Uncovering the Complexity of Single Neurons: A Look into Homoclinic Orbits

[mrspeedybob]

A brief internet search revealed that the number of neurons in a human brain is in the 85 - 100 billion ballpark. (reference) What I could not find was any clear indication of how complex a single neuron is. Is the brain like a network of 85 - 100 billion transistors or 85 - 100 billion super-computers?

 

[atyy]

Closer to the transistors than the supercomputers, but nonethless a single neuron does have many compartments. However, sometimes we can get away with modelling it as a single compartment, eg. http://www.ncbi.nlm.nih.gov/pubmed/10436067 treats the Purkinje cell as if it had just one compartment, although it clearly has a complex structure http://www.coloradocollege.edu/academics/dept/neuroscience/course/slides/histology-and-cellular-.dot . On the other hand, some phenomena require taking the shape of the neuron and the differences between the compartments into account, eg. http://www.ncbi.nlm.nih.gov/pubmed/20364143. (I'm using "compartments" loosely, it just means things are different in differents parts of the neuron. In general things can change smoothly, and not in discrete steps that "compartments" might imply.)

 

[Pythagorean]

A canonical model for studying the neuron is the Hodgkin Huxley model, a four dimensional system of differential equations. This is still just a single compartment, but the single compartment can do a lot of things.

Neurons can function both as https://www.tapr.org/images/dsp4.figure_3.jpg and integrators; a bit more complex than transistors, I think, but definitely not a super computer (unless maybe you buy into microtubule quantum cosnciousness, but I think that's still in crackpot realm).

 

[madness]

There's also dendritic properties and subthreshold membrane oscillations, both of which can carry out complex computations on inputs to a neuron.

 

[ManFrommars]

... not to mention neuron-glia interactions, calcium spikes, multiple synapse types, plasticity, gap junctions... plus all the complex machinery that other cells in the body have (nucleus, mitochondria etc).

It can be very useful when modelling some phenomena to abstract neurons down to very simple models (even 2-dimensional simplifications of the Hodgkin Huxley equations can exhibit many properties mentioned above like resonating, integrating, sub-threshold oscillations, bursting, see e.g. http://www.scholarpedia.org/article/Adaptive_exponential_integrate-and-fire_model ), but real biological neurons are a lot more complex than these simplifications.

 

[Pythagorean]

I would say synapses, dendrites, or gap junctions are more for network considerations.

But yeah, you can add ligand gated channels and genetic expression dynamics to make things more complicated, or even add more types of ion current like a persistent sodium channel.

The 2D models aren't able to truly burst except for in very specific parameter regions that produce a "homoclinic orbit". Other than that, bursting requires three dimensions, as far as I know.

 

[ManFrommars]

I would say synapses, dendrites, or gap junctions are more for network considerations.

Indeed (perhaps not dendrites), but I guess my point was that even if each neuron is simple, they can be connected together in different ways with complex synapse types, meaning that it's not necessarily just like connecting together billions of transistors.

The 2D models aren't able to truly burst except for in very specific parameter regions that produce a "homoclinic orbit". Other than that, bursting requires three dimensions, as far as I know.

I'm far from an expert on the mathematics of bursting (currently brushing up a bit here ;) ) - I'm not sure what you mean by 'true' bursting...

 

[Pythagorean]

Basically, the neuron makes a bunch of oscillations before returning to the resting potential (in the scholarpedia article, notice between eac spike in the burst, the neuron doesn't return to the rest potential.

This is only possible with a third dimension, as in 2D the trajectories would intersect (which can't happen in deterministic systems as differential equations model them)

An exception to this is the homoclinic orbit, where trajectories come into an equilibrium point on a stable manifold and immediately leave via the unstable manifold, giving a similar appearance to an intersection.

If you look up homoclinic orbit you may see what I mean. Am on the phone now so linking is a pain. We can develop this more later though if you have any questions.

 

[ManFrommars]

Basically, the neuron makes a bunch of oscillations before returning to the resting potential (in the scholarpedia article, notice between eac spike in the burst, the neuron doesn't return to the rest potential.

This is only possible with a third dimension, as in 2D the trajectories would intersect (which can't happen in deterministic systems as differential equations model them)

An exception to this is the homoclinic orbit, where trajectories come into an equilibrium point on a stable manifold and immediately leave via the unstable manifold, giving a similar appearance to an intersection.

If you look up homoclinic orbit you may see what I mean. Am on the phone now so linking is a pain. We can develop this more later though if you have any questions.

Ah I see what you mean. Thanks for the explanation :)