Bipolar Pulse-Induced Coexisting Firing Patterns in Two-Dimensional Hindmarsh–Rose Neuron Model

2019 ◽  
Vol 29 (01) ◽  
pp. 1950006 ◽  
Author(s):  
Han Bao ◽  
Aihuang Hu ◽  
Wenbo Liu
Keyword(s):  
Membrane Potential ◽  
Neuron Model ◽  
Periodic Variation ◽  
Constant Current ◽  
Two Dimensional ◽  
Bipolar Pulse ◽  
Firing Patterns ◽  
Current Stimulus ◽  
Unstable Node ◽  
Stimulus Effect

In this paper, a bipolar pulse (BP) current is taken to mimic a periodic stimulus effect on the membrane potential in the axon of a neuron. By introducing the BP current to substitute the externally applied constant current, a BP-forced two-dimensional Hindmarsh–Rose (HR) neuron model is proposed. Based on the proposed neuron model, the BP-switched equilibrium point and its stability evolution with the periodic variation in time are explored. Furthermore, coexisting asymmetric attractors (or coexisting firing patterns) with bistability are revealed by phase plane orbits, time sequences, and attraction basins, as well as the BP-induced coexisting asymmetric attractors’ behaviors are then elaborated through bifurcation analysis. The research results exhibit that, with the increase of the time, the stabilities of the neuron model are continually switched between an unstable node-focus and a stable point, resulting in the coexisting behaviors of numerous asymmetric attractors under the specified initials. Consequently, the newly introduced BP current stimulus, instead of the original constant current stimulus, allows the two-dimensional HR neuron model to possess complex dynamical behaviors for the membrane potential. Additionally, a hardware breadboard is fabricated and circuit experiments are carried out to validate the numerical simulations.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

scholarly journals Injection of inositol trisphosphorothioate into Limulus ventral photoreceptors causes oscillations of free cytosolic calcium.

1991 ◽  
Vol 97 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
R Payne ◽  
B V Potter
Keyword(s):  
Membrane Potential ◽  
Calcium Release ◽  
Feedback Inhibition ◽  
Initial Response ◽  
Periodic Variation ◽  
Cytosolic Calcium ◽  
Ion Concentration ◽  
Free Calcium ◽  
Calcium Stores ◽  
Calcium Ion Concentration

Limulus ventral photoreceptors contain calcium stores sensitive to release by D-myo-inositol 1,4,5 trisphosphate (InsP3) and a calcium-activated conductance that depolarizes the cell. Mechanisms that terminate the response to InsP3 were investigated using nonmetabolizable DL-myo-inositol 1,4,5 trisphosphorothioate (InsPS3). An injection of 1 mM InsPS3 into a photoreceptor's light-sensitive lobe caused an initial elevation of cytosolic free calcium ion concentration (Cai) and a depolarization lasting only 1-2 s. A period of densensitization followed, during which injections of InsPS3 were ineffective. As sensitivity recovered, oscillations of membrane potential began, continuing for many minutes with a frequency of 0.07-0.3 Hz. The activity of InsPS3 probably results from the D-stereoisomer, since L-InsP3 was much less effective than InsP3. Injections of 1 mM InsP3 caused an initial depolarization and a period of densensitization similar to that caused by 1 mM InsPS3, but no sustained oscillations of membrane potential. The initial response to InsPS3 or InsP3 may therefore be terminated by densensitization, rather than by metabolism. Metabolism of InsP3 may prevent oscillations of membrane potential after sensitivity has recovered. The InsPS3-induced oscillations of membrane potential accompanied oscillations of Cai and were abolished by injection of ethyleneglycol-bis (beta-aminoethyl ether)-N,N'-tetraacetic acid. Removal of extracellular calcium reduced the frequency of oscillation but not its amplitude. Under voltage clamp, oscillations of inward current were observed. These results indicate that periodic bursts of calcium release underly the oscillations of membrane potential. After each burst, the sensitivity of the cell to injected InsP3 was greatly reduced, recovering during the interburst interval. The oscillations may, therefore, result in part from a periodic variation in sensitivity to a constant concentration of InsPS3. Prior injection of calcium inhibited depolarization by InsPS3, suggesting that feedback inhibition of InsPS3-induced calcium release by elevated Cai may mediate desensitization between bursts and after injections of InsPS3.

Chaos in the Rulkov Neuron Model Based on Marotto’s Theorem

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Penghe Ge ◽  
Hongjun Cao
Keyword(s):  
Neuron Model ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Fast Subsystem ◽  
One Dimensional ◽  
Sensitive Dependence ◽  
The Stability ◽  
Slow Subsystem

The existence of chaos in the Rulkov neuron model is proved based on Marotto’s theorem. Firstly, the stability conditions of the model are briefly renewed through analyzing the eigenvalues of the model, which are very important preconditions for the existence of a snap-back repeller. Secondly, the Rulkov neuron model is decomposed to a one-dimensional fast subsystem and a one-dimensional slow subsystem by the fast–slow dynamics technique, in which the fast subsystem has sensitive dependence on the initial conditions and its snap-back repeller and chaos can be verified by numerical methods, such as waveforms, Lyapunov exponents, and bifurcation diagrams. Thirdly, for the two-dimensional Rulkov neuron model, it is proved that there exists a snap-back repeller under two iterations by illustrating the existence of an intersection of three surfaces, which pave a new way to identify the existence of a snap-back repeller.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Zhongquan Gao ◽  
Zhixuan Yuan ◽  
Zuo Wang ◽  
Peihua Feng
Keyword(s):  
Membrane Potential ◽  
Resting State ◽  
Electromagnetic Induction ◽  
Neuron Model ◽  
Mode Selection ◽  
Neuron Firing ◽  
Two Factors ◽  
Selection Of ◽  
Strong Ability

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.

Membrane potential oscillation from a novel combination of ion channels

1993 ◽  
Vol 265 (3) ◽  
pp. C720-C727 ◽  
Author(s):  
J. W. Stelling ◽  
T. J. Jacob
Keyword(s):  
Membrane Potential ◽  
Cell Voltage ◽  
Inward Rectifier ◽  
Constant Current ◽  
K Channel ◽  
Patch Clamp Recording ◽  
Current Clamp ◽  
Membrane Potential Oscillation ◽  
Low Amplitude

Single pigmented epithelial cells from the ciliary body of the eye were studied using the whole cell voltage and current clamp, permeabilized patch recording, and patch-clamp recording. These cells can produce two types of oscillation. Both are slow, with a period in the range of 1-2 min; one has a low amplitude and oscillates between -60 and -80 mV, and the second is larger, with biphasic hyperpolarizing and depolarizing phases. The latter was seen when the membrane potential was driven negative by a constant current and results from the interplay between the inward rectifier K+ channel and a hyperpolarizing-activated cation channel. The hyperpolarization is caused by the constant current acting on a decreasing conductance as the inward rectifier inactivates, and the depolarization drive results from the activation of cation channels. It is suggested that the constant current would be provided by the Na+ pump in vivo, and such an interplay of channels and pumps could drive the uptake of cations in absorbing epithelia or provide an increased driving force for chloride exit in secretory epithelia.

Solitary Waves of Integrate-and-Fire Neural Fields

1997 ◽  
Vol 9 (8) ◽  
pp. 1677-1690 ◽  
Author(s):  
David Horn ◽  
Irit Opher
Keyword(s):  
Solitary Waves ◽  
Two Dimensional ◽  
Neural Fields ◽  
Firing Activity ◽  
Firing Patterns ◽  
Integrate And Fire ◽  
Strong Inhomogeneity ◽  
Continuum Formulation ◽  
Dimensional Surface

Arrays of interacting identical neurons can develop coherent firing patterns, such as moving stripes that have been suggested as possible explanations of hallucinatory phenomena. Other known formations include rotating spirals and expanding concentric rings. We obtain all of them using a novel two-variable description of integrate-and-fire neurons that allows for a continuum formulation of neural fields. One of these variables distinguishes between the two different states of refractoriness and depolarization and acquires topological meaning when it is turned into a field. Hence, it leads to a topologic characterization of the ensuing solitary waves, or excitons. They are limited to pointlike excitations on a line and linear excitations, including all the examples noted above, on a two dimensional surface. A moving patch of firing activity is not an allowed solitary wave on our neural surface. Only the presence of strong inhomogeneity that destroys the neural field continuity allows for the appearance of patchy incoherent firing patterns driven by excitatory interactions.

DIFFERENT BIFURCATION SCENARIOS OF NEURAL FIRING PATTERNS OBSERVED IN THE BIOLOGICAL EXPERIMENT ON IDENTICAL PACEMAKERS

2013 ◽  
Vol 23 (12) ◽  
pp. 1350195 ◽  
Author(s):  
HUAGUANG GU
Keyword(s):  
Parameter Space ◽  
Period Doubling ◽  
Two Dimensional ◽  
Neural Firing ◽  
Biological Experiment ◽  
Dimensional Parameter ◽  
Simple Process ◽  
Firing Patterns ◽  
Complex Process ◽  
Different Levels

Two different bifurcation scenarios of spontaneous neural firing patterns with decreasing extracellular calcium concentrations were observed in the biological experiment on identical pacemakers when potassium concentrations were fixed at two different levels. Six typical experimental scenarios manifesting dynamics closely matching those previously simulated using the Hindmarsh–Rose model and Chay model are provided as representative examples. Bifurcation scenarios from period-1 bursting to period-1 spiking via a complex process and via a simple process, period-doubling bifurcation to chaos, period-adding bifurcation with chaos, and period-adding bifurcation with stochastic burstings were identified. The results not only reveal that an experimental neural pacemaker is capable of generating different bifurcation scenarios but also provide a basic framework for bifurcations in neural firing patterns in a two-dimensional parameter space.

scholarly journals RECTIFICATION AND INDUCTANCE IN THE SQUID GIANT AXON

1941 ◽  
Vol 25 (1) ◽  
pp. 29-51 ◽  
Author(s):  
Kenneth S. Cole
Keyword(s):  
Membrane Potential ◽  
Electrical Properties ◽  
Giant Axon ◽  
Squid Giant Axon ◽  
Constant Current ◽  
Ion Permeability ◽  
Squid Axon ◽  
Axon Membrane ◽  
Piezoelectric Structure ◽  
In Series

Previous measurements have shown that the electrical properties of the squid axon membrane are approximately equivalent to those of a circuit containing a capacity shunted by an inductance and a rectifier in series. Selective ion permeability of a membrane separating two electrolytes may be expected to give rise to the rectification. A quasi-crystalline piezoelectric structure of the membrane is a plausible explanation of the inductance. Some approximate calculations of behavior of an axon with these membrane characteristics have been made. Fair agreement is obtained with the observed constant current subthreshold potential and impedance during the foot of the action potential. In a simple case a formal analogy is found between the calculated membrane potential and the excitability defined by the two factor formulations of excitation. Several excitation phenomena may then be explained semi-quantitatively by further assuming the excitability proportional to the membrane potential. Some previous measurements and subthreshold potential and excitability observations are not consistent with the circuit considered and indicate that this circuit is only approximately equivalent to the membrane.

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