The neuronal and synaptic dynamics underlying post-inhibitory rebound burst related to major depressive disorder in the lateral habenula neuron model.

A burst behavior observed in the lateral habenula (LHb) neuron related to major depressive disorder has attracted much attention. The burst is induced from silence by the excitatory N-methyl-D-aspartate (NMDA) synapse or by the inhibitory stimulation, i.e., a post-inhibitory rebound (PIR) burst, which has not been explained clearly. In the present paper, the neuronal and synaptic dynamics for the PIR burst are acquired in a theoretical neuron model. At first, dynamic cooperations between the fast rise of inhibitory γ-aminobutyric acid (GABA) synapse, slow rise of NMDA synapse, and T-type calcium current to evoke the PIR burst are obtained. Similar to the inhibitory pulse stimulation, fast rising GABA current can reduce the membrane potential to a level low enough to de-inactivate the low threshold T-type calcium current to evoke a PIR spike, which can enhance the slow rising NMDA current activated at a time before or after the PIR spike. The NMDA current following the PIR spike exhibits slow decay to induce multiple spikes to form the PIR burst. Such results present a theoretical explanation and a candidate for the PIR burst in real LHb neurons. Then, the dynamical mechanism for the PIR spike mediated by the T-type calcium channel is obtained. At large conductance of T-type calcium channel, the resting state corresponds to a stable focus near Hopf bifurcation and exhibits an "uncommon" threshold curve with membrane potential much lower than the resting membrane potential. Inhibitory modulation induces membrane potential decreased to run across the threshold curve to evoke the PIR spike. At small conductance of the T-type calcium channel, a stable node appears and manifests a common threshold curve with higher membrane potential, resulting in non-PIR phenomenon. The results present the dynamic cooperations between neuronal dynamics and fast/slow dynamics of different synapses for the PIR burst observed in the LHb neuron, which is helpful for the modulations to major depressive disorder.

Complex dynamics of hair bundle of auditory nervous system (I): spontaneous oscillations and two cases of steady states.

The hair bundles of inner hair cells in the auditory nervous exhibit spontaneous oscillations, which is the prerequisite for an important auditory function to enhance the sensitivity of inner ear to weak sounds, otoacoustic emission. In the present paper, the dynamics of spontaneous oscillations and relationships to steady state are acquired in a two-dimensional model with fast variable X (displacement of hair bundles) and slow variable X a . The spontaneous oscillations are derived from negative stiffness modulated by two biological factors (S and D) and are identified to appear in multiple two-dimensional parameter planes. In (S, D) plane, comprehensive bifurcations including 4 types of codimension-2 bifurcation and 5 types of codimension-1 bifurcation related to the spontaneous oscillations are acquired. The spontaneous oscillations are surrounded by supercritical and subcritical Hopf bifurcation curves, and outside of the curves are two cases of steady state. Case-1 and Case-2 steady states exhibit Z-shaped (coexistence of X) and N-shaped (coexistence of X a ) X-nullclines, respectively. In (S, D) plane, left and right to the spontaneous oscillations are two subcases of Case-1, which exhibit the stable equilibrium point locating on the upper and lower branches of X-nullcline, respectively, resembling that of the neuron. Lower to the spontaneous oscillations are 3 subcases of Case-2 from left to right, which manifest stable equilibrium point locating on left, middle, and right branches of X-nullcline, respectively, differing from that of the neuron. The phase plane for steady state is divided into four parts by nullclines, which manifest different vector fields. The phase trajectory of transient behavior beginning from a phase point in the four regions to the stable equilibrium point exhibits different dynamics determined by the vector fields, which is the basis to identify dynamical mechanism of complex forced oscillations induced by external signal. The results present comprehensive viewpoint and deep understanding for dynamics of the spontaneous oscillations and steady states of hair bundles, which can be used to well explain the experimental observations and to modulate functions of spontaneous oscillations.