Bressloff Paul C
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112 USA.
J Math Neurosci. 2015 Feb 27;5:4. doi: 10.1186/s13408-014-0016-z. eCollection 2015.
We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.
我们考虑将路径积分方法应用于一个随机混合模型的分析，该模型表示一个由突触耦合的脉冲发放神经元群体网络。每个局部群体的状态由两个随机变量描述，一个连续的突触变量和一个离散的活动变量。突触变量根据分段确定性动力学演化，在群体水平上描述由脉冲发放活动驱动的突触。突触电流的动力学方程仅在脉冲发放活动的跳跃之间有效，而后者由一个跳跃马尔可夫过程描述，其转移速率取决于突触变量。我们假设在具有时间常数[公式：见正文]的快速脉冲发放动力学和具有时间常数τ的较慢突触动力学之间存在时间尺度分离。这自然地引入了一个小的正参数[公式：见正文]，它可用于开发随机动力学相应路径积分表示的各种渐近展开。首先，我们推导了一个从亚稳态逃逸的最大似然路径的变分原理（在小噪声极限[公式：见正文]下的大偏差）。然后，我们展示了路径积分如何为小ϵ的混合系统获得扩散近似提供一种有效方法。由此产生的朗之万方程可用于分析亚稳态吸引盆内波动的影响，即忽略大偏差的影响。我们通过使用朗之万近似来分析内在噪声对空间结构混合网络中模式形成的影响来说明这一点。特别是，我们展示了噪声如何以类似于偏微分方程的方式扩大出现模式的参数范围。最后，我们对路径积分进行[公式：见正文]圈展开，并使用此来推导基于电压的平均场方程的修正，类似于从神经主方程生成的基于活动的修正方程。
