The sum of the natural numbers is -1/12?

This wonderfully entertaining video giving a proof for why the sum of the natural numbers  is -1/12 has been viewed over 1.5 million times. It just shows that there is a hunger for interesting and well explained math and science content out there. Now, we all know that the sum of all the natural numbers is infinite but the beauty (insidiousness) of infinite numbers is that they can be assigned to virtually anything. The proof for this particular assignment considers the subtraction of the divergent oscillating sum S_1=1-2+3-4+5 \dots from the divergent sum of the natural numbers S = 1 + 2 + 3+4+5\dots to obtain 4S.  Then by similar trickery it assigns S_1=1/4. Solving for S gives you the result S = -1/12.  Hence, what you are essentially doing is dividing infinity by infinity and that as any school child should know, can be anything you want. The most astounding thing to me about the video was learning that this assignment was used in string theory, which makes me wonder if the calculations would differ if I chose a different assignment.

Addendum: Terence Tao has a nice blog post on evaluating such sums.  In a “smoothed” version of the sum, it can be thought of as the “constant” in front of an asymptotic divergent term.  This constant is equivalent to the analytic continuation of the Riemann zeta function. Anyway, the -1/12 seems to be a natural way to assign a value to the divergent sum of the natural numbers.

Talk in Taiwan

I’m currently at the National Center for Theoretical Sciences, Math Division, on the campus of the National Tsing Hua University, Hsinchu for the 2013 Conference on Mathematical Physiology.  The NCTS is perhaps the best run institution I’ve ever visited. They have made my stay extremely comfortable and convenient.

Here are the slides for my talk on Correlations, Fluctuations, and Finite Size Effects in Neural Networks.  Here is a list of references that go with the talk

E. Hildebrand, M.A. Buice, and C.C. Chow, `Kinetic theory of coupled oscillators,’ Physical Review Letters 98 , 054101 (2007) [PRL Online] [PDF]

M.A. Buice and C.C. Chow, `Correlations, fluctuations and stability of a finite-size network of coupled oscillators’. Phys. Rev. E 76 031118 (2007) [PDF]

M.A. Buice, J.D. Cowan, and C.C. Chow, ‘Systematic Fluctuation Expansion for Neural Network Activity Equations’, Neural Comp., 22:377-426 (2010) [PDF]

C.C. Chow and M.A. Buice, ‘Path integral methods for stochastic differential equations’, arXiv:1009.5966 (2010).

M.A. Buice and C.C. Chow, `Effective stochastic behavior in dynamical systems with incomplete incomplete information.’ Phys. Rev. E 84:051120 (2011).

MA Buice and CC Chow. Dynamic finite size effects in spiking neural networks. PLoS Comp Bio 9:e1002872 (2013).

MA Buice and CC Chow. Generalized activity equations for spiking neural networks. Front. Comput. Neurosci. 7:162. doi: 10.3389/fncom.2013.00162, arXiv:1310.6934.

Here is the link to relevant posts on the topic.

New paper on neural networks

Michael Buice and I have a new paper in Frontiers in Computational Neuroscience as well as on the arXiv (the arXiv version has fewer typos at this point). This paper partially completes the series of papers Michael and I have written about developing generalized activity equations that include the effects of correlations for spiking neural networks. It combines two separate formalisms we have pursued over the past several years. The first was a way to compute finite size effects in a network of coupled deterministic oscillators (e.g. see here, herehere and here).  The second was to derive a set of generalized Wilson-Cowan equations that includes correlation dynamics (e.g. see here, here, and here ). Although both formalisms utilize path integrals, they are actually conceptually quite different. The first formalism adapted kinetic theory of plasmas to coupled dynamical systems. The second used ideas from field theory (i.e. a two-particle irreducible effective action) to compute self-consistent moment hierarchies for a stochastic system. This paper merges the two ideas to generate generalized activity equations for a set of deterministic spiking neurons.

Richard Azuma, 1930 – 2013

I was saddened to learn that Richard “Dick” Azuma, who was a professor in the University of Toronto Physics department from 1961 to 1994 and emeritus after that, passed yesterday. He was a nuclear physicist par excellence and chair of the department when I was there as an undergraduate in the early 80’s. I was in the Engineering Science (physics option) program, which was an enriched engineering program at UofT. I took a class in nuclear physics with Professor Azuma during my third year. He brought great energy and intuition to the topic. He was one of the few professors I would talk to outside of class and one day I asked if he had any open summer jobs. He went out of his way to secure a position for me at the nuclear physics laboratory TRIUMF in Vancouver in 1984. That was the best summer of my life. The lab was full of students from all over Canada and I remain good friends with many of them today. I worked on a meson scattering experiment and although I wasn’t of much use to the experiment I did get to see first hand what happens in a lab. I wrote a 4th year thesis on some of the results from that experiment. I last saw Dick in 2010 when I went to Toronto to give a physics colloquium. He was still very energetic and as engaged in physics as ever. We will all miss him greatly.

New paper on neural networks

Michael Buice and I have just published a review paper of our work on how to go beyond mean field theory for systems of coupled neurons. The paper can be obtained here. Michael and I actually pursued two lines of thought on how to go beyond mean field theory and we show how the two are related in this review. The first line started in trying to understand how to create a dynamic statistical theory of a high dimensional fully deterministic system. We first applied the method to the Kuramoto system of coupled oscillators but the formalism could apply to any system. Our recent paper in PLoS Computational Biology was an application for a network of synaptically coupled spiking neurons. I’ve written about this work multiple times (e.g. here,  here, and here). In this series of papers, we looked at how you can compute fluctuations around the infinite system size limit, which defines mean field theory for the system, when you have a finite number of neurons. We used the inverse number of neurons as a perturbative expansion parameter but the formalism could be generalized to expand in any small parameter, such as the inverse of a slow time scale.

The second line of thought was with regards to the question of how to generalize the Wilson-Cowan equation, which is a phenomenological population activity equation for a set of neurons, which I summarized here. That paper built upon the work that Michael had started in his PhD thesis with Jack Cowan. The Wilson-Cowan equation is a mean field theory of some system but it does not specify what that system is. Michael considered the variable in the Wilson-Cowan equation to be the rate (stochastic intensity) of a Poisson process and prescribed a microscopic stochastic system, dubbed the spike model, that was consistent with the Wilson-Cowan equation. He then considered deviations away from pure Poisson statistics. The expansion parameter in this case was more obscure. Away from a bifurcation (i.e. critical point) the statistics of firing would be pure Poisson but they would deviate near the critical point, so the small parameter was the inverse distance to criticality. Michael, Jack and I then derived a set of self-consistent set of equations for the mean rate and rate correlations that generalized the Wilson-Cowan equation.

The unifying theme of both approaches is that these systems can be described by either a hierarchy of moment equations or equivalently as a functional or path integral. This all boils down to the fact that any stochastic system is equivalently described by a distribution function or the moments of the distribution. Generally, it is impossible to explicitly calculate or compute these quantities but one can apply perturbation theory to extract meaningful quantities. For a path integral, this involves using Laplace’s method or the method of steepest descents to approximate an integral and in the moment hierarchy method it involves finding ways to truncate or close the system. These methods are also directly related to WKB expansion, but I’ll leave that connection to another post.

Mass

Since the putative discovery of the Higgs boson this past summer, I have heard and read multiple attempts at explaining what exactly this discovery means. They usually go along the lines of “The Higgs mechanism gives mass to particles by acting like molasses in which particles move around …” More sophisticated accounts will then attempt to explain that the Higgs boson is an excitation in the Higgs field. However, most of the explanations I have encountered assume that most people already know what mass actually is and why particles need to be endowed with it. Given that my seventh grade science teacher didn’t really understand what mass was, I have a feeling that most nonphysicists don’t really have a full appreciation of mass.

To start out, there are actually two kinds of mass. There is inertial mass, which is the resistance to acceleration and is mass that goes into Newton’s second law of  F = m a and then there is gravitational mass which is like the “charge” of gravity. The more gravitational mass you have the stronger the gravitational force. Although they didn’t need to be, these two masses happen to be the same.  The equivalence of inertial and gravitational mass is one of the deepest facts of the universe and is the reason that all objects fall at the same rate. Galileo’s apocryphal Leaning Tower of Pisa experiment was a proof that the two masses are the same. You can see this by noting that the gravitational force is given by

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New paper on finite size effects in spiking neural networks

Michael Buice and I have finally published our paper entitled “Dynamic finite size effects in spiking neural networks” in PLoS Computational Biology (link here). Finishing this paper seemed like a Sisyphean ordeal and it is only the first of a series of papers that we hope to eventually publish. This paper outlines a systematic perturbative formalism to compute fluctuations and correlations in a coupled network of a finite but large number of spiking neurons. The formalism borrows heavily from the kinetic theory of plasmas and statistical field theory and is similar to what we used in our previous work on the Kuramoto model (see here and  here) and the “Spike model” (see here).  Our heuristic paper on path integral methods is  here.  Some recent talks and summaries can be found here and here.

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