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→Temporal filtering
$\widehat{\rho}$ is the Fourier transform of $\rho$, and $I_0$ and $x_0$ are the stationary values of $I$ and $x$, respectively [see Eq. \ref{stationary} with $u_0 = U$].
$\widehat{\rho}$ is the Fourier transform of $\rho$, and $I_0$ and $x_0$ are the stationary values of $I$ and $x$, respectively [see Eq. \ref{stationary} with $u_0 = U$].
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.
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.
$\widehat{\rho}$ 是 $\rho$ 的傅里叶变换，$I_0$ 和 $x_0$ 分别是 $I$ 和 $x$ 的平稳值 [参见方程。 \ref{stationary} 与 $u_0 = U$]。 滤波器的幅度 [math]\displaystyle{ |\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 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 (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 (Mejías 08, 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]]).
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 (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（财富 01）。
===Gain control===
===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.
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（Abbott 97, Abbott 04 , 库克 03)。如果一个稳定的突触前放电率 [math]\displaystyle{ R }[/math] 突然改变了 [math]\displaystyle{ \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]\displaystyle{ E }[/math]。因此，突触输入的瞬时增加将与 [math]\displaystyle{ \Delta R E(R) }[/math] 成正比，这与 [math]\displaystyle{ \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]\displaystyle{ \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: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: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: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]] <br />
[[Image:Fig2D_short_term_plasticity.png|300px|链接=Special:FilePath/Fig2D_short_term_plasticity.png]] <br />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. \ref{eq:chihat}]. (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.
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. \ref{eq:chihat}]. (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]\displaystyle{ ux }[/math] 和 [math]\displaystyle{ uxR }[/ 数学]，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]\displaystyle{ |\widehat{\chi}(w)| 衡量的 STD 主导突触的过滤特性 }[/数学] [等式。 \ref{eq:chihat}]。 (D) 对突然输入变化的神经反应 [math]\displaystyle{ \Delta R}[/math] 与 STD 主导突触的稳定速率值。 [数学]\displaystyle{ \Delta R=5 }[/math]Hz. 参数与图 1B 相同。
==Effects on network dynamics==
==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.
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。
===Prolongation of neural responses to transient inputs ===
===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 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]]).
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 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]]).
延长对瞬态输入的神经反应
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]\displaystyle{ 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，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（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（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 (Mongillo 08)。
===Modulation of network responses to external input===
===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.
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.
调制网络对外部输入的响应
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（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（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。
===Induction of instability or mobility of network state===
===Induction of instability or mobility of network state===
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.
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.
诱导网络状态的不稳定性或移动性
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（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。
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（Torres 07，Katori 11，Igarashi 12）。这种特性与皮层神经元上下状态之间的自发转换（Holcman 06）、双眼竞争现象（Kilpatrick 10）以及增强的叠加模糊输入的辨别能力（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 (Melamed 08)。
已经研究了 STD 和 STF 对经典 Hopfield 模型的记忆容量的联合影响 (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。
===Enrichment of attractor dynamics===
===Enrichment of attractor dynamics===
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.
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.
吸引子动力学的丰富
连续吸引子神经网络 (CANN)，也称为神经场模型或环模型 (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。
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（York 09、Fung 12、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（图 3D；Fung 12）。
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 (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。
==Appendix A: Derivation of a temporal filter for short-term depression==
==Appendix A: Derivation of a temporal filter for short-term depression==
We consider the rate-based dynamics in Eq. \ref{poisson} for depression-dominated synapses (<math>u^+ \approx U</math>) and for synaptic responses that are much faster than the depression dynamics ($\tau_s \ll \tau_d$):
We consider the rate-based dynamics in Eq. \ref{poisson} for depression-dominated synapses (<math>u^+ \approx U</math>) and for synaptic responses that are much faster than the depression dynamics ($\tau_s \ll \tau_d$):
我们考虑方程式中基于速率的动态。 \ref{poisson} 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 ($\tau_s \ll \tau_d$ ):
<math>
<math>
The aim is to derive a filter $\chi$ that relates the output synaptic current $I$ to the input rate $R$.
The aim is to derive a filter $\chi$ that relates the output synaptic current $I$ to the input rate $R$.
Note that because the input rate $R$ 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 $R_1 \rho(t)$ of the firing rate $R(t)$ around a constant rate $R_0$, that is,
Note that because the input rate $R$ 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 $R_1 \rho(t)$ of the firing rate $R(t)$ around a constant rate $R_0$, that is,
<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>
目的是导出一个过滤器 $\chi$，它将输出突触电流 $I$ 与输入速率 $R$ 联系起来。请注意，由于输入速率 $R$ 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 $R_0$ 附近的发射率 $R(t)$ 的小扰动 $R_1 \rho(t)$，即
<math>
<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 \, .
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 \, .
Therefore the output current $I$ is the sum of the steady-state current $I_0$ and the filtered perturbation $\frac{I_0 R_1}{R_0} \int {\rm d}\tau \, \chi(\tau) \rho(t-\tau)$ where $\chi$ is the filter we are interested in.
Therefore the output current $I$ is the sum of the steady-state current $I_0$ and the filtered perturbation $\frac{I_0 R_1}{R_0} \int {\rm d}\tau \, \chi(\tau) \rho(t-\tau)$ where $\chi$ is the filter we are interested in.
==References==
<nowiki>我们假设 $R$ 中的这种小扰动会在变量 $x$ 中围绕其稳态值 $x_0>0$ 产生小的扰动： [math]\displaystyle{ 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]</nowiki>
我们现在可以通过近似乘积将 $x(t)$ 的动态线性化为围绕稳态值 $x_0$
[数学]\displaystyle{ \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\\ &\大约& 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]
在方程式中的位置。 \ref{eq:appA_rx} 我们删除了二阶项 $x_1 R_1\rho$，因为我们假设 $R_1\ll R_0$ 和 $|x_1|\ll x_0$。堵塞方程式。 \ref{eq:appA_rx} 转化为等式。 \ref{eq:appA_x} 产生
[数学]\displaystyle{ \begin{eqnarray} {\frac<nowiki>{{\rm d} x}{{\rm d}t}} = \frac{1-x}{\tau_{d}}</nowiki> - U R_0 x - U x_0 R + U x_0 R_0\,.\label{eq:appA_xlin} \end{eqnarray} }[/math]
<nowiki>我们现在对等式两边进行傅里叶变换。 \ref{eq:appA_xlin} [数学]\displaystyle{ \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]我们定义了傅里叶变换对 [math]\displaystyle{ \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] 和 $j=\sqrt{-1}$ 是虚数单位。求解方程。 \ref{eq:appA_xhat0} 对于变量 $\widehat{x}$，我们找到 [math]\displaystyle{ \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] 从哪里方程。 \ref{eq:appA_x01} 我们使用了 $U R_0 \tau_{d}=1/x_0 - 1$。</nowiki>
接下来，我们插入方程式。 \ref{eq:appA_rx} 转化为等式。 \ref{eq:appA_I} 线性化突触电流的动态
[数学]\displaystyle{ \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] 围绕稳态值 $I_0 = \tau_{s}AU x_0 R_0$。
<nowiki>通过对等式两边进行傅里叶变换。 \ref{eq:appA_Ilin}，使用等式。 \ref{eq:appA_xhat}，我们得到 [math]\displaystyle{ \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] 我们定义了过滤器 [math]\displaystyle{ \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]</nowiki>
为了解释结果，我们插入方程式。 \ref{eq:appA_Ihat} 傅里叶变换 $\widehat{R}=R_0\delta(\omega)+R_1 \widehat{\rho}$，产生
[数学]\displaystyle{ \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]
<nowiki>最后，等式的傅里叶逆变换。 \ref{eq:appA_Ihat_final} 读取 [math]\displaystyle{ \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 [math]\displaystyle{ \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\lt 0 \end{cases}\,. \label{eq:appA_chi_final} \end{eqnarray} }[/math]</nowiki>
因此输出电流 $I$ 是稳态电流 $I_0$ 和滤波后的扰动 $\frac{I_0 R_1}{R_0} \int {\rm d}\tau \, \chi(\tau) 之和\rho(t-\tau)$ 其中 $\chi$ 是过滤器
==参考文献References==
*<span id="ResearchTopic" /> '''Research Topic''': ''Neural Information Processing with Dynamical Synapses''. S. Wu, K. Y. Michael Wong and M. Tsodyks. ''Frontiers in Computational Neuroscience'', 2013 [http://www.frontiersin.org/Computational_Neuroscience/researchtopics/Neural_Information_Processing_/821 link]
*<span id="ResearchTopic" /> '''Research Topic''': ''Neural Information Processing with Dynamical Synapses''. S. Wu, K. Y. Michael Wong and M. Tsodyks. ''Frontiers in Computational Neuroscience'', 2013 [http://www.frontiersin.org/Computational_Neuroscience/researchtopics/Neural_Information_Processing_/821 link]