• 대한수학회논문집
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• 제37권3호
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• pp.693-703
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• 2022

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$(1){\hspace{32}}{\ddot{u}}(t)+q(t){\dot{u}}(t)-L(t)u(t)+{\nabla}W(t,u(t))=0,\;{\forall}t{\in}{\mathbb{R}},$$ where q ∈ C(ℝ, ℝ), L ∈ C(ℝ, ${\mathbb{R}}^{N^2}$) is a symmetric and positive definite matix-valued function and W ∈ C¹(ℝ×ℝ^N, ℝ). The novelty of this paper is that, assuming that L is bounded from below unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

• Journal of Korean Neurosurgical Society
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• 제42권3호
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• pp.205-210
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• 2007

Objective : Hyperpolarization-activated cyclic nucleotide-gated (HCN) channels mediate the hyperpolarization-activated currents (Ih) that participate in regulating neuronal membrane potential and contribute critically to pacemaker activity, promoting synchronization of neuronal networks. However, distinct regional and cellular localization of HCN channels in the brain have not been precisely defined. Aim of this study was to verify the precise cellular location of HCN1 channels in rat cerebellum to better understand the physiological role these channels play in synaptic transmission between CNS neurons. Methods : HCN1 expression in rat brain was analyzed using immunohistochemistry and electron-microscopic observations. Postsynaptic density-95 (PSD-95), otherwise known as locating and clustering protein, was also examined to clarify its role in the subcellular location of HCN1 channels. In addition, to presume the binding of HCN1 channels with PSD-95, putative binding motifs in these channels were investigated using software-searching method. Results : HCN1 channels were locally distributed at the presynaptic terminal of basket cell and exactly corresponded with the location of PSD-95. Moreover, nine putative SH3 domain of PSD-95 binding motifs were discovered in HCN1 channels from motif analysis. Conclusion : Distinct localization of HCN1 channels in rat cerebellum is possible, especially when analyzed in conjunction with the SH3 domain of PSD-95. Considering that HCN1 channels contribute to spontaneous rhythmic action potentials, it is suggested that HCN1 channels located at the presynaptic terminal of neurons may play an important role in synaptic plasticity.