Summation spatial

Explain how temporal summation and spatial summation take place... 

As with temporal summation, this example has been simplified to illustrate the concept clearly. In actuality, a large number of excitatory inputs from different presynaptic neurons are necessary to depolarize the postsynaptic neuron to threshold. Because a typical neuronal cell body receives thousands of presynaptic inputs, spatial summation also occurs quite readily. The number of presynaptic neurons that are active simultaneously therefore influences the strength of the signal to the postsynaptic neuron. Under normal physiological conditions, temporal summation and spatial summation may occur concurrently. 

Figure 5.3 Spatial summation. Multiple excitatory postsynaptic potentials (EPSPs) or inhibitory postsynaptic potentials (IPSPs) produced by many presynaptic neurons simultaneously may add together to alter the membrane potential of the postsynaptic neuron. Sufficient excitatory input (A and B) will depolarize the membrane to threshold and generate an action potential. The simultaneous arrival of excitatory and inhibitory inputs (A and C) may cancel each other out so that the membrane potential does not change.

Convergence occurs when the axon terminals of many presynaptic neurons all synapse with a single postsynaptic neuron. As discussed previously, spatial summation of nerve impulses relies on the presence of convergence. Divergence occurs when the axon of a single presynaptic neuron branches and synapses with multiple postsynaptic neurons. In this way, activity in a... 

Both temporal and spatial summation of subthreshold potentials are possible, (a) Subthreshold, no summation. (b) Temporal summation, (c) Spatial summation, (d) Spatial summation of EPSP and IPSP. (From Campbell, N.A. et al.. Biology, 5th edn., Addison Wesley Longman, Menlo Park, CA, 1999. With permission.)... 

A spatial summation effect in the nervous system. The current density is reduced when the same current is spread by a larger electrode, but at the same time a larger number of nerve endings are excited. Consequently, the current threshold is not so much altered when electrode area is changed... 

Eqns. (21)-(23), (25), (27), and (30) contain summations of quantities associated with all grid points used in the finite element discretization of These spatial summations (integrations) are only performed once, producing time invariant coefficients to the temporal quantities. 

T is the free energy fiinctional, for which one can use equation (A3.3.52). The summation above corresponds to both the sum over the semi-macroscopic variables and an integration over the spatial variableThe mobility matrix consists of a synnnetric dissipative part and an antisyimnetric non-dissipative part. The syimnetric part corresponds to a set of generalized Onsager coefficients. 

Since the spatial locations r,- of active droplets are not correlated, we can replace the summation over the droplets by a continuous integral, assuming at the same time that the ripplon frequency corresponding to co/ varies from droplet to droplet within a (normalized) distribution Vi (oi) centered around co/ and having a characteristic width 8co/, whose value will be discussed shortly. There is no reason to believe that the frequency and location of the tunneling centers are correlated therefore one obtains... 

The independent variables on which fJK depends are k and t. The principal advantage of using this formulation is that spatial derivatives become summations over wavenumber space. The resulting numerical solutions have higher accuracy compared with finite-difference methods using the same number of grid points. 

The method used for the localization of the orbitals is to be carefully chosen. It is natural to expect that if the orbitals are localized into different spatial regions, for the matrix elements ij kf) the zero differential approximation can be applied all terms containing at least one factor ij kl) in which tj/itj/,-and/or are localized to different spatial regions can be neglected. Thus the summation in a closed loop in evaluating a perturbation correction should only be extended over indices of orbitals which are localized into the same region of space. 

These equations are superficially similar to Eq. (1.1). The main difference is that the Kohn-Sham equations are missing the summations that appear inside the full Schrodinger equation [Eq. (1.1)]. This is because the solution of the Kohn-Sham equations are single-electron wave functions that depend on only three spatial variables, ij ,(r). On the left-hand side of the Kohn-Sham equations there are three potentials, V, VH, and Vxc- The first... 

Simultaneously td is also the characteristic time of migrational relaxation of a spatially uniform space charge. Indeed, assume for simplicity that diffusivities of all ionic species are equal to >o. Assume further that concentrations Ci(x,t) are spatially constant, dependent on time only. Then from (1.1), (1.3), after multiplication by Z F, summation over all i, and use of (1.4), we get... 

The symbol JVu will denote integration over the full range of the spatial coordinates only, without summation over spin variables. 

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