and unbranched dendritic structure between synapses, and their vicinity, on average, to the Ca 2+ hotspot better promotes supralinear summing of NMDA-mediated EPSPs and Ca 2+ spike generation, thus generating more bursts and more spikes per burst. Inhibition was introduced by activating 20 GABA A synapses at one of the numbered dendritic locations at each experimental condition (#1 -#9, Fig 4A). Varying the location of the activated inhibitory synapses produced the respective voltage traces in Fig 4B. The control case without inhibition is shown in Fig 4C. When compared to the control case without inhibition (Fig 4C), we found that perisomatic apical trunk inhibition (at location #1) suppresses the first somatic spike (blue trace in Fig  4B1) but does not affect the dendritic Ca 2+ spike (orange trace) nor the NMDA spike in the apical tuft (green trace). Distal trunk inhibition does not affect the Ca 2+ spike, nor any somatic spikes (compare Fig 4B location #4 to Fig 4C top-no inhibition). Inhibition centered at the Ca 2+ hotspot (#6) or adjacent to it, at about 100 μm distal ("off path"; #7) suppressed the Ca 2+ spike and, consequently, the two latter somatic Na + spikes, thus abolishing the burst. The impact of "off path" (rather than "on path") inhibition on dendritic excitability was demonstrated and discussed by Gidon and Segev [39]. Finally, inhibition acting on the distal tuft hardly modulates Ca 2+ or somatic Na + spiking (compare locations #8 and #9 to Fig 4C top), but only alters local NMDA spikes (green traces).
The effect of the dendritic location of inhibitory synapses on burst generation is summarized in Fig 4D, showing the mean number of somatic spikes per burst for each inhibition locus (numbered as in Fig 4A and 4B). Clearly, the most effective inhibition on burst generation is located 200-400 μm distal to the apical nexus (points #6 and #7, respectively). Without excitation, distal inhibition hyperpolarized the distal dendritic terminal by~2 mV (right inset trace). Inhibition in the hotspot did similarly, but also in the nexus (and the soma very slightly, bottom left trace). Inhibtion between the hotspot and the distal branches created the largest hyperpolarization of~5 mV in the distal terminal (top left trace) which did not spread effectively beyond the tuft compartments. We varied the temporal order and separation (Δt inh ) of excitation and inhibition and found no differential effect between dendritic locations, except for the effective time window width for inhibition. The inhibition with the largest effect (in location 6) disrupts spiking when activated even ±15 ms before or after excitation. Inhibition with a smaller effect (at locations #5, #8) contributes only when activated synchronously with excitation (data supplied in repository).
Next, we tested the case where the excitatory synapses were distributed over the entire apical tuft. Inhibitory synapses were distributed at all branches with a fixed distance from the soma (Fig 5A). The traces in Fig 5B show the influence of inhibition on somatic (blue) and dendritic tuft (orange red and green) voltage during burst generation. Each row relates to a single distance of inhibitory synapses: 100 μm from soma in bottom row, 950 μm in center, and 1150 μm in top. Each column depicts inhibition at a different timing condition: left Δt inh = 0, center Δt inh = -10 ms (inhibition before excitation), and right Δt inh = +10 ms (inhibition after excitation). Surprisingly, this experiment reveals a novel inhibitory time dependence (Fig 5B columns  and 5D), by which all the aforementioned inhibitory consequences appear at Δt inh = 0 ( Fig 5B  left column), and at either Δt inh = -10 ms (center column; inhibition before excitation), effective perisomatic (location #1) and distal inhibition (#3), or at Δt inh = +10 ms (right column; inhibition after excitation), effective inhibition around the Ca 2+ hotspot (#2). Only this last delayed inhibition effect resembles modelling and experimental results [40,41]. A comparison between inhibition effectiveness at various locations and under these three timing conditions (Fig 5D) exhibits preference for concurrent inhibition (Δt inh = 0; green), then inhibition Voltage traces from a somatic (blue) and two dendritic electrodes (orange and green) shown in a, during the activation of inhibition in a single locus (1)(2)(3)(4)(5)(6)(7)(8)(9) in the apical trunk (1,4) and apical tuft (6)(7)(8)(9), indexed as that of the synaptic location in a. Proximal dendritic inhibition at location (1) suppresses the first Na + spike (blue trace; compare to c, top trace) whereas inhibition at locations (6,7) abolishes the Ca 2+ spike (orange) and NMDA spikes (green) and, consequently, the latter two spikes in the burst. Distal inhibition (8)(9) only attenuates the terminal-branch NMDA spikes (green trace) but does not significantly affect the Ca 2+ spike (orange trace) nor the somatic burst. c. Voltage traces without inhibition for basal and tuft excitation (top) and for basal-only excitation (bottom) for comparison. d. Mean # of spikes per burst as a function of location of inhibitory synapses (as in a). In all cases 70 basal and 40 tuft excitatory synapses were activated simultaneously (Δt = 0) with σ = 10 ms. For simulating inhibitory input, 20 inhibitory synapses were uniformly distributed up to 100 μm from each location (1-9) as marked in a; the peak conductance per inhibitory synapse was 1 nS (see Methods). Inset: Voltage traces of inhibition alone, in locations 6, 7 and 9, as pointed by the arrows. https://doi.org/10.1371/journal.pcbi.1009558.g004 following excitation (Δt inh = +10 ms; yellow) in intermediate tuft and preceding excitation (blue) in distal terminals.
The blue traces in Fig 5B show somatic spiking (intra-burst spike-count lower for both proximal trunk and tuft inhibition, see Fig 5D), and some additional depolarization from perisomatic (basal) current sources. Orange traces signify the Ca 2+ spike at the hotspot (above the nexus) occurring for all tuft activations except nexus-proximal tuft inhibition (Fig 5B central row; decoupling NMDA spikes from the hotspot), either after voltage accumulation during extensive NMDA spikes (right) or concurrently with a somatic bAP (left). Red traces exhibit mid-branch NMDA spikes, and green their distal parallels, both eliminated by distal inhibition (Fig 5B top row). Each spike form and locus are manipulated by inhibition at a location ineffective for the other spike types, and only in part of the delay conditions.
Proceeding with whole-tuft excitation configuration, but slightly changing the balance of excitatory synapses between dendrites to 80 basal and 30 apical synapses (from 70 and 40 in Fig 5) creates a relatively "tuft-independent" burst generating scheme, where even strong (20 nS) tuft inhibition distributed on all branches of any one fixed distance from the soma (and nexus) does not prevent initiation of a Ca 2+ spike and a burst. However, bursting generated in these conditions is affected by inhibition at the mid-upper apical trunk, decoupling the Ca 2+ hotspot from its igniting bAP which did not happen in Fig 4 or Fig 5. We determined optimal inhibition spatial extent by adjusting the uniform synaptic distribution to a variable portion of the apical tuft, the same as previously described for excitation, and presented together (S1 Fig). The optimal spatial dispersion for 20 inhibitory synapses with 1 nS peak conductance each (see Methods), which maximally decreases the number of spikes per burst, was 2.5-5% of the total apical dendritic length (5-10% of tuft length), or 185-370 μm. As illustrated in S1 Fig, this dispersion is at half or less the optimal extent of excitatory synapses (measured inversely, by finding maximal spikes per burst). We therefore fixed the Normalized distribution of synaptic activation times separated for excitatory and inhibitory inputs, with different delays Δt inh for the three columns shown. b2-b4. Voltage traces for inhibition at distances 1, 2 and 3, respectively. c1-c3. Control cases. c1. Forty excitatory synapses activated at the tuft triggers NMDA spikes but no dendritic Ca +2 spike nor somatic spikes. c2. Excitatory input to both basal (70) and tuft (40) dendrites generates an NMDA-(red/green) and a Ca 2+ spike (orange) and, in turn, a somatic burst of Na + spikes (blue). c3. Seventy excitatory synapses activated at basal dendrites generate a Na + spike. dendritic length containing inhibition at 200 μm, i.e., a mean synaptic density of one inhibitory synapse in every~10 μm, in line with experimental data [37], thus determining the specific locations most susceptible to inhibition of bursting.

Impact of inhibiting bursts on the plasticity of excitatory synapses
Recent studies associate inhibition of bursts and of dendritic Ca 2+ spikes with restriction of plasticity to specific connections for efficient learning [42][43][44]. To study how inhibition of bursts is associated with synaptic plasticity in the apical tuft, we utilized an established Ca 2 + -dependent plasticity model [45,46] (Eqs (1) and (2) in Methods, and Fig 6A1 and 6A21). Low Ca 2+ concentration, [Ca 2+ ] i , values result in "protected" (unchanged) efficacy of the excitatory synapses (Fig 6A), higher [Ca 2+ ] i produce long-term depression (LTD, Fig 6A1, blue), and even higher values elicit potentiation (LTP, Fig 6A1, pink) [45]. A Ca 2+ -dependent learning rate, η, (Fig 6A2) multiplies changes in efficacy (Eq (2) in Methods) to fit experimental findings. Using our model, we tested how L5PC excitatory synapses act under this plasticity rule, and explicitly how inhibition of dendritic Na + , Ca 2+ and NMDA spikes (determined by location and Δt inh ) controls the manifestation of this plasticity rule. Note that this maximum value of [Ca 2+ ] i does not fully represent whether an excitatory synapse at the location will be potentiated or depressed, as the time that [Ca 2+ ] i lasts at any given concentration will eventually determine the sign of plasticity. When [Ca 2+ ] i rises from baseline to LTP levels, it goes through LTD levels, so the time above the respective thresholds (together with the learning rate) determines the final synaptic weight (the plasticity sign and magnitude). I.e., if [Ca 2+ ] i crosses θ d for a long duration but briefly reaches a maximum over θ p , then Fig  6C suggests LTP while actually the synapse would undergo LTD. Fig 6C1 depicts the case in which the inhibition is proximal to the Ca 2+ hotspot (at location #1) on the apical trunk. The maximum [Ca 2+ ] i at the hotspot is high and nearly invariant to inhibition timing at this inhibitory location (orange line in Fig 6C1), resulting with LTP for all Δt inh values for synapses at the hotspot. The maximum [Ca 2+ ] i at the more distal locations is lower but above the potentiation threshold θ p , so weaker LTP is observed there (Fig 6C1). Fig  6D1 depicts the same case of proximal inhibition, specifically for Δt inh = 0, by showing superimposed excitatory synapses on the dendritic tree, color coded by their respective synaptic weights after plasticity, normalized to initial values. Ten input repetitions were introduced in succession (but nonoverlapping) for a magnified effect on synaptic weights. For this proximal inhibition at location #1 the somatic burst is unaffected and therefore so is the Ca +2 spike at the hotspot (blue trace) and the distal NMDA spike (purple trace). Fig 6C2 and 6D2 depicts the case in which the inhibition is 100 μm distal to the Ca 2+ hotspot (at location #2). Well-timed inhibition (|dt inh |< 17 ms) at this location suppressed dendritic spikes, allowing Ca 2+ influx only into the distal tuft (purple line in Fig 6C2). Without closely timed inhibition each Ca 2+ spike promotes tuft-wide LTP. This observation suggests that any circuit that elicits frequent Ca 2+ spikes and bursts must also incorporate some stabilizing inhibitory synapses to avoid uncontrolled potentiation. With inhibition activated in this location, the synapses proximal to inhibition including the hotspot are "protected", and distal to it undergo LTD; only for the large positive or negative Δt inh values (|Δt inh | > 17 ms) do synapses undergo LTP (Fig 6C2). At the distal dendritic locations (indicated by the red and purple electrodes in Fig 6B), the [Ca 2+ ] i (corresponding red and purple lines in Fig 6C2) is more sensitive to the timing of inhibition. This may be explained by the fact that the voltage response is all-or-none in the hotspot (blue inset traces in Fig 6D), while the distal branches (purple) may exhibit more variable levels of [Ca 2+ ] I due to the synergistic interplay between NMDA current and VGCCs voltage dependence. For this inhibition location for Δt inh = 0, the somatic burst is suppressed and so are the Ca 2+ spike at the hotspot and the distal NMDA spike (Fig 6D2). This severe dendritic voltage attenuation will lower the maximum distal [Ca 2+ ] i only slightly if active after excitation (Fig 6C2 purple line, similarly to NMDA spike suppression in [40]), but enough to switch between LTP and LTD. following the input as a function of timing, Δt inh , between excitation and inhibition, for each inhibitory location (#1-#3 in b). Dashed lines mark threshold values, θ p and θ d , as in a1. Blue shading-LTD; pink-LTP; whiteprotected. d1. Up: Twenty excitatory synapses are superimposed on the dendritic tree, color coded by their respective synaptic weights after activation of inhibition at location #1. The weights are normalized to their initial value (1). Changes are due to the plasticity protocol during 10 input repetitions for the case of Δt inh = 0. Lower traces: Somatic (black); nexus Ca 2+ spike (blue trace at respective blue electrode in b) and distal NMDA spike (purple at respective purple electrode in b) for inhibition location #1. d2. As in d1, with inhibition at location #2-burst and Ca +2 spike suppressed. d3. As in d1, with inhibition at location #3-distal NMDA spike suppressed, burst is unaffected. e. Changes in synaptic weights following the plasticity protocol as a function of synapse location along the tuft branch, for each of the three inhibitory locations.  Fig 6C3). For distal inhibition the proximal synapses undergo LTP (as without inhibition) whereas synapses distal to the inhibition undergo LTD (Fig 6D3). The somatic burst and the Ca 2+ spike are unaffected whereas the distal NMDA spike is attenuated severely by inhibition activated before excitation, shunting the backpropagating Ca 2+ spike (inset voltage traces in Fig 6D3).
A summary plot of the change in the efficacy of the excitatory synapses along the excited dendritic branch for each inhibition location and Δt inh = 0 is presented in Fig 6E. Inhibition in locations #1 and #3 (pink and purple lines in Fig 6E, respectively) create similar plasticity (LTP) patterns. The patterns differ considerably only at the distal terminal, proximal inhibition allows LTP and distal inhibition promotes LTD. Inhibition in location #2 (green line in Fig  6E) suppresses almost all plasticity, allowing mostly LTD in synapses distal to it. Note that the synaptic efficacies following the plasticity protocol are highly sensitive to the choice of potentiation and depression threshold (θ p,d ) values for plasticity. In Fig 6 we fixed the thresholds at values conforming with previous modelling studies, but we provide supplementary results varying θ p,d (S4 Fig). We briefly summarize the key results of this section by noting that (1) inhibition located at the hotspot or immediately distal to it suppressed the Ca 2+ spike and the branch-wide LTP; (2) inhibition of local NMDA spikes controls branch-specific plasticity by transforming LTP to local LTD or to the protected regime, and (3) following the association of bursts, Ca 2+ spikes and inhibition with plasticity (see Discussion), we find a correlation between inhibition of bursts and limiting plasticity. The last point is in line with recent experimental studies, showing that limiting plasticity to specific connections relies on inhibition of bursts [42][43][44].
Finally, we tested for diverse outcomes of plasticity by the different burst classes and their corresponding inhibition, by utilizing the same Ca 2+ -dependent plasticity rule with whole-tuft excitation as in Fig 5A. We associate this scenario to "coincidence bursts", in contrast to bursts resulting from putative NMDA spikes described in Figs 4 and 6. Full tuft excitation was mainly associated with low [Ca 2+ ] i outside the hotspot, meaning minor plasticity effects beyond LTP at the hotspot in the burst-suppressed dendritic tuft (S5 Fig). With these final results, our plasticity findings ultimately connect temporal (Figs 1 and 2) and spatial (S1 Fig) conditions for bursting with class separation (Fig 3), now evidently suggesting a functional basis (NMDAburst dependent plasticity), through differential inhibition (Figs 4 and 5) and an implementation for restricting and tuning Ca 2+ -dependent plasticity (Figs 6 and S5).

Discussion
Several studies have demonstrated the importance of spike bursts in pyramidal neurons in neural coding schemes [47], plasticity [48,49], and information flow [20,21]. Our understanding of the burst phenomenon as well as of dendritic Ca 2+ and NMDA spikes mechanistically was well-established using simplified neuron models, dynamical systems and phase plane analysis [40,[50][51][52]. In the present study, rather than simplifying the model in order to make it analytically tractable, we used the full nonlinear complexity of the modeled neuron together with detailed simulations to study the connection between the various dendritic spikes and spatiotemporal excitatory and inhibitory synaptic activation and somatic output bursts.

Excitatory conditions for burst generation: Two types of bursts
We showed that somatic bursts arise either from a bidirectional stream of excitatory inputs combining basal and tuft synapses, or by generation of a local dendritic voltage plateau due to clustered excitatory input to the apical tuft alone (Figs 1 and S1). Bursts that result from a coincidence of basal and apical synaptic activation consist of 3-4 spikes fired at a frequency of~200 Hz (Fig 3), whereas the activation of > 30 excitatory synapses on a length of 350 μm-750 μm of the dendritic tuft (S1 Fig) generates NMDA spikes at several apical loci [53] initiating a Ca 2+ spike at the main tuft and a high-frequency burst of~2 spikes. The number of tuft synapses required for burst generation is less than half that of basal synapses (Figs 3 and S3). This finding is counterintuitive, as tuft synapses are thought to be less potent in generating somatic spikes, due to steep voltage attenuation expected in a length of~1 mm along the apical dendrite towards the soma. Several factors compensate for the attenuation and make tuft synapses more likely to than basal synapses initiate bursts: clustered distribution (S1 Fig), high excitability of thin tuft terminals [53], and proximity of tuft synapses to the Ca 2+ hotspot which promotes generation of Ca 2+ spikes. Further experiments should be conducted to explore whether these two burst types exist in cortical L5 pyramidal neurons, and to reveal the implications of the different types for neuronal computation and for plasticity-related processes (see below).
In agreement with Larkum et al. [30], we found similar conditions on the timing of synaptic activation for burst generation when basal-then-apical (Δt < 0) synapses are activated (Fig 2). However, by varying the standard deviation, σ, of the apical and basal synapses activation times (Fig 2) we found that, for small σ (< 10 ms) burst generation is also enabled in the reverse tuft-to-basal order of activation (0 ms < Δt < 30 ms; Fig 2). This tuft-to-basal order was found favorable for Ca 2+ -dependent LTP in spines, resembling the classical asymmetric spike-timing dependent plasticity rules (STDP) [54]. This resulted from the long timescales and boosting effect of NMDA potentials that benefited from the bAP's arrival after initial voltage buildup at the apical dendrites. Removing NMDA conductance disabled burst generation with Δt > 10 ms (S3D Fig). This prediction still awaits experimental validation.

Inhibitory control of bursts
Next, we examined how the location and timing of dendritic inhibition 'edits' the somatic bursts. To the best of our knowledge, this is the first direct theoretical study of this question. For excitation on a single tuft branch, perisomatic apical trunk inhibition (e.g., via basket cells) abolished the first somatic spike but left the dendritic spikes unchanged and allowed for burst initiation (Fig 4B1). Inhibition in or immediately distal to the Ca 2+ hotspot ("off path" condition) [39] disrupted the Ca 2+ spike and suppressed the somatic burst (Fig 4B6 and 4B7 and 4D), whereas more distal inhibition attenuated local NMDA spikes without affecting the burst (Fig 4B8 and 4B9).
A burst induced by whole-tuft excitation was suppressed by inhibition distributed at all branches of a fixed distance distal and adjacent to the hotspot, for Δt inh � 0 (inhibition following excitation; Fig 5). Δt inh � 0 is more effective in disrupting burst generation when inhibition is activated in distal locations, by suppressing NMDA spikes there. Contrary to our results, theoretical and experimental studies found the most effective timing of inhibition for suppression of NMDA spikes to be after excitation [40,41]. This is because they use basal activation and set the criterion for efficiency of inhibition as the reduction in the voltage integral, rather than the strong instantaneous onset of NMDA spikes that is crucial for burst formation. Overall, we showed that by controlling local dendritic excitability, in particular the NMDA and/or Ca 2+ spikes, local dendritic inhibition can finely edit the output bursts.

Effect of inhibition on dendritic plasticity: Relationship to burst control
Notably, intracellular Ca 2+ concentration is implicated in long-term synaptic plasticity [55,56]. Utilizing the calcium-dependent synaptic plasticity model [45], our study showed that inhibition, during somatic bursts and correlated Ca 2+ spikes, enables diverse plasticity modification maps. Due to well-located and timed dendritic inhibition, local Ca 2+ concentration in dendrites could be finely tuned, resulting in nearby synapses undergoing LTP or LTD, or left 'protected' from plasticity (Fig 6). These effects of synaptic inhibition on local dendritic excitability (modifying local dendritic NMDA and Ca 2+ spikes) and, consequently, on synaptic plasticity are correlated with the impact of dendritic inhibition on burst activity. It seems that dendritic inhibition might simultaneously control local synaptic plasticity and global somatic burst activity (see also [42]).
In the case of single-branch activation whole-tuft LTP is very robust to the location and timing of inhibition (Fig 6D1 and 6D3). Furthermore, different inhibitory locations generate varied plasticity maps: segmenting the branch to LTP and LTD (distal tuft inhibition; Fig 6D3), or to LTP, LTD and protection of weights (intermediate tuft inhibition; Fig 6D2). However, for the case of whole-tuft activation, all inhibition locations distal to the intermediate trunk with correct timing reduce the tree-wide LTP to highly localized LTD (trunk or distal inhibition; S5C1 and S5C3 Fig We showed that pairing EPSPs in the tuft with somatic bursting generally produces LTP (Fig 6D1). This prediction is supported by several experimental results [35,57,58] but disagrees with others [29]. In another paper, Owen et al. [42] suggested a role for burst inhibition that is compatible with our findings: limiting excitatory synaptic plasticity for efficient and stimulusspecialized implementation of learning. That SOM inhibition of bursts leads to ineffective plasticity was shown in the hippocampus [48] but not directly and causally in the cortex.

Related studies
In their theoretical study, Shai et al. [38] showed that the intraburst firing frequency is best approximated by a composite sigmoidal function of the number of basal, and apical synapses, with apical number modulating the threshold of basal number required for bursting. This simplification is challenged by our findings, because it doesn't generalize to the new class of bursts generated by single tuft branch activation (Fig 3). Specifically, their results do not show bursting for tuft only input, even at high numbers of synapses. Our findings require additional nonlinearities to account first for NMDA spikes and then for the non-monotonic transition in frequency between burst classes. The non-monotony of intraburst frequency is expressed in the corresponding heatmap (Fig 3B right). First more synapses mean higher frequency, but then it means shifting to coincidence-burst with lower frequency (compare to Shai's [38] Fig  4A left). This discrepancy arises from our clustering of the apical synapses compared to their whole-apical dendrite distribution, leading to stronger interactions (more spikes, S2 Fig) and a basal-independent Ca 2+ spike (Fig 1C and top green region in Fig 3B). However, our findings support Shai's [38] result of bursting with basal only input (Fig 3; see also [29]), challenging the notion that bursts depend on Ca 2+ spike firing [59,60].
Combined with our burst classes findings, we predict that input patterns generating coincidence many-spikes high frequency bursts (in which suppression of plasticity is more common) will also activate local SOM+ interneurons, and they would supply feedback inhibition near Ca 2+ hotspot to withhold excessive LTP [23]. This could be tested by recording SOM-L5PC pairs verified for feedback inhibitory connections, possibly with excitation of presynaptic axonal projections. Analyzing their activity during L5PC burst/BAC-firing and using long-term plasticity indicators (e.g., AMPA/NMDA ratios, spine sizes) would validate our predictions.

Implications of burst control on perception
Larkum et al. [19,20]suggested that the BAC firing mechanism implements coincidence detection between bottom-up sensory stream and top-down context modulation. Other studies made even stronger claims implicating bursts to be involved, via BAC firing, with conscious perception [61], visual illusions [62] and attentional modulation of activity [63]. In view of our findings of two types of bursts, these effects could be reexamined, and the impact of dendritic inhibition of L5 pyramidal neurons on such high-level processing explored. Particularly, Takahashi et al. [6,64]showed that optogenetic stimulation of L5bPC dendritic tufts enhances tactile stimuli detection, whereas blocking these cells' outputs to subcortical target regions suppressed stimuli detection. We predict that distal tuft inhibition (e.g., by feedback via Martinotti cells) will attenuate local dendritic excitability and, in turn, decouple between the bottom-up and top-down input streams and, thus, disable conscious perception. As overreaching as the claims mentioned above seem to be, it is exciting to connect biophysical mechanisms such as specific dendritic inhibition to our subjective experience.

Methods
All simulations were run using NEURON 7.7 [65] and Python 2.7.16 (NumPy v1.16.2), initially on Windows/Linux PC, and for final plots on a parallel processing cluster unit.
Code and generated data for obtaining the results are available at github.com/EilamLeleo/burst.

Model cell
We used the established compartmental model of rat thick-tufted L5bPC developed by Hay et al. [33], including modifications of voltage gated calcium channel densities as in Shai et al. (2015) [38] and I h channel density distribution as in Labarrera et al. (2018) [66]. The two similarly functioning reconstructed morphologies were used to verify our findings (see Fig 6 in [33]), though plots were generated with the first for convenience and consistency. This model consists of 10 different active membrane conductances, internal Ca 2+ dynamics and hundreds of compartments arranged in 4 main section types: somatic, axonal and apical and basal dendritic. Hay et al. used a genetic algorithm in a procedure called multiple-objective optimization (MOO) to create thousands of working models combining all participating conductances in the different section types, with values within some experimentally constrained range. The resulting parameters span a subspace of continuous ranges which recreates the physiological voltage measurements. The greatness of their work is that unlike other compartmental models, they succeeded in fitting perisomatic and dendritic electrogenesis, and their interaction (i.e., backpropagation and critical frequency). Viewing our simulated voltage traces, we noted a biologically unrealistic amplitude and prolonged duration of after depolarization (ADP) in the somatic Na + spikes. Scanning our parameter space, we decided to change a single maximal conductance variable, so as not to significantly affect our results and the main findings of previous publications-that of the somatic calcium-dependent potassium current (SKv3_1). The peak conductance value of the SK channel was thus increased by 1.5-fold compared to that of Hay et al. [33] such that somatic g SK = 3380 � 1.5 = 5070 pS/μm 2 .

Simulations
Electric activity of the neuron was simulated for 600 ms at each instantiation with simulation dt = 25 μs. Simulation temperature was 34˚C as previously suggested [67], and initial voltage was -76 mV.
We save all simulation data to Python NumPy arrays, initializing simulation of any experiment with each parameter combination at 100/200 random synaptic instantiations using a parallel processing unit (cluster). Randomly drawn properties were both spatial-site of synapse impinging on the dendrite, and temporal-activation time, as described in the following Input Distributions subsection. For most extensive data summarizing plots, we save only inter-spike intervals (ISIs), for calculation of spikes per burst and firing rate.

Synapse models
Excitatory synapse model chosen (ProbAMPANMDA2.mod), implemented by Ramaswamy et al. [68] and modified by Hay et al. [33], combines a fixed ratio (of equal weights) of fast-AMPA (decay time constant τ AMPA = 1.7 ms) and slower-NMDA (decay τ NMDA = 43 ms) ionotropic receptors. Reversal potential for both was e = 0 mV. V rest = -80 mV. Peak conductance gmax was fixed at 0.4 nS for both AMPA-and NMDA-synaptic conductances. Contemporary studies use a higher decay τ NMDA = 70 ms, which is likely to make our findings of different classes more pronounced, as more current will be recruited by NMDA spikes. In the AMPA-only control (S3 Fig) we exchange ProbAMPANMDA2.mod with ProbUDFsyn2.mod which includes the same AMPA mechanism and parameters, but not the NMDA parallelseffectively setting g NMDA = 0.

Input distributions
Synaptic locations and activation times were randomly drawn from spatially uniform and temporally normal distributions, separated for basal and apical (excitatory) populations, and for inhibitory. For simplicity and lack of additional preliminary findings, all temporal distributions of any single simulation are identical in variance.
We note three main types of input patterns by how they spread on the dendritic tree: basal, apical or both. Basal branches are plentiful, interact at the soma and are less prone to NMDAspike generation. Of course, some may be active spontaneously on the same branch and would contribute to an NMDA-spike formation, and a few of those will allow swift somatic APs or a burst [69]. However, the apical tuft allows this more readily by having long thin branches that create NMDA-spikes by fewer synapses [70], and by the Ca 2+ hotspot transforming these into prolonged depolarizations at the soma. This lower threshold also means uniform distribution of synapses on the tuft will cause a burst before it would on the basal tree. Nevertheless, a bAP from the soma will lower the threshold for Ca 2+ -spike firing by the tuft and a burst, so the main options for bursting are tuft alone or both tuft and basal. How feasible, abundant and distinguishable are both input types? We show they are very much so.
For generating Figs 1-3, excitatory synapses were scattered on the entire basal tree, and on a randomly drawn continuous 750 μm stretch of the apical dendritic tuft, which is about 1/10 of the entire apical length or 1/5 of the tuft-equivalent to a single offshoot of the tuft from nexus to all distal terminals.
Various values of σ were used in Figs 1 and 2 (σ = 3,6,9,12 ms) to examine the implications of temporal jitter of synaptic activation on burst output. In Fig 3 we used σ = 10 ms as a single round parameter to test activation of many synapses on the different dendrites. We chose this value both for a more realistic (dispersed) activation, and for comparison with Shai et al. [38], that obtained some different results.
The delay described by Larkum et al. [30] from somatic spiking to apical EPSP isn't the same as from basal to tuft synaptic activation, but we note a~3 ms delay of both-somatic spiking from mean basal activation and apical EPSP peak from tuft activation, so this equivalence is accounted for. Synaptic noise was introduced for in-vivo like state by input of the same magnitude, but uniformly distributed on the dendrites and in simulation-time (600 ms starting 100 ms before targeted synaptic activity). Results were insignificantly different (except for a minor reduction in apical synapse number threshold for bursting in Fig 3 from 50 to 40).
For plotting Figs 4-6, inhibitory synapses were introduced on the apical tree. In all three, they were manipulated by distance from soma (20 synapses in < 200 μm disparity). Temporal jitter was allowed with the same parameters as excitation, excitation on both trees were activated simultaneously (Δt = 0), and Δt inh was introduced to separate between excitation to inhibition mean activation times. For Figs 4 and 6, the number of excitatory synapses was increased from 50 basal and 30 tuft in Fig 1 to 70 basal and 40 tuft synapses, owing to the higher bursting threshold at σ = 10 ms. Tuft excitation was restricted to a single continuous branch of length 600 μm from nexus to the most distal terminal, inhibition is only on the trunk or the same branch. In Fig 5 excitation is on the entire tuft, and inhibition is at a fixed distance, but in all oblique or tuft branches sharing this distance from the soma.

Data analysis
Most initial and direct analysis of electrodynamics in the model L5PC was calculated online on Python after each run of NEURON. Gathering of all data in any single experiment for plotting results and drawing conclusions was generally executed manually.

Spike detection
We set the spike detection somatic voltage threshold at 0, though it essentially does not differ by setting it at -20 mV or +20 mV, nor by detecting it from the voltage trace at the axon initial segment. Spike width did not exceed 2 ms, and minimal ISI was above 3 ms, so effectively the soma was instantly hyperpolarized below detection threshold right after spiking, thus allowing for the next to be detected even at high intraburst (between spikes in a burst) firing rates of 300 Hz.

Number of spikes and firing rate calculation
The definition of bursts relates to statistical deviation of firing rate from random Poisson firing. A cell is considered bursting if either the coefficient of variation (CV = standard deviation/mean) or the Fano factor (= variance/mean) of ISIs over some time interval, is higher than 1 (Poisson). Relying on mean firing rates of pyramidal cells, we allow ourselves to group spikes as part of a burst by their ISI alone, < 20 ms (> 50 Hz): A burst of > 50 Hz will occur by chance for a Poisson neuron with a high 5 Hz mean firing rate (FR) approximately once every 10 15 seconds. So, we set ISI � 20 ms as our burst-grouping interval, and we note that our generated cell intraburst FR exceeds 100 Hz significantly. If no two spikes arrive within 20 ms of one another, then we count only one spike. By this criterion we count and average the number of spikes per burst (at each simulation instance), even if isolated spikes precede the burst. The Δt range in Fig 2F is defined by > 95% chance for bursting and shown in Fig 2D as > 2 spike per trial average. Firing rate in Fig 3B  and 3C is the mean over all spikes of all bursts in the same parameter combination, taking isolated spikes as a 0 Hz burst. Measuring intraburst FR for < 2 mean spikes per burst seems peculiar, because at least 2 spikes are needed in order to measure it, still it proves useful to keep values not only from bursts. We tested for big variations inside a single burst, that would discredit our conclusions-a burst of one ISI at 4 ms and another at 6 ms is not firing at 200 Hz but rather in the 170-250 Hz range. We find no substantial (1.5-fold) differences occurring in over 5% of instances at any parameter combination.

Generating plots
All voltage traces represent a single representative simulation. Input pattern was kept fixed for parameters not probed at the particular experiment, i.e., changes in activation times do not alter spatial locations of synapses in Figs 1 and 2, and variations in inhibition do not modify excitatory pattern in Figs 4-6. In contrast, each value on the summarizing graphs and heatmaps (in Fig 3) averages 100-1000 random spatiotemporal synaptic patterns, with matching distribution parameters. Error bars are generally missing from all graphs that plot more than a single line. In Fig 2F and 2G they reflect sampling resolution. In calculating mean spikes per burst, standard deviation values are large, as any mean close to half a whole spike would indicate > 0.5 standard deviation. This will not reflect the consistency of our measurement, so instead we use (in Figs 4D and 5D) the standard deviation of batch means (10, each from 20 repetitions). Hence, the values are very generalizable and will occlude a possible variation within the fixed-parameter data.
The scatterplot in Fig 3 transforms the input-aligned picture on the heatmaps, into a mapping of each mean spike per burst value with its corresponding FR (each point represents a spike # from Fig 3B left and its collocated FR to the right). There we directly assess their correlations, and cluster output parameters pairs to three dense regions, high-rate low-spike #, low-rate high-spike # and zero spikes. These clusters in-turn map to correlating histograms, showing both parameter ranges attributed to each kind, when separated by input synapse numbers: purple generated by > 40 tuft and < 120 basal synapses; green by > 150 basal and < 100 tuft synapses.

Calcium dependent plasticity
The learning rule applied is summarized by the two equations below (Eqs (1) and (2)) and graphs (Fig 6A) shown. The equations define the plasticity function O and the resulting learning rule as the change in synaptic strength (w j ). θ p,d are concentration thresholds for potentiation and depression respectively (Fig 6A1 and 6C), and η the learning rate ( Fig 6A2).
O ¼ 0:25 þ e 80ð½Ca 2þ � i À y p Þ 1 þ e 80ð½Ca 2þ � i À y p Þ À 0:25e 80ð½Ca 2þ � i À y d Þ 1 þ e 80ð½Ca 2þ � i À y d Þ ð1Þ Graphs are created for maximal [Ca 2+ ] i (concentrations) during the simulation at different inhibitory locations and times (Fig 6C). Threshold was slightly shifted from the previous studies [45,46] to θ p = 0.4 μM, θ d = 0.25 μM (originally 0.5 and 0.3), which creates a more variable and realistic range of phenomena (LTP/D & protected). To generate synaptic weight changes in Fig 6D, the duration over or under any threshold is multiplied by a learning factor chosen as ten repetitions for effect size, as physiological [Ca 2+ ] i measurements are difficult to control for in these relatively short simulations. Synapses on trees represent predicted weights after learning by this rule. Heatmap of mean number of spikes per burst as a function of the number of activated synapses on the basal and apical trees, with AMPA-only synapses. The apical-only burst class vanishes with zero NMDA conductance. c. Same as b but as a control experiment with both AMPA and NMDA conductances (from Fig 2B). d. Mean number of somatic spikes per burst for a range of Δt values, without NMDA. Colored lines correspond to different σ values. e. Same as d but for control with NMDA (from Fig 2D). Note that the bursting window's center is negative (basal input following apical input) without NMDA for small σ.