A discovery reported in last Friday's Science Magazine offers a new unifying principle for how pyramidal cells perform their function in the neocortex, as well as other cortical areas: Synaptic Integration in Tuft Dendrites of Layer 5 Pyramidal Neurons: A New Unifying Principle (by Matthew E. Larkum, Thomas Nevian, Maya Sandler, Alon Polsky, and Jackie Schiller), unfortunately behind a paywall. What this paper does is demonstrate that a type of neural activity called an NMDA spike (see below), already known to occur in the basal branches of the dendrites of pyramidal cells, also occur in the branches in the apical tuft, the arborization that occurs at the top of the apical dendrite, the main dendrite that reaches from the soma (body) up to the top layer (Layer I) where the majority of synaptic connections are made between incoming axons (from other brain regions) and pyramidal cells (which produce outgoing axons to other brain regions).
As of late 2007, "NMDA spikes have been observed in basal dendrites but not apical dendrites", but now:
We report the existence of N-methyl-D-aspartate (NMDA) spikes in the fine distal tuft dendrites that otherwise did not support the initiation of calcium spikes.  Both direct measurements and computer simulations showed that NMDA spikes are the dominant mechanism by which distal synaptic input leads to firing of the neuron and provide the substrate for complex parallel processing of top-down input arriving at the tuft.Before we look at what these NMDA spikes are, let's look briefly at the implications of this discovery:
Integrative Calculations in Pyramidal Neurons
Figure 1: Structures of selected pyramidal neurons from different cortical areas.  Click here to see full image and original caption.  (From Reference 4, figure 1..)
Figure 2: Simplified (!) diagram of the calculating modules according to the "New Unifying Principle" in Reference 1. Each box represents an integrative calculating unit, capable of performing a large variety of calculations with its inputs. The circles represent input information coming via synapses. Note that the logical organization here doesn't map exactly to the physical organization of the neuron: the non-linear voltage responses of the synapses, the locations of the synapses on the local branches, and the non-linear voltage responses of the dendritic membrane of the local branches should all be considered part of the calculation box. The input, then, consists of the flow of neurotransmitters across the synapse. The timing of the arrival of the action potential, and any calculations that take place in the pre-synaptic neuron, aren't included in this diagram. Click on image to see larger version. (Original. You may link to, copy, and/or modify this image, as long as you give credit with a link to this post.)
Looking at Figure 2, we can see that there are a number of modules, potentially nested, which perform semi-independent calculations.  They feed their results to modules progressively closer (and ultimately identical) to the soma, which (along with the axon hillock and the first 50-100 microns of the axon) performs the final calculation regarding whether, and when, to fire an action potential.  Prior to this research, the modules in the apical tuft (those feeding the proximate apical dendrite, see also figure 1) were regarded as different in kind from those in the basal branches. What this research has (tentatively) demonstrated is that the integrating modules called "Fine Dendrite Branches" in Figure 2 appear to act in very similar fashions, although they provide their outputs to different places.
The outputs of from the tuft branches feed the integrating/calculating module represented by the proximate apical dendrite, which performs a calculation that involves a process called a "calcium spike" (see below), which had been thought to be caused in the tuft branches, but with this research are (tentatively) shown not to be caused in the branches, but only in the proximate apical dendrite. This permits the "New Unifying Principle":
The thin distal tuft and basal dendrites of pyramidal neurons, which receive the overwhelming majority of synaptic inputs ([ref]), appear to constitute a class of dendrite in which NMDA spikes are the predominant regenerative events summing synaptic inputs in semi-independent compartments.  The output of each subunit in this class of dendrite is passed on to the major sites of integration at the axon and apical calcium initiation zones, which can all interact via actively propagated signals ([ref]), enabling the interactions between top-down and bottom-up information.This represents a major change to how pyramidal cells should be viewed, offering different pictures of their modular breakdown and evolution. ... We can reasonably suppose that the evolution of the neocortex involved, among other things, some improvements and refinements to the integrating calculation process of the apical calcium initiation zones in the proximate apical dendrite.  Pyramidal neurons "are abundant in the cerebral cortex of virtually every mammal that has ever been studied, as well as in those of birds, fish and reptiles, but not amphibians." They are:
found in most mammalian forebrain structures, including the cerebral cortex, the hippocampus and the amygdala, but not the olfactory bulb, the striatum, the midbrain, the hindbrain or the spinal cord.  Thus, they are found primarily in structures that are associated with advanced cognitive functions[.]We can reasonably suppose that the original version, developed either by early amniotes (and independently by fish), or much earlier in vertebrate evolution (and lost by ancestral amphibians), was only capable of supporting the three-layer cortical structure of reptiles and the allocortex of mammals.
Currents, Spikes, and Action Potentials
The functioning of a neuron is very complex, with a very large number of interacting "moving parts".  Any effort to simplify the picture to the point that it can be discussed in a blog post will inevitably lose critical details, as well as producing a picture that fails to apply to most types of these cells.  Nevertheless, we can look at a few general processes that are common to most (if not all) neurons.
The first thing we need to consider is the cell membrane, and various differences between the inside and outside.  The two critical differences for our purposes are the voltage across the membrane, and the different concentrations of certain critical ions: sodium (Na+), calcium (Ca+2), potassium (K+), and chloride (Cl-).  Both the concentration differences and the voltage tend to drive a movement of ions across the membrane, however they can't get through the membrane proper without the help of ion channels, of which every type of neuron is provided with a large number of many types.  In a typical resting state, Na+ is being "pulled" into the cell by both concentration difference and voltage, while K+ is closer to equilibrium, with a smaller "push" outwards because the resting voltage across the membrane isn't quite enough to balance the difference in concentrations.  Ca+2 is being "pulled" inwards even harder than Na+, while Cl- is a special case, in some neuron membranes at rest seeing a slight "pull" inwards, while in others there's a slight "push" outwards.  (I won't be covering Cl- in this post, this is a subject for a future post.) Each ion has an equilibrium voltage which will exactly balance the concentration difference.
Figure 3: Equilibrium Voltages and the Effect of Various Ion Currents on Membrane Potential.  Note that the equilibrium voltages (ENa+, ECa2+, EK+, and ECl-) are actually ranges, depending on the specific ion concentrations inside and out.  This is especially important in the case of Cl-, where variations in concentration can move the equilibrium voltage to either side of the Resting Membrane Potential.  (Based on Reference 24 Figure 2.2, p41.  You may link to, copy, and/or modify this image.)
Now, the actual voltage across the cell membrane will be determined by the relative concentrations of all (charged) ions, along with more transient effects created by various currents.  These currents are caused by the flow of charged ions through ion channels and ion pumps.  Ion pumps tend to run at a (roughly) constant rate, so their currents are roughly constant, balanced by typically small currents through ion channels that are slightly open at the resting voltage.  This balance creates the resting voltage, and can be changed by long-term changes to ion channels that occur e.g. between various states of consciousness.
It's very important to realize that the membrane voltage, and concentration differences across the membrane with their associated equilibrium voltages, can vary between cells, between different parts of the cell, and over time even for the same part of the cell.  A large number of factors go into determining the voltage, and even the concentration differences are subject to a number of interacting controls, not all of them part of the specific cell in question. 
Currents, both resting and transitory, are produced and controlled by a large number of ion channels, many of which have very complex and often time-dependent reactions to voltage, ion concentrations (including those that they don't pass, for instance there are two K+ currents, well enough known to be named, that are controlled by the internal Ca+2 concentration: IC and IAHP, produced and controlled by calcium activated ion channels), and the concentrations of a large variety of messenger molecules and neurotransmitters, both inside and outside the cell.
A spike is a sudden change in voltage caused by a large transitory current of some sort.  The best-known spike is the action potential, which is a regenerative positive-feedback current of sodium and/or calcium produced by voltage-gated channels for these ions.  We need to consider what the word "regenerative" means in this context: when the voltage across the membrane drops to and past zero at one point on the membrane, it causes the voltage nearby also to drop, which in turn activates voltage-gated channels in these nearby positions, causing the positive feedback process that creates the action potential.  (The voltage-gated channels usually shut after a short time (around a millisecond (ms)), especially the sodium channels that mediate the action potential in the axon, the best known.) This is the "regeneration": one process causes the voltage to drop at a specific point on the membrane, and when it passes a threshold a new, voltage-dependent process regenerates a further voltage drop, amplifying the initial impulse.
There are other types of spikes: localized non-propagating Na+ and Ca2+ regenerative currents that amplify local voltage changes caused by other currents.  Sodium and Calcium are the principle spiking ions, because of the strong forces pulling them into the cell.  In principle, K+ could also cause a spike, with the voltage change going in the other direction, but this doesn't seem to happen in nature.  However, there are many types of spikes where channels open for both Na+ and K+, going in opposite directions and creating currents opposed to each other, where the much larger driving force on Na+ overwhelms the K+ force, producing a net inwards current.
We need, now, to take a closer look at the voltages involved.  As you can see from figure 3, membrane voltage at any point can be moved by changes to any of the various currents.  When a burst of neurotransmitters from an action potential arrives at a synapse, they will have various effects on the ion channels that are present, causing various currents.  The effect of the currents is to either depolarize or hyperpolarize the membrane, bringing its voltage closer or farther away from the action potential threshold.
Figure 4: Action Potential Threshold and Directions of Depolarization and Hyperpolarization Relative to Figure 3.  (Based on Reference 24 Figure 2.2, p41.  You may link to, copy, and/or modify this image.)
Now, in an action potential, voltage-gated Na+ and/or Ca2+ channels open when the voltage falls below their threshold, causing a spike where the voltage continues to fall.  Because there is a large number of such channels present throughout the local membrane, the spike at one point drags down the voltage at nearby points, causing the channels there to open.  Thus the spike propagates itself at nearby points on the membrane, and moves away from its initiation point along the axon or back from the soma along the major dendrites, if they possess the right mix of voltage-gated channels.
Suppose, however, that there's just a large concentration of a certain type of voltage-gated channel at one point.  In this case, the current created by ion channels responding to a burst of neurotransmitters (from an action potential in another neuron) might bring the voltage to the threshold, causing the concentration of channels to open and amplify the original current with a regenerative current of their own.
There is a constant rain of these bursts of neurotransmitters on various synapses at various locations on the dendritic arbor.  Each creates currents, some depolarizing, some hyperpolarizing, and the effects spread from the synapse along the cell membrane.  The original current fades as the neurotransmitters are taken up and their effects pass, meanwhile the voltage at the synapse decays as the charges transferred spread out along the membrane.  Other synapses also see a voltage bump, with size and timing depending on their distance along the dendrite from the location of the arriving neurotransmitter burst, as well as the conduction qualities of the intervening membrane.
If an exitory (depolarizing) action potential arrives at a second (nearby) synapse while it is already somewhat depolarized from the earlier exitory action potential at the first, the current caused by the burst will start at a lower (more depolarized) voltage and thus drag the voltage at that point lower than it would if the membrane was at normal resting potential.  If this lower voltage crosses the threshold for some concentration of a certain type of voltage-gated channels, they will open, creating a spike that wouldn't have happened in response to either action potential by itself.
In the dendrites, calcium spikes seem to be more common than sodium, although given its higher driving force channels that allow both through will probably show a bigger calcium effect than sodium.  Calcium is useful in the dendrites, because the recovery from these spikes requires considerable energy and the elevated Ca+2 levels inside the cell stimulate the mitochondria to start producing more energy immediately, rather than waiting for the energy use to draw down the ATP/ADP ratio (as I've discussed previously).
The effects of all these various currents, as spread while decaying to the soma, axon hillock, and first part of the axon, determines whether and when an action potential will fire.  In order for that to happen, the voltage at the critical point (depending on the cell type) must fall below the threshold as a result of the combination of the effects of the various exitory (depolarizing) and inhibitory (hyperpolarizing) currents at various synapses.  Large spikes in the dendrites can have a disproportionately large effect here, touching off action potentials when the combination of the original currents would never have caused this.
Types of Spikes in Pyramidal Cell Dendrites
There are two major types of spikes that participate in dendritic calculation: calcium spikes and NMDA spikes.  Calcium spikes often propagate in a regenerative process similar to the action potential, at least in the apical dendrite (see Figure 1).  They carry information generated in the apical tuft to the soma, where they can participate in the final decision whether/when to fire an action potential.  They are similar to sodium-driven action potentials, in that when a certain threshold depolarization is reached voltage-driven calcium channels open, causing a positive feedback.  The typical recovery is mediated by calcium activated K+ channels which tend to be somewhat slower than those involved in the Na+-based action potential seen in the axon.  Once these channels open, the K+ current tends to repolarize and hyperpolarize the local membrane before they close, because unlike the voltage-gated K+ channels involved in the classic action potential they don't respond to changes in voltage, but stay open until the uptake of Ca2+ by the mitochondria is complete.
NMDA spikes are different.  NMDA receptors are for glutamate, the most common exitory neurotransmitter in the brain.  It is not the only one, the major fast response to glutamate from an action potential is mediated by AMPA receptors.  AMPA receptors are pretty much voltage independent: they translate a certain sized burst of glutamate into a specific current of sodium and perhaps calcium (depending on the type).  NMDA receptors, OTOH, are both voltage sensitive and in need of some quantities of glycine, an otherwise inhibitory neurotransmitter released by other types of cells in the brain.  NMDA receptors are also subject to modulation by a wide variety of different substances, which can modify their response to glutamate as well as the threshold voltage for opening.
This threshold is caused by the tendency of extracellular Mg2+ to block the pore through which ions travel at higher (more polarized) voltages.  Thus a burst of glutamate can arrive at a synapse and cause only a small (or non-existent) current from AMPA receptors, while a much larger number of NMDA receptors stay silent, or perhaps only join in when the voltage is at its peak of depolarization.  However, if the synapse is already depolarized somewhat from other effects, including currents from action potentials at other synapses, nearby spikes (calcium and NMDA), and back-propagating action potentials form the soma, the arrival of the burst of glutamate could instead cause a much larger current: an NMDA spike. (We should note that NMDA receptors are pretty non-specific regarding which positive ions they will pass: Na+, K+, and Ca2+ will all have currents.)
These spikes tend to support very localized calculations.  Inhibitory synapses, using either GABA or glycine, tend to produce Cl- currents, which in the dendrites are usually hyperpolarizing and also tend to resist the depolarizing effect of other exitory effects in their immediate vicinity.  These synapses, then, have a much greater effect in suppressing NMDA spikes in nearby exitory (depolarizing) synapses than their effect on the final voltage as felt at the soma.  The localized interactions tend to convert the fine dendritic branches of the basal dendrites into semi-independent calculating modules, and the implication of the paper being covered here is that the same is true of the fine dendritic branches of the apical tuft.  Thus, the exact same mechanism is used in both types of calculating module. We should also note that the ability of so many substances to modulate the threshold voltage of the NMDA receptor means that chemically mediated information is input directly into the most basic and fundamental system of dendritic calculation in the neocortex, as well as the rest of the "cerebral cortex, the hippocampus and the amygdala [...] primarily in structures that are associated with advanced cognitive functions". We should also note that these local calculations aren't limited to the dendrites of the cell ultimately firing the action potential: the immediate chemical environment of both the NMDA receptors in the synapse and the various voltage-gated channels in the local membrane between them is subject to influences with sub-second timing from local membranes of many other nearby cells, both neurons and glia, as I've discussed before.
Contra the previous understanding, calcium spikes in the studied pyramidal neurons appear to be limited to the proximate apical dendrite (see Figures 1&2), where "[s]everal properties, including the existence of local NMDA spikes and the weak electrogenesis of both sodium and calcium spikes, suggest that the relationship of the fine distal tuft branches to the apical Ca2+ initiation zone is similar to the relationship of the basal dendrites to the axosomatic initiation zone". In Figure 2, then, we can see that the proximate apical dendrite serves as a "smart" relay, integrating the calculations made in the distal branches of the apical tuft and passing along the result to the soma (and axon hillock and early axon) where it is integrated with the results of calculations made (using the same or similar process) in the distal branches of the basal dendrites.
While there appear to be some minor differences between the apical and basal mechanisms, such as the "presence of hyperpolarization-activated current (Ih)" in apical dendrites, these can probably be regarded as "fine-tuning" of the working of the integration process, perhaps to adapt it to the different functions of the apical and basal dendrites.  Overall, the similarity of function presented in this paper supports a much more unified paradigm for how calculations in pyramidal cells actually work.
Larkum, M., Nevian, T., Sandler, M., Polsky, A., & Schiller, J. (2009). Synaptic Integration in Tuft Dendrites of Layer 5 Pyramidal Neurons: A New Unifying Principle Science, 325 (5941), 756-760 DOI: 10.1126/science.1171958
Links: I've included only the links called out in this leader.Only a few of the links here are called out in the text. Far too many are behind paywalls; I've mentioned where links are formally open access. Use the back key if you came by clicking a footnote.
1. Synaptic Integration in Tuft Dendrites of Layer 5 Pyramidal Neurons: A New Unifying Principle paywall
2. In vivo two-photon voltage-sensitive dye imaging reveals top-down control of cortical layers 1 and 2 during wakefulness
3. Thalamic Input to Distal Apical Dendrites in Neocortical Layer 1 Is Massive and Highly Convergent paywall
4. Pyramidal neurons: dendritic structure and synaptic integration
5. Properties of basal dendrites of layer 5 pyramidal neurons: a direct patch-clamp recording study
6. Calcium electrogenesis in distal apical dendrites of layer 5 pyramidal cells at a critical frequency of back-propagating action potentials Open Access
7. Computational subunits in thin dendrites of pyramidal cells
8. Ca2+ accumulations in dendrites of neocortical pyramidal neurons: An apical band and evidence for two functional compartments paywall
9. Signaling of Layer 1 and Whisker-Evoked Ca2+ and Na+ Action Potentials in Distal and Terminal Dendrites of Rat Neocortical Pyramidal Neurons In Vitro and In Vivo Open Access
10. Apical tuft input efficacy in layer 5 pyramidal cells from rat visual cortex Open Access
11. Role of dendritic synapse location in the control of action potential output
12. The Properties and Implications of NMDA Spikes in Neocortical Pyramidal Cells Open Access
13. Spatiotemporally Graded NMDA Spike/Plateau Potentials in Basal Dendrites of Neocortical Pyramidal Neurons Open Access
14. Astrocytic control of synaptic NMDA receptors Open Access
15. NMDA receptor-mediated dendritic spikes and coincident signal amplification paywall
16. Diversity and Dynamics of Dendritic Signaling
17. Dendritic arithmetic
18. Polarized and compartment-dependent distribution of HCN1 in pyramidal cell dendrites
19. Dendritic Computation
20. Integrative Properties of Radial Oblique Dendrites in Hippocampal CA1 Pyramidal Neurons Open Access
21. Dendrites: bug or feature?
22. Excitatory effect of GABAergic axo-axonic cells in cortical microcircuits requires free registration
23. Cation–chloride co-transporters in neuronal communication, development and trauma paywall
24. The synaptic organization of the brain Edited by Gordon M. Shepherd