
#37
Sep810, 08:24 PM

PF Gold
P: 4,182

This is anecdotal, but from my experience, one of the great things about defining a system with a set of nonlinear differential equations, is that (because of the nature of nonlinearity) you no longer need to use algorithms to describe different behavior.
That is, you don't need to make a bunch of if statements when you're organizing the behavioral structure of a system. Instead, bifurcations already exist in the equation themselves. All relevant behaviors are contained in the system of equations and it's a matter of what parameter space you're in, so all you have to do is adjust the proper parameter values and the appropriate behavior is described by the equation. We've already gained a lot of ground (in terms of elegance and simplicity) by avoiding algorithms (which, to me, are patchwork... you can describe nearly anything with a long list of conditional algorithms, but it's not as intuitive or easy to manage as a system of two or three differential equations that can be written in two or three lines). 



#38
Sep810, 08:31 PM

PF Gold
P: 4,182

oh, by the way, here's the weblog of Markus Dahlem, who uses the nonlinear approach to understanding migraine's in terms of volume transmission:
http://mdlabblog.blogspot.com/ Volume transmission is an extracellular interaction between neurons that don't utilize synapses. Some examples would be electromagnetic field effects between neurons and neurotransmitter concentrations. 



#39
Sep1810, 07:23 AM

PF Gold
P: 4,182

An article of Izhikevich (I have his book, Dynamical Systems in Neuroscience)
http://ieeexplore.ieee.org/stamp/sta...number=1333071 a Quote from the conclusion: 



#40
Sep2510, 04:00 AM

PF Gold
P: 4,182

Network Modeling of Epileptic Seizure Genesis in Hippocampus
Somayeh Raiesdana, S. Mohammad R. Hashemi Golpayegani, Member, IEEE, and S. Mohammad P. Firoozabadi Proceedings of the 4th International SaD1.24 IEEE EMBS Conference on Neural Engineering Antalya, Turkey, April 29  May 2, 2009 



#41
Aug1811, 10:38 PM

PF Gold
P: 4,182

"Lectures in Supercomputational Neuroscience Dynamics in Complex Brain Networks"
From Series: "Understanding Complex Systems" http://www.springerlink.com/content/...ontmatter.pdf 



#42
Aug1811, 10:42 PM

PF Gold
P: 4,182




#43
Sep1111, 11:26 PM

PF Gold
P: 4,182

Subdiffusion and Superdiffusion in the biological sciences:
http://www.cell.com/biophysj/abstrac...495(09)009837 http://en.wikipedia.org/wiki/Anomalous_diffusion http://www.cell.com/biophysj/abstrac...495(11)008770 Sub/superdiffusive systems are fractal objects, their derivatives being noninteger. This also means they don't depend just on nearest neighbors, but all other members of the ensemble! Very expected from a dynamical systems perspective! http://arxiv.org/ftp/mathph/papers/0311/0311047.pdf I wonder if these could be used to model modulation in neural networks: http://arxiv.org/abs/0805.3769v1 



#44
Dec1811, 12:55 AM

P: 1,306

You probably know this but your wiki article is no longer there. Why?




#45
Dec1811, 03:26 PM

PF Gold
P: 4,182

Instead I decided to contribute at a lower level, so I made this page: http://en.wikipedia.org/wiki/Morris%E2%80%93Lecar_model This is a popular model in computational neuroscience, based on the physics of real neurons; the Hodgkin Huxley model is the original, but this model reduces the dimensions in half for faster computation at the cost of simplifying assumptions. 



#46
Dec1911, 12:10 PM

P: 1,306





#47
Dec1911, 03:42 PM

PF Gold
P: 4,182

A large part of dynamical systems approach is studying quantities geometrically (i.e you draw the average trajectories of your system and you begin to see structures that have functional meaning in the statespace of the system). You generally measure quantities of the system like: the lyapunov constant, the natural measure, the basin of attraciton, etc. 



#48
Dec1911, 04:26 PM

Sci Advisor
P: 8,005

Hmmm, but is there such a thing as nondynamical neuroscience?
I can imagine that some of the "optimization" approaches are considered nondynamical by some. But that would be like saying statistical mechanics which maximizes free energy is nondynamical, and place Boltzmann's work on kinetic theory out of the realm of statistical mechanics, which is strictly correct, but surely not your intention. Also many dynamical systems can be described as optimal (but not uniquely) through the use of Lagrangians (maybe take a look at Enzo Tonti's work for how far this can go). Even if you consider anatomy as nondynamical, I'm sure most anatomists do their work because they know how it fits in with physiology. Similarly, most physiologists know how important network topology (ie. anatomy) is for interpreting physiology. It's the same at a lower level in chemistry, where no one would interpret the diagram A+B→C as nondynamical because the rate constants were not explicitly included. Incidentally, I have read books where it is said that control or systems theory goes beyond dynamics, in the strictly true sense that most dynamics deals with autonomous equations. But I'm sure you'd disagree with that! 



#49
Dec1911, 04:56 PM

PF Gold
P: 2,432

The same issue arise in biology, with the fundamental division between genes and organisms, or replication and metabolism. And there have been continuing efforts to marry the two sides, as in systems biology, relational biology, evodevo, biosemiotics, etc. 



#50
Dec1911, 11:37 PM

PF Gold
P: 4,182

dynamics in this sense refers to the dynamical systems theory (DST) approach, which entails nonlinear equations that generally have no analytical solution.
Much quantitative scientific work relies on the expectation of equilibrium which allows for linear equations. Thus, you can take the system apart into its components, solve each one, and put them back together, because they obey the superposition principle. Nonlinear systems are more general (if you make particular terms first order or particular constants zero, you can reduce the equations to linear). They do not obey superposition principle (so you will often hear the sum of the parts is not equal to the whole). So naturally, dynamical systems are closer to the reality, since they make less simplifying assumptions (particularly the assumption of equilibrium and superposition) but previously to the computer age scientists would have had to derive, literally, thousands of equations... the "accounting" errors associated with this kind of work with paper and pen wouldn't even make it worth it. And these solution would not be analytical, they would be numerical. Poincare discovered a way to geometrically assert things about the system (it's stable points, where it attracts solutions in state space, where it repels solutions, etc) without explicitly finding solutions. So this is what was done before computers with systems that had low enough dimension that you could visualize it geometrically. So dynamical systems theorists are often sometimes called "geometers" because a major emphasis is visualizing the system in state space. The actual "dynamical" word only serves the purpose of a coup against the equilibrium assumption. But yeah, there is no panacea. No one approach will tell the whole story of anything, ever. So there's no reason to jump in the DST bucket and ignore the rest of the world. 



#51
Dec2011, 04:09 AM

Sci Advisor
P: 8,005

Also, maybe the attractor is irrelevant http://prl.aps.org/abstract/PRL/v60/i26/p2715_1 



#52
Dec2011, 06:14 AM

Sci Advisor
P: 8,005

I guess what you are saying, which is true, is that the differential geometric (or differential topological) viewpoint alone isn't so useful for defining useful emergent variables. For example, in certain variables, the "attractor" could be a limit cycle, while in "coarse grained" variables, the same "attractor" would be described by a fixed point. Also, one may choose to discretize time and use a generating partition or markov partition to make a link to symbolic dynamics. And that's of course just the beginning. So perhaps one could say that dynamics is everything, but so is emergence. How's that for an attempt to paraphrase your "higher view" 



#53
Dec2011, 06:30 AM

PF Gold
P: 4,182

But lets say I use this linearization to find the fixed points of my system. Then I run the actual numerical simulation. The simulation does not rely on the linearized fixed point. I would overlay the two different sources in a plot to make qualitative assertions about the behavior of the system. DST is a powerful and versatile tool. I'm often tempted by the idea that DST will help bridge quantum and classical through quantum chaos. But I also don't hold my breath, because people have been really excited about DST for 40 or so years now. I don't know whether Hodgkins and Huxley were dynamical systems theorists. I don't think they were; I was under the impression they were just modeling currents and recorded what they got. The equations just happened to be nonlinear. It appears to me that it was dynamical systems theorists who picked up the empirical model and ran the barrage of dynamical tests on it, and what they found was that the system was really quite fitting to all the language that had been developed and found that the Hodgkin Huxley system was chaotic (which had a lot of implications for irregularity and diversity in biological systems). 



#54
Dec2011, 07:52 AM

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P: 8,005




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