Volterra

Linz, P. Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985). 

Volterra integral equations have an integral with a variable limit. The Volterra equation of the second land is... 

Equations of the first land are very sensitive to solution errors so that they present severe numerical problems. Volterra equations are similar to initial value problems. 

This integral equation is a Volterra equation of the second land. Thus the initial-value problem is eqmvalent to a Volterra integral equation of the second kind. 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

FIGURE 2.6 Population dynamics predicted by the Lotka-Volterra model for an initial population of 100 rabbits and 10 lynx. 

The Lotka-Volterra reaction described in Section 2.5.4 has three initial conditions—one each for grass, rabbits, and lynx—all of which must be positive. There are three rate constants assuming the supply of grass is not depleted. Use dimensionless variables to reduce the number of independent parameters to four. Pick values for these that lead to a sustained oscillation. Then, vary the parameter governing the grass supply and determine how this affects the period and amplitude of the solution. 

From the overall data, it is becoming inaeasingly apparent that astrocytes, by releasing chemical transmitters (the so called gliotrasmitters , Bezzi and Volterra 2001), now appears to be aitically implicated in rapid conununication systems in the brain. 

Bezzi P, Carmignoto G, Pasti L, Vesce S, Rossi D, Rizzini BL, Pozzan T, Volterra A (1998) Prostaglandins stimulate calcium-dependent glutamate release in astrocytes. Nature 391 281-285... 

Domercq M, BrambiUa L, PUati E, Marchalaud J, Volterra A, Bezzi P (2006) P2Y1 receptor-evoked glutamate exocytosis from astrocytes coutrol by tumor uecrosis factor-alpha aud prostaglaudius. J Biol Chem 281 30684-30696... 

Jourdain P, Bergersen LH, BhaukauraUy K, Bezzi P, Santello M, Domercq M, Matute C, Tonello F, Gundersen V, Volterra A (2007) Glutamate exocytosis from astrocytes controls synaptic strength. Nat Neurosci 10 331-339... 

Fasti L, Volterra A, Pozzan T, Carmignoto G (1997) Intracellular calcium oscillations in astrocytes a highly plastic, bidirectional form of communication between neurons and astrocytes in situ. J Neurosci 17 7817-7830... 

Rossi D, BrambiUa L, Valori CF, Crugnola A, Giaccone G, Capobianco R, Mangier M, Kingston AE, Bloc A, Bezzi P, Volterra A (2005) Defective tumor necrosis factor-alpha-dependent control of astrocyte glutamate release in a transgenic mouse model of Alzheimer disease. J Biol Chem 280 42088-42096... 

Saez ET, Pehar M, Vargas MR, Barbeito L, Maccioni RB (2006) Production of nerve growth factor by beta-amyloid-stimulated astrocytes induces p75NTR-dependent tau hyperphosphorylation in cultured hippocampal neurons. J Neurosci Res 84 1098-1106 Sala C, Roussignol G, Meldolesi J, Fagni L (2005) Key role of the postsynaptic density scaffold proteins Shank and Homer in the functional architecture of Ca homeostasis at dendritic spines in hippocampal neurons. J Neurosci 25 4587 592 Santello M, Volterra A (2008) Synaptic modulation by astrocytes via Ca(2-l-)-dependent glutamate release. Neuroscience 158 253-9... 

Vesce S, Rossi D, BrambiUa L, Volterra A (2007) Glutamate release from astrocytes in physiological conditions and in neurodegenerative disorders characterized by neuroinflammation. Int Rev Neurobiol 82 57-71... 

VUaM, Jackson-Lewis V, Guegan C, Wu DC, TeismannP, Choi DK, Tieu K, Przedborski S (2001) The role of gUal cells in Parkinson s disease. Curr Opin Neurol 14 483-489 Volterra A, Meldolesi J (2005) Astrocytes, from brain glue to communication elements the revolution continues. Nat Rev Neurosci 6 626-640... 

References Courant, R., and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience, New York (1953) Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM Publications, Philadelphia (1985) Porter, D., and D. S. G. Stirling, Integral Equations A Practical Treatment from Spectral Theory to Applications, Cambridge University Press (1990) Statgold, I., Greens Functions and Boundary Value Problems, 2d ed., Interscience, New York (1997). 

Solutions for Volterra equations are done in a similar fashion, except that the solution can proceed point by point, or in small groups of points depending on the quadrature scheme. See Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia (1985). There are methods that are analogous to the usual methods for... 

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