# Mathematical Modelling

• Erick Arguello asked a question:
New
How to model a tonically-active neuron?

Some kind of neurons show a tonically active behavior in absence of stimulation, either by firing action potentials or releasing neurotransmitter (e.g. inhibitory Off-cells from rostral-ventromedial medulla and photoreceptors). Can the current spiking neuron models (e.g. Hodgkin & Huxley, Izhikevich, FitzHugh-Nagumo, ...) be used to model this particular kind of neuron?

10
Is There a Mathematical Model of a Plastic/Adaptive Humain Brain?

I am working on Brain Computer Interface and would like to ask the scientific community if there is an existing mathematical model that can simulate neural adatation of the brain more importantly for application in Sensorimotor rythms based Brain Computer Interface.

Your Input will be of great help.

Thanks.

Thank you Loopinder Sood,

I will have check on the links you have provided.

Thank you for these ressources.

4
Prodromic and early Alzheimer´s disease detection using MRI...What do you think?
Creating a mathematical model of the brain changes across time from the evolution of 116 anatomical structures from more than 2100 MRI studies of Alzheimer´s disease patients (CDR=0.5) and healthy subjects showing the possibility of an early detection of this disease since it is possible to describe the neurodegenerative pattern.

The latest test for early detection

Two studies explore potential new blood tests for Alzheimer's disease
Published on November 10, 2015 at 1:25 AM · No Comments

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There is increasing evidence that the brain changes of Alzheimer's disease begin decades before memory and thinking problems occur, prompting the need for better methods of early detection for this progressive, fatal brain disease. Consequently, there is a growing school of thought that the most effective future Alzheimer's drug therapies will be administered to those who are at high risk of the disease before cognitive symptoms appear.

To bolster development of a simple, inexpensive, noninvasive test that can detect the risk of Alzheimer's disease, the Alzheimer's Association, the Crnic Institute for Down Syndrome, and the Global Down Syndrome Foundation ("Global") are funding two studies of potential new blood tests for Alzheimer's, including one that uses just one drop of blood:

One study will evaluate whether examining changes in ribonucleic acid (RNA) found in one drop of blood can accurately identify people who will develop Alzheimer's in individuals with Down syndrome who are at high risk for the disease. The study is being led by Marwan Sabbagh, M.D., Director of the Alzheimer's and Memory Disorders Division at the Barrow Neurological Institute in Phoenix, and Matt Huentelman, Ph.D., Associate Professor in the Neurogenomics Division Unit at the Translational Genomics Research Institute in Phoenix.
Another study will test whether a specific set of blood proteins can identify who is at risk for developing Alzheimer's in a unique, high -risk population, individuals with Down syndrome. The study is being led by Nicole Schupf, Ph.D., M.P.H., Dr.Ph.H., Professor of Epidemiology at Columbia University Medical Center in New York City, and Sid O'Bryant, Ph.D., Director of the Center for Alzheimer's and Neurodegenerative Disease Research at the University of North Texas Health Science Center in Fort Worth.
"Prevention of Alzheimer's dementia may be more effective and easily achieved than attempting to treat the disease once symptoms already exist and irreversible damage to the brain has already occurred," says Dean Hartley, Ph.D., Director of Science Initiatives for the Alzheimer's Association. "For this approach to be successful, we must be able to simply and accurately assess risk early in the disease process. The Alzheimer's Association and the Global Down Syndrome Foundation hope that these two exciting projects drive that effort forward."

"Autopsy is still the only way to definitively diagnose Alzheimer's disease," said Michelle Sie Whitten, President and CEO of the Global Down Syndrome Foundation. "If these researchers are successful we will be one step closer to catching Alzheimer's in its early stages and hopefully then be able to treat people with the disease earlier and actually prevent dementia from occurring, when new treatment options become available."

The grant awards are part of \$1 million in new funding for Down syndrome-related Alzheimer's disease research. Four projects will receive \$250,000 each through the joint funding effort.

Nearly all adults with Down syndrome begin developing the brain changes of Alzheimer's in their 30s. By age 55 or 60, it is estimated 55-70% will develop dementia. Because people with Down syndrome are at high risk for Alzheimer's, answers to important research questions about the disease may be developed more quickly in this population than by studying people with sporadic, late-onset Alzheimer's, where symptoms appear most often after age 65 - and in many cases not until the 70s or 80s.

"It used to be common for individuals with Down syndrome to die in their 30s, but because of medical advances they are now regularly living into their 50s and 60s. The irony is that they are now facing dementia due to Alzheimer's disease," says Huntington Potter, Ph.D., Director of Alzheimer's Research at the Crnic Institute for Down Syndrome and a Professor of Neurology at the University of Colorado, Denver. "At the same time, questions about Alzheimer's may be answered more quickly by studying this disease in people with Down syndrome because of their high risk for Alzheimer's and the earlier onset. Through this approach, people with Down syndrome have the opportunity to further our understanding of Alzheimer's disease and we have the opportunity to help this population."

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Scientists are not sure exactly why individuals with Down syndrome are at high risk for Alzheimer's disease but past research shows that a gene on chromosome 21 codes for the amyloid precursor protein (APP) that gets cut into fragments that accumulate into the hallmark amyloid brain plaques of Alzheimer's. People with Down syndrome are born with an extra copy of chromosome 21.

"The hope for our study is that the identification of RNA biomarkers for Alzheimer's could be used in a non-invasive blood test that requires just one drop of blood to assess an individual's risk of developing the disease, similar to the way a person with diabetes checks their blood sugar," says Sabbagh. "If we can learn early on that a person is at risk, the goal would be to start preventative therapies immediately. This could be a game changer."

"Our research could provide new information about potential biomarkers, including protein changes detected in blood, that could more accurately and easily predict the risk for Alzheimer's disease in people with Down syndrome," says Schupf. "If successful, we believe there is a chance that these biomarkers could also be used to assess Alzheimer's risk in all groups of people."

The Alzheimer's Association is the largest nonprofit funder of Alzheimer's research, having awarded more than \$350 million to over 2,300 projects since 1982. The Association currently supports more than 350 ongoing research projects in 21 countries totaling more than \$82 million.

The Global Down Syndrome Foundation raises funds for the Crnic Institute for Down Syndrome to underwrite critical research benefiting people with Down syndrome. To date, \$5.7 million in research grants has been given to 33 investigators.

The two other research projects the Alzheimer's Association and Global are funding through the joint grant award effort are:

A test of a potential Alzheimer's drug treatment that reduces levels of toxic protein fragments in the brain of a mouse model of Down syndrome. The project is led by William Mobley, M.D., Ph.D., Chair of the Department of Neurosciences at the University of California, San Diego (UCSD) and Executive Director of UCSD's Down Syndrome Center for Research and Treatment.
A study to determine whether a protein called Dyrk1A influences the build-up of brain proteins that lead to the formation of plaques and tangles that are key features of Alzheimer's in a mouse model of Down syndrome. Dyrk1A is created by one of the genes on chromosome 21 and is overabundant in the brains of people with Down syndrome. The study is led by Fei Liu, Ph.D., Head of Molecular Neuroscience for the Research Foundation for Mental Hygiene, Inc. at the New York State Institute for Basic Research in Staten Island.
Source:
Alzheimer's Association

15
What is the most reliable package for c++ which calculates eigen vectors?
I need to calculate the left eigen vector of a Matrix...is Eigen a good package?

Hi,

I've started using the Eigen.  I was wondering if it is possible to find the left eigenvector corresponding to the largest eigenvalue of a matrix? Thanks

9
Can anyone help with a Particle Swarm Optimization algorithm?
Can anyone help me with a PSO algorithm? Is there any c/c++ programme available?

Can anyone help me with a PSO algorithm? Mainly in c++. Thank you :)

1
Can we write requirements through Simulink Specification Models? If so, can they be used as inputs to Simulink design models?

Models are considered as requirements.  A new set of models known as specification models in Simulink by mathworks are introduced. My question is can we use these specification models as requirements rather than having requirements in the form of text.

It is may be easier to script the requirements as a configuration file and load the variables to workspace and then proceed to simulink.

2
Defining Poisson distribution via GAMS software?

I would appreciate it if anyone could help me with this problem of mine, I have a random variable N which is following Poisson distribution and another Variable named M which is calculated based on other things and I know it is being calculated correctly cause I solved the example by hand. The problem is the part where GAMS wants to calculate the probability of N

The CODE:

*Number of emergency patients arriving at time period t on day d
Table N(t,d)
1 2
1 1 1
2 1 1
3 1 1;

file emp / '%emp.info%' /;put emp '* problem %gams.i%'/;
\$onput
randvar N(1,1) poisson Ltotal
randvar N(2,1) poisson Ltotal
randvar N(3,1) poisson Ltotal
randvar N(1,2) poisson Ltotal
randvar N(2,2) poisson Ltotal
randvar N(3,2) poisson Ltotal
\$offput

put "randvar N(1,1) poisson Ltotal", N('1','1')/;
put "randvar N(2,1) poisson Ltotal", N('2','1')/;
put "randvar N(3,1) poisson Ltotal", N('3','1')/;
put "randvar N(1,2) poisson Ltotal", N('1','2')/;
put "randvar N(2,2) poisson Ltotal", N('2','2')/;
put "randvar N(3,2) poisson Ltotal", N('3','2')/;

file emp1 / '%emp.info%' /;put emp '* problem %gams.i%'/;
\$onput
chance (N(t,d)<=M(t,d)) Re(t,d)
\$offput

Dear Changtong,

I really appreciate that. Your comment and sent file are really useful but the main problem is in calculating cumulative distribution function of Poisson variable not generating a random variate with Poisson distribution.

I wrote the following code too to calculate the CDF of a Poisson variable (P(N<M)) but I get the Unknown symbol error for M which is a variable that has been calculated from some other things in my model.

CODE:

*Re(t,d)=Reliability of accepting emergency patients arriving at time period t on day d

\$funclibin stolib stodclib
function cdfpoisson /stolib.CDFPoisson/;
Re(t,d)=cdfpoisson(M(t,d),Ltotal);

8
Does the following problem have an exact closed form solution?

Consider a rectangular elastic isotropic clamped plate with Kirchhoff-Love theory. Does the vibration modes of this plate have any exact closed form solution?
The mathematical formulation of the problem is in the attached picture.

Dear Marcus

Thanks for the references. I am just looking for exact closed form solution not an approximate one. I knew that this problem has many approximate closed form solutions by Ritz and Galerkin methods.

11
Why do ordinary differential equation (ODE) models of cancer suggest different behaviors for cancer cells?

For validation part of my study, I need a comparison between my model of ductal carcinoma in situ (DCIS) and ODE models of this area. But, I’m really confused because ordinary differential equation (ODE) models of cancer have suggested different behaviors for tumor and immune cell populations. For example, the below behaviors are reported by the survey of Eftimie et al. (2011) [1]:

• Tumor size decreases exponentially through interactions with the immune cells.
• Tumor size decreases at first. Then, the decay of immune cells leads to an exponentially increase in it again.
• Tumor size decays in an oscillatory manner.
• Tumor size grows in an oscillatory manner.

I don’t understand the reason of the difference! And, I don’t know which behavior is right. Could anyone possibly help me, please?

------------------------------

[1] Eftimie, Raluca, Jonathan L. Bramson, and David JD Earn. "Interactions between the immune system and cancer: a brief review of non-spatial mathematical models." Bulletin of Mathematical Biology 73.1 (2011): 2-32.

I agree with Joseph Malinzi. It might depend on the chosen parameters to obtain a different behavior of course, and certainly you can use a logistic or Gompertz growth term in your model equation, but as Dr. Malinzi says, this has nothing to do with the interaction with the T cells.

5
How to model mathematically plate and shell structural damage subject to blast and explosion loading?

I find some paper for this topic in low-velocity dynamic loading on stiffened plate(like as grounding of ship bottom hull),but can not find in high-velocity range like explosion(internal or external). all of this re souls are in experimental or numerical method. How to model mathematically plate and shell structural damage subject to blast and explosion loading?

beat regard SAM

3
How is the evolution of contact between the asperities of two contacting bodies?

Our team is studying the problem of thermal contact resistance which depends on the contact between asperities. We are trying to capture the transient phase before a steady state is reached. So we are interested in results of analytical or numerical or experimental investigations into the deformation of asperities just after the contact is established and until the final contact patch is formed. Geometry or material of the two bodies is no constraint. Thanks.

In a very interesting problem in the field of fluid mechanics, published in the Journal of Fluid Mechanics studied the problem of two approaching spheres with different asperities. The contact between these rough particles induced a hydrodynamic diffusive coefficient. Maybe you could find some interesting references there. Here is the link of this manuscript:

http://www.damtp.cam.ac.uk/user/hinch/publications/JFM309_211.pdf

18
Which is the most efficient algorithm/package to solve delay differential equations?

I wish to know the methods of solving a system of equations that require, and those which don't require computer algebra. Is there a method, which is both theoretically and computationally efficient?

"to solve" could mean algebraic, numerical, or approximate.  If you want to construct approximate emergent models near bifurcations (pitchfork, Hopf, higher order), you can use my web service that constructs a center manifold model of ODEs or DDEs for systems a user enters.  My web page gives five examples you can try including a Double Hopf bifurcation in a delay DE.

10
What is the best time complexity of this case?

Assume that the integer numbers s, K, and ui, for i=1,2, ..., m, and ai, for i=1,2, ..., m, are given. I would like to know what is the time complexity of finding out that if the following system has an integer solution, say X=(x1x2, ..., xm), or not. Actually, it does not matter to find all the solutions, and I only want to know whether there is any integer solution for the system or not.

The system:

1. x1 + x2 + ... + xm = s
2. xi <= ui,  for i = 1, 2, ..., m
3. a1x1 + a2x2 + ... + amxm <= K

We know that one can find all the solutions of the (1) and (2), and then check each solution for the inequality (3). However, we think this is not that much efficient. I highly appreciate your consideration in advance.

4
Does anyone know the mathematical model/transfer function of the 3-phase inverter?
Even if you know for single phase, its fine.
I need to develop a controller for three phase inverter. To design controller, first I need to know the mathematical model/transfer function of the 3-phase inverter.

The transfer function of three phase inverter is available in Electric motor drives by R.Krishnan, Chapter 8, pp.495-496

18
Do we need a new definition of fractals for big data? Or must fractals be based on power laws?

So far definitions of fractals are mainly from mathematical point of view for the purpose of generating fractal sets or patterns, either strictly or statistically; see illustrations below (Figure 1 for strict fractals, while Figure 2 for statistical fractals; Fig. 4 for fractals emerged from big data):

Big data are likely to show fractal because of the underlying heterogeneity and diversity. I re-defined fractal as a set or pattern in which the scaling pattern of far more small things than large ones recurs multiple times, at least twice with ht-index being 3. I show below how geographic forms or patterns generated from twitter geolocation data bear the same scaling property as the generative fractal snowflake.

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541, Preprint: http://arxiv.org/ftp/arxiv/papers/1305/1305.0883.pdf
Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint: http://arxiv.org/ftp/arxiv/papers/1501/1501.03046.pdf

The new definition of fractals enables us to see the fractals emerged from big data. The answer to the question seems obvious. Yes, we need the new definition. BUT, some of my colleagues argued that the newly defined fractals are not fractal anymore, because they do not follow power laws.

+ 2 more attachments

I understand, so a new definition is in order. The metric aspect is less important.

40
Why are physicists stuck with Fortran and not willing to move to Python with NumPy and Scipy?

Nowadays all of the major Fortran related numerical calculus have exactly mapped equivalent libraries in more modern language framework like Numerical Python (NumPy) and Scientific Python (SciPy).

What keeps physicists stuck with Fortran?

Performance?

Portability?

Scientific evidence?

+ 1 more attachment

Btw, I like the approach of Software Carpentry web site on one of their Python online course for scientists:

the time to arrive to a solution is given by:  Ts = Td + Te

where

Ts : time to the solution

Td: time to focus on the problem, think of an algorithm, write a program, debug it

Te: time of execution

Nowdays Td>>Te, that is why i am definitively stuck with Python/Numpy/Scipy.

Cheers!

9
Does anyone have done a Co-IP for galpha protein with a GPCR?

I cannot detect my galpha protein when I immunoprecipitate it with the receptor.

See lysis buffer and IP conditions in:

Gu YZ, Schonbrunn A 1997 Coupling specificity between somatostatin receptor sst2A and G proteins: isolation of the receptor-G protein complex with a receptor antibody. Mol Endocrinol 11:527-537

2
What is the physical interpretation of local and global stability of a disease free or endemic equilibrium in disease modelling?
How can you explain the local and global stability of DFE or EE to a non-mathematician?

Thank you Konstantin K. Avilov

3
Why the dualization algorithm are not much used in traffic assignment problem with the static demand ?

I want to know why the direct methods are more qualified than the dual methods for solving the traffic assignment problem with the static demand

I need advice, what direction I should take in my search?

Here is the local RG link:

• Source
##### Article: Primal convergence from dual subgradient methods for convex optimization
[Hide abstract]
ABSTRACT: When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.
Mathematical Programming 05/2014; 150(2). DOI:10.1007/s10107-014-0772-2
8
How to check whether a system is Linear or Nonlinear?
System is a black box.

Very stimulating question!

I would visualize it in order to see whether it is nonlinear or not, e.g., urban growth and evolution of social media.

Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint: http://arxiv.org/abs/1501.03046

Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886,  Preprint: http://arxiv.org/abs/1401.6756

+ 1 more attachment

4
Does anyone know a good source on scaling-up models/techniques for chemical pilot plants to industrial scale?
I am trying build a process model for a bioenergy system. In the literature, there are sufficient lab scale and pilot plant models for this process, but much less industrial (commercial) level data. I am looking for a book or a paper discussing what options we have to mathematically scale up a pilot plant model to industrial level. thanks!

hi there

you think you could find some details on the book titled "Marco Zlokamik; Scale Up in Chemical Engineering".  more over there is another good as well as more complete on scale up, titled "Applied Dimensional Analysis and Modeling".  best

2
How do I obtain the duality system of a Caputo fractional-order distributed parameter system?

For integer-order distributed parameter system, we can get the dual system by utilizing the duality theory. However, for frational order distributed parameter system. the results is few

ok， Thanks very much .  Prof.Grabowski.   I will check them soon----

4
How can data envelopment analysis (DEA) theory be modeled in nonlinear instead of linear one?
Is it possible to develop a model of DEA in nonlinear functions instead of linear?
5
Even when using regularization techniques in linear inverse problems is it recommendable to reduce the condition number previously?

I found empirically that previous reduction of the condition number is useful even when using for instance Tikhonov-Philllips regularization, however I don't have theoretical backup on this. Has anyone a book or paper which can support this? Thank you in advance

Try the other way around. Regularize first and then reduce the condition number. It makes more sense to me.

50
What are the limits of measurement in science?
When I was in high school Bohr's atom of shells, s and p orbitals was introduced in chemistry. Realization was automatic that the world was explained according to theory that was verified by experiment. Through college and graduate school, looking for more complete explanation, theory is challanged but it is not brought to question "what is an electron or proton, if they have mass but are visible only in the sense that they emit light energy as photons that also have mass, "spots of light in orbit around nuclei?, the atom a solar system in minature"? Physicists will say this is not the picture they have evolved, but all that remains is the image of equations on a chalkboard, at best 'the image of things of a particle nature in alteration with things of a light nature'. Can a pieced-together stepwise reality of this nature be accepted? In the Feyman quote below pieces are added that can break any of the established laws "they are not directly observeable" or affect "causality". In this same meaning though neither electrons, protons, photons or atoms are observable and their causal effects are but a matter of humanly constructed theory and similarly based experimental apparatus. The possibility exists that theory and theory based apparatus entail one another and all that might be gotten is that the real universe is identical in this respect...i.e. existence entails the experienced universe and visa-verse.
"You found out in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light! It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c." These virtual photons, however, do not violate causality or special relativity, as they are not directly observable and information cannot be transmitted causally in the theory." (from "Varying c in quantum theory" http://www.researchgate.net/go.Deref.html?url=http%3A%2F%2Fen.wikipedia.org%2Fwiki%2FVariable_speed_of_light)

The limits of measurement keep on changing, and sometimes the errors of measurement are not too good.

99
How can I mathematically model the behaviour of the humans?

Any suggestion/resources are appreciated.

3
Which quadrature rule can be used in 3D BEM to evaluate hypersingular (1/r^2) Hadamard integrals?

I am developing a 3D boundary element code for problems of elastostatics. This is not the focus of my research, I just need it as an instrument. I look  for a simple way of numerical evaluation of Hadamard finite part integrals, that occur in 3D fundamental solutions for tractions. These can be either evaluated on a triangular element or boiled down to 1D integrals of a kind: Integral from 0 to 1 f(x) dx / x^2. Papers on the topic that I've found seem to be messy and contradictory, nobody gives Gauss-like quadrature with points and weights. Any help is greatly appreciated. Thanks!

There is no need to evaluate the hypersingular integral in order to get the stresses at the boundary. You can use a technique called "stress recovery". The method uses the nodal displacement tangential derivatives of a boundary element to determine the tangential strains at the point of interest. Then, by using Hooke’s law and the equilibrium condition at the boundary, the local stresses can be recovered which then have to be transformed back to the global coordinate system.

1
Which is the best model to analyse the dynamic behavior of Magnetorhological Elastomer?

I am in need of Mathematical model of MRE to study its dynamic behavior under varying magnetic field to design MRE based Dynamic vibration absorber

5
What are the best ways to do the mathematical modelling of Notch and Wnt pathway?

Need your opinion related to algorithms like Genetic Algorithm and others......

Thank You so much for informative discussions...

9
Is there any software to determine the number of codewords with fixed GC-content in DNA codes?
If so where can I find it?

hi again

cloud you please help me with writting C-programming in order to compiling number of G and C, as you mentioned to me.

I myself have problems by C-programming and cannot write a construction algorithm for compute direct content 01.

best regards