The peridynamic theory of solid mechanics is a nonlocal reformulation of the classical continuum mechanics theory. At the continuum level, it has been demonstrated that classical (local) elasticity is a special case of peridynamics. Such a connection between these theories has not been extensively explored at the discrete level. This paper investigates the consistency between nearest-neighbor discretizations of linear elastic peridynamic models and finite difference discretizations of the Navier–Cauchy equation of classical elasticity. Although nearest-neighbor discretizations in peridynamics have been numerically observed to present grid-dependent crack paths or spurious microcracks, this paper focuses on a different, analytical aspect of such discretizations. We demonstrate that, even in the absence of cracks, such discretizations may be problematic unless a proper selection of weights is used. Specifically, we demonstrate that using the standard meshfree approach in peridynamics, nearest-neighbor discretizations do not reduce, in general, to discretizations of corresponding classical models. We study nodal-based quadratures for the discretization of peridynamic models, and we derive quadrature weights that result in consistency between nearest-neighbor discretizations of peridynamic models and discretized classical models. The quadrature weights that lead to such consistency are, however, model-/discretization-dependent. We motivate the choice of those quadrature weights through a quadratic approximation of displacement fields. The stability of nearest-neighbor peridynamic schemes is demonstrated through a Fourier mode analysis. Finally, an approach based on a normalization of peridynamic constitutive constants at the discrete level is explored. This approach results in the desired consistency for one-dimensional models, but does not work in higher dimensions. The results of the work presented in this paper suggest that even though nearest-neighbor discretizations should be avoided in peridynamic simulations involving cracks, such discretizations are viable, for example for verification or validation purposes, in problems characterized by smooth deformations. Moreover, we demonstrate that better quadrature rules in peridynamics can be obtained based on the functional form of solutions.
Through various means of structural and synaptic plasticity enabling online learning, neural networks are constantly reconfiguring their computational functionality. Neural information content is embodied within the configurations, representations, and computations of neural networks. To explore neural information content, we have developed metrics and computational paradigms to quantify neural information content. We have observed that conventional compression methods may help overcome some of the limiting factors of standard information theoretic techniques employed in neuroscience, and allows us to approximate information in neural data. To do so we have used compressibility as a measure of complexity in order to estimate entropy to quantitatively assess information content of neural ensembles. Using Lempel-Ziv compression we are able to assess the rate of generation of new patterns across a neural ensemble's firing activity over time to approximate the information content encoded by a neural circuit. As a specific case study, we have been investigating the effect of neural mixed coding schemes due to hippocampal adult neurogenesis.