Localized nonlinear excitations in diffusive Hindmarsh-Rose neural networks

Effect of the electromagnetic induction in the electrical activity of the Kazantsev model of inferior Olive Neuron model

In this paper, based on the four variables Kazantsev et al. inferior olive neuron (ION) dynamic equations, a five variables neuron model is designed to describe the effect of electromagnetic induction in ION activities. Within the new ION model, the effect of magnetic flow on membrane potential is described by imposing additive memristive current in the master block of the Kasantsev et al. neuron model. The impact of magnetic flux on the stability of equilibrium point is studied. Hopf bifurcation and bifurcation diagram indicated that, as the electromagnetic field strength parameter changes, the value of the critical point also changes. Furthermore, as the electromagnetic induction is increasing, there is appearance of bursting dynamic in the slave subsystem and an increase in the spike amplitude of the master subsystem. In addition, the analog circuit of the master block confirms the observed results from numerical simulation.

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

In this article, we report on the generation and propagation of traveling pulses in a homogeneous network of diffusively coupled, excitable, slow-fast dynamical neurons. The spatially extended system is modeled using the nearest neighbor coupling theory, in which the diffusion part measures the spatial distribution of coupling topology. We derive analytically the conditions for traveling wave profiles that allow the construction of the shape of traveling nerve impulses. The analytical and numerical results are used to explore the nature of propagating pulses. The symmetric or asymmetric nature of traveling pulses is characterized, and the wave velocity is derived as a function of system parameters. Moreover, we present our results for an extended excitable medium by considering a slow-fast biophysical model with a homogeneous, diffusive coupling that can exhibit various traveling pulses. The appearance of series of pulses is an interesting phenomenon from biophysical and dynamical perspective. Varying the perturbation and coupling parameters, we observe the propagation of activities with various amplitude modulations and transition phases of different wave profiles that affect the speed of pulses in certain parameter regimes. We observe different types of traveling pulses, such as envelope solitons and multi-bump solutions, and show how system parameters and coupling play a major role in the formation of different traveling pulses. Finally, we obtain the conditions for stable and unstable plane waves.

Neuromechanical modulation of transmembrane voltage in a model of a nerve

Despite substantial evidence that mechanical variables play a crucial role in transmembrane voltage regulation, most research efforts focus mostly on the nerve cell's biochemical or electrophysiological activities. We propose an electromechanical model of a nerve in order to advance our understanding of how mechanical forces and thermodynamics also regulate neural electrical activities. We explore the spatiotemporal dynamics of the transmembrane potential using the proposed nonlinear model with a sinusoid as the initial transmembrane potential and periodic boundary conditions. The localized wave from our numerical simulation and transmembrane potentials in nerves are solitary and show the three stages of action potential (depolarization, repolarization, and hyperpolarization), as well as threshold and saturation effects. We show that the mechanical properties of membranes affect the localization of the transmembrane potential. According to simulation data, mechanical pulses of sufficient magnitude can modulate a transmembrane voltage. The current model could be used to describe the dynamics of a transmembrane potential modulated by sound. Mechanical perturbations that modulate an electrical signal have a lot of clinical potential for treating pain and other neurological diseases.

Pattern selection in coupled neurons under high-low frequency electric field

Biological neurons exposed to an external electric field can induce polarization and charge fluctuation. Indeed, the exchange of calcium, sodium and potassium ions across the cell membrane can induce an electric field as a result of a time-varying electromagnetic field set up. This field could further modulate the cell electrical activity by inducing multiple firing modes. Based on the physical law of electric field, an improved model which includes an additive electrical field variable is constructed for a network of neurons to study wave propagation and mode transition by exploring the longtime dynamics of slightly perturbed plane waves in the network. Wave pattern and mode transition dependence on the different parameters of external electric field are discussed. It is found that the plane wave propagating in the coupled system breaks down to localized structures under the activation of modulational instability. The network under high-low external electric field supports bursting synchronization. This could be a fruitful avenue to discern the occurrence of paroxysmal epilepsy.

Diffusion dynamics of a conductance-based neuronal population

We study the spatiotemporal dynamics of a conductance-based neuronal cable. The processes of one-dimensional (1D) and 2D diffusion are considered for a single variable, which is the membrane voltage. A 2D Morris-Lecar (ML) model is introduced to investigate the nonlinear responses of an excitable conductance-based neuronal cable. We explore the parameter space of the uncoupled ML model and, based on the bifurcation diagram (as a function of stimulus current), we analyze the 1D diffusion dynamics in three regimes: phasic spiking, coexistence states (tonic spiking and phasic spiking exist together), and a quiescent state. We show (depending on parameters) that the diffusive system may generate regular and irregular bursting or spiking behavior. Further, we explore a 2D diffusion acting on the membrane voltage, where striped and hexagonlike patterns can be observed. To validate our numerical results and check the stability of the existing patterns generated by 2D diffusion, we use amplitude equations based on multiple-scale analysis. We incorporate 1D diffusion in an extended 3D version of the ML model, in which irregular bursting emerges for a certain diffusion strength. The generated patterns may have potential applications in nonlinear neuronal responses and signal transmission.

Periodic soliton trains and informational code structures in an improved soliton model for biomembranes and nerves

Many experiments have shown that the action potential propagating in a nerve fiber is an electromechanical density pulse. A mathematical model proposed by Heimburg and Jackson is an important step in explaining the propagation of electromechanical pulses in nerves. In this work, we consider the dynamics of modulated waves in an improved soliton model for nerve pulses. Application of the reductive perturbation method on the resulting generalized Boussinesq equation in the low-amplitude and weak damping limit yields a damped nonlinear Schrödinger equation that is shown to admit soliton trains. This solution contains an undershoot beneath the baseline ("hyperpolarization") and a "refractory period," i.e., a minimum distance between pulses, and therefore it represents typical nerve profiles. Likewise, the linear stability of wave trains is analyzed. It is shown that the amplitude of the fourth-order mixed dispersive term introduced here can be used to control the amount of information transmitted along the nerve fiber. The results from the linear stability analysis show that, in addition to the main periodic wave trains observed in most nerve experiments, five other localized background modes can copropagate along the nerve. These modes could eventually be responsible for various fundamental processes in the nerve, such as phase transitions and electrical and mechanical changes. Furthermore, analytical and numerical analyses show that increasing the fourth-order mixed dispersion coefficient improves the stability of the nerve signal.

Dynamics of coupled mode solitons in bursting neural networks

Using an electrically coupled chain of Hindmarsh-Rose neural models, we analytically derived the nonlinearly coupled complex Ginzburg-Landau equations. This is realized by superimposing the lower and upper cutoff modes of wave propagation and by employing the multiple scale expansions in the semidiscrete approximation. We explore the modified Hirota method to analytically obtain the bright-bright pulse soliton solutions of our nonlinearly coupled equations. With these bright solitons as initial conditions of our numerical scheme, and knowing that electrical signals are the basis of information transfer in the nervous system, it is found that prior to collisions at the boundaries of the network, neural information is purely conveyed by bisolitons at lower cutoff mode. After collision, the bisolitons are completely annihilated and neural information is now relayed by the upper cutoff mode via the propagation of plane waves. It is also shown that the linear gain of the system is inextricably linked to the complex physiological mechanisms of ion mobility, since the speeds and spatial profiles of the coupled nerve impulses vary with the gain. A linear stability analysis performed on the coupled system mainly confirms the instability of plane waves in the neural network, with a glaring example of the transition of weak plane waves into a dark soliton and then static kinks. Numerical simulations have confirmed the annihilation phenomenon subsequent to collision in neural systems. They equally showed that the symmetry breaking of the pulse solution of the system leaves in the network static internal modes, sometime referred to as Goldstone modes.

Ionic wave propagation and collision in an excitable circuit model of microtubules

In this paper, we report the propensity to excitability of the internal structure of cellular microtubules, modelled as a relatively large one-dimensional spatial array of electrical units with nonlinear resistive features. We propose a model mimicking the dynamics of a large set of such intracellular dynamical entities as an excitable medium. We show that the behavior of such lattices can be described by a complex Ginzburg-Landau equation, which admits several wave solutions, including the plane waves paradigm. A stability analysis of the plane waves solutions of our dynamical system is conducted both analytically and numerically. It is observed that perturbed plane waves will always evolve toward promoting the generation of localized periodic waves trains. These modes include both stationary and travelling spatial excitations. They encompass, on one hand, localized structures such as solitary waves embracing bright solitons, dark solitons, and bisolitonic impulses with head-on collisions phenomena, and on the other hand, the appearance of both spatially homogeneous and spatially inhomogeneous stationary patterns. This ability exhibited by our array of proteinic elements to display several states of excitability exposes their stunning biological and physical complexity and is of high relevance in the description of the developmental and informative processes occurring on the subcellular scale.

Breathing pulses in the damped-soliton model for nerves

Unlike the Hodgkin-Huxley picture in which the nerve impulse results from ion exchanges across the cell membrane through ion-gate channels, in the so-called soliton model the impulse is seen as an electromechanical process related to thermodynamical phenomena accompanying the generation of the action potential. In this work, account is taken of the effects of damping on the nerve impulse propagation, within the framework of the soliton model. Applying the reductive perturbation expansion on the resulting KdV-Burgers equation, a damped nonlinear Schrödinger equation is derived and shown to admit breathing-type solitary wave solutions. Under specific constraints, these breathing pulse solitons become self-trapped structures in which the damping is balanced by nonlinearity such that the pulse amplitude remains unchanged even in the presence of damping.