Dynamical Neuroscience

atyy

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

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.


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.


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

Yes:)

Well, it has a terrible nonlinearity that makes it infinite dimensional. Yet it can be obtained as an approximation of the HH equations.

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

rhody

Pythagorean,

An aside, sort of reminds you of particle physics now, doesn't it ?

Rhody...

Pythagorean

Poincare really developed the first tools at the turn of the 19th century.

apeiron

I agree that non-linearity is the generalisation of linearity, and so any possible linearity can be treated as emergent. Dynamics must be basic in that sense.

But then there is the aspect of living systems which a dynamical description simply does not see in a useful theory sense, even if it may be able to replicate or simulate it fully (and "unknowingly).

The analogy is the hardware and software of a computer. The hardware is a material system. It changes state in some fashion. You could completely describe all that activity in material/dynamical language. You would be correct and complete in some sense. But you would not have a model that can take one state of a finite state automaton and predict its next state. It is the logic embedded in the software that is causing the state mapping. The material/dynamical description just cannot see the rules and data values that determine the running of the program.

So dynamics can describe spikes, but what describes what the spikes mean? The processes generating the spikes may be material, but the processes regulating the spikes may be informational.

The problem for neuroscience is whether to just model the informational view, just model the material view, model both as two distinct disciplines, or model both in some proper connected way.

It is a tricky business because the hardware and software of a computer are pretty easy to distinguish (OK, with microcode, it gets fuzzy). But with neurons, columns and cortical areas, meaning and medium are thoroughly mixed. As in a neural network, but far more so. You need some real strong principles to get in there and dissect apart the two aspects of what is going on.

So there is no doubt that a spike, for example, is a dynamical event. But it is just as clearly an informational event. Do you then seek to (1) ignore one of these aspects, (2) unify them in a single description, or (3) formalise the relationship between them in a way that is itself maximally general and thus "a law of nature"?

Pythagorean

The idea that atyy proposes is that you use Markov partitions and symbolic dynamics to represent the more abstract semiotics; they would represent your informational classification of dynamical events.

apeiron

This is an informational way of modelling dynamical processes. So not what I am talking about.

atyy

Hmmm, that's a very narrow definition of dynamical systems theory. It's morally ok in some sense, since Poincare is rightly regarded as the father of the topological approach to differentiable dynamics. While acknowledging you have a point, it does boggle my mind that you could exlcude Newton. Even KAM theory had its roots in the Hamilton-Jacobi formulation of mechanics, and whether action-angle variables (invariant tori in the modern language) exist.

I was really taking the particle physics point of view, as Rhody says!

Basically, there are not just 2 domains of description, but many. Each domain has its regime of validity, and degrees of freedom. A domain is always defined by subjective human interaction. This is true in thermodynamics, where the time scale of observation enters fundamentally in whether we accept something as in equilibrium or changing. It is also true in music which has no meaning played to a hydrogen atom, but does when played to a human being who uses emergent degrees of freedom such as pitch, rhythm, harmony, sonata form, expectation, frustration, resolution. The point regarding markov partitions was not to be over generalized, it simply meant that the relationship between two domains, in which one is a dynamical system describable by a diffeomorphism, is not necessarily a restriction of the system to a submanifold. As an analogy, Kadanoff-Wilson coarse graining provides one type of emergence in particle physics, but does not (in its simplest form) include other types such as holographic emergence.

apeiron

On pragmatic grounds, yes, we are allowed to create as many modelling paradigms as we wish. Models are free inventions of the human mind, so there is no limit on how creative we can get, or how finely we wish to divide the cake.

But on fundamental grounds - which I thought we were debating - in fact the reductionist goal is to reduce everything in reality to a single common basis (a TOE), and the rejoinder from a systems perspective is in fact that instead we always seem to end up with dichotomies, two polar alternatives that seem have equal pull on our imaginations.

So should we reduce all neuroscience to dynamics, or to computation? Or should we unite the two by honouring their fundamental differences?

In theoretical biology, the systems view is understood. In theoretical neuroscience, not so much .

That does not mean there are not in fact multiple modelling paradigms. Just that a modelling fundamentalist would expect them to be arranged in a hierarchy so that they would still all "talk to each other". And then a reductionist would expect this hierarchy to work bottom-up - from some actual physical/material/dynamical TOE. While a systems thinker accepts that this hierarchy has in fact its two poles - so the semiotic/formal/computational is also fundamental in the way it anchors the other end of the spectrum.

In this way, we have both your "many models" as the stuff which fills the spectrum, and then the two fundamental poles needed to anchor that hierarchy.

The alternative view would be that of extremist social constructionism - models are just all human inventions, none with any more claim to fundamentality than any others. We would have a patternless mosaic, a space of modelling fragments each with local application but no global coherence.

So be careful what you wish for!

atyy

I was hoping for the last view, but also with global coherence.

apeiron

OK, so what is the nature of that coherence exactly?

Both the conventional reductionist and the systems view would expect coherence from a hierarchical arrangement of models that all "talk to each other" across their levels.

That in itself implies a common language - and information theory is emerging as that standard coin of exchange between theory domains. (Whereas more traditionally, a scientific coherence was claimed because "everything was made of the same kind of ultimate stuff" - science being a materialistic discourse.)

So you have the differentiation of models into levels of a hierarchy, and the integration of these models through some common language, some standard unit of exchange. How it works out in all its gory details is still debatable, but the general model of how global coherence would be achieved by the scientific enterprise seems both explicit and widely accepted. Witness the angry rejection of PoMo commentaries in the Philosophy of Science.

So if you are not taking this hierarchical approach to a universe of models, then exactly how do you imagine a coherence being achieved?

And further, are you claiming that the current patchwork of models is not actually connected in this fashion - if albeit loosely and imperfectly?

Pythagorean

We can agree on all kinds of observations, but where we divide and categorize sets of observations is where we have conflicts ("It's QM", "no, it's CM!") or ("it's blue", "no, it's indigo!").

I don't consider Einstein a quantum physicist either; I think Newton and Einstein are both unique cases. They are pretty much our (i.e. society's) ideal vision of a scientist as you really can't box them up as this or that. Of course, I feel the same way about people like Poincare and Erdos :) they're just not as popular to the general public.

Well, I guess to me, symbolic dynamics means you take a particular state of the whole system of particles to be an emergent qualitative state. And while the dynamical system really has infinite states, you could (as an example) partition the phase volume into two and call one state "1" and the other state "0".

But I have no experience actually handling Markov partitions, so this is just my impression from reading literature that's full of cumbersome jargon.

atyy

Well, what I'm saying is morally related to hierarchical thinking - but with no model being fundamental, and no hierarchy - more a patchwork of coordinate charts - but even then not quite since there is no standard unit of exchange (except the human mind).

Pythagorean

That is a more of a distracting coincidence, I was actually referring to the subjectivity allowed of the investigator to define the partitions of the system himself. The investigator is free to implement a hierarchical approach... and for particular kinds of systems (at least) if we define the partition around the bifurcations of the system, we cannot even avoid adhering to heirarchy (the bifurcation branches) and its relationship to scale (the bifurcation parameter).

apeiron

But is this your goal, or just a description of best likely outcome? We were talking about goals (even you expressed coherence as a hope of yours).

And your comment about there being no unit of exchange apart from the human mind is baffling. Units of exchange are what a modelling mind would create, not what they would "be".

It might help if you could supply references to your brand of epistemology here.

For instance, an example of the adoption of information as the new universal coin of modelling is...http://en.wikipedia.org/wiki/Digital_physics

Well, actually, that is an example of people jumping from epistemology to ontology. They don't just believe physics can be modelled in the standard language of information theory, they claim it actually is just all information!

So this is an illustration of the perils of orthodox reductionism - going overboard in just one direction. But it also shows that the other pole of description exists even at the "lowest level" of material physics.

There is a battle of views going on that is framed dichotomistically - substance vs form, matter vs information.

The strings/TOE debate is another example. Shall we model reality in terms of its fundamental degrees of freedom or its fundamental constraints? The expectation of the TOE camp is that degrees of freedom are infinite, but only one form of constraint (the string theory that works) is actually possible. So then everything (even the fundamental constants, fingers crossed) will be "explained by mathematics".

So again, no quarrel that science is pragmatically formed by a ragged patchwork of modelling domains. But at the same time, the same basic fundamental division infects/unites science at its every level.

Charts can create their own co-ordinates. But generally they are in fact all trying to orientate themselves along the same general compass setting that points north to form/information, and south to sustance/matter.

Neuroscience is just another example. And the best neuroscience - like Grossberg with his plasticity~stability dilemma, or Friston with his Bayesian brain - is focused on finding the appropriate balance between the informational and material view.

apeiron

Again, you are making my point for me. If it is a subjective work-around, it is not an objective consequence of the model.

Yes, we can get away with doing things simply - either pretending reality is just dynamics, or just computation. We can rely on our informal, subjective, knowledge to avoid misusing models based on those reductionist assumptions.

But that is not the same thing as having a formal basis to a domain of knowledge. It does not address the issue of what is fundamental.

You can then respond, the fundamental doesn't actually matter if we can get by on pragmatics. And again, for some people - many probably - this is indeed enough to satisfy their personal interests.

But for science itself, it does matter. The enterprise of science does have to ensure that all the local domains of modelling connect up objectively - even just pragmatically! - somehow. And a hierarchy of modelling is the way this is being done. Which in turn means extracting the fundamental co-ordinates of this hierarchy (so as to give all the specialised sub-domains some bearings to steer by).