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Recently, numerous attempts have been made to understand the dynamic behavior of complex brain systems using neural network models. The fluctuations in blood-oxygen-level-dependent (BOLD) brain signals at less than 0.1 Hz have been observed by functional magnetic resonance imaging (fMRI) for subjects in a resting state. This phenomenon is referred to as a "default-mode brain network." In this study, we model the default-mode brain network by functionally connecting neural communities composed of spiking neurons in a complex network. Through computational simulations of the model, including transmission delays and complex connectivity, the network dynamics of the neural system and its behavior are discussed. The results show that the power spectrum of the modeled fluctuations in the neuron firing patterns is consistent with the default-mode brain network's BOLD signals when transmission delays, a characteristic property of the brain, have finite values in a given range.

This study introduces a new Generalized Leaky Integrate-and-Fire (GLIF) neuron model with variable leaking resistor and bias current in order to reproduce accurately the membrane voltage dynamics of a biological neuron. The accuracy of this model is ensured by adjusting its parameters to the statistical properties of the Hodgkin–Huxley model outputs; while the speed is enhanced by introducing a Generalized Exponential Moving Average method that converts the parameterized kernel functions into pre-calculated lookup tables based on an analytic solution of the dynamic equations of the GLIF model.

Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

A simplified neuron model is introduced to mimic the action potential generated by the famous Hodgkin–Huxley equations by using the genetic optimization algorithm. Comparison with different neuron models is investigated, and it is confirmed that the sodium and potassium channels in our simplified neuron model are made out of memristors. In addition, the channel equations in the simplified model may be adjusted to introduce a simplified memristor model that is in accordance with the theoretical conditions of the memristive systems.

In this paper, the concept of hyperneuron is introduced. The main properties and dynamical characteristics of this class of hypersystems will be studied by numerical analysis. The intrinsic robustness of the proposed model is further investigated by proposing a silicon implementation of the hyperneuron, reporting the corresponding experimental results. The study aims to present a novel approach to in silico implementations of large-scale networks of neurons with increased robustness and fault tolerance.

The main objective of this study is to investigate the chaotic dynamics and analog circuit implementation of a nonautonomous single inertial neuron. First, the dimensionless mathematical model of such neuron is established. The equilibria with three kinds of stabilities are then depicted, and system symmetry and Hamilton energy function are analyzed, respectively. In numerical simulations, by using the two-dimensional/one-dimensional bifurcation diagrams, Lyapunov exponent spectra, phase plots, and basins of attraction, bifurcation of coexisting bubbles as well as bi-stable patterns are revealed, where the generated coexisting attractors are symmetric about the origin. In addition, we confirm that the complexity of the basins of attraction deteriorates with the increase of the frequency parameter. Finally, analog circuit experiments with few analog circuit elements are deployed, via which the bi-stable patterns of the model are well verified.

The nonlinear response of the dynamics of the stochastic neuronal activity based on the leaky integrate-and-fire model was reformulated using the Loewner theory. We observed that the signal-to-noise ratio (SNR), which is an indicator of stochastic resonance (SR), and fluctuation-dissipation relation (FDR) for the neuronal dynamics were newly obtained in the theoretical framework of the Loewner evolution. Particularly, we focus on the role of the Loewner entropy $S_{\mathrm{\text{Loew}}}$, defined as the entropy of the Loewner driving force, while showing the efficacy of Loewner time conversion to theorize the nonlinear characteristic of the neuronal dynamics. The present results indicate a possible approach to the novel formulation of biophysical models.

In developing neuromimetic engineering systems, the choice of neuron-like element is critical, because this element determines the functional ability of the system. According to recent neuroscience research, a neuron is expected to perform sophisticated information processing, making use of the complex physiological properties of its dendrites. In this chapter, the computational function of neuronal dendrites is explored based on mathematical models. Theoretical analyses show that a passive dendrite is capable of complex logic operations by means of nonlinear synaptic interactions. Computer simulations show that active dendrites integrate synaptic inputs locally and then hierarchically according to synaptic organization and dendritic geometry. Based on these results, a formal neuron is proposed which simply and sufficiently describes the actual properties of the dendrites. This model is therefore suitable for constructing a large-scale network without the loss of the active dendrites' essential responsiveness. These modeling studies provide novel perspectives on the computation in a single neuron with a complex dendritic tree.