synchronisation between the Drive Beam and the Main Beam. Figure 1.49 depicts
a possible synchronisation solution. A controller measures the phases of the
Drive and the Main Beam with respect to a reference line, and uses that
information to control the amplitude of a kicker pulse which modifies the
trajectory of the Drive Beam in order to keep it well synchronised with the
Main Beam. The required precision of this alignment is around 40 fs [301], so
the timing reference precision clearly needs to be better than that. The
bandwidth of the kickers, in the several MHz region, would set a natural upper
limit for integration of phase noise.

Figure 1.49: Timing reference line usage for inter-beam synchronisation
(adapted from Ref. [301]).

####  1.16.3 Technical Solutions

There are two main types of solutions currently developed and tested for the
distribution of precise timing signals in the femtosecond realm [302]. Both
types have achieved synchronisations better than 20 fs over distances of
several hundreds of metres.

In the Continuous-Wave (CW) variant, a laser modulated in amplitude by the RF
or microwave signal to be transmitted. A fraction of the light reaching the
destination bounces back from a Faraday Rotator Mirror (FRM) and is, in the
process, shifted in frequency using a Frequency Shifter (FS). Another FRM
reflects a sample of the light signal as it leaves the source. These two
signals, upon mixing together, produce a beat whose phase can be easily
measured. The key feature of this system is that the heterodyning process
preserves phases, so a phase shift induced in the optical frequency by a
change of length in the fibre will show up as exactly the same phase shift at
the RF frequency, which is much easier to measure. Once the phase of the beat
signal is detected, it can be used to digitally shift the phase of the
recovered RF signal at the receiving end of the link. It is important to note
that what is really measured in this method is the phase delay, not the group
delay of the modulation which is ultimately what we are interested in. In
order to compensate for group delay, a first-order correction — based on
actual measurements of a given fibre type for a range of temperatures — is
applied to the raw phase delay measurements.

In the pulsed variant, a mode-locked laser is synchronised with the external
RF signal, which determines its repetition rate. Narrow light pulses come out
of the laser and travel through a dispersion-compensated fibre to the
receiver, where a fraction of the light is reflected back towards the emitter.
An optical cross-correlator measures the degree of coincidence of the two
pulse trains and the result is used to control a piezo actuator that changes
the fibre length so as to keep a constant group delay. This mechanical
actuator is unavoidable because the cross-correlator needs the pulses to
overlap, at least partially, in order to give a meaningful reading. On the
other hand, the pulsed system controls group delay directly, so no ad hoc
conversion between phase delay and group delay is needed.

####  1.16.4 Technical Issues

Both types of systems have been successfully deployed over distances of
several hundred metres, so the 6 km needed by CLIC Stage 1 is unknown
territory. In the case of the CW system, one potential source of concern is
the need for a model of phase-group delay corrections vs. temperature. If the
temperature is not uniform over the complete length of the fibre, as can
easily be the case in CLIC, a solution will need to be found.

Another potential issue is Brillouin scattering, a non-linear effect which is
especially strong in optical fibres, due to the large optical intensities in
the fibre core. Brillouin scattering in fibres leads to a limit of the optical
power which can be transmitted, since above a certain power threshold, most of
the light is scattered or reflected. This is especially a problem for narrow-
band optical signals, and for long fibre lengths. In the pulsed system,
Brillouin scattering is not much of an issue, since the power spectral
density, which is the important quantity, is much smaller than for narrowband
CW systems.

For the pulsed system, a major difficulty will be the dispersion control of
the optical fibres, as the fibre length increases. Additional problems due to
non-linearities are not expected to be any more significant than with shorter
fibre links.

Finally, one very important difference between the solutions currently
deployed and the CLIC scenario is the scale of the project. Current
implementations do not scale very well beyond some tens of destinations. This
is mainly due to the number of ports typically available in optical
components, such as splitters/combiners. For CLIC, one of the challenges will
be to explore in detail the real needs of each destination and come up with a
strategy for partitioning the system in such a way that a reasonable
compromise between performance and cost can be found.

###  1.17 Machine Protection

####  1.17.1 introduction

In the CDR [330] there is a detailed discussion of the destructive capacity of
the CLIC beams. Although the beam power is impressive (Main Beam: 14 MW and
Drive Beam for a single decelerator: 2.9 MW) the energy per pulse 2.8 MJ and
0.58 MJ respectively, is more than two orders of magnitude lower compared to
the LHC beam of 300 MJ. Nevertheless, the energy density due to the extremely
low beam emittance and spot sizes is very critical. Both beam power and energy
density have to be addressed with the machine protection system to allow for
safe and reliable operation.

As described in the CDR, the CLIC Machine Protection policy deploys different
strategies depending on the time domain of the failures. In the following
subsections, the status and implications for each of these strategies is
discussed in more detail. The description will start from the interlock
protection, which provides real-time protection for equipment failures up
about a millisecond before beam passage, and then continues with the
strategies to mitigate failures occurring at shorter time scales. Then we
discuss the strategies for failures occurring over longer time scales.

####  1.17.2 Machine Interlock System

The CLIC machine interlock system will assure that all equipment failures,
detected in time before the next beam is committed, revoke the “next beam
permit” and inhibit delivery of otherwise ill-controlled beams.

At CERN, and most notably for the LHC, a lot of experience with interlock
systems has been build up. Although there may be many similarities, there is
also a substantial number of differences to the CLIC machine. Firstly, whilst
the LHC interlock surveys the circulating beams and may demand a beam dump on
the next beam passage, in CLIC there is only a single beam passage. Hence, the
interlock system only controls the permission for the next pulse. Technically
speaking, the CLIC interlock system should be compared with the LHC injection
interlock. A second difference is that the CLIC cycle time (20 ms) is by far
not comparable with the LHC inter-cycle time (over 20 minutes). Hence, the
CLIC interlock system must be able to rearm much faster and rely on automated
analysis to determine whether the next cycle can safely be committed. Finally,
the CLIC interlock system has to deal with multiple beams (Main Beam, Drive
Beam) of various possibly beam intensities (i.e. during the intensity ramp
up), that are traversing a multitude of accelerators and transfer lines.

Due to this complexity, the CLIC interlock system must be closely integrated
with the beam scheduling system. The beam scheduling system will send out a
token with the next pulse configuration that circulates through the CLIC
complex. All interlock managing stations in the CLIC accelerator chain have to
add their acceptance signature to the token for the next pulse to be executed.

####  1.17.3 Safe by Design

The interlock system intercepts any equipment failure that occurs during the
inter-beam period. However, as the interlock system has a finite signal-
handling time, the interlock system will only cope with equipment errors that
are detected up to 1$\sim$2 ms before the next pulse permit is given. This
implies that all the equipment circuits must be designed such that there is
enough inertia in the system for the equipment settings to stay within the
tolerance required for safe beam passage during 2 ms following the onset of a
failure.

This strategy is not new, and is already deployed in many of the CERN transfer
lines which carry high-intensity beams.

####  1.17.4 In-flight Protection

The very short pulse length (156 ns for the Main Beam and 244 ns for the Drive
Beam) makes it practically impossible to correct real-time errors. Once a
pulse has been committed, there are practically no means to intercept safely
the beam in the event that beam losses are detected (note that the CDR
discusses a few exceptions where theoretically an interlock system may act on
the beam in progress). Hence, for so-called “in-flight” failures, the machine
protection system has to rely on fixed protections by masks and collimators.
Most notably, masks are needed to protect extraction and transfer channels
that follow any fast extraction or injection element. The design of the mask
will be a challenging task as the charge density of the extracted beam could
be high enough to damage the mask.

The interesting concept of reusable surfaces (i.e. where, in case of damage,
the collimator surface can be displaced parallel to the beam) which exists in
the LHC should also be deployed in CLIC. Furthermore, special attention is
given to improving the reliability of the kicker systems through the
deployment of inductive adders to pulse the kicker systems.

####  1.17.5 Beam Quality Checks and Next Pulse Permit

The strategy to deal with medium term failures is based on the concept of a
“next pulse permit”. Upon completion of every cycle, the hardware permit of
the next pulse will automatically be revoked. Only after a number of
predefined automated beam quality checks on the actual pulse have successfully
passed, will the permit for the next pulse be re-established.

Conceptually this strategy is not new, beam quality checks are also applied in
LHC. However, while in the LHC many post-cycle checks are performed under
operator/expert supervision, in CLIC this decision has to be executed within
the 20 ms cycle, imposing further constraints on the data collection and
analysis of the beam instrumentation equipment.

The most important beam quality checks are based on beam losses, beam
trajectory information and beam-based feedback/feedforward performance.

The beam quality check will work in close conjunction with the beam scheduling
system, which orchestrates the beam intensity ramp-up after a beam interlock.
Hence, the next pulse permit decision is not just a go/no-go, but it also
encompasses the decision on the beam intensities.

####  1.17.6 Long-Term Protection

Long-term protection has to ensure that the machine stays in a healthy state
over longer periods. The most important ageing factor of many components is
radiation related. Continuous machine optimisation with the aim to reduce
radiation losses is important in this context. Radiation levels will be
monitored and, where needed, the impact can be reduced by radiation shielding.

###  1.18 Beam Interception Devices

####  1.18.1 Main-Beam Dump

In order to cope with the challenge of absorbing the CLIC Main Beam, a water
Beam Dump at the end of the CLIC post-collision line is proposed. It consists
of a 10 m long cylinder, filled with water, pressurized at 10 bar, and with a
diameter of 1.8 m, surrounded by a 15 mm thick titanium vessel of the same
shape. For the present study a circular window has been considered, 30 cm in
diameter, 1 mm thick, made of a titanium alloy (Ti-6Al-4V, i.e., ASTM G5, or
UNS R56400), and directly cooled by the circulating water inside the dump.

The energy/power deposition by the primary beam on the water dump was
calculated using the FLUKA Monte Carlo simulation code [303]. Two different
beam scenarios were considered: the 1.5 TeV uncollided electron beam with a
transverse Gaussian beam profile centred on the dump axis, with $\sigma$hor =
1.79 mm and $\sigma$ver = 3.15 mm. For the collided beam the secondary
particles produced in the electron–positron interactions at 3 TeV centre-of-
mass energy were simulated with the GUINEA-PIG package [304] and were
transported along the post-collision line up to the dump with the DIMAD
tracking code [305].

Table 1.19 shows the peak energy and total power on the water dump and on the
titanium vessel/window for the uncollided and the collided beam scenarios. As
expected, the case of uncollided beam is more severe than the one of collided
beam.

Table 1.19: Peak energy and total power on the water dump and on the titanium vessel/window for the un-collided and the collided beam scenarios. Statistical uncertainties on the total values are below 0.1%. | Max | Total power
---|---|---
| [J cm-3 per bunch train] | [W]
| uncollided | collided | uncollided | collided
H2O | 230 $\pm$ 1 | 9.10 $\pm$ 0.01 | 13.8 M | 13.4 M
Ti window | 4.35 $\pm$ 0.36 | 83.8 $\pm$ 1.0 m | 6.24 | 4.91
Ti vessel (side) | 569 $\pm$ 18 $\mu$ | 903 $\pm$ 29 $\mu$ | 8.45 k | 9.07 k
Ti vessel (upstr. face) | 32.0 $\pm$ 13.2$\,\mu$ | 2.15 $\pm$ 0.13 m | 9.07 | 45.7
Ti vessel (dwnstr. face) | 245 $\pm$ 6 m | 39.1 $\pm$ 6.2 m | 1.12 k | 944

The induced nuclide production in the dump was estimated; in case of
uncollided beam, the short lived radio-nuclides are (with half-lives in
brackets) 15O (2 minutes), 13N (10 minutes) and 11C (20 minutes) are produced
with rates of 1.19$\times$1015 s-1, 5.51$\times$1013 s-1 and 3.39$\times$1014
s-1, respectively. The long-lived nuclides 7Be (53.6 days) and 3H (12.3 years)
have a production rate of 1.14$\times$1014 s-1 and 3.12$\times$1014 s-1,
respectively.

Water circulates in the dump in a closed loop, externally cooled. A continuous
water flow of at least 25–30 litre/s, at an average speed of 1.5 m/s, is
required to remove the power deposited in the innermost part of the dump and
maintain the peak temperature of water slightly below its boiling point. A
safety factor of at least two should be applied to these values.

In case of inefficient heat removal, the sudden increase of the water
temperature is about $\Delta T$$\sim$55 K per pulse [311]. Water reaches the
boiling point after few bunch trains and the beam must be interlocked. A
safety system must be activated to evacuate the vapour pressure generated in
the tank. The initial pressurization at 10 bar moves the boiling point of
water to 180∘C. allowing a larger margin of manoeuvre, but AUTODYN®[307]
simulations show that the pressure wave causes an overstress on the dump walls
and window at each bunch train, to be added to the stresses due to the 10 bar
hydrostatic pressure. Stiffeners are required, and possibly some type of shock
absorbers in critical locations, in order to guarantee the dump structural
integrity. The use of gas–water mixtures for a more effective damping of the
pressure wave is not a priori excluded.

####  1.18.2 Collimation System

The CLIC collimation system consists of pairs of spoilers and absorbers in
order to remove the unwanted halo of the beam. The CLIC collimation system is
driven by their cleaning capabilities and its ability to survive a possible
accident scenario. The system was simulated by means of FLUKA [303] for a
possible beam accident scenario, as well as for normal cleaning operation.

For the accident scenario it was assumed that the full train of 312 pulses
with intensity of 3.72$\times$109 electrons at 1.5 TeV hits the front face of
the energy spoiler(ESP) and, with lower probability, the horizontal or
vertical spoilers (XSP/YSP). For the simulation, a shallow depth of the beam
was used with an impact parameter of 2 mm assuming a beam spread of
780$\times$22 $\mu$m2 in horizontal $\times$ vertical for the energy spoiler
ESP and 8$\times$1 $\mu$m2 for the transverse spoiler XSP.

During normal operation all spoilers should see only a fraction of beam (of
the order of 10-6). For the sake of simulation, we assumed a beam spread
similar to that the accident case.

Figure 1.50 shows the energy deposition map on a beryllium ESP spoiler for the
accident scenario with 2 mm impact parameter. Although the total beam energy
is equivalent to 280 kJ, the spoiler stops only a tiny fraction of the beam
close to 6 J, about 36.5 MeV out of 1.5 TeV for every primary electron hitting
the spoiler. This energy deposition produces a peak energy density of the
order of 600 J/cm3, which will generate, assuming adiabatic conditions, an
instantaneous temperature rise of 450 K. The ESP spoiler will survive the
accident scenario, since it is practically transparent to the beam, and only a
tiny fraction of 2.4$\times$10-5 will be stopped on the jaws. The remaining
energy will continue downstream, therefore a dedicated study will be required
to ensure the protection of the downstream elements that will be exposed on
the accidental beam.

Figure 1.50: Energy density map in the case of the accident scenario with 2 mm
impact parameter. Left plot shows ESP collimator, right plot XSP collimator

In the case of the XSP (Figs. 1.50 the considerably smaller beam spread as
well as the heavier materials used (titanium for the jaws and copper for
coating), results in a very high energy deposition with a peak-energy density
of 30 kJ/cm3. Obviously, the jaw material and the coating will not survive
this huge energy density therefore a replacement of the spoiler after such a
failure scenario would be required.

The main mechanism of secondary particle production on the collimator is the
indirect interaction of the electron beam with the spoiler through photo-
nuclear interactions from the bremsstrahlung photons. Muons, being minimum-
ionizing particles, are those which have a higher probability of reaching the
experiments at the interaction point. The correct evaluation of the muon
spectrum reaching the experiments will strongly depend on the material present
in the flight path of the muons. A possible remedy would be additional
shielding after the spoiler in order to allow the charged mesons to interact
before they decay.

According to the design parameters, the ESP should be exposed to fractions of
10-6 of the full beam, with a very small impact parameter. Assuming an
operation of 200 days per year with a pulse train of
312$\times$3.72$\times$109 electrons every 20 ms, and a damage threshold for
beryllium of 31 eV, the maximum DPA obtained on the ESP spoiler is of the
order of 2.5$\times$10-6/year. Using the same parameters, the core of the ESP
spoiler will reach a maximum activity of 200 mSv/h during operation, which
very quickly will drop to values below 0.1 mSv/h after one hour of cooling due
to the very light and short-lived residual nuclei produced.

####  1.18.3 Photon Absorbers in the Damping Rings

The twenty six wigglers installed in the CLIC damping rings will generate
synchrotron radiation with a critical energy of 9.62 keV with a spectrum given
in Fig. 1.51 and emitted in a concentrated light cone with a small opening
angle of $\sim$3.2 mrad. As shown on the plot the maximum energy of these
photons is of the order of 50 keV, resulting in an attenuation length of about
19 mm on copper. The total power to be absorbed per wiggler is of the order of
8 kW.

Figure 1.51: Synchrotron radiation photon spectrum of 2.424 GeV electrons with
a 2.5 T bending magnet

Vertical and horizontal absorbers made of copper are foreseen after each
wiggler. Ray tracing simulations [308] were performed to estimate the energy
density seen by the wiggler absorbers’ This gave a total power load of 8 kW
with 95 W/mm2 maximum power density. Owing to the high surface density of
deposited power a water cooling system is foreseen on the upper and lower
plates of the absorber.

The line of absorbers in the focusing and defocusing quadrupoles of the
wiggler section intercepts around 210 kW of radiation power. The remaining 90
kW (for the ideal on-axis trajectory) should be stopped by a copper water-
cooled absorber downstream of the first bending magnet after the wiggler
section. The optimization shows that the most advantageous shape of the final
absorber is trapezoidal: in this case we can reduce the power density and
improve heat transfer from the illuminated strip on the absorber body to the
cooling tubes. The total length of the final absorber is about 5 m.

6.00-CEIS/6.00-Chapter-head 7.00-Implementation/7.00-Chapter-head
8.00-Performance/8.00-Chapter-head

##  Chapter 2 Conclusion

###  2.1 Summary and Outlook

In this document, the research and development on the Compact Linear Collider
is summarised, with emphasis on recent studies and R&D for the CLIC
accelerator complex. CLIC is foreseen to be built and operated in stages. This
report provides details of an updated staging scenario, which is optimised for
physics performance, and contains assumptions about commissioning and running
time per year which were recently harmonised with those of other future CERN
projects. For the first stage with a centre-of-mass energy of 380 GeV an
integrated luminosity of 1 ab-1 is foreseen. This is followed by operation at
1.5 TeV with 2.5 ab-1, and by a third stage at 3 TeV with 5 ab-1 integrated
luminosity. This CLIC physics programme spans over 25–30 years. The updated
baseline also specifies $\pm 80$% electron polarisation, with the sharing
between the two longitudinal polarisation states optimised for the best
physics reach at each energy stage.

The construction and operation of CLIC is described, with the Two-Beam
acceleration scheme as baseline scenario. Normal-conducting high-gradient 12
GHz accelerating structures are powered via a high-current drive beam. The
accelerating structures will be operated in the range of 70 to 100 MV/m,
resulting in a total accelerator length of 11 km for the 380 GeV stage and 50
km for 3 TeV. For the first energy stage, an alternative scenario is
presented, with X-band klystrons powering the Main-Beam accelerating
structures. Details of an implementation of CLIC near CERN are described,
generally with emphasis on the 380 GeV stage. These include results on civil
engineering studies, construction and upgrade schedules, electrical networks,
cooling and ventilation, transport and safety aspects.

Beam experiments and hardware tests described in this report demonstrate that
the CLIC performance goals can be met. For instance, accelerating gradients of
up to 145 MV/m are reached with the Two-Beam concept at CTF3, and breakdown
rates of the accelerating structures well below the limit of $3\times
10^{7}\text{m}^{-1}$ are stably achieved at X-band test platforms. High
luminosities can be achieved by using nanometre beam sizes. This requires low-
emittance beams as well as novel alignment and stabilisation techniques. There
is substantial progress in all of these domains: performances as needed for
the CLIC damping rings are achieved by modern synchrotron light sources;
special alignment procedures for the main linac are now available; sub-
nanometre stabilisation of the final focus quadrupoles is demonstrated. In
general, beam physics studies, technical developments and system tests for
CLIC resulted in significant progress in recent years. Reductions in cost and
energy consumption have been among the main objectives of these developments,
resulting in a better energy efficiency of the 380 GeV stage, with power
around 165 MW, together with a lower estimated cost, now around 6 billion CHF.

CLIC accelerator technology has reached a mature state and is increasingly
being put to use in accelerator projects around the globe. A detector design
concept exists, and technology demonstrators for the sub-detectors have been
built. A work-plan for the preparation phase towards building CLIC is outlined
in this report. The CLIC accelerator and detector can be ready for a
construction start around 2026. First collisions at the 380 GeV energy stage
would then take place towards 2035. CLIC provides a broad set of precision
Standard Model physics measurements, reaching well beyond the projections for
HL-LHC. In addition, searches for new physics at the energy frontier can be
performed. In summary, CLIC represents a compelling opportunity for the post-
LHC era.

## Acknowledgements

This work benefited from services provided by the ILC Virtual Organisation,
supported by the national resource providers of the EGI Federation. This
research was done using resources provided by the Open Science Grid, which is
supported by the National Science Foundation and the U.S. Department of
Energy’s Office of Science. This work was supported by the European Union’s
Horizon 2020 Research and Innovation programme under Grant Agreement No.
654168 (AIDA-2020); the European Union’s Horizon 2020 Research and Innovation
programme under Grant Agreement No. 777431 (CompactLight); the European
Union’s Horizon 2020 Marie Sklodowska-Curie Research and Innovation Staff
Exchange programme under Grant Agreement No. 645479 (E-JADE); the National
Commission for Scientific and Technological Research (CONICYT), Chile; the DFG
cluster of excellence “Origin and Structure of the Universe”, Germany; the
Federal Ministry of Education and Research (BMBF), Germany under Grant
Agreement No. 05H18VKRD1; the Israel Science Foundation (ISF); the I-CORE
Program, Israel; the Israel Academy of Sciences; the Programma per Giovani
Ricercatori “Rita Levi Montalcini” of the Ministero dell’Istruzione,
dell’Università e della Ricerca (MIUR), Italy; the Research Council of Norway;
the National Science Centre, Poland, HARMONIA project under contract
UMO-2015/18/M/ST2/00518 and OPUS project under contract
UMO-2017/25/B/ST2/00496; the Polish Ministry of Science and Higher Education
under contract No. 3501/H2020/2016/2 and 3812/H2020/2017/2; the Ministry of
Education, Science and Technological Development of the Republic of Serbia
under contract No. OI171012; the Spanish Ministry of Economy, Industry and
Competitiveness under projects MINEICO/FEDER-UE, FPA2015-65652-C4-3-R,
FPA2015-71292-C2-1-P and FPA2015-71956-REDT; the Generalitat Valenciana under
grant PROMETEO/2018/060, Spain; the IFIC, IFCA, IFT and CIEMAT grants under
the Centro de Excelencia Severo Ochoa and Maria de Maeztu programs,
SEV-2014-0398, MDM-2017-0765, SEV-2016-059, MDM-2015-0509, Spain; the Swedish
Research Council; the Swiss National Science Foundation FLARE and FORCE grants
147463, 141146, 135012, 131428, 125272 and 126838; the Scientific and
Technological Research Council of Turkey (TUBITAK) under grant number 118F333;
the UK Science and Technology Facilities Council (STFC), United Kingdom; and
the U.S. Department of Energy, Office of Science under contract DE-
AC02-06CH11357.


# Typical one-parameter bifurcations of gradient flows with at most six
singular points on the 2-sphere with holes

Svitlana Bilun, Maria Loseva, Olena Myshnova, Alexandr Prishlyak

###### Abstract

We describe all possible topological structures of typical one-parameter
bifurcations of gradient flows on the 2-sphere with holes in the case that the
number of singular point of flows is at most six. To describe structures, we
separatrix diagrams of flows. The saddle-node singularity is specified by
selecting a separatrix in the diagram of the flow befor the bifurcation and
the saddle connection is specified by a separatrix, which conect two saddles.

## Introduction

In this paper, we consider gradient flows on a sphere with holes. Since the
function increases along each trajectory, the flow has no cycles and
polycycles. In general position, a typical gradient flow is a Morse flow
(Morse-Smale flow without closed trajectories). In typical one-parameter
families of gradient flows, two types of bifurcations are possible: saddle-
node and saddle connection. The corresponding vector fields at the time of the
bifurcation are fields of codimension 1. In our case, they completely
determine the topological type of the bifurcation. To classify Morse flows, a
separatrix diagram is often used, in which separetrices are trajectories that
belonging to one-dimensiona stable or unstable manifolds. We apply this
approach to the classification of typical bifurcation.

Without loss of generality, we assume that under bifurcation (as the parameter
increases), the number of singular points does not increase. The saddle-node
bifurcation is defined by a separatrix, which is contructed to a point. We
mark this separatrix on the diagram. A saddle connection bifurcation in the
diagram corresponds to a separatrix, which conect two saddles.

We colar stable separatrices in red, unstable separatrices in green and saddle
connections in black.

Kronrod [14] and Reeb [39] construct topological invariants of functions
oriented 2-maniofolds. It was generlized in [17] for non-orientable two-
dimensional manifolds and in [8, 12, 13] for manifolds with boundary, in [33]
for non-compact manifolds.

In general, Morse flows are gradient flows of Morse functions. If we fix the
value of functions in singular points the flow determinate the topological
structure of the function [17, 41]. Therefore, Morse–Smale flows
classification is closely related to the classification of the functions.

Topological classification of smooth function on closed 2-manifolds was
obtained in [6, 13, 12, 33, 32, 37, 17, 22, 47, 44, 2, 40], on 2-manifolds
with the boundary in [10, 13, 11] and on closed 3- and 4-manifolds in [31,
21].

In [7, 15, 18, 19, 20, 38, 1, 27, 30, 42, 46, 28, 29, 36, 26], the
classifications of flows on closed 2- manifolds and [5, 16, 30, 27, 23, 35,
20, 36] on manifolds with the boundary were obtained. Topological properties
of Morse-Smale flows on 3-manifolds was investigated in [43, 45, 34, 22, 46,
24, 25, 9, 4, 3].

The purpose of this paper is to describe all possible topological structures
of the Morse flows and typical bifurcations with no more than six singular
points (a saddle-node point we consider as two points) on a sphere with holes.

##  1 Typical one-parameter bifurcations of gradient flows on a sphere with
holes

Typical vector fields on compact 2-manifolds are Morse-Smale fields. Morse
fields or Morse-Smale gradient-like fields that do not contain closed
trajectories are tipical among the gradient fields. They satisfy three
properties:

1) singular points are nondegenerate;

2) there are no separatric connections;

3) $\alpha$-limiting ($\omega$-limiting) set of each trajectory is a singular
point.

In typical one-parameter field families, one of these conditions is violated.
Violation of the first condition leads to a saddle-node bifurcation, and the
second to the appearance of a saddle connection. The third condition cannot be
violated, because according to the Poincaré-Bendixon theorem, the
$\alpha$-limit ($\omega$-limit) set of every trajectory on the sphere is
either a singular point, or a cycle, or a polycycle. Since gradient fields
have no cycles and polycycles, this set is a singular point.

###  1.1 Internal bifurcations

According to the theory of bifurcations, there are only two typical
bifurcations of gradient flows: a saddle-node and a saddle connection.

####  1.1.1 Saddle-node bifurcation

The saddle-node bifurcation, when the node is the source, is shown in Fig. 1.

Figure 1: SN – saddle-node bifurcation

It can be described by the equation $V(x,y,a)=\\{x,y^{2}+a\\}$ if the node is
the source and the equation $V(x,y,a)=\\{-x,-y^{2}a\\}$ if the node is a sink.
Here, $a$ is a parameter. If $a<0$ we get the flow before the bifurcation, if
$a>0$ we get the flow after the bifurcation, and if $a=0$ we get the flow at
the moment of the bifurcation (the flow of codimensionality 1). The saddle-
sink bifurcation can be obtained from the saddle-source by changing the
direction of all trajectories, that is, by taking the reversed flow. Note that
the saddle-node bifurcation corresponds to a typical one-parameter catastrophe
(deformation) of functions: $f(x,y,a)=x^{2}+y^{3}+ay$.

Consider a stable manifold (red separatrix in Fig. 1 for a singular saddle-
source point. In order to determine the saddle-node bifurcation it is
necessary to select a separatrix on the seperatrix diagram.

Note that the bifurcation of the saddle flow in Morse’s theory corresponds to
the operation of reducing points or reducing complementary handles.

####  1.1.2 Saddle connection

The bifurcation of the saddle connection is shown in Fig. 2. It can be
described by the equation $V(x,y,a)=\\{x^{2}-y^{2}-1,-2xy+a\\}$.

Figure 2: SC – saddle connection

We describe one of the possible situations when it appears in typical one-
parameter families of gradient fields. Let $p,q$ be saddle critical points of
the function $f$, $f(p)<f(q)$,
$[\frac{f(p)+f(q)}{2}-\varepsilon,\frac{f(p)+f(q)}{2}+\varepsilon]$ does not
contain critical values of $f$, $L$ of the line component of level
$f^{(}-1)(\frac{f(p)+f(q)}{2})$, $u$ is the separatrix of $p$, $v$ is the
separatrix of $q$, which intersect $L$. Then according to Morse theory,
$f^{-1}([\frac{f(p)+f(q)}{2}-\varepsilon,\frac{f(p)+f(q)}{2}+\varepsilon])$ is
homeomorphic to the cylinder $S^{1}\times[0,1]$ and in the coordinates $(s,t)$
($s$–polar angle on the circle) on it, the trajectories are given by segments
$s=\text{const}$. Consider the twisting of a cylinder given by the formula
$(s,t)\to(s+a,t)$ ($a$– deformation parameter). At the same time, the
trajectories turn into helical lines. With a continuous change of the
parameter $a$, the point of intersection of the trajectory $u$ with the circle
$frac{f(p)+f(q)}{2}+\varepsilon$ continuously rotate along it. The
intersection point of the trajectory $v$ with this circle remain unchanged. By
the intermediate value theorem, there is a moment in time $a$ at which these
points coincide. It is at this point in time that the bifurcation of the
saddle connection occur. Note that the construction described above can be
implemented by changing the Riemannian metric (changing angles considered
perpendicular), without changing the function itself. The saddle connection
will be indicated by black separatrix in the separatrix diagram.

In Morse’s theory, bifurcation of the saddle connection corresponds to the
operation of adding handles (sliding one handle over another handle of the
same index).

###  1.2 Bifurcations of singular points on the boundary

With such bifurcations, two cases are possible: 1) the topological structure
of the flow restriction to the boundary changes, 2) this structure does not
change.

Using the Poincaré-Hopf theorem for the doubling of the flow, we can conclude
that the case when one point on the boundary disappears or two points on the
boundary are replaced by one is not possible, because if this is the case,
then the parity of the sum of the indices of the flow on the doubling is
violated. Therefore, the only way to change the structure of the flow at the
boundary in a typical family is the coalescence of two points at the boundary,
which after the bifurcation disappear or are replaced by one internal singular
point.

In the second case, as for the internal saddle-node bifurcation, the
separatrix connecting them is compressed to a point. At the same time, one of
the two points before the bifurcation (saddle or node) lies on the boundary.
Both points cannot lie on the boundary, because then the structure of the
manifold at the point obtained after the compression of the separatrix will be
disturbed after compression.

Bifurcations that do not change the flow at the boundary also include
bifurcations of a saddle connection, when one or both saddles lie on the
boundary. The only thing that should be taken into account here is that these
saddles cannot be adjacent points on the boundary (there are other singular
points between them). Otherwise, the trajectory at the boundary between them,
together with the separatrix connection at the moment of bifurcation, form a
polycycle of length 2, which is not possible for gradient flows.

Depending on the types of points that stick together, different options for
bifurcations are possible.

Figure 3:  BSN – saddle-node (source) bifurcation at the surface boundary

1) In the first case, the sourse and saddle are glueded together at a point.
In Fig. 3 a) we show the flow before the bifurcation ($a=-1$), in Fig. 3 b) –
flow at the moment of bifurcation ($a=0$), in Fig. 3 c) – flow after
bifurcation ($a=1$).

2) In the second case, the saddle and the sink merge into a point, which then
disappears (Fig.4).

Figure 4:  BSN – saddle-node bifurcation (sink) at the surface boundary

If one of the two saddle-node bifurcation points is internal, and the other
lies on the boundary, then we have two types of semi-boundary saddle-node
bifurcations: at the moment of bifurcation, a saddle (HS) or a node (HN).

Figure 5:  HS – semi-boundary saddle bifurcation

Figure 6: HN – semi-boundary node bifurcation

In addition, the following options are possible:

3) both points that stick together are saddles: $\\{x^{2}-y^{2}+a,-2xy\\}$
(Fig. 7),

Figure 7: BDS – double saddle bifurcation at the boundary

4) source and sink stick together, an internal sink appears
$\\{x^{2}-y^{2}+a,2xy+ay\\}$ (Fig. 8)

Figure 8:  source-sink bifurcation – sink

5) source and sink stick together, an internal source appears
$\\{x^{2}-y^{2}+a,2xy-ay\\}$ (Fig. 9)

Figure 9:  source-sink bifurcation – source

In the last two variants, an internal singular point appears, since when
doubling the vector field on the sphere according to the Poincaré-Hopf
theorem, the sum of the Poincaré indices does not change during the
bifurcation, then two saddles on the boundary (index -1) will give rise to one
saddle in each of the half-planes, and source and sink (index +1) source or
sink in each half-plane. Therefore, only the saddle-node bifurcation does not
lead to the appearance of internal singular points. Note that all these
bifurcations at the moment $a=0$ define a flow with fixed points on the
boundary.

As for manifolds without a boundary, in addition to local bifurcations, in
typical one-parameter families of vector fields, only one type of global
bifurcations is possible - a saddle connection

We will show that the described local bifurcations occur in typical one-
parametric families of flows. Since the saddle-node bifurcation occurs in
typical one-parameter families of flows on a plane, when it is restricted to a
half-plane, it will give a typical one-parameter family on a half-plane.

Consider the situation of the merger of two saddles. The flow on the line
$y=0$ has a sink at $x=-1$ and a source at $x=1$.

It continues to the following saddles: $\\{x^{2}-1,y\\}$ in the vicinity of
$(-1,0)$ and $\\{x^{2}-1,-y\\}$ in the vicinity of $(1,0)$.

The simplest function that makes a continuous transition from 1 (at $x=-1$) to
-1 (at $x=1$) there is a function $-x$. Therefore, the flow formula will be
written as $\\{x^{2}-1,-xy\\}$. If we now consider the bifurcation
$\\{x^{2}-a,-xy\\}$, then at $a=0$ a whole straight line $x=0$ will arise from
zeros of the flow, which is not possible according to our assumptions. To
prevent this from happening, let’s add to the first coordinate a function that
rotates to zero on the line $y=0$, the simplest such functions are $y$ and
$-y$, and for symmetric fields with respect to the horizontal axis, $y^{2}$
and $-y^{2}$. Direct inspection shows that the first functions lead to an
interior singular point on the upper half-plane. So, the simplest bifurcation
has the form $\\{x^{2}-y^{2}+a,-2xy\\}$ (for convenience, we stretched the
field twice as much along the y axis). In the general case, we can also change
the second coordinate of the vector field, but considering that this
coordinate is zero at $y=0$, the changes will be written as
$\\{x^{2}-y^{2}+a,-2xy+cay\\}$ for of some constant $c$. But for all $c$ we
will get the same bifurcations.

When the source and the sink stick together, we will perform the same
considerations and obtain a bifurcation of a typical one-parametric family
according to the formula $\\{x^{2}-y^{2}+a,2xy+cay\\}$. Positive and negative
values of $c$ give two different bifurcations (described above), and with
$c=0$ we get a non-typical family of fields.

###### Theorem 1

For typical one-parametric families of flows without closed trajectories and
with fixed points on the boundary, all bifurcation points are isolated.

Proof. In the flat case, the bifurcation point can be the point of
accumulation of bifurcations, if the cycle breaks up at the bifurcation - a
closed curve or a homoclinic trajectory. Since we assume that the fields do
not have closed trajectories, the first case is not possible. In the second
case, the homo clinical trajectory begins and ends in the saddle. This is also
not possible for a saddle on the boundary, for example, if the saddle has the
form $\\{x,-y\\},y\geq 0$, then the trajectories leaving the saddle belong to
the boundary and cannot enter this saddle, since it includes only one
trajectory, which is internal. Therefore, both cases of accumulation of
bifurcation points are not possible. The theorem is proved.

Therefore, for each bifurcation, all flows at $a<0$, as well as all flows at
$a>0$, are topologically equivalent to each other. Without limiting the
generality, we will assume that the number of flow zeros at a<0 is not less
than at $a>0$. The stream at $a=-1$ will be called the initial stream.

###### Theorem 2

On a set of flows with fixed points on the boundary and without closed
trajectories, the bifurcation in a typical family is given either by the
initial flow and a compressible trajectory for local bifurcations on the
boundary or by the flow at the moment of bifurcation in the case of a saddle
connection. In case of merging of source and sink after bifurcation, a flow
with internal zero occurs. To determine such a bifurcation, it is necessary to
additionally specify whether a source or a sink is formed.

###### Theorem 3

The first three bifurcations can be realized in the space of gradient flows,
and the last two do not allow such realizations.

Proof. The one-parameter family of functions (catastrophe) corresponding to
the first bifurcation has the form $f(x,y,a)=x^{3}/3+ax+y^{2}/2$, and for the
second - - $f(x,y,a)=x^{3}/3+ax-y^{2}/2$. For the third bifurcation, the
family of functions is given by the formula $f(x,y,a)=x^{3}/3+ax-xy^{2}$.

For the third and fourth bifurcations at $a=0$, the flow consists of loops,
which is not possible for the gradient flow, since the value of the function
increases along each trajectory of the gradient flow.

Moreover, on the set of gradient flows there is no bifurcation between the
initial and final values ($a=-1$ and $a=1$) of the indicated bifurcations.
Indeed, suppose that such a bifurcation exists. Let us continue the flow at
$a=-1$ to the optimal flow of the Morse flow on a compact surface as in Losev-
Pryshlyak. This flow has a single sink and a single source. The corresponding
Morse function will have one local minimum and one local maximum. If we assume
that there is a bifurcation of the general position, then as a result we will
get a function without a local minimum or a local maximum on a compact
surface, which is not possible.

Therefore, the following types of gradient bifurcations are possible on
spheres with holes:

SN – internal saddle node;

SC – internal saddle connection;

BSN - border saddle node;

BDS – limit double saddle;

HN – semi-boundary saddle node (node);

HS – semi-boundary saddle-node (saddle);

HSC – semimarginal saddle connection;

BSC – a bundle of saddles on the border.

In the case of saddle-node bifurcations, such a bifurcation is given by a
separatrix diagram to the bifurcation, on which the trajectory (separatrix)
between the saddle and the node is highlighted, which is compressed to a
point. To specify the bifurcation of the saddle connection, a separatrix
diagram at the moment of bifurcation is sufficient.

##  2 The structure of typical flows and bifurcations on a 2-disk

In Fig. 10, we show all possible (with accuracy to homeomorphism) separatrix
flow diagrams Morse on 2-disc with no more than 4 singular points.

Figure 10: Morse flows on 2-disc with no more than 4 singular points

At the same time, diagrams 2) and 3), 5) and 6), as well as 7) and 8) are
mutually inverted. In the following figures, we will depict only one of such
diagrams (the one with the largest number of leaks). As can be seen from the
figures, only diagram 4) contains two saddles. Therefore, bifurcation of
saddle connections is impossible for flows with one saddle, and it is also not
possible for flow 4) because the saddles are adjacent on the boundary.

For saddle-node bifurcations, it is only necessary to note how many different
separatrixes and limit trajectories connecting a saddle and a node exist (with
homeomorphism accuracy) on each diagram.

Bifurcations are not possible for the first diagram. For diagrams 2) and 3) we
have one bifurcation (HN) each, for 4) two such bifurcations (HN), for 5) and
6) one SN each, two HS each. Boundary (both points on the boundary) for the
indicated diagrams are not possible, because if one of the two arcs at the
moment of bifurcation contracts to a point, then the other will form a loop,
which is not possible for gradient flows.

For diagrams 7) and 8) we have one BSN bifurcation each (arcs connecting the
saddle and the node on the boundary).

Next, we consider flows with five singular points (Fig. 11).

Figure 11: Morse flows with 5 singu;ar point on the 2-disk

The following saddle-node bifurcations are possible for these flows:

1) 2 SN, 1 HN, 1 HS;

2) 2 SN, 1 HN, 1 HS;

3) 1 BSN, 1 HN;

4) 2 HS.

Note that only 4) of the considered diagrams will turn into itself when the
flow is reversed. Therefore, the total number of bifurcations in the first
three cases is multiplied by 2, and for the last one we leave it unchanged.

In fig. 12 shows all possible separatrix flow diagrams with 6 singular points.

Figure 12: Morse streams on 2-disk with 6 singular points

Depending on the number of separatrixes on each of the diagrams, we get the
following possible bifurcations for each of them:

1) 3 SN, 2 HN;

2) 3 SN, 3 HS;

3) 3 SN, 3 HS;

4) 1 SN, 3 HS;

5) 4 SN, 4 HS;

6) 1 HN, 1 BDS;

7) 2 HN, 1 BDS, 1 BSN;

8) 1 SN, 2 BSN, 2 HS;

9) 1 SN, 3 HS, 2 BSN;

10) 1 SN, 3 HS, 1 HN, 2 BSN;

11) 1 BDS, 2 BSN;

12) 2 BSN.

When changing the direction of the flow, diagrams 5) and 11) will not change,
therefore, in the general calculation of the number of bifurcations, the above
values will not change. For the rest of the charts, these values will be
doubled.

All possible separatrix connections on a two-dimensional disc with no more
than 6 singular points are shown in fig. 13.

Figure 13: Separatrix connections on a 2-disc with no more than 6 singular
points

We went through all the possible options, and therefore it is fair

###### Theorem 4

The following possible structures of typical one-parameter gradient saddle-
node bifurcations exist on a two-dimensional disk:

  * •

with two singular points at the moment of bifurcation - two bifurcations of
HN;

  * •

with three singular points at the moment of bifurcation - two bifurcations HN,
two SN, four HS and two BSN;

  * •

with four singular points at the moment of bifurcation -8 SN, 6 HN, 2 BSN, 8
HS;

  * •

with five singular points at the moment of bifurcation - 30 SN, 12 HN, 38 HS,
22 BSN, 5 BDS.

There are the following bifurcations of the saddle connection:

  * •

with five singular points - four HSC bifurcations ( 1)$\times$2 and
2)$\times$2 in fig. 13);

  * •

seven SC (4)$\times$2,5)$\times$2,6),7)$\times$2), six HSC (3)$\times$2,
8$\times$2, 9)$\times$2 ) and two BSC (10)$\times$2).

##  3 Structure of typical flows and bifurcations on a cylinder

Since there must be at least two singular points on each component of the
boundary, then if the total number of such points is 4, then all of them lie
on the boundary.

If the total number of singular points is 5, then one of them is internal, and
two singular points lie on each component of the boundary (on each component
of the boundary there is an even number of points, because when limiting the
flow to the boundary, sources and sinks alternate).

Figure 14: Morse flows on a cylinder with 4 and 5 singular points

All possible separatrix diagrams of Morse flows with 4 and 5 singular points
are shown in Fig. 14. For the first two diagrams, saddle-node bifurcations are
not possible, since imposing an arbitrary separatrix on them will change the
topological type of the surface. The following saddle-node bifurcations are
possible on other diagrams:

3) 2 HN;

4) 2 HN;

5) 3 HN;

6) 3 HN.

In fig. 15. shows all the separatrix diagrams of Morse flows on a cylinder
with six singular points. For each pair of flow and its inverse, we have
depicted only one diagram.

Figure 15: Morse flows on a cylinder with 6 singular points

The following bifurcations are possible for them:

1) 1 SN, 2 HS, 1HN;

2) 1 SN, 2 HS, 1HN;

3) 4 HS;

4) 6 HS;

5) 2 BSN, 1 BDS;

6) 2 BSN;

7) 1 BSN, 1 BDS;

8) 2 BSN, 1 BDS;

Note that only the third and fourth diagrams will turn into themselves when
the flow orientation is changed.

All possible saddle connections of threads with 4 and 5 singular points on the
cylinder are shown in fig. 16.

Figure 16: saddle connections on a cylinder with 4 and 5 singular points

Flows with 6 singular points and one saddle connection are shown in fig. 17.

Figure 17: saddle connections on a cylinder with 6 singular points

We went through all the possible options, and therefore it is fair

###### Theorem 5

The following possible structures of typical one-parameter gradient saddle-
node bifurcations exist on the cylinder:

  * •

with four singular points at the moment of bifurcation - ten bifurcations of
the HN type;

  * •

with five singular points at the moment of bifurcation - 4 SN, 4 HN, 18 HS, 14
BSN, 6 BDS.

There are the following bifurcations of the saddle connection:

  * •

with four singular points one BSC ( 1) in fig. 16);

  * •

with five singular points – two BSC ( 2) and reversed in fig. 16) and two HSCs
( 3) and inverted in fig. 16);

  * •

with six singular points (Fig. 17) – two SC (3),4)), ten HSC (5)$\times$2,
6)$\times$2, 7)$\times$2, 8)$\times$2, 9)$\times$2) and nine BSC (1,2)
$\times$2,10) $\times$2, 11)$\times$2, 12) $\times$2).

##  4 Structure of typical flows and bifurcations on a sphere with three holes

A sphere with three holes will be considered as a three-connected region on
the plane (a 2-disk with two holes).

Since there must be two singular points on each of the holes, streams with 6
singular points have no other singular points. Therefore, there will be no
internal or semi-boundary bifurcations in this case.

The doubling of a sphere with three holes is a surface of genus 2. Except for
one source and one sink, all other points are saddle points (otherwise the
Poincaré-Hopf theorem on the sum of indices is violated). There are only two
possible structures of Morse streams in this case. In the first case, the
source and the sink lie on one component of the boundary, and in the second -
on different components (see Fig. 18).

Figure 18: Morse flows with 6 singular points on a sphere with three holes

Since only two singular points lie on each boundary component and none of the
separatrixes can be compressed without changing the topological type of the
surface, saddle-node bifurcations do not exist in this case.

Bifurcations of the saddle connection are obtained when one separatrix slides
over another. All possible 4 cases are shown in fig. 19.

Figure 19: saddle connections on a sphere with three holes

Therefore, the following statement is true.

###### Theorem 6

On a sphere with three holes, there are no gradient saddle-node bifurcations
with 5 singular points at the moment of bifurcation, but there are four
boundary saddle bifurcations with 6 singular points.

## Conclusion

All possible structures of Morse flows and typical one-parameter bifurcations
on spheres with holes in which no more than six singular points are found (see
Table 1). We hope that the research carried out in this work can be extended
to other surfaces and with a larger number of singular points.

Number of points | Morse | SN | SC | BSN | BDS | HN | HS | HSC | BSC
---|---|---|---|---|---|---|---|---|---
3 on $D^{2}$ | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0
4 on $D^{2}$ | 5 | 2 | 0 | 2 | 0 | 0 | 2 | 4 | 0
5 on $D^{2}$ | 7 | 8 | 0 | 2 | 0 | 6 | 8 | 4 | 0
6 on $D^{2}$ | 22 | 30 | 7 | 22 | 5 | 12 | 38 | 6 | 2
4 on $S^{1}\times I$ | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
5 on $S^{1}\times I$ | 4 | 0 | 0 | 0 | 10 | 0 | 0 | 2 | 2
6 on $S^{1}\times I$ | 14 | 4 | 2 | 14 | 6 | 4 | 18 | 10 | 9
6 on $F_{0,3}$ | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4
Table 1: Number of Morse flows and bifurcations on $S^{2}$ with holes (number
of points befor bifurcation)

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Taras Shevchenko National University of Kyiv

Svitlana Bilun Email address<EMAIL_ADDRESS>Orcid ID: 0000-0003-2925-5392

Maria Loseva Email<EMAIL_ADDRESS>Orcid ID: 0000-0002-2282-206X

Olena Myshnova Email<EMAIL_ADDRESS>Orcid ID: 0009-0001-8808-0038

Alexandr Prishlyak Email address<EMAIL_ADDRESS>Orcid ID:
0000-0002-7164-807X


# Training Robust Spiking Neural Networks on Neuromorphic Data with
Spatiotemporal Fragments

###### Abstract

Neuromorphic vision sensors (event cameras) are inherently suitable for
spiking neural networks (SNNs) and provide novel neuromorphic vision data for
this biomimetic model. Due to the spatiotemporal characteristics, novel data
augmentations are required to process the unconventional visual signals of
these cameras. In this paper, we propose a novel Event SpatioTemporal
Fragments (ESTF) augmentation method. It preserves the continuity of
neuromorphic data by drifting or inverting fragments of the spatiotemporal
event stream to simulate the disturbance of brightness variations, leading to
more robust spiking neural networks. Extensive experiments are performed on
prevailing neuromorphic datasets. It turns out that ESTF provides substantial
improvements over pure geometric transformations and outperforms other event
data augmentation methods. It is worth noting that the SNNs with ESTF achieve
the state-of-the-art accuracy of 83.9% on the CIFAR10-DVS dataset.

Index Terms—  Spiking Neural Networks, Event Spatiotemporal Fragment
Augmentation, Neuromorphic Data

##  1 Introduction

Event cameras [1, 2] are bio-inspired vision sensors that operate in a
completely different way from traditional cameras [3, 4]. Instead of capturing
images at a fixed rate, the cameras asynchronously measure per-pixel
brightness changes and output a stream of events that encode the time,
location, and sign of the brightness changes. Since both events and spikes are
modeled from neural signals, event cameras are inherently suitable for spiking
neural networks (SNNs), which are considered promising models for artificial
intelligence (AI) and theoretical neuroscience [5].

Due to the event characteristics, brightness variances tend to disrupt the
spatiotemporal positions of the events, posing a great challenge to the
robustness of SNNs. However, traditional data augmentations are fundamentally
designed for RGB data and lack exploration of neuromorphic events, such as
mixup [6] and random erase [7]. Therefore, novel data augmentations are
required to process the unconventional visual signals of these cameras.

In this paper, we propose a novel Event SpatioTemporal Fragments (ESTF)
augmentation method to model the effects of brightness variances on
neuromorphic data. ESTF can be simply summarized into two components,
inverting event fragments in the spatiotemporal and polarity domains, and
drift events in the spatiotemporal domain. ESTF preserves the original
spatiotemporal structure through transformed fragments and increases the
diversity of neuromorphic data. Extensive experiments are conducted on
prevailing neuromorphic datasets. It turns out that ESTF improves the
robustness of SNNs by simulating the effect of brightness variances on
neuromorphic data. Furthermore, we analyze the insightful superiority of ESTF
compared with other event data augmentation methods. It is worth noting that
ESTF is super effective on both SNNs and convolutional neural networks (CNNs).
For example, the SNNs with ESTF yield a substantial improvement over previous
state-of-the-art results and achieve an accuracy of 83.9% on the CIFAR10-DVS
dataset, which illustrates the importance of event spatiotemporal fragment
augmentation.

In addition, while this work is related to EventDrop [8] and NDA [9], there
are some distinct differences. For example, NDA is a purely global geometric
transformation, while ESTF transforms partial event stream in the
spatiotemporal or polarity domain. EventDrop introduces noise by dropping
events, resulting in poor sample continuity. However, ESTF also considers the
brightness variances in neuromorphic data and preserves the original
spatiotemporal structure of events.

##  2 Method

###  2.1 Event Generation Model

The event generation model [3, 4] is abstracted from dynamic vision sensors
[2]. Each pixel of the event camera responds to changes in its logarithmic
photocurrent $L=\log(I)$. Specifically, in a noise-free scenario, an event
$e_{k}=(x_{k},y_{k},t_{k},p_{k})$ is triggered at pixel $X_{k}=(y_{k},x_{k})$
and at time $t_{k}$ as soon as the brightness variation $|\Delta L|$ reaches a
temporal contrast threshold $C$ since the last event at the pixel. The event
generation model can be expressed by the following formula:

$\displaystyle\Delta L(X_{k},t_{k})=L(X_{k},t_{k})-L(X_{k},t_{k}-\Delta
t_{k})=p_{k}C$ (1)

where $C>0$, $\Delta t_{k}$ is the time elapsed since the last event at the
same pixel, and the polarity $p_{k}\in\\{+1,-1\\}$ is the sign of the
brightness change. During a period, the event camera triggers event stream
$\mathcal{E}$:

$\mathcal{E}=\left\\{e_{k}\right\\}_{k=1}^{N}=\left\\{\left(X_{k},t_{k},p_{k}\right)\right\\}_{k=1}^{N}$
(2)

where $N$ represents the number of events in the set $\mathcal{E}$.

Further, inspired by previous work [3], an event field is used to represent
the discrete $\mathcal{E}$ by replacing each event of the spatiotemporal
stream with a Dirac spike, resulting in a continuous representation of events.
For convenience, we define the complete set of transformation domains as
$D=\\{X,p,t\\}$, and map the polarity $p$ from $\\{-1,+1\\}$ to $\\{0,1\\}$.
The continuous event filed $S$ can be expressed as the following formula:

$\displaystyle S^{\mathcal{E}}(X,p,t)$
$\displaystyle=\sum_{e_{k}\in\mathcal{E}}\delta(X-X_{k})\delta(t-t_{k})\delta(p-p_{k})$
(3) $\displaystyle=\sum_{e_{k}\in\mathcal{E}}\prod_{i\in
D}\delta\left(i-i_{k}\right)$

The event generation model is shown in Fig. 1(a), an event is generated each
time the brightness variances reach the threshold, and then $|\Delta L|$ is
cleared. Note that due to the single-ended inverting amplifier in the sensor
differential circuit, the threshold for brightness changes in the circuit is
one sign different from the threshold in the conceptual model. In addition,
the short segment following the firing event represents the corresponding
refractory period in biological neurons.

###  2.2 Motivation

This work stems from three observations. First, the mammalian brain can make
correct judgments based on fragments of time and space, while SNNs are less
robust. In addition, it can be known from the event generation model that
events asynchronously record brightness variances in microseconds. Thus, even
slight brightness variances can disrupt the time or polarity of events,
leading to cascading changes in subsequent events. As shown in Figures 1(b),
1(c), and 1(d), microsecond-level fluctuations in brightness can disrupt the
spatiotemporal position and even the polarity of event fragments. Furthermore,
sensor noise not only discards events [8], but may also disrupt the
spatiotemporal location of event fragments. For example, jitter in the step
response may introduce sensor delays [2], resulting in delayed brightness
variances. The relative mismatch between the reset level of the differential
circuit and the comparator threshold may lead to a mismatch in the dynamic
threshold $C$ [2], causing the effect shown in Fig. 1(d). These observations
inspire us to improve the robustness of SNNs by artificially generating event
spatiotemporal fragments.

(a) Event generation model.

(b) Opposite light-dark processes.

(c) Inverted brightness variations.

(d) Delayed brightness variations.

Fig. 1: The relationship between logarithmic intensity and events generation
for each pixel. (a) is the initial brightness and (b)(c)(d) is the changed
brightness.

###  2.3 Event SpatioTemporal Fragment Augmentation

To obtain the transformed event spatiotemporal fragments, the ESTF method is
divided into two parts, inverting event fragments in spatiotemporal or
polarity domains (ISTP), and drifting event fragments in spatiotemporal domain
(DST).

ISTP refers to inverting certain event fragments $\mathcal{E}_{c}$ on
spatiotemporal or polarity domains $d$. The transformed event field $S_{ISTP}$
can be formulated as:

$\displaystyle S^{\mathcal{E}_{c}}_{ISTP}(X,p,t)$
$\displaystyle=\sum_{e_{k}\in\mathcal{E}_{c}}(\delta\left(d+d_{k}-R_{d}\right)\prod_{i\in
D\\{d\\}}\delta\left(i-i_{k}\right))$ (4)

where $d\in D$, $R_{d}$ represents the resolution of the domain $d$. Event
fragments are inverted from $d_{k}$ to their symmetrical positions
$R_{d}-d_{k}$ in the event stream. When $d$ is the time domain $t$, $R_{d}$
represents the largest timestamp, when $d$ is the polarity $p$, $R_{d}$
represents 1, and when $d$ is the space coordinate $X$, $R_{d}$ represents the
image resolution. $D\\{d\\}$ is the complement of $d$ with respect to $D$.

Table 1: Performance of ESTF method and the SOTAs on CIFAR10-DVS, N-Caltech101 and N-CARS datasets.  Network | Method | Reference | Model | CIFAR10-DVS | N-Caltech101 | N-CARS
---|---|---|---|---|---|---
CNNs-based | RG-CNNs [10] | TIP 2020 | Graph-CNN | 54.00 | 61.70 | 91.40
SlideGCN [11] | ICCV 2021 | Graph-CNN | 68.00 | 76.10 | 93.10
EventDrop [8] | IJCAI 2021 | Mobilenet-V2 | - | 87.14 | 94.55
ECSNet [12] | T-CSVT 2022 | LEE$\rightarrow$MA | 72.70 | 69.30 | 94.60
EV-VGCNN [13] | CVPR 2022 | EV-VGCNN | 67.00 | 74.80 | 95.30
NDA [9] | ECCV 2022 | Resnet-34 | - | 81.20 | 95.50
Ours(w/o. ESTF) | - | Resnet-34 | 77.15 | 90.86 | 93.80
Ours(w/. ESTF) | - | Resnet-34 | 83.12+5.97 | 93.15+2.29 | 96.00+2.20
SNNs-based | HATS[14] | CVPR 2018 | HATS-SVM | 52.40 | 64.20 | 81.0
Dart[15] | TPAMI 2020 | SPM-SVM | 65.80 | 66.80 | -
STBP [16] | AAAI 2021 | Resnet-19 | 67.80 | - | -
Dspike [17] | NeurIPS 2021 | ResNet-18 | 75.40 | - | -
AutoSNN [18] | ICML 2022 | - | 72.50 | - | -
RecDis [19] | CVPR 2022 | Resnet-19 | 72.42 | - | -
DSR [20] | CVPR 2022 | VGG-11 | 77.27 | - | -
NDA [9] | ECCV 2022 | VGG-11 | 81.70 | 78.20 | 90.1
Ours(w/o. ESTF) | - | VGG-9 | 67.70 | 67.32 | 91.65
Ours(w/. ESTF) | - | VGG-9 | 83.90+16.20 | 80.07+12.75 | 95.64+3.99

DST drifts events $\mathcal{E}_{c}$ a certain distance on spatiotemporal
domain, which can be represented as a convolution kernel $k_{DST}$:

$k_{DST}(X,p,t)=\delta\left(d-r_{d}\right)\prod_{i\in
D\\{d\\}}\delta\left(i\right)$ (5)

where $d\in\\{X,t\\}$, $r_{d}$ represents the moving distance in the domain
$d$. The transformed event field $S_{DST}$ can be obtained by convolving
$k_{DST}$ with the event field $S$:

$\displaystyle S^{\mathcal{E}_{c}}_{DST}(X,p,t)$
$\displaystyle=\left(k_{DST}*S\right)(X,p,t)$ (6)
$\displaystyle=\sum_{e_{k}\in\mathcal{E}_{c}}k_{DST}(X-X_{k},p-p_{k},t-t_{k})$
$\displaystyle=\sum_{e_{k}\in\mathcal{E}_{c}}(\delta\left(d-d_{k}-r_{d}\right)\prod_{i\in
D\\{d\\}}\delta\left(i-i_{k}\right))$

where $d\in\\{X,t\\}$ and $\mathcal{E}_{c}$ represents events to be drifted.
DST drifts fragments $\mathcal{E}_{c}$ a distance $r$ in the domain $d$, and
the part beyond borders will be discarded. It is worth noting that drifted
samples of different scales are generated by different distances $r$, which
obey a uniform distribution specified by a hyperparameter. In addition, the
event field $S_{ESTF}$ that applies both ISTP and DST can be obtained by
convolving $k_{DST}$ and $S_{ISTP}$:

$S^{\mathcal{E}_{c}}_{ESTF}(X,p,t)=\left(k_{DST}*S^{\mathcal{E}_{c}}_{ISTP}\right)(X,p,t)$
(7)

The pseudocode of ESTF is given by Algorithm 1.

Algorithm 1 ESTF Procedure

Input: Original events $S=\left\\{e_{i}\right\\}_{i=1}^{N}$
Parameter: Fragments ratio $c_{istp}$ and $c_{dst}$; Domain: $d_{istp}$ and
$d_{dst}$; Drift ratio: $r$; Resolution: $R_{d}$.
Output: Transformed event fragments $\mathcal{S}_{ESTF}$

1: $begin\leftarrow Rand(0,1-c_{istp})$

2: $\mathcal{E}_{i}\leftarrow\\{e_{i}\\}_{i\in[N*begin,N*(begin+c)]}$

3:
$S_{ISTP}=Eq4(\mathcal{E}_{i},d_{istp},R_{d})+\mathcal{E}\\{\mathcal{E}_{i}\\}$

4: $begin\leftarrow Rand(0,1-c_{dst})$

5: $\hat{r}\leftarrow Randint[-r*R_{d},r*R_{d}]$

6: $\mathcal{E}_{d}\leftarrow\\{e_{i}\\}_{i\in[N*begin,N*(begin+c)]}$

7:
$S_{ESTF}=Eq7(S^{\mathcal{E}_{d}}_{ISTP},d_{dst},\hat{r})+\mathcal{E}\\{\mathcal{E}_{d}\\}$

8: return $S_{ESTF}$

##  3 Experiments

###  3.1 Implementation

Extensive experiments are performed to demonstrate the superiority of our ESTF
method on the prevailing neuromorphic datasets, including CIFAR10-DVS [21],
N-Caltech101 [22], N-CARS [14] datasets. For the convenience of comparison,
the model with the same parameters without ESTF is used as the baseline. STBP
[16] method is used to train SNNs, and other parameters mainly refer to NDA
[9]. For example, the initial learning rate is $1e-3$, the neuron threshold
and leakage coefficient are 1 and 0.5, respectively. Furthermore, we also
evaluate the performance of ESTF on CNNs, a popular network for processing
neuromorphic data. The EST [3] representation method is used to convert
neuromorphic data into a form compatible with CNNs. Adam optimizer is used
with an initial learning rate of $1e-4$.

###  3.2 Compared with SOTAs

ESTF outperforms previous SOTAs on three prevailing and challenging
neuromorphic datasets. As shown in Tab. 1, ESTF achieves substantial
improvements on SNNs through spatiotemporal fragments. In particular, ESTF
achieves a 16.2% improvement on the CIFAR10-DVS dataset, reaching a state-of-
the-art accuracy of 83.9%. In addition, ESTF also has significant improvements
over CNNs. Since ESTF is orthogonal to most training algorithms, it can
provide a better baseline and improve the performance of existing models.

Table 2: Performance of ESTF compared with other event data augmentations on the N-Caltech101 dataset. Methods | Accuracy(%)
---|---
VGG-9 | VGG-19 | Resnet-34
(SNNs) | (ANNs) | (ANNs)
Baseline | 67.32 | 87.30 | 90.86
EventDrop | 68.69 | 88.31 | 92.06
NDA∗ | 77.44 | 88.84 | 92.42
ESTF | 80.07+12.75 | 91.29+3.99 | 93.15+2.29

(a) ISTP and DST.

(b) Different domains.

Fig. 2: Ablation studies on ESTF.

###  3.3 Compared with Event Data Augmentations

As shown in Tab. 2, ESTF is compared with other reproduced event data
augmentations at the same baseline. EventDrop [8] is similar to Cutout on
event data and achieves remarkable success on CNNs. NDA∗ represents applying
the two naive geometric transformations flipping and translation on event
data. It turns out that ESTF outperforms EventDrop and NDA∗ in multiple
networks, covering both SNNs and ANNs. It is worth noting that the performance
of EventDrop in SNNs may lag far behind ESTF because deletion destroys the
continuity of neuromorphic data and may cause the “dead neuron” problem. In
addition, ESTF is more stable and more effective than geometric
transformations. It illustrates the superiority of spatiotemporal fragment
augmentation.

###  3.4 Ablation Studies on ESTF

Performance of ISTP and DST. As shown in Fig. 2(a), the performance of ISTP
and DST are evaluated on the DVS-CIFAR10 dataset. It turns out that both ISTP
and DST have significant improvements. Notably, the improvement of DST is more
pronounced since drifting events are more common than inverting events.

Performance of Different Domains. As shown in Fig. 2(b), the performance of
ESTF in different domains is evaluated on the N-CARS dataset. It turns out
that ESTF is effective in the time, spatial, and polarity domains. It is worth
noting that the ESTF performs better in the temporal and polarity domains than
in the spatial domain, which indicates that the event spatiotemporal fragments
are critical to improving the robustness of the model.

###  3.5 Analysis of ESTF

(a) Inverted brightness variations.

(b) Delayed brightness variations.

Fig. 3: Performance under different brightness interferences.

There are at least three insightful reasons for the superior performance of
the event spatiotemporal augmentation (ESTF). Intuitively, ESTF transforms in
the time, space, and polarity domains, while NDA only transforms in space. Fig
2(b) shows the performance of ESTF in different domains. It is worth noting
that the effects of transformations in the time and polarity domains are
significant. Considering that the performance of NDA∗ geometric
transformations is already close to SOTAs, there is still more than 2%
improvement in ESTF, which is quite impressive, as shown in Tab. 2. In
addition, ESTF preserves the continuity of neuromorphic data through
spatiotemporal fragments, maintaining the original local features. The
comparison with EventDrop in Tab. 2 illustrates the importance of continuity.
Furthermore, the transformations of ISTP and DST in time and polarity domains
essentially improve the robustness of SNNs to brightness variances. Note that
event samples may only last 100ms (N-CARS), and slight brightness fluctuations
can bring brightness variances similar to Fig. 1. As shown in Fig. 3, we
further evaluate the robustness of SNNs against brightness variances. It turns
out that the model with ESTF is significantly more robust under various
degrees of brightness interference.

##  4 Conclusion

In this paper, we propose event spatiotemporal fragment augmentation. It
preserves the continuity of the data, transforms events in the spatiotemporal
and polarity domains, and simulates the influence of brightness variances on
neuromorphic data. It makes up for the omission of spatiotemporal fragment
augmentation in the past, resulting in superior improvements and SOTA
performance on prevailing neuromorphic datasets.

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