1. Synapse and Neuron

The basic units of a neural network are synapses and neurons. We use one PCM device acting as one synapse. A leaky-integrate-and-fire (LIF) circuit (as shown in Fig. 2(a)) is used to represent one neuron. It collects weighted signals and aggregates them in the V$_{\mathrm{mem}}$. As shown in Fig. 2(c), when the V$_{\mathrm{mem}}$ exceeds a threshold (V$_{\mathrm{TH}}$), the V$_{\mathrm{mem}}$ is reset to zero and the pulse generator releases a square pulse as a post-synaptic spike. A square pulse with low amplitude, which below the crystallizing voltage (V$_{\mathrm{c}}$), acts as a pre-synaptic spike (V$_{\mathrm{PRE}}$). Such pulses can be weighted and transmitted by synapses without causing a change in synaptic weight. A square pulse with high amplitude, which over the melting voltage (V$_{\mathrm{m}}$), acts as a post-synaptic spike (V$_{\mathrm{POST}}$). The overlap of V$_{\mathrm{PRE}}$ and V$_{\mathrm{POST}}$ set the voltage through a PCM synapse between V$_{\mathrm{c}}$ and V$_{\mathrm{m}}$, so that an LTP operation is performed (as shown in Fig. 2(b)). The LTD operation is performed if only V$_{\mathrm{POST}}$ exists when the voltage across the PCM synapse exceeds V$_{\mathrm{m}}$. This weight updating rule can be understood as the simplified STDP ^(11-17).

Fig. 3. The input patterns (a) for original pattern rebuilding, (b) for common feature learning.

Fig. 4. (a) Conceptual design of neuron circuit for APS, (b) The pulses scheme for APS.

2. Neural Network

In this paper, we built a two-layer fully-connected neural network to perform the feature extraction. For simplicity, the output layer contains only one LIF neuron. The input layer contains 784 units which are connecting to the pixels of the input pattern. The input pattern is a binary MNIST pattern whose white part corresponds to on-pixels and the black part corresponds to off-pixels. The input neuron releases a pre-synaptic spike if its correlating pixel is an on-pixel. By contrast, the neuron does not release a pre-synaptic spike if its correlating pixel is an off-pixel. The learning mechanism is based on Hebbian theory. The synapses that exhibit causal relationships are potentiated. In contrast, the synapses that exhibited the anti-causal relationships are depressed.

3. Feature Extraction Tasks

Feature extraction is one of the important characteristics of the neural network. We performed two tasks about feature extraction. One of the tasks is the original pattern rebuilding from the contaminated pattern. A pattern labeled “1” is contaminated by different random noises (as shown in Fig. 3(a)). Here the “contaminated” means the pixels with noise are inversed. Our goal is to extract the original pattern from the contaminated pattern. The other task is common feature learning. Different patterns with the same label “5” (as shown in Fig. 3(b)) are provided to the neural network. And our goal is to extract their common features for learning.

4. Influence of Asymmetry

Theoretically, the two tasks of feature extraction can be performed perfectly if the weight updates caused by LTP and LTD are exactly fully-matched. However, due to the mismatch between LTP and LTD, the result of feature extraction may be unexpected.

For the original pattern rebuilding task, Fig. 6(a) shows the synaptic weights after the 100$^{\mathrm{th}}$ learning operation for mismatched cases. With the increase of mismatches, we can see that the image becomes fuzzy when ${∆}$W$_{\mathrm{LTD}}$ > 5/50. In addition, the average weight difference (AWD) is shown in Fig. 6(c) and Table 1. The AWD normalizes the difference between the actual weight value and the expected weight value. The AWD after the 100$^{\mathrm{th}}$ learning operation is less than 1% for cases with ${∆}$W$_{\mathrm{LTD }}$ < 5/50. With the mismatch increasing, the final learning result is far from the target.

The result of the common feature learning operation is shown in Fig. 7. The evolution of the synaptic weights for the fully-match (${∆}$W$_{\mathrm{LTD}}$ = 1/50) and mismatch (${∆}$W$_{\mathrm{LTD}}$ = 4/50) cases are shown in Fig. 7(a) and (b), respectively. With the limitation of the minimum weight value, some information from the trained patterns is removed for the mismatched case after a majority of the synaptic weights decrease to the minimum value. Furthermore, the final weight values correlating to the overlap of on-pixels for mismatched cases (range from 0 to 0.2) are much lower than those for the fully-matched case (range from 0 to 1). In Fig. 7(d) and (e), we selected four representative weights and showed the trace of weights evolution in the case of fully-match and mismatch, respectively. For the fully-match case, the weights correlating to pixel (12,12) and pixel (12,13) finally approach the maximum value with more LTP operations, while the weights correlating to pixel (23,15) and pixel (23,16) are relatively lower with less LTP operations. However, for the mismatch case, all of these weights approach the minimum value.

Fig. 5. Flowchart of the previous scheme and proposed APS.

Fig. 6. Synaptic weights after 100$^{\mathrm{th}}$ learning operation for (a) mismatch, (b) compensation cases. The average weight difference for (c) mismatch, (d) compensation cases.

1. Concept of APS

The asymmetry between the LTP and LTD operations is an intrinsic property of a PCM device. The average mismatch “m”, which is defined as the rate of ${∆}$W$_{\mathrm{LTD}}$ to ${∆}$W$_{\mathrm{LTP}}$ (m = ${∆}$W$_{\mathrm{LTD}}$/${∆}$W$_{\mathrm{LTP}}$), can be pre-measured. A counter is added in the neuron circuit (as shown in Fig. 4(a)) to indicate how many times the neuron fires, and the pulse generator is designed to fire two kinds of pulses: pulse(1) only performs LTP operations without performing LTD operations; pulse(2) is the normal pulse for performing both LTP and LTD operations. The working principles of the pulses are shown in Fig. 4(b). The overlap of the pulse(1) can only drive the PCM synapses to set the voltage (LTP operations), and the working principle of the pulse(2) is explained in Section III. The flowchart of the previous scheme and the proposed APS are shown in Fig. 5. The counter number is set to zero initially, and it increments by one after each complete integrate-and-fire operation of a neuron. Pulse(1), which prohibits the LTD operation, is provided before the counter number reaches m. Once the counter number reaches m, pulse(2) is fired instead of the pulse(1). By performing the APS, an LTD operation is only performed once while LTP operations are performed m times. This scheme roughly compensates for the mismatch of the asymmetry property.

2. Effects of APS on Original Pattern Rebuilding

To show the effects of the APS on original pattern rebuilding, we compared the learning result for mismatched cases and for compensation cases (as shown in Fig. 6). The AWD values after the 100$^{\mathrm{th}}$ learning operation are shown in Table 1. For the cases with ${∆}$W$_{\mathrm{LTD}}$ > 8/50, the learning results by the APS more closely approach the fully-matched case than those of the mismatched cases with the same conditions. The AWDs for compensation cases are much more reduced than the mismatched cases and the patterns are much clearer. For the compensation cases with ${∆}$W$_{\mathrm{LTD }}$ < 15/50, the AWD is reduced to less than 0.6% by the APS. For the compensation case with ${∆}$W$_{\mathrm{LTD}}$ = 20/50, the AWD is as large as 1.658 due to the influence of noise in the background, but the pattern is clear enough.

3. Effects of APS on Common Feature Learning

Fig. 7. Synaptic weights for (a) fully-match, (b) mismatch, (c) compensation cases. The trace of specific synaptic weights for (d) fully-match, (e) mismatch, (f) compensation cases.

Fig. 8. (a) The synaptic weights after 5000th learning for fully match, mismatch and compensation cases, (b) The difference ratio, (c) The summation of weights.

In Fig. 7, we compare the weight evolutions and the trace of specific synaptic weights for fully-matched, mismatched and compensation cases. For the mismatched and compensation cases, the changes in the synaptic weights for the LTP and LTD operations are set as ${∆}$W$_{\mathrm{LTP}}$=1/50 and ${∆}$W$_{\mathrm{LTD}}$=4/50. In contrast with the mismatched case, the evolution process for the compensation case more closely approaches the fully-matched case. The learning results for the compensation case are the common properties of the trained patterns instead of the properties of the last training pattern for the mismatched case. The trace of the synaptic weight evolution for the compensation case is similar to that for the fully-matched case, while the trace for the mismatched case is in the lower range.

The learning results after the 5000$^{\mathrm{th}}$ learning operation shown in Fig. 8(a). All of the learning results for the compensation cases are much clearer than the mismatched cases. We statistically analyzed the difference ratio and summation of the synaptic weights in Fig. 8(b) and (c), respectively. The difference increases with the mismatch between LTP and LTD until a saturation value of about 8%. And the summation of the normalized synaptic weights is below 5 after the ratio of LTD to LTP becomes greater than 500%. This means that the learning result is merely the last training pattern and the integrated weighted signal is very weak. Compared with the mismatched cases, the difference ratios for the corresponding compensation cases are reduced and the summation of the normalized synaptic weights has a similar value to the fully-matched case. This means that the learning results are the common properties of trained patterns, just as in the fully-matched case, and the signal of the integrated weighted signal is strong enough to drive the integrate-and-fire operation of the LIF neuron.

4. Effects of APS on Energy Efficiency

An additional effect of APS is the reduction of energy consumption in the learning process. As the energy consumption for unit reset operation on the PCM device is several times larger than that for unit set operation. The total energy consumption is reduced by reducing the number of RESET operations in the APS. The operation numbers and energy consumption are shown in Table 2 to compare the previous scheme and the APS (W$_{\mathrm{LTD}}$=4/50). Here, the energy consumption for unit LTP and LTD operation is based on the data from ⁽¹³⁾. For the previous scheme, the number of RESET operation (N$_{\mathrm{RESET}}$) equals the number of LTD (N$_{\mathrm{LTD}}$). For the APS, N$_{\mathrm{RESET}}$ equals a quarter of N$_{\mathrm{LTD}}$. As a result, the total energy consumption in APS is only about 26.43% of that in the previous scheme.