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短期突触可塑性

2024-02-21T03:41:29Z

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短期突触可塑性（Short-term synaptic plasticity，STP），也称为动态突触，指的是突触效能随时间变化的现象，这种变化反映了前突触活动的历史。在实验中观察到两种具有相反效果的STP，它们被称为短期抑制（STD）和短期促进（STF）。STD是由于在突触信号传递过程中消耗的神经递质在前突触神经元的轴突末端耗尽所致，而STF是由于在产生动作电位后钙离子流入轴突末端，增加了神经递质释放的概率。STP在各种皮层区域中都有发现，并且在属性上展现出巨大的多样性。不同皮层区域的突触可以具有不同形式的可塑性，要么以STD为主，要么以STF为主，或是两种形式的混合。

与长期可塑性相比，后者被假设为体验依赖性神经回路修改的神经基质，STP具有更短的时间尺度，通常在数百到数千毫秒的范围内。它对突触效能的修改是暂时的。如果没有持续的前突触活动，突触效能将迅速回到基线水平。

尽管STP似乎是突触生理学的不可避免后果，理论研究表明，其在大脑功能中的作用可能是深远的。从计算角度看，STP的时间尺度位于快速神经信号（以毫秒计）和经验诱导的学习（以分钟或更长时间计）之间。这是日常生活中许多过程发生的时间尺度，例如运动控制、语音识别和工作记忆。因此，STP很可能作为处理相关时间尺度上的时间信息的神经基质。STP意味着后突触神经元的反应依赖于前突触活动的历史，创造了原则上可以提取和使用的信息。在大型网络中，STP可以极大地丰富网络的动态行为，赋予神经系统使用静态连接难以实现的信息处理能力。这些可能性引起了计算神经科学领域对STP计算功能的显著兴趣。

== 现象学模型 ==

STP背后的生物物理过程十分复杂。研究STP的计算角色依赖于简化的现象学模型的创建。

在Tsodyks和Markram提出的模型中，STD效应通过一个归一化变量<math>x</math>（<math>0\leq x \leq1</math>）来模拟，表示在神经递质耗尽后仍然可用的资源比例。STF效应通过利用参数<math>u</math>来模拟，代表准备使用的可用资源比例（释放概率）。在发生动作电位后，（i）由于动作电位引起的钙离子流入前突触末端，<math>u</math>增加，之后（ii）一部分<math>u</math>的可用资源被消耗以产生突触后电流。在动作电位之间，<math>u</math>随时间常数<math>\tau_f</math>衰减回零，<math>x</math>随时间常数<math>\tau_d </math>恢复到1。总之，STP的动力学由下式给出

:<math>\begin{aligned}
\frac{du}{dt} & = & -\frac{u}{\tau_f}+U(1-u^-)\delta(t-t_{sp}),\nonumber \\ \frac{dx}{dt} & = & \frac{1-x}{\tau_d}-u^+x^-\delta(t-t_{sp}), \\
\frac{dI}{dt} & = & -\frac{I}{\tau_s} + Au^+x^-\delta(t-t_{sp}), \nonumber
\label{model}\end{aligned}</math>

其中<math>t_{sp}</math>表示动作电位时间，<math>U</math>是由动作电位产生的<math>u</math>的增量。我们用<math>u^-, x^-</math>表示动作电位到达前的相应变量，<math>u^+</math>指的是动作电位之后的时刻。从第一个方程，<math>u^+ = u^- + U(1-u^-)</math>。然后，由于在<math>t_{sp}</math>时刻到达的动作电位在突触处产生的突触电流由下式给出

:<math>\Delta I(t_{sp}) = Au^+x^-,
\label{current}</math>

其中<math>A</math>表示由所有神经递质的完全释放（<math>u=x=1</math>）产生的反应幅度，称为连接的绝对突触效能（见图1A）。

<math>u</math>和<math>x</math>的动态之间的相互作用决定了<math>ux</math>的联合效应是由抑制还是促进所主导。在<math>\tau_d\gg \tau_f</math>和大<math>U</math>的参数区域，一个初始动作电位会导致<math>x</math>大幅下降，需要很长时间恢复；因此，突触是以STD为主（图1B）。在<math>\tau_f \gg \tau_d</math>和小<math>U</math>的参数区域，突触效能会逐渐通过动作电位增加，因此突触以STF为主（图1C）。这个现象学模型成功地再现了在许多皮层区域观察到的抑制和促进突触的动力学。

[[Image:Fig1A_short_term_plasticity.png|400px]]
[[Image:Fig1B_short_term_plasticity.png|350px]]
[[Image:Fig1C_short_term_plasticity.png|350px]] 
图1. (A) 由Eqs.\ref{model}和\ref{current}给出的STP的现象学模型。 (B) 由STD主导的突触产生的突触后电流。神经元发射率<math>R=15</math>Hz。参数<math>A=1</math>，<math>U=0.45</math>，<math>\tau_s=20</math>ms，<math>\tau_d=750</math>ms，和<math>\tau_f=50</math>ms。 (C) STF主导的突触的动态。参数<math>U=0.15</math>，<math>\tau_f=750</math>ms，和<math>\tau_d=50</math>ms。

==对信息传输的影响Effects on information transmission==

Because STP modifies synaptic efficacy based on the history of presynaptic activity, it can alter neural information transmission ([[#Abbott97|Abbott 97]], [[#Tsodyks97|Tsodyks 97]], [[#Fuhrmann02|Fuhrmann 02]], [[#Rotman11|Rotman 11]], [[#Rosenbaum12|Rosenbaum 12]]). In general, an STD-dominated synapse favors information transfer for low firing rates, since high-frequency spikes rapidly deactivate the synapse. An STF-dominated synapse, however, tends to optimize information transfer for high-frequency bursts, which increase the synaptic strength.

因为 STP 根据突触前活动的历史来修改突触功效，所以它可以改变神经信息传递([[#Abbott97|Abbott 97]], [[#Tsodyks97|Tsodyks 97]], [[#Fuhrmann02|Fuhrmann 02]], [[#Rotman11|Rotman 11]], [[#Rosenbaum12|Rosenbaum 12]])。一般来说，以 STD 为主的突触有利于低发射率的信息传递，因为高频尖峰会迅速使突触失活。然而，以 STF 为主的突触倾向于优化高频突发的信息传递，从而增加突触强度。

Firing-rate-dependent transmission via dynamic synapses can be analyzed by examining the transmission of uncorrelated Poisson spike trains from a large neuronal population with global firing rate <math>R(t)</math>. The time evolution for the postsynaptic current <math>I(t)</math> can be obtained by averaging Eq. （1）over different realization of Poisson processes corresponding to different spike trains ([[#Tsodyks98|Tsodyks 98]]):

可以通过检查来自具有全局放电率 <math>R(t)</math>的大型神经元群体的不相关 Poisson 尖峰序列的传输来分析通过动态突触的放电率依赖性传输。突触后电流<math>I(t)</math>的时间演化可以通过对等式求平均来获得。等式（1）对应于不同尖峰序列的泊松过程的不同实现([[#Tsodyks98|Tsodyks 98]])：

:<math>\begin{aligned}
\frac{du}{dt} & = & -\frac{u}{\tau_f} + U(1-u^-)R(t),\nonumber \\
\frac{dx}{dt} & = & \frac{1-x}{\tau_d}-u^+xR(t), \\
I(t) &= & \tau_s Au^+xR(t), \nonumber
\label{poisson}\end{aligned} </math>

where again <math>u^+ = u^- + U(1-u^-)</math> and we neglect time scales on the order of the synaptic time constant. For the stationary rate, <math>R(t) \equiv R_0</math>, we obtain

其中<math>u^+ = u^- + U(1-u^-)</math>我们忽略了突触时间常数阶的时间尺度。对于固定速率，<math>R(t) \equiv R_0</math>，我们得到

:<math>\begin{aligned}
u^+=u_0 & \equiv & U\frac{1+\tau_fR_0}{1+U\tau_fR_0}, \nonumber \\
x=x_0 & \equiv & \frac{1}{1+u_0\tau_d R_0},\\
I=I_0 & \equiv & \tau_s Au_0x_0 R_0, \nonumber \label{stationary} \end{aligned}</math>

which is shown in Fig. 2A,B. In particular, for depression-dominated synapses (<math>u^+ \approx U</math>), the average synaptic efficacy <math>E=Au^+x</math> decays inversely with the rate, and the stationary synaptic current saturates at the limiting frequency <math>\lambda \sim \frac{1}{U\tau_d}</math>, above which dynamic synapses cannot transmit information about the stationary firing rate (Fig. 2A). On the other hand, facilitating synapses can be tuned for a particular presynaptic rate that depends on STP parameters (Fig. 2B).

如图 2A、B 所示。特别是，对于抑郁为主的突触 <math>u^+ \approx U</math>，平均突触效能<math>E=Au^+x</math>衰减与速率成反比，静态突触电流在极限频率<math>\lambda \sim \frac{1}{U\tau_d}</math>处饱和，高于该频率的动态突触不能传输有关固定发射率（图 2A）。另一方面，促进突触可以针对取决于 STP 参数的特定突触前速率进行调整（图 2B）。

===时间过滤Temporal filtering===

The above analysis only describes neural population firing with stationary firing rates. Eq. （3）can be used to derive the filtering properties of dynamic synapses when the presynaptic population firing rate changes arbitrarily with time. In [[#Appendix A: Derivation of a temporal filter for short-term depression|Appendix A]] we present the corresponding calculation for depression-dominated synapses (<math>u^+ \approx U</math>). By considering small perturbations<math>R(t):=R_0 + R_1 \rho (t)</math>with <math>R_1\ll R_0</math>around the constant rate <math>R_0>0</math>, the Fourier transform of the synaptic current <math>I</math>is approximated by

上述分析仅描述了具有固定放电率的神经群体放电。当突触前群体放电率随时间任意变化时，方程（3）可用于推导动态突触的过滤特性。在附录 A 中，我们给出了抑郁支配突触的相应计算 (<math>u^+ \approx U</math>)。通过考虑小扰动<math>R(t):=R_0 + R_1 \rho (t)</math>和<math>R_1\ll R_0</math>在恒定速率<math>R_0>0</math>附近，突触电流 <math>I</math>的傅里叶变换近似为

<math>
\begin{eqnarray}
\widehat{I}(\omega) \approx I_0 \delta(\omega) + \frac{I_0 R_1}{R_0} \widehat{\chi}(\omega) \widehat{\rho}(\omega)
\label{eq:Ihat_final}
\end{eqnarray}
</math>

where we defined the filter其中我们定义了过滤器

<math>
\begin{eqnarray}
\widehat{\chi}(\omega) := 1- \frac{1/x_0 -1}{1/x_0 + j\omega \tau_{d}} = \frac{1+(\tau_{d}\omega)^2x_0+j\omega\tau_{d}(1-x_0)}{1/x_0+(\tau_{d}\omega)^2 x_0}\,,
\label{eq:chihat}
\end{eqnarray}
</math>

<math>\widehat{\rho}</math>is the Fourier transform of <math>\rho</math>, and<math>I_0</math>and<math>x_0</math>are the stationary values of<math>I</math>and<math>x</math>, respectively [see Eq. （4） with <math>u_0 = U</math>].
The amplitude of the filter <math>|\widehat{\chi}(w)|</math> is shown in Fig. 2C, illustrating the high-pass filter properties of depressing synapses. In other words, fast changes in presynaptic firing rates are faithfully transmitted to the postsynaptic targets, while slow changes are attenuated by depression.

<math>\widehat{\rho}</math>是<math>\rho</math>的傅里叶变换，<math>I_0</math>和<math>x_0</math>分别是<math>I</math>和<math>x</math>的平稳值 [参见方程（4）与 <math>u_0 = U</math>]。滤波器的幅度<math>|\widehat{\chi}(w)|</math>如图 2C 所示，说明了抑制突触的高通滤波器特性。换句话说，突触前放电率的快速变化忠实地传递到突触后目标，而缓慢的变化则因抑郁而减弱。

STP can also regulate information transmission in other ways. For instance, STD may contribute to remove auto-correlation in temporal inputs, since temporally proximal spikes tend to magnify the depression effect and hence reduce the output correlation of the post-synaptic potential ([[#Goldman02|Goldman 02]]). On the other hand, STF, whose effect is enlarged by temporally proximal spikes, improves the sensitivity of a post-synaptic neuron to temporally correlated inputs ([[#Mejías08|Mejías 08]], [[#Bourjaily12|Bourjaily 12]]).

STP 还可以通过其他方式规范信息传输。例如，STD 可能有助于消除时间输入中的自相关，因为时间近端尖峰倾向于放大抑郁效应，从而降低突触后电位的输出相关性 ([[#Goldman02|Goldman 02]])。另一方面，STF 的效果因时间近端尖峰而扩大，提高了突触后神经元对时间相关输入的敏感性([[#Mejías08|Mejías 08]], [[#Bourjaily12|Bourjaily 12]])。

By combining STD and STF, neural information transmission could be further improved. For example, by combining STF-dominated excitatory and STD-dominated inhibitory synapses, the detection of high-frequency epochs by a postsynaptic neuron can be enhanced ([[#Klyachko06|Klyachko 06]]). In a postsynaptic neuron receiving both STD-dominated and STF-dominated inputs, the neural response can show both low- and high-pass filtering properties ([[#Fortune01|Fortune 01]]).

通过结合STD和STF，可以进一步改善神经信息传输。例如，通过结合 STF 主导的兴奋性突触和 STD 主导的抑制性突触，可以增强突触后神经元对高频时期的检测([[#Klyachko06|Klyachko 06]])。在同时接收 STD 主导和 STF 主导输入的突触后神经元中，神经反应可以显示低通和高通滤波特性（[[#Fortune01|Fortune 01]]）。

===增益控制Gain control===

Since STD suppresses synaptic efficacy in a frequency-dependent manner, it has been suggested that STD provides an automatic mechanism to achieve gain control, namely, by assigning high gain to slowly firing afferents and low gain to rapidly firing afferents ([[#Abbott97|Abbott 97]], [[#Abbott04|Abbott 04]], [[#Cook03|Cook 03]]). If a steady presynaptic firing rate <math>R</math> changes abruptly by an amount <math>\Delta R</math>, the first spike at the new rate will be transmitted with the efficacy <math>E</math> before the synapse is further depressed. Thus, the transient increase in synaptic input will be proportional to <math>\Delta R E(R)</math>, which is approximately proportional to <math>\Delta R/R</math> for large rates (see above). This is reminiscent of Weber’s law, which states that a transient synaptic response is roughly proportional to the percentage change of the input firing rate. Fig. 2D shows that for a fixed-size rate change <math>\Delta R</math>, the response decreases as a function of the steady input value; whereas without STD, the response would be constant for a fixed-size rate change.

由于 STD 以频率依赖性方式抑制突触功效，因此有人提出 STD 提供了一种自动机制来实现增益控制，即<math>\Delta R</math>通过将高增益分配给缓慢放电的传入神经并将低增益分配给快速放电的传入神经（[[#Abbott97|Abbott 97]], [[#Abbott04|Abbott 04]], [[#Cook03|Cook 03]])。如果一个稳定的突触前放电率 <math>R</math>突然改变了<math>\Delta R</math>的量，那么新的突触前放电率将与在突触被进一步抑制之前的功效 <math>E</math>。因此，突触输入的瞬时增加将与 <math>\Delta R E(R)</math>成正比，这与<math>\Delta R/R</math>大致成正比] 对于大利率（见上文）。这让人想起韦伯定律，该定律指出瞬态突触反应大致与输入放电率的百分比变化成正比。图 2D 显示对于固定大小的速率变化<math>\Delta R</math>，响应随着稳定输入值的变化而减小；而在没有 STD 的情况下，对于固定大小的速率变化，响应将是恒定的。

[[Image:Fig2A_short_term_plasticity.png|300px|链接=Special:FilePath/Fig2A_short_term_plasticity.png]]
[[Image:Fig2B_short_term_plasticity.png|300px|链接=Special:FilePath/Fig2B_short_term_plasticity.png]]
[[Image:Fig2C_short_term_plasticity.png|300px|链接=Special:FilePath/Fig2C_short_term_plasticity.png]]
[[Image:Fig2D_short_term_plasticity.png|300px|链接=Special:FilePath/Fig2D_short_term_plasticity.png]] Figure 2. (A) The steady values of the efficacy of an STD-dominated synapse and the postsynaptic currents it generates, measured by <math>ux</math> and <math>uxR</math>, respectively. The parameters are the same as in Fig.1B. (B) Same as (A) for an STF-dominated synapse. The parameters are the same as in Fig. 1C. (C) The filtering properties of an STD-dominated synapse, measured by <math>|\widehat{\chi}(w)|</math> [Eq.（6）]. (D) The neural response to an abrupt input change <math>\Delta R</math> vs. the steady rate value for a STD-dominating synapse. <math>\Delta R=5</math>Hz. The parameters are the same as in Fig.1B.

图 2. (A) 由 <math>ux</math>和 <math>uxR</math>，分别。参数与图 1B 相同。 (B) 对于 STF 主导的突触，与 (A) 相同。参数与图 1C 中的相同。 (C) 以 <math>|\widehat{\chi}(w)|</math>衡量的 STD 主导突触的过滤特性[等式（6）]。 (D) 对突然输入变化的神经反应<math>\Delta R=5</math>与 STD 主导突触的稳定速率值。 <math>\Delta R=5</math>Hz. 参数与图 1B 相同。

==对网络动态的影响Effects on network dynamics==

In addition to feedforward and feedback transmission, neural circuits generate recurrent interactions between neurons. With STP included in the recurrent interactions, the network dynamics exhibits many new interesting behaviors that do not arise with purely static synapses. These new dynamical properties could therefore implement STP-mediated network computation.

除了前馈和反馈传输之外，神经回路还会在神经元之间产生循环交互。由于 STP 包含在循环交互中，网络动力学表现出许多新的有趣行为，这些行为不会出现在纯静态突触中。因此，这些新的动态特性可以实现 STP 介导的网络计算。

===延长对瞬态输入的神经反应Prolongation of neural responses to transient inputs===

Since STP has a much longer time scale than that of single neuron dynamics (the latter is typically in the time order of <math>10-20</math> milliseconds), a new feature STP can bring to the network dynamics is prolongation of neural responses to a transient input. This stimulus-induced residual activity therefore holds a memory trace of the input, lasting up to several hundred milliseconds in a large-size network, and can serve as a buffer for information processing. For example, it has been<math>10-20</math> shown that STD-mediated residual activity can cause a neural system to discriminate between rhythmic inputs of different periods ([[#Karmorkar07|Karmorkar 07]]). STP also plays an important role in a general computation framework called a reservoir network. In this framework, STP, together with other dynamical elements of a large-size network, effectively map the input features from a low-dimensional space to the high-dimensional state space of the network that includes both active (neural) and hidden (synaptic) components, so that the input information can be more easily read out ([[#Buonomano09|Buonomano 09]]). In a recent development it was proposed that STF-enhanced synapses themselves can hold the memory trace of an input without recruiting persistent firing of neurons, potentially providing the most economical and robust way to implement working memory ([[#Mongillo08|Mongillo 08]]).

由于 STP 的时间尺度比单神经元动力学要长得多（后者的时间顺序通常为 <math>10-20</math>毫秒），因此 STP 可以为网络带来一个新功能动力学是对瞬态输入的神经反应的延长。因此，这种刺激引起的残余活动保留了输入的记忆轨迹，在大型网络中持续长达数百毫秒，并且可以作为信息处理的缓冲区。例如，<math>10-20</math>已经表明 STD 介导的残余活动可以导致神经系统区分不同时期的节律输入（[[#Karmorkar07|Karmorkar 07]]）。STP 在称为水库网络的通用计算框架中也起着重要作用。在这个框架中，STP 与大型网络的其他动态元素一起，有效地将输入特征从低维空间映射到网络的高维状态空间，包括活动（神经）和隐藏（突触） ) 组件，从而可以更轻松地读出输入信息（[[#Buonomano09|Buonomano 09]]）。在最近的一项发展中，有人提出 STF 增强的突触本身可以保持输入的记忆轨迹，而无需招募神经元的持续放电，这可能为实现工作记忆提供最经济和最稳健的方式 ([[#Mongillo08|Mongillo 08]])。

===调制网络对外部输入的响应Modulation of network responses to external input===

Since STP modifies synaptic efficacy instantly, it can modulate the network response to sustained external inputs. An example of this is bursty synchronous firing in an STD-dominated network, either spontaneously or in response to external inputs. The resulting bursts of activity are called population spikes ([[#Loebel02|Loebel 02]]). To understand this effect, consider a network with strong recurrent interactions between neurons. When a sufficiently large group of neurons fire together, e.g. triggered by external stimulus, they can recruit other neurons via an avalanche-like process. However, after a large synchronous burst of activity, the synapses are weakened by STD, reducing the recurrent currents rapidly, and consequently the network activity returns to baseline. The network will not be activated again until the synapses are sufficiently recovered from depression. Therefore, the rate of population spikes is determined by the time constant of STD (Fig.3A,B). STF can also modulate the network response to external inputs, but in a very different manner ([[#Barak07|Barak 07]]). The varied response properties mediated by STP may provide different ways of representing and conveying the stimulus information in a network.

由于 STP 会立即修改突触功效，因此它可以调节网络对持续外部输入的响应。这方面的一个例子是 STD 主导网络中的突发同步触发，无论是自发地还是响应外部输入。由此产生的活动爆发称为人口高峰（[[#Loebel02|Loebel 02]]）。要理解这种效应，请考虑一个神经元之间具有强循环交互的网络。当足够大的一组神经元一起发射时，例如由外部刺激触发，它们可以通过类似雪崩的过程招募其他神经元。然而，在大量同步突发活动之后，突触被 STD 削弱，快速减少循环电流，因此网络活动恢复到基线。在突触从抑郁症中充分恢复之前，网络不会再次被激活。因此，人口峰值的速率由 STD 的时间常数决定（图 3A，B）。STF 还可以调制网络对外部输入的响应，但方式非常不同（([[#Barak07|Barak 07]]）。由 STP 介导的不同响应属性可以提供在网络中表示和传达刺激信息的不同方式。

===诱导网络状态的不稳定性或移动性Induction of instability or mobility of network state===

Persistent firing, referring to situations in which a group of neurons continue firing without external drive, is widely regarded as a neural substrate for information representation ([[#Fuster71|Fuster 71]]). To maintain persistent activity in a network, strong excitatory recurrent interactions between neurons are needed to establish a positive-feedback loop sustaining neuronal responses. Mathematically, persistent activity is often modeled as an active stationary state (attractor) of the network. Since STD weakens synaptic efficacy depending on the level of neuronal activity, it can suppress an attractor state. This property, however, can be used to carry out valuable computations.

持续放电，指的是一组神经元在没有外部驱动的情况下继续放电的情况，被广泛认为是信息表示的神经基质（[[#Fuster71|Fuster 71]]）。为了维持网络中的持续活动，需要神经元之间强烈的兴奋性反复相互作用来建立维持神经元反应的正反馈回路。在数学上，持续活动通常被建模为网络的活动静止状态（吸引子）。由于 STD 会根据神经元活动的水平削弱突触的功效，因此它可以抑制吸引子状态。但是，此属性可用于执行有价值的计算。

Consider a network that holds multiple attractor states competing with each other, STD destabilizing one of them can incur the network to switch to another attractor state ([[#Torres07|Torres 07]], [[#Katori11|Katori 11]], [[#Igarashi12|Igarashi 12]]). This property has been linked to spontaneous transition between up and down states of cortical neurons ([[#Holcman06|Holcman 06]]), to the binocular rivalry phenomenon ([[#Kilpatrick10|Kilpatrick 10]]), and to enhanced discrimination capacity for superimposed ambiguous inputs ([[#Fung13|Fung 13]]). STF can also induce state switching, but this is achieved in an indirect way through facilitating the excitatory synapses to interneurons, with the latter in turn suppressing excitatory neurons ([[#Melamed08|Melamed 08]]).

考虑一个拥有多个相互竞争的吸引子状态的网络，STD 破坏其中一个可能会导致网络切换到另一个吸引子状态 ([[#Torres07|Torres 07]], [[#Katori11|Katori 11]], [[#Igarashi12|Igarashi 12]])。这种特性与皮层神经元上下状态之间的自发转换 ([[#Holcman06|Holcman 06]])、双眼竞争现象 ([[#Kilpatrick10|Kilpatrick 10]])以及增强的叠加模糊输入的辨别能力([[#Fung13|Fung 13]])有关。 STF 也可以诱导状态转换，但这是通过促进中间神经元的兴奋性突触以间接方式实现的，后者反过来抑制兴奋性神经元 ([[#Melamed08|Melamed 08]])。

The joint effect of STD and STF on the memory capacity of the classical Hopfield model has been investigated ([[#Mejías09|Mejías 09]]). It was found that STD degrades the memory capacity of the network, but induces a novel computationally desirable property, that is, the network can hop among memory states, which could be useful for memory searching. Interestingly, STF can compensate for the lost memory capacity caused by STD.

已经研究了 STD 和 STF 对经典 Hopfield 模型的记忆容量的联合影响([[#Mejías09|Mejías 09]])。研究发现，STD 会降低网络的记忆容量，但会产生一种新的计算上理想的特性，即网络可以在记忆状态之间跳跃，这可能对记忆搜索很有用。有趣的是，STF 可以弥补 STD 造成的内存容量损失。

===吸引子动力学的丰富Enrichment of attractor dynamics===

Continuous Attractor Neural Networks (CANNs), also called neural field models or ring models ([[#Amari77|Amari 77]]), have been widely used to describe the encoding of continuous stimuli in the neural system, such as for head-direction, orientation, movement direction, and spatial location of objects. A CANN, due to its translation-invariant recurrent interactions between neurons, holds a continuous family of localized stationary states, called bumps. These stationary states form a subspace on which the network is neutrally stable, enabling the network to track time-varying stimuli smoothly.

连续吸引子神经网络 (CANN)，也称为神经场模型或环模型 ([[#Amari77|Amari 77]])，已广泛用于描述神经系统中连续刺激的编码，例如头部方向、方向、运动方向和物体的空间位置。由于神经元之间的平移不变循环交互，CANN 拥有一系列连续的局部静止状态，称为颠簸。这些静止状态形成了一个子空间，网络在该子空间上是中性稳定的，使网络能够平滑地跟踪随时间变化的刺激。

With STP included, a CANN displays new interesting dynamical behaviors. One of them is a spontaneous traveling wave phenomenon ([[#York09|York 09]], [[#Fung12a|Fung 12]], [[#Bressloff12|Bressloff 12]]) (Fig.3C). Consider a network that is initially in a localized bump state. Because of STD, the neural interactions in the bump region are weakened. As a result of competition from neighboring attractor states, a small displacement will push the bump away, and it will continue to move in that direction due to the STD effect. If the network is driven by a continuously moving input, in a proper parameter regime the bump movement can even lead the external drive by a constant time irrespective to the input moving speed, achieving an anticipative behavior that is reminiscent to the predictive responses of head-direction neurons in rodents (Fig.3D; [[#Fung12b|Fung 12]]).

在包含 STP 的情况下，CANN 会显示出新的有趣的动态行为。其中之一是自发行波现象 ([[#York09|York 09]], [[#Fung12a|Fung 12]], [[#Bressloff12|Bressloff 12]])（图 3C）。考虑一个最初处于局部碰撞状态的网络。由于 STD，凹凸区域的神经相互作用被削弱。由于来自相邻吸引子状态的竞争，一个小的位移会将凸起推开，并且由于 STD 效应，它将继续朝那个方向移动。如果网络由连续移动的输入驱动，则在适当的参数状态下，无论输入移动速度如何，颠簸运动甚至可以引导外部驱动一段恒定的时间，从而实现让人联想到头部预测响应的预期行为。啮齿动物中的方向神经元(Fig.3D; [[#Fung12b|Fung 12]])。

[[Image:Fig3AB_short_term_plasticity.png|700px|链接=Special:FilePath/Fig3AB_short_term_plasticity.png]]
[[Image:Fig3C-TravellingWave.gif|链接=Special:FilePath/Fig3C-TravellingWave.gif]]
[[Image:Fig3D-Leading.gif|链接=Special:FilePath/Fig3D-Leading.gif]] 
Figure 3. (A,B) Population spikes generated by a STD-dominating network in response to external excitatory pulses. When the presentation rate of the pulses is low (A), the network responds to each one of them. For higher presentation rate (B), the network only responds to a fraction of the inputs. Adapted from ([[#Loebel02|Loebel 02]]). (C) The traveling wave generated by STD in a CANN. (D) The anticipative tracking behavior of a CANN with STD.

图 3. (A,B) 以 STD 为主的网络响应外部兴奋性脉冲而产生的人口峰值。当脉冲的呈现率低 (A) 时，网络对它们中的每一个做出响应。对于更高的呈现率 (B)，网络仅响应一小部分输入。改编自([[#Loebel02|Loebel 02]])。 (C) STD 在 CANN 中产生的行波。 (D) 具有 STD 的 CANN 的预期跟踪行为。

==附录A:短期抑郁的临时过滤器的推导Appendix A: Derivation of a temporal filter for short-term depression ==

We consider the rate-based dynamics in Eq. （3）for depression-dominated synapses (<math>u^+ \approx U</math>) and for synaptic responses that are much faster than the depression dynamics (<math>\tau_s \ll \tau_d</math>):

我们考虑方程式中基于速率的动态。等式（3）用于抑郁症主导的突触 <math>u^+ \approx U</math>和比抑郁症动力学快得多的突触反应 (<math>\tau_s \ll \tau_d</math>):

<math>
\begin{eqnarray}
{\frac{{\rm d} x}{{\rm d}t}}&=&\frac{1-x}{\tau_{d}}-Ux R(t) \label{eq:appA_x}\\
I(t) &= & \tau_{s} AU x R(t) \label{eq:appA_I} \,.
\end{eqnarray}
</math>

The aim is to derive a filter<math>\chi</math>that relates the output synaptic current <math>I</math>to the input rate<math>R</math>.
Note that because the input rate<math>R</math>enters the equations in a multiplicative fashion the input-output transfer function is non linear. Yet a linear filter can be derived by considering small perturbations <math>R_1 \rho(t)</math>of the firing rate <math>R(t)</math>around a constant rate <math>R_0</math>, that is,

目的是导出一个过滤器 <math>\chi</math>，它将输出突触电流 <math>I</math>与输入速率 <math>R</math> 联系起来。请注意，由于输入速率 <math>R</math>以乘法方式进入方程，因此输入-输出传递函数是非线性的。然而，线性滤波器可以通过考虑在恒定速率 <math>R_0</math>附近的发射率<math>R(t)</math>的小扰动 <math>R_1 \rho(t)</math>，即

<math>
R(t):=R_0 + R_1 \rho (t)\, \quad\text{with}\quad R_0,R_1>0 \quad\text{and}\quad R_1\ll R_0 \, .
\label{eq:appA_pert}
</math>

We assume that such small perturbations in <math>R</math>produce small perturbations in the variable<math>x</math>around its steady state value<math>x_0>0</math>

我们假设 <math>R</math>中的这种小扰动会在变量<math>x</math>中围绕其稳态值<math>x_0>0</math>产生小的扰动:

<math>
x(t) = x_0 + x_1(t)\quad\text{with}\quad x_0 = \frac{1}{1+UR_0\tau_{d}} \quad\text{and}\quad |x_1(t)| \ll x_0 \, .
\label{eq:appA_x01}
</math>

We can now linearize the dynamics of <math>x(t)</math> around the steady-state value <math>x_0</math>by approximating the product

我们现在可以通过近似乘积将 <math>x(t)</math>的动态线性化为围绕稳态值<math>x_0</math>

<math>
\begin{eqnarray}
xR &=& (x_0+x_1)(R_0+R_1\rho)\\
&=& x_0 R_0 + x_0 R_1 \rho + x_1 R_0+ x_1 R_1\rho\\
&\approx& x_0 R_0 + x_0 R_1 \rho + x_1 R_0\\
&\approx& R_0 x+ x_0R -x_0 R_0 \label{eq:appA_rx}
\end{eqnarray}
</math>

where in Eq. （11）we dropped the second-order term <math>x_1 R_1\rho</math> because we assumed<math>R_1\ll R_0</math>and<math>|x_1|\ll x_0</math>. Plugging Eq. （11） into Eq. （7） yields

在等式（11）中，我们删除了二阶项 <math>x_1 R_1\rho</math>，因为，我们假设<math>R_1\ll R_0</math>和 <math>|x_1|\ll x_0</math>，将等式（11）插入等式（7）产生,

<math>
\begin{eqnarray}
{\frac{{\rm d} x}{{\rm d}t}} = \frac{1-x}{\tau_{d}} - U R_0 x - U x_0 R + U x_0 R_0\,.\label{eq:appA_xlin}
\end{eqnarray}
</math>

We now take the Fourier transform at both sides of Eq. （12）

我们现在对等式两边进行傅里叶变换。

<math>
\begin{eqnarray}
j\omega \tau_{d} \widehat{x} = -\widehat{x} - U R_0 \tau_{d} \widehat{x} - U x_0 \tau_{d}\widehat{R} + (1+ U R_0 \tau_{d} x_0) \delta(\omega)
\label{eq:appA_xhat0}
\end{eqnarray}
</math>

where we defined the Fourier transform pair

这里我们定义了傅里叶变换对

<math>
\begin{eqnarray}
\widehat{x}(\omega) := \int \!{\rm d}{t}\, x(t) \exp(-j\omega t ) \quad ; \quad x(t) = \frac{1}{2\pi}\int \!{\rm d}\omega\, \widehat{x}(\omega) \exp(j\omega t)
\label{eq:appA_ft}
\end{eqnarray}
</math>
and<math>
j=\sqrt{-1}
</math>is the imaginary unit. Solving Eq. （13） for the variable <math>
\widehat{x}
</math>, we find
<math>
\begin{eqnarray}
\widehat{x} = -\frac{U\tau_{d}x_0}{1/x_0 + j \omega \tau_{d}} \widehat{R} + x_0 (2-x_0) \delta(\omega) \label{eq:appA_xhat}
\end{eqnarray}
</math>

<math>
j=\sqrt{-1}
</math>是虚数单位。求解等式（13）的对应变量 <math>
\widehat{x}
</math>，我们得到

where from Eq. （10） we used <math>
U R_0 \tau_{d}=1/x_0 - 1
</math>.

其中从等式（10）中，我们使用了<math>
U R_0 \tau_{d}=1/x_0 - 1
</math>。

Next, we plug Eq. （11）into Eq. （8） to linearize the dynamics of the synaptic current

接下来，我们方程式（11）插入方程式（8）线性化突触电流的动态性

<math>
\begin{eqnarray}
I &=& \tau_{s}AU (R_0x+x_0R-x_0R_0)\\
&=& I_0 \left( \frac{x}{x_0}+ \frac{R}{R_0}-1\right) \label{eq:appA_Ilin}
\end{eqnarray}
</math>

around the steady-state value围绕静态值

<math>
I_0 = \tau_{s}AU x_0 R_0
</math>.

By taking the Fourier transform at both sides of Eq. （16）, using Eq. （15）, we obtain

通过对等式（16）两边进行傅里叶变换。使用等式（15），我们得到

<math>
\begin{eqnarray}
\widehat{I} &=& I_0 \frac{\widehat{x}}{x_0} + I_0 \frac{\widehat{R}}{R_0} - I_0 \delta(\omega) \\
&=& \frac{I_0}{R_0} \widehat{\chi} \widehat{R} + I_0(1-x_0) \delta(\omega)
\label{eq:appA_Ihat}
\end{eqnarray}
</math>

where we defined the filter其中我们定义了过滤器

<math>
\begin{eqnarray}
\widehat{\chi}(\omega) := 1- \frac{1/x_0 -1}{1/x_0 + j\omega \tau_{d}} = \frac{1+(\tau_{d}\omega)^2x_0+j\omega\tau_{d}(1-x_0)}{1/x_0+(\tau_{d}\omega)^2 x_0}\,.
\label{eq:appA_chihat}
\end{eqnarray}
</math>

To interpret the result, we plug into Eq.（17）the Fourier transform

为了解释结果，我们插入等式（17）傅里叶变换

<math>
\widehat{R}=R_0\delta(\omega)+R_1 \widehat{\rho}
</math>,

which yields产生

<math>
\begin{eqnarray}
\widehat{I}(\omega) = I_0 \delta(\omega) + \frac{I_0 R_1}{R_0} \widehat{\chi}(\omega) \widehat{\rho}(\omega)\,.
\label{eq:appA_Ihat_final}
\end{eqnarray}
</math>

Finally, the inverse Fourier transform of Eq.（19）reads

最后，等式（19）的傅里叶逆变换为

<math>
\begin{eqnarray}
I(t) = I_0 + \frac{I_0 R_1}{R_0} \int {\rm d}\tau \, \chi(\tau) \rho(t-\tau)
\label{eq:appA_I_final}
\end{eqnarray}
</math>

with以及

<math>
\begin{eqnarray}
\chi(t)=\delta(t) - \frac{1/x_0-1}{\tau_{d}} \begin{cases} \displaystyle {\exp\left(-\frac{t}{x_0\tau_{d}}\right)} & \text{for}\quad t\ge0 \\ 0 & \text{for}\quad t<0 \end{cases}\,.
\label{eq:appA_chi_final}
\end{eqnarray}
</math>

Therefore the output current <math>
I
</math>is the sum of the steady-state current <math>
I_0
</math> and the filtered perturbation <math>
\frac{I_0 R_1}{R_0} \int {\rm d}\tau \, \chi(\tau) \rho(t-\tau)
</math> where <math>
\chi
</math>is the filter we are interested in.

因此，输出电流<math>
I
</math> 是稳态电流 <math>
I_0
</math>和滤波后的扰动之和，其中<math>
\chi
</math>是过滤器。

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