SCIENCE CHINA Information Sciences, Volume 63 , Issue 2 : 122402(2020) https://doi.org/10.1007/s11432-019-1468-0

Deterministic conversion rule for CNNs toefficient spiking convolutional neural networks

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  • ReceivedFeb 28, 2019
  • AcceptedJul 8, 2019
  • PublishedJan 15, 2020



This work was supported by National Natural Science Foundation of China (Grant Nos. 61704167, 61434004), Beijing Municipal Science and Technology Project (Grant No. Z181100008918009), Youth Innovation Promotion Association Program, Chinese Academy of Sciences (Grant No. 2016107), and Strategic Priority Research Program of Chinese Academy of Science (Grant No. XDB32050200).



Proof of the conversion theory

We aim to derive the relation between the firing rate $r_i^l(T)$ of the IF neuron in spiking CNN and the real-value activation $a_i^l$ of the corresponding neuron in CNN. Starting from the total residual shown in 5, the firing rate of neuron $i$ in layer $l$ over time $T{\rm~d}t$ can be rewritten with 5, then Eq. 4 is transferred into \begin{equation} r_i^l(T)=\frac{N_i^l(T)}{T} r_{\rm max}=\frac{\sum_{t=1}^T Z_i^l(t)-\varepsilon_i^l(T)}{\tau^l T} r_{\rm max}, \tag{18}\end{equation} where $\sum_{t=1}^T~Z_i^l(t)$ is the total weighted inputs received over time $T{\rm~d}t$. Because the accumulation of weighted input spikes in time and spatial scale is not correlative, the accumulation of input spike over $T{\rm~d}t$ can be calculate first. So $\sum_{t=1}^T~Z_i^l(t)$ is transferred into the sum up of weighted total number of spikes emitted by presynaptic neurons: \begin{equation} \sum_{t=1}^T Z_i^l(t)=\sum_{t=1}^T \sum_{j=1}^{J^{l-1}} W_{ij}^l\sum_{s\in S_j}\delta_j^{l-1}(t-s) =\sum_{j=1}^{J^{l-1}} W_{ij}^l \sum_{t=1}^T \sum_{s\in S_j}\delta_j^{l-1}(t-s) =\sum_{j=1}^{J^{l-1}}W_{ij}^l N_j^{l-1}(T). \tag{19}\end{equation} Then, according to 4, the total number of spikes $N_j^{l-1}(T)$ can be rewritten into the form containing firing rate, like \begin{equation} N_j^{l-1}(T)=T \frac{r_j^{l-1}(T)}{r_{\rm max}}. \tag{20}\end{equation} Substituting 20 into 19, $\sum_{t=1}^T~Z_i^l(t)$ are ultimately denoted as an equation containing the firing rate of the neurons in previous layer: \begin{equation} \sum_{t=1}^T Z_i^l(t)=\frac{T}{r_{\rm max}}\sum_{j=1}^{J^{l-1}}W_{ij}^l r_j^{l-1}(T). \tag{21}\end{equation} Replacing the $\sum_{t=1}^T~Z_i^l(t)$ in 18 with 21 and rearranging, the relationship about firing rate of IF neurons in two connected layers can be gotten: \begin{equation} r_i^l(T)=\frac{1}{\tau^l} \sum_{j=1}^{J^{l-1}} W_{ij}^l r_j^{l-1}(T)-\frac{\varepsilon_i^l(T)}{\tau^l T}r_{\rm max}. \tag{22}\end{equation} Eq. 22 is a recursive expression and it can be unfolded by inserting the firing rate of neurons in previous layers iteratively until the known firing rate of input spike trains $r_i^0$: \begin{equation} r_i^l(T)=\frac{1}{\tau^l} \sum_{j=1}^{J^{l-1}}W_{ij}^l \left ( \frac{1}{\tau^{l-1}} \sum_{j=1}^{J^{l-2}} W_{ij}^{l-1} \cdots\left( \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 r_j^0(T)-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max} \right)\cdots -\frac{\varepsilon_i^{l-1}(T)}{\tau^{l-1}T}r_{\rm max} \right ) -\frac{\varepsilon_i^l(T)}{\tau^lT}r_{\rm max}. \tag{23}\end{equation} According to 22, the term within the innermost brackets of 23 is the firing rate of $i$th IF neuron in layer $l=1$, which is solved by substituting 7 into 23: \begin{equation} \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 r_j^0(T)-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max} = r_i^1(T) = \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 a_j^0 r_{\rm max}-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max}. \tag{24}\end{equation}

Case 1: If $\sum_{j=1}^{J^0}~W_{ij}^1~a_j^0~\geqslant~0$, the term in the right of equation (24) can be rewritten as the form that contains ReLU function, and according to (2), it transfer into \begin{equation} r_i^1(T) = \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 a_j^0 r_{\rm max}-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max} = \frac{r_{\rm max}}{\tau^1}\left( {\rm max}\left(0, \sum_{j=1}^{J^0} W_{ij}^1 a_j^0\right)-\frac{\varepsilon_i^1(T)}{T} \right) =\frac{r_{\rm max}}{\tau^1}\left(a_i^1 - \frac{\varepsilon_i^1(T)}{T}\right). \tag{25}\end{equation}

Case 2: If $\sum_{j=1}^{J^0}~W_{ij}^1~a_j^0~<~0$, then ${\rm~max}(0,~\sum_{j=1}^{J^0}~W_{ij}^1~a_j^0)=0$. And Eq. (24) can be written as \begin{equation} r_i^1(T) = \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 a_j^0 r_{\rm max}-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max} = \frac{r_{\rm max}}{\tau^1} \left( \sum_{j=1}^{J^0} W_{ij}^1 a_j^0(T)-\frac{\varepsilon_{ia}^1(T)}{T} \right) - \frac{\varepsilon_{ib}^1(T)}{\tau^1 T}r_{\rm max}, \tag{26}\end{equation} where $\sum_{j=1}^{J^0}~W_{ij}^1~a_j^0(T)-\varepsilon_{ia}^1(T)/T=0$ and $\varepsilon_i^1(T)=\varepsilon_{ia}^1(T)+\varepsilon_{ib}^1(T)$. Therefore, Eq. (26) can be rewritten as the form which also contains ReLU function, and according to (2), it transfer into \begin{equation} r_i^1(T) = \frac{1}{\tau^1} \sum_{j=1}^{J^0} W_{ij}^1 a_j^0 r_{\rm max}-\frac{\varepsilon_i^1(T)}{\tau^1T}r_{\rm max} = \frac{r_{\rm max}}{\tau^1}\left( {\rm max}\left(0, \sum_{j=1}^{J^0} W_{ij}^1 a_j^0\right)-\frac{\varepsilon_{ib}^1(T)}{T} \right) =\frac{r_{\rm max}}{\tau^1}\left( a_i^1 - \frac{\varepsilon_{ib}^1(T)}{T} \right). \tag{27}\end{equation} According to (24) and (26), $\varepsilon_{ia}^1$ can be solved, and based on (4) and (19), it can be further rewritten as \begin{equation}\varepsilon_{ia}^1 = \frac{T}{r_{\rm max}}\sum_{j=1}^{J^0} W_{ij}^1 r_j^0(T) = \frac{T}{r_{\rm max}}\sum_{j=1}^{J^0} W_{ij}^1 \frac{N_j^0(T)}{T{\rm d}t}, \tag{28}\end{equation} and $\varepsilon_{ia}^1$ equals \begin{equation}\varepsilon_{ia}^1=\sum_{t=1}^T Z_i^1(t). \tag{29}\end{equation} For $\varepsilon_{ib}^1$, it acts like the total residual membrane potential. When $\sum_{j=1}^{J^0}~W_{ij}^1~a_j^0~<~0$, the result of input pixels' value $a_j^0$ convoluted by $W_{ij}^1$ after ReLU function becomes the activation $a_i^1$ of $i$th neuron in first hidden layer, which equals to zero. So, the firing rate $r_i^1$ of corresponding IF neuron in spiking CNN should be equal to zero too. In this case, according to (27), $\varepsilon_{ib}$ is the part of membrane potential used to generate spikes, thereby causing approximation error: \begin{equation}r_i^1(T)=\frac{N_i^1(T)}{Tdt}=- \frac{\varepsilon_{ib}^1(T)}{\tau^1T}r_{\rm max}, \tag{30}\end{equation} and $\varepsilon_{ib}^1$ equals \begin{equation}\varepsilon_{ib}^1(T) = -N_i^1(T)\tau^1. \tag{31}\end{equation} Comparing the results of (25) and (27), we found that they are in similar form that the activation of corresponding neuron in CNN along with an additional error. When putting the result of (25) or (27) into (23) during each iteration, the general form of the innermost term is listed in the following, which is similar with the cases we discuss above when $l=1$. So the general result for arbitrary layer $l$ is \begin{eqnarray}&&{r_i^l(T) = \frac{1}{\tau^l} \sum_{j=1}^{J^{l-1}} W_{ij}^l a_j^{l-1}(T)r_{\rm max}-\frac{\varepsilon_i^l(T)}{\tau^lT}r_{\rm max} =} \frac{r_{\rm max}}{\tau^l}\left( a_i^l - \frac{\varepsilon_i^l(T)}{T} \right), & if $\sum_{j=1}^{J^{l-1}} W_{ij}^l a_j^{l-1} \geqslant 0$, \tag{32} \\ && \frac{r_{\rm max}}{\tau^l}\left( a_i^l - \frac{\varepsilon_{ib}^l(T)}{T}\right ),& if $\sum_{j=1}^{J^{l-1}} W_{ij}^l a_j^{l-1} < 0$, \tag{33} \end{eqnarray} where $\varepsilon_{ib}^l=-N_i^l(T)\tau^l$. Solving the recursive expression by inserting (32) and (33) iteratively, we finally obtain a general description of the relationship between firing rate $r_i^l(T)$ and real-value activation $a_i^l$: \begin{equation} r_i^l(T)=\frac{ a_i^l}{\tau^l\tau^{l-1}\cdots\tau^1}r_{\rm max} - \Delta\varepsilon_i^l, \tag{34}\end{equation} where $\Delta\varepsilon_i^l$ is the approximation error.

Case 1: When $\sum_{j=1}^{J^{l-1}}~W_{ij}^l~a_j^{l-1}~\geqslant~0$ for all layers $l=\{1,2,\ldots,L\}$, approximation error equals to \begin{equation} \Delta\varepsilon_i^l = \frac{r_{\rm max}}{T}\bigg( \frac{\varepsilon_i^l(T)}{\tau^l} + \frac{1}{\tau^l \tau^{l-1}}\sum_{j=1}^{J^{l-1}} W_{ij}^l \varepsilon_j^{l-1}(T)+\cdots+\frac{1}{\tau^l \tau^{l-1}..\tau^1}\sum_{j=1}^{J^{l-1}}W_{ij}^l\cdots\sum_{j=1}^{J^1}W_{ij}^2 \varepsilon_j^1(T)\bigg). \tag{35}\end{equation}

Case 2: When $\sum_{j=1}^{J^{l-1}}~W_{ij}^l~a_j^{l-1}~<~0$ for some layers $l$, approximation error equals to \begin{equation} \Delta\varepsilon_i^l = \frac{r_{\rm max}}{T}\bigg( \frac{\varepsilon_i^l(T)}{\tau^l} + \frac{1}{\tau^l \tau^{l-1}}\sum_{j=1}^{J^{l-1}} W_{ij}^l \varepsilon_{jb}^{l-1}(T)+\cdots+\frac{1}{\tau^l \tau^{l-1}..\tau^1}\sum_{j=1}^{J^{l-1}}W_{ij}^l\cdots\sum_{j=1}^{J^1}W_{ij}^2 \varepsilon_{jb}^1(T)\bigg). \tag{36}\end{equation} Though there are some $\varepsilon_{jb}^l(T)$ existed in (36), the result of two above cases are in same form. So, we adopt the result shown in (35) as the approximation error $\Delta\varepsilon_i^l$ with the total residual membrane potential $\varepsilon_i^l(T)$ of each IF neuron within $T{\rm~d}t$ equaling: \begin{eqnarray}&&{\varepsilon_i^l(T)=} \sum_{s\in S_i}(V_i^l(s)-\tau^l)+\sum_{u\in U_i}(V_i^l(u)-V_{\rm min}^l)+V_i^l(T), \tag{37} \\ &&-N_i^l(T)\tau^l. \tag{38} \end{eqnarray} And the final relation between the firing rate $r_i^l(T)$ of the IF neuron in spiking CNN and the real-value activation $a_i^l$ of the corresponding neuron in CNN is described as (8) and (9).

Pseudocode of conversion rule


Conversion rules. $L$ is the number of layers. ${\rm~conv}(W,a)$ is a function which represents that activation map $a$ is convoluted by weight matrix $W$. ${\rm~avg\_pool}(a)$ is a function which represents that activation map $a$ is pooling with the average value. ${\rm~layer}(l).{\rm~type}$ indicates the type of layer $l$. KwDataWeight matrix $W_{ij}^l$ from a well-trained CNN with biases fixed to 0. Normalized training set as input $a^0$. KwResultObtain parameters $f^l$, scaled weights $W_{ij}^{l'}$ and $V_{\rm~min}$. $f^0=1$; $l=1~{\rm~to}~L$


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  • Table 1   Comparison between CNN and spiking CNN
    CNN Spiking CNN
    5*makecellNetworkarchitectureNumber of convolution layers$n$ $n$
    Number of pooling layers$n$ $n$
    Additional layers$\times$ Spike counter
    InputAn image Series of binary map
    OutputReal-value Num of spikes
    5*makecellNeuronmodel Weights$W_{ij}$ $W&apos;_{ij}$
    Firing threshold ($\tau$)$\times$ $\circ$
    $V_{\rm~min}$$\times$ $\circ$
    Bias$\circ$ $\times$
    Notion of time$\times$ $\circ$
    4*makecellProcessingflow Information domainReal-value Spike (rate coding)
    3*makecell Operation Convolution
    Average pooling
    Activation ReLU Firing operation
  • Table 2   Architecture of CNNs for MNIST, SVHN and CIFAR-10 dataset
    Input $28\times28$ gray image Input $32\times32$ RGB image Input $24\times24$ RGB image
    conv5-12, ReLU conv5-32, ReLU conv5-64, ReLU
    avg-pool/2 avg-pool/2 avg-pool/2
    conv5-64, ReLU conv5-96, ReLU conv5-64, ReLU
    avg-pool/2 avg-pool/2 avg-pool/2
    3*FC-10 conv3-64, ReLU conv3-64, ReLU
    avg-pool/3 conv1-64, ReLU
  • Table 3   Classification accuracy of different types of SNNs on three datasets$^{\rm~a)}$
    Dataset Network-type Method Accuracy
    MNIST[38] SNN Balanced learning + STDP 98.64%
    MNIST[39] SNN NormAD 98.17%
    MNIST[31] Converted spiking CNN Modified IF neuron 99.44% ($\approx$0%)
    MNIST[29] Converted spiking CNN Weight Normalization 99.11% (0.04%)
    MNIST Converted spiking CNN (this paper) Conversion rule 99.09% (0.02%)
    SVHN[33] Converted spiking CNN Threshold rescaling 93.66% (2.27%)
    SVHN[30] Spiking CNN Synaptic filter 93.92% (0.43%)
    SVHN Converted spiking CNN (this paper) Conversion rule 96.33% (0.27%)
    CIFAR-10[28] Spiking CNN None 77.43% (1.66%)
    CIFAR-10[30] Spiking CNN Synaptic filter 83.54% (2.43%)
    CIFAR-10 Converted spiking CNN (this paper) Conversion rule 80.41% (0.40%)


  • Table 4   Stable accuracy and approximation error $\Delta~\varepsilon$ of spiking CNNs with different $n$-values on MNIST, SVHN, and CIFAR-10 dataset, respectively
    Accuracy (%)$\Delta~\varepsilon$ Accuracy (%) $\Delta~\varepsilon$ Accuracy (%) $\Delta~\varepsilon$
    0.1 98.75 0.582 95.13 0.100 76.10 0.278
    0.2 99.06 0.138 96.04 0.035 79.42 0.079
    0.3 99.12 0.066 96.10 0.021 79.82 0.040
    0.4 99.12 0.042 96.19 0.015 80.08 0.026
    0.5 99.13 0.031 96.30 0.012 80.14 0.019
    0.6 99.10 0.024 96.33 0.010 80.38 0.014
    0.7 99.13 0.020 96.30 0.009 80.46 0.012
    0.8 99.10 0.017 96.35 0.007 80.53 0.010
    0.9 99.15 0.015 96.43 0.007 80.55 0.008
    1.0 99.13 0.013 96.31 0.006 80.29 0.007
  • Table 5   Accuracy loss on three datasets when images are distorted by different degrees of AWGN and SP noise
    Additive white Gaussian noise (SNR)Salt-and-pepper noise (density)
    $\Delta~$Acc30 25 20 15 10 5 3% 6% 9% 12% 15%
    MNIST (%) 0.03 0.07 0.04 0.02 0.10 0.31 0.10 0.31 0.44 0.68 1.14
    SVHN (%) 0.37 0.35 0.39 0.37 0.49 0.80 0.41 0.47 0.63 0.76 0.89
    CIFAR-10 (%) 0.61 0.39 $-$0.52 $-$1.63 $-$1.46 $-$0.79 0.18 0.37 0.35 0.36 0.19