# SMS: Spiking Marching Scheme for Efficient Long Time Integration of
Differential Equations

Qian Zhang , Adar Kahana∗, George Em Karniadakis
Division of Applied Mathematics
Brown University
Providence, RI
{qian_zhang1, adar_kahana<EMAIL_ADDRESS>
&Panos Stinis
Advanced Computing, Mathematics and Data Division
Pacific Northwest National Laboratory
Richland, WA
<EMAIL_ADDRESS>
The first and second author contributed equally.

###### Abstract

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for
long time integration of time-dependent Ordinary and Partial Differential
Equations (ODEs, PDEs). The core element of the method is a SNN, trained to
use spike-encoded information about the solution at previous timesteps to
predict spike-encoded information at the next timestep. After the network has
been trained, it operates as an explicit numerical scheme that can be used to
compute the solution at future timesteps, given a spike-encoded initial
condition. A decoder is used to transform the evolved spiking-encoded solution
back to function values. We present results from numerical experiments of
using the proposed method for ODEs and PDEs of varying complexity.

_K_ eywords efficient solver $\cdot$ differential equations $\cdot$ spiking
neural networks $\cdot$ machine intelligence $\cdot$ time marching

##  1 Introduction

Spiking Neural Networks (SNNs), named “the third generation of neural
networks” [1], are an emerging trend in the scientific community and they are
becoming the center of attention of many research groups. A SNN operates on
input spiking data (spike-encoded images, text, continuous values, etc.) and
produces output spiking predictions. The spiking framework makes the
implementation and interpretation of SNNs much different from their Artificial
Neural Networks (ANNs) counterparts. The incentive for exploring the use of
SNNs is that the human brain, arguably the most sophisticated learning machine
in existence, uses much less power than popular hardware used for machine
learning (Core and Graphics Processing Units, CPUs and GPUs). The human brain
stands as a good example for learning algorithms that are both more accurate,
as well as efficient. However, creating such biologically plausible learning
algorithms is a challenging task. Many authors are investigating this topic
[2, 3, 4, 5], but in most cases the methods developed are not fully
satisfactory yet in terms of the accuracy/efficiency trade-off.

On the hardware side, neuromorphic processing units [6, 7] have been developed
to perform computations using spike trains, similar to what happens in the
brain. These chips can be analog or digital (or mixed), and consist of many
small units that carry a signal, either 0 or 1. These small units mimic the
neurons in the brain, and the chips perform Very Large Scale Integration
(VLSI) to perform computations using these units. A layer of logical gates
allows the chips to control these units, and using these one can implement
algorithms on neuromorphic chips. SNNs are meant to run on neuromorphic chips
in order to advance towards the full potential of biologically inspired
machine learning. In this work, we focus on the algorithmic side. The method
proposed herein can, in principle, be implemented to run on a neuromorphic
chip with much higher efficiency.

SNNs are often used for classification, since the output of a SNN is usually a
few neurons that have either 0 or 1 values, which resembles a one-hot encoded
vector in classification tasks [8, 9, 10]. On the other hand, in Scientific
Machine Learning (SciML), we are interested in using SNNs for regression,
which includes approximating continuous functions, solutions of ODEs and PDEs,
and operators. In [11], we presented the first SNN-based method involving
operator regression [12] to approximate functions and solutions of PDEs. In
this paper, we aim to extend some of these ideas to time-dependent problems
using a new approach.

To approximate solutions of time-dependent differential equations, one usually
employs numerical methods [13, 14, 15]. There is an obvious trade-off between
the accuracy and the efficiency of a method, since higher accuracy requires
more computational resources. Most numerical methods can be classified as
either implicit or explicit. Explicit methods approximate the solution at the
next timestep from the solution at the current timestep, while implicit
methods obtain the approximate solution by solving an equation involving the
solution at both the current and the next timesteps. All explicit methods are
conditionally stable, meaning that if the stability condition is not met, the
approximate solution accumulates catastrophically large errors even within a
few timesteps. In the context of time-dependent PDEs, popular explicit
numerical schemes are, e.g., those based on finite differences. The idea is to
define a spatio-temporal mesh, and by approximating the derivatives of the
solution on that mesh using values at the current (and previous) timestep, one
can create an explicit marching scheme and use it to advance an approximation
of the PDE solution over time. Explicit schemes are usually straightforward in
terms of interpretation and implementation. However, they are conditionally
stable, for example, for advection-dominated problems they are characterized
by the Courant-Friedrichs-Lewy (CFL) condition [16] while for the diffusion-
dominated problems they are characterized by the Diffusion condition. The CFL
and Diffusion conditions couple the spatial and temporal discretizations. In
particular, finer spatial discretization has to be matched by a proportionally
finer temporal discretization. This can become a serious bottleneck for the
simulation of phenomena that require long time integration.

Figure 1: Illustration of the proposed approach. First, we encode the initial
condition into a spike train. Then, we feed the spiking data into a trained
SNN and use it iteratively to advance the solution for many time steps.
Finally, we decode the output to get the non-spiking solution.

In this work, we develop a SNN based method to construct numerical integrators
for ODEs and PDEs. In the proposed construction, the trained SNN acts as an
explicit method. It receives as input an approximate solution at a previous
timestep and predicts the approximate solution at the next step. The input and
output of the SNN are in the form of spikes, and are able to utilize the
efficiency advantages of SNNs. To this end, we encode the continuous inputs
into spiking data, and after advancing them in time through the use of the
SNN, we decode the output from spikes back into function values. After the
network has been trained, one encodes an initial condition into spikes,
applies the SNN recursively to advance the solution for many timesteps with
low computational cost, thus approximating the solution for long times with
high accuracy (if fine temporal discretization is used), and decodes the
output. This process is illustrated in Fig. 1. Through numerical experiments,
we show that the proposed approach has no stability constraints and even when
not complying with the CFL or Diffusion conditions it is still able to make
accurate predictions of the solution for long times. The full potential of the
proposed approach in terms of efficiency requires implementation on a
neuromorphic chip, which we have not pursued here.

##  2 Related work

In recent years, biologically plausible machine learning (ML) methods have
been attracting increasing interest. The three popular directions are: a)
converting a pre-trained artificial neural network (ANN) to a SNN [17, 18]; b)
building surrogate models so that back propagation can be implemented in SNNs
[19, 20]; and c) using a pure SNN with only biologically plausible components
[8]. In [21], the authors present a review of a large variety of SNN based
methods and compare them. Since most SNN architectures are designed for
classification, specifically classifying handwritten digits [8], they can be
compared in terms of accuracy and efficiency on this specific task. There has
also been work on the problem of classifying images [22, 23].

With respect to the three popular directions mentioned above, each has
advantages and disadvantages. The first is about converting a trained ANN into
an SNN. Two drawbacks of this approach are that it is hard to convert complex
ANN architectures into SNNs (for example, supporting all kinds of activation
functions such as hyperbolic tangent, sinusoidal, etc.), and also training the
ANN still requires a lot of computational effort. The second approach
discusses building surrogate models to train a SNN. These methods are as
accurate and efficient as the developed surrogates. Lastly, there are SNNs
that use Hebbian learning rules [4] for training, which however are highly
unstable. Even if one manages to stabilize them and train them correctly, they
currently underperform in terms of accuracy, and cannot compete with current
state-of-the-art ANNs.

Scientific Machine Learning (SciML) is becoming increasingly popular for
problems in computational science and engineering. For example, Physics-
Informed Neural Networks (PINNs) [24] have been applied to various physical
and biological systems with considerable success. However, like other learning
methods, they consume a lot of computational resources and require a lot of
data for training, especially for complex problems (high dimensions, high
frequencies, etc.). There are some works on computing forward gradients [25]
and eliminating the need for back-propagation [26], which is the most
expensive part of training the neural network.

We began exploring this field with the goal of using SNNs to enhance the
training of machine learning based methods for solving differential equations.
The first step was to use SNNs for regression [11]. We proposed a method to
encode input numeric data ($x$) and train a SNN to infer the values of a
function ($y(x)$). We also proposed using a deep neural operator, DeepONet
[12], for the same function regression task, essentially approximating a
spiking (and not continuous) operator. We used back-propagation for training
the networks, as a surrogate for a synapse. The proposed method operates on
structured grids, and while we managed to show interpolation capabilities,
extrapolation was not investigated in that work.

Time marching schemes are essential in solving time-dependent PDEs. They have
been developed for a long time to achieve improved efficiency and accuracy
[15]. Explicit schemes are usually faster, but less accurate and only
conditionally stable, meaning that without satisfying the CFL or Diffusion
conditions the solution may diverge after just a few time integration steps.
On the other hand, implicit schemes are often more accurate and without
timestep constraints (up to second-order), but require a much larger
computational effort. There are also Implicit-Explicit schemes (IMEX) that
combine both approaches to achieve better performance [27].

ML based methods for long time integration have also been proposed. Properly
trained, PINNs have shown an ability to extrapolate for longer times [28, 29].
In addition, DeepONets can be trained for this task as well [30]. In [31] the
authors proposed a ML based explicit scheme that is stable even when violating
the stability conditions. A noticeable drawback of such methods is that they
take long time to train, and the PINN based methods require training for each
problem being investigated. A less noticeable drawback is that machine
learning methods often reach a limit of accuracy. Whereas numerical solvers
may achieve an error norm of, e.g., $10^{-10}$ per step, ML based methods may
only achieve at best a $10^{-5}$ error norm, resulting in large error
accumulation over time. As a result, they are able to forecast accurately only
for relatively short time intervals.

On the hardware side, neuromorphic chips have been developed for fast
implementation of spiking neural networks. (e.g., Intel’s Loihi 2 [32, 33]).
These chips are using either an analog or a digital interface, and their
architecture consists of many small units that can either have a current
flowing through or be idle (spiking or not). Software packages have been
developed to interact with such chips, defining the logical gates that switch
the currents between the tiny units, enabling the users to train SNNs. In
addition, simulators (such as LAVA [34] for Loihi 2) have been made available,
enabling researchers to develop code that can be implemented on the
neuromorphic chip, while testing it first on non-neuromorphic machines.

##  3 Spiking Marching Scheme

###  3.1 Formulation as a data-driven problem

We propose using a machine learning based method, which utilizes SNNs for the
time integration of ODEs and PDEs. We employ supervised learning to train the
neural networks. To do that, we create a dataset of samples consisting of
inputs and outputs. The goal is to create a SNN based numerical scheme, where
the input is the solution at the current (and if needed previous) timestep(s),
and the output is the solution at the next time step. In the case of ODEs, the
solution at the current timestep is specified by the state vector of the
system. In the case of PDEs, the solution at the current timestep is specified
on a pre-defined spatial grid.

Without loss of generality we can define a PDE as follows:

$\begin{aligned}
\mathcal{L}_{\textbf{x},t}(u;k)&=f,\quad\textbf{x}\in\Omega,\quad t\in[0,T]\\\
\mathcal{IC}_{\textbf{x},0}(u)&=u_{0},\quad\textbf{x}\in\Omega\\\
\mathcal{B}_{\textbf{x},t}(u)&=u_{b},\quad\textbf{x}\in\partial\Omega,\quad
t\in[0,T]\end{aligned}\qquad,$ (1)

where x is the spatial coordinate, $t$ is the temporal coordinate, $\Omega$ is
a domain in $\mathbb{R}^{d}$. The terms $\mathcal{L}_{\textbf{x},t}$,
$\mathcal{IC}_{\textbf{x},0}$ and $\mathcal{B}_{\textbf{x},t}$ represent the
partial differential operator, the initial condition, and the boundary
operator, respectively. The quantities $k(\textbf{x})$ reflect parameters that
the differential operator can depend on, and $f$, $u_{0}$ and $u_{b}$ are the
forcing term, the initial condition and boundary condition, respectively. The
quantity $u(\textbf{x},t)$ is the solution of the PDE. We present a general
formulation of differential operators since the method proposed in this paper
can be applied to a variety of problems of this formulation with few
adjustments only. In this work, we focus on a small set of problems with
specific parameters, initial and boundary conditions. The exact problem
parameters are given in their specific subsections of Sec. 4. We note that an
ODE can be defined as a special case of (1) by focusing only on one
independent variable (either the spatial or temporal).

We discretize the problem in space and time (for simplicity, we use a uniform
grid). In the case of one spatial dimension and one temporal dimension we use
the following notation: $u^{n}_{i}=u(x_{0}+i\Delta x;n\Delta t)$,
$i=0,...,N_{x}$ and $n=0,...,N_{t}$, where $N_{x}$ and $N_{t}$ are the number
of points in the spatial and temporal discretizations, respectively. Given
this notation, a couple of examples of finite difference approximations (of
second-order accuracy in $\Delta t$) for the first and second temporal
derivatives are:

$\begin{aligned} u_{t}&\approx\frac{u^{n+1}_{i}-u^{n-1}_{i}}{2\Delta t}\\\
u_{tt}&\approx\frac{u^{n+1}_{i}-2u^{n}_{i}+u^{n-1}_{i}}{\Delta
t^{2}}\end{aligned}\qquad,$ (2)

with similar expressions for spatial derivatives. For example, if the PDE
contains up to second-order temporal derivatives, we can create an explicit
time marching scheme by using (2) to re-write the approximation of the PDE as
$u^{n+1}=\mathcal{L}_{*}(u^{n},u^{n-1})$ (for ease of notation we have omitted
the spatial index). The expression $\mathcal{L}_{*}$ is often referred to as
the scheme or a kernel, and its exact form depends on the order of the method
and the form of the PDE. Using the initial conditions, given usually as
$u^{0},u^{1}$, one can compute $u^{2}$ and iteratively continue until the
terminal timestep number $N_{t}$ is reached.

To formulate this process as a data-driven problem, a possible approach is to
use $u^{n},u^{n-1}$ as input data, and train a learning algorithm to infer
$u^{n+1}$ as the output data. In this work, we also consider using only
$u^{n}$ to infer $u^{n+1}$. In addition, we also tried marching with larger
timesteps by using $u^{mn}$ to infer $u^{m(n+1)}$, where $m$ is a positive
integer number larger than one. The data used to train the SNN based model
were created using appropriately accurate existing time marching schemes for
ODEs and PDEs.

###  3.2 SNN Formulation

####  3.2.1 Spike encoding-decoding

We encode the input data into spikes. To encode the data we introduce a new
axis for the temporal behavior of the spikes, and the continuous function
values are encoded into binary spikes. We use an approach inspired by the
lower triangular encoding presented in [11]. Suppose that we want to encode a
function $f(x)$ which is defined on the spatial interval $[0,1].$ To encode
the function, first, we consider a discretization of the spatial axis. Second,
we define a temporal axis and we use the number of temporal nodes chosen for
the new temporal axis to discretize the function values. The upper and lower
limits of this discretization are chosen based on the function values and may
very between functions. Then, we add spikes based on the function value. For
example, if the number of nodes in the temporal axis is $10$ and the function
value at a point $x$ is $f(x)=0.4$ (assuming the function values are between
$0$ and $1$), the spike encoding for that value is $(1111000000)$, so it has
four ones as the spikes. We do this for every value of the function and get a
two dimensional array, where the first dimension is the location on the
spatial axis and the second is the location on the temporal axis. The data is
binary as desired. In this way we exploit the sparsity benefit of SNNs. The
drawback of this encoding is that we lose some information. After encoding, we
cannot reconstruct the original signal, and the loss of information depends on
the discretization of the temporal axis. A more detailed description of the
encoding mechanism can be found in [11]. An example of encoding a function is
given in Fig. 2. The function is encoded using 30 steps in the spikes train
(the horizontal axis of Fig. 2(b)). A similar approach is applied for the
cases presented in Sec. 4.

(a) A sample function $f(x)$.

(b) The sample function $f(x)$ encoded into spikes. Pixels with value 1 are in
yellow and pixels with value 0 are in dark blue. The vertical axis corresponds
to the spatial discretization ($j=1,...,N_{x}$) and the horizontal axis
corresponds to the timesteps in the spikes train ($k=1,...,\\#T$).

Figure 2: Spike encoding illustration.

In some of the numerical experiments we also consider other encoding methods.
There are two reasons for this: a) we would like to show that the spiking
marching scheme can operate with other encoding methods, in other words, it is
not limited to the aforementioned encoding method, and b) we are interested in
the effect of different encoding methods on the performance of the spiking
marching scheme. The second encoding method we use is the popular rate
encoding approach, where one normalizes the data points (values of the
function) into the $[0,1]$ interval and for each value, creates Poisson trials
with expected value set as the normalized function value. The number of trials
is set to the number of time steps of the spikes train, which is denoted by
$\\#T$. In that way, the function representation is expanded into a
$N_{x}\times\\#T$ binary images. The third encoding method we use is
representing the number using the float32 representation of it, which is
indeed binary and in this case the number of steps in the spikes train is set
to 32. For more information about these encoding methods we refer the readers
to [11]. To decode the data (at the end of the process) we use the exact
opposite of the encoding procedures mentioned here.

We compare the performance of the spiking marching scheme with lower-
triangular encoding to that with either rate encoding or the float32 encoding.
We point out which encoding method was used for each of the experimental
setups in Sec. 4. The main motivation is showing that the spiking marching
scheme is not encoding-specific, and can work with different types of encoding
methods. Also, we are interested in choosing the encoding method that gives
the best overall performance of the spiking marching scheme (most accurate and
efficient).

Encoding methods induce a certain error for the computation. When encoding the
signal, we use a fixed number of spike train steps ($\\#T$), and the smaller
this number is, the more information is lost. If one wishes to reconstruct the
encoded signal accurately, a large $\\#T$ should be used which then requires
more computational resources to train the network. Facing this trade-off
between efficiency and accuracy, a fixed $\\#T$ is chosen for each problem in
the numerical tests. We investigate the impact of the encoding error by taking
a reference signal, encoding it using each of the three encoding methods,
decoding it back and computing the $L_{2}$ error. The reference signal taken
is
$\eta_{ref}(x_{j})=e^{-\frac{1}{2}}\left(\frac{x_{j}-\mu}{\sigma}\right)^{2},x_{j}\in\\{j\Delta
x\\}_{j=0}^{N_{x}}$ (linearly spaced $[0,1]$ interval with $\Delta x=0.01$),
with $\mu=\frac{1}{2}$ and $\sigma=0.1$. The error is computed by:
$||\eta_{encoded}(x)-\eta_{ref}(x)||_{L_{2}}=\frac{1}{N_{x}}\sum_{j=1}^{N_{x}}|\eta_{encoded}(x_{j})-\eta_{ref}(x_{j})|^{2}$.
The error comparison is given in Table 1. We can observe that the error for
the lower triangular encoding is much lower than the error for the rate
encoding methods. The float32 encoding error is, as expected, the lowest and
in fact it is at the level of machine zero. However, it does not necessarily
translate to better performance of the spiking marching scheme. We explore in
Sec. 4 the impact of the encoding methods on the prediction errors for various
systems.

| $L_{2}$ | Relative $L_{2}$
---|---|---
Lower Triangular | 2.1195e-05 | 3.3822e-04
Rate | 5.8053e-03 | 9.2640e-02
Float32 | 2.8336e-17 | 4.5218e-16
Table 1: Encoding errors comparison. The $L_{2}$ and relative $L_{2}$ errors
of encoding and decoding a reference signal using the three encoding methods
employed in this study (see text for details).

The quality of the decoding (sharpness of the recovered signal) is also
determined by the number of time steps we choose to encode the data. There is
a trade-off between the accuracy (finer spike discretization is more accurate)
and efficiency (more spikes require more computations). We experimented with
several values of spike numbers, eventually choosing one that is suitable for
each problem by trial-and-error.

####  3.2.2 Network architecture

For the SNN architecture we use a relatively simple model, with a Leaky
Integrate and Fire (LIF) layer followed by two dense layers. We use $\\#T$
steps for number of time steps of the SNN. Note that $\\#T$ is different from
the number of timesteps of the physical problem $N_{t}$. The former regards
the number of steps in the spikes train (see a detailed explanation in [11]).
$N_{x}$ is the number of points in the spatial discretization, and the first
dense layer is of size $N_{H}$ (chosen equal to $500$ for all of the numerical
experiments in the current work). The second dense layer has the same size
($\\#T\cdot N_{x}$) as the output layer. Both dense layers are followed by a
Rectified Linear Unit (ReLU) activation. Note that we flatten the inputs
(noted by $\\#T\times N_{x}$ and $\\#T\cdot N_{x}$) in order to apply the
intermediate dense layers. $N_{b}$ is the size of the batch (number of
samples). A schematic of the architecture used in this work is given in Fig.
3. The loss function we use for training the model is the Sigmoid Cross
Entropy (SCE) loss [35].

Figure 3: Schematic of the network architecture, consisting of an input layer
followed by a LIF layer. Then, we flatten the output and use two dense layers
that are trainable (marked in dashed lines). The output layer has the same
size as the input layer.

The approach proposed here is proof-of-concept for the use of SNNs to
construct numerical integrators for differential equations. Thus, we are fully
aware that the simple architecture we propose can be improved. Some studies
use an architecture that mixes LIF layers with dense layers. For example, one
can start with a dense layer, followed by a LIF layer, followed by a dense
layer, followed by another LIF layer (see e.g. tutorials of SNNTorch, a
popular package for training SNNs [36]). A different architecture is proposed
in the tutorial of the LAVA [34] framework, in particular an architecture
consisting of three successive (LIF layer - dense layer) pairs. Because the
LIF layer has a thresholding operation inside of it, performing back-
propagation requires computing gradients through the use of a surrogate model,
which can introduce errors. We chose to use a LIF layer followed by two dense
layers because it does not require a surrogate model for the computation of
back-propagation gradients (the thresholding is done before the dense layers),
and is sufficiently expressive for presenting the capabilities of the proposed
method.

###  3.3 Accuracy metrics

To assess the accuracy of the proposed approach, we use two metrics. The first
is the error committed during a one timestep prediction (inference). We
compute the $L_{2}$ error of the estimate of the solution at a certain
timestep given by the proposed approach with respect to the reference
solution, given as input the reference solution values. Specifically, we use
the reference solution $u^{mn}$ (or both $u^{m(n-1)},u^{mn}$) as input to the
SNN to estimate $u^{m(n+1)}$. This is what the network is trained to do and
the errors are expected to be relatively small. This evaluation serves as a
sanity check, and if it fails it signifies a problem with the training of the
model.

The second metric is what we call cascade prediction (inference). We start
from the solution at the initial timestep and predict the solution at the next
timestep. We then use the predicted solution timestep (not the reference
solution like before) to predict the solution at the next timestep, and so on.
Since every prediction commits an error, the errors can accumulate and cause
divergence of the predicted solution from the reference one. This is a more
challenging accuracy metric, and is the main criterion used to monitor the
accuracy of the proposed approach.

For both the one-step and cascade metrics, we are interested in observing the
ability of the spiking marching scheme to make correct predictions: a) for
times inside the temporal interval used for training (interpolation), and b)
for times outside the temporal interval the network was trained on
(extrapolation). All marching schemes suffer from accumulating numerical
errors over time, and seek to keep the cascade inference error from growing
too fast.

##  4 Results

###  4.1 ODE Examples

We have applied the proposed approach to solve the initial-value problem for
two ODE systems. The first is the Van der Pol oscillator [37] while the second
is the Lorenz system, which is chaotic and thus presents a challenge for any
time marching scheme [38].

####  4.1.1 Van der Pol oscillator

The Van der Pol equation is a mathematical model for the motion of an
oscillator, which is nonlinearly damped. It is a second-order ODE, which can
be rewritten as a system of two first order ODEs:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=y$ $\displaystyle\frac{dy}{dt}$
$\displaystyle=\mu(1-x^{2})y-x$

where $x$ and $y$ are the position and velocity of the oscillator,
respectively, $t$ is the temporal variable, and $\mu$ is a parameter. We solve
the Van der Pol system in the interval $[a,b]$. For this test, we choose
$\mu=2,a=0,b=10$ with initial condition $x(a)=1$ and $y(a)=0.$

To create the dataset for training we use a numerical initial value problem
ODE integrator in Python (from the SciPy library [39]). We compute the
solution for $N_{t}=2,500$ steps and sub-sample every 10th step. This means
that in the new dataset, we effectively use a $10$ times larger $\Delta t$,
ending up with $250$ steps for training and testing. The first 200 steps are
used for training (interpolation) and the last 50 for testing (extrapolation).
We use both previous and current time steps to infer the next time step, so
each sample is of size $4$ (the $x,y$ components of both previous and current
steps), which substitutes in the ODE setting the notion of spatial
discretization (see $N_{x}$ in Fig. 3). We then encode the data using the
lower triangular encoding with $\\#T=100$ spike train steps. The model is
trained with $5000$ epochs. Fig. 4(a) shows the trajectory prediction using
SMS for $N_{t}=250$ timesteps with both cascade and non-cascade predictions.
The evolution of the error with time is given in Fig. 4(b).

(a) Trajectory prediction.

(b) Error evolution over timesteps. The first 200 timesteps show interpolation
and the remaining 50 show extrapolation. Here the error at timestep $n$ is
defined as $\frac{1}{2}((x^{n}-\tilde{x}^{n})^{2}+(y^{n}-\tilde{y}^{n})^{2})$,
where $x,y$ are the prediction and $\tilde{x},\tilde{y}$ are the reference
solution.

Figure 4: Van der Pol oscillator. Encoding method: Lower triangular encoding.

We observe the highly accurate one-step and cascade predictions in the first
100 steps (within the interpolation interval). Also, the error accumulates
slowly outside of the training interval (beyond 200 steps) where the marching
scheme performs extrapolation. We note that the accuracy of the trained model
is promising, given that it takes steps which are 10 times larger than the ODE
integrator used to produce the reference solution.

####  4.1.2 Lorenz system

The Lorenz system is given by:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=\sigma(y-x)$
$\displaystyle\frac{dy}{dt}$ $\displaystyle=x(\rho-z)-y$
$\displaystyle\frac{dz}{dt}$ $\displaystyle=xy-\beta z$

where $x,y,z$ describe the state of the system, $t$ is the temporal variable
and $\sigma,\rho$, and $\beta$ are parameters. For this experiment we choose
$\sigma=10,\rho=28$, and $\beta=\frac{8}{3}$, which are common choices for
this system and lead to chaotic behavior. The initial state is set to
$(x_{a},y_{a},z_{a})=(1,1,1)$ and the time interval is set to $[a,b]=[0,40]$
with $N_{t}=4,000$ timesteps. We then sub-sample the dataset and keep only 400
timesteps by using every 10th timestep in the original dataset. In this case
we use both previous and current time steps to infer the next time step, so
each sample is of size $6$ (the $x,y,z$ components of both previous and
current steps), which substitutes in the ODE setting the notion of spatial
discretization (see $N_{x}$ in Fig. 3). For the Lorenz system we also
experiment with using only the current time step to infer the next time step
as mentioned in Sec. 3.1. We achieve good results for the interpolation part,
but for extrapolation the network struggles to compete with the two steps
method. We encode the dataset using the lower triangular encoding, this time,
with $\\#T=1,000$ steps. The network is trained for $5000$ epochs on the first
$320$ timesteps. To show the extrapolation capabilities, we also predict the
solution for the remaining $80$ timesteps. The results are given in Fig.
5(a-b).

In this case, while the interpolation is accurate, extrapolation is rather
difficult for the SMS. However, the error is still relatively small, and is
not exploding by many orders of magnitude every iteration as often experienced
with certain explicit numerical methods (demonstrated in [31]).

For the Lorenz system we also demonstrate (see Fig. 5(c-f)) the performance of
the rate and float32 encoders which were defined in 3.2.1. We observe that for
the rate encoding, the results suffer from large errors (yet not exponentially
increasing in time). For the float32 encoding, we observe that the network was
able to accurately infer the solution for both cascade and non-cascade
predictions, but extrapolation induces an error that is about an order of
magnitude larger than the one for the lower triangular encoding.

(a) Trajectory prediction, lower triangular encoding.

(b) Evolution of error over timesteps. The first 320 timesteps show
interpolation and the remaining 80 show extrapolation. Here the error at
timestep $n$ is defined as
$\frac{1}{3}((x^{n}-\tilde{x}^{n})^{2}+(y^{n}-\tilde{y}^{n})^{2}+(z^{n}-\tilde{z}^{n})^{2})$,
where $x,y,z$ are the prediction and $\tilde{x},\tilde{y},\tilde{z}$ are the
reference solution.

(c) Trajectory prediction, rate encoding.

(d) Similar configuration as in (b) but with rate encoding.

(e) Trajectory prediction, float32 encoding.

(f) Similar configuration as in (b) but with float32 encoding.

Figure 5: Lorenz system. Results for various encoding schemes.

###  4.2 PDE Examples

We have also applied the proposed approach to the wave and heat equations in
one spatial dimension. While these PDEs are straightforward to analyze, both
have interesting features that provide insight into the performance of the
SMS.

####  4.2.1 Wave equation

The one dimensional wave equation we solve is given by:

$\displaystyle\frac{\partial^{2}u}{\partial
t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$ $\displaystyle,\qquad
t\in[a,b],\quad x\in[x_{min},x_{max}],$ $\displaystyle u(0,x)=u_{0}(x)$
$\displaystyle,\qquad x\in[x_{min},x_{max}],$ $\displaystyle\frac{\partial
u(0,x)}{\partial t}=v_{0}(x)$ $\displaystyle,\qquad x\in[x_{min},x_{max}],$
$\displaystyle u(t,x_{min})=u(t,x_{max})=0$ $\displaystyle,\qquad t\in[a,b],$

where $u(t,x)$ is the wave pressure, $x$ is the spatial variable, $t$ is the
temporal variable, $u_{0}(x)$ is the initial wave pressure, $v_{0}(x)$ is the
initial wave velocity, chosen as $v_{0}(x)\equiv 0$ in this experiment. The
zero Dirichlet boundary condition set in the last equation causes reflective
scattering from the edges of the domain (set by $x_{min}$ and $x_{max}$). For
this simulation we choose $x_{min}=0$, $x_{max}=1$, and the spatial increments
are $\Delta x=0.01$. The wave velocity $c$ is chosen to be 1. The initial time
is set to $a=0$ and the temporal increments are chosen to satisfy the Courant-
Friedrichs-Lewy condition: $\Delta t=\frac{\Delta x}{c}=0.01$. We integrate
for $N_{t}=1,000$ steps so $b=\Delta tN_{t}=10$, and use both previous and
current time steps to predict the next time step. We do not sub-sample the
dataset in this case, and choose $\\#T=100$ for encoding the data. We choose
as initial condition the function:

$\displaystyle
f(x)=e^{-\left(\frac{x-\overline{x}}{0.05}\right)^{2}},\quad\overline{x}=\frac{x_{max}-x_{min}}{2}$

We train the network for 5000 epochs, using the first 800 timesteps for
training (interpolation) and the remaining 200 for extrapolation. The results
are given in Fig. 6. From the results we observe that SMS works well for the
wave propagation problem. For this problem we also investigate the effect of
the additional two encoding methods. We observe a similar behavior to the one
observed in the Lorenz system results. The rate encoding induces a much higher
error and produces a completely wrong solution. However, the float32 encoder
is performing significantly well, achieving machine zero error for the cascade
prediction at interpolation and the extrapolation intervals.

(a) The prediction snapshot at timestep 980 which is in the temporal
extrapolation regime, lower triangular encoding.

(b) Evolution of error over time-steps. The first 800 timesteps show
interpolation and the remaining 200 show extrapolation. Here the error at
timestep $n$ is defined as
$\frac{1}{N_{x}}\sum_{j=1}^{N_{x}}(u_{j}^{n}-\tilde{u}_{j}^{n})^{2}$, where
$u$ is the prediction and $\tilde{u}$ is the reference solution.

(c) The prediction snapshot at timestep 980 which is in the temporal
extrapolation regime, rate encoding.

(d) Similar configuration as in (b) but with rate encoding.

(e) The prediction snapshot at timestep 980 which is in the temporal
extrapolation regime, float32 encoding.

(f) Similar configuration as in (b) but with float32 encoding.

Figure 6: Wave equation. Results for various encoding schemes.

####  4.2.2 Heat equation

The one-dimensional heat problem we solve is given by:

$\displaystyle\frac{\partial u}{\partial
t}=\alpha\frac{\partial^{2}u}{\partial x^{2}}$ $\displaystyle,\qquad
t\in[a,b],\quad x\in[x_{min},x_{max}]$ $\displaystyle u(0,x)=u_{0}(x)$
$\displaystyle,\qquad x\in[x_{min},x_{max}]$

where $u(t,x)$ is the temperature, $x$ is the spatial variable, $t$ is the
temporal variable, and $u_{0}(x)$ is the initial temperature distribution. We
use Dirichlet boundary condition set to 0. For this simulation we choose
$x_{min}=0$, $x_{max}=1$, and the spatial increments are $\Delta x=0.01$. The
diffusion term $\alpha$ is chosen to be 1. The temporal domain is set to $a=0$
and the temporal increments are chosen to satisfy the condition: $\Delta
t=\frac{0.4*\Delta x^{2}}{\alpha}=0.00004$. We integrate for $N_{t}=1,000$
steps so $b=0.04$, and use both previous and current time steps to predict the
next time step. The number $\\#T$ of encoding steps is chosen to be 100 . We
sub-sample the data keeping every 10th timestep, so that we have 100 timesteps
in total. As an initial condition we choose the function:

$\displaystyle f(x)=\begin{cases}0&,\quad 0\leq x<0.45\\\ 20(x-0.45)&,\quad
0.45\leq x<0.5\\\ 20(0.55-x)&,\quad 0.5\leq x<0.55\\\ 0&,\quad 0.55\leq x\leq
1\\\ \end{cases}.$

We train the network for 5000 epochs and split the data into 80 timesteps used
for training and the remaining 20 used for evaluating the extrapolation
capabilities. The results are given in Figure 7.

(a) The prediction snapshot at timestep 90 which is in the extrapolation
regime.

(b) Evolution of error over timesteps. The first 80 time-steps show
interpolation and the remaining 20 show extrapolation. Here the error at
timestep $n$ is defined as
$\frac{1}{N_{x}}\sum_{j=1}^{N_{x}}(u_{j}^{n}-\tilde{u}_{j}^{n})^{2}$, where
$u$ is the prediction and $\tilde{u}$ is the reference solution.

Figure 7: Heat equation. Encoding method: Lower triangular encoding.

We observe that SMS is able to capture the fine changes in the heat diffusion
correctly and extrapolate as well (note the small error magnitude).

##  5 Discussion and future work

We have presented a novel method for long time integration using a SNN based
marching scheme. We formulated the time integration problem as a data-driven
problem and trained a SNN to be an explicit numerical scheme, receiving
information about the solution from previous timesteps as input and predicting
(inferring) the solution at the next timestep. We conducted several numerical
experiments that showed the ability of the proposed method to interpolate and
extrapolate, concluding that it is indeed an explicit numerical scheme that
can be used for time marching. We measured the error of this spiking marching
scheme and showed that the errors stay relatively small, allowing for accurate
predictions for long time intervals. To exploit the full potential of the
proposed approach, one can implement this on a neuromorphic chip and make the
SNN inference very efficient, thus enabling very fine timestepping, leading to
highly accurate solutions over time. This can be done in future work when
Loihi-2 will be broadly available.

##  6 Acknowledgements

This work was supported by the Vannevar Bush Faculty Fellowship award (GEK)
from ONR (N00014-22-1-2795). The work of PS and GEK is supported by the U.S.
Department of Energy, Advanced Scientific Computing Research program, under
the Scalable, Efficient and Accelerated Causal Reasoning Operators, Graphs and
Spikes for Earth and Embedded Systems (SEA-CROGS) project, DE-SC0023191.
Pacific Northwest National Laboratory (PNNL) is a multi-program national
laboratory operated for the U.S. Department of Energy (DOE) by Battelle
Memorial Institute under Contract No. DE-AC05-76RL01830.

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D. Giljanović, N. Godinovic, D. Lelas, I. Puljak, T. Sculac University of
Split, Faculty of Science, Split, Croatia
Z. Antunovic, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia
V. Brigljevic, S. Ceci, D. Ferencek, K. Kadija, B. Mesic, M. Roguljic, A.
Starodumov9, T. Susa University of Cyprus, Nicosia, Cyprus
M.W. Ather, A. Attikis, E. Erodotou, A. Ioannou, M. Kolosova, S. Konstantinou,
G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A. Razis, H.
Rykaczewski, D. Tsiakkouri Charles University, Prague, Czech Republic
M. Finger10, M. Finger Jr.10, A. Kveton, J. Tomsa Escuela Politecnica
Nacional, Quito, Ecuador
E. Ayala Universidad San Francisco de Quito, Quito, Ecuador
E. Carrera Jarrin Academy of Scientific Research and Technology of the Arab
Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt
S. Abu Zeid11, S. Khalil12 National Institute of Chemical Physics and
Biophysics, Tallinn, Estonia
S. Bhowmik, A. Carvalho Antunes De Oliveira, R.K. Dewanjee, K. Ehataht, M.
Kadastik, M. Raidal, C. Veelken Department of Physics, University of Helsinki,
Helsinki, Finland
P. Eerola, L. Forthomme, H. Kirschenmann, K. Osterberg, M. Voutilainen
Helsinki Institute of Physics, Helsinki, Finland
F. Garcia, J. Havukainen, J.K. Heikkilä, T. Järvinen, V. Karimäki, M.S. Kim,
R. Kinnunen, T. Lampén, K. Lassila-Perini, S. Laurila, S. Lehti, T. Lindén, P.
Luukka, T. Mäenpää, H. Siikonen, E. Tuominen, J. Tuominiemi Lappeenranta
University of Technology, Lappeenranta, Finland
T. Tuuva IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
M. Besancon, F. Couderc, M. Dejardin, D. Denegri, B. Fabbro, J.L. Faure, F.
Ferri, S. Ganjour, A. Givernaud, P. Gras, G. Hamel de Monchenault, P. Jarry,
C. Leloup, E. Locci, J. Malcles, J. Rander, A. Rosowsky, M.Ö. Sahin, A. Savoy-
Navarro13, M. Titov Laboratoire Leprince-Ringuet, Ecole polytechnique,
CNRS/IN2P3, Université Paris-Saclay, Palaiseau, France
S. Ahuja, C. Amendola, F. Beaudette, P. Busson, C. Charlot, B. Diab, G.
Falmagne, R. Granier de Cassagnac, I. Kucher, A. Lobanov, C. Martin Perez, M.
Nguyen, C. Ochando, P. Paganini, J. Rembser, R. Salerno, J.B. Sauvan, Y.
Sirois, A. Zabi, A. Zghiche Université de Strasbourg, CNRS, IPHC UMR 7178,
Strasbourg, France
J.-L. Agram14, J. Andrea, D. Bloch, G. Bourgatte, J.-M. Brom, E.C. Chabert, C.
Collard, E. Conte14, J.-C. Fontaine14, D. Gelé, U. Goerlach, M. Jansová, A.-C.
Le Bihan, N. Tonon, P. Van Hove Centre de Calcul de l’Institut National de
Physique Nucleaire et de Physique des Particules, CNRS/IN2P3, Villeurbanne,
France
S. Gadrat Université de Lyon, Université Claude Bernard Lyon 1, CNRS-IN2P3,
Institut de Physique Nucléaire de Lyon, Villeurbanne, France
S. Beauceron, C. Bernet, G. Boudoul, C. Camen, A. Carle, N. Chanon, R.
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Gouzevitch, B. Ille, Sa. Jain, F. Lagarde, I.B. Laktineh, H. Lattaud, A.
Lesauvage, M. Lethuillier, L. Mirabito, S. Perries, V. Sordini, L. Torterotot,
G. Touquet, M. Vander Donckt, S. Viret Georgian Technical University, Tbilisi,
Georgia
A. Khvedelidze10 Tbilisi State University, Tbilisi, Georgia
Z. Tsamalaidze10 RWTH Aachen University, I. Physikalisches Institut, Aachen,
Germany
C. Autermann, L. Feld, M.K. Kiesel, K. Klein, M. Lipinski, D. Meuser, A.
Pauls, M. Preuten, M.P. Rauch, J. Schulz, M. Teroerde, B. Wittmer RWTH Aachen
University, III. Physikalisches Institut A, Aachen, Germany
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B, Aachen, Germany
G. Flügge, W. Haj Ahmad15, O. Hlushchenko, T. Kress, T. Müller, A. Nehrkorn,
A. Nowack, C. Pistone, O. Pooth, D. Roy, H. Sert, A. Stahl16 Deutsches
Elektronen-Synchrotron, Hamburg, Germany
M. Aldaya Martin, P. Asmuss, I. Babounikau, H. Bakhshiansohi, K. Beernaert, O.
Behnke, A. Bermúdez Martínez, D. Bertsche, A.A. Bin Anuar, K. Borras17, V.
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P. Schütze, C. Schwanenberger, R. Shevchenko, A. Singh, H. Tholen, O. Turkot,
A. Vagnerini, M. Van De Klundert, R. Walsh, Y. Wen, K. Wichmann, C. Wissing,
O. Zenaiev, R. Zlebcik University of Hamburg, Hamburg, Germany
R. Aggleton, S. Bein, L. Benato, A. Benecke, V. Blobel, T. Dreyer, A.
Ebrahimi, F. Feindt, A. Fröhlich, C. Garbers, E. Garutti, D. Gonzalez, P.
Gunnellini, J. Haller, A. Hinzmann, A. Karavdina, G. Kasieczka, R. Klanner, R.
Kogler, N. Kovalchuk, S. Kurz, V. Kutzner, J. Lange, T. Lange, A. Malara, J.
Multhaup, C.E.N. Niemeyer, A. Perieanu, A. Reimers, O. Rieger, C. Scharf, P.
Schleper, S. Schumann, J. Schwandt, J. Sonneveld, H. Stadie, G. Steinbrück,
F.M. Stober, B. Vormwald, I. Zoi Karlsruher Institut fuer Technologie,
Karlsruhe, Germany
M. Akbiyik, C. Barth, M. Baselga, S. Baur, T. Berger, E. Butz, R. Caspart, T.
Chwalek, W. De Boer, A. Dierlamm, K. El Morabit, N. Faltermann, M. Giffels, P.
Goldenzweig, A. Gottmann, M.A. Harrendorf, F. Hartmann16, U. Husemann, S.
Kudella, S. Mitra, M.U. Mozer, D. Müller, Th. Müller, M. Musich, A. Nürnberg,
G. Quast, K. Rabbertz, M. Schröder, I. Shvetsov, H.J. Simonis, R. Ulrich, M.
Wassmer, M. Weber, C. Wöhrmann, R. Wolf Institute of Nuclear and Particle
Physics (INPP), NCSR Demokritos, Aghia Paraskevi, Greece
G. Anagnostou, P. Asenov, G. Daskalakis, T. Geralis, A. Kyriakis, D. Loukas,
G. Paspalaki National and Kapodistrian University of Athens, Athens, Greece
M. Diamantopoulou, G. Karathanasis, P. Kontaxakis, A. Manousakis-katsikakis,
A. Panagiotou, I. Papavergou, N. Saoulidou, A. Stakia, K. Theofilatos, K.
Vellidis, E. Vourliotis National Technical University of Athens, Athens,
Greece
G. Bakas, K. Kousouris, I. Papakrivopoulos, G. Tsipolitis University of
Ioánnina, Ioánnina, Greece
I. Evangelou, C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, S. Mallios, K.
Manitara, N. Manthos, I. Papadopoulos, J. Strologas, F.A. Triantis, D.
Tsitsonis MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary
M. Bartók20, R. Chudasama, M. Csanad, P. Major, K. Mandal, A. Mehta, M.I.
Nagy, G. Pasztor, O. Surányi, G.I. Veres Wigner Research Centre for Physics,
Budapest, Hungary
G. Bencze, C. Hajdu, D. Horvath21, F. Sikler, T.Á. Vámi, V. Veszpremi, G.
Vesztergombi${}^{\textrm{\textdagger}}$ Institute of Nuclear Research ATOMKI,
Debrecen, Hungary
N. Beni, S. Czellar, J. Karancsi20, A. Makovec, J. Molnar, Z. Szillasi
Institute of Physics, University of Debrecen, Debrecen, Hungary
P. Raics, D. Teyssier, Z.L. Trocsanyi, B. Ujvari Eszterhazy Karoly University,
Karoly Robert Campus, Gyongyos, Hungary
T. Csorgo, W.J. Metzger, F. Nemes, T. Novak Indian Institute of Science
(IISc), Bangalore, India
S. Choudhury, J.R. Komaragiri, P.C. Tiwari National Institute of Science
Education and Research, HBNI, Bhubaneswar, India
S. Bahinipati23, C. Kar, G. Kole, P. Mal, T. Mishra, V.K. Muraleedharan Nair
Bindhu, A. Nayak24, D.K. Sahoo23, S.K. Swain Panjab University, Chandigarh,
India
S. Bansal, S.B. Beri, V. Bhatnagar, S. Chauhan, R. Chawla, N. Dhingra, R.
Gupta, A. Kaur, M. Kaur, S. Kaur, P. Kumari, M. Lohan, M. Meena, K. Sandeep,
S. Sharma, J.B. Singh, A.K. Virdi, G. Walia University of Delhi, Delhi, India
A. Bhardwaj, B.C. Choudhary, R.B. Garg, M. Gola, S. Keshri, Ashok Kumar, M.
Naimuddin, P. Priyanka, K. Ranjan, Aashaq Shah, R. Sharma Saha Institute of
Nuclear Physics, HBNI, Kolkata, India
R. Bhardwaj25, M. Bharti25, R. Bhattacharya, S. Bhattacharya, U. Bhawandeep25,
D. Bhowmik, S. Dutta, S. Ghosh, M. Maity26, K. Mondal, S. Nandan, A. Purohit,
P.K. Rout, G. Saha, S. Sarkar, T. Sarkar26, M. Sharan, B. Singh25, S. Thakur25
Indian Institute of Technology Madras, Madras, India
P.K. Behera, P. Kalbhor, A. Muhammad, P.R. Pujahari, A. Sharma, A.K. Sikdar
Bhabha Atomic Research Centre, Mumbai, India
D. Dutta, V. Jha, V. Kumar, D.K. Mishra, P.K. Netrakanti, L.M. Pant, P. Shukla
Tata Institute of Fundamental Research-A, Mumbai, India
T. Aziz, M.A. Bhat, S. Dugad, G.B. Mohanty, N. Sur, RavindraKumar Verma Tata
Institute of Fundamental Research-B, Mumbai, India
S. Banerjee, S. Bhattacharya, S. Chatterjee, P. Das, M. Guchait, S. Karmakar,
S. Kumar, G. Majumder, K. Mazumdar, N. Sahoo, S. Sawant Indian Institute of
Science Education and Research (IISER), Pune, India
S. Chauhan, S. Dube, V. Hegde, B. Kansal, A. Kapoor, K. Kothekar, S. Pandey,
A. Rane, A. Rastogi, S. Sharma Institute for Research in Fundamental Sciences
(IPM), Tehran, Iran
S. Chenarani27, E. Eskandari Tadavani, S.M. Etesami27, M. Khakzad, M.
Mohammadi Najafabadi, M. Naseri, F. Rezaei Hosseinabadi University College
Dublin, Dublin, Ireland
M. Felcini, M. Grunewald INFN Sezione di Bari a, Università di Bari b,
Politecnico di Bari c, Bari, Italy
M. Abbresciaa,b, R. Alya,b,28, C. Calabriaa,b, A. Colaleoa, D. Creanzaa,c, L.
Cristellaa,b, N. De Filippisa,c, M. De Palmaa,b, A. Di Florioa,b, W.
Elmetenaweea,b, L. Fiorea, A. Gelmia,b, G. Iasellia,c, M. Incea,b, S.
Lezkia,b, G. Maggia,c, M. Maggia, G. Minielloa,b, S. Mya,b, S. Nuzzoa,b, A.
Pompilia,b, G. Pugliesea,c, R. Radognaa, A. Ranieria, G. Selvaggia,b, L.
Silvestrisa, F.M. Simonea, R. Vendittia, P. Verwilligena INFN Sezione di
Bologna a, Università di Bologna b, Bologna, Italy
G. Abbiendia, C. Battilanaa,b, D. Bonacorsia,b, L. Borgonovia,b, S. Braibant-
Giacomellia,b, R. Campaninia,b, P. Capiluppia,b, A. Castroa,b, F.R. Cavalloa,
C. Cioccaa, G. Codispotia,b, M. Cuffiania,b, G.M. Dallavallea, F. Fabbria, A.
Fanfania,b, E. Fontanesia,b, P. Giacomellia, C. Grandia, L. Guiduccia,b, F.
Iemmia,b, S. Lo Meoa,29, S. Marcellinia, G. Masettia, F.L. Navarriaa,b, A.
Perrottaa, F. Primaveraa,b, A.M. Rossia,b, T. Rovellia,b, G.P. Sirolia,b, N.
Tosia INFN Sezione di Catania a, Università di Catania b, Catania, Italy
S. Albergoa,b,30, S. Costaa,b, A. Di Mattiaa, R. Potenzaa,b, A. Tricomia,b,30,
C. Tuvea,b INFN Sezione di Firenze a, Università di Firenze b, Firenze, Italy
G. Barbaglia, A. Cassese, R. Ceccarelli, V. Ciullia,b, C. Civininia, R.
D’Alessandroa,b, E. Focardia,b, G. Latinoa,b, P. Lenzia,b, M. Meschinia, S.
Paolettia, G. Sguazzonia, L. Viliania INFN Laboratori Nazionali di Frascati,
Frascati, Italy
L. Benussi, S. Bianco, D. Piccolo INFN Sezione di Genova a, Università di
Genova b, Genova, Italy
M. Bozzoa,b, F. Ferroa, R. Mulargiaa,b, E. Robuttia, S. Tosia,b INFN Sezione
di Milano-Bicocca a, Università di Milano-Bicocca b, Milano, Italy
A. Benagliaa, A. Beschia,b, F. Brivioa,b, V. Cirioloa,b,16, S. Di Guidaa,b,16,
M.E. Dinardoa,b, P. Dinia, S. Gennaia, A. Ghezzia,b, P. Govonia,b, L.
Guzzia,b, M. Malbertia, S. Malvezzia, D. Menascea, F. Montia,b, L. Moronia, M.
Paganonia,b, D. Pedrinia, S. Ragazzia,b, T. Tabarelli de Fatisa,b, D. Zuoloa,b
INFN Sezione di Napoli a, Università di Napoli ’Federico II’ b, Napoli, Italy,
Università della Basilicata c, Potenza, Italy, Università G. Marconi d, Roma,
Italy
S. Buontempoa, N. Cavalloa,c, A. De Iorioa,b, A. Di Crescenzoa,b, F.
Fabozzia,c, F. Fiengaa, G. Galatia, A.O.M. Iorioa,b, L. Listaa,b, S.
Meolaa,d,16, P. Paoluccia,16, B. Rossia, C. Sciaccaa,b, E. Voevodinaa,b INFN
Sezione di Padova a, Università di Padova b, Padova, Italy, Università di
Trento c, Trento, Italy
P. Azzia, N. Bacchettaa, D. Biselloa,b, A. Bolettia,b, A. Bragagnoloa,b, R.
Carlina,b, P. Checchiaa, P. De Castro Manzanoa, T. Dorigoa, U. Dossellia, F.
Gasparinia,b, U. Gasparinia,b, A. Gozzelinoa, S.Y. Hoha,b, P. Lujana, M.
Margonia,b, A.T. Meneguzzoa,b, J. Pazzinia,b, M. Presillab, P. Ronchesea,b, R.
Rossina,b, F. Simonettoa,b, A. Tikoa, M. Tosia,b, M. Zanettia,b, P. Zottoa,b,
G. Zumerlea,b INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy
A. Braghieria, D. Fiorinaa,b, P. Montagnaa,b, S.P. Rattia,b, V. Rea, M.
Ressegottia,b, C. Riccardia,b, P. Salvinia, I. Vaia, P. Vituloa,b INFN Sezione
di Perugia a, Università di Perugia b, Perugia, Italy
M. Biasinia,b, G.M. Bileia, D. Ciangottinia,b, L. Fanòa,b, P. Laricciaa,b, R.
Leonardia,b, E. Manonia, G. Mantovania,b, V. Mariania,b, M. Menichellia, A.
Rossia,b, A. Santocchiaa,b, D. Spigaa INFN Sezione di Pisa a, Università di
Pisa b, Scuola Normale Superiore di Pisa c, Pisa, Italy
K. Androsova, P. Azzurria, G. Bagliesia, V. Bertacchia,c, L. Bianchinia, T.
Boccalia, R. Castaldia, M.A. Cioccia,b, R. Dell’Orsoa, G. Fedia, L.
Gianninia,c, A. Giassia, M.T. Grippoa, F. Ligabuea,c, E. Mancaa,c, G.
Mandorlia,c, A. Messineoa,b, F. Pallaa, A. Rizzia,b, G. Rolandi31, S. Roy
Chowdhury, A. Scribanoa, P. Spagnoloa, R. Tenchinia, G. Tonellia,b, N. Turini,
A. Venturia, P.G. Verdinia INFN Sezione di Roma a, Sapienza Università di Roma
b, Rome, Italy
F. Cavallaria, M. Cipriania,b, D. Del Rea,b, E. Di Marcoa,b, M. Diemoza, E.
Longoa,b, P. Meridiania, G. Organtinia,b, F. Pandolfia, R. Paramattia,b, C.
Quarantaa,b, S. Rahatloua,b, C. Rovellia, F. Santanastasioa,b, L. Soffia,b
INFN Sezione di Torino a, Università di Torino b, Torino, Italy, Università
del Piemonte Orientale c, Novara, Italy
N. Amapanea,b, R. Arcidiaconoa,c, S. Argiroa,b, M. Arneodoa,c, N. Bartosika,
R. Bellana,b, A. Bellora, C. Biinoa, A. Cappatia,b, N. Cartigliaa, S.
Comettia, M. Costaa,b, R. Covarellia,b, N. Demariaa, B. Kiania,b, C.
Mariottia, S. Masellia, E. Migliorea,b, V. Monacoa,b, E. Monteila,b, M.
Montenoa, M.M. Obertinoa,b, G. Ortonaa,b, L. Pachera,b, N. Pastronea, M.
Pelliccionia, G.L. Pinna Angionia,b, A. Romeroa,b, M. Ruspaa,c, R.
Salvaticoa,b, V. Solaa, A. Solanoa,b, D. Soldia,b, A. Staianoa INFN Sezione di
Trieste a, Università di Trieste b, Trieste, Italy
S. Belfortea, V. Candelisea,b, M. Casarsaa, F. Cossuttia, A. Da Rolda,b, G.
Della Riccaa,b, F. Vazzolera,b, A. Zanettia Kyungpook National University,
Daegu, Korea
B. Kim, D.H. Kim, G.N. Kim, J. Lee, S.W. Lee, C.S. Moon, Y.D. Oh, S.I. Pak, S.
Sekmen, D.C. Son, Y.C. Yang Chonnam National University, Institute for
Universe and Elementary Particles, Kwangju, Korea
H. Kim, D.H. Moon, G. Oh Hanyang University, Seoul, Korea
B. Francois, T.J. Kim, J. Park Korea University, Seoul, Korea
S. Cho, S. Choi, Y. Go, D. Gyun, S. Ha, B. Hong, K. Lee, K.S. Lee, J. Lim, J.
Park, S.K. Park, Y. Roh, J. Yoo Kyung Hee University, Department of Physics
J. Goh Sejong University, Seoul, Korea
H.S. Kim Seoul National University, Seoul, Korea
J. Almond, J.H. Bhyun, J. Choi, S. Jeon, J. Kim, J.S. Kim, H. Lee, K. Lee, S.
Lee, K. Nam, M. Oh, S.B. Oh, B.C. Radburn-Smith, U.K. Yang, H.D. Yoo, I. Yoon,
G.B. Yu University of Seoul, Seoul, Korea
D. Jeon, H. Kim, J.H. Kim, J.S.H. Lee, I.C. Park, I.J Watson Sungkyunkwan
University, Suwon, Korea
Y. Choi, C. Hwang, Y. Jeong, J. Lee, Y. Lee, I. Yu Riga Technical University,
Riga, Latvia
V. Veckalns32 Vilnius University, Vilnius, Lithuania
V. Dudenas, A. Juodagalvis, G. Tamulaitis, J. Vaitkus National Centre for
Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia
Z.A. Ibrahim, F. Mohamad Idris33, W.A.T. Wan Abdullah, M.N. Yusli, Z. Zolkapli
Universidad de Sonora (UNISON), Hermosillo, Mexico
J.F. Benitez, A. Castaneda Hernandez, J.A. Murillo Quijada, L. Valencia Palomo
Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico
H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-De La Cruz34, R. Lopez-
Fernandez, A. Sanchez-Hernandez Universidad Iberoamericana, Mexico City,
Mexico
S. Carrillo Moreno, C. Oropeza Barrera, M. Ramirez-Garcia, F. Vazquez Valencia
Benemerita Universidad Autonoma de Puebla, Puebla, Mexico
J. Eysermans, I. Pedraza, H.A. Salazar Ibarguen, C. Uribe Estrada Universidad
Autónoma de San Luis Potosí, San Luis Potosí, Mexico
A. Morelos Pineda University of Montenegro, Podgorica, Montenegro
J. Mijuskovic, N. Raicevic University of Auckland, Auckland, New Zealand
D. Krofcheck University of Canterbury, Christchurch, New Zealand
S. Bheesette, P.H. Butler National Centre for Physics, Quaid-I-Azam
University, Islamabad, Pakistan
A. Ahmad, M. Ahmad, Q. Hassan, H.R. Hoorani, W.A. Khan, M.A. Shah, M. Shoaib,
M. Waqas AGH University of Science and Technology Faculty of Computer Science,
Electronics and Telecommunications, Krakow, Poland
V. Avati, L. Grzanka, M. Malawski National Centre for Nuclear Research,
Swierk, Poland
H. Bialkowska, M. Bluj, B. Boimska, M. Górski, M. Kazana, M. Szleper, P.
Zalewski Institute of Experimental Physics, Faculty of Physics, University of
Warsaw, Warsaw, Poland
K. Bunkowski, A. Byszuk35, K. Doroba, A. Kalinowski, M. Konecki, J.
Krolikowski, M. Misiura, M. Olszewski, M. Walczak Laboratório de
Instrumentação e Física Experimental de Partículas, Lisboa, Portugal
M. Araujo, P. Bargassa, D. Bastos, A. Di Francesco, P. Faccioli, B. Galinhas,
M. Gallinaro, J. Hollar, N. Leonardo, T.S. Niknejad, J. Seixas, K. Shchelina,
G. Strong, O. Toldaiev, J. Varela Joint Institute for Nuclear Research, Dubna,
Russia
S. Afanasiev, P. Bunin, M. Gavrilenko, I. Golutvin, I. Gorbunov, A. Kamenev,
V. Karjavine, A. Lanev, A. Malakhov, V. Matveev36,37, P. Moisenz, V. Palichik,
V. Perelygin, M. Savina, S. Shmatov, S. Shulha, N. Skatchkov, V. Smirnov, N.
Voytishin, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St.
Petersburg), Russia
L. Chtchipounov, V. Golovtcov, Y. Ivanov, V. Kim38, E. Kuznetsova39, P.
Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, D. Sosnov, V. Sulimov, L.
Uvarov, A. Vorobyev Institute for Nuclear Research, Moscow, Russia
Yu. Andreev, A. Dermenev, S. Gninenko, N. Golubev, A. Karneyeu, M. Kirsanov,
N. Krasnikov, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical
and Experimental Physics named by A.I. Alikhanov of NRC ‘Kurchatov Institute’,
Moscow, Russia
V. Epshteyn, V. Gavrilov, N. Lychkovskaya, A. Nikitenko40, V. Popov, I.
Pozdnyakov, G. Safronov, A. Spiridonov, A. Stepennov, M. Toms, E. Vlasov, A.
Zhokin Moscow Institute of Physics and Technology, Moscow, Russia
T. Aushev National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
O. Bychkova, R. Chistov41, M. Danilov41, S. Polikarpov41, E. Tarkovskii P.N.
Lebedev Physical Institute, Moscow, Russia
V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Terkulov Skobeltsyn
Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow,
Russia
A. Belyaev, E. Boos, M. Dubinin42, L. Dudko, A. Ershov, A. Gribushin, V.
Klyukhin, O. Kodolova, I. Lokhtin, S. Obraztsov, S. Petrushanko, V. Savrin, A.
Snigirev Novosibirsk State University (NSU), Novosibirsk, Russia
A. Barnyakov43, V. Blinov43, T. Dimova43, L. Kardapoltsev43, Y. Skovpen43
Institute for High Energy Physics of National Research Centre ‘Kurchatov
Institute’, Protvino, Russia
I. Azhgirey, I. Bayshev, S. Bitioukov, V. Kachanov, D. Konstantinov, P.
Mandrik, V. Petrov, R. Ryutin, S. Slabospitskii, A. Sobol, S. Troshin, N.
Tyurin, A. Uzunian, A. Volkov National Research Tomsk Polytechnic University,
Tomsk, Russia
A. Babaev, A. Iuzhakov, V. Okhotnikov Tomsk State University, Tomsk, Russia
V. Borchsh, V. Ivanchenko, E. Tcherniaev University of Belgrade: Faculty of
Physics and VINCA Institute of Nuclear Sciences
P. Adzic44, P. Cirkovic, D. Devetak, M. Dordevic, P. Milenovic, J. Milosevic,
M. Stojanovic Centro de Investigaciones Energéticas Medioambientales y
Tecnológicas (CIEMAT), Madrid, Spain
M. Aguilar-Benitez, J. Alcaraz Maestre, A. Álvarez Fernández, I. Bachiller, M.
Barrio Luna, J.A. Brochero Cifuentes, C.A. Carrillo Montoya, M. Cepeda, M.
Cerrada, N. Colino, B. De La Cruz, A. Delgado Peris, C. Fernandez Bedoya, J.P.
Fernández Ramos, J. Flix, M.C. Fouz, O. Gonzalez Lopez, S. Goy Lopez, J.M.
Hernandez, M.I. Josa, D. Moran, Á. Navarro Tobar, A. Pérez-Calero Yzquierdo,
J. Puerta Pelayo, I. Redondo, L. Romero, S. Sánchez Navas, M.S. Soares, A.
Triossi, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain
C. Albajar, J.F. de Trocóniz, R. Reyes-Almanza Universidad de Oviedo,
Instituto Universitario de Ciencias y Tecnologías Espaciales de Asturias
(ICTEA), Oviedo, Spain
B. Alvarez Gonzalez, J. Cuevas, C. Erice, J. Fernandez Menendez, S. Folgueras,
I. Gonzalez Caballero, J.R. González Fernández, E. Palencia Cortezon, V.
Rodríguez Bouza, S. Sanchez Cruz Instituto de Física de Cantabria (IFCA),
CSIC-Universidad de Cantabria, Santander, Spain
I.J. Cabrillo, A. Calderon, B. Chazin Quero, J. Duarte Campderros, M.
Fernandez, P.J. Fernández Manteca, A. García Alonso, G. Gomez, C. Martinez
Rivero, P. Martinez Ruiz del Arbol, F. Matorras, J. Piedra Gomez, C. Prieels,
T. Rodrigo, A. Ruiz-Jimeno, L. Russo45, L. Scodellaro, N. Trevisani, I. Vila,
J.M. Vizan Garcia University of Colombo, Colombo, Sri Lanka
K. Malagalage University of Ruhuna, Department of Physics, Matara, Sri Lanka
W.G.D. Dharmaratna, N. Wickramage CERN, European Organization for Nuclear
Research, Geneva, Switzerland
D. Abbaneo, B. Akgun, E. Auffray, G. Auzinger, J. Baechler, P. Baillon, A.H.
Ball, D. Barney, J. Bendavid, M. Bianco, A. Bocci, P. Bortignon, E. Bossini,
C. Botta, E. Brondolin, T. Camporesi, A. Caratelli, G. Cerminara, E. Chapon,
G. Cucciati, D. d’Enterria, A. Dabrowski, N. Daci, V. Daponte, A. David, O.
Davignon, A. De Roeck, M. Deile, M. Dobson, M. Dünser, N. Dupont, A. Elliott-
Peisert, N. Emriskova, F. Fallavollita46, D. Fasanella, S. Fiorendi, G.
Franzoni, J. Fulcher, W. Funk, S. Giani, D. Gigi, A. Gilbert, K. Gill, F.
Glege, M. Gruchala, M. Guilbaud, D. Gulhan, J. Hegeman, C. Heidegger, Y.
Iiyama, V. Innocente, P. Janot, O. Karacheban19, J. Kaspar, J. Kieseler, M.
Krammer1, N. Kratochwil, C. Lange, P. Lecoq, C. Lourenço, L. Malgeri, M.
Mannelli, A. Massironi, F. Meijers, J.A. Merlin, S. Mersi, E. Meschi, F.
Moortgat, M. Mulders, J. Ngadiuba, J. Niedziela, S. Nourbakhsh, S. Orfanelli,
L. Orsini, F. Pantaleo16, L. Pape, E. Perez, M. Peruzzi, A. Petrilli, G.
Petrucciani, A. Pfeiffer, M. Pierini, F.M. Pitters, D. Rabady, A. Racz, M.
Rieger, M. Rovere, H. Sakulin, C. Schäfer, C. Schwick, M. Selvaggi, A. Sharma,
P. Silva, W. Snoeys, P. Sphicas47, J. Steggemann, S. Summers, V.R. Tavolaro,
D. Treille, A. Tsirou, G.P. Van Onsem, A. Vartak, M. Verzetti, W.D. Zeuner
Paul Scherrer Institut, Villigen, Switzerland
L. Caminada48, K. Deiters, W. Erdmann, R. Horisberger, Q. Ingram, H.C.
Kaestli, D. Kotlinski, U. Langenegger, T. Rohe, S.A. Wiederkehr ETH Zurich -
Institute for Particle Physics and Astrophysics (IPA), Zurich, Switzerland
M. Backhaus, P. Berger, N. Chernyavskaya, G. Dissertori, M. Dittmar, M.
Donegà, C. Dorfer, T.A. Gómez Espinosa, C. Grab, D. Hits, T. Klijnsma, W.
Lustermann, R.A. Manzoni, M. Marionneau, M.T. Meinhard, F. Micheli, P.
Musella, F. Nessi-Tedaldi, F. Pauss, G. Perrin, L. Perrozzi, S. Pigazzini,
M.G. Ratti, M. Reichmann, C. Reissel, T. Reitenspiess, D. Ruini, D.A. Sanz
Becerra, M. Schönenberger, L. Shchutska, M.L. Vesterbacka Olsson, R. Wallny,
D.H. Zhu Universität Zürich, Zurich, Switzerland
T.K. Aarrestad, C. Amsler49, D. Brzhechko, M.F. Canelli, A. De Cosa, R. Del
Burgo, S. Donato, B. Kilminster, S. Leontsinis, V.M. Mikuni, I. Neutelings, G.
Rauco, P. Robmann, D. Salerno, K. Schweiger, C. Seitz, Y. Takahashi, S. Wertz,
A. Zucchetta National Central University, Chung-Li, Taiwan
T.H. Doan, C.M. Kuo, W. Lin, A. Roy, S.S. Yu National Taiwan University (NTU),
Taipei, Taiwan
P. Chang, Y. Chao, K.F. Chen, P.H. Chen, W.-S. Hou, Y.y. Li, R.-S. Lu, E.
Paganis, A. Psallidas, A. Steen Chulalongkorn University, Faculty of Science,
Department of Physics, Bangkok, Thailand
B. Asavapibhop, C. Asawatangtrakuldee, N. Srimanobhas, N. Suwonjandee Çukurova
University, Physics Department, Science and Art Faculty, Adana, Turkey
A. Bat, F. Boran, A. Celik50, S. Cerci51, S. Damarseckin52, Z.S. Demiroglu, F.
Dolek, C. Dozen53, I. Dumanoglu, G. Gokbulut, EmineGurpinar Guler54, Y. Guler,
I. Hos55, C. Isik, E.E. Kangal56, O. Kara, A. Kayis Topaksu, U. Kiminsu, G.
Onengut, K. Ozdemir57, S. Ozturk58, A.E. Simsek, D. Sunar Cerci51, U.G. Tok,
S. Turkcapar, I.S. Zorbakir, C. Zorbilmez Middle East Technical University,
Physics Department, Ankara, Turkey
B. Isildak59, G. Karapinar60, M. Yalvac Bogazici University, Istanbul, Turkey
I.O. Atakisi, E. Gülmez, M. Kaya61, O. Kaya62, Ö. Özçelik, S. Tekten, E.A.
Yetkin63 Istanbul Technical University, Istanbul, Turkey
A. Cakir, K. Cankocak, Y. Komurcu, S. Sen64 Istanbul University, Istanbul,
Turkey
B. Kaynak, S. Ozkorucuklu Institute for Scintillation Materials of National
Academy of Science of Ukraine, Kharkov, Ukraine
B. Grynyov National Scientific Center, Kharkov Institute of Physics and
Technology, Kharkov, Ukraine
L. Levchuk University of Bristol, Bristol, United Kingdom
E. Bhal, S. Bologna, J.J. Brooke, D. Burns65, E. Clement, D. Cussans, H.
Flacher, J. Goldstein, G.P. Heath, H.F. Heath, L. Kreczko, S. Paramesvaran, B.
Penning, T. Sakuma, S. Seif El Nasr-Storey, V.J. Smith, J. Taylor, A.
Titterton Rutherford Appleton Laboratory, Didcot, United Kingdom
K.W. Bell, A. Belyaev66, C. Brew, R.M. Brown, D. Cieri, D.J.A. Cockerill, J.A.
Coughlan, K. Harder, S. Harper, J. Linacre, K. Manolopoulos, D.M. Newbold, E.
Olaiya, D. Petyt, T. Reis, T. Schuh, C.H. Shepherd-Themistocleous, A. Thea,
I.R. Tomalin, T. Williams, W.J. Womersley Imperial College, London, United
Kingdom
R. Bainbridge, P. Bloch, J. Borg, S. Breeze, O. Buchmuller, A. Bundock,
GurpreetSingh CHAHAL67, D. Colling, P. Dauncey, G. Davies, M. Della Negra, R.
Di Maria, P. Everaerts, G. Hall, G. Iles, T. James, M. Komm, C. Laner, L.
Lyons, A.-M. Magnan, S. Malik, A. Martelli, V. Milosevic, J. Nash68, V.
Palladino, M. Pesaresi, D.M. Raymond, A. Richards, A. Rose, E. Scott, C. Seez,
A. Shtipliyski, M. Stoye, T. Strebler, A. Tapper, K. Uchida, T. Virdee16, N.
Wardle, D. Winterbottom, J. Wright, A.G. Zecchinelli, S.C. Zenz Brunel
University, Uxbridge, United Kingdom
J.E. Cole, P.R. Hobson, A. Khan, P. Kyberd, C.K. Mackay, A. Morton, I.D. Reid,
L. Teodorescu, S. Zahid Baylor University, Waco, USA
K. Call, B. Caraway, J. Dittmann, K. Hatakeyama, C. Madrid, B. McMaster, N.
Pastika, C. Smith Catholic University of America, Washington, DC, USA
R. Bartek, A. Dominguez, R. Uniyal, A.M. Vargas Hernandez The University of
Alabama, Tuscaloosa, USA
A. Buccilli, S.I. Cooper, C. Henderson, P. Rumerio, C. West Boston University,
Boston, USA
D. Arcaro, Z. Demiragli, D. Gastler, C. Richardson, J. Rohlf, D. Sperka, I.
Suarez, L. Sulak, D. Zou Brown University, Providence, USA
G. Benelli, B. Burkle, X. Coubez17, D. Cutts, Y.t. Duh, M. Hadley, J. Hakala,
U. Heintz, J.M. Hogan69, K.H.M. Kwok, E. Laird, G. Landsberg, J. Lee, Z. Mao,
M. Narain, S. Sagir70, R. Syarif, E. Usai, D. Yu, W. Zhang University of
California, Davis, Davis, USA
R. Band, C. Brainerd, R. Breedon, M. Calderon De La Barca Sanchez, M. Chertok,
J. Conway, R. Conway, P.T. Cox, R. Erbacher, C. Flores, G. Funk, F. Jensen, W.
Ko, O. Kukral, R. Lander, M. Mulhearn, D. Pellett, J. Pilot, M. Shi, D.
Taylor, K. Tos, M. Tripathi, Z. Wang, F. Zhang University of California, Los
Angeles, USA
M. Bachtis, C. Bravo, R. Cousins, A. Dasgupta, A. Florent, J. Hauser, M.
Ignatenko, N. Mccoll, W.A. Nash, S. Regnard, D. Saltzberg, C. Schnaible, B.
Stone, V. Valuev University of California, Riverside, Riverside, USA
K. Burt, Y. Chen, R. Clare, J.W. Gary, S.M.A. Ghiasi Shirazi, G. Hanson, G.
Karapostoli, E. Kennedy, O.R. Long, M. Olmedo Negrete, M.I. Paneva, W. Si, L.
Wang, S. Wimpenny, B.R. Yates, Y. Zhang University of California, San Diego,
La Jolla, USA
J.G. Branson, P. Chang, S. Cittolin, S. Cooperstein, N. Deelen, M. Derdzinski,
R. Gerosa, D. Gilbert, B. Hashemi, D. Klein, V. Krutelyov, J. Letts, M.
Masciovecchio, S. May, S. Padhi, M. Pieri, V. Sharma, M. Tadel, F. Würthwein,
A. Yagil, G. Zevi Della Porta University of California, Santa Barbara -
Department of Physics, Santa Barbara, USA
N. Amin, R. Bhandari, C. Campagnari, M. Citron, V. Dutta, M. Franco Sevilla,
L. Gouskos, J. Incandela, B. Marsh, H. Mei, A. Ovcharova, H. Qu, J. Richman,
U. Sarica, D. Stuart, S. Wang California Institute of Technology, Pasadena,
USA
D. Anderson, A. Bornheim, O. Cerri, I. Dutta, J.M. Lawhorn, N. Lu, J. Mao,
H.B. Newman, T.Q. Nguyen, J. Pata, M. Spiropulu, J.R. Vlimant, S. Xie, Z.
Zhang, R.Y. Zhu Carnegie Mellon University, Pittsburgh, USA
M.B. Andrews, T. Ferguson, T. Mudholkar, M. Paulini, M. Sun, I. Vorobiev, M.
Weinberg University of Colorado Boulder, Boulder, USA
J.P. Cumalat, W.T. Ford, A. Johnson, E. MacDonald, T. Mulholland, R. Patel, A.
Perloff, K. Stenson, K.A. Ulmer, S.R. Wagner Cornell University, Ithaca, USA
J. Alexander, J. Chaves, Y. Cheng, J. Chu, A. Datta, A. Frankenthal, K.
Mcdermott, J.R. Patterson, D. Quach, A. Rinkevicius71, A. Ryd, S.M. Tan, Z.
Tao, J. Thom, P. Wittich, M. Zientek Fermi National Accelerator Laboratory,
Batavia, USA
S. Abdullin, M. Albrow, M. Alyari, G. Apollinari, A. Apresyan, A. Apyan, S.
Banerjee, L.A.T. Bauerdick, A. Beretvas, D. Berry, J. Berryhill, P.C. Bhat, K.
Burkett, J.N. Butler, A. Canepa, G.B. Cerati, H.W.K. Cheung, F. Chlebana, M.
Cremonesi, J. Duarte, V.D. Elvira, J. Freeman, Z. Gecse, E. Gottschalk, L.
Gray, D. Green, S. Grünendahl, O. Gutsche, AllisonReinsvold Hall, J. Hanlon,
R.M. Harris, S. Hasegawa, R. Heller, J. Hirschauer, B. Jayatilaka, S.
Jindariani, M. Johnson, U. Joshi, B. Klima, M.J. Kortelainen, B. Kreis, S.
Lammel, J. Lewis, D. Lincoln, R. Lipton, M. Liu, T. Liu, J. Lykken, K.
Maeshima, J.M. Marraffino, D. Mason, P. McBride, P. Merkel, S. Mrenna, S.
Nahn, V. O’Dell, V. Papadimitriou, K. Pedro, C. Pena, G. Rakness, F. Ravera,
L. Ristori, B. Schneider, E. Sexton-Kennedy, N. Smith, A. Soha, W.J. Spalding,
L. Spiegel, S. Stoynev, J. Strait, N. Strobbe, L. Taylor, S. Tkaczyk, N.V.
Tran, L. Uplegger, E.W. Vaandering, C. Vernieri, R. Vidal, M. Wang, H.A. Weber
University of Florida, Gainesville, USA
D. Acosta, P. Avery, D. Bourilkov, A. Brinkerhoff, L. Cadamuro, A. Carnes, V.
Cherepanov, F. Errico, R.D. Field, S.V. Gleyzer, B.M. Joshi, M. Kim, J.
Konigsberg, A. Korytov, K.H. Lo, P. Ma, K. Matchev, N. Menendez, G.
Mitselmakher, D. Rosenzweig, K. Shi, J. Wang, S. Wang, X. Zuo Florida
International University, Miami, USA
Y.R. Joshi Florida State University, Tallahassee, USA
T. Adams, A. Askew, S. Hagopian, V. Hagopian, K.F. Johnson, R. Khurana, T.
Kolberg, G. Martinez, T. Perry, H. Prosper, C. Schiber, R. Yohay, J. Zhang
Florida Institute of Technology, Melbourne, USA
M.M. Baarmand, M. Hohlmann, D. Noonan, M. Rahmani, M. Saunders, F. Yumiceva
University of Illinois at Chicago (UIC), Chicago, USA
M.R. Adams, L. Apanasevich, R.R. Betts, R. Cavanaugh, X. Chen, S. Dittmer, O.
Evdokimov, C.E. Gerber, D.A. Hangal, D.J. Hofman, K. Jung, C. Mills, T. Roy,
M.B. Tonjes, N. Varelas, J. Viinikainen, H. Wang, X. Wang, Z. Wu The
University of Iowa, Iowa City, USA
M. Alhusseini, B. Bilki54, W. Clarida, K. Dilsiz72, S. Durgut, R.P.
Gandrajula, M. Haytmyradov, V. Khristenko, O.K. Köseyan, J.-P. Merlo, A.
Mestvirishvili73, A. Moeller, J. Nachtman, H. Ogul74, Y. Onel, F. Ozok75, A.
Penzo, C. Snyder, E. Tiras, J. Wetzel Johns Hopkins University, Baltimore, USA
B. Blumenfeld, A. Cocoros, N. Eminizer, A.V. Gritsan, W.T. Hung, S. Kyriacou,
P. Maksimovic, J. Roskes, M. Swartz The University of Kansas, Lawrence, USA
C. Baldenegro Barrera, P. Baringer, A. Bean, S. Boren, J. Bowen, A. Bylinkin,
T. Isidori, S. Khalil, J. King, G. Krintiras, A. Kropivnitskaya, C. Lindsey,
D. Majumder, W. Mcbrayer, N. Minafra, M. Murray, C. Rogan, C. Royon, S.
Sanders, E. Schmitz, J.D. Tapia Takaki, Q. Wang, J. Williams, G. Wilson Kansas
State University, Manhattan, USA
S. Duric, A. Ivanov, K. Kaadze, D. Kim, Y. Maravin, D.R. Mendis, T. Mitchell,
A. Modak, A. Mohammadi Lawrence Livermore National Laboratory, Livermore, USA
F. Rebassoo, D. Wright University of Maryland, College Park, USA
A. Baden, O. Baron, A. Belloni, S.C. Eno, Y. Feng, N.J. Hadley, S. Jabeen,
G.Y. Jeng, R.G. Kellogg, J. Kunkle, A.C. Mignerey, S. Nabili, F. Ricci-Tam, M.
Seidel, Y.H. Shin, A. Skuja, S.C. Tonwar, K. Wong Massachusetts Institute of
Technology, Cambridge, USA
D. Abercrombie, B. Allen, A. Baty, R. Bi, S. Brandt, W. Busza, I.A. Cali, M.
D’Alfonso, G. Gomez Ceballos, M. Goncharov, P. Harris, D. Hsu, M. Hu, M.
Klute, D. Kovalskyi, Y.-J. Lee, P.D. Luckey, B. Maier, A.C. Marini, C. Mcginn,
C. Mironov, S. Narayanan, X. Niu, C. Paus, D. Rankin, C. Roland, G. Roland, Z.
Shi, G.S.F. Stephans, K. Sumorok, K. Tatar, D. Velicanu, J. Wang, T.W. Wang,
B. Wyslouch University of Minnesota, Minneapolis, USA
R.M. Chatterjee, A. Evans, S. Guts, P. Hansen, J. Hiltbrand, Y. Kubota, Z.
Lesko, J. Mans, R. Rusack, M.A. Wadud University of Mississippi, Oxford, USA
J.G. Acosta, S. Oliveros University of Nebraska-Lincoln, Lincoln, USA
K. Bloom, D.R. Claes, C. Fangmeier, L. Finco, F. Golf, R. Kamalieddin, I.
Kravchenko, J.E. Siado, G.R. Snow${}^{\textrm{\textdagger}}$, B. Stieger, W.
Tabb State University of New York at Buffalo, Buffalo, USA
G. Agarwal, C. Harrington, I. Iashvili, A. Kharchilava, C. McLean, D. Nguyen,
A. Parker, J. Pekkanen, S. Rappoccio, B. Roozbahani Northeastern University,
Boston, USA
G. Alverson, E. Barberis, C. Freer, Y. Haddad, A. Hortiangtham, G. Madigan, B.
Marzocchi, D.M. Morse, T. Orimoto, L. Skinnari, A. Tishelman-Charny, T.
Wamorkar, B. Wang, A. Wisecarver, D. Wood Northwestern University, Evanston,
USA
S. Bhattacharya, J. Bueghly, T. Gunter, K.A. Hahn, N. Odell, M.H. Schmitt, K.
Sung, M. Trovato, M. Velasco University of Notre Dame, Notre Dame, USA
R. Bucci, N. Dev, R. Goldouzian, M. Hildreth, K. Hurtado Anampa, C. Jessop,
D.J. Karmgard, K. Lannon, W. Li, N. Loukas, N. Marinelli, I. Mcalister, F.
Meng, C. Mueller, Y. Musienko36, M. Planer, R. Ruchti, P. Siddireddy, G.
Smith, S. Taroni, M. Wayne, A. Wightman, M. Wolf, A. Woodard The Ohio State
University, Columbus, USA
J. Alimena, B. Bylsma, L.S. Durkin, S. Flowers, B. Francis, C. Hill, W. Ji, A.
Lefeld, T.Y. Ling, B.L. Winer Princeton University, Princeton, USA
G. Dezoort, P. Elmer, J. Hardenbrook, N. Haubrich, S. Higginbotham, A.
Kalogeropoulos, S. Kwan, D. Lange, M.T. Lucchini, J. Luo, D. Marlow, K. Mei,
I. Ojalvo, J. Olsen, C. Palmer, P. Piroué, J. Salfeld-Nebgen, D. Stickland, C.
Tully, Z. Wang University of Puerto Rico, Mayaguez, USA
S. Malik, S. Norberg Purdue University, West Lafayette, USA
A. Barker, V.E. Barnes, S. Das, L. Gutay, M. Jones, A.W. Jung, A. Khatiwada,
B. Mahakud, D.H. Miller, G. Negro, N. Neumeister, C.C. Peng, S. Piperov, H.
Qiu, J.F. Schulte, J. Sun, F. Wang, R. Xiao, W. Xie Purdue University
Northwest, Hammond, USA
T. Cheng, J. Dolen, N. Parashar Rice University, Houston, USA
U. Behrens, K.M. Ecklund, S. Freed, F.J.M. Geurts, M. Kilpatrick, Arun Kumar,
W. Li, B.P. Padley, R. Redjimi, J. Roberts, J. Rorie, W. Shi, A.G. Stahl
Leiton, Z. Tu, A. Zhang University of Rochester, Rochester, USA
A. Bodek, P. de Barbaro, R. Demina, J.L. Dulemba, C. Fallon, T. Ferbel, M.
Galanti, A. Garcia-Bellido, O. Hindrichs, A. Khukhunaishvili, E. Ranken, R.
Taus Rutgers, The State University of New Jersey, Piscataway, USA
B. Chiarito, J.P. Chou, A. Gandrakota, Y. Gershtein, E. Halkiadakis, A. Hart,
M. Heindl, E. Hughes, S. Kaplan, I. Laflotte, A. Lath, R. Montalvo, K. Nash,
M. Osherson, H. Saka, S. Salur, S. Schnetzer, S. Somalwar, R. Stone, S. Thomas
University of Tennessee, Knoxville, USA
H. Acharya, A.G. Delannoy, G. Riley, S. Spanier Texas A&M University, College
Station, USA
O. Bouhali76, M. Dalchenko, M. De Mattia, A. Delgado, S. Dildick, R. Eusebi,
J. Gilmore, T. Huang, T. Kamon77, S. Luo, S. Malhotra, D. Marley, R. Mueller,
D. Overton, L. Perniè, D. Rathjens, A. Safonov Texas Tech University, Lubbock,
USA
N. Akchurin, J. Damgov, F. De Guio, S. Kunori, K. Lamichhane, S.W. Lee, T.
Mengke, S. Muthumuni, T. Peltola, S. Undleeb, I. Volobouev, Z. Wang, A.
Whitbeck Vanderbilt University, Nashville, USA
S. Greene, A. Gurrola, R. Janjam, W. Johns, C. Maguire, A. Melo, H. Ni, K.
Padeken, F. Romeo, P. Sheldon, S. Tuo, J. Velkovska, M. Verweij University of
Virginia, Charlottesville, USA
M.W. Arenton, P. Barria, B. Cox, G. Cummings, R. Hirosky, M. Joyce, A.
Ledovskoy, C. Neu, B. Tannenwald, Y. Wang, E. Wolfe, F. Xia Wayne State
University, Detroit, USA
R. Harr, P.E. Karchin, N. Poudyal, J. Sturdy, P. Thapa University of Wisconsin
- Madison, Madison, WI, USA
T. Bose, J. Buchanan, C. Caillol, D. Carlsmith, S. Dasu, I. De Bruyn, L. Dodd,
F. Fiori, C. Galloni, B. Gomber78, H. He, M. Herndon, A. Hervé, U. Hussain, P.
Klabbers, A. Lanaro, A. Loeliger, K. Long, R. Loveless, J. Madhusudanan
Sreekala, D. Pinna, T. Ruggles, A. Savin, V. Sharma, W.H. Smith, D. Teague, S.
Trembath-reichert, N. Woods †: Deceased
1: Also at Vienna University of Technology, Vienna, Austria
2: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France
3: Also at Universidade Estadual de Campinas, Campinas, Brazil
4: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil
5: Also at UFMS, Nova Andradina, Brazil
6: Also at Universidade Federal de Pelotas, Pelotas, Brazil
7: Also at Université Libre de Bruxelles, Bruxelles, Belgium
8: Also at University of Chinese Academy of Sciences, Beijing, China
9: Also at Institute for Theoretical and Experimental Physics named by A.I.
Alikhanov of NRC ‘Kurchatov Institute’, Moscow, Russia
10: Also at Joint Institute for Nuclear Research, Dubna, Russia
11: Also at Ain Shams University, Cairo, Egypt
12: Also at Zewail City of Science and Technology, Zewail, Egypt
13: Also at Purdue University, West Lafayette, USA
14: Also at Université de Haute Alsace, Mulhouse, France
15: Also at Erzincan Binali Yildirim University, Erzincan, Turkey
16: Also at CERN, European Organization for Nuclear Research, Geneva,
Switzerland
17: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen,
Germany
18: Also at University of Hamburg, Hamburg, Germany
19: Also at Brandenburg University of Technology, Cottbus, Germany
20: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary,
Debrecen, Hungary
21: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary
22: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös
Loránd University, Budapest, Hungary, Budapest, Hungary
23: Also at IIT Bhubaneswar, Bhubaneswar, India, Bhubaneswar, India
24: Also at Institute of Physics, Bhubaneswar, India
25: Also at Shoolini University, Solan, India
26: Also at University of Visva-Bharati, Santiniketan, India
27: Also at Isfahan University of Technology, Isfahan, Iran
28: Now at INFN Sezione di Bari a, Università di Bari b, Politecnico di Bari
c, Bari, Italy
29: Also at Italian National Agency for New Technologies, Energy and
Sustainable Economic Development, Bologna, Italy
30: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia,
Catania, Italy
31: Also at Scuola Normale e Sezione dell’INFN, Pisa, Italy
32: Also at Riga Technical University, Riga, Latvia, Riga, Latvia
33: Also at Malaysian Nuclear Agency, MOSTI, Kajang, Malaysia
34: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico
35: Also at Warsaw University of Technology, Institute of Electronic Systems,
Warsaw, Poland
36: Also at Institute for Nuclear Research, Moscow, Russia
37: Now at National Research Nuclear University ’Moscow Engineering Physics
Institute’ (MEPhI), Moscow, Russia
38: Also at St. Petersburg State Polytechnical University, St. Petersburg,
Russia
39: Also at University of Florida, Gainesville, USA
40: Also at Imperial College, London, United Kingdom
41: Also at P.N. Lebedev Physical Institute, Moscow, Russia
42: Also at California Institute of Technology, Pasadena, USA
43: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia
44: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia
45: Also at Università degli Studi di Siena, Siena, Italy
46: Also at INFN Sezione di Pavia a, Università di Pavia b, Pavia, Italy,
Pavia, Italy
47: Also at National and Kapodistrian University of Athens, Athens, Greece
48: Also at Universität Zürich, Zurich, Switzerland
49: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria,
Vienna, Austria
50: Also at Burdur Mehmet Akif Ersoy University, BURDUR, Turkey
51: Also at Adiyaman University, Adiyaman, Turkey
52: Also at Şırnak University, Sirnak, Turkey
53: Also at Tsinghua University, Beijing, China
54: Also at Beykent University, Istanbul, Turkey, Istanbul, Turkey
55: Also at Istanbul Aydin University, Istanbul, Turkey
56: Also at Mersin University, Mersin, Turkey
57: Also at Piri Reis University, Istanbul, Turkey
58: Also at Gaziosmanpasa University, Tokat, Turkey
59: Also at Ozyegin University, Istanbul, Turkey
60: Also at Izmir Institute of Technology, Izmir, Turkey
61: Also at Marmara University, Istanbul, Turkey
62: Also at Kafkas University, Kars, Turkey
63: Also at Istanbul Bilgi University, Istanbul, Turkey
64: Also at Hacettepe University, Ankara, Turkey
65: Also at Vrije Universiteit Brussel, Brussel, Belgium
66: Also at School of Physics and Astronomy, University of Southampton,
Southampton, United Kingdom
67: Also at IPPP Durham University, Durham, United Kingdom
68: Also at Monash University, Faculty of Science, Clayton, Australia
69: Also at Bethel University, St. Paul, Minneapolis, USA, St. Paul, USA
70: Also at Karamanoğlu Mehmetbey University, Karaman, Turkey
71: Also at Vilnius University, Vilnius, Lithuania
72: Also at Bingol University, Bingol, Turkey
73: Also at Georgian Technical University, Tbilisi, Georgia
74: Also at Sinop University, Sinop, Turkey
75: Also at Mimar Sinan University, Istanbul, Istanbul, Turkey
76: Also at Texas A&M University at Qatar, Doha, Qatar
77: Also at Kyungpook National University, Daegu, Korea, Daegu, Korea
78: Also at University of Hyderabad, Hyderabad, India