Mathematics and biology, my story

The interest for both biology and mathematics came to me from my teachers, my readings and my friends. When I was 6 I moved with my family, first to France (till 10) and then to Mexico. In the primary school my teacher Gallo showed me the Pythagoras theorem and a cross section ob the spinal chord. Both things did dazzle me. In my secondary studies the professors Ramirez and Navarro opened the curtains of botanics and zoology. For some years I was a passionate collector of insects with Gonzalo Halfter and other companions: the shapes and colors of some insects show more esthetic value that the works of Gaudi, Picasso, Dali, and Miró. You only have to look at a Cayena Arlequin (Acrocinus Longimanus) or at some Phaneus beetle. On the other direction, Ricardo Vinós and Marcel Santaló awakened on me the interest in mathematics and physics. (1939-1955).

In spite of such interests, I chose to study electrical and mechanical engineering, owing to a kind of familiar legend, but at the end of the first year my friend “Quico” Tomas encouraged me to study also the career of mathematics. While I was studying I met August
Pi- Sunyer, the grandfather of my girl-friend at the time, who was a specialist in neurophysiology and lived in Venezuela (as my family we were refugees of the Spanish civil war) He had published s book where I saw for the first time the drawings of the neural tissue on the brain cortex. Captivating. In finishing the undergraduate studies of mathematics, Professor Solomon Lefschetz of Princeton University offered me to go to his university for my doctorate (in topology!). I declined saying that I preferred finishing my engineering studies and dirtying a little bit my hands, and so I did for seven years working on the erection of thermo electrical power plants. While I was working as an engineer I kept going to the university, both lecturing and attending doctoral courses. One of these courses with Professor Martone taught me what the innards of a digital computer were: the logical gates, the flip flops, and the whole von Neumann architecture with its arithmetic unit, its control unit and its memory. The course interested me so much that I started reading papers on control and automatics. In the Automata Studies of Princeton I found a paper by von Neumann that dealt with logical circuits, with faulty gates (as long as they were more than fifty per cent reliable), which could be made, through redundancy, as precise as the brain of a mathematician!
Of course all this was building up in my mind a very foggy idea of how the brain works. (1956-1962).

My job as an engineer ended when the Construction Department of the Mexican Light and Power Company (the Mexican branch of the company that had been responsible of the electrification of Catalonia, the “Canadenca”), where I worked, was suppressed at the nationalization of the company. I went looking for a job at the Centro de Investigaciones y Estudios Avanzados whose director was Arturo Rosenblueth, who worked with Walter Cannon on chemical transmission along nervous elements and wrote with Norbert Wiener and Julian Bigelow “Behaviour, purpose and teleology” and of whom it is said that suggested the name “Cybernetics” for Wiener’s book. Rosenblueth had dealt with nervous impulse transmisions, synaptic transmissions, the control of blood circulation and the physiology of brain cortex. When I exposed to him my interest to work on something you would nowadays qualify as neural systems, Rosenblueth said he would give me a job if I obtained a doctorate.

I went back to Professor Lefschetz, who was no more at Princeton: he had been appointed the direction of the mathematical department of RIAS, the Research Institute for Advanced Studies that the Martin Co. had created in Baltimore. The institute was part of the effort of the USA to not be left behind the Soviet Union in the conquest of the extraterrestrial space. Lefschetz was no more in topology, but in the qualitative theory of differential equations. He gave me a place at RIAS to work in what had to be my thesis. At the beginning I had the idea of working with Rudy Kalman in control theory: observability and controllability, but there all was linear, so I found my place with Jack Hale who worked on differential equations with time lag, following the path of Krasnoselsky, Shimanov, and other Russian mathematicians. I suppose the interest in time lag obeys to the fact the signals to and from outer space take some time. As a matter of fact that happened also with the control by compressed air in the steam boilers in the power plants I had been helping to build. Of course nowadays compressed air is not used for control of boilers anymore!

I wrote my thesis, I got a doctorate on applied mathematics from Brown University and I went back to Mexico. There, good at his word, Rosenblueth gave me a research job at the CIEA. There I didn’t make good my expressed interest for neural systems, in spite of the fact that there were people studying them (but not with mathematical models). I remember watching the electrical spikes travelling along the axons of the neurons in the brain of a cat (which reminds me that Rosenblueth, had said, referring to the complexity and variability of living systems something like “the best model of a cat is a cat, especially the same cat!”). I didn’t keep a long time working at delay equations; instead I was captivated by the dynamical systems of Stephen Smale and his school, with their topology, their strange attractors and their chaos. Later I got into area preserving mappings, (where I lost the opportunity of publishing a paper because I wasn’t fast enough and someone advanced me in the publication of the results) and Hamiltonian systems, with their Arnold tori, the series with small divisors and the strong implicit functions theorems of John F. Nash and Jacob T. Schwartz. (1962-1972).

I came back to Catalonia 34 years after having left it, to the UAB, carrying with me my working themes. Eventually, at this seminar we dealt with population dynamics (Calsina, Saldaña, Aldama, Madrid...), the uncoupling of the slow and fast dynamics, which I named “dynamical dynamics”, i.e. metabolism against growth (Bonet). Around 1977 Xavier Mora presented us with the paper of L.A.Segel on the acrasial amoebae (slime moulds), unicellular individuals which group in the shape of hexagonal tiles on the horizontal support, and grow a vertical talus which develop a capsule at its extremity that fills up with the spores for new individuals. Segel included in the paper the reaction and diffusion equations proposed by Alan Turing to explain morphogenetic phenomena. These equations admit a spatially constant solution that may become unstable with the change of the parameter. I studied for some time the mathematical aspects of the bifurcation: the values of the parameter for the losing of stability of the homogeneous constant solution and the shape of the bifurcating new equilibria.

In a book by J.D.Murray on nonlinear differential equations in biology, the reaction and diffusion equations are used to explain the dots and stripes in the animal skins and a prey-predator system in a one dimensional region. The same kind of equations were used by Nicolis and Prigogine and by Gierer and Meinhardt to explain self-organizing phenomena in biology, in social systems and even in the evolution of consciousness.

For years I was thinking in reaction and diffusion equations as the mechanism to model the division of the fertilized ovules. As a matter of fact it could have been any cell, but I spoke always of the egg! I considered it as a vessel with different reacting substances in a liquid medium (water?), whose concentrations would obey the equations. The unstabilizing parameter would be the size of the egg that was supposed growing (in spite of the fact that I was supposing a Newmann condition at the border!); at some critical size a bifurcation would ensue concentrating the substances in the so called “poles” of the cell, according to the eigenfunctions of the Laplace operator in a first approximation, promoting its division. The mathematics was (and is) simple and exciting. What a pity that the whole conception was utterly wrong! In a meeting in Lisbon, when I spoke to Dan Henry about my view, he told me that cell division was more mechanical than a question of continuous media. I didn’t hear him, or was it that I didn’t want to hear him?

Quite recently a book fell in my hands that dealt with biological morphogenesis. I was quite sceptical, thinking that the author was not acquainted with Turing’s conception. But no, he was quite acquainted and he mentioned that Turing’s scheme had not explained any single morphogenetic event in biology. And then I remembered my lessons of biology at the secondary school. I remembered the Golgi system with its microtubules, the chromosomes, and more impressively, the genome. What a different landscape! Transported by the magic carpet of mathematics I had overlooked what is obvious: That the cell is not a chemical reactor, but rather a complicated system where the proteins that are being synthesized thanks to the genome constitute the building elements of the structure of the body. Have you heard about tensegrity, or about the protein engines?

So, it is quite recently that I corrected (I hope) my view. Even the segmentation of the body is produced, not by reaction and diffusion, but through the Hox genes, for instance. Murray explained the marks in the skin of leopards with reaction and diffusion; I doubt you can treat the skin of a leopard as a chemical reactor. I had had some lack of respect for treating these formations of stains on the skin as the result of a molecular (or cell) automaton. Now I wouldn’t laugh at it.

Around 1989, we dealt with ecological population dynamics (natural selection) plus mutation, that is, Darwinian evolution. In the model a population with the same genetic pool was considered and the mutation of some hereditary characteristics was modelled as a diffusion of some of these characteristics. The selection term came from the ecological medium where the characteristic was relevant for the survival of the population.

In different models we could have an internal competition for the resources within the same population or we could consider a competition for resources among different populations, and also some pray predator interactions. With Calsina we worked on this; there are theorems on existence and uniqueness as well as on the existence of fixed points and their stability and on the asymptotic behaviour of solutions. Everything expressed in the frame of semi linear parabolic systems with non negative solutions.

The model on which we wrote a paper had some shortcomings: if a mutation appeared in one individual its effect on the rest of the population was instantaneous. Moreover the mutations were commanded by the Laplacian operator and they couldn’t admit other distributions. In the thesis of Manuel Sanchón (2003) we corrected these shortcomings putting an integral operator instead of the Laplacian and admitting a time lag to take into account a maturation age, that is, the time elapsed between birth and reproduction on the same individual.

Of this model we obtained theorems of existence, uniqueness, continuity and continuation for all time. We could find the equilibrium points, study their stability and the asymptotic behavior of solutions. With Angel Calsina and Manuel Sanchon we worked on it and we wrote a paper which was not accepted for publication in the journal where we sent it (!). May be we shall send it somewhere else, but time goes by….

We did not occupy ourselves on the special (geographical) distribution of the populations, which sounds very interesting to me. Anyhow, something will be said by someone, some time, on the subject. Now Calsina and his collaborators are modeling the control of bacterial infections with mutating viruses: a useful accomplishment.

In recent time the development of the whole cellular system begins to be understood; and the first that has been understood is that it is very complex: from the workings of the genoma to the protein engines, from the microtubules to tensegrity, from the “hox” genes to the segmentation of the body of the phenotype, everything is under scrutiny. There are millions of entrances through Google on these subjects.

What I have not found is much mathematics, at least not the mathematics that explains and models the observed phenomena; and hence neither the numerical simulations. Everything is so complex! Probability and statistics may show regularities of formation or behaviour, but not how things work. Things are so precise! Descendants resemble their parents and may be almost identical at a first glance in the case of twins from the same ovule. Some more generations of scientists and mathematicians will be required.

Alan Turing besides introducing the reaction and diffusion equations in order to explain the morphogenetic processes was also interested in the working of the brain, and as a result he invented the Turing machine, the grandfather of our computers.

On the neural system and the mind a lot has been studied and a lot has been written, mainly lately. There are models of the brain, or parts of it, that consider it a sort of neural system, and there are studies on the transmission of signals along the axons (Hodgkin-Huxley, Fitz Hugh-Nagumo), and the ionic channels on the neurons. As a matter of fact there are all kinds of studies on the neural system and its function;

If one looks at Google for Computational Neuroscience or Computational Intelligence one is snowed from the amount of contributions. Some mathematics is used on all this studies in the shape, mainly, of differential equations that portray the electrical activity of the neurons and their connections. For instance the Blue brain project is studying the workings of columns of neurons with the unavoidable help of the IBM Blue Gene Supercomputer (may be they would need a HAL!). Close to us we have the Computational Neurosciences Group that has organised meetings in 2005, 2007 and this year, I gather.

What is still lacking is an understanding of thought; and I am not referring only to how the electrical spikes travel along the axons or what are the voltage or chemical changes in the neuron, or how the synapses work; but to what is the mechanism of thought. Or how does our memory work, or how concepts and words are handled, or just thinking, reasoning or what is consciousness, and more enigmatically, what is self-consciousness or what is the process of taking decisions. There is not a lack of thinkers that have looked for answers to these questions. Roger Penrose thought that the secret of consciousness had to be looked for in the microtubules of the neurons and the gravitons. Crick looked for it in the synchronized action of columns of neurons. Gerald Edelman tries an explanation through the changes that the brain suffers in its evolution, through experience, modifying synapses for instance, and creating configurations that respond to concepts, may be associated to words (qualia), and also to instructions, syllogisms, recipes. And also could produce some drug, prize or punishment for some action, and hence creating values. Consciousness would be this set of brain charts with which we “see” the world, including ourselves.

Edelman looks at the formation of the brain along the ages and to the development of the brain in each individual as a Darwinian process, that is, changes which favour survival are adopted. In any case Edelman doesn’t look the human brain as a digital computer with von Neumann architecture. The brain is evolving with experience, and not all are electrical impulses.

Certainly in the lectures that will follow we are going to hear a lot about these subjects. And may be we shall hear about circuitry that incorporates the nemristor of Leon Chua, a resistor that remembers the current that has gone through it by changing its resistance, and which some think that it could be incorporated to circuits that emulate some workings of the brain. HP is working on it.

How can we cut through such complexity? Well, as usual, with time, interest, work and serendipity, and then, mathematics; although not everything has to be reduced to equations and numerical methods! Mathematics and science evolve together (like fashion) in a sort of Darwinian process: which has been shown by history to be quite effective.