Monday, 21 March 2011

Independent model proposes spherical harmonics for lateral ventricle development (Monica K. Hurdal and Deborah A. Striegel 2009)

Neurodevelopmental mathematicians Monica K. Hurdal and Deborah A. Striegel have produced a paper in September 2009 that proposes the lateral ventricles can be described in neurodevelopment by spherical harmonics. Paper is called "Chemically Based Mathematical Model for Development of Cerebral Cortical Folding Patterns"

http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1000524
http://www.math.fsu.edu/~mhurdal/posters/hbm10prolate36x48.pdf

The approach is clearly "structuralist"..from the paper.

One argument that has been presented against the intermediate progenitor (traditional) model is that an ‘‘elaborately choreographed set of developmental instructions [regulating the production of IP cells] would be required to account for the tremendous complexity of human cortical convolutions’’ [7]. The beauty of the GIP global intermediate progenitor (GIP) model model is that it provides an uncomplicated approach that relates to a biologically plausible mechanism of pattern formation. It uses chemical morphogens that may be governed by specific genes to control IP cell production, resulting in the ability to predict the placement and directionality of sulcal pattern formation.








Basically we see that by using the spherical harmonic for the lateral ventricle they then predict the main cortical features. This is of course consistent with the dipole neurology 2003 Limbic EMS (electromagnetic multipole solution) model that is based on spherical harmonics for the lateral ventricles as well as all limbic system appendages. But the similarity ends there. We disagree on where the prolate poles are taken from and my proposal that all the limbic morphology is the result of my proposed Magnetohydrodynamic model. I think at the current stage of knowledge a more comprehensive prediction for the correct spherical harmonics should be based on morphology and biophysical mechanisms before proceeding to mathematical modelling. i.e. to start with the approach.




Observable analysis of primary limbic /cortical system regions when mirrored and rebuilt using spherical harmonic components. The entire limbic system can be modelled from variations of spherical harmonics the components of electromagnetic multipole expansions. (a) This shows that the thalamus has a basic hourglass lobe form, and has a third ventricle between it, which is a toroid. (b) The third ventricles and caudate nucleus are mirrored produce sheared discs. Hippocampus is similar but has been left out for simplicity. These are added to thalamus third ventricle from (a). (c) the induseum griseum of the corpus callosum, a continuation of the hippocampus has toroidal structures. These are added to (a) and (b). (d) (a,b,c) are enclosed in the cortex which is a large lobe structure. (cortical surface folds left out due to modelling restrictions.) Again a toroidal structure occurs in midline. (see section 1.2) Consistent with the concept that it is only the limbic system which can be modelled in this manner the induseum griseum and cortex although possessing some of the structures (toroid’s and lobes) don’t appear to possess an overall pattern in line with any of the spherical harmonics. The indiseum griseum has four toroid’s and cortex structure overall cannot be approximated. Figure References (a Adapted from Best, B., 2009) (b Adapted from Sundsten,, 2009) (c Adapted from Nieuwenhuys et al., 1988) (d Adapted from Williams and Gluhbegovic,1980)

The point of the above exercise was to show how the limbic system when not intruded by the brainstem and skull has a spherical harmonic type morphology. If you read the rest of mypaper (chapter 3) I then try to illustrate the respective regular oscillations, delta, alpha, theta and lower beta range operate in the developed brain using a harmonic mode principle and this is a predicted mechanism and mathematical description for subconscious information sorting and pattern overlay. Its basically a fourier/wavelet like information distribution using time dependent multiplexing through axons that is predicted as the deeper method for limbic system function. i.e Importantly we are discussing here the predicted reason why its conserved in terms of information processing.

Such processing structure allows "holistic" or model free patterns to be represented in line with a Pribaum like holographic scheme.  This will be scale free in terms of keeping connections in sync. i.e. Integrating spikes to tonic timings across wide regions of the brain.  Its function  computationally can be described in many ways. At the highest level its not hard to derive a model for creative cross associations that iterate episodic memories into autobiographic narrative across synchronized limbic areas. i.e. Imagination. This is highly powerful from a computational perspective as many cutting edge Artificial Intelligence programmers look to uncover the basis for human creativity. The Spherical harmonics processor is the part of ourselves which seeks to remain whole and integral so patterns can overlay without interference.


Wavlets spectra (linear) arise when we analyse internet communication systems, and as would be expected easily destroyed by non linear inputs.  Taken from (stoev 2005,On the wavelet spectrum diagnostic for Hurst parameter estimation in the analysis of Internet traffic). It is predicted that the developed limbic system is multiplexing in this manner but the details are beyond the scope of this posting which concerns developmental morphology.

The location for the predicted multiplexing using continuous harmonic modes of Delta, Alpha, theta and low beta is predicted for thalamus, medial components of the basal ganglia and the septal (medial) end of the hippocampus (there is more on the computation in section 3 and 4 of my paper). Why exclude lateral limbic areas from this modelling ?  Basically my reasoning is that where limbic components are lateral and integrate with the cortex this is where i propose a developmental boundary interaction between MHD Linear quadrupole and regular MHD dipole, and so at the boundary the structure is an integration of two different types of MHD functions which is yet to be clarified mathematically, and I would consider hard to integrate such inverse functions without computational modelling.

The first predicted harmonic models I propose for the lateral and 3rd ventricles as well as the development of the medial limbic system neuron and axon configurations that follow (staying in harmonic configuration through the first prenatal oscillations). These were chosen to accommodate the development of four ventricles but a 3 mode series may be used if we discover the cerebellum (4th ventricle) is not time synchronized with the development of 3rd and the two lateral ventricles.


To make it visually clearer how we transfer this to a 3rd ventricle.


(1 and 2 above), take in a toroidal morphology to a modeller (2 can be extended so the centre closes)  then compress it as occurs in a spherical harmonic system (3), then shear and cut (3 and 4) to accommodate the fact all this evolved over the confines of jaw intrusion.  From simple physical constraints as would occur in evolving systems MHD harmonic forces can transform to 3rd and lateral ventricle structures.

These would be described by colatidue and longitude, y1/3,


with the mode function of the poles, derived as a Longitudinal linear waves in the case of lateral ventricles as Y2/3 or  Y1/3 for 3rd and lateral ventricle mode interactions.   To simplify it would be predicted the linear waves in the following diagram, if we are looking at the ventricular zone on the sagittal plane. This linear function will then break down and fragment at its boundary before it reaches the cortex surface, which i will explain next.


BTW its should be noted deriving harmonic modes from MHD is is common for high frequencies as the application for most MHD is astrophysics,

http://iopscience.iop.org/0295-5075/94/1/10005

and this can be applied apparently to lower energy systems.

http://en.wikipedia.org/wiki/Magnetohydrodynamics

it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is relatively simple and captures many of the important properties of plasma dynamics it is often qualitatively accurate and is almost invariably the first model tried.

The point is the solution offered by myself is relatively painless from a mathematical and biophysics perspective, and relies only on simple re-arrangements of spherical functions derived from Calcium ion flow through radial glia and the proliferation zones. The power of a multipole expansion is that it allows nature a relatively simple mechanism to layer associative multiplexing, by using principles of the harmonic quantum oscillator in development to emerge as a classical quantum oscillator in the developed brain.

As to the actual cortical folding itself these would be predicted not as a harmonic (as stated in the Hurdal and Streigel paper) but a boundary effect resulting from mode breakdown (at the harmonic boundary) i.e. The turing like patterns of the cortex surface I propose are magnetic dipole domains (the ising ferroelectric models for this have a similar mathematical description to turings reaction diffusion). This is where I veer of from the current accepted work in neurodevelopment. I substitute MHD functions as the overarching mechanics. However this is still compatible with previous positions based on turing models.




Figure 6: (a and c) shows thin layer ferromagnetic and fluid patterns are similar to cortical folds and turing patterns. Underneath the thin layer patterns columns form similar to cortical columns.(c) The brain veins are precisely aligned to these folds, more so than arteries. (d) As veins carry paramagnetic blood are these aligned due to magnetic forces ? (see section 2.5)

These cortical patterns (which are sometimes random) are predicted to result from an inversion of the underlying harmonic mode. My reasoning is that calcium ion flow is forced to evolve a flow which can push against the earths static field. To achieve this field strength, ion flow becomes fragmented and time independent (meaning it loses the harmonic modes) inverting itself into more powerful dipole pulses (see work by fleury above). This is (one reason) why the cortex has magnetic dipole type structure.  

For skeptics who claim the magnetic dipole appearance is a co-incidence, it should be borne in mind that the cortex surface is predicted latterly (in past decade) to have magnetic mechanisms at play, as remnants of the radial glial (astrocyes) are being proposed as the missing key to where short term memory is sustainted  (Ingber & Nunez, 2010 , Bokkon & Banaclocha, 2010 , Pereira & Furlan, 2010, Størmer & Laane, 2009). The key point is we are not without the required radial glia magnetic mechanism here, and by studying morphology further we can further differentiate between a cortical dipole formation which arises due to electro-osmosis or MHD force.

Some questions for the authors (They received my communication in 2009). Their mathematics involves recruiting a classic wave mechanic equation (Helmholtz) and timelimiting its function for the prolate spheroidal wave functions,



As mentioned at the end of this post here, I would advocate the use of harmonic wavefront analysis for the limbic systems development. However in this paper the explanation for how such truncation of time maps or bandpassing these functions maps out to developmental time have not been worked to remain consistent with the limbic / cortical development relationship famously mapped out to a statistical relationship across many species by Finlay / Darlington.   As I predict the harmonic modes will map out in the manner i state above, perhaps future research can clarify which mathematical model is more consistent with Finlay / Darlingtons amazingly wide ranging statistical maps.

The authors will not communicate with me on this matter, so I can only guess that Hurdal and Streigel either do not accord with my approach here and/or competitive elements are at play giving rise to constraints. I mentioned  to them I had been working on this since 2003 (accepted to NQ archives May 2009), while their paper states they are the first to reach this conclusion in september 2009.   I have been about 50/50% successful co-operation wise with other scientists so far, which is not a bad result considering how competitive this field is.

NOTE:  This model I propose here Limbic EMS and harmonics in general was never a hypothesis I had been comfortable with anyway,  when it popped out the result I found in 2003. I had come in strongly from the cortex dipole end to break the brain system apart and there was nothing at all going for the concept of spherical harmonics except it appeared like a good fit.  However it was important to publish this in 2009 with its problems, as the limbic system would have to remain consistent with the approach taken for Cortical EMS and the MHD models.  It is more reassuring that independent work now exists coming from a more rigorous maths and developmental view which tries to build a case for Limbic models derived from spherical harmonics.


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