Kotani Kiyoshi, Yamaguchi Ikuhiro, Yoshida Lui, Jimbo Yasuhiko, Ermentrout G Bard
Graduate School of Frontier Science, University of Tokyo, , Chiba, Japan.
J R Soc Interface. 2014 Mar 19;11(95):20140058. doi: 10.1098/rsif.2014.0058. Print 2014 Jun 6.
Gamma oscillations of the local field potential are organized by collective dynamics of numerous neurons and have many functional roles in cognition and/or attention. To mathematically and physiologically analyse relationships between individual inhibitory neurons and macroscopic oscillations, we derive a modification of the theta model, which possesses voltage-dependent dynamics with appropriate synaptic interactions. Bifurcation analysis of the corresponding Fokker-Planck equation (FPE) enables us to consider how synaptic interactions organize collective oscillations. We also develop the adjoint method (infinitesimal phase resetting curve) for simultaneous equations consisting of ordinary differential equations representing synaptic dynamics and a partial differential equation for determining the probability distribution of the membrane potential. This method provides a macroscopic phase response function (PRF), which gives insights into how it is modulated by external perturbation or internal changes of parameters. We investigate the effects of synaptic time constants and shunting inhibition on these gamma oscillations. The sensitivity of rising and decaying time constants is analysed in the oscillatory parameter regions; we find that these sensitivities are not largely dependent on rate of synaptic coupling but, rather, on current and noise intensity. Analyses of shunting inhibition reveal that it can affect both promotion and elimination of gamma oscillations. When the macroscopic oscillation is far from the bifurcation, shunting promotes the gamma oscillations and the PRF becomes flatter as the reversal potential of the synapse increases, indicating the insensitivity of gamma oscillations to perturbations. By contrast, when the macroscopic oscillation is near the bifurcation, shunting eliminates gamma oscillations and a stable firing state appears. More interestingly, under appropriate balance of parameters, two branches of bifurcation are found in our analysis of the FPE. In this case, shunting inhibition can effect both promotion and elimination of the gamma oscillation depending only on the reversal potential.
局部场电位的伽马振荡由众多神经元的集体动力学组织而成，在认知和/或注意力方面具有多种功能作用。为了从数学和生理学角度分析单个抑制性神经元与宏观振荡之间的关系，我们推导了theta模型的一种修正形式，该模型具有依赖电压的动力学以及适当的突触相互作用。对相应的福克 - 普朗克方程（FPE）进行分岔分析，使我们能够考虑突触相互作用如何组织集体振荡。我们还针对由表示突触动力学的常微分方程和用于确定膜电位概率分布的偏微分方程组成的联立方程，开发了伴随方法（无穷小相位重置曲线）。该方法提供了一个宏观相位响应函数（PRF），它能深入了解其如何受到外部扰动或参数内部变化的调制。我们研究了突触时间常数和分流抑制对这些伽马振荡的影响。在振荡参数区域分析了上升和衰减时间常数的敏感性；我们发现这些敏感性在很大程度上不依赖于突触耦合速率，而是取决于电流和噪声强度。对分流抑制的分析表明，它既能促进也能消除伽马振荡。当宏观振荡远离分岔点时，分流促进伽马振荡，并且随着突触反转电位的增加，PRF变得更平坦，这表明伽马振荡对扰动不敏感。相比之下，当宏观振荡接近分岔点时，分流消除伽马振荡并出现稳定的放电状态。更有趣的是，在适当的参数平衡下，我们对FPE的分析中发现了分岔的两个分支。在这种情况下，分流抑制仅取决于反转电位，既能促进也能消除伽马振荡。
