The internal simulation time step was taken as 0. Input patterns were conveyed to the network by the collective firing activity of presynaptic neurons, where a pattern consisted of a single spike at each neuron. Presynaptic spikes were uniformly distributed over the pattern duration, and selected independently for each neuron.
The choice of single rather than multiple input spikes to form pattern representations proved to be more amenable to the subsequent analysis of gathered results. In all cases, an arbitrary realisation of each pattern was used at the start of each simulation run, which was then held fixed thereafter.
By this method, a total number p of unique patterns were generated. The postsynaptic neuron was trained to reproduce an arbitrary target output spike train in response to each of the p input patterns through synaptic weight modifications in the network, using either the INST, FILT or CHRON learning rules. In this way, the network learned to perform precise temporal encoding of input patterns. During training, all p input patterns were sequentially presented to the network in batches, where the completion of a batch corresponded to one epoch of learning.
Resulting synaptic weight changes computed for each of the individually presented input patterns or each trial were accumulated, and applied at the end of an epoch. As shall be shown in our simulation results, it was indicated that all of the learning rules shared a similar, optimal value for the learning rate.
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For demonstrative purposes, we first applied the INST and FILT learning rules to training the network to perform a mapping between a single, fixed input spike pattern and a target output spike train containing four spikes. The network contained presynaptic neurons, and the target output spikes were equally spaced out with timings of 40; 80; ; ms. These wide separations were selected to avoid excessive nonlinear accumulation of error due to interactions between postsynaptic spikes during learning.
Simulations for the learning rules were run over epochs, where each epoch corresponded to one, repeated, presentation of the pattern.
The Learning Network
Hence, a single simulation run reflected 40s of biological time. Shown in Fig 5A is a spike raster of an arbitrarily generated input pattern, consisting of a single input spike at each presynaptic neuron. In this example, two postsynaptic neurons were tasked with transforming the input pattern into the target output spike train through synaptic weight modifications, as determined by either the INST or FILT learning rule. From the actual output spike rasters depicted in panel B, it can be seen that both postsynaptic neurons learned to rapidly match their target responses during learning.
Despite this, persistent fluctuations in the timings of actual output spikes were associated with just the INST rule, while the FILT displayed stability over the remaining epochs. Finally, panel C shows the accuracy of each learning rule, given as the average vRD see Eq 21 plotted as a function of the number of learning epochs.
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With respect to the INST rule, it can be seen the vRD failed to reach zero and was subject to a high degree of variance, as reflected by the corresponding spike raster in panel B; its final, convergent vRD value was 0. A A spike raster of an arbitrarily generated input pattern, lasting ms, where each dot represents a spike.
Target output spike times are indicated by crosses. C The evolution of the vRD for each learning rule, taken as a moving average over 40 independent simulation runs. The shaded regions show the standard deviation. In plotting Fig 6 , synaptic weights were sorted in chronological order with respect to their associated presynaptic firing times; for example, the height of a bar at 40ms reflects the average value of a synaptic weight from a presynaptic neuron which transmitted a spike at 40ms.
The gold overlaid lines correspond to the previously defined target output spike timings of 40; 80; ; ms. The input synaptic weight values are plotted in chronological order, with respect to their associated firing time. A The distribution of weights before learning. The gold coloured vertical lines indicate the target postsynaptic firing times.
Note the different scales of A, B and C. Results were averaged based on 40 independent runs. The design of this figure is inspired from [ 9 ].
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From these two panels, a rapid increase in the synaptic weight values preceding the target output spike timings can be seen, which then proceeded to fall off. Furthermore, only the INST rule resulted in negatively-valued weights, which is especially noticeable for weights associated with input spikes immediately following the target output spike timings.
In effect, these sharp depressions offset the relatively strong input drive received by the postsynaptic neuron just before the target output spike timings, which is indicative of the unstable nature of the INST learning rule. With respect to the experimental setup, the network consisted of presynaptic neurons and a single postsynaptic neuron, and was tasked with learning to map a total of 10 different input patterns to the same, single target output spike with a timing of ms.
In this case learning took place over epochs. In every instance, a network containing presynaptic neurons and a single postsynaptic neuron was tasked with mapping 10 arbitrary input patterns to the same target output spike with a timing of ms. Learning took place over epochs, and results were averaged over 40 independent runs. To summarise, these results support our choice of an identical learning rate for all three learning rules as used in the subsequent learning tasks of this section.
Additional, more exhaustive parameter sweeps from further simulations conclusively demonstrated that the learning rates for all three learning rules shared the same inverse proportionality with the number of presynaptic neurons, patterns and target output spikes. An important characteristic of a neural network is the maximum number of patterns it can learn to reliably memorise, as well the time taken to train it.
Therefore, we tested the performance of the network on a generic classification task, where input patterns belonging to different classes were identified by the precise timings of individual postsynaptic spikes. We first determine the performance of a network when trained to identify separate classes of input patterns based on the precise timing of a single postsynaptic spike, and then later consider identifications based on multiple postsynaptic spike timings.
The network was tasked with learning to classify p arbitrarily generated input patterns into five separate classes through hetero-association: an equal number of patterns were randomly assigned to each class, and all patterns belonging to the same class were identified using a shared target output spike timing. For each input class a target output spike time was randomly generated according to a uniform distribution that ranged in value between 40 and ms; the lower bound of 40ms was enforced, given previous evidence indicating that smaller values are harder to reproduce by an SNN [ 9 , 11 ].
To ensure input classes were uniquely identified, target output spikes were distanced from each other by a vRD of at least 0. Shown in the left column of Fig 8 is the performance of a network containing either , or presynaptic neurons, as a function of the number of input patterns to be classified. Hence, in order to determine the maximum number of patterns memorisable by the network, we took an averaged performance level as our cut-off point when deciding whether all of the patterns were classified with sufficient reliability; this criterion was also used to determine the minimum number of epochs taken by the network to learn all the patterns, and is plotted in the right column of this figure.
Epoch values not plotted for an increased number of patterns reflected an inability of the network to learn every pattern within epochs. Each input class was identified using a single, unique target output spike timing, which a single postsynaptic neuron had to learn to match to within 1ms. More than epochs was considered a failure by the network to learn all the patterns at the required performance level. Results were averaged over 20 independent runs, and error bars show the standard deviation.
As expected, Fig 8 demonstrates a decrease in the classification performance as the number of input patterns presented to the network was increased, with a clear dependence on the number of presynaptic neurons contained in the network. Finally, it is evident from this figure that both FILT and CHRON shared roughly the same performance levels over the entire range of input patterns and network structures considered.
This difference in the training time became more pronounced as both the number of input patterns and presynaptic neurons were increased. By contrast, FILT still maintained a memory capacity close to 0. As a validation of our method, we note that our measured memory capacity for CHRON at a timing precision of 1ms is in close agreement with that determined originally in Fig 9A of [ 11 ]: with a value close to 0.
The network contained a single postsynaptic neuron, and was trained to classify input patterns into five separate classes within epochs. Results were averaged over 20 independent runs.protimreapar.tk
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Finally, we examine the performance of the learning rules when input patterns are identified by the timings of multiple postsynaptic spikes. In this case, the network contained presynaptic neurons and a single postsynaptic neuron, and was trained to classify a total of 10 input patterns into five separate classes, with two patterns belonging to each class. For each input class, target output spikes were randomly generated according to a uniform distribution bound between 40 and ms, as used previously. A minimum inter-spike interval of 10ms was enforced to minimise interactions between output spikes.
Because the learning rate was inversely proportional to the number of target spikes, we extended the maximum number of epochs to to ensure the convergence of each rule.
The network was tasked when classifying 10 input patterns into 5 separate classes. Correct classifications were considered when the number of actual output spikes fired by a single postsynaptic neuron matched that of its target, and each actual spike fell within 1ms of its corresponding target timing. In this case, a network containing presynaptic neurons was trained over an extended epochs to allow for decreased learning speed, and results were averaged over 20 independent runs. Taken together, the experimental results of this section demonstrate a similarity in the performance between the FILT and CHRON rules under most circumstances, except when applied to learning multiple target output spikes for which the CHRON rule was best suited.
The INST rule, however, performed worst in all cases, and in particular displayed difficulties when classifying input patterns with increasingly fine temporal precision. This disparity between INST and the other two rules is explained by its unstable behaviour, since it essentially fails to account for the temporal proximity of neighbouring target and actual postsynaptic spikes.
As was predicted in our earlier analysis, this instability gave rise to fluctuating postsynaptic spikes close to their target timings see Fig 5. Hence, it is evident that exponentially filtering postsynaptic spikes in order to drive more gradual synaptic weight modifications confers a strong advantage when temporally precise encoding of input patterns is desired. From the experiment concerning pattern classifications based on multiple output spike timings, it was found for each of the learning rules that the performance decreased with the number of target output spikes.
This is not surprising given that the network needed to match every one of its targets with the same level of temporal precision, effectively increasing the synaptic load of the network during learning. We have studied the conditions under which supervised synaptic plasticity can most effectively be applied to training SNN to learn precise temporal encoding of input patterns. For this purpose, we have derived two supervised learning rules, termed INST and FILT, and analysed the validity of their solutions on several, generic, input-output spike timing association tasks.
In order to benchmark the performance of our proposed rules, we also implemented the previously established E-learning CHRON rule. From our simulations, we found FILT approached the high performance level of CHRON: relating to its ability to smoothly converge towards stable, desired solutions by account of its exponential filtering of postsynaptic spike trains.