Morris-Lecar Research Articles

Memristive biophysical neuron models forming an excitatory–inhibitory neural network for modeling PING rhythm generation

SPICE models are constructed for memristive devices to form associated biophysical neuron circuit models such as the Hodgkin–Huxley (HH) type II excitability neuron circuit model, the HH type III excitability neuron circuit model, the simplified HH neuron circuit model, the Morris–Lecar neuron circuit model, and the memristive based direct-current (DC) circuit model. Rigorous nonlinear circuit-theoretic principles are also applied to analyze the different behaviors of the generic memristor Na $$^{+}$$ -ion, K $$^{+}$$ -ion, and Ca $$^{++}$$ -ion channels forming these biophysical neuron circuit models. Detailed explanations and clarifications are presented on the memristive HH type II and HH type III axonal excitabilities based on mathematical analysis as well as the circuit models. This is done from the perspective of the spike patterns generated by both of these biophysical neuron circuit models. Moreover, various experimental studies have revealed a synchronous brain state known as gamma rhythms that are responsible for sensory, memory, and motor processes. This suggests that understanding how the gamma oscillation (30–100 Hz) is generated in the brain will be extremely important to unravel the link between the activity of an individual neuron and the cognitive processing achieved by a population of networked neurons. We thus also study the dynamics of an interconnected excitatory–inhibitory (E–I) network population, which is ubiquitous in the brain. Utilizing biophysical models of the E–I network, we investigate the generation of pyramidal-interneuronal network gamma (PING) rhythms caused by the external input to the network and the connectivity heterogeneities. The results reveal that synchronous strong PING and sparsely firing weak PING rhythms are generated based on the network connectivities and external input heterogeneities in simulations of 100 memristive HH type II excitability neurons forming an E–I network.

Paradoxical reduction and the bifurcations of neuronal bursting activity modulated by positive self-feedback

Paradoxical enhancement rather than reduction in firing activity induced by inhibitory effect is very important for both nonlinear dynamics and neuroscience. In the present paper, to build a more comprehensive viewpoint related to excitatory effect, a paradoxical phenomenon modulated by the excitatory self-feedback of autapse is extended to bursting activity in a modified Morris–Lecar model. With increasing the strength of excitatory autapse, the bursting patterns exhibit inverse period-adding bifurcations to spiking pattern via chaos, and the burst duration, the spike number per burst, and the firing frequency of bursting decrease, which is different from the traditional view that firing frequency should be enhanced. Furthermore, the paradoxical reduction in bursting activity is well explained with the nonlinear dynamics of the fast subsystem. The burst begins from neighborhood of a fold bifurcation of equilibrium point and terminates at a saddle homoclinic (SH) bifurcation. With increasing conductance of autapse, the fold bifurcation point remains unchanged, which is due to too small autaptic current near the fold point to influence the dynamics, and the SH point shifts left, which follows from the enhanced potassium current at the minimal value of the stable limit cycle. Therefore, the range between the fold and SH point becomes narrower to shorten the burst duration to reduce spike number per burst and firing frequency. The novel example of paradoxical reduction modulated by positive self-feedback and the associated bifurcation mechanism enrich the phenomena of nonlinear science and present the potential functions of excitatory autapse in the brain neurons with bursting behavior.

A LSTM based prediction model for nonlinear dynamical systems with chaotic itinerancy

The prediction for chaotic trajectory from the measured data of time history, without prior knowledge of underlying dynamical model, is a challenging task in the data-driven analysis, due to its sensitivity to initial conditions. In this paper, the Long Short-Term Memory Network (LSTM) with the merge layer is proposed to predict the future states of the coupled Morris-Lecar (M-L) system with the chaotic itinerancy responses. Here, the two LSTM models with single-branch and multi-branch are constructed respectively to carry out the predictions in the multivariate loading conditions. By comparison to the network model with single-branch, the multi-branch model with adding merge layer can provide a high utilization of weights to reduce training cost greatly and receive a low prediction error, which make the multi-layer LSTM promising to estimate a high-dimensional complex dynamical behavior like transient chaotic itinerancy.

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.

A phase model with large time delayed coupling

We consider two identical oscillators with weak, time delayed coupling. We start with a general system of delay differential equations then reduce it to a phase model. With the assumption of large time delay, the resulting phase model has an explicit delay and phase shift in the argument of the phases and connection function, respectively. Using the phase model, we prove that for any type of oscillators and any coupling, the in-phase and anti-phase phase-locked solutions always exist and give conditions for their stability. We show that for small delay these solutions are unique, but with large enough delay multiple solutions of each type with different frequencies may occur. We give conditions for the existence and stability of other types of phase-locked solutions. We discuss the various bifurcations that can occur in the phase model as the time delay is varied. The results of the phase model analysis are applied to Morris–Lecar oscillators with diffusive coupling and compared with numerical studies of the full system of delay differential equations. We also consider the case of small time delay and compare the results with the existing ones in the literature.

Wave propagation in a network of extended Morris–Lecar neurons with electromagnetic induction and its local kinetics

An extended Morris–Lecar neuron model incorporating electromagnetic flux coupling and external excitation is proposed. The complete dynamical behavior of the new model is investigated as autonomous and non-autonomous forms. Multistability and coexisting attractors are shown for the new neuron model. It is shown that when external excitation is introduced, the complete dynamical behavior of the system changes compared to the non-excited version. To study the wave propagation pattern, we construct a network of the extended neuron model and study the generation of spiral waves under various conditions of coupling strength, stimuli parameters and system parameters. We have also studied the wave propagation pattern considering the time-based chaotic-periodic neurons in the network.

Bifurcations and excitability in the temperature-sensitive Morris–Lecar neuron

The neuronal excitability related to the transition between firing and resting states is a basic and important dynamic behavior which has different bifurcation characteristics and spiking frequency responses. In this work, we study the response dynamics and the excitability of a Morris–Lecar neuron for two temperature-sensitive ion channels, calcium and leak current, respectively. The codimension-1 bifurcations and the frequency–response curves for different temperatures show that the neuronal excitability is from Class II to I with increasing temperature in the case of the temperature-sensitive calcium current but is from Class I to II for the case of the leak current. The further extensive codimension-2 bifurcations uncover that the neurons undergo different routes of the neuronal dynamics under increasing external current for different temperatures even the same neuronal excitability. These results provide insight into understanding the effect of temperature on the neuronal dynamics and response behaviors with the diversity of temperature-sensitive ion channels.

Memristor Synapse-Based Morris–Lecar Model: Bifurcation Analyses and FPGA-Based Validations for Periodic and Chaotic Bursting/Spiking Firings

Electromagnetic induction current sensed by the membrane potential in biological neurons can be characterized with a memristor synapse, which can be employed to demonstrate the real oscillating voltage patterns of Barnacle muscle fibers. This paper presents a 3D autonomous memristor synapse-based Morris–Lecar (abbreviated as m-ML) model, which is implemented through introducing a memristor synapse-based induction current to substitute the externally applied current in an existing 2D nonautonomous Morris–Lecar model. Making use of one- and two-parameter bifurcation plots and time-domain representations, diverse period-adding bifurcations as well as abundant periodic and chaotic burst firings are demonstrated. Through constructing the fold and Hopf bifurcation sets of fast spiking subsystem, bifurcation analyses of these chaotic and periodic burst firings are carried out. Moreover, the periodic and chaotic spiking firings and coexisting firing behaviors are illustrated by using one- and two-parameter bifurcation plots and local attraction basins. Finally, based on a field programmable gate array (FPGA) board, a compact digital electronic neuron is fabricated for the 3D m-ML model, from which periodic and chaotic bursting/spiking firings are experimentally measured to verify the results of the numerical simulations.

The noise cancelation effects caused by spike-frequency adaptation in single neurons

Spike-frequency adaptation, which reduces the firing rate of a neuron during a constant stimulus, is a prominent property observed in many neurons. In this work, we studied the effects of the $$I_{\mathrm{AHP}}$$ spiking adaptation on the information transmission and efficiency of the Morris–Lecar (ML) neuron model in the spike-timing coding scheme. We showed that this kind of adaptation caused non-trivial spiking dynamics when the input rate was high. Under the stimulation of high-rate inputs, although $$I_{\mathrm{AHP}}$$ adaptation neurons could not outperform non-adaptation neurons in terms of information rate, $$I_{\mathrm{AHP}}$$ adaptation neurons yielded higher coding efficiency than non-adaptation neurons. We also found that the noise could enlarge the range of input rates that the adaptation takes effect to enhance coding efficiency. Increasing the calcium-activated $$\hbox {K}^{+}$$ current could also extend the range of input rates in which adaptation takes effect. Therefore, we argue that the $$I_{\mathrm{AHP}}$$ adaptation mechanism may play a role of adaptive noise cancelation mechanism in neuronal information processing, i.e., it maintains information transmission when neurons receive low-rate inputs but substantially enhancing coding efficiency for high-rate inputs and highly noise environments by suppressing the spike train variability caused by the noise.

Chimera states in hybrid coupled neuron populations

Here we study the emergence of chimera states, a recently reported phenomenon referring to the coexistence of synchronized and unsynchronized dynamical units, in a population of Morris–Lecar neurons which are coupled by both electrical and chemical synapses, constituting a hybrid synaptic architecture, as in actual brain connectivity. This scheme consists of a nonlocal network where the nearest neighbor neurons are coupled by electrical synapses, while the synapses from more distant neurons are of the chemical type. We demonstrate that peculiar dynamical behaviors, including chimera state and traveling wave, exist in such a hybrid coupled neural system, and analyze how the relative abundance of chemical and electrical synapses affects the features of chimera and different synchrony states (i.e. incoherent, traveling wave and coherent) and the regions in the space of relevant parameters for their emergence. Additionally, we show that, when the relative population of chemical synapses increases further, a new intriguing chaotic dynamical behavior appears above the region for chimera states. This is characterized by the coexistence of two distinct synchronized states with different amplitude, and an unsynchronized state, that we denote as a chaotic amplitude chimera. We also discuss about the computational implications of such state.

State transitions in the Morris-Lecar model under stable Lévy noise

This paper considers the state transition of the stochastic Morris-Lecar neuronal model driven by symmetric $\alpha$-stable L\'evy noise. The considered system is bistable: a stable fixed point (resting state) and a stable limit cycle (oscillating state), and there is an unstable limit cycle (borderline state) between them. Small disturbances may cause a transition between the two stable states, thus a deterministic quantity, namely the maximal likely trajectory, is used to analyze the transition phenomena in non-Gaussian stochastic environment. According to the numerical experiment, we find that smaller jumps of the L\'evy motion and smaller noise intensity can promote such transition from the sustained oscillating state to the resting state. It also can be seen that larger jumps of the L\'evy motion and higher noise intensity are conducive for the transition from the borderline state to the sustained oscillating state. As a comparison, Brownian motion is also taken into account. The results show that whether it is the oscillating state or the borderline state, the system disturbed by Brownian motion will be transferred to the resting state under the selected noise intensity.

Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris–Lecar model

AbstractIn the first part of this paper we study approximations of trajectories of piecewise deterministic processes (PDPs) when the flow is not given explicitly by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for piecewise deterministic Markov processes (PDMPs). These estimates are the building blocks of the multilevel Monte Carlo (MLMC) method, which we study in the second part. The coupling required by the MLMC is based on the thinning procedure. In the third part we apply these results to a two-dimensional Morris–Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms classical Monte Carlo.

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.

Impact of Chloride Channel on Spiking Patterns of Morris-Lecar Model

In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl- fluctuations on firing patterns of this neuron model. After adding Cl- channel in the original Morris-Lecar model, the dynamics of the original model such as its bifurcations of equilibrium points would be changed and they occurred at different values compared to the primary model. We discover these qualitative changes in the point of dynamical systems and neuroscience. We will conduct the co-dimension two bifurcations analysis with respect to different control parameters to explore the complicated behaviors for this new neuron model.

The Phenomenon of Neural Bursting and Spiking in Neurons: Morris-Lecar Model

In this study, we explore the interesting phenomenon of firing spikes and complex dynamics of Morris-Lecar model. We consider a set of parameters such that the model exhibits a wide range of phenomenons. We investigate the influences of injected current and temperature on the spiking dynamics of Morris-Lecar model. Moreover, we study bifurcations, and computational properties of this neuron model. Also, we define a bound (Max and Min voltage) for membrane potential and a certain voltage value or threshold for firing the spikes. Studying the two co-dimension bifurcations demonstrates much more complicated behaviors for this single neuron model. We also describe the phenomenon of neural bursting, and investigate the dynamics of Morris-Lecar model as a square-wave burster, elliptic burster and parabolic burster.

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.

Initial-induced coexisting and synchronous firing activities in memristor synapse-coupled Morris–Lecar bi-neuron network

A memristor synapse with threshold memductance is employed to couple two neurons for representation of the electromagnetic induction effect triggered by their membrane potential difference. This paper presents a memristor synapse-coupled bi-neuron network by bidirectionally coupling two three-dimensional heterogeneous or homogeneous Morris–Lecar neurons with such a memristor synapse. The memristive bi-neuron network possesses a line equilibrium set with its stability related to the induction coefficient and memristor initial value. Coexisting firing activities in the heterogeneous memristive bi-neuron network are explored using bifurcation plots, phase plots, and time sequences, upon which the initial-induced infinitely many firing patterns including hyperchaotic, chaotic, and periodic bursting and tonic-spiking patterns are disclosed, indicating the emergence of the initial-induced extreme multistability. Furthermore, synchronous firing activities in homogeneous memristive bi-neuron network are investigated using the time sequences, synchronization transition states, and mean synchronized errors. The results demonstrate that the synchronous firing activities are associated with the induction coefficient and specially associated with the initial values of memristor synapse and coupling neurons. Finally, an FPGA-based electronic bi-neuron network is designed to experimentally confirm the memristor initial-induced coexisting firing activities.

Chaotic Bursting Dynamics and Coexisting Multistable Firing Patterns in 3D Autonomous Morris–Lecar Model and Microcontroller-Based Validations

A three-dimensional (3D) autonomous Morris–Lecar (simplified as M–L) neuron model with fast and slow structures was proposed to generate periodic bursting behaviors. However, chaotic bursting dynamics and coexisting multistable firing patterns have been rarely discussed in such a 3D M–L neuron model. For some specified model parameters, MATLAB numerical plots are executed by bifurcation plots, time sequences, phase plane plots, and 0–1 tests, from which diverse forms of chaotic bursting, chaotic tonic-spiking, and periodic bursting behaviors are uncovered in the 3D M–L neuron model. Furthermore, based on the theoretically constructing fold/Hopf bifurcation sets of the fast subsystem, the bifurcation mechanism for the chaotic bursting behaviors is thereby expounded qualitatively. Particularly, through numerically plotting the attraction basins related to the initial states under two sets of specific parameters, coexisting multistable firing patterns are demonstrated in the 3D M–L neuron model also. Finally, a digitally circuit-implemented electronic neuron is generated based on a low-power microcontroller and its experimentally captured results faultlessly validate the numerical plots.

Multi-scale modeling of the circadian modulation of learning and memory.

We propose a multi-scale model to explain the time-of-day effects on learning and memory. We specifically model the circadian variation of hippocampus (HC) dependent long-term potentiation (LTP), depression (LTD), and the fear conditioning paradigm in amygdala. The model we built has both Goodwin type circadian gene regulatory network (GRN) and the conductance model of Morris-Lecar (ML) type to explain the spontaneous firing patterns (SFR) in suprachiasmatic nucleus (SCN). In the conductance model, we also include N-Methyl-D-aspartic acid receptor (NMDAR) to study the circadian dependent changes in LTP/LTD in hippocampus and include both NMDAR and α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptor (AMPAR) dynamics to explain the circadian modulation of fear conditioning paradigm in memory acquisition, recall, and extinction as seen in amygdala. Our multi-scale model captures the essential dynamics seen in the experiments and strongly supports the circadian time-of-the-day effects on learning and memory.

Autaptic Effects on Synchronization and Phase Response Curves of Neurons with a Chemical Synapse

Previously, we reported the effects of an autapse, a time-delayed self-feedback synapse, on synchronization of Morris-Lecar (ML) neurons with an electrical synapse [J. Korean Phys. Soc. 71, 63 (2017)]. Because communication between neurons via a chemical synapse is more common than that via a electrical synapse, the desire to study the autaptic effect on synchronization of neurons with a chemical synapse is natural. In this paper, we present the results for the autaptic effects on synchronization of neurons with a chemical synapse. We show that the phase response curves of a single neuron and a pair of neurons are useful for understanding the autaptic effects on synchronization. We introduce a model for a theoretical analysis of the autaptic effect on synchronization of parameter-mismatched neurons and find that the model qualitatively explains that effect well.