Bifurcations in Morris–Lecar neuron model

Wave propagation in a light-temperature neural network under adaptive local energy balance

External electric and mechanical stimuli can induce shape deformation in excitable media because of its intrinsic flexible property. When the signals propagation in the media is described by a neural network, creation of heterogeneity or defect is considered as the effect of shape deformation due to accumulation or release of energy in the media. In this paper, a temperature-light sensitive neuron model is developed from a nonlinear circuit composed of a phototube and a thermistor, and the physical energy is kept in capacitive and inductive terms. Furthermore, the Hamilton energy for this function neuron is obtained in theoretical way. A regular neural network is built on a square array by activating electric synapse between adjacent neurons, and a few of neurons in local area is excited by noisy disturbance, which induces local energy diversity, and continuous coupling enables energy propagation and diffusion. Initially, the Hamilton energy function for a temperature-light sensitive neuron can be obtained. Then, the finite neurons are applied noise to obtain energy diversity to explore the energy spread between neurons in the network. For keeping local energy balance, one intrinsic parameter is regulated adaptively until energy diversity in this local area is decreased greatly. Regular pattern formation indicates that local energy balance creates heterogeneity or defects and a few of neurons show continuous parameter shift for keeping energy balance in a local area, which supports gradient energy distribution for propagating waves in the network.

Modeling and simulation for prediction of multiple sclerosis progression

In light of extensive work that has created a wide range of techniques for predicting the course of multiple sclerosis (MS) disease, this paper attempts to provide an overview of these approaches and put forth an alternative way to predict the disease progression. For this purpose, the existing methods for estimating and predicting the course of the disease have been categorized into clinical, radiological, biological, and computational or artificial intelligence-based markers. Weighing the weaknesses and strengths of these prognostic groups is a profound method that is yet in need and works directly at the level of diseased connectivity. Therefore, we propose using the computational models in combination with established connectomes as a predictive tool for MS disease trajectories. The fundamental conduction-based Hodgkin-Huxley model emerged as promising from examining these studies. The advantage of the Hodgkin-Huxley model is that certain properties of connectomes, such as neuronal connection weights, spatial distances, and adjustments of signal transmission rates, can be taken into account. It is precisely these properties that are particularly altered in MS and that have strong implications for processing, transmission, and interactions of neuronal signaling patterns. The Hodgkin-Huxley (HH) equations as a point-neuron model are used for signal propagation inside a small network. The objective is to change the conduction parameter of the neuron model, replicate the changes in myelin properties in MS and observe the dynamics of the signal propagation across the network. The model is initially validated for different lengths, conduction values, and connection weights through three nodal connections. Later, these individual factors are incorporated into a small network and simulated to mimic the condition of MS. The signal propagation pattern is observed after inducing changes in conduction parameters at certain nodes in the network and compared against a control model pattern obtained before the changes are applied to the network. The signal propagation pattern varies as expected by adapting to the input conditions. Similarly, when the model is applied to a connectome, the pattern changes could give an insight into disease progression. This approach has opened up a new path to explore the progression of the disease in MS. The work is in its preliminary state, but with a future vision to apply this method in a connectome, providing a better clinical tool.

Weak coupling of neurons enables very high-frequency and ultra-fast oscillations through the interplay of synchronized phase shifts

Recently, in the past decade, high-frequency oscillations (HFOs), very high-frequency oscillations (VHFOs), and ultra-fast oscillations (UFOs) were reported in epileptic patients with drug-resistant epilepsy. However, to this day, the physiological origin of these events has yet to be understood. Our study establishes a mathematical framework based on bifurcation theory for investigating the occurrence of VHFOs and UFOs in depth EEG signals of patients with focal epilepsy, focusing on the potential role of reduced connection strength between neurons in an epileptic focus. We demonstrate that synchronization of a weakly coupled network can generate very and ultra high-frequency signals detectable by nearby microelectrodes. In particular, we show that a bistability region enables the persistence of phase-shift synchronized clusters of neurons. This phenomenon is observed for different hippocampal neuron models, including Morris-Lecar, Destexhe-Paré, and an interneuron model. The mechanism seems to be robust for small coupling, and it also persists with random noise affecting the external current. Our findings suggest that weakened neuronal connections could contribute to the production of oscillations with frequencies above 1000 Hz, which could advance our understanding of epilepsy pathology and potentially improve treatment strategies. However, further exploration of various coupling types and complex network models is needed.

Data-driven control of oscillator networks with population-level measurement

Controlling complex networks of nonlinear limit-cycle oscillators is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of oscillator populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without prior model assumptions or pre-existing data remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's current dynamics to iteratively learn an appropriate control online without constructing a global model of the system. We illustrate through a range of numerical simulations that the proposed technique can effectively regulate synchrony in various oscillator networks after a small number of trials using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach, illustrate its robustness to system variations, and compare its performance with existing model-based and data-driven approaches.

How neuronal morphology impacts the synchronisation state of neuronal networks

The biophysical properties of neurons not only affect how information is processed within cells, they can also impact the dynamical states of the network. Specifically, the cellular dynamics of action-potential generation have shown relevance for setting the (de)synchronisation state of the network. The dynamics of tonically spiking neurons typically fall into one of three qualitatively distinct types that arise from distinct mathematical bifurcations of voltage dynamics at the onset of spiking. Accordingly, changes in ion channel composition or even external factors, like temperature, have been demonstrated to switch network behaviour via changes in the spike onset bifurcation and hence its associated dynamical type. A thus far less addressed modulator of neuronal dynamics is cellular morphology. Based on simplified and anatomically realistic mathematical neuron models, we show here that the extent of dendritic arborisation has an influence on the neuronal dynamical spiking type and therefore on the (de)synchronisation state of the network. Specifically, larger dendritic trees prime neuronal dynamics for in-phase-synchronised or splayed-out activity in weakly coupled networks, in contrast to cells with otherwise identical properties yet smaller dendrites. Our biophysical insights hold for generic multicompartmental classes of spiking neuron models (from ball-and-stick-type to anatomically reconstructed models) and establish a connection between neuronal morphology and the susceptibility of neural tissue to synchronisation in health and disease.

Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

Complex networks have been programmed to mimic the input and output functions in multiple biophysical algorithms of cortical neurons at spiking resolution. Prior research has demonstrated that the ineffectual features of membranes can be taken into account by discrete fractional commensurate, non-commensurate and variable-order patterns, which may generate multiple kinds of memory-dependent behaviour. Firing structures involving regular resonator chattering, fast, chaotic spiking and chaotic bursts play important roles in cortical nerve cell insights and execution. Yet, it is unclear how extensively the behaviour of discrete fractional-order excited mechanisms can modify firing cell attributes. It is illustrated that the discrete fractional behaviour of the Izhikevich neuron framework can generate an assortment of resonances for cortical activity via the aforesaid scheme. We analyze the bifurcation using fragmenting periodic solutions to demonstrate the evolution of periods in the framework's behaviour. We investigate various bursting trends both conceptually and computationally with the fractional difference equation. Additionally, the consequences of an excitable and inhibited Izhikevich neuron network (INN) utilizing a regulated factor set exhibit distinctive dynamic actions depending on fractional exponents regulating over extended exchanges. Ultimately, dynamic controllers for stabilizing and synchronizing the suggested framework are shown. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

Role of Fast and Slow Inhibitors in Oscillatory Rhythm Design

In biological or abiotic systems, rhythms occur, owing to the coupling between positive and negative feedback loops in a reaction network. Using the Semenov-Whitesides oscillatory network for thioester hydrolysis as a prototype, we experimentally and theoretically analyzed the role of fast and slow inhibitors in oscillatory reaction networks. In the presence of positive feedback, a single fast inhibitor generates a time delay, resulting in two saddle-node bifurcations and bistability in a continuously stirred tank reactor. A slow inhibitor produces a node-focus bifurcation, resulting in damped oscillations. With both fast and slow inhibitors present, the node-focus bifurcation repeatedly modulates the saddle-node bifurcations, producing stable periodic oscillations. These fast and slow inhibitions result in a pair of time delays between steeply ascending and descending dynamics, which originate from the positive and negative feedbacks, respectively. This pattern can be identified in many chemical relaxation oscillators and oscillatory models, e.g., the bromate-sulfite pH oscillatory system, the Belousov-Zhabotinsky reaction, the trypsin oscillatory system, and the Boissonade-De Kepper model. This study provides a novel understanding of chemical and biochemical rhythms and suggests an approach to designing such behavior.

Global dynamics, thermodynamics and non-equilibrium origin of bifurcations for single neuron dynamics

The understanding of neural excitability and oscillations in single neuron dynamics remains incomplete in terms of global stabilities and the underlying mechanisms for phase formation and associated phase transitions. In this study, we investigate the mechanism of single neuron excitability and spontaneous oscillations by analyzing the potential landscape and curl flux. The topological features of the landscape play a crucial role in assessing the stability of resting states and the robustness/coherence of oscillations. We analyze the excitation characteristics in Class I and Class II neurons and establish their relation to biological function. Our findings reveal that the average curl flux and associated entropy production exhibit significant changes near bifurcation or phase transition points. Moreover, the curl flux and entropy production offer insights into the dynamical and thermodynamical origins of nonequilibrium phase transitions and exhibit distinct behaviors in Class I and Class II neurons. Additionally, we quantify time irreversibility through the difference in cross-correlation functions in both forward and backward time, providing potential indicators for the emergence of nonequilibrium phase transitions in single neurons.

Research on constrained localization of ultrasound geometric distribution based on FHN neurons

The autopilot positioning process is mainly affected by three aspects: the first is the spatial geometric distribution of positioning sensors; the second is the screening of spurious observations; and the third is the equivalent ranging error. A constrained positioning method based on the geometric distribution of FitzHugh-Nagumo (FHN) neurons is proposed. To reduce the geometric accuracy factor, a Horizontal Dilution Of Precision value algorithm with a weight factor was proposed by considering the spatial geometric distribution of base stations and the geometric relationship of anchor points. This paper proposes a geometric constraint data processing method for the error of the pseudo-observation value. Finally, considering the significant weak signal perception ability of the biological nervous system, and the stochastic resonance phenomenon caused by noise can enhance the ability of the neuronal system to detect weak signals, an ultrasonic receiving method based on the stochastic resonance characteristics of FHN neuronal system is proposed, to enhance the signal and reduce noise. The results show that under the optimized base station layout and data geometric constraint processing, the ultrasonic wave based on FHN neuron improves the accuracy of spurious observations, reduces the calculation amount of geometric constraint processing, and reduces the positioning error by 66.67%, which provides a new direction for improving the positioning accuracy.

Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback

We study the dynamics and bifurcations of temporal dissipative solitons in an excitable system under time-delayed feedback. As a prototypical model displaying different types of excitability, we use the Morris-Lecar model. In the limit of large delay, soliton like solutions of delay-differential equations can be treated as homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of pulse solutions and to explain their dependence on the system parameters. In particular, we show how a homoclinic orbit flip of a single-pulse soliton leads to the destabilization of equidistant multi-pulse solutions and to the emergence of stable pulse packages. It turns out that this transition is induced by a heteroclinic orbit flip in the system without feedback, which is related to the excitability properties of the Morris-Lecar model.

Spiking Neuron Mathematical Models: A Compact Overview

The features of the main models of spiking neurons are discussed in this review. We focus on the dynamical behaviors of five paradigmatic spiking neuron models and present recent literature studies on the topic, classifying the contributions based on the most-studied items. The aim of this review is to provide the reader with fundamental details related to spiking neurons from a dynamical systems point-of-view.

Interplay of different synchronization modes and synaptic plasticity in a system of class I neurons

We analyze the effect of spike-timing-dependent plasticity (STDP) on a system of pulse-coupled class I neurons. Our research begins with a system of two mutually connected quadratic integrate-and-fire (QIF) neurons, which are canonical representatives of class I neurons. Along with various asymptotic modes previously observed in other neuronal models with plastic synapses, we found a stable synchronous mode characterized by unidirectional link from a slower neuron to a faster neuron. In this frequency-locked mode, the faster neuron emits multiple spikes per cycle of the slower neuron. We analytically obtain the Arnold tongues for this mode without STDP and with STDP. We also consider larger plastic networks of QIF neurons and show that the detected mode can manifest itself in such a way that slow neurons become pacemakers. As a result, slow and fast neurons can form large synchronous clusters that generate low-frequency oscillations. We demonstrate the generality of the results obtained with two connected QIF neurons using Wang-Buzsáki and Morris-Lecar biophysically plausible class I neuron models.

Noise-induced collective dynamics in the small-world network of photosensitive neurons

Photosensitive neurons can capture and convert external optical signals, and then realize the encoding signal. It is confirmed that a variety of firing modes could be induced under optical stimuli. As a result, it is interesting to explore the mode transitions of collective dynamics in the photosensitive neuron network under external stimuli. In this work, the collective dynamics of photosensitive neurons in a small-world network with non-synaptic coupling will be discussed with spatial diversity of noise and uniform noise applied on, respectively. The results prove that a variety of different collective electrical activities could be induced under different conditions. Under spatial diversity of noise applied on, a chimera state could be observed in the evolution, and steady cluster synchronization could be detected in the end; even the nodes in each cluster depend on the degree of each node. Under uniform noise applied on, the complete synchronization window could be observed alternately in the transient process, and steady complete synchronization could be detected finally. The potential mechanism is that continuous energy is pumped in the phototubes, and energy exchange and balance between neurons to form the resonance synchronization in the network with different noise applied on. Furthermore, it is confirmed that the evolution of collective dynamical behaviors in the network depends on the external stimuli on each node. Moreover, the bifurcation analysis for the single neuron model is calculated, and the results confirm that the electrical activities of single neuron are sensitive to different kinds of noise.

Hamiltonian energy and coexistence of hidden firing patterns from bidirectional coupling between two different neurons

In this paper, bidirectional-coupled neurons through an asymmetric electrical synapse are investigated. These coupled neurons involve 2D Hindmarsh-Rose (HR) and 2D FitzHugh-Nagumo (FN) neurons. The equilibria of the coupled neurons model are investigated, and their stabilities have revealed that, for some values of the electrical synaptic weight, the model under consideration can display either self-excited or hidden firing patterns. In addition, the hidden coexistence of chaotic bursting with periodic spiking, chaotic spiking with period spiking, chaotic bursting with a resting pattern, and the coexistence of chaotic spiking with a resting pattern are also found for some sets of electrical synaptic coupling. For all the investigated phenomena, the Hamiltonian energy of the model is computed. It enables the estimation of the amount of energy released during the transition between the various electrical activities. Pspice simulations are carried out based on the analog circuit of the coupled neurons to support our numerical results. Finally, an STM32F407ZE microcontroller development board is exploited for the digital implementation of the proposed coupled neurons model.

Unraveling the stochastic transition mechanism between oscillation states by landscape and minimum action path theory

Cell fate transitions have been studied from various perspectives, such as the transition between stable states, or the transition between stable states and oscillation states. However, there is a lack...

Effects of degree distributions in random networks of type-I neurons

We consider large networks of theta neurons and use the Ott-Antonsen ansatz to derive degree-based mean-field equations governing the expected dynamics of the networks. Assuming random connectivity, we investigate the effects of varying the widths of the in- and out-degree distributions on the dynamics of excitatory or inhibitory synaptically coupled networks and gap junction coupled networks. For synaptically coupled networks, the dynamics are independent of the out-degree distribution. Broadening the in-degree distribution destroys oscillations in inhibitory networks and decreases the range of bistability in excitatory networks. For gap junction coupled neurons, broadening the degree distribution varies the values of parameters at which there is an onset of collective oscillations. Many of the results are shown to also occur in networks of more realistic neurons.

M-current induced Bogdanov-Takens bifurcation and switching of neuron excitability class

In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current [Formula: see text], the maximal conductance of the M-current [Formula: see text] and the conductance of the leak current [Formula: see text]. We give precise conditions for the model that ensure the existence of a Bogdanov-Takens (BT) point and show that such a point can occur by varying [Formula: see text] and [Formula: see text]. We discuss the case when the BT point becomes a Bogdanov-Takens-cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

A Simplified Model for Simulating Population Responses of Tactile Afferents and Receptors in the Skin

Tactile information about an object can only be extracted from population responses of tactile receptors and their afferents. Thus, to best control tactile information in robots, neuroprostheses or haptic devices, inputs should represent responses from full populations of afferents. Here, we describe a simplified model that recreates afferent population responses of thousands of tactile afferents in a personal computer. The whole model includes a resistance network model to simplify the skin mechanics and an improved version of a single unit model that we have previously described. The whole model was implemented by short and efficient python code. The parameters of the model were fit based on a simple vibrating stimulus, but the simulated outputs generalize to match receptive field sizes, edge enhancement, and neurophysiological responses to dot textures, embossed letters and curved surfaces. We discuss how to use this work to model haptic perception and provide guidance in designing and controlling highly realistic tactile interfaces in robots, neural prostheses and haptic devices.

Excitability and Threshold Mechanism for Enhanced Neuronal Response Induced by Inhibition Preceding Excitation

Postinhibitory facilitation (PIF) of neural firing presents a paradoxical phenomenon that the inhibitory effect induces enhancement instead of reduction of the firing activity, which plays important roles in sound location of the auditory nervous system, awaited theoretical explanations. In the present paper, excitability and threshold mechanism for the PIF phenomenon is presented in the Morris-Lecar model with type I, II, and III excitabilities. Firstly, compared with the purely excitatory stimulations applied to the steady state, the inhibitory preceding excitatory stimulation to form pairs induces the firing rate increased for type II and III excitabilities instead of type I excitability, when the interval between the inhibitory and excitatory stimulation within each pair is suitable. Secondly, the threshold mechanism for the PIF phenomenon is acquired. For type II and III excitabilities, the inhibitory stimulation induces subthreshold oscillations around the steady state. During the middle and ending phase of the ascending part and the beginning phase of the descending part within a period of the subthreshold oscillations, the threshold to evoke an action potential by an excitatory stimulation becomes weaker, which is the cause for the PIF phenomenon. Last, a theoretical estimation for the range of the interval between the inhibitory and excitatory stimulation for the PIF phenomenon is acquired, which approximates half of the intrinsic period of the subthreshold oscillations for the relatively strong stimulations and becomes narrower for the relatively weak stimulations. The interval for the PIF phenomenon is much shorter for type III excitability, which is closer to the experiment observation, due to the shorter period of the subthreshold oscillations. The results present the excitability and threshold mechanism for the PIF phenomenon, which provide comprehensive and deep explanations to the PIF phenomenon.

Deep Learning of Biological Models from Data: Applications to ODE Models

Mathematical equations are often used to model biological processes. However, for many systems, determining analytically the underlying equations is highly challenging due to the complexity and unknown factors involved in the biological processes. In this work, we present a numerical procedure to discover dynamical physical laws behind biological data. The method utilizes deep learning methods based on neural networks, particularly residual networks. It is also based on recently developed mathematical tools of flow-map learning for dynamical systems. We demonstrate that with the proposed method, one can accurately construct numerical biological models for unknown governing equations behind measurement data. Moreover, the deep learning model can also incorporate unknown parameters in the biological process. A successfully trained deep neural network model can then be used as a predictive tool to produce system predictions of different settings and allows one to conduct detailed analysis of the underlying biological process. In this paper, we use three biological models-SEIR model, Morris-Lecar model and the Hodgkin-Huxley model-to show the capability of our proposed method.

Emergence and stability of periodic two-cluster states for ensembles of excitable units

We study dynamics in ensembles of identical excitable units with global repulsive interaction. Starting from active rotators with additional higher order Fourier modes in on-site dynamics, we observe, at sufficiently strong repulsive coupling, large-scale collective oscillations in which the elements form two separate clusters. Transitions from quiescence to clustered oscillations are caused by global bifurcations involving the unstable clustered steady states. For clusters of equal size, the scenarios evolve either through simultaneous formation of two heteroclinic trajectories or through two simultaneous saddle-node bifurcations on invariant circles. If the sizes of clusters differ, two global bifurcations are separated in the parameter space. Stability of clusters with respect to splitting perturbations depends on the kind of higher order corrections to on-site dynamics; we show that for periodic oscillations of two equal clusters the Watanabe-Strogatz integrability marks a change of stability. By extending our studies to ensembles of voltage-coupled Morris-Lecar neurons, we demonstrate that similar bifurcations and switches in stability occur also for more elaborate models in higher dimensions.

Modulation of Astrocytes on Mode Selection of Neuron Firing Driven by Electromagnetic Induction

Both of astrocytes and electromagnetic induction are magnificent to modulate neuron firing by introducing feedback currents to membrane potential. An improved astro-neuron model considering both of the two factors is employed to investigate their different roles in modulation. The mixing mode, defined by combination of period bursting and depolarization blockage, characterizes the effect of astrocytes. Mixing mode and period bursting alternatively appear in parameter space with respect to the amplitude of feedback current on neuron from astrocyte modulation. However, magnetic flux obviously plays a role of neuron firing inhibition. It not only repels the mixing mode but also suppresses period bursting. The mixing mode becomes period bursting mode and even resting state when astrocytes are hyperexcitable. Abnormal activities of astrocytes are capable to induce depolarization blockage to compose the mixing mode together with bursting mode. But electromagnetic induction shows its strong ability of inhibition of neuron firing, which is also illustrated in the bifurcation diagram. Indeed, the combination of the two factors and appropriate choice of parameters show the great potential to control disorder of neuron firing like epilepsy.

Numerical Bifurcation Analysis of Pacemaker Dynamics in a Model of Smooth Muscle Cells

Evidence from experimental studies shows that oscillations due to electro-mechanical coupling can be generated spontaneously in smooth muscle cells. Such cellular dynamics are known as pacemaker dynamics. In this article, we address pacemaker dynamics associated with the interaction of [Formula: see text] and [Formula: see text] fluxes in the cell membrane of a smooth muscle cell. First we reduce a pacemaker model to a two-dimensional system equivalent to the reduced Morris-Lecar model and then perform a detailed numerical bifurcation analysis of the reduced model. Existing bifurcation analyses of the Morris-Lecar model concentrate on external applied current, whereas we focus on parameters that model the response of the cell to changes in transmural pressure. We reveal a transition between Type I and Type II excitabilities with no external current required. We also compute a two-parameter bifurcation diagram and show how the transition is explained by the bifurcation structure.

The effects of within-neuron degree correlations in networks of spiking neurons

We consider the effects of correlations between the in- and out-degrees of individual neurons on the dynamics of a network of neurons. By using theta neurons, we can derive a set of coupled differential equations for the expected dynamics of neurons with the same in-degree. A Gaussian copula is used to introduce correlations between a neuron's in- and out-degree, and numerical bifurcation analysis is used determine the effects of these correlations on the network's dynamics. For excitatory coupling, we find that inducing positive correlations has a similar effect to increasing the coupling strength between neurons, while for inhibitory coupling it has the opposite effect. We also determine the propensity of various two- and three-neuron motifs to occur as correlations are varied and give a plausible explanation for the observed changes in dynamics.

Dynamics of a neuron-glia system: the occurrence of seizures and the influence of electroconvulsive stimuli : A mathematical and numerical study

In this paper, we investigate the dynamics of a neuron-glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro-scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single-cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters.

Different dynamical behaviors induced by slow excitatory feedback for type II and III excitabilities

Neuronal excitability is classified as type I, II, or III, according to the responses of electronic activities, which play different roles. In the present paper, the effect of an excitatory autapse on type III excitability is investigated and compared to type II excitability in the Morris-Lecar model, based on Hopf bifurcation and characteristics of the nullcline. The autaptic current of a fast-decay autapse produces periodic stimulations, and that of a slow-decay autapse highly resembles sustained stimulations. Thus, both fast- and slow-decay autapses can induce a resting state for type II excitability that changes to repetitive firing. However, for type III excitability, a fast-decay autapse can induce a resting state to change to repetitive firing, while a slow-decay autapse can induce a resting state to change to a resting state following a transient spike instead of repetitive spiking, which shows the abnormal phenomenon that a stronger excitatory effect of a slow-decay autapse just induces weaker responses. Our results uncover a novel paradoxical phenomenon of the excitatory effect, and we present potential functions of fast- and slow-decay autapses that are helpful for the alteration and maintenance of type III excitability in the real nervous system related to neuropathic pain or sound localization.

Hidden Bursting Firings and Bifurcation Mechanisms in Memristive Neuron Model With Threshold Electromagnetic Induction

Memristors can be employed to mimic biological neural synapses or to describe electromagnetic induction effects. To exhibit the threshold effect of electromagnetic induction, this paper presents a threshold flux-controlled memristor and examines its frequency-dependent pinched hysteresis loops. Using an electromagnetic induction current generated by the threshold memristor to replace the external current in 2-D Hindmarsh-Rose (HR) neuron model, a 3-D memristive HR (mHR) neuron model with global hidden oscillations is established and the corresponding numerical simulations are performed. It is found that due to no equilibrium point, the obtained mHR neuron model always operates in hidden bursting firing patterns, including coexisting hidden bursting firing patterns with bistability also. In addition, the model exhibits complex dynamics of the actual neuron electrical activities, which acts like the 3-D HR neuron model, indicating its feasibility. In particular, by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem, the bifurcation mechanisms of hidden bursting firings are expounded. Finally, circuit experiments on hardware breadboards are deployed and the captured results well match with the numerical results, validating the physical mechanism of biological neuron and the reliability of electronic neuron.

Characterizing signal encoding and transmission in class I and class II neurons via ordinal time-series analysis

Neurons encode and transmit information in spike sequences. However, despite the effort devoted to understand the encoding and transmission of information, the mechanisms underlying the neuronal encoding are not yet fully understood. Here, we use a nonlinear method of time-series analysis (known as ordinal analysis) to compare the statistics of spike sequences generated by applying an input signal to the neuronal model of Morris-Lecar. In particular, we consider two different regimes for the neurons which lead to two classes of excitability: class I, where the frequency-current curve is continuous and class II, where the frequency-current curve is discontinuous. By applying ordinal analysis to sequences of inter-spike-intervals (ISIs) our goals are (1) to investigate if different neuron types can generate spike sequences which have similar symbolic properties; (2) to get deeper understanding on the effects that electrical (diffusive) and excitatory chemical (i.e., excitatory synapse) couplings have; and (3) to compare, when a small-amplitude periodic signal is applied to one of the neurons, how the signal features (amplitude and frequency) are encoded and transmitted in the generated ISI sequences for both class I and class II type neurons and electrical or chemical couplings. We find that depending on the frequency, specific combinations of neuron/class and coupling-type allow a more effective encoding, or a more effective transmission of the signal.

Transition between multimode oscillations in a loaded hair bundle

In this paper, we study the dynamics of an autonomous system for a hair bundle subject to mechanical load. We demonstrated the spontaneous oscillations that arise owing to interactions between the linear stiffness and the adapting stiffness. It is found that by varying the linear stiffness, the system can induce a weakly chaotic attractor in a certain region where the stable periodic orbit is infinitely close to a parabolic curve composed of unstable equilibrium points. By altering the adapting stiffness associated with the calcium concentration, the system is able to trigger the transition from the bistable resting state, through a pair of symmetric Hopf bifurcation, into the bistable limit cycle, even to the chaotic attractor. At a negative adapting stiffness, the system exhibits a double-scroll chaotic attractor. According to the method of qualitative theory of fast-slow decomposition, the trajectory of a double-scroll chaotic attractor in the whole system depends upon the symmetric fold/fold bifurcation in a fast system. Furthermore, the control of the adapting stiffness in the improved system with two slow variables can trigger a new transition from the bistable resting state into the chaotic attractor, even to the hyperchaotic attractor by observing the Lyapunov exponent.

Alternating activity patterns and a chimeralike state in a network of globally coupled excitable Morris-Lecar neurons

Spatiotemporal chaos collapses to either a rest state or a propagating pulse in a ring network of diffusively coupled, excitable Morris-Lecar neurons. Adding global varying synaptic coupling to the ring network reveals complex transient behavior. Spatiotemporal chaos collapses into a transient pulse that reinitiates spatiotemporal chaos to allow sequential pattern switching until a collapse to the rest state. A domain of irregular neuron activity coexists with a domain of inactive neurons forming a transient chimeralike state. Transient spatial localization of the chimeralike state is observed for stronger synapses.

Neural network model for path-finding problems with the self-recovery property

The large-scale synchronization of neural oscillations is crucial in the functional integration of brain modules, but the combination of modules changes depending on the task. A mathematical description of this flexibility is a key to elucidating the mechanism of such spontaneous neural activity. We present a model that finds the loop structure of a network whose nodes are connected by unidirectional links. Using this model, we propose a path-finding system that spontaneously finds a path connecting two specified nodes. The solution path is represented by phase-synchronized oscillatory solutions. The model has the self-recovery property: that is, it is a system with the ability to find a new path when one of the connections in the existing path is suddenly removed. We show that the model construction procedure is applicable to a wide class of nonlinear systems arising in chemical reactions and neural networks.

A Python Code for Simulating Single Tactile Receptors and the Spiking Responses of Their Afferents

This work presents a pieces of Python code to rapidly simulate the spiking responses of large numbers of single cutaneous tactile afferents with millisecond precision. To simulate the spike responses of all the major types of cutaneous tactile afferents, we proposed an electromechanical circuit model, in which a two-channel filter was developed to characterize the mechanical selectivity of tactile receptors, and a spike synthesizer was designed to recreate the action potentials evoked in afferents. The parameters of this model were fitted using previous neurophysiological datasets. Several simulation examples were presented in this paper to reproduce action potentials, sensory adaptation, frequency characteristics and spiking timing for each afferent type. The results indicated that the simulated responses matched previous neurophysiological recordings well. The model allows for a real-time reproduction of the spiking responses of about 4,000 tactile units with a timing precision of <6 ms. The current work provides a valuable guidance to designing highly realistic tactile interfaces such as neuroprosthesis and haptic devices.

A biophysical model explains the spontaneous bursting behavior in the developing retina

During early development, waves of activity propagate across the retina and play a key role in the proper wiring of the early visual system. During a particular phase of the retina development (stage II) these waves are triggered by a transient network of neurons, called Starburst Amacrine Cells (SACs), showing a bursting activity which disappears upon further maturation. The underlying mechanisms of the spontaneous bursting and the transient excitability of immature SACs are not completely clear yet. While several models have attempted to reproduce retinal waves, none of them is able to mimic the rhythmic autonomous bursting of individual SACs and reveal how these cells change their intrinsic properties during development. Here, we introduce a mathematical model, grounded on biophysics, which enables us to reproduce the bursting activity of SACs and to propose a plausible, generic and robust, mechanism that generates it. The core parameters controlling repetitive firing are fast depolarizing V-gated calcium channels and hyperpolarizing V-gated potassium channels. The quiescent phase of bursting is controlled by a slow after hyperpolarization (sAHP), mediated by calcium-dependent potassium channels. Based on a bifurcation analysis we show how biophysical parameters, regulating calcium and potassium activity, control the spontaneously occurring fast oscillatory activity followed by long refractory periods in individual SACs. We make a testable experimental prediction on the role of voltage-dependent potassium channels on the excitability properties of SACs and on the evolution of this excitability along development. We also propose an explanation on how SACs can exhibit a large variability in their bursting periods, as observed experimentally within a SACs network as well as across different species, yet based on a simple, unique, mechanism. As we discuss, these observations at the cellular level have a deep impact on the retinal waves description.

Effect of external periodic signals and electromagnetic radiation on autaptic regulation of neuronal firing

An improved Hindmarsh-Rose (HR) neuron model, where the memristor is a bridge between membrane potential and magnetic flux, can be used to investigate the effect of periodic signals on autaptic regulation of neurons under electromagnetic radiation. Based on the improved HR model driven by periodic high-low-frequency current and electromagnetic radiation, the responses of electrical autaptic regulation with diverse high-low-frequency signals are investigated using bifurcation analysis. It is found that the electrical modes of neurons are determined by the selecting parameters of both periodic high and low-frequency current and electromagnetic radiation, and the Hamiltonian energy depends on the neuronal firing modes. The effects of Gaussian white noise on the membrane potential are discussed using numerical simulations. It is demonstrated that external high-low-frequency stimulus plays a significant role in the autaptic regulation of neural firing mode, and the electrical mode of neurons can be affected by the angular frequency of both high-low-frequency forcing current and electromagnetic radiation. The mechanism of neuronal firing regulated by high-low-frequency signal and electromagnetic radiation discussed here could be applied to research neuronal networks and synchronisation modes.

Synchronization by uncorrelated noise: interacting rhythms in interconnected oscillator networks

Oscillators coupled in a network can synchronize with each other to yield a coherent population rhythm. How do multiple such rhythms interact with each other? Do these collective oscillations synchronize like individual oscillators? We show that this is not the case: for strong, inhibitory coupling rhythms can become synchronized by noise. In contrast to stochastic synchronization, this new mechanism synchronizes the rhythms even if the noisy inputs to different oscillators are completely uncorrelated. Key for the synchrony across networks is the reduced synchrony within the networks: it substantially increases the frequency range across which the networks can be entrained by other networks or by periodic pacemaker-like inputs. We demonstrate this type of robust synchronization for different classes of oscillators and network connectivities. The synchronization of different population rhythms is expected to be relevant for brain rhythms.

Minimum Action Path Theory Reveals the Details of Stochastic Transitions Out of Oscillatory States

Cell state determination is the outcome of intrinsically stochastic biochemical reactions. Transitions between such states are studied as noise-driven escape problems in the chemical species space. Escape can occur via multiple possible multidimensional paths, with probabilities depending nonlocally on the noise. Here we characterize the escape from an oscillatory biochemical state by minimizing the Freidlin-Wentzell action, deriving from it the stochastic spiral exit path from the limit cycle. We also use the minimized action to infer the escape time probability density function.

Dynamical Mechanism of Hyperpolarization-Activated Non-specific Cation Current Induced Resonance and Spike-Timing Precision in a Neuronal Model

Hyperpolarization-activated cyclic nucleotide-gated cation current (I_h) plays important roles in the achievement of many physiological/pathological functions in the nervous system by modulating the electrophysiological activities, such as the rebound (spike) to hyperpolarization stimulations, subthreshold membrane resonance to sinusoidal currents, and spike-timing precision to stochastic factors. In the present paper, with increasing g_h (conductance of I_h), the rebound (spike) and subthreshold resonance appear and become stronger, and the variability of the interspike intervals (ISIs) becomes lower, i.e., the enhancement of spike-timing precision, which are simulated in a conductance-based theoretical model and well explained by the nonlinear concept of bifurcation. With increasing g_h, the stable node to stable focus, to coexistence behavior, and to firing via the codimension-1 bifurcations (Hopf bifurcation, saddle-node bifurcation, saddle-node bifurcations on an invariant circle, and saddle homoclinic orbit) and codimension-2 bifurcations such as Bogdanov-Takens (BT) point related to the transition between saddle-node and Hopf bifurcations, are acquired with 1- and 2-parameter bifurcation analysis. The decrease of variability of ISIs with increasing g_h is induced by the fast decrease of the standard deviation of ISIs, which is related to the increase of the capacity of resisting noisy disturbance due to the firing becomes far away from the bifurcation point. The enhancement of the rebound (spike) with increasing g_h builds up a relationship to the decrease of the capacity of resisting disturbance like the hyperpolarization stimulus as the resting state approaches the bifurcation point. The “typical”-resonance and non-resonance appear in the parameter region of the stable focus and node far away from the bifurcation points, respectively. The complex or “strange” dynamics, such as the “weak”-resonance for the stable node near the transition point between the stable node and focus and the non-resonance for the stable focus close to the codimension-1 and −2 bifurcation points, are discussed.

Transient chaos and associated system-intrinsic switching of spacetime patterns in two synaptically coupled layers of Morris-Lecar neurons

Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a single layer of diffusively coupled, excitable Morris-Lecar neurons. Weak synaptic coupling of two such layers reveals system intrinsic switching of spatiotemporal activity patterns within and between the layers at irregular times. Within a layer, switching sequences include spatiotemporal chaos, erratic and regular pulse propagation, spontaneous network wide neuron activity, and rest state. A momentary substantial reduction in neuron activity in one layer can reinitiate transient spatiotemporal chaos in the other layer, which can induce a swap of spatiotemporal chaos with a pulse state between the layers. Presynaptic input maximizes the distance between propagating pulses, in contrast to pulse merging in the absence of synapses.

Routes to Chaos Induced by a Discontinuous Resetting Process in a Hybrid Spiking Neuron Model

Several hybrid spiking neuron models combining continuous spike generation mechanisms and discontinuous resetting processes following spiking have been proposed. The Izhikevich neuron model, for example, can reproduce many spiking patterns. This model clearly possesses various types of bifurcations and routes to chaos under the effect of a state-dependent jump in the resetting process. In this study, we focus further on the relation between chaotic behaviour and the state-dependent jump, approaching the subject by comparing spiking neuron model versions with and without the resetting process. We first adopt a continuous two-dimensional spiking neuron model in which the orbit in the spiking state does not exhibit divergent behaviour. We then insert the resetting process into the model. An evaluation using the Lyapunov exponent with a saltation matrix and a characteristic multiplier of the Poincar'e map reveals that two types of chaotic behaviour (i.e. bursting chaotic spikes and near-period-two chaotic spikes) are induced by the resetting process. In addition, we confirm that this chaotic bursting state is generated from the periodic spiking state because of the slow- and fast-scale dynamics that arise when jumping to the hyperpolarization and depolarization regions, respectively.

Response of Electrical Activity in an Improved Neuron Model under Electromagnetic Radiation and Noise

Electrical activities are ubiquitous neuronal bioelectric phenomena, which have many different modes to encode the expression of biological information, and constitute the whole process of signal propagation between neurons. Therefore, we focus on the electrical activities of neurons, which is also causing widespread concern among neuroscientists. In this paper, we mainly investigate the electrical activities of the Morris-Lecar (M-L) model with electromagnetic radiation or Gaussian white noise, which can restore the authenticity of neurons in realistic neural network. First, we explore dynamical response of the whole system with electromagnetic induction (EMI) and Gaussian white noise. We find that there are slight differences in the discharge behaviors via comparing the response of original system with that of improved system, and electromagnetic induction can transform bursting or spiking state to quiescent state and vice versa. Furthermore, we research bursting transition mode and the corresponding periodic solution mechanism for the isolated neuron model with electromagnetic induction by using one-parameter and bi-parameters bifurcation analysis. Finally, we analyze the effects of Gaussian white noise on the original system and coupled system, which is conducive to understand the actual discharge properties of realistic neurons.

Transitions between classes of neuronal excitability and bifurcations induced by autapse

Neuronal excitabilities behave as the basic and important dynamics related to the transitions between firing and resting states, and are characterized by distinct bifurcation types and spiking frequency responses. Switches between class I and II excitabilities induced by modulations outside the neuron (for example, modulation to M-type potassium current) have been one of the most concerning issues in both electrophysiology and nonlinear dynamics. In the present paper, we identified switches between 2 classes of excitability and firing frequency responses when an autapse, which widely exists in real nervous systems and plays important roles via self-feedback, is introduced into the Morris-Lecar (ML) model neuron. The transition from class I to class II excitability and from class II to class I spiking frequency responses were respectively induced by the inhibitory and excitatory autapse, which are characterized by changes of bifurcations, frequency responses, steady-state current-potential curves, and nullclines. Furthermore, we identified codimension-1 and -2 bifurcations and the characteristics of the current-potential curve that determine the transitions. Our results presented a comprehensive relationship between 2 classes of neuronal excitability/spiking characterized by different types of bifurcations, along with a novel possible function of autapse or self-feedback control on modulating neuronal excitability.

A Consistent Definition of Phase Resetting Using Hilbert Transform

A phase resetting curve (PRC) measures the transient change in the phase of a neural oscillator subject to an external perturbation. The PRC encapsulates the dynamical response of a neural oscillator and, as a result, it is often used for predicting phase-locked modes in neural networks. While phase is a fundamental concept, it has multiple definitions that may lead to contradictory results. We used the Hilbert Transform (HT) to define the phase of the membrane potential oscillations and HT amplitude to estimate the PRC of a single neural oscillator. We found that HT's amplitude and its corresponding instantaneous frequency are very sensitive to membrane potential perturbations. We also found that the phase shift of HT amplitude between the pre- and poststimulus cycles gives an accurate estimate of the PRC. Moreover, HT phase does not suffer from the shortcomings of voltage threshold or isochrone methods and, as a result, gives accurate and reliable estimations of phase resetting.

Autaptic regulation of electrical activities in neuron under electromagnetic induction

Realistic neurons may hold complex anatomical structure, for example, autapse connection to some internuncial neurons, which this specific synapse can connect to its body via a close loop. Continuous exchanges of charged ions across the membrane can induce complex distribution fluctuation of intracellular and extracellular charged ions of cell, and a time-varying electromagnetic field is set to modulate the membrane potential of neuron. In this paper, an autapse-modulated neuron model is presented and the effect of electromagnetic induction is considered by using magnetic flux. Bifurcation analysis and sampled time series for membrane potentials are calculated to investigate the mode transition in electrical activities and the biological function of autapse connection is discussed. Furthermore, the Gaussian white noise and electromagnetic radiation are considered on the improved neuron model, it is found appropriate setting and selection for feedback gain and time delay in autapse can suppress the bursting in neuronal behaviors. It indicates the formation of autapse can enhance the self-adaption of neuron so that appropriate response to external forcing can be selected, this biological function is helpful for encoding and signal propagation of neurons. It can be useful for investigation about collective behaviors in neuronal networks exposed to electromagnetic radiation.

Neural Excitability and Singular Bifurcations

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

Artificial Electrical Morris-Lecar Neuron

In this paper, an experimental electronic neuron based on a complete Morris-Lecar model is presented, which is able to become an experimental unit tool to study collective association of coupled neurons. The circuit design is given according to the ionic currents of this model. The experimental results are compared with the theoretical prediction, leading to a good agreement between them, which therefore validate the circuit. The use of some parts of the circuit is also possible for other neurons models, namely for those based on ionic currents.

Bursting dynamics in a population of oscillatory and excitable Josephson junctions

We report an emergent bursting dynamics in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducting device is considered for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. For a coupling value above a threshold, the network splits into two clusters when a reductionism approach is applied to reproduce the bursting behavior of the large network. The excitable junctions effectively induce a slow dynamics on the oscillatory units to generate parabolic bursting in a broad parameter space. We reproduce the bursting dynamics in a mixed population of dynamical nodes of the Morris-Lecar model.