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Conceived and designed the experiments: RAJvE AvO. Performed the experiments: RAJvE. Analyzed the data: RAJvE AvO. Wrote the paper: RAJvE AvO.

The authors have declared that no competing interests exist.

Neurons display a wide range of intrinsic firing patterns. A particularly relevant pattern for neuronal signaling and synaptic plasticity is burst firing, the generation of clusters of action potentials with short interspike intervals. Besides ion-channel composition, dendritic morphology appears to be an important factor modulating firing pattern. However, the underlying mechanisms are poorly understood, and the impact of morphology on burst firing remains insufficiently known. Dendritic morphology is not fixed but can undergo significant changes in many pathological conditions. Using computational models of neocortical pyramidal cells, we here show that not only the total length of the apical dendrite but also the topological structure of its branching pattern markedly influences inter- and intraburst spike intervals and even determines whether or not a cell exhibits burst firing. We found that there is only a range of dendritic sizes that supports burst firing, and that this range is modulated by dendritic topology. Either reducing or enlarging the dendritic tree, or merely modifying its topological structure without changing total dendritic length, can transform a cell's firing pattern from bursting to tonic firing. Interestingly, the results are largely independent of whether the cells are stimulated by current injection at the soma or by synapses distributed over the dendritic tree. By means of a novel measure called mean electrotonic path length, we show that the influence of dendritic morphology on burst firing is attributable to the effect both dendritic size and dendritic topology have, not on somatic input conductance, but on the average spatial extent of the dendritic tree and the spatiotemporal dynamics of the dendritic membrane potential. Our results suggest that alterations in size or topology of pyramidal cell morphology, such as observed in Alzheimer's disease, mental retardation, epilepsy, and chronic stress, could change neuronal burst firing and thus ultimately affect information processing and cognition.

Neurons possess highly branched extensions, called dendrites, which form characteristic tree-like structures. The morphology of these dendritic arborizations can undergo significant changes in many pathological conditions. It is still poorly known, however, how alterations in dendritic morphology affect neuronal activity. Using computational models of pyramidal cells, we study the influence of dendritic tree size and branching structure on burst firing. Burst firing is the generation of two or more action potentials in close succession, a form of neuronal activity that is critically involved in neuronal signaling and synaptic plasticity. We found that there is only a range of dendritic tree sizes that supports burst firing, and that this range is modulated by the branching structure of the tree. We show that shortening as well as lengthening the dendritic tree, or even just modifying the pattern in which the branches in the tree are connected, can shift the cell's firing pattern from bursting to tonic firing, as a consequence of changes in the spatiotemporal dynamics of the dendritic membrane potential. Our results suggest that alterations in pyramidal cell morphology could, via their effect on burst firing, ultimately affect cognition.

Neurons exhibit a wide range of intrinsic firing patterns with respect to both spike frequency and spike pattern

Electrophysiology, in combination with computational modeling, has elucidated the ionic mechanisms underlying intrinsic neuronal burst firing. Two main classes of mechanisms have been distinguished ^{+} and K^{+} channels, which promote propagation of action potentials from the soma into the dendrites, cause the dendrites to be depolarized when, at the end of a somatic spike, the soma is hyperpolarized, leading to a return current from dendrites to soma. The return current gives rise to a depolarizing afterpotential at the soma, which, if strong enough, produces another somatic spike

Although ion channels play a pivotal role in burst firing, dendritic morphology also appears to be an important factor. In many cell types, including neocortical and hippocampal pyramidal cells

Considering that dendritic morphology can undergo significant changes in many pathological conditions, such as Alzheimer

We use a morphologically and biophysically realistic model of a bursting layer 5 pyramidal cell from cat visual cortex (^{−2}) are as follows: a fast sodium current, ^{−2}, the axial resistance

^{2}, a total volume of 13292 µm^{3}, a root segment with a diameter of 8.5 µm, and 41 terminal segments with diameters in the range 0.30–1.33 µm. The 10 basal dendrites have a total length of 10232 µm, a total surface area of 27396 µm^{2}, a total volume of 7650 µm^{3}, root segments with diameters in the range 1–4 µm, and in total 35 terminal segments with diameters in the range 0.59–1.31 µm. The MEP value (in units of the electrotonic length constant) of the apical dendrite is 0.74. The input conductance of the cell without basal dendrites is 7.9 µS and with basal dendrites 19.9 µS.

The pyramidal cell is activated by either somatic or dendritic stimulation. For somatic stimulation, the cell is continuously stimulated with a fixed current injection of 0.2 nA at the soma. For dendritic stimulation, the cell is stimulated by synapses that are regularly distributed across the apical dendrite, with a density of 1 synapse per 20 µm^{2}. For this synaptic density, the total input current, based on the current transfer at a single synapse, is approximately the same as with somatic stimulation.

The excitatory synaptic input is mediated by AMPA receptors. The time course of conductance changes follows an alpha function

The firing patterns were recorded from the soma. Each simulation lasted 10000 ms, of which the first 1000 ms were discarded in the analysis in order to remove possible transient firing patterns.

In studying the influence of pyramidal cell morphology on burst firing, we distinguish between dendritic size and dendritic topology. The size of a dendritic tree is the total length of all its dendritic segments. The segment between the soma and the first branch point is called the root segment (see

To investigate how the dendritic size of the pyramidal cell influences burst firing, we varied the total length of the cell's apical dendrite according to two methods. In the first method, we successively pruned terminal segments from the apical dendritic tree. Starting with the full pyramidal cell morphology, in each round of pruning we randomly removed a number of terminal segments from the apical dendritic tree. Each terminal segment had a chance of 0.3 to be removed. From the reduced dendritic tree, we again randomly cut terminal segments, and so on, until in principle the whole apical dendrite was eliminated. This whole procedure was repeated 20 times. The density of synapses was kept constant during pruning, so with dendritic stimulation pruning also changed the total input to the cell. With somatic simulation, the total input to the cell did not change when the apical dendrite was pruned.

In the second method, we kept the dendritic arborization intact and changed the size of the apical dendrite by multiplying the lengths of all its segments by the same factor. Thus in this way the entire apical dendritic tree was compressed or expanded. For dendritic stimulation, we kept the total synaptic input to the cell constant by adapting the density of the synapses. So, both with somatic and dendritic stimulation, the total input to the cell did not change when the size of the apical dendrite was modified.

To examine the impact of the cell's dendritic branching structure on burst firing, we varied the topology of the apical dendritic tree by swapping branches within the tree. The apical dendritic trees that were generated in this way all have exactly the same total dendritic length and other metrical properties such as total dendritic surface area and differ only in their topological structure. The total input to the cell, both with somatic and dendritic stimulation, did not change when the topological structure was altered.

To facilitate a systematic analysis of the role of dendritic size and dendritic topology in shaping burst firing, we also use a set of morphologically simplified neurons. The neurophysiological complexity of these neurons is similar to that of the full pyramidal cell model. For a systematic study, one must use trees with a relatively small number of terminal segments, because otherwise the number of topologically different trees becomes so large that simulating all of them becomes impossible. For a tree with only 12 terminal segments, for example, there already exist as many as 451 different tree topologies

All segments in the tree (intermediate and terminal segments; see

The ion channel types and densities are based on Mainen and Sejnowski's ^{−2}). The dendrites contain a fast sodium current, ^{−2} and the axial resistance

As in the pyramidal cell model with full morphological complexity, the neurons are activated by either somatic or dendritic stimulation. All the tree topologies receive the same input. For somatic stimulation, the neurons are continuously stimulated with a fixed current injection of 0.03 nA (0.1 nA for the non-Rall neurons, in which segment diameter is equal throughout the dendritic tree). For dendritic stimulation, the cells are stimulated by 600 synapses, with on each terminal or intermediate segment (in total 15 segments for a tree with 8 terminal segments) 40 uniformly distributed synapses. With this number of synapses, the total input current, based on the current transfer at a single synapse, is approximately the same as with somatic stimulation. The synaptic input is mediated by AMPA receptors, with the same parameters as in the full pyramidal cell model. Also as in the full pyramidal cell model, each synapse is randomly activated according to a Poisson process, with a mean activation frequency of 1 Hz.

The simulations were performed in NEURON

To examine how the size of the dendritic tree influences firing pattern, we changed the total dendritic length of a given tree topology by multiplying the lengths of all its segments by the same factor. For dendritic stimulation, the number of synapses on the tree was thereby kept constant. So, both with somatic and dendritic stimulation, the total input to the cell did not change when the size of the dendritic tree was modified.

For presenting the firing patterns from the different tree topologies, we ordered the trees according to the degree of symmetry of their branching structure. To do this, we used a variant of the ranking scheme proposed by Harding

To obtain a unique notation, we applied the following two rules. First, if the subtrees at a particular node have a different size, the largest subtree is put left of the comma. So we write 3(2(1,1),1) instead of 3(1,2(1,1)). Second, to order (sub)trees of equal size, we consider to be larger the tree that has the highest number at the first figure in which the tree descriptions differ. So of the following two trees, 4(2(1,1),2(1,1)) and 4(3(2,1),1), the second one is considered to be larger (since 3>2). Thus, in a description of an 8-terminal tree of which these two trees are the subtrees, the second subtree is put first, i.e., 8(4(3(2,1),1),4(2(1,1),2(1,1))).

Once all tree topologies were written in a unique form, they were ordered according to their size (in the extended sense, as described above), whereby the largest one was put first in the list. Since two trees can now be ordered simply by looking at the first figure in which their descriptions differ, this ordering is called a reverse lexicographical ordering. Applied to trees of the same size, it puts trees in order of symmetry, with the most asymmetrical tree first and the most symmetrical tree last.

Bursting is defined as the occurrence of two or more successive spikes with short interspike intervals followed by a relatively long interspike interval. To quantify bursting, we used the burst measure developed in

In

The input conductance of a pyramidal cell with a given dendritic morphology was determined by applying a static, subthreshold current injection at the soma. The ratio of the magnitude of the injected current to the resulting change in membrane potential at the soma is defined as the input conductance of the cell

To quantify the electrotonic extent of a dendritic tree, we introduce a new measure called mean electrotonic path length (MEP). For a given terminal segment (see

The analysis and model code for this paper including a tool for NEURON parameter scanning is available from ModelDB at

Employing a standard model of a bursting pyramidal cell

To investigate how pyramidal cell size influences burst firing, we changed the total length of the apical dendrite according to two methods. In the first method, we successively pruned terminal branches of the apical dendrite. Regression of pyramidal apical dendrities has been observed in response to, for example, chronic stress

In the second method, we kept the dendritic arborization intact and varied the size of the apical dendritic tree by multiplying the lengths of all its segments by the same factor. Thus, the entire apical dendritic tree was compressed or expanded. Both with somatic and with dendritic stimulation, and in line with the previous results, burst firing disappears as the total dendritic length is decreased (

Using the cell from

With somatic stimulation, the transitions from tonic firing to bursting and from bursting to tonic firing occur quite abruptly as dendritic length is modified. As with pruning, this implies that a small change in dendritic length can have a large effect on the firing state of the cell. Because of the stochastic nature of the activation of synapses, the shifts in firing state are more gradual with dendritic than with somatic stimulation, especially from bursting to tonic firing. However, the results obtained under both stimulation regimes are very much comparable, with even the onset and cessation of bursting taking place at approximately the same dendritic sizes.

To examine whether dendritic branching structure, or topology, could influence burst firing, we varied the topology of the apical dendritic tree by swapping branches within the tree. Thus all the dendritic trees that were generated in this way have exactly the same total length and other metrical properties such as total surface area and differ only in the way their branches are connected. In this set of pyramidal cells, we find cells that produce firing patterns ranging from tonic firing to strongly bursting (

Using the cell from

To facilitate a better understanding of our findings obtained with the pyramidal cell model and to analyse more precisely the role of dendritic morphology in shaping burst firing, we also investigated a set of 23 morphologically simplified neurons consisting of all the topologically different trees with 8 terminal segments. Because the cells have relatively few terminal segments, the impact of dendritic topology on burst firing can be studied in a systematic way.

To show how both dendritic length and dendritic topology affect bursting, we plotted in

The segment diameters in the trees obey Rall's power law.

Because different tree topologies start and stop bursting at different total lengths, a change in the topology of the tree, while keeping the size of the tree the same (i.e., a change along a vertical line through

As with somatic stimulation, with dendritic stimulation there is also only a range of tree sizes where bursting occurs and this range depends on topological structure, with asymmetric trees starting bursting at lower total lengths than symmetric trees (

The segment diameters in the trees obey Rall's power law.

To analyse further how dendritic morphology controls burst firing, we also studied the set of 23 tree topologies with all dendritic segments of a tree having exactly the same diameter. If dendritic diameters obey Rall's power law, as they do in

Thus, all the 23 tree topologies have the same total dendritic surface area as well the same total dendritic length.

Although in

Except for the white contour lines, the left and right panels are identical and show the degree of burst firing (color coded) as a function of both dendritic topology and total dendritic length. In the left panel, contour lines of equal input conductance (IC, in µS) are superimposed. These contour lines show all the combinations of dendritic length and dendritic topology that result in a given input conductance. Similarly, in the right panel, contour lines of equal mean electrotonic path length (MEP, in units of the electrotonic length constant) are superimposed. These contour lines show all the combinations of dendritic length and dendritic topology that result in a given MEP value. The onset and cessation of bursting are strongly associated with mean electrotonic path length, but not with input conductance.

What instead appears to be a critical factor is the electrotonic extent of the dendritic tree, as measured by the mean electrotonic path length. The mean electrotonic path length of a dendritic tree is the sum of all electrotonic dendritic path lengths measured from the tip of a terminal segment to the soma divided by the total number of terminal segments (see

We now return to the set of tree topologies in which the segment diameters obey Rall's power law, to see whether mean electrotonic path length is also a determining factor for dendritic trees with a more realistic distribution of diameter sizes. For both somatic and dendritic stimulation,

The degree of burst firing (color coded) is shown as a function of both dendritic topology and total dendritic length, under somatic stimulation (

We next provide an explanation for the importance of mean electrotonic path length for bursting, which is further supported in ^{+} conductance is still high, it will be difficult for the soma to depolarize and produce a spike. Since the propagation velocity of voltages and currents is proportional to the electrotonic length constant ^{+} conductance is still high, so that it cannot produce another spike—that is, no bursting.

^{+} channels are still open, preventing the generation of a second spike. See further

Furthermore, if the electrotonic distance between soma and distal dendrites is too small, the large conductive coupling will lead to currents that quickly annul membrane potential differences between soma and distal dendrites. This prohibits a strong and long-lasting differentiation in membrane potential between soma and distal dendrites, which is the generator of the return current that lies at the heart of the ping-pong mechanism of bursting. In the limiting case, with very small electrotonic distance, soma and dendrites can be considered as a single compartment with a uniform potential.

If the electrotonic distance between soma and distal dendrites is too large, bursting will also not occur. Note that even in the absence of a return current, the cell will generate a next spike as a result of the external (somatic or dendritic) stimulation. So, what the return current in fact does when it causes bursting is to advance the timing of the next spike. If the electronic distance is too large, the return current will arrive too late—that is, not before the external stimulation has already caused the cell to spike. Furthermore, if the electrotonic distance is too large, the potential gradient between distal dendrites and soma will become too shallow for a strong return current. In summary, for the soma and distal dendrites to engage in a ping-pong interaction, the electrotonic distance between the two should be neither too small nor too large.

Thus, when the mean electrotonic path length becomes too small or too large as the total size of the dendritic tree is varied, as in

To illustrate the importance of electrotonic distance for bursting and the impact of topological structure, we show in ^{+} channels are still open, hampering somatic depolarization. Consequently, the membrane potential is not raised enough to trigger another spike, whereas in the asymmetrical tree it is (between

We now return to the pyramidal cells with full morphological complexity, to see whether burst firing is also there related to mean electrotonic path length. In

(In the calculation of the correlation, we excluded the two MEP values around 0.84, thus ignoring the part of the curve that is clearly flat). The letters A–F correspond to those in

In general, the results obtained with dendritic stimulation are comparable to those produced with somatic stimulation (

Given the crucial role of bursts of action potentials in synaptic plasticity and neuronal signaling, it is important to determine what factors influence their generation. Using a standard compartmental model of a reconstructed pyramidal cell

We have shown that either shortening or lengthening the apical dendrite tree beyond a certain range can transform a bursting pyramidal cell into a tonically firing cell. Remarkably, altering only the topology of the dendritic tree, whereby the total length of the tree remains unchanged, can likewise shift the firing pattern from bursting to non-bursting or vice versa. Moreover, both dendritic size and dendritic topology not only influence whether a cell is bursting or not, but also affect the number of spikes per burst and the interspike intervals between and within bursts.

The influence of dendritic morphology on burst firing is attributable to the effect dendritic length and dendritic topology have, not on input conductance, but on the spatial extent of the dendritic tree, as measured by the mean electrotonic path length between soma and distal dendrites. For the spatiotemporal dynamics of dendritic membrane potential to generate burst firing, this electrotonic distance should be neither too small nor too large. Because the degree of symmetry of the dendritic tree also determines mean electrotonic path length, with asymmetrical trees having larger mean path lengths than symmetrical trees, dendritic topology as well as dendritic size affects the occurrence of burst firing.

In Mainen and Sejnowski's

The effect of dendritic size and topology on burst firing and the correlation of burst firing region with mean electrotonic path length are robust to changes in model properties, including morphology, strength of input stimulus, ion channel densities, and keeping the number of ion channels constant as morphology is changed. The specific range of dendritic sizes that supports burst firing, as well as the impact of dendritic topology, does not strongly depend on the strength of the input stimulus, especially with somatic stimulation (

In changing dendritic size or topology, we held the density of ion channels constant (i.e., the conductances were fixed), which implies that the total number of ion channels also changed when dendritic morphology was varied. Keeping the conductances fixed seems biologically the most appropriate choice, since removing membrane to shrink the dendritic tree (as well as adding membrane to enlarge it) will include the membrane's ion channels and is therefore not expected to affect ion channel density. But even if we hold the number of ion channels constant, by adjusting the values of the conductances as the surface area of the dendritic tree is changed when dendritic topology or total length is varied, we obtain surprisingly similar results (

Since recent studies have shown that the same firing patterns can be produced by different combinations of conductances

Since it has experimentally been shown that removal of the apical dendrite abolishes bursting in layer 5 pyramidal cells

Compared with other modeling studies investigating the relationship between dendritic morphology and firing pattern

We stimulated the cells either by a current injection at the soma, as is done in most experimental and modeling studies

Our study confirms a suggestion by Krichmar et al.

Our results are in accord with empirical observations suggesting that pyramidal cells should have reached a minimal size to be capable of burst firing. In weakly electric fish, the tendency of pyramidal cells to fire bursts is positively correlated with the size of the cell's apical dendritic tree

In addition, the developmental time course of bursting shows similarities with that of dendritic morphology. In rat sensorimotor cortex, the proportion of bursting pyramidal cells progressively increases from postnatal day 7 onwards, while at the same time the dendritic arborizations become more complex

Direct experimental testing of the influence of dendritic morphology on burst firing could be done by physically manipulating the shape or size of the dendritic tree, e.g., by using techniques developed by Bekkers and Häusser

Dendritic morphology can undergo significant alterations in many pathological conditions, including chronic stress

Chronic stress, as well as daily administration of corticosterone, induces extensive regression of pyramidal apical dendrites in hippocampus

With regard to epilepsy, a significant decrease in total dendritic length and number of branches has been found in pyramidal cells following neocortical kindling

In Alzheimer's disease, various aberrations in dendritic morphology have been observed— including a reduction in total dendritic length and number of dendritic branches

Since firing patterns characteristic of different classes of neurons may in part be determined by total dendritic length, we expect on the basis of our results that a neuron may try to keep its dendritic size within a restricted range in order to maintain functional performance. Indeed, Samsonovich and Ascoli

We predict that dendritic topology may similarly be protected from large variations. In fact, there could be a trade-off between dendritic size and dendritic topology. In a set of bursting pyramidal cells, we expect that apical dendritic trees with a lower degree of symmetry are shorter in terms of total dendritic length or have thicker dendrites to reduce electrotonic length than those with a higher degree of symmetry.

Although changes in dendrite morphology of pyramidal cells have been observed in response to environmental enrichment

An intriguing possibility is that firing pattern and dendritic morphology could mutually tune each other during development, as a result of a reciprocal influence between dendritic growth and neuronal activity. Dendritic morphology affects firing pattern, and neuronal activity in turn is known to modulate dendritic growth and branching

As our study underscores, differences in neuronal firing properties may not necessarily reflect differences in ion channel composition. In some cases, variability in dendritic morphology may even have a relatively bigger effect on firing pattern than variability in membrane conductances

The region of burst firing is relatively insensitive to the strength of somatic stimulation. For three different tree topologies of the morphologically simplified cells (see

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The region of burst firing is relatively insensitive to the strength of dendritic stimulation. For three different tree topologies of the morphologically simplified cells (see

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The mean electrotonic path length correlates with the region of burst firing also when the number of ion channels is kept constant as the topology or total length of the tree is changed. To implement a constant number of ion channels, we decreased (increased) the ion channel densities (i.e., maximal conductances, expressed in pS µm∧−2) as the total surface area of the dendritic tree increased (decreased). The total dendritic surface area of the fully symmetrical tree (topology 23) at dendritic length 2500 µm was thereby taken as reference. Thus, g_x new = g_x * (surface area of the symmetrical tree at 2500 µm)/(surface area of the tree under consideration), where g_x is the standard maximal conductance as given in

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The influence of dendritic size and topology on burst firing and the importance of mean electrotonic path length are robust to changes in ion channel densities. For a wide range of dendritic ion channel densities, the mean electrotonic path length correlates with the region of burst firing. The maximal conductance of Na is 90% of the standard value (see

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The influence of dendritic size and topology on burst firing and the importance of mean electrotonic path length are robust to changes in ion channel densities. For a wide range of dendritic ion channel densities, the mean electrotonic path length correlates with the region of burst firing. The maximal conductance of Na is 110% of the standard value (see

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The influence of dendritic size and topology on burst firing and the importance of mean electrotonic path length are robust to changes in ion channel densities. For a wide range of dendritic ion channel densities, the mean electrotonic path length correlates with the region of burst firing. The maximal conductance of Na is 90% of the standard value (see

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The influence of dendritic size and topology on burst firing and the importance of mean electrotonic path length are robust to changes in ion channel densities. For a wide range of dendritic ion channel densities, the mean electrotonic path length correlates with the region of burst firing. The maximal conductance of Na is 110% of the standard value (see

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Interspike-interval (ISI) distributions, together with burst measure values (B), in the experiment in which the total length of the pyramidal cell was gradually reduced by pruning the apical dendrite (see

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The burst measure and the derivation of its expected value for a periodic spike train with two-spike bursts.

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Burst firing described as a semi-Markov process and generalization of the burst measure to n-spike bursts

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^{+}channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons.