Local mean field potential

The implications of individual neuron dynamics on neuronal network synchronization is evident. In Fig. 7.9 (from Schneider et al., unpublished data) this is demonstrated with network simulations (10 x 10 neurons) of nearest neighbor gap-junction coupling. It is illustrated in quite a simple form which, in a similar way, can also be experimentally used with the local mean field potential (LFP). In the simulations LFP simply is the mean potential value of all neurons. In the nonsynchronized state LFP shows tiny, random fluctuations. In the completely in-phase synchronized states the spikes should peak out to their full height... 

Fig. 7.9 Computer simulations of a neuronal network (lOx 10 neurons) with nearest neighbor gap-junction coupling (see equations). The diagrams show the local mean field potentials (LFP) which are calculated as the mean potential values V of all neurons during continuously increasing coupling strength gc- Insets illustrate the different types of patterns in which the neurons originally operate (with randomly set initial values), (a) The network of tonic firing...

The free energy formulation in Equations 1-4 implies that the local volume fraction (p t) and the local mean-field potential iy(r) are not independent towards one another. However, in a thermodynamical equilibrium the relationship between these two order parameters can be written based on a so-called saddle-point approximation, i.e.,... 

Fig. 6. Free energy (at zero pressure) for a-nitrogen and /3-nitrogen, in different mean field models (closed lines). The dashed free rotor curve has been calculated from the isotropic (/, 12, ly) - (0,0, 0) term of the ab initio potential by adding the free rotor expression for the free energy. The dashed jump model curve has been obtained from the localized mean field solution (with the full anisotropic potential) by adding an entropy term -kBT In 6 (see the text).

In a mean field approach, the interaction between one chain and all the other chains of the adsorbed layer is approximated by a mean field potential proportional to the local concentration U(z) = kTvc(z) and the average conformation of one chain is calculated in this potential. The conformation of one chain is characterized by the partition function... 

The mean-field potential describes a coupling between the surface element da and the local director n, and it can be described as a coupling... 

Hartree s original idea of the self-consistent field involved only the direct Coulomb interaction between electrons. This is not inconsistent with variational theory [163], but requires an essential modification in order to correspond to the true physics of electrons. In neglecting electronic exchange, the pure Coulombic Hartree mean field inherently allowed an electron to interact with itself, one of the most unsatisfactory aspects of pre-quantum theories. Hartree simply removed the self-interaction by fiat, at the cost of making the mean field different for each electron. Orbital orthogonality, necessary to the concept of independent electrons, could only be imposed by an artificial variational constraint. The need for an ad hoc self-interaction correction (SIC) persists in recent theories based on approximate local exchange potentials. 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

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