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

Abstract

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.

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

“…The PIR spike identified in the nervous system with I h current is involved in the locomotor rhythms [3,26], "short-term memory" mechanism in the lateral pyloric neuron of the stomatogastric ganglion [11]. Recently, I h current is identified to induce the SN bifurcation changed to the Hopf bifurcation [35], which seems to follow that the PIR spike induced by I h current in neurophysiology and by Hopf bifurcation in nonlinear dynamics are consistent with each other. Therefore, it is often thought that the PIR spike in a neuron model is mainly related to the stable focus near the Hopf bifurcation with type II excitability in nonlinear dynamics [2,16] or I h current in neurophysiology [1,3,11,26].…”

“…Conclusion and discussion. The nonlinear concept of threshold and postinhibitory rebound spike are the fundamental conceptions in both nonlinear dynamics and neurophysiology, which are helpful to identify the physiological functions and modulation measures to firing pattern related to threshold or PIR spike [12,16,18,35]. In the present paper, we obtain the threshold curve for spike or spiking evoked from stable node and focus near bifurcation of big homoclinic (BHom) orbit in a modified FHN model without I h current, which confirms the conjecture in Ref [16].…”

“…In most of the previous investigations, PIR spike in a neuron is related to either Hopf bifurcation or I h current [2,16]. Such a viewpoint is strengthened due to that I h current induces SN bifurcation changed to Hopf bifurcation point [35]. Furthermore, the condition for PIR spike is extended to be evoked from stable node near SNIC in system with I h current [12] and SN bifurcation near a Bogdanov-Takens bifurcation in system without I h current [18].…”

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

“…For example, another important example that the inhibitory stimulation can evoke an action potential from the steady state in the nervous system is called postinhibitory rebound (PIR), which is always observed in the nervous system with a hyperpolarization active caution current (Ih) [ 36 , 37 ]. In theory, the PIR phenomenon has been built a relationship to the Hopf bifurcation or type II excitability [ 12 , 38 ]. Recently, the PIR phenomenon has been associated with the SNIC bifurcation and type I excitability in a model with Ih current, due to the changes of the threshold surface induced by the Ih current [ 39 ].…”

Multiple and Complex Codimension-2 Bifurcations underlying Post-inhibitory Rebound Spike and Excitability Transition

“…However, no PIR spike is evoked due to the absence of down-left part of the threshold curve for the SNIC with type I excitability. Recently, I h current induces SNIC changed to Hopf bifurcation, which further implies that the PIR spike induced by I h current is related to the Hopf bifurcation [57]. More recently, in a three-dimensional neuronal model with I h current, the PIR spike appears near SNIC because the threshold surface changes for stronger I h current [58].…”