I am a bit confused between single neuron models and population of neuron models. If we have single neuron models (Integrate and fire model etc) then why we want to develop models for population of neurons? Although single neuron models are ordinary differential equations (easy to analyze) and population models are complex partial differential equations difficult to analyze. What are the advantages of population of neuron models over single neuron models?
Interactions that emerge from neuron networks are distinct from the dynamics of a single neuron. Determining brain function has a lot do with how the neurons are coupled together. Concepts like feedback and entrainment, the "bump" in bump attractor networks.
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 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'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.
if you look at this article they modeled the population of neurons by developing a pde model: 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.
First of all, thanks for the thanks, uetmathematics. I'm trying to catch up to WannabeNewton, but have a ways to go. 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
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.
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?
I'm guessing atyy interpreted your sentence as a generalization of all network models. That's how I interpreted it.