Archive for June, 2009

The mass of humanity

June 26, 2009

Ever since Malthus, there has been a concern about overpopulation.  I thought it would be an interesting excercise to see how much space the human population actually takes up.  For example, how many oil tankers would it take to carry around the volume of humanity if converted to liquid.  Let’s say there are 6 billion people on the planet and the average mass per person is 100 kg (this is an overestimate).  Hence, the upper bound on the mass of humanity is 10 ^{12} kg, or a billion metric tons.  Given that we are mostly water, we can assume that this is about 10^{12} litres.  Taking the cube root gives 10^4\times .1 metres or a kilometre.  Thus, if we liquefied the mass of all humans, it would fit in a cube whose sides are a kilometre long.   The largest oil tankers can carry about five hundred thousand metric tons, so two thousand oil tankers could cart around all of humanity.  To put that into perspective, according to Wikipedia, the current fleet of oil tankers moves around 2 billion metric tons a year, so half the world’s fleet could carry around the world’s population.

Now, how much area would we take up if we were to stand side by side.  Let’s say 6 people can fit into a square metre of space, then we would all be able to fit into a billion square metres or 1000 square kilometres, about the size of  Hong Kong (according to Wolfram Alpha), or we could all fit 4 to a square metre  onto the island of Oahu in Hawaii.  If we each wanted about 100 square metres of space, then we would take up about a million square kilometres or about twice the area of France.  Wolfram Alpha also tells me that there is about 1.5\times 10^7 square kilometres of arable land in the world.  If we assume that a square kilometre can feed 1000 people (10 people per hectare), then that puts the capacity of the earth at 15 billion people.

New paper on liver regeneration

June 19, 2009

I have a new paper that has just appeared in Biophysical Journal entitled “A model of liver regeneration ” by Furchgott, Chow and Periwal.  The liver has this remarkable property that if a portion of it is removed up to a critical fraction, it will grow back to approximately 10% of its original size.  The restoration does not recreate the original morphology of the liver but it does restore function.  Although this has been known since ancient times, it is still a puzzle as to exactly how the liver does this, especially how it knows to stop growing when it is back to its original size and why it does not oscillate.  Our paper proposes a simple model to explain it.

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Math and biology (and what an educated citizen should know)

June 13, 2009

At the NJIT conference two weeks ago, I was asked to sit on a panel on “Future roles for mathematics and statistics within the biological sciences”.  To start the discussion, the moderator Farzan Nadim asked two questions.  The first was on whether biology needed mathematics (the implication being that mathematics was not the basis of biology as it is for physics) and the second was how to educate biology students in mathematics and statistics. Mathematical ecologist Stuart Pimm of Duke, immediately jumped in to protest that in evolutionary biology and ecology, mathematics has always played a dominant role.  He also made the tongue-in-cheek quip that we didn’t need to train Americans in mathematics or science because we could always import them.  Robert Miura followed with some personal observations that there was a cultural divide between mathematicians and biologists, one example being that mathematicians look for dimensionless quantities to determine what’s big and small (e.g. important and unimportant) whereas biologists think in terms of dimensional quantities that make sense experimentally (e.g. it’s important to know if the current you are measuring is in picoamps or microamps).

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Talk at NJIT

June 3, 2009

I was at the FACM ’09 conference held at the New Jersey Institute of Technology the past two days.  I gave a talk on “Effective theories for neural networks”.  The slides are here.  This was an unsatisfying talk on two accounts.  The first was that I didn’t internalize how soon this talk came after the Snowbird conference and so I didn’t have enough time to properly prepare.   I thus ended up giving a talk that provided enough information to be confusing and hopefully thought provoking but not enough to be understood.   The second problem was that there is a flaw in what I presented.

I’ll give a brief backdrop to the talk for those unfamiliar with neuroscience.  The brain is composed of interconnected neurons and as a proxy for understanding the brain, computational neuroscientists try to understand what a collection of coupled neurons will do.   The state of a neuron is characterized by the voltage across its membrane and the state of its membrane ion channels.  When a neuron is given enough input,  there can be a  massive change of voltage and flow of ions called an action potential.  One of the ions that flows into the cell is calcium, which can trigger the release of neurotransmitter to influence other neurons.  Thus, neuroscientists are highly focused on how and when action potentials or spikes occur.

We can thus model a neural network at many levels.  At the bottom level, there is what I will call a microscopic description where we write down equations for the dynamics of the voltage and ion channels for each neuron.  These neuron models are sometimes called conductance-based neurons and the Hodgkin-Huxley neuron is the first and most famous of them.  They usually consist of two to four differential equations and can easily be a lot more.  On the other hand, if one is more interested in just the spiking rate,  then there is a reduced description for that.  In fact, much of the early progress in mathematically understanding neural networks used rate equations, examples being Wison and Cowan, Grossberg, Hopfield and Amari.  The question that I have always had was what is the precise connection between a microscopic description and a spike rate or activity description.  If I start with a network of conductance-based neurons can I derive the appropriate activity based description?

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