scholarly journals Spontaneous Fluctuations and Non-linear Ignitions: Two Dynamic Faces of Cortical Recurrent Loops

Neuron ◽  
2015 ◽  
Vol 88 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Clément Moutard ◽  
Stanislas Dehaene ◽  
Rafael Malach

The spontaneous fluctuations of a system with a non-linear decay law are investigated analytically by means of a model which is a significant generalization of the ‘Rayleigh particle’. Our model consists of a piston, whose velocity fluctuates owing to the collisions of two different gas mixtures on each side. This model involves a large number of parameters in a non-trivial way, and moreover it is not symmetrical as regards positive and negative piston velocities. Special care is taken in the analysis to ensure that all neglected effects can in principle be made arbitrarily small. It is then possible to write the exact ‘Master equation’ for the probability distribution of the velocity of the piston. This exact master equation is then expanded systematically in reciprocal powers of the piston mass. The lowest approximation is Rayleigh’s equation (or the linear Fokker-Planck equation). The next approximation describes the first-order effect of non-linearity on the fluctuations. Higher approximations still are not investigated here. An alternative expansion of the master equation establishes the direct connexion with the (non-linear) phenomenological damping law of the system to this order of approximation. Several questions concerning non-linear fluctuations can be answered on this model. (i) An asymmetrical phenomenological law is not incompatible with the Gaussian equilibrium distribution, (ii) It is actually possible to utilize the fact that the equilibrium distribution must be a solution, for the purpose of obtaining a priori relations between the coefficients of the equation, provided that proper attention is paid to the order of the several terms. A more powerful tool, however, is the principle of microscopic reversibility, (iii) The equation governing the spontaneous fluctuations to the first order of non-linearity is uniquely determined by this means, together with the additional information provided by the phenomenological damping law.


Soft Matter ◽  
2011 ◽  
Vol 7 (5) ◽  
pp. 2042-2051 ◽  
Author(s):  
Young Zoon Yoon ◽  
Jurij Kotar ◽  
Aidan T. Brown ◽  
Pietro Cicuta

1967 ◽  
Vol 28 ◽  
pp. 105-176
Author(s):  
Robert F. Christy

(Ed. note: The custom in these Symposia has been to have a summary-introductory presentation which lasts about 1 to 1.5 hours, during which discussion from the floor is minor and usually directed at technical clarification. The remainder of the session is then devoted to discussion of the whole subject, oriented around the summary-introduction. The preceding session, I-A, at Nice, followed this pattern. Christy suggested that we might experiment in his presentation with a much more informal approach, allowing considerable discussion of the points raised in the summary-introduction during its presentation, with perhaps the entire morning spent in this way, reserving the afternoon session for discussion only. At Varenna, in the Fourth Symposium, several of the summaryintroductory papers presented from the astronomical viewpoint had been so full of concepts unfamiliar to a number of the aerodynamicists-physicists present, that a major part of the following discussion session had been devoted to simply clarifying concepts and then repeating a considerable amount of what had been summarized. So, always looking for alternatives which help to increase the understanding between the different disciplines by introducing clarification of concept as expeditiously as possible, we tried Christy's suggestion. Thus you will find the pattern of the following different from that in session I-A. I am much indebted to Christy for extensive collaboration in editing the resulting combined presentation and discussion. As always, however, I have taken upon myself the responsibility for the final editing, and so all shortcomings are on my head.)


Optimization ◽  
1975 ◽  
Vol 6 (4) ◽  
pp. 549-559
Author(s):  
L. Gerencsér

1979 ◽  
Author(s):  
George W. Howe ◽  
James H. Dalton ◽  
Maurice J. Elias
Keyword(s):  

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