Mathematical Methods in Biology and Neurobiology (Universitext)
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This book helps readers develop mathematical tools required for modern biology. It offers a survey of mathematics, from stochastic processes to pattern formation, and contains biological examples from the molecular to the evolutionary and ecological levels.
to control the largest eigenvalue, see . The spectrum of the graph Laplacian is a useful tool to analyze biological networks, see [6, 8, 9, 10]. Whereas most computational problems on graphs are NP-hard or even NP-complete, and hence require a number of steps that grows exponentially with the number of vertices, the computation of the spectrum proceeds by linear algebra. Therefore, there exist algorithms that grow only like a polynomial of low order in the number of vertices. With current
ancestral population will then have no descendents from an uninterrupted female line in the present populations; of course they may or may not have descendents from some lineages that include some males. As already described above, we can essentialize the graph by eliminating all females without female descendents in an iterative manner so that only those remain that have an uninterrupted line of female descendents down to the present sample. When we do coalescence theory, that is, follow the
distribution of the time interval between events. The combination of these two points of view, that is, counting the number of events having occured until time vs. recording the temporal distance between subsequent events, will prove quite insightful.—If these time intervals are independently and identically distributed (a point to be returned to shortly), the process is called a renewal process. That is, the th event occurs at time where are independent positive random variables that are
having offspring is , this means that . Let the random variable denote the size of generation . One usually assumes that the process starts with a single individual in generation 0, that is, . Let be the generating function of . Lemma 3.4.1 is the -th fold iterate of , (3.4.1) and thus also for (3.4.2) Proof We shall show (3.4.2) which easily implies (3.4.1) by iteration. Let the random variable denote the number of members of the th generation that derive from member of the th one.
Hodgkin-Huxley model, see [83, 92] and, for new mathematical aspects of it, . The Hodgkin-Huxley model is analyzed with the tools of dynamical systems in much detail in [40, 64], and these are useful references for methods of dynamical systems theory in the neurosciences in general. It is important to note that already a relatively small or short external current that is barely able to increase by about 5 mV suffices to trigger the spiking of the neuron, that is, an increase of by about