Synergism and antagonism of neurons caused by an electrical synapse (original) (raw)

Emergence of bursting in two coupled neurons of different types of excitability

Chaos, Solitons & Fractals, 2020

In this manuscript, a spiking neuron of type I excitability and a silent neuron of type II excitability are coupled through a gap junction with unequal coupling strengths, and none of the coupled neurons can burst intrinsically. By applying the theory of dynamical systems (e.g. bifurcation theory), we investigate how the coupling strength affects the dynamics of the neurons, when one of the coupling strengths is fixed and the other varies. We report four different regimes of oscillations as the coupling strength increases. (1) Spike-Spike Phase-Locking, where both neurons are in tonic spiking mode but with different frequencies; (2) Spike-Burst mode, where the type II neuron bursts while the type I neuron remains in tonic spiking mode; (3) Burst-Burst synchronization, where the neurons burst synchronously, i.e., both neurons enter and exit the active phase almost together; (4) Spike-Spike Synchronization, where the neurons synchronize as two oscillators, i.e., they oscillate with equal time period and fraquency. An interesting finding is that there exist two different synchronous behaviours, one of them corresponds to 1 −burst synchronization of the neurons and the other corresponds to the synchronizations of 1 −bursting oscillations in type II neuron and tonic spiking oscillations in type I neuron. Finally it should be pointed out that all through increasing the coupling strength we observe sequences of intermittency in the neurons, which is an abrupt and irregular transition between periodic oscillations and chaotic dynamics.

Dynamic Hopf Bifurcation in Spatially Extended Excitable Systems from Neuroscience

2012

One explanation for membrane accommodation in response to a slowly rising current, and the phenomenon underlying the dynamics of elliptic bursting in nerves, is the mathematical problem of dynamic Hopf bifurcation. This problem has been studied extensively for linear (deterministic and stochastic) current ramps, nonlinear ramps, and elliptic bursting. These studies primarily investigated dynamic Hopf bifurcation in space-clamped excitable cells. In this study we introduce a new phenomenon associated with dynamic Hopf bifurcation. We show that for excitable spiny cables injected at one end with a slow current ramp, the generation of oscillations may occur an order one distance away from the current injection site. The phenomenon is significant since in the model the geometric and electrical parameters, as well as the ion channels, are uniformly distributed. In addition to demonstrating the phenomenon computationally, we analyze the problem using a singular perturbation method that provides a way to predict when and where the onset will occur in response to the input stimulus. We do not see this phenomenon for excitable cables in which the ion channels are embedded in the cable membrane itself, suggesting that it is essential for the channels to be isolated in the spines.

Bifurcation transitions in gap-junction-coupled neurons

Physical review, 2016

Here we investigate transitions occurring in the dynamical states of pairs of distinct neurons electrically coupled, with one neuron tonic and the other bursting. Depending on the dynamics of the individual neurons, and for strong enough coupling, they synchronize either in a tonic or a bursting regime, or initially tonic transitioning to bursting via a period doubling cascade. Certain intrinsic properties of the individual neurons such as minimum firing rates are carried over into the dynamics of the coupled neurons affecting their ultimate synchronous state.

Dynamical behavior of the firings in a coupled neuronal system

Physical Review E, 1993

The time-interval sequences and the spatiotemporal patterns of the firings of a coupled neuronal network are investigated in this paper. For a single neuron stimulated by an external stimulus I, the timeinterval sequences show a low-frequency firing of bursts of spikes and a reversed period-doubling cascade to a high-frequency repetitive firing state as the stimulus Iis increased. For two neurons coupled to each other through the firing of the spikes, the complexity of the time-interval sequences becomes simple as the coupling strength increases. A network with a large number of neurons shows a complex spatiotemporal pattern structure. As the coupling strength increases, the number of phase-locked neurons increases and the time-interval diagram shows temporal chaos and a bifurcation in the space. The dynamical behavior is also verified by the behavior of the Lyapunov exponent.

Bifurcation analysis and diverse firing activities of a modified excitable neuron model

Cognitive Neurodynamics, 2019

Electrical activities of excitable cells produce diverse spiking-bursting patterns. The dynamics of the neuronal responses can be changed due to the variations of ionic concentrations between outside and inside the cell membrane. We investigate such type of spiking-bursting patterns under the effect of an electromagnetic induction on an excitable neuron model. The effect of electromagnetic induction across the membrane potential can be considered to analyze the collective behavior for signal processing. The paper addresses the issue of the electromagnetic flow on a modified Hindmarsh-Rose model (H-R) which preserves biophysical neurocomputational properties of a class of neuron models. The different types of firing activities such as square wave bursting, chattering, fast spiking, periodic spiking, mixed-mode oscillations etc. can be observed using different injected current stimulus. The improved version of the model includes more parameter sets and the multiple electrical activities are exhibited in different parameter regimes. We perform the bifurcation analysis analytically and numerically with respect to the key parameters which reveals the properties of the fast-slow system for neuronal responses. The firing activities can be suppressed/enhanced using the different external stimulus current and by allowing a noise induced current. To study the electrical activities of neural computation, the improved neuron model is suitable for further investigation.

Hopfield Model of a Neuron Action under Dynamical Thresholds

International Journal of Computer Applications, 2011

In this paper we present Hopfield model of a neuron dynamics given by the neuronic equation. In the first model second order neuronic equation describe the behavior of a neuron in the presence of some local positive feedback. The second model portray two neurons in which first order neuronic equation represents dynamics of the second neuron in the presence of a discharged pulse coded signal function from the first neuron. We have shown that the solution is bounded and the paths surrounding the equilibrium point are not closed curves in the phase plane. Some conditions ensuring the existence and uniqueness of the equilibrium point are derived.

Modèles mathématiques pour l'étude des phénomènes de synchronisation dans les réseaux neuronaux

2006

The spike train, i.e. the sequence of the action potential timings of a single unit, is the usual data that is analyzed in electrophysiological recordings for the description of the firing pattern which is supposed to characterize a certain type of cell.. We present the results obtained describing the firing activity of a small network of neurons with a mathematical jump diffusion model. That is the membrane potential as a function of time is given by the sum of a stochastic diffusion process and two counting processes that provoke jumps of constant sizes at discrete random times. Different distributions are considered for such processes. Two main results emerge. The first one is that interspike intervals (ISI) histograms show more than one peak (multimodality) and exhibit a resonant like behavior. This fact suggests that in correspondence of each mode (i.e. the lag of the maxima) the cell has a higher probability of firing such that the the lags become characteristic times of the c...

On the dynamics of a pair of coupled neurons subject to alternating input rates

Biosystems, 2005

We present a statistical analysis of the firing activity of two coupled neuronal units that interact according to a 'sendingreceiving' model. The membrane potential's behavior of both units is described by the Stein equations under the additional assumption that the spikes released by the sending neuron constitute an extra excitation for the receiving one. We also assume the presence of an alternating behavior for the rates of inputs to the sending neuron. By means of ad hoc simulations, we obtain, and then discuss, some statistical results concerning the spike production times of the units within the subintervals of the alternating inputs, as well as the reaction times of the receiving neuron.

Hopf bifurcations in dynamics of excitable systems

Ricerche di Matematica

A general FitzHugh–Rinzel model, able to describe several neuronal phenomena, is considered. Linear stability and Hopf bifurcations are investigated by means of the spectral equation for the ternary autonomous dynamical system and the analysis is driven by both an admissible critical point and a parameter which characterizes the system.

An Analytic Picture of Neuron Oscillations

International Journal of Bifurcation and Chaos, 2004

Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrai...