- #1

- 6

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter uetmathematics
- Start date

- #1

- 6

- 0

- #2

Pythagorean

Gold Member

- 4,217

- 276

- #3

- 1,189

- 512

Although single neuron models are ordinary differential equations (easy to analyze) and population models are complex partial differential equations difficult to analyze

That's actually incorrect, uetmathematics. We model population dynamics of neurons using ODE's, not PDE's. They are coupled sets of non-linear ODE's, that are essentially modeled as coupled oscillators. And they're not easy to analyze, I don't know who told you that. If you're Matlab savvy I think my friend Robert Kozma has a Matlab toolbox he could direct you too if you wanna play around with it. Hit him up here: http://memphis.edu/clion/members/index.php

If we have single neuron models (Integrate and fire model etc) then why we want to develop models for population of neurons?

Well, obviously we want to develop models for populations of neurons because it cuts down on computing time. It's takes essentially the same amount of processing power to solve for one neuron what we could use to solve for a 10,000 neuron "node." So why wouldn't we want to do that?

I am a bit confused between single neuron models and population of neuron models.

I'd recommend this this article if you're really interested:TUTORIAL ON NEUROBIOLOGY:

FROM SINGLE NEURONS TO BRAIN CHAOS

You can find it on this website: http://sulcus.berkeley.edu/

Unfortunately, I can't provide a direct link, but the whole article is there, just scroll down and look for it.

Once you click on the article, scroll down to figure 10. That will answer your question. The main difference between the single neuron model and the population model is that the population model posits a nonlinear gain function to drive information flow which takes the shape of a sigmoid curve, while the single neuron "integrate and fire" model is more linear. It's basically a summation of dendritic pulses that add at the axon hillock, or as Freeman calls it the trigger zone. It's all there in the article, though. Happy searching.

- #4

- 6

- 0

A principled dimension-reduction method for the population density approach to modeling networks of neurons with synaptic dynamics. (http://www.ncbi.nlm.nih.gov/pubmed/23777517)

and all the references in this article also modeled using pde.

- #5

- 1,189

- 512

if you look at this article they modeled the population of neurons by developing a pde model:

Unfortunately, I can only access the abstract from that post, so I can't really comment on that right now. However, what I can say is that, obviously, there are many different groups working on brain modeling, so there's accordingly going to be many different approaches. We found that there really only was a single variable we needed to use to accurately model what we were seeing in raw EEG tracings, and that was how the difference in the voltage potential crossed over each modeled node as a function of time. It's a second order ODE, nonlinear because you have to account for feedback not only between each oscillator, but also between the glutamate, etc. excitatory cells and the GABA-ergic inhibitory interneurons which drive the oscillations.

Plus, you have several nested layers of gradiently distributed feedback from progressively removed cortical regions. These regions disrupt the periodic oscillations in the target cortex which lead to aperiodicity, more specifically they show up as chaotic attractors on phase portraits which themselves oscillate between more chaotic forms and near limit cycle attractors. Interestingly, it kind of works like a Carnot cycle. Check out this article: http://www.ncbi.nlm.nih.gov/pubmed/?term=freeman+carnot

Getting back to your original question, though, it doesn't add anything to try to model these networks as PDE's. It's enough of a pain that the equations are nonlinear, but thankfully we have Matlab and Runge-Kutta to help us out with that.

You might also want to check out this article:http://www.ncbi.nlm.nih.gov/pubmed/19395236

- #6

atyy

Science Advisor

- 14,440

- 2,738

uetmathematics is correct that some model networks of neurons have collective behaviours that can be described by partial diferential equations. Here are some examples in addition to the reference he provided.

http://www.math.pitt.edu/~bard/pubs/nnetrev.pdf

Neural networks as spatio-temporal pattern-forming systems

Reports on Progress in Physics, 61:353-430, 1998

Ermentrout B

http://galton.uchicago.edu/~nbrunel/pdfs/brunel00JCNS.pdf

http://www.ncbi.nlm.nih.gov/pubmed/10809012

J Comput Neurosci. 2000 May-Jun;8(3):183-208.

Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.

Brunel N.

http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002872

http://www.ncbi.nlm.nih.gov/pubmed/23359258

PLoS Comput Biol. 2013;9(1):e1002872. doi: 10.1371/journal.pcbi.1002872. Epub 2013 Jan 24.

Dynamic finite size effects in spiking neural networks.

Buice MA, Chow CC

http://www.math.pitt.edu/~bard/pubs/nnetrev.pdf

Neural networks as spatio-temporal pattern-forming systems

Reports on Progress in Physics, 61:353-430, 1998

Ermentrout B

http://galton.uchicago.edu/~nbrunel/pdfs/brunel00JCNS.pdf

http://www.ncbi.nlm.nih.gov/pubmed/10809012

J Comput Neurosci. 2000 May-Jun;8(3):183-208.

Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.

Brunel N.

http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002872

http://www.ncbi.nlm.nih.gov/pubmed/23359258

PLoS Comput Biol. 2013;9(1):e1002872. doi: 10.1371/journal.pcbi.1002872. Epub 2013 Jan 24.

Dynamic finite size effects in spiking neural networks.

Buice MA, Chow CC

Last edited:

- #7

- 1,189

- 512

uetmathematics is correct that some model networks of neurons have collective behaviours that can be described by partial diferential equations. Here are some examples in addition to the reference he provided.

http://www.math.pitt.edu/~bard/pubs/nnetrev.pdf

Neural networks as spatio-temporal pattern-forming systems

Reports on Progress in Physics, 61:353-430, 1998

Ermentrout B

http://galton.uchicago.edu/~nbrunel/pdfs/brunel00JCNS.pdf

http://www.ncbi.nlm.nih.gov/pubmed/10809012

J Comput Neurosci. 2000 May-Jun;8(3):183-208.

Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons.

Brunel N.

http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002872

http://www.ncbi.nlm.nih.gov/pubmed/23359258

PLoS Comput Biol. 2013;9(1):e1002872. doi: 10.1371/journal.pcbi.1002872. Epub 2013 Jan 24.

Dynamic finite size effects in spiking neural networks.

Buice MA, Chow CC

Thanks for the references, atty. I typically referee 2-3 articles a month and don't get paid for it so I need to concentrate on doing those justice because I take that seriously and it is time consuming. So please don't just post a bunch of articles, if you have a specific question, just ask it.

- #8

atyy

Science Advisor

- 14,440

- 2,738

Getting back to your original question, though, it doesn't add anything to try to model these networks as PDE's. It's enough of a pain that the equations are nonlinear, but thankfully we have Matlab and Runge-Kutta to help us out with that.

Thanks for the references, atty. I typically referee 2-3 articles a month and don't get paid for it so I need to concentrate on doing those justice because I take that seriously and it is time consuming. So please don't just post a bunch of articles, if you have a specific question, just ask it.

Your assertion that pdes don't add anything is wrong.

- #9

- 1,189

- 512

Your assertion that pdes don't add anything is wrong.

I have absolutely no idea what you are talking about atyy. You need to give me more than just that kind of blanket statement. I'll make this really simple. If you're going to challenge our model, provide a disqualification and a suitable alternative. Don't just say I'm wrong. I can't babysit everyone's pet project brain model. The way science works is that you write and publish, and you battle it out during the annual conference, although you can do that online 24/7 these days. But you're not giving me anything to argue against with that kind of a blanket statement.

In that spirit, I will give you something specific to read and comment on. Not only that, there's some source code you can play around with. Scroll down to section 3.2., entitled "ODE based approach to neural populations." http://www.sciencedirect.com/science/article/pii/S0893608009000434

This model is also a DARPA funded project designed to work towards the creation of autonomous rovers:http://www-robotics.jpl.nasa.gov/publications/Terrance_Huntsberger/srr2k-AdvRob-2.pdf

If you want to challenge that model, then do so. Otherwise, I don't know what your point is?

- #10

Pythagorean

Gold Member

- 4,217

- 276

Share: