短期突触可塑性

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Short-term synaptic plasticity

短期的突触可塑性

Post-publication activity

Curator: Si Wu

  • Dr. Misha Tsodyks, Weizmann Institute, Rehovot, Israel
  • Prof. Si Wu, State Key Lab of Cognitive Neuroscience and Learning & IDG/McGovern Institute for Brain Research, Beijing Normal University, Beijing, China

Short-term plasticity (STP) (Stevens 95, Markram 96, Abbott 97, Zucker 02, Abbott 04), also called dynamical synapses, refers to a phenomenon in which synaptic efficacy changes over time in a way that reflects the history of presynaptic activity. Two types of STP, with opposite effects on synaptic efficacy, have been observed in experiments. They are known as Short-Term Depression (STD) and Short-Term Facilitation (STF). STD is caused by depletion of neurotransmitters consumed during the synaptic signaling process at the axon terminal of a pre-synaptic neuron, whereas STF is caused by influx of calcium into the axon terminal after spike generation, which increases the release probability of neurotransmitters. STP has been found in various cortical regions and exhibits great diversity in properties (Markram 98, Dittman 00, Wang 06). Synapses in different cortical areas can have varied forms of plasticity, being either STD-dominated, STF-dominated, or showing a mixture of both forms.

Compared with long-term plasticity (Bi 01), which is hypothesized as the neural substrate for experience-dependent modification of neural circuit, STP has a shorter time scale, typically on the order of hundreds to thousands of milliseconds. The modification it induces to synaptic efficacy is temporary. Without continued presynaptic activity, the synaptic efficacy will quickly return to its baseline level.

Although STP appears to be an unavoidable consequence of synaptic physiology, theoretical studies suggest that its role in brain functions can be profound (see, e.g., publications in (Research Topic) and the references therein). From a computational point of view, the time scale of STP lies between fast neural signaling (on the order of milliseconds) and experience-induced learning (on the order of minutes or more). This is the time scale of many processes that occur in daily life, for example motor control, speech recognition and working memory. It is therefore plausible that STP might serve as a neural substrate for processing of temporal information on the relevant time scales. STP implies that the response of a post-synaptic neuron depends of the history of presynaptic activity, creating information that in principle can be extracted and used. In a large-size network, STP can greatly enrich the network's dynamical behaviors, endowing the neural system with information processing capacities that would be difficult to implement using static connections. These possibilities have led to significant interest in the computational functions of STP within the field of Computational Neuroscience.

吴馆长:如果

Misha Tsodyks博士，以色列雷霍沃特魏茨曼研究所

吴思，北京师范大学麦戈文大脑研究所/认知神经科学与学习国家重点实验室，中国北京


短期可塑性(STP) (Stevens 95, Markram 96, Abbott 97, Zucker 02, Abbott 04)，也被称为动态突触，是指突触效能随着时间的推移而改变，以某种方式反映突触前活动的历史的一种现象。在实验中观察到两种类型的STP对突触效能有相反的影响。它们被称为短期抑郁(STD)和短期促进(STF)。STD是由于突触前神经元轴突末端突触信号传导过程中神经递质消耗殆尽而引起的，而STF是由于刺突产生后钙流入轴突末端引起的，这增加了神经递质释放的概率。STP存在于不同的皮层区域，并表现出不同的特性(Markram 98, Dittman 00, Wang 06)。不同皮层区域的突触可以有不同形式的可塑性，要么是std主导，要么是stf主导，或者是两种形式的混合。


与长期可塑性(Bi 01)相比，STP具有更短的时间尺度，通常在数百至数千毫秒之间。它对突触功效的影响是暂时的。如果没有持续的突触前活动，突触效能会很快恢复到基线水平。


虽然STP似乎是突触生理学不可避免的结果，但理论研究表明，它在大脑功能中的作用可能是深刻的(参见，例如，在(研究主题)中的出版物和其中的参考文献)。从计算的角度来看，STP的时间尺度介于快速神经信号(毫秒量级)和经验诱导学习(分钟量级或更多)之间。这是发生在日常生活中的许多过程的时间尺度，例如运动控制、语音识别和工作记忆。因此，STP可能作为处理相关时间尺度上的时间信息的神经基质是合理的。STP意味着突触后神经元的反应取决于突触前活动的历史，从而产生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，赋予神经系统利用静态连接难以实现的信息处理能力。这些可能性导致了在计算神经科学领域内STP的计算功能的重大兴趣。

Contents

[hide]

  • 1 Phenomenological model
  • 2 Effects on information transmission
    • 2.1 Temporal filtering
    • 2.2 Gain control
  • 3 Effects on network dynamics
    • 3.1 Prolongation of neural responses to transient inputs
    • 3.2 Modulation of network responses to external input
    • 3.3 Induction of instability or mobility of network state
    • 3.4 Enrichment of attractor dynamics
  • 4 Appendix A: Derivation of a temporal filter for short-term depression
  • 5 References
  • [隐藏] 1现象学模型
  • 2对信息传递的影响
    • 2.1时间过滤
    • 2.2增益控制
  • 3对网络动态的影响
    • 3.1延长神经对瞬态输入的反应
    • 3.2网络对外部输入响应的调制
    • 3.3网络状态不稳定或迁移的诱导
    • 3.4吸引子动力学的富集
  • 4附录A:短期抑郁的时间过滤器的推导
  • 5引用

Phenomenological model

The biophysical processes underlying STP are complex. Studies of the computational roles of STP have relied on the creation of simplified phenomenological models (Abbott 97,Markram 98,Tsodyks 98).

In the model proposed by Tsodyks and Markram (Tsodyks 98), the STD effect is modeled by a normalized variable  (), denoting the fraction of resources that remain available after neurotransmitter depletion. The STF effect is modeled by a utilization parameter , representing the fraction of available resources ready for use (release probability). Following a spike, (i)  increases due to spike-induced calcium influx to the presynaptic terminal, after which (ii) a fraction  of available resources is consumed to produce the post-synaptic current. Between spikes,  decays back to zero with time constant  and  recovers to 1 with time constant . In summary, the dynamics of STP is given by

where  denotes the spike time and  is the increment of  produced by a spike. We denote as  the corresponding variables just before the arrival of the spike, and  refers to the moment just after the spike. From the first equation, . The synaptic current generated at the synapse by the spike arriving at  is then given by

where  denotes the response amplitude that would be produced by total release of all the neurotransmitter (), called absolute synaptic efficacy of the connections (see Fig. 1A).

The interplay between the dynamics of  and  determines whether the joint effect of  is dominated by depression or facilitation. In the parameter regime of  and large , an initial spike incurs a large drop in  that takes a long time to recover; therefore the synapse is STD-dominated (Fig.1B). In the regime of  and small , the synaptic efficacy is increased gradually by spikes, and consequently the synapse is STF-dominated (Fig.1C). This phenomenological model successfully reproduces the kinetic dynamics of depressed and facilitated synapses observed in many cortical areas.

 

Figure 1. (A) The phenomenological model for STP given by Eqs.(1) and (2). (B) The post-synaptic current generated by an STD-dominated synapse. The neuronal firing rate Hz. The parameters , , ms, ms, and ms. (C) The dynamics of a STF-dominating synapse. The parameters , ms, and ms.


STP的生物物理过程是复杂的。STP的计算作用的研究依赖于简化现象学模型的创建(Abbott 97,Markram 98,Tsodyks 98)。

在Tsodyks和Markram (Tsodyks 98)提出的模型中，STD效应用一个归一化变量()来建模，表示在神经递质耗尽后仍然可用的资源的比例。STF效应由利用率参数建模，表示可供使用的可用资源的比例(释放概率)。在突波之后，(i)由于突波诱导的钙内流到突触前末端而增加，之后(ii)消耗一部分可用资源来产生突触后电流。在峰值之间，随时间常数衰减回零，随时间常数恢复到1。综上所述，STP的动态特性由

式中为峰值时间，为峰值产生的增量。我们表示为峰值到来之前的对应变量，指的是峰值之后的时刻。从第一个方程。由到达的脉冲在突触处产生的突触电流由

其中表示所有神经递质的总释放所产生的反应振幅()，称为连接的绝对突触效能(见图1A)。

二者之间的动态相互作用决定了二者的共同作用是以抑郁为主还是以促进为主。在和大的参数范围内，初始峰值会引起大的下降，需要很长时间才能恢复;因此突触以std为主(图1b)。在and small的机制下，突触效能通过突刺逐渐增加，因此突触以stf为主(Fig.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突触抑制和促进的动力学。

 

图1所示。(A)由式(1)和式(2)给出的STP现象学模型。(B)由std主导的突触产生的突触后电流。神经元放电频率Hz。参数，ms, ms, and ms. (C) stf支配突触的动态。参数:ms、ms。

Effects on information transmission

Because STP modifies synaptic efficacy based on the history of presynaptic activity, it can alter neural information transmission (Abbott 97, Tsodyks 97, Fuhrmann 02, Rotman 11, 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.

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 . The time evolution for the postsynaptic current  can be obtained by averaging Eq. (1) over different realization of Poisson processes corresponding to different spike trains (Tsodyks 98):

where again  and we neglect time scales on the order of the synaptic time constant. For the stationary rate, , we obtain

which is shown in Fig. 2A,B. In particular, for depression-dominated synapses (), the average synaptic efficacy  decays inversely with the rate, and the stationary synaptic current saturates at the limiting frequency , 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).

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 we present the corresponding calculation for depression-dominated synapses (). By considering small perturbations  with  around the constant rate , the Fourier transform of the synaptic current  is approximated by

where we defined the filter

is the Fourier transform of , and  and  are the stationary values of  and , respectively [see Eq. (4) with ]. The amplitude of the filter  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.

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 (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 (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 (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 (Fortune 01).


因为STP会根据突触前活动的历史改变突触的功效，因此它可以改变神经信息的传递(Abbott 97, Tsodyks 97, Fuhrmann 02, Rotman 11, Rosenbaum 12)。一般来说，以性传播疾病为主的突触更倾向于低放电率的信息传递，因为高频率的峰值会迅速使突触失去活性。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。

通过检测来自具有全局放电速率的大神经元群的不相关泊松峰序列的传输，可以分析放电速率依赖于动态突触的传输。突触后电流的时间演化可以通过将Eq.(1)对不同实现的对应于不同峰值序列的泊松过程(Tsodyks 98)进行平均得到:

这里我们再次忽略时间尺度按照突触时间常数的顺序。对于平稳速率，我们得到

如图2A,B。特别是对于以抑郁为主的突触()，突触的平均效能与速率成反比衰减，静止的突触电流在极限频率处饱和，超过此频率，动态突触无法传递静止放电速率的信息(图2A)。另一方面，促进突触可以根据特定的突触前速率进行调整，这取决于STP参数(图2B)。

时间过滤

上述分析仅描述了固定放电速率下的神经群放电。通过式(3)可以推导出突触前种群放电率随时间任意变化时动态突触的滤波特性。在附录A中，我们给出了抑郁症为主突触的相应计算()。通过考虑恒定速率附近的小扰动，突触电流的傅里叶变换近似为

我们在哪里定义了过滤器

为的傅里叶变换，和分别为和的平稳值[见式(4)和]。滤波器的振幅如图2C所示，说明了抑制突触的高通滤波器特性。换句话说，突触前放电速率的快速变化会忠实地传递给突触后的目标，而缓慢的变化会被抑郁减弱。

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性，因为时间近端峰值倾向于放大抑制效应，从而降低突触后电位的输出相关性(Goldman 02)。另一方面，STF的效应被时间近端峰值放大，它提高了突触后神经元对时间相关输入的敏感性(Mejías 08, Bourjaily 12)。

通过将STD和STF相结合，可以进一步提高神经信息的传输。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代的检测(Klyachko 06)。在接受std为主和stf为主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(Fortune 01)。

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 (Abbott 97, Abbott 04, Cook 03). If a steady presynaptic firing rate  changes abruptly by an amount , the first spike at the new rate will be transmitted with the efficacy  before the synapse is further depressed. Thus, the transient increase in synaptic input will be proportional to , which is approximately proportional to  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 , 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.

 

Figure 2. (A) The steady values of the efficacy of an STD-dominated synapse and the postsynaptic currents it generates, measured by  and , 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  [Eq. (6)]. (D) The neural response to an abrupt input change  vs. the steady rate value for a STD-dominating synapse. Hz. The parameters are the same as in Fig.1B.

由于STD以一种频率依赖的方式抑制突触效能，因此有人认为STD提供了一种实现增益控制的自动机制，即，将高增益分配给慢发射传入信号，而将低增益分配给快速发射传入信号(Abbott 97, Abbott 04, Cook 03)。如果一个稳定的突触前放电速率突然发生一定程度的变化，那么在突触进一步被抑制之前，新的放电速率的第一个尖峰就会与效力一起传递。因此，突触输入的短暂增加将与成比例，这与大的速率近似成比例(见上文)。这让人想起韦伯定律，即瞬时突触反应大致与输入放电率的百分比变化成正比。从图2D可以看出，对于固定尺寸的速率变化，响应随输入值的稳定而减小;而如果没有STD，对于固定大小的速率变化，响应将是不变的。

 

图2。(A)一个以std为主的突触的有效性和它产生的突触后电流的稳定值，分别由和测量。参数如图1b所示。(B)对于stf为主的突触，与(A)相同。参数与图1C相同。(C) std主导突触的过滤特性，由[Eq.(6)]测量。(D)对突然输入变化的神经反应vs. std主导突触的稳定速率值。赫兹。参数如图1b所示。

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.

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  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 (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 (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 (Mongillo 08).

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 (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 (Barak 07). The varied response properties mediated by STP may provide different ways of representing and conveying the stimulus information in a network.


除了前馈和反馈传输，神经回路在神经元之间产生周期性的相互作用。随着STP被包括在循环的相互作用中，网络动态表现出许多新的有趣的行为，这是纯静态突触所没有的。因此，这些新的动态特性可以实现stp介导的网络计算。

对瞬态输入的神经反应延长

由于STP比单神经元动力学具有更长的时间尺度(后者的时间顺序通常为毫秒)，STP可以给网络动力学带来的一个新特征是延长神经对瞬态输入的响应。因此，这种刺激引起的残余活动保存着输入的记忆痕迹，在一个大型网络中持续长达几百毫秒，并可以作为信息处理的缓冲区。例如，已有研究表明，std介导的残留活性可以使神经系统区分不同时期的节律性输入(Karmorkar 07)。STP在水库网络的通用计算框架中也扮演着重要的角色。在此框架下，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，包括主动(神经)和隐藏(突触)成分，从而更容易读出输入信息(Buonomano 09)。在最近的一项研究中，有人提出，stf增强的突触本身可以保存输入的记忆痕迹，而无需招募持续放电的神经元，这可能提供了实现工作记忆的最经济和最稳健的方法(Mongillo 08)。

调制网络对外部输入的响应

由于STP可以瞬间改变突触的效能，因此它可以调节网络对持续的外部输入的反应。这方面的一个例子是在一个以性病为主的网络中，突发的同步放电，要么是自发的，要么是对外部输入的响应。由此产生的活动爆发被称为人口峰值(Loebel 02)。为了理解这种效应，考虑一个神经元之间具有强周期性相互作用的网络。当一个足够大的神经元群一起放电时，例如由外部刺激触发，它们可以通过雪崩过程招募其他神经元。然而，在一个大的同步活动爆发后，突触被STD削弱，快速减少循环电流，因此网络活动返回基线。直到突触从抑郁中充分恢复，网络才会再次被激活。因此，种群峰值率由STD的时间常数决定(图3a,B)。STF还可以调节网络对外部输入的响应，但方式非常不同(巴拉克07)。STP所介导的不同的响应特性为网络中刺激信息的表达和传递提供了不同的方式。

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 (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.

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 (Torres 07, Katori 11, Igarashi 12). This property has been linked to spontaneous transition between up and down states of cortical neurons (Holcman 06), to the binocular rivalry phenomenon (Kilpatrick 10), and to enhanced discrimination capacity for superimposed ambiguous inputs (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 (Melamed 08).

The joint effect of STD and STF on the memory capacity of the classical Hopfield model has been investigated (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.

Enrichment of attractor dynamics

Continuous Attractor Neural Networks (CANNs), also called neural field models or ring models (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.

With STP included, a CANN displays new interesting dynamical behaviors. One of them is a spontaneous traveling wave phenomenon (York 09, Fung 12, 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; Fung 12).

 

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 (Loebel 02). (C) The traveling wave generated by STD in a CANN. (D) The anticipative tracking behavior of a CANN with STD.


网络状态的不稳定性或移动性的诱导

持续性放电是指一组神经元在没有外部驱动的情况下持续放电，被广泛认为是信息表示的神经基质(Fuster 71)。为了维持网络的持续活动，神经元之间需要强烈的兴奋性循环相互作用来建立一个支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一个活跃的固定状态(吸引子)。由于STD减弱突触效能取决于神经元活动的水平，它可以抑制吸引子状态。然而，这个属性可以用于进行有价值的计算。

假设一个网络拥有多个相互竞争的吸引子状态，使其中一个状态不稳定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有关(Holcman 06)，与双眼竞争现象有关(Kilpatrick 10)，与对叠加的模糊输入的识别能力增强有关(Fung 13)。STF也可以诱导状态切换，但这是通过间接的方式实现的，通过促进兴奋性突触到中间神经元，后者反过来抑制兴奋性神经元(Melamed 08)。

研究了STD和STF对经典Hopfield模型记忆容量的共同影响(Mejías 09)。研究发现，STD降低了网络的存储容量，但产生了一种新的计算性能，即网络可以在不同的存储状态之间跳跃，这对内存搜索很有帮助。有趣的是，STF可以弥补STD造成的记忆容量损失。

吸引子动力学的富集

连续吸引子神经网络(CANNs)，也被称为神经场模型或环模型(Amari 77)，被广泛用于描述神经系统中连续刺激的编码，如物体的头部方向、方向、运动方向和空间位置。一个CANN，由于其在神经元之间的平移不变循环相互作用，拥有一个连续的局部稳定状态家族，称为凸点。这些稳态形成了一个子空间，网络在这个子空间上是中立稳定的，使网络能够平滑地跟踪时变刺激。

随着STP的加入，CANN显示了新的有趣的动态行为。其中一种是自发行波现象(York 09, Fung 12, Bressloff 12)(图3c)。考虑一个最初处于局域凹凸状态的网络。由于STD，肿块区域的神经相互作用减弱。由于邻近吸引子状态的竞争，一个小的位移会把凸起推开，并且由于STD效应，它会继续朝那个方向移动。如果网络是由连续移动的输入驱动的，在适当的参数范围内，凹凸运动甚至可以与输入移动速度无关，以恒定的时间引导外部驱动器，实现一种预期行为，使人想起啮齿类动物头部方向神经元的预测响应(图3d;Fung 12)。

 

图3。(A,B)由std主导的网络对外部兴奋脉冲产生的群体峰值。当脉冲呈现率较低(A)时，网络对每一个脉冲都做出响应。对于更高的呈现率(B)，网络只对输入的一小部分做出响应。改编自(洛贝尔02)。(C) CANN中STD产生的行波。(D)带有STD的CANN的预期跟踪行为。

Appendix A: Derivation of a temporal filter for short-term depression

We consider the rate-based dynamics in Eq. (3) for depression-dominated synapses () and for synaptic responses that are much faster than the depression dynamics ()

The aim is to derive a filter  that relates the output synaptic current  to the input rate . Note that because the input rate  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  of the firing rate  around a constant rate , that is,

We assume that such small perturbations in  produce small perturbations in the variable  around its steady state value  

We can now linearize the dynamics of  around the steady-state value  by approximating the product

where in Eq. (11) we dropped the second-order term  because we assumed  and . Plugging Eq. (11) into Eq. (7) yields

We now take the Fourier transform at both sides of Eq. (12)  where we defined the Fourier transform pair  and  is the imaginary unit. Solving Eq. (13) for the variable , we find  where from Eq. (10) we used .

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

around the steady-state value .

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

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

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

Therefore the output current  is the sum of the steady-state current  and the filtered perturbation  where  is the filter we are interested in.

我们考虑了Eq.(3)中以速率为基础的动态，用于抑郁为主的突触()，以及比抑郁动力学快得多的突触反应()

目的是推导出一个将输出突触电流与输入速率联系起来的滤波器。注意，由于输入速率以乘法的方式进入方程，因此输入-输出传递函数是非线性的。然而，可以通过考虑射速在恒定速率周围的小扰动来导出线性滤波器，即:

我们假定这样的小扰动在其稳态值周围产生了变量的小扰动

我们现在可以通过近似乘积来将稳态值周围的动力学线性化

式(11)中我们去掉了二阶项，因为我们假设和。将式(11)插入式(7)得到

现在我们对等式(12)两边进行傅里叶变换这里我们定义了傅里叶变换对是虚单位。通过求解变量的Eq.(13)，我们可以从Eq.(10)中找到我们使用的公式。

接下来，我们将Eq.(11)插入Eq.(8)中，以线性化突触电流的动力学

在稳态值附近。

通过在Eq.(16)两边进行傅里叶变换，利用Eq.(15)，我们得到我们定义滤波器的位置

为了解释结果，我们将傅里叶变换代入式(17)，得到

最后，Eq.(19)的傅里叶反变换等于

因此输出电流是稳态电流和滤波扰动的总和其中是我们感兴趣的滤波器。

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References

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  • Goldman, Mark S.; Maldonado, Pedro and Abbott, L. F. (2002). Redundancy Reduction and Sustained Firing with Stochastic Depressing Synapses The Journal of Neuroscience 22(2): 584-591.
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Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Reviewed by: Prof. Boris Gutkin, (1) Group for Neural Theory, LNC INSERM U960, Département d'Études Cognitives, École Normale Supérieure, Paris, France; (2) Faculty of Psychology, HIgher Shcool of Economics, Moscow, Russia
Reviewed by: Dr. Stefano Fusi, Institute of Neuroinformatics, University of Zurich, Switzerland
Accepted on: 2013-10-11 01:17:19 GMT


参考http://www.scholarpedia.org/article/Short-term_synaptic_plasticity

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