Bifurcation study of neuron firing activity of the modified Hindmarsh–Rose model

Hamiltonian energy in a modified Hindmarsh-Rose model

This paper investigates the Hamiltonian energy of a modified Hindmarsh-Rose (HR) model to observe its effect on short-term memory. A Hamiltonian energy function and its variable function are given in the reduced system with a single node according to Helmholtz's theorem. We consider the role of the coupling strength and the links between neurons in the pattern formation to show that the coupling and cooperative neurons are necessary for generating the fire or a clear short-term memory when all the neurons are in sync. Then, we consider the effect of the degree and external stimulus from other neurons on the emergence and disappearance of short-term memory, which illustrates that generating short-term memory requires much energy, and the coupling strength could further reduce energy consumption. Finally, the dynamical mechanisms of the generation of short-term memory are concluded.

Coexisting Firing Patterns in an Improved Memristive Hindmarsh-Rose Neuron Model with Multi-Frequency Alternating Current Injection

With the development of memristor theory, the application of memristor in the field of the nervous system has achieved remarkable results and has bright development prospects. Flux-controlled memristor can be used to describe the magnetic induction effect of the neuron. Based on the Hindmarsh-Rose (HR) neuron model, a new HR neuron model is proposed by introducing a flux-controlled memristor and a multi-frequency excitation with high-low frequency current superimposed. Various firing patterns under single and multiple stimuli are investigated. The model can exhibit different coexisting firing patterns. In addition, when the memristor coupling strength changes, the multiple stability of the model is eliminated, which is a rare phenomenon. Moreover, an analog circuit is built to verify the numerical simulation results.

Topology identification and dynamical pattern recognition for Hindmarsh-Rose neuron model via deterministic learning

Studies have shown that Parkinson's, epilepsy and other brain deficits are closely related to the ability of neurons to synchronize with their neighbors. Therefore, the neurobiological mechanism and synchronization behavior of neurons has attracted much attention in recent years. In this contribution, it is numerically investigated the complex nonlinear behaviour of the Hindmarsh-Rose neuron system through the time responses, system bifurcation diagram and Lyapunov exponent under different system parameters. The system presents different and complex dynamic behaviors with the variation of parameter. Then, the identification of the nonlinear dynamics and topologies of the Hindmarsh-Rose neural networks under unknown dynamical environment is discussed. By using the deterministic learning algorithm, the unknown dynamics and topologies of the Hindmarsh-Rose system are locally accurately identified. Additionally, the identified system dynamics can be stored and represented in the form of constant neural networks due to the convergence of system parameters. Finally, based on the time-invariant representation of system dynamics, a fast dynamical pattern recognition method via system synchronization is constructed. The achievements of this work will provide more incentives and possibilities for biological experiments and medical treatment as well as other related clinical researches, such as the quantifying and explaining of neurobiological mechanism, early diagnosis, classification and control (treatment) of neurologic diseases, such as Parkinson's and epilepsy. Simulations are included to verify the effectiveness of the proposed method.

Spatiotemporal Patterns in a General Networked Hindmarsh-Rose Model

Neuron modelling helps to understand the brain behavior through the interaction between neurons, but its mechanism remains unclear. In this paper, the spatiotemporal patterns is investigated in a general networked Hindmarsh-Rose (HR) model. The stability of the network-organized system without delay is analyzed to show the effect of the network on Turing instability through the Hurwitz criterion, and the conditions of Turing instability are obtained. Once the analysis of the zero-delayed system is completed, the critical value of the delay is derived to illustrate the profound impact of the given network on the collected behaviors. It is found that the difference between the collected current and the outgoing current plays a crucial role in neuronal activity, which can be used to explain the generation mechanism of the short-term memory. Finally, the numerical simulation is presented to verify the proposed theoretical results.

When Noise meets Chaos: Stochastic Resonance in Neurochaos Learning

Chaos and Noise are ubiquitous in the Brain. Inspired by the chaotic firing of neurons and the constructive role of noise in neuronal models, we for the first time connect chaos, noise and learning. In this paper, we demonstrate Stochastic Resonance (SR) phenomenon in Neurochaos Learning (NL). SR manifests at the level of a single neuron of NL and enables efficient subthreshold signal detection. Furthermore, SR is shown to occur in single and multiple neuronal NL architecture for classification tasks - both on simulated and real-world spoken digit datasets, and in architectures with 1D chaotic maps as well as Hindmarsh-Rose spiking neurons. Intermediate levels of noise in neurochaos learning enable peak performance in classification tasks thus highlighting the role of SR in AI applications, especially in brain inspired learning architectures.

Anti-control of periodic firing in HR model in the aspects of position, amplitude and frequency

This paper proposes a novel controller to control position, amplitude and frequency of periodic firing activity in Hindmarsh-Rose model based on Hopf bifurcation theory which is composed of linear control gain and nonlinear control gain. First, we select the activation of the fast ion channel as control parameter. Based on explicit criterion of Hopf bifurcation, a series of conditions are obtained to derive the linear gains of controller responsible for control of the location where the periodic firing activity occurs. Then, based on the control parameter, a series of conditions are obtained to derive the nonlinear gains of controller responsible for controlling the amplitude and frequency of periodic firing activity by using center manifold and normal form. Finally, the numerical experiments show that our controller can make the periodic firing activity occur at designed value and control the amplitude and frequency of periodic firing activity by adjusting nonlinear control gain of controller.

Various firing activities and finite-time synchronization of an improved Hindmarsh-Rose neuron model under electric field effect

Nowadays, it is important to realize systems that can model the electrical activity of neurons taking into account almost all the properties of the intracellular and extracellular environment in which they are located. It is in this sense that we propose in this paper, the improved model of Hindmarsh-Rose (HR) which takes into account the fluctuation of the membrane potential created by the variation of the ion concentration in the cell. Considering the effect of the electric field that is produced on the dynamic behavior of neurons, the essential properties of the model such as equilibrium point and its stability, bifurcation diagrams, Lyapunov spectrum, frequency spectra, time series of the membrane potential and phase portraits are thoroughly investigated. We thus prove that Hopf bifurcation occurs in this system when the parameters are chosen appropriately. We also observe that by varying specific parameters of the electric field, the model presents a very rich and striking event, namely hysteresis phenomenon, which justifies the coexistence of multiple attractors. Besides, by applying a suitable sinusoidal excitation current, we prove that the neuron under electric field effect can present several important electrical activities including quiescent, spiking, bursting and even chaos. We propose the improved HR model under electric field effect (mHR) to study the finite-time synchronization between two neurons when performing synapse coupling across the membrane potential and the electric field coupling. As a result, we find that the synchronization between the two neurons is weakly influenced by the variation of the intensity of the electric field coupling while it is strongly impacted when the intensity of the synapse coupling is modified. From these results, it is obvious that the electric field can be another effective bridge connection to encourage the exchange and coding of the signal. Using the finite-time synchronization algorithm, we theoretically quantify the synchronization time between these neurons. Finally, Pspice simulations are presented to show the feasibility of the proposed model as well as that of the developed synchronization strategy.

Spiral waves in externally excited neuronal network: Solvable model with a monotonically differentiable magnetic flux

Spiral waves are particular spatiotemporal patterns connected to specific phase singularities representing topological wave dislocations or nodes of zero amplitude, witnessed in a wide range of complex systems such as neuronal networks. The appearance of these waves is linked to the network structure as well as the diffusion dynamics of its blocks. We report a novel form of the Hindmarsh-Rose neuron model utilized as a square neuronal network, showing the remarkable multistructure of dynamical patterns ranging from characteristic spiral wave domains of spatiotemporal phase coherence to regions of hyperchaos. The proposed model comprises a hyperbolic memductance function as the monotone differentiable magnetic flux. Hindmarsh-Rose neurons with an external electromagnetic excitation are considered in three different cases: no excitation, periodic excitation, and quasiperiodic excitation. We performed an extensive study of the neuronal dynamics including calculation of equilibrium points, bifurcation analysis, and Lyapunov spectrum. We have found the property of antimonotonicity in bifurcation scenarios with no excitation or periodic excitation and identified wide regions of hyperchaos in the case of quasiperiodic excitation. Furthermore, the formation and elimination of the spiral waves in each case of external excitation with respect to stimuli parameters are investigated. We have identified novel forms of Hindmarsh-Rose bursting dynamics. Our findings reveal multipartite spiral wave formations and symmetry breaking spatiotemporal dynamics of the neuronal model that may find broad practical applications.