# Spiking neural networks trained via proxy

Saeed Reza Kheradpisheh1,111Corresponding Author
Email addresses:
<EMAIL_ADDRESS>(FA),
<EMAIL_ADDRESS>(SA),
<EMAIL_ADDRESS>(TA) 1 Department of Computer Science, Faculty of
Mathematical Sciences, Shahid Beheshti University, Tehran, Iran Maryam
Mirsadeghi2 1 Department of Computer Science, Faculty of Mathematical
Sciences, Shahid Beheshti University, Tehran, Iran Timothée Masquelier3 1
Department of Computer Science, Faculty of Mathematical Sciences, Shahid
Beheshti University, Tehran, Iran

###### Abstract

We propose a new learning algorithm to train spiking neural networks (SNN)
using conventional artificial neural networks (ANN) as proxy. We couple two
SNN and ANN networks, respectively, made of integrate-and-fire (IF) and ReLU
neurons with the same network architectures and shared synaptic weights. The
forward passes of the two networks are totally independent. By assuming IF
neuron with rate-coding as an approximation of ReLU, we backpropagate the
error of the SNN in the proxy ANN to update the shared weights, simply by
replacing the ANN final output with that of the SNN. We applied the proposed
proxy learning to deep convolutional SNNs and evaluated it on two benchmarked
datasets of Fahion-MNIST and Cifar10 with 94.56% and 93.11% classification
accuracy, respectively. The proposed networks could outperform other deep SNNs
trained with tandem learning, surrogate gradient learning, or converted from
deep ANNs. Converted SNNs require long simulation times to reach reasonable
accuracies while our proxy learning leads to efficient SNNs with much shorter
simulation times.

##  1 Introduction

Spiking neural networks (SNNs) are known to be suitable for hardware
implementation specially on neuromorphic devices [1, 2]. However, the temporal
dynamics of SNNs both in neuron and network levels along with the non-
differentiabilty of spike functions have made it difficult to train efficient
SNNs [3]. Different studies with different approaches have tried to adapt
backpropagation based supervised learning algorithms to SNNs [4].

The first approach is to train an artificial neural network (ANN) and then
convert it to an equivalent SNN [5, 6, 7, 8, 9]. Although converted SNNs could
be applied to deep architectures and reached reasonable accuracies, they
totally neglect the temporal nature of SNNs and usually are inefficient in
terms of the number of spikes/time-steps. The second approach is to directly
apply backpropagation on SNNs. Their main challenge is to overcome the non-
differentiblity of spike functions required in error-backpropagation
algorithm. To solve this problem, some studies propose to use smoothed spike
functions with true gradients [10] and others use surrogate gradients for non-
differentible discrete spike functions [11, 12, 13, 14, 15, 16, 17, 18].The
main issue with this approach is the use of backpropagation through time which
makes it too costly and faces it with vanishing/exploding gradient problem,
especially for longer simulation times.

In the third approach, known as latency learning, the idea is to define the
neuron activity as a function of its firing time [19, 20, 21, 22, 23, 24, 25].
In other words, neurons fire at most once and stronger outputs correspond to
shorter spike delays. To apply backpropagation to such SNNs, one should define
the neuron firing time as a differentiable function of its membrane potential
or the firing times of its afferents [26, 19]. As an advantage, latency
learning does not need backpropagation through time, however, it is more
difficult to train outperforming SNNs with latency learning.

The forth approach is the tandem learning which is consists of an SNN and an
ANN coupled layer-wise through weight sharing [27, 28]. The auxiliary ANN is
used to facilitate the error backpropagation for the training of the SNN at
the spike-train level, while the SNN is used to derive the exact spiking
neural representation. Literally, in the forward pass, each ANN layer receives
its input as the spike counts of the previous SNN layer, and consequently, in
the backward pass, each ANN layer computes the gradient of its output with
respect to the shared weights based on these input spike counts. Regarding
this layer-wised coupling, the learning can be also done in a progressive
layer by layer manner.

In this paper, we propose a new learning method to train an SNN via a proxy
ANN. To do so, we make an ANN (consists of ReLU neurons) structurally
equivalent to the SNN (made of integrate-and-fire (IF) neurons). The two
network share their synaptic weights, however, IF neurons in the SNN work with
temporally distributed spikes, while, neurons in ANN work with floating points
process their input instantly. Contrary to tandem learning, the forward pass
of the two networks are totally independent with no interference. By
considering IF with rate coding as an approximation of ReLU, we replace the
final output of the SNN into that of the ANN, and therefore, we update the
shared weights by backpropagating the SNN error in the ANN model. One of the
main challenges in conversion and tendem learning methods is the approximation
of the ANN max-pooling layers in the corresponding SNN, hence, they exclude
pooling layers and use convolutional layers with stride of 2. Here, we used
spike-pooling layers to mimic the behaviour of max-pooling layers in the
corresponding ANN. We evaluated the proposed proxy learning on Cifar10 and
Fashion-MNIST dataset with different deep convolutional architectures and
outperformed the currently existing conversion and tandem learning methods.

##  2 Spiking neural networks

###  2.1 Spiking neuron

Artificial neurons are simply implemented by a linear combination of inputs
followed by a non-linear activation function. Different activation functions
have different mathematical properties and choosing the right function can
largely impact the learning speed and efficiency of the whole network. The
revolution of deep learning was accompanied by the use of Rectified Linear
Unit (ReLU) instead of prior activation functions such as sigmoid and
hyperbolic tangent. An artificial neuron $j$ with ReLU activation can be
formulated as follows,

$z_{j}=\sum_{i}w_{ji}x_{i},$ (1) $y_{j}=ReLU(z)=max(0,z),$ (2)

where $x$ and $w$ are respectively the input and weight vectors, and $y_{j}$
is the neuron output.

Contrary to the artificial neuron models which work with synchronous instant
inputs, spiking neurons have a temporal dynamics through which the neuron’s
internal membrane potential changes in time by receiving asynchronous incoming
spikes. The complexity of this neural dynamics can largely impact the
computational cost and learning efficiency of SNNs. Integrate-and-fire (IF) is
the simplest spiking neuron model in which the membrane potential only changes
when receiving an input spike from a presynaptic neuron by an amount
proportional to the synaptic weight.

The membrane potential $u_{j}$ of an IF neuron $j$ is updated at each time
step $t$ by the input current $I_{j}(t)$ caused by the spike train $s_{i}(t)$
received from each presynaptice neuron $i$ through a synapse with the weight
$w_{ji}$,

$u_{j}(t)=u_{j}(t-1)+RI_{j}(t),$ (3) $I_{j}(t)=\sum_{i}w_{ji}s_{i}(t),$ (4)

where $s_{i}(t)=1$ if presynaptic neuron $i$ has fired at time $t$ and it is
zero elsewhere. We set the membrane resistance $R$ to unitary (i. e., $R=1$).

The IF neuron emits an output spike whenever its membrane potential crosses a
certain threshold $\theta$,

$s_{j}(t)=\begin{cases}1&\quad\text{if }V_{j}(t)\geq\theta,\\\
0&\quad\mathrm{otherwise},\end{cases}$ (5)

and then resets its membrane potential to zero as $u_{j}=0$ to be ready for
the forthcoming input spikes.

###  2.2 Information encoding

ANNs work with decimal values, therefore, their inputs are represented by
vectors, matrices, and tensors of floating-point numbers. However, in SNNs,
neurons talk through spikes, hence, information needs to be somehow encoded in
the spike trains. In other words, the analog input signal should be converted
into an equivalent spike train in the entry layer of the network. Different
coding schemes are suggested to be used in SNNs ranging from heavy rate codes
to economical temporal codes with single spikes.

Another spike-free input coding approach is to use constant input currents
(aka direct input coding) applied to input neurons. This way, input neurons
with higher input currents will fire with higher rates than others. In other
words, contrary to the neurons in the middle and output layers, the input
current to the IF neurons in the first layer is proportional to the input
signal. Consider an input image $x$, the constant input current to an input IF
neuron $j$ is computed as

$I_{j}(t)=\sum_{i}w_{ji}x_{i},$ (6)

where $x_{i}$ is the $i^{th}$ input pixels inside the receptive field of
neuron $j$ and $w_{ji}x_{i}$ is the constant input current from $x_{i}$ to
$j$.

###  2.3 Convolutional SNN

Convolutional ANNs (CANNs), largely inspired by the hierarchical object
recognition process in visual cortex, are comprised of a cascade of
interlaying convolution and pooling layers to extract descriptive visual
features followed by a stack of fully connected layers to make the final
decision. Neural processing in CANNs are performed in a layer by layer
fashion, neurons in each layer receive their whole input from the previous
layer at once, and instantly send their output to the next layer. The neural
processing in convolutional SNNs (CSNNs) is different as neurons work in
temporal domain and information is encoded in asynchronous spike trains.

Each convolutional layer of a CSNN is made of IF neurons organized in numerous
feature maps. At every time step, each neuron receives the total input current
from the afferents inside its receptive window (Eq. 6 for input layer and Eq.
4 for other layers), updates its membrane potential (Eq. 3) and fires whenever
reaches to the threshold (Eq. 5). Neurons in the same feature map share their
synaptic weights, and hence, look for the same feature but in different
locations.

At each time step, neurons in spike-pooling layers of the CSNN simply emit an
spike if there is at least one spike in their input window. Hence, if
different neurons inside the receptive window of a spike-pooling neuron fire
at several different times, the spike-pooling neuron will also fire at each of
those time steps. Spike-pooling neurons can simply be implemented by IF
neurons with synaptic weights and threshold of one.

Fully-connected layers of the CSNN, including the readout layer, are all made
of IF neurons. The last convolution or spike-pooling layer is flattened and
its spikes are fed to the first fully-connected layer. The readout layer
includes one neuron for each class, and the neuron with the maximum number of
spikes determines the winner class.

##  3 IF approximating ReLU

Although the input encoding, internal mechanism, and output decoding of IF are
different from those of ReLU neuron model, several studies from different
perspectives have shown that an equivalent IF (or Leaky-IF) neuron can fairly
approximate the activation of the ReLU neuron. For time-to-first-spike coding,
the firing time of IF neuron is inversely proportional to the output of ReLU
[19, 5]. In short, the IF neuron remains silent if the output of ReLU is zero,
and it will fire with shorter delay for larger ReLU outputs. For rate coding,
the higher ReLU outputs correspond to higher firing rates in IF neuron [27,
28, 29].

Here, we consider that neural information is encoded in the spike rate of
neurons. Let $r_{i}$ be the input spike rate received from the $i^{th}$
afferent to neuron $j$,

$r_{i}=\frac{\sum_{t}s_{i}(t)}{T},$ (7)

where $T$ is the maximum simulation time. For simplicity, we assume that these
input spikes are uniformly distributed in time and the sudden current caused
by each spike is evenly and continuously delivered during the inter-spike
intervals, then according to Eq. 4, the input current to neuron $j$ is
constant in time and is calculated as,

$I_{j}=\sum_{i}w_{ji}r_{i}.$ (8)

If we apply this constant input current to IF neuron in Eq. 3 with the
thresholding function in Eq. 5, the firing rate of IF neuron $j$ can be
calculated as,

$r_{j}=ReLU(\frac{RI_{j}}{\theta})=\frac{R}{\theta}\>ReLU(\sum_{i}w_{ji}r_{i}).$
(9)

The use of ReLU is necessary here, since the IF neuron will not fire at all
for negative input currents. As shown in Eq. 9, the firing rate of the IF
neuron $j$ can be expressed by applying ReLU on the weighted summation of the
input firing rates from its afferents.

##  4 Training via proxy

The general idea of the proposed learning method is illustrated in Figure 1.
Here we have a proxy CANN coupled with an equivalent CSNN having same
architectures with $L$ layers. The weights in all layers are shared between
the two networks. However, the CANN model is made of artificial neurons with
ReLU activation and the CSNN model is made of IF neurons and works in the
temporal domain. The input image is fed to both model and their outputs are
obtained at their softmax layer. Since the thresholding function of IF neurons
is not differentiable, it can not be directly trained. We discard the output
of CANN (the proxy network) and replace it with the CSNN output, then, we
backpropagate the CSNN error in CANN to update the the weights.

###  4.1 Forward pass

During the forward pass the input image is fed to both networks. The first
layer of CANN simply convolves its filter kernels over the input image and
sends it output, $y^{1}$ to the next layer by applying the ReLU activation
function. The following pooling and convolutional layers apply the max-pooling
and convolution operations on their inputs from the previous layer and send
their output to the next layer. The fully-connected layers on top receive
inputs from all neurons in their previous layer through synaptic weights and
send their output to the next layer. Note that we use ReLU activation function
in all convolutional and fully-connected layers. At the end, CANN applies
softmax activation function over the output of the last layer, $y^{L}$, to
obtain the final output, $O^{A}$,

$O_{k}^{A}=\frac{e^{y_{k}^{L}}}{\sum_{j}e^{y_{j}^{L}}}.$ (10)

The first layer of the CSNN model obeys the input encoding scheme explained in
Section 2.2. The CSNN process each input image in $T$ simulation time step. At
every time step, the constant input current to IF neurons in the first layer
of the CSNN model is computed by convolving the input image with the
corresponding filter kernel (see Eq. 6). These IF neurons will emit spikes
whenever their membrane potential crosses the threshold and send spike train
$s^{1}$ to the next layer. As explained in Section 2.3, in any time step,
spike-pooling neurons will fire a spike if there is at least one spike in
their receptive window. Spiking convolutional IF neurons integrate incoming
spikes inside their receptive window through weighted synapses and fire when
the threshold is reached. It is the same for IF neurons in fully-connected
layers but they do not have restricted receptive window and receive spikes
from all neurons in the previous layer. To compute the output of CSNN, we
count the number of spikes for each neuron in the output layer, $C_{k}^{L}$,
and send it to a softmax layer to obtain final output, $O^{S}$,

$c_{k}^{L}=\sum_{t}s_{k}^{L}(t),$ (11)
$O_{k}^{S}=\frac{e^{c_{k}^{L}}}{\sum_{j}e^{c_{j}^{L}}}.$ (12)

Figure 1: The proposed proxy learning that is comprised of a CSNN coupled with
a CANN through the shared weights. The $s^{l}$ and $y^{l}$ are respectively
the spike train and the output of the $l^{th}$ CSNN and CANN layers during the
forward pass. The final output of the CANN is replaced with the output of the
CSNN model. During the Backward pass, the error of the CSNN is backpropagated
through the CANN network to update the shared weights.

###  4.2 Backward pass

As mentioned earlier, to update the weights of the CSNN model we use the
corresponding shared weights in CANN. To do so, the softmax output of the CANN
model is replaced by the softmax output of the CSNN model. By comparing it to
the target values in the loss function of the CANN model, we are literally
computing the error and the loss of the CSNN model. This loss is then
backpropagated through the CANN model to update the shared weights. As
explained in Section 3, we assume that the input and output of the ReLU neuron
in the CANN model is approximated by the input and output firing rates of the
corresponding IF neuron in the CSNN model

Let assume that $Y_{k}$ is the target of the $k^{th}$ output in the CANN
model. The cross-entropy loss function for the CANN model is defined as

$L=-\sum_{k}Y_{k}ln(O^{A}_{k})\simeq-\sum_{k}Y_{k}ln(O^{S}_{k}).$ (13)

To update an arbitrary shared weight $w^{l}_{ji}$ in the $l^{th}$ layer
through the CANN model, we have

$\Delta w^{l}_{ji}=w^{l}_{ji}-\eta\frac{\partial L}{\partial w^{l}_{ji}}$ (14)

where $\eta$ is the learning rate. Instead of using the true gradient of the
CANN model which is computed as

$\frac{\partial L}{\partial w^{l}_{ji}}=\sum_{k}\frac{\partial L}{\partial
O^{A}_{k}}\sum_{d}\frac{\partial O^{A}_{k}}{\partial y_{d}^{L}}\frac{\partial
y_{d}^{L}}{\partial w^{l}_{ij}},$ (15)

we use the following approximated gradient which is obtained by replacing the
output of the CSNN into the output of the CANN model,

$\frac{\partial L}{\partial w^{l}_{ji}}\simeq\sum_{k}\frac{\partial
L}{\partial O^{S}_{k}}\sum_{d}\frac{\partial O^{S}_{k}}{\partial
y_{d}^{L}}\frac{\partial y_{d}^{L}}{\partial w^{l}_{ij}}.$ (16)

Indeed, we are backpropagating the error of the CSNN model in the CANN model
to update the shared weights.

Table 1: Network architecture and parameter setting for Fashion-MNIST and
Cifar10 datasets.

Dataset | Architecture | $\theta$ | $T$ | $\eta$ | $\beta_{1}$ | $\beta_{2}$ | $\epsilon$ | $\lambda$
---|---|---|---|---|---|---|---|---
Fashion | 128C3-128C3-P2-128C3-P2-1024F-256F-10F | 2 | 50 | $10^{-3}$ | 0.8 | 0.99 | $10^{-7}$ | $10^{-5}$
Cifar10 | 256C3-512C3-P2-512C3-512C3-512C3-P2-256C3-1024F-512F-256F-10F | 3 | 60 | $10^{-3}$ | 0.8 | 0.99 | $10^{-7}$ | $10^{-5}$

Table 2: Classification accuracies of different CSNNs with different learning rules on Fashion-MNIST. $T$ is the simulation time. The STDBP and STiDi-BP terms stand for spike-time-dependent error backpropagation and spike-time-displacement-based error backpropagation, respectively. Model | Acc (%) | Network | $T$ | Learning
---|---|---|---|---
Cheng et al. (2020) [30] | 92.07 | 4-layer CSNN | 20 | Surrogate Gradient Learning
Fang et al. (2021) [31] | 94.36 | 6-layer CSNN | 8 | Surrogate Gradient Learning
Yu et al. (2021) [32] | 92.11 | 4-layer CSNN | 220 | ANN-SNN Conversion
W. Zhang et al. (2020) [22] | 92.83 | 4-layer CSNN | 5 | Spike Sequence Learning
M. Zhang et al. (2020) [33] | 90.1 | 5-layer CSNN | - | STDBP
Mirsadeghi et al. (2021) [34] | 92.8 | 4-layer CSNN | 100 | STiDi-BP
This work (CANN) | 94.60 | 6-layer CANN | - | Backpropagation
This work (CSNN) | 84.63 | 6-layer CSNN | 50 | ANN-to-SNN Conversion
This work (CSNN) | 93.12 | 6-layer CSNN | 100 | ANN-to-SNN Conversion
This work (CSNN) | 94.50 | 6-layer CSNN | 200 | ANN-to-SNN Conversion
This work (CSNN) | 94.41 | 6-layer CSNN | 50 | Surrogate Gradient Learning
This work (CSNN) | 94.56 | 6-layer CSNN | 50 | Proxy Learning

##  5 Results

To evaluate the proposed proxy learning method, we performed experiments on
two benchmark datasets of Fashion-MNIST and Cifar10. For both datasets, we
used deep convolutional networks with architectural details and parameter
settings provided in Table 1. We use Adam optimizer with parameters
$\beta_{1}$, $\beta_{2}$, and $\epsilon$ and L2-norm regularization with
parameter $\lambda$. Our proxy learning outperforms conversion and tandem
learning methods by reaching 94.5% and 92.50% on Fashion-MNIST and Cifar10
datasets, respectively. In the following subsections, results on both datasets
are provided in more details.

###  5.1 Fashion-MNIST

Fashion-MNIST [35] is a fashion product image dataset with 10 classes
(T-shirt, Trouser, Pullover, Dress, Coat, Sandal, Shirt, Sneaker, Bag, and
Ankle boot). Images are gathered from the thumbnails of the clothing products
on an online shopping website. The Fashion-MNIST dataset contains 60,000
images of size $28\times 28$ pixels as the train set. The test set contains
10,000 images ($1000$ images per class). As presented in Table 1, the proposed
network is comprised of three convolutional, two pooling, and three fully
connected layer.

Table 2 provides the categorization accuracy of the proposed network with
proxy learning along with the accuracy of other recent spiking neural networks
on Fashion-MNIST dataset. Using a 6-layer CSNN architecture our proxy learning
method could reach 94.56% with $T=50$ accuracy and outperform other CSNNs with
different learning methods. Interestingly, our network with proxy learning
could surpass other networks trained with surrogate gradient learning (SGL).
To do a fair comparison, we trained the same CSNN as ours using surrogate
gradient learning method (with arc-tangent surrogate function [31]) that
reached to the best accuracy of 94.41% with $T=50$. We also trained a CANN
with the same architecture to our CSNN using backpropagation that reached to
94.60% accuracy best (it is 0.04% better than proxy learning). We also
converted this ANN to an SNN with IF neurons (using the conversion method in
[36]) and evaluated it on Fashion-MNIST with different simulation times. As
mentioned before, conversion methods require long simulation times to reach
acceptable accuracies. With 50 time steps the converted CSNN could only reach
to 84.63% accuracy and it required more 200 time steps for 94.50% accuracy.

Figure 2: The categorization accuracy and MSSE of the proposed CSNN with proxy
learning and simulation time of $T=50$ over testing set of Fashion-MNIST.
Figure 3: Classification accuracy of the proposed CSNN trained via proxy
learning on Fashion-MNIST with maximum simulation time varying from $T=10$ to
$T=60$. The accuracy increases by $T$ and reaches to 94.56 at $T=50$.

The categorization accuracy and the mean sum of squared error (MSSE) of the
proposed CSNN with the simulation time of $T=50$ over the test set of Fashion-
MNIST dataset is provided in Figure 2. Only 30 epochs are enough to reach an
accuracy above 94.0% and a MSSE lower than 1.5.

As explained in section 3, we assume that IF neurons in CSNN approximate ReLU
neurons in the proxy CANN. However, it is expected that this approximation
should get more accurate by increasing the maximum simulation time, $T$. To
verify this expectation, we evaluated the recognition accuracy of the CSNN
model (see Figure 3) trained with proxy learning on Fashion-MNIST dataset with
the simulation time varying from $T=10$ to $T=60$. As seen in Figure 3, the
recognition accuracy increases as $T$ is increased. The model reached to the
reasonable accuracy of 94.26% from $T=15$ and then ascends up to 94.56% with
$T=50$.

Figure 4: The output feature maps of five randomly selected filters in different convolutional layers of both CSNN and CANN networks over a randomly selected image from Fashion-MNIST. The feature maps of the CSNN network are obtained by computing the spike counts of IF neurons in each map. The firing rates of IF neurons can well approximate the activations of corresponding ReLU neurons. Table 3: Classification accuracies of different CSNNs with different learning rules on Cifar10. $T$ is the simulation time. The STDB and SGL terms stand for spike-time-dependent backpropagation and surrogate gradient learning, respectively. Model | Acc (%) | Network | $T$ | Learning
---|---|---|---|---
Y. Wu et al. (2019) [37] | 90.53 | 8-layer CSNN | 12 | Surrogate Gradient Learning
Syed et al. (2021) [38] | 91.58 | VGG-13 (CSNN) | 15 | Surrogate Gradient Learning
Fang et al. (2021) [31] | 93.50 | 9-layer CSNN | 8 | Surrogate Gradient Learning
Rueckauer et al. (2017) [39] | 90.85 | VGG-16 (CSNN) | 400 | ANN-to-SNN conversion
Rathi et al. (2020) [6] | 92.22 | Resnet-20 (SNN) | 250 | ANN-to-SNN conversion
Rathi et al. (2020) [6] | 91.13 | VGG-16 (CSNN) | 100 | ANN-to-SNN conversion + STDB
Sengupta et al. (2019) [7] | 91.46 | VGG-16 (CSNN) | 2500 | ANN-to-SNN conversion
Lee et al. (2019) [40] | 91.59 | ResNet-11 (CSNN) | 3000 | ANN-to-SNN conversion
Rathi et al. (2020) [41] | 92.70 | VGG-16 (CSNN) | 5 | ANN-to-SNN conversion + SGL
J. Wu et al. (2021) [27] | 90.98 | CifarNet (CSNN) | 8 | Tandem Learning
J. Wu et al. (2020) [28] | 91.24 | VGG-11 (CSNN) | 16 | Progressive Tandem Learning
This work (CANN) | 93.20 | 10-layer CANN | - | Backpropagation
This work (CSNN) | 89.14 | 10-layer CSNN | 60 | ANN-to-SNN conversion
This work (CSNN) | 92.91 | 10-layer CSNN | 120 | ANN-to-SNN conversion
This work (CSNN) | 93.16 | 10-layer CSNN | 240 | ANN-to-SNN conversion
This work (CSNN) | 92.85 | 10-layer CSNN | 60 | Surrogate Gradient Learning
This work (CSNN) | 93.11 | 10-layer CSNN | 60 | Proxy Learning

In Figure 4 we plotted the output feature maps of five randomly selected
convolutional filters in different layers of the CANN model and their
corresponding in the CSNN model. Subfigures 4A, 4B, and 4C respectively show
the selected feature maps in the first, second, and third convolutional
layers. In each subfigure, the top (bottom) row belongs to the feature maps of
the CANN (CSNN) model. The feature maps of the CSNN model are obtained by
computing the spike count (i.e., firing rate) of each IF neuron. As seen, the
activation of ReLU neurons in CANN layers is well approximated by the firing
rates of corresponding IF neurons in the CSNN model.

###  5.2 Cifar10

Cifar10 is a widely-used benchmark dataset in deep learning and suitable for
evaluating spiking neural network on natural image classification tasks.
Cifar10 consists color images from 10 different classes, with a standard split
of 50,000 and 10,000 for training and testing, respectively. To solve Cifiar10
classification task, we developed a 10-layer CSNN trained via proxy.
Architectural details of the proposed network are provided in Table 1.

The classification accuracy of the proposed network along with those of other
CSNNs trained by different learning strategies including surrogate gradient
learning, ANN-to-SNN conversion, and tandem learning are presented in Table 3.
Our proposed network could reach 93.11% categorization accuracy on Cifar10
with $T=60$ and outperform any other CSNN trained listed in Table 3, except
Fang et al. (2021) [31] that use surrogate gradient in CSNNs with Leaky-IF
neurons with trainable membrane time constants (i.e., each spiking neuron
layer has an independent and trainable membrane time constant). Although, they
reached 0.04% better accuracy than us, implementing large CSNNs with Leaky-IF
neurons having different time constants, independent of the implementation
platform, is highly expensive in terms of memory and computation. However, we
use simple IF neurons with no leak and no need for extra parameters, that is
easy to implement and has low memory and computational costs.

Interestingly, our proposed CSNN with proxy learning could significantly
outperform CSNNs with tandem learning rule [27, 28]. This might be due to
inconsistency between the forward and backward passes of tandem learning. In
our proxy learning method, only the final output of the CANN is replaced with
that of the CSNN, and hence, the forward pass of the two networks are totally
independent. However, in the forward pass of tandem learning, CANN layers are
disconnected from each other and receive the spike counts of the previous CSNN
layer as their input, while in the backward pass, the CSNN error
backpropagates through the CANN layers and based on their true outputs,
without the intervene of CSNN layers.

Our proposed CSNN has also outperformed CSNNs converted from CANNs with even
deeper architectures. In proxy learning, the final error is computed based on
the spike counts of the output layer of the CSNN, while in conversion methods,
the training phase is totally independent of the CSNN. This shows that being
aware of the quantized CSNN activations (the spike counts) in our proxy
learning, which are ignored by conversion methods, can lead to CSNNs with
higher classification accuracy.

Also, we developed a CANN with ReLU neurons and same architecture to our CSNN
and trained it using backpropagation with ADAM optimizer (the learning
parameters and conditions were the same as the CSNN model in Table 1). The
CANN reached the best accuracy of 93.20% that is only 0.09% better than our
CSNN with proxy learning. We then converted this ANN to an SNN with IF neurons
using the conversion method in [36]. With 60 time steps, that our proposed
CSNN had its best accuracy, the converted CSNN could only reach to 89.15%
accuracy and it required 240 time steps to reach 93.16% accuracy. Again,
conversion methods require long simulation times to reach reasonable
accuracies.

In the last comparison, we trained a CSNN with same architecture and learning
parameters to ours using surrogate gradient learning. As presented in Table 3,
it reached the classification accuracy of 92.85% at its best. Note that
contrary to other CSNNs in Table 3 that are trained with surrogate gradient
learning, ours is consists of pure IF neurons without leakage.

Figure 5: Classification accuracy of the proposed CSNN trained via proxy
learning on Cifar10 with maximum simulation time varying from $T=10$ to
$T=70$. The accuracy increases by $T$ and reaches to 93.11% at $T=60$.

In another experiment, we varied the maximum simulation time from $T=10$ to
$T=70$ and evaluated the classification accuracy of the proposed CSNN. As
depicted in Figure 5, there is a trade-off between the simulation time and the
classification accuracy. The classification accuracy starts from $88.35\%$
with $T=10$, steadily increases with $T$, and culminates at $93.11\%$ with
$T=60$.

##  6 Discussion

In this paper we proposed a proxy learning method for deep convolutional
spiking neural networks. The main idea is that IF neurons with rate coding can
approximate the activation of ReLU neurons. To do so, first, we build a proxy
CANN with ReLU neurons and the same architecture as the CSNN with IF neurons.
Then, we feed the input image to both networks and the forward pass is done in
both networks independently. Decision in CSNN network is made by applying a
softmax on the spike counts of the ouput neurons. Finally, the error of the
CSNN model is backpropagated in the CANN model by replacing its output with
that of the CSNN.

Our proxy learning method reached 94.56% on Fashion-MNIST and 93.11% on
Cifar10 datasets and outperformed other ANN-to-SNN conversion and tandem
learning methods (see Tables 2 and 3). The main issue with the conversion
methods is neglecting the temporal nature of spiking neural networks [19].
Another limitation of conversion methods is a trade-off between the inference
speed and classification accuracy [37]. To reach the optimal classification
accuracy, they usually require at least several hundred of inference time
steps. Tandem learning methods [27, 28] could resolve these issues in
conversion methods. Same as our proxy learning, they assume that IF neurons
with rate-code approximate artificial neurons with ReLU activation. Hence,
they connect the CANN and CSNN layers in tandem and feed the spike counts of
CSNN layers (not the output of the previous CANN layer) as the input of the
next CANN layer. This breaking of the forward pass in the CANN, with
approximated inputs from the CSNN layers, can attenuate the cohesion of its
backward pass. This problem is solved in proxy learning by separating the
forward pass of the two networks.

Although, surrogate gradient learning is one of the best direct learning
algorithms for spiking neural networks [11], but it suffers from the
challenges of backpropagation through time, especially for longer simulation
times, including vanishing/exploding gradients and high computational cost and
memory demand. However, in our proxy learning method we do not confront with
such issues, as the backpropagation is done with the proxy CANN that is a
time-free network.

An important aspect of the proposed CSNN is the use of IF neuron model with no
leak. IF neurons are pure integrators and have the simplest neuronal dynamics
compared to any other spiking neuron model. For instance in leaky-IF neurons,
in case of no other input spikes, at every time step, the neuron membrane
potential exponentially decays with a time constant, while in pure IF neuron
model, the membrane potential is updated simply by increasing or decreasing it
just at the arrival of input spikes and according to the input synaptic
weights. Hence, IF neurons are much simpler to be implemented on different
hardware and neuromorphic platforms [42, 43], especially, in large and deep
networks.

The proposed proxy learning is based on the approximation of ReLU with rate-
coded IF neurons. Rate-coding is the mostly used coding scheme in SNNs,
however, other coding schemes such as temporal coding and rank-order coding
are more efficient in terms of the number of spikes [26, 33, 44, 45, 46, 47].
Extending the proxy learning to CSNNs with temporal coding in future studies
would lead to accurate and low-cost CSNNs.

## Acknowledgments

The authors would like to thank Mr. Wei Fang who helped us in the
implementation of our idea with his Spikingjelly package designed for
developing deep spiking neural networks available at
https://github.com/fangwei123456/spikingjelly.

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# Symplectic foliations induced by harmonic forms on 3-manifolds

Romero Solha  Department of Geometry and Topology, Faculty of Sciences,
University of Granada. Postal address: Departamento de Geometría y Topología,
Facultad de Ciencias - Universidad de Granada, Avenida Fuente Nueva S.N.,
Granada, Spain (postal code 18071). email<EMAIL_ADDRESS>
###### Abstract

This article details a construction of symplectic foliations on
$3$-dimensional orientable riemannian manifolds from harmonic forms; and how
it suggests a topological approach to Poisson’s equation and newtonian
gravity.

##  I Nonvanishing harmonic 2-form

On a $3$-dimensional orientable riemannian manifold $(M,\mathrm{g})$ the Hodge
star operators $\star:\Omega^{k}(M)\to\Omega^{3-k}(M)$ satisfy
$\star^{2}=\mathds{1}$ and take $2$-forms to $1$-forms. Using them, one can
express the riemannian volume by $\star 1\in\Omega^{3}(M)$.

######  Lemma 1.

Given any $\alpha\in\Omega^{2}(M)$, and the unique solution
$X\in\mathfrak{X}(M)$ to

$\mathrm{g}(X,\boldsymbol{\cdot})=\star\alpha\ ,$

it holds

$\alpha\wedge\star\alpha=\mathrm{g}(X,X)\star 1$

and

$\alpha=\imath_{X}\star 1\ .$

###### Proof.

By the definition of the Hodge star operator applied to $1$-forms, for any
$\beta\in\Omega^{1}(M)$,

$\beta(X)\star 1=\beta\wedge\star(\star\alpha)=\beta\wedge\alpha\ .$

Which implies

$\mathrm{g}(X,X)\star 1=(\star\alpha)\wedge\alpha=\alpha\wedge\star\alpha\ ,$

and, since $\beta\wedge\star 1$ must be zero for it is a $4$-form,

$0=\imath_{X}(\beta\wedge\star 1)=\beta(X)\star 1-\beta\wedge\imath_{X}\star
1=\beta\wedge(\alpha-\imath_{X}\star 1)\ .$

∎

If $\omega\in\Omega^{2}(M)$ is both closed and coclosed,

$\left\\{\begin{array}[]{c}\mathrm{d}\omega=0\\\
\star\mathrm{d}\star\omega=0\end{array}\right.\ ,$

and nonvanishing, then it provides a symplectic foliation on $M$.
Unfortunately, even under this strong hypothesis ($\omega$ never vanishing
pointwise), there is no consensus in the literature regarding the nomenclature
of this geometric structure. Some authors call it cosymplectic, others might
treat it as a symplectic vector bundle or a Poisson manifold. To avoid
adhering to a nomenclature, the article limits itself to the description of
features of the underlying geometry when needed; the reader might want to
consult GMP11 .

######  Lemma 2.

Assuming $\omega\in\Omega^{2}(M)$ to be closed, coclosed, and nonvanishing,
the kernel of $\star\omega$ defines a foliation in which $\omega$ restricts to
a symplectic structure on each leaf. And together with the kernel of $\omega$,

$\mathrm{ker}(\omega):=\mathrm{ker}\\{\mathfrak{X}(M)\ni
X\mapsto\imath_{X}\omega\in\Omega^{1}(M)\\}\ ,$

the following orthogonal decomposition is true:

$\mathfrak{X}(M)=\mathrm{ker}(\omega)\oplus\mathrm{ker}(\star\omega)\ .$

###### Proof.

Considering the dual to $-\star\omega$ with respect to $\mathrm{g}$, the
unique $R\in\mathfrak{X}(M)$ solving

$\mathrm{g}(R,\boldsymbol{\cdot})=-\star\omega\ ,$

lemma 1 together with the hypothesis of $\omega$ does not vanishing pointwise
yield that the same holds for $\star\omega$, $R$, and
$\omega\wedge\star\omega$. They also assure that the rank of the kernel of
both forms are constant, $R\in\mathrm{ker}(\omega)$, and

$\mathrm{ker}(\star\omega)=\\{Y\in\mathfrak{X}(M)\ ;\ \mathrm{g}(R,Y)=0\\}\ .$

Because $\star\omega$ is closed, for any pair of vector fields
$X,Y\in\mathrm{ker}(\star\omega)$,

$\displaystyle 0=\mathrm{d}\star\omega(X,Y)$
$\displaystyle=X(\star\omega(Y))-Y(\star\omega(X))-\star\omega([X,Y])$
$\displaystyle=-\star\omega([X,Y])\ ,$

showing that $\mathrm{ker}(\star\omega)$ is integrable and defines a foliation
which the leaves are surfaces (because it is defined by the kernel of a
nonvanishing $1$-form on a $3$-dimensional manifold).

As for the symplectic nature of the leaves, assuming
$X\in\mathrm{ker}(\star\omega)$ to be such that, for all
$Y\in\mathrm{ker}(\star\omega)$, it holds $\omega(X,Y)=0$; thus

$\displaystyle\imath_{Y}\circ\imath_{X}(\omega\wedge\star\omega)$
$\displaystyle=\imath_{Y}((\imath_{X}\omega)\wedge\star\omega+\star\omega(X)\omega)$
$\displaystyle=\imath_{Y}((\imath_{X}\omega)\wedge\star\omega)$
$\displaystyle=\omega(X,Y)\star\omega-\star\omega(Y)\imath_{X}\omega=0\ ,$

making it impossible to exist $Z\in\mathfrak{X}(M)$ satisfying

$\omega\wedge\star\omega(X,Y,Z)\neq 0\ .$

Ergo, $X$ must be the zero vector field and
$\omega|_{\mathrm{ker}(\star\omega)}$ is nondegenerate. Finally, the
hypothesis of $\omega$ being closed secures that it restricts to a symplectic
structure on each leaf of the foliation defined by the kernel of its Hodge
dual.

What is missing is the direct sum condition of the last claim. If
$X\in\mathfrak{X}(M)$ belongs to the intersection
$\mathrm{ker}(\omega)\cap\mathrm{ker}(\star\omega)$, the nondegeneracy of
$\omega|_{\mathrm{ker}(\star\omega)}$ implies that $X=0$, as well as that the
rank of $\mathrm{ker}(\omega)$ equals $1$ (because it is defined by the kernel
of a nonvanishing $2$-form which is nondegenerate when restricted to
$2$-dimensional submanifolds foliating a $3$-dimensional manifold). ∎

##  II Symplectic mapping torus

An application of the previous lemma shows that a compact orientable
$3$-dimensional manifold $M$ fibres over $S^{1}$ if and only if $S^{1}\times
M$ has an $S^{1}$-invariant symplectic structure (which is related to Taubes’s
conjecture Taubesconjecture ), or it is a symplectic mapping torus; i.e. there
exist a $2$-dimensional symplectic manifold $(\Sigma,\sigma)$ and a
diffeomorphism $\psi:\Sigma\to\Sigma$ such that $\psi^{*}(\sigma)=\sigma$, and
$M$ is $\Sigma\times[0,1]$ after identifying $(p,0)\in\Sigma\times[0,1]$ to
$(\psi(p),1)\in\Sigma\times[0,1]$.

Any generator of $H^{1}(S^{1})$ can be pulled back to a nonvanishing closed
$1$-form $\theta$, by the projection mapping of the fibration
$M\mathrel{\mathrlap{\rightarrow}\mkern 1.0mu\rightarrow}S^{1}$. Since a
nonvanishing $1$-form is transitive, Calabi Calabiharmonic  provides a
riemannian structure such that $\star\theta$ is a closed $2$-form (it
satisfies the hypothesis of lemma 2); and results from Liarticle  justify the
claim.

##  III Area and mean curvature

Unless otherwise stated, the hypothesis of lemma 2 are assumed hereafter.

######  Proposition 1.

If $\Sigma\subset M$ is a leaf of $\mathrm{ker}(\star\omega)$, then $\omega$
restricted to it coincides with the induced area form from the restriction of
$\mathrm{g}$ to $\Sigma$ multiplied by the norm of $R$ (which is equivalent to
the norm of $\omega$).

###### Proof.

Lemma 1 provides $\omega\wedge\star\omega=\mathrm{g}(R,R)\star 1$ as well as
$\omega=-\imath_{R}\star 1$; therefore, the normal vector field to $\Sigma$
inducing the same orientation as $\omega$ is $-\mathrm{g}(R,R)^{-1/2}R$,
whilst the induced area form from the restriction of $\mathrm{g}$ to $\Sigma$
is $-\mathrm{g}(R,R)^{-1/2}\imath_{R}\star 1$.∎

######  Corollary 1.

Any compact (without boundary) and oriented leaf $\Sigma\subset M$ of
$\mathrm{ker}(\star\omega)$ will satisfy $\int_{\Sigma}\omega>0$.

###### Proof.

Exact symplectic manifolds cannot be compact, $\Sigma$ is positively oriented
with respect to the orientation induced by $\omega$, and the norm of $\omega$
is a positive function. ∎

Because $-\mathrm{g}(R,R)^{-1/2}\imath_{R}\star 1$ is symplectic when
restricted to $\Sigma$, and the mapping $\mathrm{j}$ defined by

$\mathrm{g}\circ(\mathrm{j}\oplus\mathds{1})=-\frac{\imath_{R}\star
1}{\sqrt{\mathrm{g}(R,R)}}$

is a complex structure when restricted to the leaves of
$\mathrm{ker}(\star\omega)$, one has a Kähler structure on $\Sigma$.

As for the mean curvature of the leaves, they can be expressed in terms of
$\omega$.

######  Proposition 2.

The mean curvature of a leaf $\Sigma\subset M$ of $\mathrm{ker}(\star\omega)$
is
$-\tfrac{1}{2}\star(\omega\wedge\mathrm{d}(\star(\omega\wedge\star\omega))^{-1/2})$.

###### Proof.

By computing the mean curvature as the divergence of a normal vector field to
the leaf, and using $\mathrm{d}\omega=0$ and
$\mathrm{g}(R,R)=\star(\omega\wedge\star\omega)$ (cf. lemma 1),

$\displaystyle-\frac{1}{2}\mathrm{div}\left(\frac{-R}{\sqrt{\mathrm{g}(R,R)}}\right)$
$\displaystyle=-\frac{1}{2}\star\mathrm{d}\left(\frac{-\imath_{R}\star
1}{\sqrt{\mathrm{g}(R,R)}}\right)$
$\displaystyle=-\frac{1}{2}\star\mathrm{d}\left(\frac{\omega}{\sqrt{\star(\omega\wedge\star\omega)}}\right)$
$\displaystyle=-\frac{1}{2}\star((\mathrm{d}(\star(\omega\wedge\star\omega))^{-1/2})\wedge\omega)\
.$

∎

##  IV Harmonic function

When the Hodge dual $\star\omega$ is exact, the vector field dual to it is the
gradient of a harmonic function.

######  Lemma 3.

If there exists a function $\phi\in C^{\infty}(M)$ such that
$\star\omega=\mathrm{d}\phi$, then $R\in\mathfrak{X}(M)$ solving

$\mathrm{g}(R,\boldsymbol{\cdot})=-\star\omega$

is the gradient of a harmonic function.

###### Proof.

On the one hand,

$\mathrm{g}(R,\boldsymbol{\cdot})=-\star\omega=\mathrm{d}(-\phi)\ .$

On the other hand,

$\displaystyle\Delta\phi$
$\displaystyle=(\star\mathrm{d}\star)\circ\mathrm{d}\phi=(\star\mathrm{d}\star)(\star\omega)$
$\displaystyle=(\star\mathrm{d})\star^{2}\omega=\star\mathrm{d}\omega=0\ .$

∎

As a consequence, level sets of harmonic functions inherit a symplectic
structure.

######  Proposition 3.

The kernel of $\star\omega$ is the orthogonal complement of the distribution
generated by the vector field $R$ (dual to $-\star\omega$ with respect to
$\mathrm{g}$), and its symplectic leaves are the level sets of any harmonic
function which the gradient is $R$.

###### Proof.

From the previous definitions,

$\displaystyle\mathrm{ker}(\star\omega)$
$\displaystyle:=\\{Y\in\mathfrak{X}(M)\ ;\ \star\omega(Y)=0\\}$
$\displaystyle=\\{Y\in\mathfrak{X}(M)\ ;\ \mathrm{g}(R,Y)=0\\}$
$\displaystyle=\\{Y\in\mathfrak{X}(M)\ ;\ \mathrm{d}\phi(Y)=0\\}\ .$

∎

This is a good opportunity to summarise the results. Starting with a
$3$-dimensional orientable riemannian manifold $(M,\mathrm{g})$, any harmonic
function $\phi\in C^{\infty}(M)$ has the property that its level sets are
symplectic manifolds with respect to the $2$-form $\star\mathrm{d}\phi$, cf.
lemma 2 and proposition 3; moreover, $\star\mathrm{d}\phi$ induces an
orientation on each level set, and its integral over the compact ones must be
positive (cf. proposition 1). Those results are even more relevant when
considering solutions of Poisson’s equation, instead of Laplace’s equation.

##  V Poisson’s equation

Let $\rho\in C^{\infty}(M)$ (or a distribution), then $\phi\in C^{\infty}(M)$
(which may not be smooth over the whole of $M$) satisfies the Poisson’s
equation with source $\rho$ whenever

$\Delta\phi=\rho\ .$

By definition $\Delta\phi=\star\mathrm{d}(\star\mathrm{d}\phi)$; thus,
$\rho\star 1=\mathrm{d}(\star\mathrm{d}\phi)$ and Stokes’s theorem gives, for
any $3$-chain $V\subset M$,

$\int_{V}\rho\star 1=\int_{V}\mathrm{d}(\star\mathrm{d}\phi)=\int_{\partial
V}\star\mathrm{d}\phi\ .$

Setting $J\subset M$ as the closure of the set where $\rho$ does not vanish
(its support) and $M-J$ (the interior of $\rho^{-1}(\\{0\\})$), if
$\Sigma\subset M-J$ is a $2$-cycle satisfying $\Sigma=\partial V$ on $M$, then

$\int_{V}\rho\star 1=\int_{\Sigma}\star\mathrm{d}\phi\ .$

Implying that the de Rham class $[\star\mathrm{d}\phi]\in H^{2}(M-J)$ is
predetermined by integrals of the source $\rho\in C^{\infty}(M)$ with respect
to the volume $\star 1$.

This suggests another formulation to Poisson’s equation. Instead of the
initial data being a $3$-dimensional orientable riemannian manifold
$(M,\mathrm{g})$ and a source function $\rho\in C^{\infty}(M)$ (or
distribution), the information contained in $\rho$ is translated into a closed
subset $J\subset M$ and an element of $H^{2}(M-J)$. And rather than looking
for a function $\phi\in C^{\infty}(M)$ satisfying $\Delta\phi=\rho$, a
solution would be a closed and coclosed form $\omega\in\Omega^{2}(M-J)$ such
that $\star\omega$ is exact and $[\omega]\in H^{2}(M-J)$ coincides with the
given de Rham class. The advantage of this formulation is apparent when $\rho$
is a distribution: for a solution $\phi$ of $\Delta\phi=\rho$ is not well
defined over the support of $\rho$, whilst $\omega$ captures all pertinent
information.

######  Remark 1.

The subset where $(\star\mathrm{d}\phi)\wedge(\mathrm{d}\phi)$ fails to be a
volume form needs not to be empty. Indeed, the $3$-form vanishes where the
derivative of the harmonic function is zero; therefore, if
$\star\mathrm{d}\phi$ vanishes on an open subset, the harmonic function will
be constant there, and $\star\mathrm{d}\phi$ must be zero over the whole
connected component containing it. Yet, lemma 2 and proposition 1 are still
valid for the points where $\omega\wedge\star\omega$ defines a volume form.

##  VI Newtonian gravity

Since newtonian gravity can be described in terms of a gravitational vector
field determined by a function solving Poisson’s equation, it is possible to
exploit this alternative formulation to Poisson’s equation to obtain an
equivalent theory.

Over a spacelike region described by a $3$-dimensional orientable riemannian
manifold $(M,\mathrm{g})$ with gravitational sources located on a closed (and
possibly not connected) subset $J\subset M$, the gravitational field is a
closed differential $2$-form $\omega\in\Omega^{2}(M-J)$ such that, for every
smooth singular 1-cycle $\gamma\subset M-J$, $\int_{\gamma}\star\omega=0$. The
gravitational mass inside a $3$-dimensional spacelike region bounded by any
gaussian surface $\Sigma$ is $\int_{\Sigma}\omega$.

A gaussian surface $\Sigma\subset M-J$ is a $2$-dimensional compact, without
boundary, and oriented submanifold such that $[\Sigma]\in H_{2}(M)$ is
trivial. If $V\subset M$ is some $3$-dimensional submanifold satisfying
$\partial V=\Sigma$, then the orientation of $\Sigma$ is the one induced by
the orientation of the restriction of $\star 1$ to $V$ and used in Stokes
theorem.

######  Remark 2.

The notion of a gaussian surface without a predetermined orientation renders
it impossible to decide if the mass of a gravitational source is positive or
negative: for a smooth singular $2$-cycle $\Sigma\subset M-J$ representing a
generator $[\Sigma]\in H_{2}(M-J)$, the element $-[\Sigma]\in H_{2}(M-J)$ is
also a generator and
$\left\langle{\emph{$[\omega]$}},{\emph{$-[\Sigma]$}}\right\rangle=-\int_{\Sigma}\omega$.

Given a gravitational source distribution $J\subset M$ and its respective
gravitational mass, an element of $H^{2}(M-J)$ is fixed, and a representative
$\omega$ will be the gravitational field for those sources whenever, for every
smooth singular $1$-cycle $\gamma$, it holds $\int_{\gamma}\star\omega=0$.

Regarding dynamics, the gravitational vector field $R$ is the unique solution
to $\mathrm{g}(R,\boldsymbol{\cdot})=-\star\omega$, using units where the
gravitational constant equals $1/4\pi$. A test particle of gravitational mass
$m$ under the influence of this gravitational field $\omega$ suffers a force
given by $mR$.

For illustration, taking $(M,\mathrm{g})$ to be the $3$-dimensional euclidean
space and $J=\\{0\\}\subset M=\mathds{R}^{3}$, using linear projections
$x,y,z\in C^{\infty}(\mathds{R}^{3})$ as coordinates

$\sigma:=\frac{1}{4\pi}\frac{z\mathrm{d}x\wedge\mathrm{d}y-y\mathrm{d}x\wedge\mathrm{d}z+x\mathrm{d}y\wedge\mathrm{d}z}{(x^{2}+y^{2}+z^{2})^{\tfrac{3}{2}}}$

is coexact and its de Rham class generates $H^{2}(\mathds{R}^{3}-\\{0\\})$.
Any element of $H^{2}(\mathds{R}^{3}-\\{0\\})$ associated to
$m_{0}\in\mathds{R}$ through the isomorphism
$H^{2}(\mathds{R}^{3}-\\{0\\})\cong\mathds{R}$ (induced by using
$\\{[\sigma]\\}\subset H^{2}(\mathds{R}^{3}-\\{0\\})$ as a basis) can be
represented by $\omega=m_{0}\sigma$, and if $\Sigma$ is any sphere centred at
the origin oriented to be a gaussian surface, $\int_{\Sigma}\omega=m_{0}$.
Supposing that $\omega$ is a gravitational field, its gravitational vector
field,

$R=-\frac{m_{0}}{4\pi}\frac{x\textstyle\frac{\partial\emph{$$}}{\partial{\emph{$x$}}}+y\textstyle\frac{\partial\emph{$$}}{\partial{\emph{$y$}}}+z\textstyle\frac{\partial\emph{$$}}{\partial{\emph{$z$}}}}{(x^{2}+y^{2}+z^{2})^{\frac{3}{2}}}
\ , \end{equation}coincides with the expected solution for the gravitational
vector field generated by a particle of gravitational mass $m_{0}$ at the
origin of the $3$-dimensional euclidean space; moreover, it is a multiple of
the radial vector field, implying that the symplectic leaves of
$\mathrm{ker}(\star\omega)$ are the spheres centred at the origin (cf.
proposition \ref{gravitationalleaves}), i.e. $|m_{0}|=\int_{\Sigma}\omega>0$,
because it is its symplectic area and the orientation of $\Sigma$ induced by
$\omega$ will coincide with its original orientation only when $m_{0}>0$.
\par\par\par\par\@@numbered@section{section}{toc}{Assumptions and predictions}
\par Here are the assumptions made in the last section. \par \begin{enumerate}
\par\enumerate@item@Physical space is described by a $3$-dimensional
orientable riemannian manifold, \par\enumerate@item@there exists a
gravitational field described by a closed and coexact differential $2$-form,
\par\enumerate@item@the gravitational field is produced by sources (having
gravitational mass) located on a closed subset of the physical space,
\par\enumerate@item@the gravitational mass ties to the differential $2$-form
via integration over special subsets of the physical space (gaussian
surfaces), \par\enumerate@item@test particles interact with the source of a
gravitational field through a vector field constructed out of the differential
$2$-form and riemannian metric. \par\end{enumerate} \par No sort of proof to
under what conditions those assumptions are true, or if those conditions
exist, is provide. Those are premises. However, they lead to predictions that
can be tested and refuted. \par \begin{enumerate} \par\enumerate@item@A test
particle is attracted by a source of gravitational field with a force
proportional to the product of their masses and inversely proportional to the
square of the distance between their centres (acting along the line
intersecting them both), \par\enumerate@item@the gravitational field is linear
with respect to gravitational sources, \par\enumerate@item@and
Newton\textquoteright s shell theorem is valid. \par\end{enumerate} \par The
first prediction follows from equation \ref{newtonsuniversallaw} (together
with the fifth assumption), the second from the linearity of exterior
derivatives, Hodge star operators, and relevant cohomology and homology
groups, and the third from the topological invariance of cohomology and
homology groups. \par Regarding the ability to test and refute those
predictions, it is clear that the first is Newton's law of universal
gravitation; hence, any experiment and observation that can test and refute
newtonian gravity applies to the predictions of those new assumptions. In
particular, said assumptions alone are not able to predict Mercury's orbit
with better precision than Newton's theory. \par Whilst newtonian gravity has
the first prediction as an assumption, the present article derives Newton's
law of universal gravitation from a different set of assumptions.
\par\begin{remark}Although tempting, substituting gravitational by electric
and mass by charge in the list of five assumptions will provide a limited
theory describing Electrostatics phenomena. The reader must bear in mind that
Electrostatics can be fully explained by Electromagnetism, a theory that must
assume the physical space to be part of a set described by a $4$-dimensional
lorentzian manifold; resulting in hyperbolic field equations ---drastically
different from the 3-dimensional riemannian case. \end{remark}
\par\begin{remark}Insisting in a parallel between Electrostatics and newtonian
gravity, one might apply the assumptions of \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{TheSolha}{\@@citephrase{(}}{\@@citephrase{)}}} to gravity
and reach the prediction that gravitational mass is an integral multiple of
some fixed constant. Since there is evidence showing that this prediction is
false, the logical conclusion is that Gravitation (being it relativistic or
not) does not satisfy the assumptions of \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{TheSolha}{\@@citephrase{(}}{\@@citephrase{)}}}, and as a
theory it must be different from Electromagnetism in its foundation.
\end{remark} \par\par\par\par\vspace{2em} \par\acknowledgments This work was
partially supported by PNPD/CAPES, the Basque government grant IT1094-16 of
Spain, and Andalusia government grant P20\textunderscore 01391 of Spain.
\endacknowledgments
\par\par\thebibliography\@@lbibitem{}\NAT@@wrout{1}{}{}{}{(1)}{}\lx@bibnewblock\addcontentsline{toc}{chapter}{References}
\@@lbibitem{Calabiharmonic}\NAT@@wrout{2}{}{}{}{(2)}{Calabiharmonic}\lx@bibnewblock\href
https://www.degruyter.com/document/doi/10.1515/9781400871230-006/html \\\
Princeton University Press, (1970).
\par\@@lbibitem{Taubesconjecture}\NAT@@wrout{3}{}{}{}{(3)}{Taubesconjecture}\lx@bibnewblock\href
https://www.jstor.org/stable/23030381?refreqid=excelsior%3A284b3e681a15f38f280d85bb3c209384
\par\@@lbibitem{GMP11}\NAT@@wrout{4}{}{}{}{(4)}{GMP11}\lx@bibnewblock\href
https://link.springer.com/article/10.1007/s00574-011-0031-6 \\\ Bull. Braz.
Math. Soc. (N.S.) {42}, 607 (2011).
\par\par\@@lbibitem{Liarticle}\NAT@@wrout{5}{}{}{}{(5)}{Liarticle}\lx@bibnewblock\href
https://projecteuclid.org/journals/asian-journal-of-
mathematics/volume-12/issue-4/Topology-of-Co-symplecticCo-K%c3%a4hler-
Manifolds/ajm/1235140172.full
\par\@@lbibitem{MarsdenAndTromba}\NAT@@wrout{6}{}{}{}{(6)}{MarsdenAndTromba}\lx@bibnewblock
Marsden and Tromba, {Vector Calculus}, \\\ W. H. Freeman and Company, fifth
edition (2003).
\par\@@lbibitem{TheSolha}\NAT@@wrout{7}{}{}{}{(7)}{TheSolha}\lx@bibnewblock\href
http://www.sciencedirect.com/science/article/pii/S0393044015002284?via%3Dihub
\par\par\par\endthebibliography
\par\par\@add@PDF@RDFa@triples\par\end{document}$ (1)


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LHCb collaboration

R. Aaij38, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, S. Akar6,
J. Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P.
Alvarez Cartelle53, A.A. Alves Jr57, S. Amato2, S. Amerio22, Y. Amhis7, L.
An3, L. Anderlini17, J. Anderson40, G. Andreassi39, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, P.
d’Argent11, A. Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,m, M.
Baalouch5, S. Bachmann11, J.J. Back48, A. Badalov36, C. Baesso60, W.
Baldini16,38, R.J. Barlow54, C. Barschel38, S. Barsuk7, W. Barter38, V.
Batozskaya28, V. Battista39, A. Bay39, L. Beaucourt4, J. Beddow51, F.
Bedeschi23, I. Bediaga1, L.J. Bel41, V. Bellee39, N. Belloli20, I. Belyaev31,
E. Ben-Haim8, G. Bencivenni18, S. Benson38, J. Benton46, A. Berezhnoy32, R.
Bernet40, A. Bertolin22, M.-O. Bettler38, M. van Beuzekom41, A. Bien11, S.
Bifani45, P. Billoir8, T. Bird54, A. Birnkraut9, A. Bizzeti17,h, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38,
W. Bonivento15, S. Borghi54, M. Borsato7, T.J.V. Bowcock52, E. Bowen40, C.
Bozzi16, S. Braun11, M. Britsch10, T. Britton59, J. Brodzicka54, N.H. Brook46,
A. Bursche40, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,j, M.
Calvo Gomez36,o, P. Campana18, D. Campora Perez38, L. Capriotti54, A.
Carbone14,d, G. Carboni24,k, R. Cardinale19,i, A. Cardini15, P. Carniti20, L.
Carson50, K. Carvalho Akiba2,38, G. Casse52, L. Cassina20,j, L. Castillo
Garcia38, M. Cattaneo38, Ch. Cauet9, G. Cavallero19, R. Cenci23,s, M.
Charles8, Ph. Charpentier38, M. Chefdeville4, S. Chen54, S.-F. Cheung55, N.
Chiapolini40, M. Chrzaszcz40, X. Cid Vidal38, G. Ciezarek41, P.E.L. Clarke50,
M. Clemencic38, H.V. Cliff47, J. Closier38, V. Coco38, J. Cogan6, E.
Cogneras5, V. Cogoni15,e, L. Cojocariu29, G. Collazuol22, P. Collins38, A.
Comerma-Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, M. Corvo16,f, B. Couturier38, G.A. Cowan50, D.C. Craik48, A.
Crocombe48, M. Cruz Torres60, S. Cunliffe53, R. Currie53, C. D’Ambrosio38, E.
Dall’Occo41, J. Dalseno46, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, P. De Simone18, C.-T.
Dean51, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, M. Demmer9, D.
Derkach55, O. Deschamps5, F. Dettori38, B. Dey21, A. Di Canto38, F. Di
Ruscio24, H. Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil
Suárez37, D. Dossett48, A. Dovbnya43, K. Dreimanis52, L. Dufour41, G.
Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51, I. El Rifai5, Ch.
Elsasser40, S. Ely59, S. Esen11, H.M. Evans47, T. Evans55, A. Falabella14, C.
Färber38, N. Farley45, S. Farry52, R. Fay52, D. Ferguson50, V. Fernandez
Albor37, F. Ferrari14, F. Ferreira Rodrigues1, M. Ferro-Luzzi38, S.
Filippov33, M. Fiore16,38,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick39, T.
Fiutowski27, K. Fohl38, P. Fol53, M. Fontana15, F. Fontanelli19,i, R. Forty38,
O. Francisco2, M. Frank38, C. Frei38, M. Frosini17, J. Fu21, E. Furfaro24,k,
A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,38, S. Gambetta50, M.
Gandelman2, P. Gandini55, Y. Gao3, J. García Pardiñas37, J. Garra Tico47, L.
Garrido36, D. Gascon36, C. Gaspar38, R. Gauld55, L. Gavardi9, G. Gazzoni5, A.
Geraci21,u, D. Gerick11, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph.
Ghez4, A. Gianelle22, S. Gianì39, V. Gibson47, O. G. Girard39, L. Giubega29,
V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a,
C. Gotti20,j, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado
Cardoso38, E. Graugés36, E. Graverini40, G. Graziani17, A. Grecu29, E.
Greening55, S. Gregson47, P. Griffith45, L. Grillo11, O. Grünberg63, B. Gui59,
E. Gushchin33, Yu. Guz35,38, T. Gys38, T. Hadavizadeh55, C. Hadjivasiliou59,
G. Haefeli39, C. Haen38, S.C. Haines47, S. Hall53, B. Hamilton58, X. Han11, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, J. He38, T.
Head39, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando
Morata37, E. van Herwijnen38, M. Heß63, A. Hicheur2, D. Hill55, M. Hoballah5,
C. Hombach54, W. Hulsbergen41, T. Humair53, N. Hussain55, D. Hutchcroft52, D.
Hynds51, M. Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E.
Jans41, A. Jawahery58, F. Jing3, M. John55, D. Johnson38, C.R. Jones47, C.
Joram38, B. Jost38, N. Jurik59, S. Kandybei43, W. Kanso6, M. Karacson38, T.M.
Karbach38,†, S. Karodia51, M. Kecke11, M. Kelsey59, I.R. Kenyon45, M.
Kenzie38, T. Ketel42, B. Khanji20,38,j, C. Khurewathanakul39, S. Klaver54, K.
Klimaszewski28, O. Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P.
Koppenburg41,38, M. Kozeiha5, L. Kravchuk33, K. Kreplin11, M. Kreps48, G.
Krocker11, P. Krokovny34, F. Kruse9, W. Krzemien28, W. Kucewicz26,n, M.
Kucharczyk26, V. Kudryavtsev34, A. K. Kuonen39, K. Kurek28, T.
Kvaratskheliya31, D. Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, G.
Lanfranchi18, C. Langenbruch48, B. Langhans38, T. Latham48, C. Lazzeroni45, R.
Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A. Leflat32,38, J.
Lefrançois7, O. Leroy6, T. Lesiak26, B. Leverington11, Y. Li7, T.
Likhomanenko65,64, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B.
Liu15, X. Liu3, D. Loh48, I. Longstaff51, J.H. Lopes2, D. Lucchesi22,q, M.
Lucio Martinez37, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, N.
Lusardi21, A. Lusiani23, F. Machefert7, F. Maciuc29, O. Maev30, K. Maguire54,
S. Malde55, A. Malinin64, G. Manca7, G. Mancinelli6, P. Manning59, A.
Mapelli38, J. Maratas5, J.F. Marchand4, U. Marconi14, C. Marin Benito36, P.
Marino23,38,s, J. Marks11, G. Martellotti25, M. Martin6, M. Martinelli39, D.
Martinez Santos37, F. Martinez Vidal66, D. Martins Tostes2, A. Massafferri1,
R. Matev38, A. Mathad48, Z. Mathe38, C. Matteuzzi20, A. Mauri40, B. Maurin39,
A. Mazurov45, M. McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B.
Meadows57, F. Meier9, M. Meissner11, D. Melnychuk28, M. Merk41, D.A.
Milanes62, M.-N. Minard4, D.S. Mitzel11, J. Molina Rodriguez60, I.A. Monroy62,
S. Monteil5, M. Morandin22, P. Morawski27, A. Mordà6, M.J. Morello23,s, J.
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V. Müller9, M. Mussini14, B. Muster39, P. Naik46, T. Nakada39, R.
Nandakumar49, A. Nandi55, I. Nasteva2, M. Needham50, N. Neri21, S. Neubert11,
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M. Patel53, C. Patrignani19,i, A. Pearce54,49, A. Pellegrino41, G. Penso25,l,
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Petridis46, A. Petrolini19,i, M. Petruzzo21, E. Picatoste Olloqui36, B.
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Plo Casasus37, T. Poikela38, F. Polci8, A. Poluektov48,34, I. Polyakov31, E.
Polycarpo2, A. Popov35, D. Popov10,38, B. Popovici29, C. Potterat2, E.
Price46, J.D. Price52, J. Prisciandaro39, A. Pritchard52, C. Prouve46, V.
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Rauschmayr38, G. Raven42, F. Redi53, S. Reichert54, M.M. Reid48, A.C. dos
Reis1, S. Ricciardi49, S. Richards46, M. Rihl38, K. Rinnert52, V. Rives
Molina36, P. Robbe7,38, A.B. Rodrigues1, E. Rodrigues54, J.A. Rodriguez
Lopez62, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
Vidal37, J. W. Ronayne12, M. Rotondo22, J. Rouvinet39, T. Ruf38, P. Ruiz
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
63Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
64National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
65Yandex School of Data Analysis, Moscow, Russia, associated to 31
66Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
67Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 41

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
uPolitecnico di Milano, Milano, Italy
†Deceased