Correction
29 Jan 2014: Yuan WJ, Zhou JF, Zhou C (2014) Correction: Network Evolution Induced by Asynchronous Stimuli through Spike-Timing-Dependent Plasticity. PLOS ONE 9(1): 10.1371/annotation/417c1eb3-1de1-4d04-8c1d-3f73ffc57f26. https://doi.org/10.1371/annotation/417c1eb3-1de1-4d04-8c1d-3f73ffc57f26 View correction
Figures
Abstract
In sensory neural system, external asynchronous stimuli play an important role in perceptual learning, associative memory and map development. However, the organization of structure and dynamics of neural networks induced by external asynchronous stimuli are not well understood. Spike-timing-dependent plasticity (STDP) is a typical synaptic plasticity that has been extensively found in the sensory systems and that has received much theoretical attention. This synaptic plasticity is highly sensitive to correlations between pre- and postsynaptic firings. Thus, STDP is expected to play an important role in response to external asynchronous stimuli, which can induce segregative pre- and postsynaptic firings. In this paper, we study the impact of external asynchronous stimuli on the organization of structure and dynamics of neural networks through STDP. We construct a two-dimensional spatial neural network model with local connectivity and sparseness, and use external currents to stimulate alternately on different spatial layers. The adopted external currents imposed alternately on spatial layers can be here regarded as external asynchronous stimuli. Through extensive numerical simulations, we focus on the effects of stimulus number and inter-stimulus timing on synaptic connecting weights and the property of propagation dynamics in the resulting network structure. Interestingly, the resulting feedforward structure induced by stimulus-dependent asynchronous firings and its propagation dynamics reflect both the underlying property of STDP. The results imply a possible important role of STDP in generating feedforward structure and collective propagation activity required for experience-dependent map plasticity in developing in vivo sensory pathways and cortices. The relevance of the results to cue-triggered recall of learned temporal sequences, an important cognitive function, is briefly discussed as well. Furthermore, this finding suggests a potential application for examining STDP by measuring neural population activity in a cultured neural network.
Citation: Yuan W-J, Zhou J-F, Zhou C (2013) Network Evolution Induced by Asynchronous Stimuli through Spike-Timing-Dependent Plasticity. PLoS ONE 8(12): e84644. https://doi.org/10.1371/journal.pone.0084644
Editor: Matjaž Perc, University of Maribor, Slovenia
Received: November 12, 2013; Accepted: November 25, 2013; Published: December 31, 2013
Copyright: © 2013 Yuan et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work is supported by the Hong Kong Research Grant Council No. HKBU202710 (CZ), the National Natural Science Foundation of China under Grant Nos. 11275027 (CZ) and 11005047 (WJY), the Young University Teacher’s Fund of Anhui Provincein China under Grant No. 2008jql071 (WJY), and the Young Fund of Huaibei Normal University under Grant No. 2013xqz17 (JFZ). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
In neural systems, an asynchronous state characterized by arbitrarily low mean spiking correlations has been observed extensively in experiments [1], [2], which offers substantial advantages for information processing and coding [2]. Especially in sensory systems, the asynchronous firings or external asynchronous stimuli (which can produce the asynchronous firings) play an important role in perceptual learning, associative memory and map development [3], [4]. It has been experimentally found that, the sensory asynchronous stimuli are used to mediate the plasticity of neural responses for learning and memory in adults and the activity-dependent development of sensory map during a critical period of early postnatal life [3]–[8]. For instance, late visual stimulus is able to serve instructive role for earlier arriving auditory input in the barn owl, a highly efficient predator [5]. Neural associations between stimuli and reward expectancy in primary visual cortex can form when adult rats experience an association between visual activity (conditioned stimulus) and subsequent reward (unconditioned stimulus) [6]. A recent study has reported that the asynchronous stimuli producing binocular retinal activities in mice during a critical period in development enhance eye-specific segregation and regulate retinotopy in the developing visual system [7].
Recently, an important cognitive function, cue-triggered recall of learned temporal sequences, has been experimentally studied [9], [10]. The cue-triggered recall of a learned temporal sequential firing has been found in neuronal ensemble of hippocampus, its surrounding cortical areas [11], [12] and visual cortical circuits [13]. Importantly, the learned sequential firing can be trained in experiments by the repeated asynchronous stimuli. For example, a recent study reported that [13], a moving spot that asynchronously stimulated the neurons whose receptive fields fall along the motion path, can evoke the sequential firing of neuronal ensemble in primary visual cortex. Shortly after repeated motion conditioning, a brief flash at the starting point of the motion path can also evoke the sequential firing similar to that evoked by the moving spot. In this experiment, the sequence learning and cue-triggered recall become more persistent with more repeats of the stimulation of the moving spot. This indicates that, asynchronous stimuli evoked by the moving spot, play an important role in the formation of the sequence learning and recall.
In general, asynchronous stimuli can be used to mediate the activity-dependent map development during a critical period of early postnatal life and the synaptic plasticity for learning and memory in adults [3], [4], [6], [8], [14]. Especially, asynchronous stimuli contribute to the formation of sequence learning and recall [11]–[13]. Many neuron scientists thought that, these functional roles result from activity-dependent synaptic plasticity between neurons [4], [5], [7], [8], [13]. However, the organization of structure and dynamics of neural networks due to synaptic plasticity are not well understood in the presence of asynchronous stimuli. It is clear that, the asynchronous stimuli can produce asynchronous, segregative firings of pre- and postsynaptic neurons. So far, experiments have extensively found a type of synaptic modification: spike-timing-dependent plasticity (STDP), which is highly sensitive to correlations between pre- and postsynaptic firings [15]–[17]. Particularly, this synaptic plasticity has been widely found in sensory systems [18]–[21], which could contribute to learning, memory and development [4], [21]–[23]. Therefore, it is natural to expect that STDP could play an important role in response to asynchronous stimuli.
In order to explore the role of STDP in the presence of external asynchronous stimuli, we construct a network model with local connectivity and sparseness depending on the spatial distance between neurons, and use external currents to stimulate alternately on different spatial layers. We here expect that, STDP can modify some synaptic strengths to form a feedforward network due to asymmetric pre-post spike orderings resulting from asynchronous stimulus pairings. In the resulting feedforward structure, a short stimulation current can be expected to produce wave propagation of neuronal firings, which could provide an alternative explanation for the cue-triggered recall of sequence learning. In addition, the results imply an important role of STDP in generating feedforward structure and collective propagation activity required for experience-dependent map plasticity in developing in vivo sensory pathways and cortices. Since the synaptic modification due to STDP is sensitive to correlations between pre- and postsynaptic firings [15], [16], the resulting feedforward structure is expected to reflect the property of STDP. In the feedforward network, directed propagation dynamics of neural activity is also expected to reflect the underlying property of STDP, which suggests an application of the analysis for examining STDP by measuring neural population activity in cultured neural network.
Materials and Methods
Spatial Neural Network Model
Recently, spatial network has been widely modeled [24]–[27]. Considering the locally-linked structure in real neural systems, we here construct a two-dimensional neural network model with local connectivity and sparseness, depending on the spatial distance between connected neurons. As shown in Fig. 1(a), neurons are located in lattice points of the two-dimensional square with
layers, where each layer has
neurons. Each pair of neurons is connected locally with probability
[28] by synapse with random-chosen initial synaptic strength from
to
, where
denotes the locally connecting strength and
is the distance between two neurons (the side length of square is regarded as
here). For simplicity but without loss of generality, the model is composed of Integrate-and-Fire type neurons with chemical couplings of
function. The dynamics of the membrane potential
of neuron
is described by
(1)
(a) The locally-linked neuronal network model on two-dimensional square having neurons and
layers. Here, we take
,
and label neural number (1 – 1000) from left to right layers. (b) To train the network, the input pulse current
with duration
is injected alternately into each pair of layers with the left-right sequence having the same inter-stimulus interval
, respectively. After each training trial, there is a long enough time
to let network activity recover to the rest state for the next training. We perform training for all the pair of layers with the same number of trials. (c) To test the resulting feedforward structure and its propagative capacity, a steady current
is injected into a certain layer (here, we choose the 3rd layer, i.e., L3).
Here, we adopted the parameter values as those in Refs. [29], [30], which model V1 cells according to empirical observations [31]. The membrane time constant equals
ms, the resting potential
is
mV, and the reversal potentials
for all the excitatory synapses are
mV. Each neuron
integrates external stimulus
and inputs coming from the connecting neurons
at spike time
. When the potential
reaches the threshold value
mV, the neuron
emits a spike, and then the membrane potential is reset to the resting potential
. Here, we do not consider the refractory period of neuronal activity in order to improve the speed of modeling. However, the qualitative results are independent of the refractory period. Also, the parameter
is constant synaptic conductance,
is the adjacency matrix (
or
) and
is the weight of synaptic strength from neurons
to
. The synaptic modifications are subject to STDP, which will be described in the following.
Spike-Timing-Dependent Plasticity
STDP has been extensively found in experiments and has received much theoretical attention in recent years [32]–[36]. The temporal order of presynaptic and postsynaptic spikes determines whether the synaptic change is potentiated or depressed. The synaptic change will be potentiated if presynaptic neuron spikes before postsynaptic neuron; otherwise, it will be depressed. The changes of synaptic strength are a function of the relative timing between presynaptic spike arrival and postsynaptic firing. The smaller the lag between the spikes, the larger the change. The modification of synaptic weight
from neuron
to
is approximated by exponential functions of the spiking time difference
between post- and presynaptic neurons [15],
(2)
The parameters ,
,
and
describe the property of STDP. Generally, STDP can be divided into two forms in numerical studies. If
and
, it is temporally symmetric form; otherwise, it is temporally asymmetric form. According to Eq. (2), the synaptic dynamics in Eq. (1) is described as,
(3)when neuron
fires. For simplicity of simulation and without loss of generality, we only consider the low-frequency stimuli. Thus, we here consider only the largest modification corresponding to last spike time
of neuron
at spiking time
of neuron
(
). Particularly, at
(i.e., synchronous spikes of neurons
and
),
and
are both equal to
. As discussed in [15],
(or
) is set to
if the change makes
(or
) and
(or
) is set to the maximal value
if the change would make
(or
).
For STDP model, the authors only consider.
External Asynchronous Stimuli
For many years, the neural-response to external stimulus has been challenging hot topics for the study of neurology [5], [7], [37], [38]. By using external asynchronous stimuli, our model produces a feedforward network due to STDP. In our model, for each pair of layers, we firstly stimulate the left layer with duration and then stimulate the corresponding right layer with the same duration
after time
by inputting pulse current
(Fig. 1(b)), where
is set as a small enough value that it cannot induce spikes of neurons in other layers. Clearly, the synapses from neurons of the left layer to those of the right layer are potentiated by STDP because spikes of the left neurons precede those of the right neurons due to interval
, while the synaptic strengths of reverse connections are depressed. After each training trial for each pair of layers, there are long enough time
to let network activity recover to the rest state for the next training. Here, we do the training with the same number of trials for each pair of layers. After many training trials for all the pairs of layers, all the synaptic weights from left layers to right layers will increase almost to
, while reverse synaptic weights will decrease approximately to
. Namely, the resulting network structure is expected to be a feedforward structure from left to right after many enough training trials. Then, to test the feedforward structure, we measure collective propagation property of neural activity by injecting a steady current
into a certain layer in the resulting network (Fig. 1(c); here, we choose the 3rd layer).
Results
Generation of Feedforward Structure
We firstly simulate the generation of feedforward structure. As shown in Fig. 2, the initially locally-linked network (Fig. 2(a)) forms a feedforward structure (Fig. 2(b)) due to STDP by the training procedure mentioned above. Moreover, we show in Fig. 2(c) the propagation activity of neural spikes by inputting steady current. Obviously, the signals from the stimulated layer 3 (neurons to
) spread only to the right layers (neurons
to
), while the left layers (neurons
to
) cannot respond. This directed propagative activity indicates the formation of feedforward structure and thus reflects the existing of STDP in this neural network.
The weights of synaptic strengths on the model before (a) and after (b) training trials. (c) The spatiotemporal pattern
of neuron spikes by injecting steady current for testing feedforward network. The system parameters are
,
and
ms. The parameters for training (Fig. 1(b)) and testing (Fig. 1(c)) are given by
,
ms,
s,
ms, and
.
Effect of Number of Training Trials
In order to further explore the impact of the training on the resulting feedforward structure, we next investigate the dependance of synaptic weights for different connection types on the number of training trials while keeping ms. Here, we divide the synaptic connections into three types: feedforward (in the training direction, from left layer to right layer in model Fig. 1(a)), feedback (the opposite training direction, from right layer to left layer) and recurrent (intra-layer) connections. Fig. 3(a) shows the situation at the beginning of our simulations, with all synaptic weights randomly chosen from 0 to 1. After 10 trials (Fig. 3(b)), the feedforward synaptic weights are potentiated by STDP because left (presynaptic) neuron spikes before right (postsynaptic) neurons. On the contrary, the feedback synaptic weights are depressed due to reverse pre-post spike ordering. In addition, we can see the strong increase of recurrent synaptic weights. This is because the linked neurons within a layer often synchronously fire due to the same driving stimulus, causing the synaptic potentiation by STDP (see Eq. 3 for
). After 20 (Fig. 3(c))and 30 (Fig. 3(d)) trials, many more feedforward synapses have been strengthened (e.g. 63% being approximate to 1 in the left panel of Fig. 3(d)) and many feedback synapses have been weakened (e.g. 63% being approximate to 0 in the middle panel of Fig. 3(d)), which makes the network form feedforward structure (Fig. 2(b)).
Distribution of synaptic weights for three different connection types: feedforward (left), feedback (middle) and recurrent (right) at the beginning of training (a) and after the training with different number of trials 10 (b), 20 (c), 30 (d). The same parameters are the same as in Fig. 2, except for ms.
Fig. 4(a) shows clearly the changes of average synaptic weights with respect to the number of training trials. Obviously, the increasing amount of synaptic modification for feedforward connections is equal to the decreasing amount of synaptic modification for feedback connections (Fig. 4(a)), which results from the temporally symmetric form of STDP adopted here (i.e., and
). For temporally asymmetric form of STDP, we also show the similar changes of average synaptic weights in Fig. 4(b). As in the symmetric case, the feedforward structure will emerge and become stronger with the increase of the number of training trials. Differently, the incremental amount of synaptic strengths for feedforward connections and reductive amount of synaptic strengths for feedback connections are not equal (Fig. 4(b)) owing to the asymmetric modification between potentiated and depressed synaptic strengths with the same pre-post firing timing.
Average synaptic weights for three different connection types: feedforward (squares), feedback (circles) and recurrent (upward triangles) as a function of the trial number for temporally symmetric form of STDP ( and
ms) (a) and for temporally asymmetric form of STDP (
and
ms) (b). The other parameters are the same as in Fig. 2 except for
ms. Also, we show the average synaptic weights for the connections including both feedforward and feedback types (downward triangles) in (a) and (b).
Effect of Inter-stimulus Interval
We also studied the effect of inter-stimulus timing (i.e., interval ) on synaptic weights for different connection types. When the interval is shortened to 0 ms (Fig. 5(a)), about 62% of feedforward synapses have been strengthened to 1 (see the left panel of Fig. 5(a)), while 62% of feedback synapses have been weakened to 0 (see the middle of Fig. 5(a)) after 20 trials, indicating a generation of strong feedforward structure (see Fig. 2(b)). With the increasing of inter-stimulus interval (Figs. 5(b)–5(d)), the resulting distributions of synaptic weights become broader, showing that the feedforward structure is weaker. For example, when the inter-stimulus interval is increased to 60 ms, there are insufficient potentiation (only 23% being approximate to 1 in the left panel of Fig. 5(d)) and insufficient depression (only 23% being approximate to 0 in the middle panel of Fig. 5(d)) after 20 trials to form feedforward structure.
Distribution of synaptic weights for different connection types: feedforward (left), feedback (middle) and recurrent (right) after the 20 training trials with different stimulus intervals ms (a),
ms (b),
ms (c),
ms (d). The other parameters are the same as in Fig. 2.
The right panels of Fig. 5 show that the synaptic distribution in recurrent connections appears to be independent of inter-stimulus interval. Since the training current is set as a small enough value that it cannot induce spikes of neurons in other layers without receiving training, the amount of recurrent synaptic modification within layer is completely determined by duration of external current stimuli imposed to the layer (i.e., the product of
and number of training trials in Fig. 1(b)), independent of the inter-stimulus interval. This result implies that the synaptic modifications for the recurrent and feedforward/feedback connections are independent. So, the effect of recurrent connections on the formation of feedforward network can be neglected in our model.
Fig. 6(a) clearly shows the changes of average synaptic weights with respect to the inter-stimulus interval. Importantly, the amount of average feedforward modification relative to its initial value 0.5 (see the value at 0 trials in Fig. 4(a)) exhibits an exponential decay with ,
with an exponent
(Fig. 6(b)), and the amount of modification of average feedback connections relative to its initial value 0.5 (see the value at 0 trials in Fig. 4(a)) is also characterized as an exponential decay
with another exponent
(Fig. 6(c)). For the temporally symmetric form of STDP, the two exponents
and
are equal (Figs. 6(b) and 6(c)) due to the equal amounts of synaptic potentiation and depression for a pre-post firing timing. The two exponents
and
are different for the temporally asymmetric form of STDP because the symmetric modification of potentiation and depression is broken, shown in Fig. 7.
Average synaptic weights for three different connection types: feedforward (squares), feedback (circles) and recurrent (upward triangles) as a function of the stimulus interval after the 20 training trials (a). The amounts of average synaptic modification relative to its initial value 0.5, (b) and
(c) for feedforward (b) and feedback (c) connections exhibit exponential falloffs as the stimulus interval
increases with exponents
(b) and
(c), respectively in linear-log scales. we also show the average synaptic weights for the connections including both feedforward and feedback types (downward triangles) in (a). The parameters are the same as in Fig. 2.
The same as in Fig. 6, but for the temporally asymmetric form of STDP. Here, the parameters are the same as in Fig. 2, except for ,
ms,
ms.
We numerically examined the relationships between the exponents and
characterizing the exponential falloffs with the exponents
and
in the STDP functions (Figs. 8(a) and 8(b)). The exponents
and
are close to
and
, respectively. Moreover,
and
are independent of
and
, respectively. It’s further found that, these equal relationships hold for networks with different connecting parameter
(results not shown). These findings suggest that the dependence of the structure modification on inter-stimulus interval
reflects the exponents of STDP. Since network structure determines dynamics of the network, a similar dependance of population dynamics on interval
is naturally expected to reflect the underlying STDP properties in the resulting feedforward networks, which is described below.
(a) vs.
for different
(b)
vs.
for different
. For comparison,
and
are shown as solid lines in (a) and (b), respectively. The other parameters are the same as in Fig. 2.
Propagation Property
In brain, information and cognitive processing requires signal propagation through multiple regions [39]. The feedforward network plays a crucial role in signal propagation [40]. Here we study the propagation property in the above feedforward network after training. In Fig. 9(a), we show the effect of inter-stimulus interval of training on the propagation ability. As we see in Fig. 3(c), the resulting network can support directed propagation of activity from left to right when the 3rd layer is stimulated, and the activity decays out. Here we do not consider further effect of STDP during propagation. We plot the number of spikes of all propagated layers (i.e., layers
to
) as a function of inter-stimulus interval
. Clearly, it exhibits an exponential falloff
with exponent
, similar to the exponential falloff of STDP in Eq. (3). The exponent
increases almost linearly with the exponent
in STDP, which is almost independent of the other exponent
(Fig. 9(b)). This is because the spike number
of propagated layers depends mainly on the amount of feedforward synaptic modification (not on that of feedback synaptic modification), determined by the exponent
of potentiation term in Eq. (3) (Fig. 8(a)). For a comparison, we find that the scaling of the propagative activity remains the same when the feedback connections are cut (Fig. 9(a) without feedback), showing that the feedback connections are so weak that their effect on neural activity can be neglected. Additionally, the propagative activity is still almost the same without recurrent connections (Fig. 9(a)), indicating that the recurrent connections within layers also have little effect on the propagative dynamics. This is because recurrent connections are very sparse with k = 0.2 (Fig. 3(b)) although recurrent connection strengths are strong. Consequently, the resulting feedforward structure governs the propagative property of the whole network in the presence of the input current. In Fig. 9(c) we show the linear relation
between the structure modification and dynamical propagation. The coefficient
depends strongly on the stimulation strength
. As shown in Fig. 9(d),
increases with
. From these simulation results, we can see that a STDP-structure-dynamics relation
(Figs. 8(a), 9(b) and 9(c)) can emerge in the feedforward network induced by asynchronous stimuli on different spatial neuronal layers.
(a) The total number of spikes of propagated layers as a function of inter-stimulus interval
in the feedforward network obtained after 20 trials of training when keeping all the connections, indicating the exponential behavior of propagation dynamics with exponent
in the linear-log scales. For comparison, the
are also plotted in (a) when feedback and recurrent connections are cut (i.e., without feedback and without recurrent), respectively. (b) The exponent
as a function of exponent
in STDP for different
. (c) The exponent
as a function of the corresponding exponent
in the feedforward network structure for different
. In (b) and (c), the linear fit lines of data are also given with slope
. (d) The coefficient
as a function of stimulation strength
. The other parameters are the same as in Fig. 2. Data of
are compiled for
ms, and averaged over
independent runs with error bar in (a).
Discussion
In experiments, cultured neuronal networks and developing neural systems can both generate spontaneous activity in the form of synchronized bursting events (SBEs) [41]–[43], short time events during which most of the recorded neurons fire rapidly, with several subgroups. Besides, neural activity often exhibits avalanches with some dynamical clusterings described by synchronous firing of neurons in vivo and in vitro [42], [44]–[46]. In SBEs and avalanche activity, the different dynamical clusterings behave inter-cluster asynchronous firings. So, the dynamical clusterings might strongly shape the network structure due to STDP, as the external asynchronous stimuli do in our model. This suggests an important role of STDP in generating feedforward structure and collective propagation activity on neural systems in vivo and in vitro [23], [41], [44], [45]. The role of STDP in the formation of feedforward structure driven by asynchronous stimuli might provide insight into important role of propagating feedforward (from low-level to high-level brain areas) information flow for experience-dependent map plasticity in the development of in vivo sensory pathways and cortices [4], [7], [21], [22].
The sequence learning and recall may result from activity-dependent synaptic plasticity. The conditioning-evoked sequential spiking of neighboring neurons is well suited for the induction of STDP [13], which selectively potentiate the directed connections between neurons and facilitate spike propagation in the same direction. Our study may provide an insight into formation mechanism of the sequence learning and recall. In our model, the resulting feedforward structure could denote sequence memory formed by the sequence learning. The spiking propagation in the direction of feedforward structure could reflect dynamical process of the cue-triggered recall. From Figs. 3 and 4, we can see the resulting feedforward structure will become clearer and so produce the more prominent propagation, as the number of our training trials increases. Indeed, with more repeats of stimulus sequence in experiment, the sequence learning and recall become more persistent [13]. Thus, our study suggests that STDP may modify neuronal structure and dynamics induced by external asynchronous stimuli to support the sequence learning and recall, which is consistent well with the experimental observation.
The amount of synaptic potentiation or depression due to STDP is typically measured by pairing a number of pre- and postsynaptic action potentials with a specified time interval between them [32], [33], [35], [47]. But, for a large-scaled neural system, it becomes very complex and difficult to directly test each synaptic modification due to spike timing and ordering. In our model, the resulting STDP-structure-dynamics relation (Figs. 8(a), 9(b) and 9(c)) suggests a potential method to examine STDP in large-scaled neural network. Our results demonstrate the propagative activity exponentially decays with exponent , which provides an application for examining STDP by measuring neural population activity in a cultured neural network trained by external asynchronous stimuli. Cultured neural networks cultivated on multi-electrode arrays provide relatively simple and well-controlled model systems for investigating long-term activity of individual neurons at different locations [33], [41], [48]–[50]. By cultivating our two-dimensional network model and adopting our training/testing method, the strength of exponent
in STDP might be qualitatively examined by measuring propagation property in the resulting feedforward network. This study could shift the attention in the measurement of synaptic plasticity from the single synapse to the collective network activity. In the future, how to design other model and method for examining more parameters (e.g.,
,
and
) of STDP is an interesting question to pursue.
To sum up, using a simple locally linked network of integrate-and-fire neurons with STDP, we showed that it can naturally evolve into a feedforward structure when a pair of external asynchronous stimuli is used to train the network. Interestingly, the amount of synaptic modification due to asynchronous stimulus pairing with a time interval falls off exponentially as the interval increases, similar to the falloff of STDP as the interspike interval increases. Moreover, exponents ( and
) of the exponential falloff are almost the same as the exponents (
and
) of STDP (i.e.,
and
). Most interestingly, propagative activity also exhibits a similar exponential falloff with exponent
in the resulting feedforward network. Thus, by measuring the exponential falloff of the propagation activity as a function the inter-stimulus intervals in the resulting trained network, we are able to extract a direct measurement of the characteristic time scale in the network’s STDP property. Generally, STDP-structure-dynamics relation
can emerge in our model. Although the formation of feedforward structure under the action of STDP has been widely studied [51], [52], the external asynchronous stimuli observed and used extensively in experiments, has not yet been investigated to induce the feedforward structure through STDP. The STDP-structure-dynamics relation is the first to be shown in the resulting feedforward network by imposing asynchronous stimuli. Our model may prove useful for understanding the relation between structure and dynamics induced by asynchronous stimuli through STDP and provide a starting point for studying the property of STDP by using external asynchronous stimuli.
Author Contributions
Conceived and designed the experiments: WJY CZ. Performed the experiments: WJY JFZ. Analyzed the data: WJY JFZ CZ. Contributed reagents/materials/analysis tools: WJY JFZ CZ. Wrote the paper: WJY JFZ CZ.
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