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.

/pdf/transitions-between-classes-of-neuronal-excitability-and-igl3fdwpy2.pdf

Transitions between classes of neuronal excitability and bifurcations induced by autapse

Neuron can receive electric signals or forcing currents from more than one channel, and these forcing currents could show some diversity Based on the Hindmarsh–Rose neuron model, mixed forcing currents, which are composed of low-frequency, high-frequency and constant signals, are imposed on the neuron, and multiple modes of electric activities could be observed alternately (in turn) from the neuron Based on the Helmholtz theorem, the Hamilton energy is calculated to discern the energy dependence on the mode selection of the electric activities of neuron It is found that the response of electrical activities much depends on the amplitude than the frequency when mixed signals are imposed on the neuron synchronously; however, the rhythm of electrical activities could be adjusted by the frequency of the periodical signals in the mixed signal It is confirmed that the energy is much dependent on the mode of electrical activities instead of the external forcing currents directly, and a smaller energy occurs under bursting states The delayed response of Hamilton energy to external forcing currents confirms that neuron contributes to energy coding These results could be helpful for further investigation on energy problems in neuronal network associated with model transition for collective behaviors

Energy dependence on modes of electric activities of neuron driven by multi-channel signals

We study an effect of random disturbances on the discrete two-dimensional Rulkov neuron model. We show that close to the Neimark–Sacker bifurcation, the increasing noise can cause the transition from the noisy quiescence with small-amplitude oscillations near the stable equilibria to the stochastic bursting with large-amplitude spikes. Mean values and variations of the interspike intervals are studied in dependence of the noise intensity. To study the noise-induced bursting, the analytical approach based on the stochastic sensitivity functions technique and confidence ellipses method is applied. On the basis of the largest Lyapunov exponents, we show how the noise-induced transition from the quiescence to stochastic bursting regime is accompanied by the transformation of dynamics from regular to chaotic.

Noise-induced bursting and chaos in the two-dimensional Rulkov model

Mode selection in a neuron driven by Josephson junction current in presence of magnetic field

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.

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

In this paper, we investigate the dynamical behaviors of a Morris---Lecar neuron model. By using bifurcation methods and numerical simulations, we examine the global structure of bifurcations of the model. Results are summarized in various two-parameter bifurcation diagrams with the stimulating current as the abscissa and the other parameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay much attention to the emergence of periodic solutions and bistability. Different membrane excitability is obtained by bifurcation analysis and frequency-current curves. The alteration of the membrane properties of the Morris---Lecar neurons is discussed.

Congmin Liu

Papers

Bifurcation analysis of a Morris---Lecar neuron model