Neuronal activity in the human brain occurs in a complex physiologic environment, and noise from all aspects in this physiologic environment affects all aspects of nervous-system function. An essential issue of neural information processing is whether the environmental noise in a neural system can be estimated and quantified in a proper way. In this paper, we calculated the neural energy to estimate the range of critical values of thermal noise intensity that markedly affect the membrane potential and the energy waveform, in order to define such a noisy environment which neuronal activity relies on.

Let $G$ be a simple graph with $n$ vertices and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be the eigenvalues of its adjacency matrix; the Estrada index $EE\left(G\right)$ of the graph $G$ is defined as the sum of the terms $e^{\lambda i}$, $i=\mathrm{1,2},\dots ,n$. The $n$-dimensional folded hypercube networks $F{Q}_n$ are an important and attractive variant of the $n$-dimensional hypercube networks $Q_n$, which are obtained from $Q_n$ by adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networks $F{Q}_n$ by deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networks $F{Q}_n$ are proposed.

This paper addresses the problem of robust $H_{\infty }$ control design via the proportional-spatial derivative (P-sD) control approach for a class of nonlinear distributed parameter systems modeled by semilinear parabolic partial differential equations (PDEs). By using the Lyapunov direct method and the technique of integration by parts, a simple linear matrix inequality (LMI) based design method of the robust $H_{\infty }$ P-sD controller is developed such that the closed-loop PDE system is exponentially stable with a given decay rate and a prescribed $H_{\infty }$ performance of disturbance attenuation. Moreover, a suboptimal $H_{\infty }$ controller is proposed to minimize the attenuation level for a given decay rate. The proposed method is successfully employed to address the control problem of the FitzHugh-Nagumo (FHN) equation, and the achieved simulation results show its effectiveness.