Evolutionary developmental biology meets the brain: The origins of the mammalian cortex

Karten, HJ. (1997)  Evolutionary developmental biology meets the brain: The origins of mammalian cortex. PNAS 94: 2800-2804.


Harvey Karten is a prof at UCSD. He's retired now, but still hangs around.

"Cortical Equivalent" Circuits in the Nonmammalian Forebrain. Suggests that nuclei in retiles/birds in DVR are equivalent to lamina in cortex.

Two steps. 1. neurons of each sensory system evolved for all vertabrates. 2. lamination of these populations occured within mammals.

The development of DVR and neocortex should thus have similar mechanisms that have been transformed through evolution. The prosomeres (the developmental zones) around the ventricular zone should then be "found in either of two configurations: (i) in birds and reptiles, reflecting the ancestral condition common to allamniotes, they contribute to the DVR... (ii) in mammals, these prosomeres are tranposed and become components of the "subventricular zones"... as areas of proliferation seperate of epndymal zone"

Radial progenitor hypothesis poses a problem: cortex is organized into columns, stemming from a single zone. Layers are secondary from this vertical migration. This clashes with idea that layers are coming from different progenitor zones.

I think what the Molnar paper will say is that the SVC does not arise from a different progenitor zone, but instead comes from IPCs originating in the same place. Things then are almost all radially distributed.

Evolution of brain development

Molnar, Z. (2010) Evolution of brain Development. Phenotype 7: 8-10.


Two major hypotheses exist to explain the increase in mammalian cortical neurons.
Karten: additional neurons are generated outside neocortex, then migrate and integrate into neocortex. Equivalent circuit hypothesis: DVR in avian and reptillian brains establish equivalent circuits as those responsible for sensory pathways in mammalian neocortex. (Review this paper further).

Cheung, Molnar: additional site is present within mammals which is responsible for production of extra cortical neurons. Assumes direct homology between reptilian dorsal cortex and mammalian neocortex.

Ventricular zone is proliferative layer of progenitor cells (radial glial cells). Early development these cells divide symmetrically producing more neuroprogenitors. Mice radial glial cells can divide asymmetrically creating a neuron or an intermediate progenitor cell (IPC). The IPC migrates to an adjacent zone - the subventricular zone, where they divide to make more IPCs or neurons. The SVZ is not present in reptiles, and only present in a specific region of avian brains. This gives more credence to the Molnar hypothesis.

The "Independent Components" of Natural Scenes are Edge Filters

Bell, AJ. Sejnowski, TJ. (1997) The "Independent Components" of Natural Scenes are Edge Filters. Vision Research 37 (23): 3327-3338.

Coding prinicipals of edge detectors in V1 - Barlow: redundancy reduction. Each feature detector is supposed to be as "statistically independent" from others as possible.

Difference between "decorrelated" filters (i.e. PCA) and "indpendent" features (i.e. ICA). ICA is equivalent to redundancy reduction problem. There are many solutions to decorrelations problem - PCA is the orthogonal solution. With stationary image statistics (pictures, not videos) the PCA filters are global Fourier filters. ZCA is another decorrelation metric, but the "polar" opposite of PCA. PCA is order according to the amplitude spectrum, and ZCA is ordered according to the phase spectrum.

ICA has a stricter requirement - that the outputs are not just decorrelated, but also statistically independent. ICA is semi-local. ICA can't be directly calculated. Use algorithm called "infomax" - maximize by stochastic gradient ascent the joint entropy.

Independence vs. sparseness. There are slight differences, but both lead to similar results. Sparseness is defined by the kurtosis of the filters. High kurtosis means a peakier middle and bigger tails - a normal distribution has kurtosis of 3 (although some people subtract 3 from kurtosis and say normal is 0, but then you could have negative kurtosis). 

Classical hebbian-style learning is correlation based, and these types of learning rules would amount the decorrelation style filter sets. Learning rules for independence should be studied. 

Hubel & Wiesel

Lets talk about the visual system, in particular what we know about visual cortex. One of the big nobel-prize winning works were by Hubel and Weisel where they stuck an extracellular electrode in V1 and made a cat watch a shiny bar. They would hold the bar in different orientations and locations and the bar could be resized. This video explains everything:

http://www.youtube.com/watch?v=8VdFf3egwfg

So beautiful.

Anyway, what they discovered was that cortical neurons have receptive fields that are like oriented bars. And different neurons would have different orientations. They also clearly show that they have on-centers and off-surrounds. I this type of on-off was known at the time as the retinal cells have circular on-off receptive fields. Here's how they thought it worked:

There's been so much work on what these receptive fields mean, and everyone tries to explain them to explain what cortex is doing.

SAILnet

I was at OCNS this weekend. There was one poster that I though was pretty interesting about sparse coding. It hasn't been published, yet, but it was really just an extension of this paper:

Zylberberg, J. Murphy, JT. DeWeese, MR. (2011) A Sparse Coding Model with Synaptically Local Plasticity and Spiking Neurons Can Account for the Diverse Shapes of V1 Simple Cell Receptive Fields. PLoS Computational Biology 7(10): e1002250.

The main update that they had made was that they seperated the excitatory and inhibitory populations, in this paper their neurons are allowed to inhibit and excite each other. There were some interesting ideas for learning rules between E->I, I->E, and I->I synapses, which could be of use later.

Notes:
The literature has shown that V1 is representing visual space through these gabor-like receptive fields of the principal neurons. I'll have to do some further reviews. Oshausen and Fields showed that by minimizing the neural activity and maintaining a full representation of the input, you could learn similar receptive fields from natural images. This is sparse coding - minimal representation of the input. There's been a ton of other work trying to reproduce V1 receptive fields, another paper to review later is one where they show that ICA (indepentent components analysis) of images also generates similar looking receptive fields.

This paper's primary contribution is to translate the sparse coding algorithms into rules for spiking neural networks that have local plasticity. Local plasticity means that there are no global learning rules - like you can't normalize all of your synapses based on the synaptic weights elsewhere in the population. A lot of computational models fail to be realistic because they use these non-local learning rules, which are physically impossible for a brain to be implementing.

Leaky integrate and fire. Foldiak learning rule - units are active for a small but non-zeros fraction of time and maintain uncorrelated activity with respect to other units:

Where W is the inhibitory weight between neurons, Q is the excitatory weights coming from the inputs, theta is the threshold of each neuron, p is the target firing rate for each neuron, and alpha, beta, and gamma are learning rates.

Here are the receptive fields that it learns, it looks pretty good compared to the standard receptive fields:

So, sparsity is interesting, and the poster I saw was basically a slight modification of those learning rules to seperate out excitatory and  inhibitory populations, but the ideas were about the same.

Figure 2 of this paper shows a reconstructed stimulus from the activity of the neurons in the network. It does a fairly good job, but the idea is that the network has come up with a complete representation. This means that it should be able to reconstruct any stimulus with the features that the network has learned. The problem/extension of models like this is that there is nothing that pushes the network to have a fully complete representation. All the neurons could learn roughly independent features, but there is nothing that is forcing the network to build a complete feature set. A new way of designing these networks such that they do build a full feature-set would be a nice step forward.

NMDA Receptor - Calcium Spikes

Once the NMDA receptor allows for there to be a large amount of Calcium influx into the cell, a protein cascade is triggered that potentiates the synapse. It is thought that there are two calcium thresholds - the upper threshold is for potentiation (the synapse gets stronger), but STDP has both potentiation and depression. There is a lower calcium threshold that triggers depression - this happens when the NMDA receptor doesn't quite flux enough calcium and other calcium channels (i.e. VGCCs) let in a small amount - if the post-synaptic AP is before the pre-synaptic AP then a slight amount of Ca gets through triggering the depression cascade.

The calcium spikes triggered ala Larkum would then be signals for potentiation. An actual Calcium spike would be much more robust as a learning signal for LTP than just a threshold. A calcium spike would be consistent and also would send a signal forward to the rest of the neuron. This would be a much more reliable situation for implementing STDP. Plus the calcium spike would more reliably trigger action-potentials.

An extra possibility of the Calcium spike is that it could be communicated to the next neuron. In cortex it is known that neurons can get in a bursting mode. It is thought that these bursts are action-potentials that are riding on-top of a calcium spike (calcium-spikes are typically longer in duration than action-potentials). What's also interesting is that bursting can influence synaptic release - a burst is typically a much more reliable signal for synaptic release than a single action-potential. It may be possible that this burst could signal to the next neuron that some learning was happening - possibly as a back-propagation signal.

NMDA Receptor - STDP mechanism

I've talked a lot about STDP and also calcium spikes - Larkum calls them NMDA spikes. I thought I'd dive into a little more detail about NMDA receptors - how they work, how they may be signaling, and how they are involved in learning.

The peculiar thing about STDP is that it almost completely reverses in a manner of milliseconds. If the action-potential is not precisely timed relative to the release of glutamate at the synapse, then STDP can completely reverse. People think that the mechanism of this is the NMDA receptor. What makes the NMDA receptor uniquely suitable as a mechanism for STDP is that it can be blocked by Magnesium. NMDA is primarily a Ca+2 transmitter, and Mg+2 - with the same charge, is pretty similar to Ca. However, Mg cannot fit through the NMDA receptor and can get stuck. If it gets stuck in the receptor it blocks any Calcium from getting through.

There is a special circumstance where the Mg block is removed. It all has to do with driving forces and electro-chemical gradients. The simplest way of understanding how ions will move across the cell membrane is to consider the voltage of the cell and the charges of the ions - if the cell is negatively charged it will attract positively charged ions. But there is another aspect: the concentration gradient. Ions will move to also equilibrate their relative concentrations. Calcium is a very special ion. The neuron has channels and calcium stores that pump out or suck up all the calcium floating in the intracellular space. There is basically no calcium in the cell. Terry Sejnowski said that they've calculated that there probably is only a single free calcium atom floating in a synaptic bouton. So calcium has a strong chemical gradient, and will still move into the cell, even if the cell is positively charged. Magnesium, however, is not pumped out, and its movement across the membrane is dominated by the electrical gradient - Mg mainly follows the electrical gradient. 

So there is a range where the cell can be at the right voltage such that Calcium is being driven into the cell, while magnesium is driven out of the cell. This period is when the NMDA receptor will flux calcium into the cell. The NMDA receptor is also a glutamate receptor. For it to conduct calcium the pre-synaptic cell must release glutamate just before the post-synaptic cell depolarizes - like from an action-potential. This sets up a precise temporal window where NMDA receptors can flux calcium, and it is thought that this triggers the cascade leading to potentiation of the synapse.

Inverse and Forward Learning

So a major aspect of cortex will be inverse and forward learning. This idea was inspired by some birdsong work, but I can't find the papers.

The big question is how does the brain do something like one-shot learning? Birds can hear something and then produce almost the identical output, but how?

The answer is that through feedback birds can learn how their motor commands map onto auditory commands and vice-versa. The idea is that young birds are babbling - randomly searching through their motor outputs. The babbles are a direct result of certain motor neurons firing. The action-potentials of these neurons are sent via efferent copy to the sensory areas. The sensory neurons at the same time receive the auditory feedback of the actual sound produced by the babble, these spikes are also sent to the motor neurons.

So with the feeback, both directions of learning can take place. The sensory neuron that responded to a particular sound can learn which motor neurons make that sound. The motor neurons that produce a particular sound can learn which sensory neurons hear that sound. So, when a bird hears a sound, the bird has already learned how sounds map onto the motor-neurons. It would then be pretty easy to activate the same motor-neurons to produce a similar sound.

The thing is, you don't even need the actual sound to do this. A spike from thalamus to cortex will activate a bunch of cortical neurons who send their spikes back. So thalamus can learn which neurons it activates. Similary a spike from cortex will activate thalamus and send spikes back. So cortex can learn its mapping back to thalamus. During the wakefulnesss, thalamus is being set by the environment - so cortex can learn features of the environment. During sleep, cortex can learn its features by activating thalamus and seeing which neurons turn on. This is essentially generative modeling, as cortex is learning what its features are in thalamus. Then you can extend this up the hierarchy.

Pattern separation, pattern completion

Leutgeb, S. Leutgeb, JK. (2007) Pattern separation, pattern completion, and new neuronal codes within a continuous CA3 map. Learning & Memory 14: 745-757.


Hippocampal CA3 cells are predominantly connected to themselves. Only 1/3rd comes from elsewhere.

"CA3 cell ensemble characteristics are consistent with completely distinguishing between two places and with making an additional distinction between related sensory configurations within each place. Although each of these two modes of processing corresponds roughly to pattern separation, they do not necessarily result in global attractor dynamics in CA3."


Locally continuous, but globally orthogonal representation.


CA1 and CA3 are primarily characterized by place-cell firing - principal cells are activated when the animal is in specific locations in the world. It forms a map of space.


Maps made by hippocampus can change depending on context - changing a black cylinder to a white one can completely change the hippocampus map (Muller and Kubie, 1987). Introducing a barrier - cells close to the barrier would remap, but distal cells would stay the same.


Visually identical enclosures: CA1 will fire similarly to similar sensory cues. CA3 will always respond differently if the location is different, even if other sensory cues are identical. CA3 has large path-integration component.

CA3 pattern completes. Spatial map can remain stable, even when some of the sensory cues are removed. So long as location remains unambiguous the CA3 map can remain stable. Recurrent activity and plasticity are necessary for this mapping. CA3 is more bound to the current sensory state, and CA1 more reliant on temporal sequence of sensory information. If sensory cues change dramatically, or the animal is moved to an entirely different location, CA3 will form a new, orthogonal map. 

TD-Gammon

Tesauro, G. (1994) TD-Gammon, A Self-Teaching Backgammon Program, Achieves Master-Level Play. Neural Computation 6, 215-219.

http://www.research.ibm.com/massive/tdl.html

Tesauro does some interesting work in teaching a neural network how to play backgammon. The basis is that he uses a multilayer perceptron network in conjunction with reinforcement learning. The idea is pretty simple. The neural network is learning a value function - it translates a board state into a prediction of reward. At the end of the game the network is given a reward, and through practice can learn which board states are good and which are bad.

The reason I'm thinking about the RL literature is that making a spiking neural network learn a value function would be extremely useful and something that we could work on. This was actually one of my projects for my first rotation. The neural network part is basically to just learn the value function, and the temporal difference algorithm is put around it. I was trying to mimic the TD-gammon idea, but with spiking neurons that are recurrently connected. And instead of backgammon my neural network was playing tic-tac-toe. TTT is still quite non-linear, so it is interesting, and it's nice because we know that there is an optimal solution. Once the neural network converges on the optimal solution we can say that it has learned and is done.

I think it would be worthwhile to make something similar. The main addition would be to have a gamma clock. There were two main problems with the network model - it became unstable quite easily (either all the weights would explode and it would go into seizure, or all the weights would dwindle to 0), and there was no synchronizing (so spike timings would drift over time, and the info spike-timing carried would be lost).

A Neural Substrate of Prediction and Reward

Schultz, W. Dayan, P. Montague, PR. (1997) A Neural Substrate of Prediction and Reward. Science 276, 1593.

Reinforcement learning is one of the primary ways that the brain interacts with its environment to learn how to behave. There is plenty of theoretical work in reinforcement learning. One case is called temporal difference learning - where a value function is learned by translating reward signals backwards in time. Initially you are randomly making state changes (choices) and you stumble upon a reward. This reward signals to reinforce the previous states, making it more likely for you to enter the same states. This can be used to learn a variety of value functions.

The key to temporal difference learning is a signal that reports an error in the prediction of reward.

Dopamine neurons in the VTA have been shown to have the properties consistent with a temporal difference learning signal.

Two types of circuits?

When you compare three-layer and six-layer cortex, it appears that there are two types of cortical microcircuits. What's interesting is that hippocampus and olfactory (piriform) cortex have two spatially seperated types of circuits, while neocortex has the two circuits stacked on top of each other.

The first type of circuit is what you see in CA3, anterior piriform cortex and the superficial layers of neocortex. The pyramidal neurons in these areas are characterized by massive recurrent connections. They receive the primary afferent inputs. It is widely believed that these recurrent circuits are "pattern completion" circuits. They learn associations and can recover the association given only partial input - like hopfield nets.

The second type of circuit is what you see in CA1, posterior piriform cortex, and the deep layers of neocortex. Here the input is mainly from the pattern completion circuit. It is thought that these circuits are then "pattern separation" - finding new features that distinguish the current state from other states. This allow for new associations to be learned.

The microcircuit concept applied to cortical evolution

Shepherd, GM. (2011) The microcircuit concept applied to cortical evolution: from three-layer to six-layer cortex. Frontiers in Neuroanatomy 5 (30).


Notes
Fish, amphibians reptiles have 3-layer cortex, mammals developed neocortex with 6-layers. Mammals have hippocampus and olfactory cortex which are still 3-layer - left relatively untouched by evolution.

Early work provided evidence that 3-layer contains basic microcircuit core that has been elaborated in 6-layer.

Table 2: Summary of elements common to cortical circuits (Shepherd, 1988).

1. Both three-layer and six-layer cortex are built on pyramidal cells with apical and basal dendritic trees.
2. Strong excitatory afferents are received in the spines of the branches of the apical and basal dendrites.
3. The spines and local branches create local sites with varying degrees of local information processing properties.
4. The spines, local branches, and main stems contain different combinations of Na, K, and Ca ionic channels, which create local sites of integration and boosting of input signals to reach the sites of action potential output in the cell body and axon hillock.
5. The pyramidal cells have well developed recurrent axon collaterals.
6. The axon collaterals give rise to two main types of intrinsic circuit.
7. One type is direct feedback and lateral recurrent excitation.
8. This excitatory recurrent system has long lateral extensions, which enable widespread recombination of intrinsic excitation with the excitatory afferent input.
9. The other type is feedback and lateral inhibition through inhibitory interneurons.
10. Inhibitory interneurons are subdivided into multiple types which target different sites and levels of the soma-dendritic extent of the pyramidal cells.
11. Cortical information processing therefore involves a continual balance between excitatory and inhibitory circuits.
12. In three-layer cortex, these intrinsic circuits are organized around a single layer of pyramidal and pyramidal like neurons.
13. Five- and six-layer neocortex appears an expansion of the three-layer microcircuit into closely integrated superficial and deep layers.
14. Cortico-cortical afferents make synapses at different levels of the pyramidal cell soma-dendirtic axis to excite, inhibit, or modulate the transfer of synaptic inputs and extent of backpropagating action  potentials in the dendritic trees.
15. Brainstem systems provide differential modulation in different layers.

"... there is an underlying logic to the construction of cortical local circuits and microcircuits that provides a common modular framework that is adapted to generate the special properties of the neocortex."

Cortical circuits can be seen to be poised on the knife edge of exciation restrained by inhbition.

In 3-layer inputs make contacts on apical branches. In neocortex inputs come up through the layers and can make contacts to any layer. 

Olfactory cortex acts as "content-addressable memory" (Haberly 1985): each site in system contains information about the entire input. "Odor images" (Xu et al 2000).  Cortex processes spatial maps - similarities between olfactory (odor images) and hippocampus (2-D spatial location). 

Superficial pyramidal cells in neocortex (L2/3) are massively recurrently connected - winner take-all/pattern completion. Deep layers have little recurrent connections, input dominated by superficial pyramids. 

Most turtle pyramids are characterized by multiple apical dendrites and no basal dendrites. Pyramids in mammal olfactory are of 2 varieties: one has multiple apical trees and no basal, called "semilunar cell". Superficial pyramids in mammals also only have apical trees.

Broad calcium spikes not seen in turtle pyramids, but seen in neocortical pyramids. These can lead to burst-mode - bursting results in combination of feed-forward basal tree activation and feed-back apacial tuft activation. Different type of signal - more confident, more potent. Could be learning signal, used for back-prop. 

Lots of stuff about the dendrites doing processing. Review more of Larkum's work.

Electrically coupled inhibitory neurons

Gibson, JR. Beierlein, M. Connors, BW. (1999) Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402: 75-79.


This is a fantastic paper showing how there are two electrically connected inhibitory circuits in the brain. One of the networks is made up of the FS cells, which are thought to be responsible for gamma. These cells are acting like a clock, and keep the pyramidal cells synchronized. The pryamidal cells will mainly only fire during the troughs of the inhibition.

Here you can see an experiment showing two cells coupled together. What's amazing is how precise the action potentials align. In the cross-correlogram the neurons have a 2ms synchrony. So the spikes align very well - especially with regard to a 25ms oscillation. 

This table shows the synaptic connectivity of a bunch of different cell types. For the FS cells 24/39 pairs have electrical synapses, so the whole network has strong coupling. These cells have chemical connections to RS (pyramidal) cells.

Spike-Timing Dependent Plasticity

Markram, H. Lubke, J. Frotscher, M. Sakmann, B. (1997) Regulation of Synaptic Efficacy by Coincidence of Postsynaptic APs and EPSPs.  Science 275, 213.

Bi, G. Poo, M. (1998) Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type. The Journal of Neuroscience 18(24): 10464-10472.


These were the first two papers that showed STDP experimentally. STDP is a learning rule that is dependent on the precise temporal order of spiking in the pre- and post-synaptic cells. Bi, Poo show it nicely:

Basically, if the spikes are temporally causal (pre before post) then the synapse will get stronger. If they are acausal (pre after post) they get weaker. Here's how Markram showed it:

There's extensive work on the mechanisms of STDP. The primary mechanism is an NMDA channel, I'll explain that in more detail later. The signal is based on Calcium - excess calcium causes a chemical cascade that recruits or removes AMPA receptors to the synapse.

These papers naively think about the neuron as a single compartment. There is much more work about how STDP is really dependent on the dendritic spike - this is a big calcium signal. Eliciting an action-potential (or burst which is sometimes necessary to get this effect), will influence the dendrites and make a dendritic spike more likely.  So the learning rule is not really based on pre-post spikes, but whether the dendrite spikes.

There are a lot of other factors that can modulate this learning rule - frequency, synaptic strength, chemical modulators. Dopamine is an especially interesting one as it has been implicated in reinforcement learning and has been shown to be able to modulate STDP.

Microcircuitry of the Neocortex

Markram, H. (2010) Microcircuitry of the Neocortex. Handbook of Brain Microcircuits. 

Markram is a really famous neuroscientist - he was one of the first people to identify spike-timing dependent plasticity. I'll talk about STDP in more detail later. Now he is working on the Blue Brain Project - which is supported by IBM, and one of the more promising groups that are out there trying to build a brain.

Notes:
Neocortex is six layer sheet of neurons. Has a functional module called a cortical column, which has diameter approximately the size of basal dendrites of L5 pyramdial neurons.

86% of synapses in column exitatory, 14% inhibitory. Excitatory synapses - 1/3 from local neurons, 1/3 from neighboring columns, 1/3 from other distal regions (cortical, opposite hemisphere, subcortical). Most inhibition is local.

Principal Neurons:

• LII/III pyramids.
• LIV - two types, classical and star. In sensory areas also spiny stellate cells (targeted by thalamus).
• LV - thin untufted: project to opposite hemisphere. Thick tufted: project subcortically.
• LVI - at least 4 types depending on projection: cortico-cortico, cortico-thalamic, cortico-callosal, cortical-claustral.
• Connectivity directional tendency within column: 4 -> 2/3 -> infragranular (5 and 6).

Interneurons:

• Tons of possible interneurons. Morphologically:
• 4 types in L1
• 9 types in L2-6: large basket, nest basket, small basket, bitufted, bipolar, neurogliaform, Martinotti, Double bouqet, chandelier.
• Each anatomical type can have different electrical behaviors. Combinatorically there could be 200 types of interneurons, also including layer differences.
• Can also be subclassified according to molecular expression.

So, you can see how incredibly complex neocortex is. One of the big mysteries is why are there so many interneurons. There's clearly some combinatorics, but we should try and think of what the different functions of all of these interneurons could be and unify them into a single theory. Neocortex just seems too complicated to make sense of the microcircuitry and figure out what kind of computation it is doing. But we know that neocortex evolved from 3-layer cortex, and so maybe we can get some insights by studying 3-layer cortex more closely.

Larkum - 2009 - Synaptic Integration in Tuft Dendrites of Layer 5 Pyramidal Neurons: A New Unifying Principle

Larkum, ME. 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, 756

The distal tuft dendrites of pyramidal neurons has been a mystery. Part of the problem is that the apical tuft is so far away and electrically uncoupled to the axon, that it is unclear how synapses here influence the axon. The key is that the dendrites are performing computation and signal the axon via their own regenerative mechanism - dendritic spiking.

The primary mediator of dendritic spiking are Calcium spikes - which are generated by NMDA receptors and Voltage-Gated Calcium Channels (VGCCs).

This paper sets up the primary idea that the different parts of the dendritic tree are acting differently and interacting. The basal tree, which branches in layer 4, is going to be primarily connected with the feed-forward inputs - thalamus, for instance, has axons that branch mainly in L4. The apical tree which branches in L1 and L2/3 is getting inputs from feed-back sources - higher cortical areas.

There is extensive evidence on the role of calcium in long-term synaptic changes. There is a ton of potential in how these calcium spikes in the apical tuft interact with the sodium spikes of action-potentials as well as the calcium spikes generated in the basal tuft. The general idea is that the apical tuft is receiving feed-back signals that are trying to predict the feed-forward signals coming in through the basal tree. If certain synapses do well at predicting the inputs then these synapses will get stronger - this could be signaled by a calcium spike in the apical tuft simultaneous or preceding an action potential.

G and H summarize it best. The apical tuft acts as its own neural network that integrates the feed-back inputs and causes a calcium spike. The basal tuft is a seperate neural network that integrates the feed-forward inputs and causes a sodium spike. These can interact and be used to implement learning rules.

Izhikevich - 2006 - Polychronization: Computation with Spikes

Izhikevich, E. (2006) Polychronization: Computation with Spikes. Neural Computation 18, 245-282.

Izhikevich is the head of brain corp, and he used to be a mathematician at the Neurosciences Institute. He did a bunch of large-scale simulations, modeling and theoretical work. Polychronization is key to coming up with a spiking neural code. So, I decided to go back and reread this paper.

Notes:
Conduction delays can vary over a wide range <1ms to 44ms. Individual delays, however, are precise and reproducible.

Basic idea: different temporal activation of neurons can activate different populations based on conduction delays. This allows for a much higher dimensional representation space, and a lot more memory capacity. Firing is not synchronous, but it is time-locked = polychrony.

STDP automatically associates the neurons in the network into polychronous groups. There can be far more polychronous groups than neurons. He lets the network just settle for 24 hours - and these groups emerge spontaneously from the random connectivity initial conditions.

He gets gamma oscillations to emerge from the network directly. Not sure how, he does not say anything about exc-inh plasticity. He uses a FS model, so these cells would be prone to firing at 40Hz, but not sure how they get synchronized. We should remake this, and get all the FS cells coupled electrically. We also need an exc-inh learning rule.

With inputs new groups emerge, and links between groups form via STDP. The state representations should be considered based on these groups and linking groups.

Expansion:
I think this is a very powerful idea. I would change how it works in several different ways.

• Gamma is not an emergent phenomenon - the interneurons are designed to sync-up and cause the gamma activity. This forces the polychronous chains to stay time-locked as the pyramidal cells can only fire during the troughs of gamma.
• The gamma inhibition should be "multiplicative". Basically this will normalize the population, and prevent the network form exploding.
• This is the neural "clock". It is necessary to keep synchrony to maintain any neural code. Noise will cause slight drift which will get compounded over time without a synchronizing force.
• We will want to consider inverse learning phases. These polychronous groups will activate to remap their representation to the inputs that activate them. This is the generative part of the learning. 
• The dendritic tree can be used to select the polychronous inputs more precisely. Can act as a full feed-forward neural network and make non-linear classifications. 
• Two dendritic trees on pyramidal cells - one for the feed-forward input, and one for feed-back/recurrent. These can interact such that the feed-back connections are predicting the feed-forward inputs. 
• Seperate inhibitory populations. Other inhibitory interneurons can contribute as the negative weights for the dendritic tree network. These would be additive, and would need their own learning rule.