I just gave a seminar in the math department at the University of Iowa today. I gave a talk that was similar to the one I gave at the Mathematical Biosciences Institute on Bayesian Inference for dynamical systems. My slides are here. I was lucky I made it to give this talk. My flight from Chicago O’Hare to Cedar Rapids, Iowa was cancelled yesterday and I was rebooked on another flight tonight, which wouldn’t have been of much use for my talk this afternoon. There was one fight left last evening to Cedar Rapids but bad weather cancelled several flights yesterday so there were many stranded travelers. I was number 38 on the standby list and thought I had no chance to make it out that night. However, on a whim I decided to wait it out and I started to move up the list because other people had evidently given up as well. To my great surprise and relief I was the last person to get on the plane. There was a brief scare when they asked me to get off because we exceeded the weight limitation but then they changed their mind and let us all fly. (Someone else had kindly volunteered to take my place). I learned two lessons. One is to keep close watch of your flight at all times so you can get on the standby list as soon as possible and two is that even number 38 on a plane that only seats 50 can still make it.
It is often recommended that when investing money in a mutual fund or security to use a strategy called dollar cost averaging. This simply means you invest a fixed amount of money at periodic intervals rather than say buy a fixed number of shares at periodic intervals or invest as a lump sum. The argument is that when the price of the fund is low you buy more shares and when it is high you buy fewer shares. In this way you can ride out the fluctuations in the stock price.
Evaluating which strategy is best is a great example of how concepts in calculus and in particular Jensen’s inequality enter every day life and more reasons for why an average person should learn more math. The question is how do the investment strategies compare over time. Let be the price of the fund at time . In dollar cost averaging, you would invest dollars at periodic intervals (say each week). So the number of shares you would buy each week is . Over a time period you will have purchased shares, which can be rewritten as , where is the expectation value or average. Since you have invested dollars, you have paid per share on average, which is also called the harmonic mean.
If on the other hand you decided to buy a fixed number of shares per week then it would cost . Over the time period you will have spend and have shares. Thus you have paid on average. Now the question is how does the mean compare to the harmonic mean . This is where Jensen’s inequality comes in, which says that for a convex function , . A convex function is a function where a straight line joining any two points on the function is greater than or equal to the function. Since is a convex function, this means that which can be rearrange to show that . So dollar cost averaging is always better than or as good as buying the same number of shares each week. Now of course if you buy on a day when the price is below the harmonic mean of a time horizon you are looking at you will always do better .
S. Vattikuti and C.C. Chow, ‘A computational model for cerebral cortical dysfunction in Autism Spectrum Disorders’, Biol Psychiatry 67:672-6798 (2010). PMID: 19880095
PDF available here.
Shashaank Vattikuti was a medical student and wanted to do a rotation in my lab. He had done some pediatric rotations and was frustrated at the lack of treatments for autistic children. He thought that a better biophysical understanding of the neural activity that caused autism was necessary to make progress. The conventional wisdom is that autism is due to some problem in global connectivity in the brain. This makes sense because neural imaging data seems to show that different regions of the brain seem to be less functionally connected in autistics. However, Shashaank thought that the deficit was probably a local microscopic one and that the global perturbations were due to the brain’s attempt to compensate for these deficits.
He found papers that showed that cortical structures called minicolums (on the hundred micron scale) were denser (closer together) in autistics. This immediately set off a flag in my head because I had previously shown for spiking neurons (e.g. Chow and Coombes) that localized persistent activity (often called a bump) was more stable if the neuron density increased. Bumps in networks of spiking neurons tended to wander around for small numbers but stabilized as the density increased. Shashaank also found genetic and physiological evidence that the synaptic balance is tilted towards an excess of excitation in autistics.
The real breakthrough in making this more than an academic exercise was that Shashaank also found a simple behavioral task to model, where subjects visually fixate on a point for a certain amount of time and then shift their gaze to a target when instructed. Most people will undershoot and this is called hypometria and make errors called dysmetria. The data was mixed but it seemed like autistics had more hypometria and dysmetria. The way we implemented this visually guided task in the model was to consider a one dimensional network of excitatory and inhibitory neurons. The parameters were tuned so that when a stimulus was applied at a given position a bump of firing neurons would form. The fixation point was represented by a stimulus applied to a location in the network. We then stimulated at another location to indicate the saccade target and tracked how the bump moved. We found that when there was an excess of excitation, hypometria and dysmetria both increased. However, when the minicolumn structure was perturbed, only hypometria increased.
Before Shashaank ran the simulations, I really wasn’t sure what would happen. Given more excitation, it is plausible that the bump would move more quickly and hence reduce hypometria with increased excitation. Instead, the excessive excitation makes bumps more persistent and stable so it is harder to move them once they are established. Hence, our result hinges on there being prior neural activity that must be moved in a saccade task. This is consistent with autistics having more difficulty switching mental tasks. Our hypothesis is that the underlying source of autistic symptoms arise from excessive local persistent activity. This excessive persistence is also why the effective connectivity between brain regions is reduced because each region is less responsive to external inputs. It also suggests that restoring synaptic balance with medication may alleviate some of these symptoms. We’re currently trying to devise ways to validate our hypothesis.
Casanova MF, van Kooten IA, Switala AE, van Engeland H, Heinsen H, Steinbusch HW, et al. (2006): Minicolumnar abnormalities in autism. Acta Neuropathol 112:287–303.
C.C. Chow and S. Coombes, ‘Existence and Wandering of bumps in a spiking neural network model’. SIAM Journal on Applied Dynamical Systems 5, 552-574 (2006) [PDF]