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Dynamical Neuroscience

by Pythagorean
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Pythagorean
#37
Sep8-10, 08:24 PM
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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).
Pythagorean
#38
Sep8-10, 08:31 PM
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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.
Pythagorean
#39
Sep18-10, 07:23 AM
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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:

As the reader can see in Fig. 2, many models of spiking neurons
have been proposed. Which one to choose? The answer depends
on the type of the problem. If the goal is to study how the
neuronal behavior depends on measurable physiological parameters,
such as the maximal conductances, steady–state (in)activation
functions and time constants, then the Hodgkin–Huxleytype
model is the best. Of course, you could simulate only tens
of coupled spiking neurons in real time.
In contrast, if you want to simulate thousands of spiking neurons
in real time with 1 ms resolution, then there are plenty of
models to choose from. The most efficient is the I&F model.
However, the model cannot exhibit even the most fundamental
properties of cortical spiking neurons, and for this reason it
should be avoided by all means. The only advantage of the I&F
model is that it is linear, and hence amenable to mathematical
analysis. If no attempts to derive analytical results are made,
then there is no excuse for using this model in simulations.
The quadratic I&F model is practically as efficient as the
linear one, and it exhibits many important properties of real neurons,
such as spikes with latencies, and bistability of resting and
tonic spiking modes. However, it is 1-D, and hence, it cannot
burst and cannot exhibit spike frequency adaptation. Thus, it can
be used in simulations of cortical neural networks only when biological
plausibility is not a great concern.
Pythagorean
#40
Sep25-10, 04:00 AM
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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

Based on the use
of mathematical nonlinear models of neuronal networks, it is
possible to formulate hypotheses concerning the
mechanisms by which a given neuronal network can switch
between qualitatively different types of oscillations. This
switching behavior is a dynamical paradigm in epileptic
seizure when e.g. an EEG characterized by alpha rhythmic
activity suddenly changes into a spike- burst pattern. This
transition, however, depends on input conditions and on
modulating parameters where often even a subtle change in
one or more parameters can cause a dramatic change in the
behavior of neural circuits. This sensitivity to initial
condition or to a small perturbation is the major hallmark of
nonlinearity and chaos. Nowadays, modeling neuronal
ensemble is one of the most rapidly developing fields of
application of nonlinear dynamics and the success of such
models depends on the universality of the underlying
dynamical principles.
Epilepsy is a brain disorder characterized by periodic and
unpredictable seizures mediated by the recurrent
synchronous firing of large groups of neurons in the cortex
and seizure represents transitions of an epileptic brain from
its normal less ordered (chaotic) interictal state to an abnormal
(more ordered) ictal state
Pythagorean
#41
Aug18-11, 10:38 PM
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"Lectures in Supercomputational Neuroscience Dynamics in Complex Brain Networks"

From Series: "Understanding Complex Systems"

Complex Systems are systems that comprise many interacting parts with the ability to generate a new quality of macroscopic collective behavior the manifestations
of which are the spontaneous formation of distinctive temporal, spatial or functional
structures. Models of such systems can be successfully mapped onto quite diverse
“real-life” situations like the climate, the coherent emission of light from lasers,
chemical reaction-diffusion systems, biological cellular networks, the dynamics of
stock markets and of the internet, earthquake statistics and prediction, freeway traf-
fic, the human brain, or the formation of opinions in social systems, to name just
some of the popular applications.
Although their scope and methodologies overlap somewhat, one can distinguish
the following main concepts and tools: self-organization, nonlinear dynamics, synergetics, turbulence, dynamical systems, catastrophes, instabilities, stochastic processes, chaos, graphs and networks, cellular automata, adaptive systems, genetic algorithms and computational intelligence.

http://www.springerlink.com/content/...ont-matter.pdf
Pythagorean
#42
Aug18-11, 10:42 PM
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"Computational Neurogenetic Modeling"

CNGM is concerned with the study and development of dynamic neuronal models for modeling brain functions with respect to genes and dynamic interactions between genes.
http://books.google.com/books/about/...d=GFdzpAasI4oC
Pythagorean
#43
Sep11-11, 11:26 PM
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Subdiffusion and Superdiffusion in the biological sciences:

http://www.cell.com/biophysj/abstrac...495(09)00983-7
http://en.wikipedia.org/wiki/Anomalous_diffusion
http://www.cell.com/biophysj/abstrac...495(11)00877-0

Sub/superdiffusive systems are fractal objects, their derivatives being non-integer. 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/math-ph/papers/0311/0311047.pdf

I wonder if these could be used to model modulation in neural networks:

http://arxiv.org/abs/0805.3769v1
Nano-Passion
#44
Dec18-11, 12:55 AM
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You probably know this but your wiki article is no longer there. Why?
Pythagorean
#45
Dec18-11, 03:26 PM
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Quote Quote by Nano-Passion View Post
You probably know this but your wiki article is no longer there. Why?
reading the posts in this thread by JFGariepy might help understand that. I think though, that I started too early. The dynamical systems approach to neuroscience is still developing and it makes it difficult to comment on. It is the physicist's approach to computational neuroscience, so it could be integrated into the computational neuroscience page.

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.
Nano-Passion
#46
Dec19-11, 12:10 PM
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Quote Quote by Pythagorean View Post
reading the posts in this thread by JFGariepy might help understand that. I think though, that I started too early. The dynamical systems approach to neuroscience is still developing and it makes it difficult to comment on. It is the physicist's approach to computational neuroscience, so it could be integrated into the computational neuroscience page.

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.
Oh.. so then what would be the difference between theoretical and dynamical neuroscience?
Pythagorean
#47
Dec19-11, 03:42 PM
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Quote Quote by Nano-Passion View Post
Oh.. so then what would be the difference between theoretical and dynamical neuroscience?
Theoretical neuroscience is an umbrella term encompassing many current approaches. Dynamical approach to neuroscience is a particular theoretical approach that utilizes time-smooth continuous equations to describe neural events as the neural systems evolve through state-space. They are structurally deterministic, but noise terms and random processes can be integrated into them. There's also symbolic dynamics which utilize the Markov partition.

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 state-space of the system). You generally measure quantities of the system like: the lyapunov constant, the natural measure, the basin of attraciton, etc.
atyy
#48
Dec19-11, 04:26 PM
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Hmmm, but is there such a thing as non-dynamical neuroscience?

I can imagine that some of the "optimization" approaches are considered non-dynamical by some. But that would be like saying statistical mechanics which maximizes free energy is non-dynamical, 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 non-dynamical, 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 non-dynamical 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!
apeiron
#49
Dec19-11, 04:56 PM
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Quote Quote by atyy View Post
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!
There is definitely a live issue here. It seems obvious both that everything is grounded in biological dynamics, yet also that dynamics is only half the story. Therefore some kind of hybrid is the "higher view".

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, evo-devo, biosemiotics, etc.
Pythagorean
#50
Dec19-11, 11:37 PM
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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.
atyy
#51
Dec20-11, 04:09 AM
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Quote Quote by Pythagorean View Post
Much quantitative scientific work relies on the expectation of equilibrium which allows for linear equations.
Quote Quote by Pythagorean View Post
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.
I'm not sure that ideas of equilibria and linearity are morally distinct from fixed points and whether they are attractive or repelling. After all, fixed points are equilibria, and whether they are attractive or repelling can often be found by linearization (one has to go to higher orders in the "marginal" cases).

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

Quote Quote by Pythagorean View Post
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.
But couldn't one say dynamics is the panacea because it includes the rest of the world? By including non-autonomous systems and Lie brackets the geometric viewpoint can be extended to control or systems theory, and there is a relationship to symbolic dynamics via generating partitions and markov partitions. Even classical mechanics has a link to probability theory via Liouville's theorem, and a link to optimality via Lagrangians. So perhaps "dynamical neuroscience" is redundant - the integrate-and-fire neuron is more than 100 years old, and the HH equations are in every textbook.
atyy
#52
Dec20-11, 06:14 AM
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Quote Quote by apeiron View Post
There is definitely a live issue here. It seems obvious both that everything is grounded in biological dynamics, yet also that dynamics is only half the story. Therefore some kind of hybrid is the "higher view".

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, evo-devo, biosemiotics, etc.
I had a much dumber idea in mind than what you are mentioning. The control or systems view merely meant including non-autonomous systems. In the continuous time and degrees of freedom case, this is still Pythagorean's differential geometric viewpoint.

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"
Pythagorean
#53
Dec20-11, 06:30 AM
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Quote Quote by atyy View Post
I'm not sure that ideas of equilibria and linearity are morally distinct from fixed points and whether they are attractive or repelling. After all, fixed points are equilibria, and whether they are attractive or repelling can often be found by linearization (one has to go to higher orders in the "marginal" cases).

Also, maybe the attractor is irrelevant http://prl.aps.org/abstract/PRL/v60/i26/p2715_1
Fixed points are equilibria, but a truly chaotic system never actually reaches the fixed points. Most interesting fixed points are wildly unstable, like a pencil standing on it's tip. And of course, as you are hinting at, the linearization is an approximation.

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.


But couldn't one say dynamics is the panacea because it includes the rest of the world? By including non-autonomous systems and Lie brackets the geometric viewpoint can be extended to control or systems theory, and there is a relationship to symbolic dynamics via generating partitions and markov partitions. Even classical mechanics has a link to probability theory via Liouville's theorem, and a link to optimality via Lagrangians.
Does that cover life, the universe, and everything, then? :)

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.


So perhaps "dynamical neuroscience" is redundant - the integrate-and-fire neuron is more than 100 years old, and the HH equations are in every textbook.
Is integrate-and-fire dynamical? I thought it was a linear superposition...?

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 non-linear.

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).
atyy
#54
Dec20-11, 07:52 AM
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Quote Quote by Pythagorean View Post
Does that cover life, the universe, and everything, then? :)
Yes:)

Quote Quote by Pythagorean View Post
Is integrate-and-fire dynamical? I thought it was a linear superposition...?
Well, it has a terrible nonlinearity that makes it infinite dimensional. Yet it can be obtained as an approximation of the HH equations.

Quote Quote by Pythagorean View Post
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 non-linear.

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).
Are you also not counting Newton as a dynamical systems theorist?


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