II, 3 II/3

# WEAKLY SUPERVISED PSEUDO-LABEL ASSISTED LEARNING FOR ALS POINT CLOUD
SEMANTIC SEGMENTATION

Puzuo Wang1 Wei Yao1 Corresponding author. 1 Department of Land Surveying and
Geo-Informatics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
\-<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Competitive point cloud semantic segmentation results usually rely on a large
amount of labeled data. However, data annotation is a time-consuming and
labor-intensive task, particularly for three-dimensional point cloud data.
Thus, obtaining accurate results with limited ground truth as training data is
considerably important. As a simple and effective method, pseudo labels can
use information from unlabeled data for training neural networks. In this
study, we propose a pseudo-label-assisted point cloud segmentation method with
very few sparsely sampled labels that are normally randomly selected for each
class. An adaptive thresholding strategy was proposed to generate a pseudo-
label based on the prediction probability. Pseudo-label learning is an
iterative process, and pseudo labels were updated solely on ground-truth weak
labels as the model converged to improve the training efficiency. Experiments
using the ISPRS 3D sematic labeling benchmark dataset indicated that our
proposed method achieved an equally competitive result compared to that using
a full supervision scheme with only up to 2$\permil$ of labeled points from
the original training set, with an overall accuracy of 83.7$\%$ and an average
F1 score of 70.2$\%$.

###### keywords:

Semantic segmentation, Pseudo labels, Weakly supervised learning, Airborne
Laser Scanning, Point clouds.

##  1 INTRODUCTION

The recent development of convolutional neural networks (CNNs) has been
followed by the progress of point cloud semantic segmentation tasks, and there
are constantly new methods that achieve state-of-the-art results. Most of
these methods focus on designing new network structures or convolution kernels
based on the characteristics of point cloud data (Hu et al.,, 2020; Thomas et
al.,, 2019; Huang et al.,, 2020; Steinsiek et al.,, 2017), whereas few focus
on data labeling. In experiments, competitive results rely on a large amount
of data annotations, which continuously require labeling operators with expert
knowledge. The labeling of point cloud data is particularly difficult, and the
operator usually needs to confirm the category of a point from multiple
perspectives. Although labeling work is difficult and time-consuming,
obtaining raw point cloud data has become easier. With the advancement of
light detection and ranging sensor technology and the diversification of data
acquisition platforms, obtaining massive unlabeled point cloud data is no
longer a problem. Thus, extracting information that helps the task of semantic
segmentation from unlabeled data is an essential issue. By using a small
amount of annotated data to achieve classification results comparable to full
supervision, the workload of data annotation would be considerably reduced and
the efficiency in related applications would be improved.

Figure 1: Comparison of (a) fully-labeled points and (b) weak-label. (b)
Situation in which only 100 points per class ($2\permil$) are labeled. The
size of labeled points in (b) is enlarged for better visualization.

Semi- and weakly supervised learning are commonly used to address situations
in which the number of labels is scarce. There are some comprehensive reviews
on these two types of supervised learning methods (Chapelle et al.,, 2010;
Zhou,, 2017; Zhu,, 2005). Choosing a suitable weak label annotation strategy
is key to balancing the labeling workload and experimental results. In image
processing, weak labels are represented as a few labeled images (Dong and
Xing,, 2018), a few labeled pixels (Bearman et al.,, 2016), and a few bounding
boxes or categories that appear on the images (Kolesnikov and Lampert,, 2016;
Papandreou et al.,, 2015). Until now, there have been very few works that used
weakly supervised methods to process point cloud data. Wei et al., (2020)
applied a point class activation map to classify point clouds using only
cloud-level labels. However, we believe that it is not practical for point
cloud data, particularly in outdoor scenes that cover a wide region, because
the complete data must be divided into many small blocks and categories
contained in each block must be specified. In comparison, assigning labels to
a few points is a more direct approach. Polewski et al., (2015) used an active
learning method to detect standing dead trees from airborne laser scanning
(ALS) point cloud data combined with infrared images. Lin et al., (2020)
proposed an active and incremental learning strategy for ALS semantic
segmentation, and manual annotation was iteratively added for training.
Nonetheless, the setting of a weak label is used to annotate all points of
tiles, and manual intervention is required during training. Through
theoretical analysis and experimental results, Xu and Lee, (2020) found that
when the number of labeled points remains constant, the spatially sparse
annotations achieve better results than those samples gathering in certain
object instances, and this weak-label setting was adopted in this work. An
illustration of the sparse weak-label situation is shown in Fig. 1. A weakly
supervised semantic point cloud segmentation framework was proposed by Xu and
Lee, (2020), and an approximate result of fully supervised learning was
obtained using 10$\%$ labels. However, the characteristic of this weak label
can be understood as being spatially continuous at a lower resolution, and the
workload of labeling is still large. Guinard and Landrieu, (2017) utilized the
point cloud segmentation method to improve the classification accuracy with
very few labels, but the result largely depends on the segmentation accuracy
and focuses on classifying the point cloud of the area where the initial weak
label is located, which is transductive learning.

Pseudo-labels, assigning annotations to unlabeled data based on the
predictions of the current model, can enable the use of unlabeled data in
parameter updates. Through the pseudo-label method, the classification model
can calculate the loss function and backpropagation from the dataset with more
labels, thereby improving the accuracy. Iscen et al., (2019) and Lee, (2013)
applied pseudo-labels in image classification, and Zou et al., (2021) designed
pseudo-labeling and data augmentation to improve the performance of image
semantic segmentation. Berthelot et al., (2019) mixed several proven semi-
supervision strategies and proposed a holistic framework. Only a few studies
have utilized pseudo-labels in point cloud processing. Yao et al., (2020)
introduced a pseudo-labeling method into point cloud semantic segmentation.
However, similar to Guinard and Landrieu, (2017), the framework is also
transductive learning , and the performance of the model has not been verified
on untrained test data.

To cooperate with pseudo-labels, it is essential to propose an effective
semantic segmentation network structure. Owing to the irregular distribution
of point clouds, the network structure is not as uniform as that of two-
dimensional (2D) CNNs. Early processing methods project point clouds onto
images and directly use mature 2D CNNs to train the model (Boulch et al.,,
2018). Another branch of methods voxelizes point clouds and proposes three-
dimensional (3D) CNNs to process voxel data (Tchapmi et al.,, 2017). The
disadvantage of these two methods is that as point clouds need to be converted
into regularized data, geometric information may be lost. PointNet and
PointNet++ (Qi et al., 2017a, ; Qi et al., 2017b, ) are pioneers in the
development of shared multilayer perceptrons (MLPs) to directly analyze point
clouds. However, the spatial distribution relationship of the point cloud
requires well-designed MLPs, which complicates network structures. Compared to
pointwise MLP networks, graph convolution networks construct a graph through
relative spatial positions between points for feature extraction and fusion
(Wang et al.,, 2019). Point convolution networks are similar to graph-based
methods, in which point kernels are designed to learn local geometric
information (Thomas et al.,, 2019).

In this study, we explored how to obtain reliable semantic segmentation
results with very few labels. A pseudo-label-assisted point cloud semantic
segmentation framework with extremely few annotations is proposed. We used
KPConv (Thomas et al.,, 2019), a point convolution network, as our encoder
network. Our weak label was defined as sparsely labeled points randomly
distributed in space. An initial model was trained using the selected weak
labels. Pseudo-labels were then generated by the trained model. Considering
that the model obtained from weak label training is underfitting to the entire
data space, an adaptive threshold is proposed to balance the number and
accuracy of pseudo-labels. The training procedure that combines the ground-
truth labels, referred to weak labels, and pseudo-labels was iteratively
performed. To accelerate the training progress and reduce the influence of
pseudo labels containing errors on the model, we updated the pseudo labels
when the model converged on the ground-truth labels. Experiments on the ALS
dataset showed the effectiveness of the method.

##  2 METHODOLOGY

An overview of the proposed method is presented in Fig. 2. There are two
stages in the framework: red lines represent incomplete training and blue
lines represent pseudo-label-assisted training. In stage 1, the initial model
is trained using sparse labeled points, based on which pseudo labels are
generated. In stage 2, the training progress is continued with a combination
of the initial ground truth and pseudo labels. Pseudo labels are iteratively
updated once the mixed trained model converges on the labels. The following
sections include the introduction of the semantic segmentation network, the
generation and update of pseudo labels, and the entire training process.

Figure 2: Architecture of the proposed method.

###  2.1 Point Cloud Semantic Segmentation Network

This section briefly reviews the architecture of the adopted network. In this
work, KPConv was used as our encoder network because of its state-of-the-art
experimental results on several public datasets. KPConv uses the idea of
convolution kernels in image convolutions and can extend kernel points to be
deformable to adapt the local features of point clouds. We chose the rigid
point convolution kernel and used the same network architecture as KPConv for
the semantic segmentation task. The encoder network contained five
convolutional layers, and the ResNet-like structure was embedded. Skip links
were used in the decoder network, and features were passed by the nearest
sampling.

###  2.2 Incomplete Supervision

In this study, weak labels are defined as limited and sparse ground-truth
labels distributed across the space, which is referred to as incomplete
supervision. In the form of manual annotation , we randomly selected several
points according to the category and assigned the ground-truth labels. Given
that the input points $P\in\mathbb{R}^{N\times D}$ consist of N points with D
dimensional features, M(M$\ll$N) points are assigned labels, denoted as
$P_{w}\in\mathbb{R}^{M\times D}$ and the corresponding labels
$L_{w}\in\mathbb{R}^{M}$.

The training process of incomplete supervision is similar to that of full
supervision, and the only difference lies in the design of the loss function.
Because only a few points have label information, we calculated the loss of
these points and performed backpropagation. We chose the softmax cross-entropy
loss function on the labeled points, denoted as:

$l_{true}=-\frac{1}{M}\sum\limits_{i}{{{\hat{y}}_{ic}}}\frac{{\exp({y_{ic}})}}{{\sum\nolimits_{c}{\exp({y_{ic}})}}},\
$ (1)

where $y_{ic}$ and ${\hat{y}}_{ic}$ are the prediction and label of point
$p_{i}$, respectively.

###  2.3 Pseudo-Label assisted Learning

In incomplete learning, model parameters are trained by partial points, and
the accuracy of prediction is significantly lower than that of fully
supervised learning. This is because limited labeled points cannot describe
the overall distribution of the data. Thus, using more training data can
achieve predictions with higher accuracy. Pseudo-labels can effectively
exploit unlabeled data as a semi-supervised learning method. The pseudo-label-
assisted learning method predicts the unlabeled data through the model
parameters obtained by the previous training and regards those labels as
ground truth in back propagation calculation (Lee,, 2013). By increasing the
number of pseudo-labels, we intended to simulate the data distribution of the
entire scene. We propose a framework for pseudo-label assisted point cloud
semantic segmentation, and the details of the generation and update of pseudo-
labels are introduced in the following two sections.

####  2.3.1 Generation of Pseudo-labels

We used the predictions at the unlabeled data of the training sets to generate
pseudo-labels, and the category of each point was determined as the one with
the maximum posterior probability:

$L_{pred}=\arg\max{p_{ic}},\ $ (2)

Because they are pseudo labels, there are inevitably some incorrect
predictions. This is because the parameters of the trained model are only
obtained from limited labeled data, which is underfitting to the entire data.
Therefore, it is necessary to perform label selection to obtain suitable
pseudo-labels. Generally, a prediction with a higher posterior probability is
more likely to be correct. Thus, a commonly used method is to set a strict
threshold and select labels with predicted probabilities above it. Although a
high threshold will result in more accurate pseudo-labels, it will lead to a
decrease in the number of labeled points. However, Yao et al., (2020) found
that the experimental results are not sensitive to the threshold setting.
Unlike using a fixed value, we developed an adaptive threshold-setting method.
After attaining the probability of points toward each predicted category, we
chose their average value as the threshold:

$N_{pseudo}=N({p_{i}}>\frac{1}{n}\sum\limits_{i}{{p_{i}})},\ $ (3)

where ${p_{i}}=\max({p_{ic}})$. N and n are the index and the number of
predictions of the training data, respectively. Predictions with a probability
above the threshold are considered reliable and are selected as pseudo labels.

####  2.3.2 Update of Pseudo Labels

We aimed to enhance the performance of the classifier through iterative
training and identify essential pseudo-labels, thus the previous ones were
discarded with the update. The initial pseudo-labels were generated by the
model obtained from incomplete supervision, and subsequent updates were
generated by pseudo-label-assisted learning. The real labels from the ground
truth are essential. Thus, to maintain their influence on the model training,
we used the loss function in Lee, (2013), which calculates the loss of ground
truth and pseudo-labels separately. A cross-entropy loss was still utilized
for pseudo-labels, denoted as:

$l_{pseudo}=-\frac{1}{{M^{\prime}}}\sum\limits_{i}{{{\hat{y}^{\prime}}_{ic}}}\frac{{exp({y_{ic}})}}{{\sum\nolimits_{c}{exp({y_{ic}})}}},\
$ (4)

where ${\hat{y}^{\prime}}_{ic}$ and $M^{\prime}$ are pseudo-labels and their
numbers, respectively. Thus, the total loss in pseudo-label-assisted learning
is defined as:

$L_{p}=l_{true}+\alpha\cdot l_{pseudo},\ $ (5)

where $\alpha$ is the weight coefficient. Although the number of ground truth
and pseudo-labels is quite different, both $l_{true}$ and $l_{pseudo}$ are
averaged loss values over all the points, thus was empirically set to 1. The
pseudo-labels were updated when the model fitted the current labels.

Unlike the setting in Yao et al., (2020), the model that converges on the
ground truth is used as the condition for updating pseudo labels. We believe
that there are some errors in pseudo labels, particularly when the scene is
complex, thus completely fitting pseudo-labels may result in error
transmission. In addition, because $l_{true}$ is retained in the entire
training process, convergence on the ground truth is sufficient to guarantee
the performance of the model. Another advantage is that it can increase the
frequency of pseudo-label updates and improve training efficiency. A
comparison is presented in the Experimental section. Convergence was
determined by the minimum training accuracy in one epoch. If the value was
above 99$\%$, the pseudo labels were updated at the end of the epoch.

Input: Point clouds ${P\in\mathbb{R}^{N\times D}}$, Labels
${L\in\mathbb{Z}^{M}}$

Output: Predictions ${Y\in\mathbb{Z}^{N}}$

Initialize ${\Theta}$ randomly;

# Incomplete supervision

for  _$epoch\leftarrow 1$ to $100$_ do

Train One Epoch:

$\Theta_{T}=\Theta_{T}-\eta\nabla l_{true}(L,{\rm{}}Y|\Theta_{T})$;

# $\Theta$ is learned parameters

# Pseudo-label assisted learning

$\Theta_{P}=\Theta_{T}$;

for  _$epoch\leftarrow 1$ to $100$_ do

Pseudo-labels $L^{\prime}=Y|(P,\Theta_{P})\cap Selection$;

repeat

Train One Epoch:

$\Theta_{P}=\Theta_{P}-\eta(\nabla l_{true}(L,Y|\Theta_{P})$

\+ $\nabla l_{pseudo}(L^{\prime},Y|\Theta_{P}))$;

until  _Convergence(L)_ ;

Algorithm 1 Pseudo-label assisted Point Cloud Semantic Segmentation

###  2.4 Training Progress

The entire training process consisted of incomplete supervised learning and
pseudo-label-assisted learning. The algorithm is detailed in Algorithm 1.
First, we trained the model with initial ground-truth labels for 100 epochs.
Then, the initial pseudo labels were generated by the trained model, and we
started pseudo-label-assisted learning. When the model converged on the ground
truth, we updated the pseudo-labels. This was repeated for 100 epochs.

Figure 3: (a) and (b) represent the color maps (IR, R, G) of training and
testing datasets, respectively, whereas (c) and (d) represent the normalized
intensity maps of the two sets.

##  3 EXPERIMENT

###  3.1 Dataset

The ISPRS benchmark dataset (Rottensteiner et al.,, 2014) was used in this
study, as illustrated in Fig. 3. The dataset contains ALS data obtained from
the Leica ALS50 system and the corresponding remote sensing image for
extracting color information. ALS data and remote sensing images were obtained
from the Stuttgart region of Germany. The point density of the data was
between 4 and 7 points per $m^{2}$. The corresponding remote sensing image
covered the entire area, and the ground sampling distance was 8 cm. There were
9 categories in the dataset, including powerlines, low vegetation, impervious
surfaces, cars, fences, building roofs, building facades, shrubs, and trees.
The data were divided into two sets for training and testing, and the number
of points for the training and testing sets was 753,859 and 411,721,
respectively. The number of point clouds of different types considerably
varied and were mainly distributed in the following four categories: low
vegetation, impervious surfaces, building roofs, and trees. This data is
listed in Table 1.

The dataset contained multiple scan data, and the distance between some points
in the overlap area was very small. To remove redundant point clouds in
overlapping ALS strips and maintain the even point cloud density, we set the
subsampling grid size to d = 0.4 m and assigned labels to deleted points
according to the nearest neighbor point in testing. After subsampling, the
number of training sets was 401892, and weak labels were selected from the
subsampled data. We followed three weak-label selection criteria:

  * $\bullet$

Weak labels were randomly selected from the training data;

  * $\bullet$

The number of weak labels in each category did not exceed $10\%$ of the number
of that category;

  * $\bullet$

Less weak-label cases were included in the more weakly labeled cases. For
instance, the selected labels in 15 weak-label settings were fully included in
the 30 weak-label settings.

Table 1: Class distribution of ISPRS 3D dataset Class | Training set | Testing set
---|---|---
Powerline | 546 | 600
Low vegetation | 180,850 | 98,690
Impervious surfaces | 193,723 | 101,986
Car | 4,614 | 3,708
Fence/hedge | 12,070 | 7,422
Roof | 152,045 | 109,048
Facade | 27,250 | 11,224
Shrub | 47,605 | 24,818
Tree | 135,173 | 54,226
Total | 753,876 | 411,722

Four parameter settings, that is, 15, 30, 60, and 100 indicating 135, 270,
512, and 832 initially labeled points, respectively were examined for weak
label numbers. The percentage of 100 weak-label settings to total points was
approximately 0.2$\%$.

Figure 4: The impact of the number of labeled points on OA and Avg. F1. Table 2: Comparison of full supervision and weak supervision Setting | Method | F1 Score | Avg. F1 | OA
---|---|---|---|---
Power |  | Low
---
veg.
| Imp.
---
Surf.
Car | Fence | Roof | Facade | Shrub | Tree
Full Sup. | NANJ2 | 62.0 | 88.8 | 91.2 | 66.7 | 40.7 | 93.6 | 42.6 | 55.9 | 82.6 | 69.3 | 85.2
WhuY4 | 42.5 | 82.7 | 91.4 | 74.7 | 53.7 | 94.3 | 53.1 | 47.9 | 82.8 | 69.2 | 84.9
KPConv | 68.7 | 81.6 | 91.0 | 75.0 | 31.5 | 95.0 | 64.2 | 44.8 | 82.2 | 70.5 | 84.2
Weak Sup. | 15 | Baseline | 20.9 | 67.4 | 82.0 | 40.7 | 21.5 | 85.4 | 42.0 | 37.6 | 69.6 | 51.9 | 70.8
PL | 68.7 | 78.4 | 89.1 | 60.9 | 25.6 | 94.2 | 54.6 | 16.0 | 80.7 | 63.1 | 80.4
30 | Baseline | 20.8 | 74.7 | 87.9 | 37.1 | 19.8 | 86.1 | 48.7 | 33.8 | 67.3 | 52.9 | 73.4
PL | 65.4 | 78.2 | 89.3 | 65.5 | 27.0 | 94.4 | 58.3 | 31.5 | 79.2 | 65.4 | 81.1
60 | Baseline | 34.3 | 76.1 | 90.2 | 55.8 | 28.8 | 89.5 | 52.2 | 37.9 | 71.8 | 59.6 | 77.4
PL | 64.0 | 79.9 | 89.9 | 69.6 | 31.0 | 94.6 | 60.1 | 46.6 | 81.3 | 68.6 | 83.0
100 | Baseline | 45.9 | 75.3 | 89.9 | 58.8 | 32.3 | 93.2 | 60.6 | 36.1 | 76.3 | 63.2 | 79.1
PL | 68.9 | 81.0 | 90.7 | 68.0 | 37.4 | 94.7 | 64.0 | 45.8 | 81.5 | 70.2 | 83.7

###  3.2 Implementation

During the training, we sliced blocks with a radius of 30 m, and each time the
selected central point was the one with the minimum number of times being
trained. The batch size and steps models were implemented in the framework of
Tensorflow and trained on a GeForce GTX 1080Ti 11 GB GPU. The entire training
process lasted approximately 3 h.

###  3.3 Evaluation

Following the evaluation criteria of the ISPRS benchmarks, we used the overall
accuracy (OA) and F1 score to evaluate the performance of our method. OA is
the percentage of predictions correctly classified. The F1 score is the
harmonic mean of the precision and recall:

$\displaystyle precision=\frac{{tp}}{{tp+fp}},$ (6) $\displaystyle
recall=\frac{{tp}}{{tp+fn}},$ $\displaystyle F1=2\times\frac{{precision\times
recall}}{{precision+recall}},$

where $tp$, $fp$, and $fn$ are true positives, false positives, and false
negatives, respectively.

Figure 5: (a) and (b) represent the classification result and the error map of
the proposed method in 100 weak-label setting, respectively.

###  3.4 Results and analysis

We first trained the model under weak-label settings, and pseudo-label-
assisted learning was proposed with trained model parameters. For a better
comparison, a result was obtained by the KPConv network under full
supervision. Figure 4 shows the OA and average F1 score (avg. F1) under the
different conditions. OA and Avg. F1 were significantly improved after pseudo-
label-assisted learning. The improvement was most evident in 15 weak-label
settings, and the OA improved by nearly 10$\%$, from 70.8$\%$ to 80.4$\%$, as
well as the Avg. F1, from 51.9$\%$ to 63.1$\%$. In the 100 weak-label setting,
the numbers for OA and Avg. F1 were 84.2$\%$ and 70.5$\%$, slightly below
those under full supervision. With pseudo-label-assisted learning, we achieved
a result comparable to that from full supervision by using approximately
0.2$\%$ of full labels in the training set only. The classification result and
error map using 100 weak-label settings are shown in Fig. 5.

Two methods with leading results were also compared in this work, both using
fully supervised learning during training, and the results are presented in
Table LABEL:Tab:table2. NANJ2 (Zhao et al.,, 2018) and WhuY4 (Yang et al.,,
2018) had better performance regarding OA, reaching 85.2$\%$ and 84.9$\%$,
respectively, whereas KPConv achieved a higher Avg. F1. This result
illustrates the effectiveness of KPConv in point cloud semantic segmentation,
and it also demonstrates that a competitive result compared to fully
supervised methods was achieved with very limited and sparse ground-truth
labels. A detailed comparison of the results before and after pseudo-label-
assisted learning is presented in Table 2. The F1 score of almost every
category was considerably improved under all weak-label settings, except for
the class of shrub, in which the number of weak labels was very small, namely
15 weak-label settings. This is a limitation of pseudo-label methods, and
leads to considerable errors in the pseudo labels generated from a model with
poor performance. Because these false predictions are treated as ground-truth
labels for training, when the error rate in certain categories is too high,
false pseudo-labels dominate during subsequent training iterations and
contribute to a side effect in the model performance with respect to these
categories, which provides overconfident predictions (Lokhande et al.,, 2020).

To verify the effectiveness of our pseudo-label update strategy, a comparison
experiment was conducted, as shown in Fig. 6. PL-all represents the strategy
in which pseudo labels are updated when the model converges on both weak and
pseudo labels, and PL is our method. As pseudo-label-assisted learning is
transductive learning in training data, we used the OA of the training data to
indicate the training process. We observed that PL achieved a better accuracy
in early epochs, which means that the model was trained at a faster speed. In
addition, the OA of the PL was higher than that of the PL-all at the end of
training. However, because the number of training epochs was fixed at 100, the
consumed time was the same for the two strategies.

Figure 6: Comparison of overall accuracy of validation on training data in 100
weak-label setting between two pseudo-label update strategies.

Figure 7: Details of predictions in training set. (a) and (b) represent the
results before and after pseudo-label assisted learning, and (c) is the ground
truth.

The effect of the pseudo-label method was similar to that of entropy
regularization (Grandvalet et al.,, 2005) by considering the probability of
predictions $P(P\leq 1)$ as 1. Therefore, the posteriori probability of
predictions will increase through the information transmission of pseudo-
labels during training. We found that it is beneficial to label smoothness and
reduce noise in the predictions. Figure 7 presents a comparison of the details
in the training set before and after pseudo-label-assisted learning. Within
the black boxes, the misclassified points of tree and low vegetation were
corrected, which indicates that the accuracy of pseudo labels generated from
the training set increased along with iterative update steps.

##  4 CONCLUSION

In this work, we proposed a pseudo-label-assisted point cloud semantic
segmentation method for ALS data in urban areas. KPConv was used as the
backbone for the feature extraction. An adaptive threshold was designed to
balance the training accuracy and the required amount of pseudo labels. In
addition, we used convergence on ground-truth labels as the requirement for
updating pseudo-labels, which considerably improved the training efficiency.
Experiments indicated that a competitive result was achieved compared to those
under the full supervision scheme with only 2$\permil$ of the original
abundant labeled points.

In the future, the limitations of the pseudo-labels mentioned in the
experimental analysis are expected to be solved. Because the recursive
generation of pseudo labels is completely determined by the probability of the
predictions, they are highly correlated with the selected weak labels. Thus,
additional constraints are required to control the generation of pseudo-
labels. In addition, an elaborated weak-label selection strategy can help
achieve better performance under the same number of weak labels, which is a
good advance for data annotation.

##  5 ACKNOWLEDGEMENTS

The dataset was provided by the German Society for Photogrammetry, Remote
Sensing, and Geoinformation (DGPF). The work described in this paper was
substantially supported by a grant from the Research Grants Council of the
Hong Kong Special Administrative Region, China (Project No. PolyU 25211819),
and was also supported by a grant from the Hong Kong Polytechnic University
(Project No. G-YBZ9).

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# A model for neural activity in the absence of external stimuli

Aline Duarte and Guilherme Ost
Universidade de São Paulo and GSSI - L’Aquila  e-mail addresses:
aline.duart@gmail and<EMAIL_ADDRESS>authors are PhD students from
the Universidade de São Paulo and this work was developed during their partial
completion doctoral program at GSSI both being fully supported by CNPq.

(November 7, 2014)

###### Abstract

We study a stochastic process describing the continuous time evolution of the
membrane potentials of finite system of neurons in the absence of external
stimuli. The values of the membrane potentials evolve under the effect of
chemical synapses, electrical synapses and a leak current. The evolution of
the process can be informally described as follows. Each neuron spikes
randomly following a point process with rate depending on its membrane
potential. When a neuron spikes, its membrane potential is immediately reset
to a resting value. Simultaneously, the membrane potential of the neurons
which are influenced by it receive an additional positive value. Furthermore,
between consecutive spikes, the system follows a deterministic motion due both
to electrical synapses and the leak current. Electrical synapses push the
system towards its average potential, while the leak current attracts the
membrane potential of each neuron to the resting value.

We show that in the absence leakage the process converges exponentially fast
to an unique invariant measure, whenever the initial configuration is non
null. More interesting, when leakage is present, we proved the system stops
spiking after a finite amount of time almost surely. This implies that the
unique invariant measure is supported only by the null configuration.

Key words : piecewise deterministic Markov process, limiting distribution,
neuronal systems, chemical synapses, electrical synapses, leak current

AMS Classification: 60K35, 60F99, 60J25

##  1 Introduction

We study the behavior of a finite number of interacting neurons in the absence
of external stimuli our goal being to determine the long-run behavior of the
process. Our system is composed of $N$ neurons whose state at time $t\geq 0$
is specified by
$\mathrm{U}(t)=\left(\mathrm{U}_{1}(t),\ldots\mathrm{U}_{N}(t)\right)$, with
$\mathrm{U}(t)\in\mathbb{R}_{+}^{N}.$ For each neuron $i=1,\ldots,N$ and each
time $t\geq 0,$ $\mathrm{U}_{i}(t)$ represents the membrane potential of
neuron $i$ at time $t$. We consider two kinds of interactions among neurons
and also a constant interplay between neurons and the environment.

More precisely the neurons interact via electrical and chemical synapses.
Electrical synapses are due to so-called gap-junction channels between neurons
which induce a constant sharing of potential, pushing the system towards its
average value. By contrast, chemical synapses are point events which can be
described as follows. Each neuron spikes randomly at rate
$\varphi(\mathrm{U})\geq 0$ which depends on its membrane potential
$\mathrm{U}$, we suppose $\varphi$ a non decreasing function, positive at
$\mathrm{U}>0$ and both integrable and vanishing at $0$ (in agreement with the
assumption of non external stimuli). When neuron $i$ spikes, its membrane
potential is immediately reset to a resting potential $0$. Simultaneously, the
neurons which are influenced by neuron $i$ receive an additional positive
value to their membrane potential. This value may vary for each pair of
neurons. Moreover, in the whole time, the neurons loose potential to the
environment, due to leakage channels which pushes down the membrane potential
of each neuron toward zero. This outgoing constant flow of potential is
defined as the leak current. For technical details we refer the reader to
Gersnter and Kistler (2002).

Our system is inspired by the one introduced in Galves and Löcherbach (2013).
This model is an example of piecewise-deterministic Markov processes (PDPs)
introduced by Davis (1984). Such processes combine random jump events, the
chemical synapses, with deterministic continuous evolutions, in our case due
both to electrical synapses and the leak current. The PDPs have been used also
to model neuronal systems by other authors, see for instance the papers by
Riedler, Thieullen and Wainrib (2012), De Masi et al. (2014), Fournier and
Löcherbach (2014) and Robert and Touboul (2014).

Chemical synapses and leakage make the system non-conservative. Moreover,
there is an evident competition between the incoming energy induced by the
spikes and the outgoing energy induced by the leak current. Therefore it is
natural to ask about the limiting behavior of the system as time
$t\rightarrow\infty$. The main results of the paper, presented in Theorem 2.3
and Theorem 2.4, provide a complete description of the asymptotic distribution
of the process. The theorem 2.3 states that under the presence of the leakage
then almost surely there are only a finite number of spikes and the system
converges to an “inactive global state” interpreted as “brain sleep”.

When the leakage is absent we prove a Harris-type condition which shows
exponential convergence to an unique non trivial invariant measure, whenever
the initial configuration is non null. This is the content of the Theorem 2.4.

Our paper is organized as follows. In section 2, we introduce the process,
prove its existence and we state the main results, Theorem 2.3 and Theorem
2.4. In Section 3, we proof the theorem 2.3, while in section 4, we prove
Theorem 2.4. In section 5, we briefly compare similar results recently
obtained by Roberto and Touboul (2014).

##  2 Model definition and main results

Let $\mathcal{N}=\\{1,\cdots,N\\}$ be a finite set of neurons, for some fixed
integer $N\geq 1$ and consider the family of non-negative synaptic weights
$\left(W_{i\rightarrow
j}\right)_{i,j\in\mathcal{N}}\in\mathbb{R}_{+}^{N\choose 2}$ such that
$W_{i\rightarrow i}=0$ for all $i\in\mathcal{N}$. The value $W_{i\rightarrow
j}$ corresponds to the value added to the membrane potential of neuron $j$
when the neuron $i$ spikes.

We consider a continuous time Markov process

$\mathrm{U}(t)=(\mathrm{U}_{1}(t),\ldots,\mathrm{U}_{N}(t)),\ t\geq 0,$

taking values in $\mathbb{R}^{N}_{+}$, whose infinitesimal generator is given
for any smooth test function $f:\mathbb{R}^{N}_{+}\rightarrow\mathbb{R}$, by

$\mathcal{L}f(u)=\sum_{i\in\mathcal{N}}\varphi(u_{i})[f(\Delta_{i}(u))-f(u)]-\lambda\sum_{i\in\mathcal{N}}\left(\frac{\partial
f}{\partial
u_{i}}(u)[u_{i}-\bar{u}]\right)-\alpha\sum_{i\in\mathcal{N}}\left(\frac{\partial
f}{\partial u_{i}}(u)[u_{i}]\right),$ (1)

where for all $i\in\mathcal{N}$,
$\Delta_{i}:\mathbb{R}^{N}_{+}\rightarrow\mathbb{R}_{+}^{N}$ is defined by

$(\Delta_{i}(u))_{j}=\left\\{\begin{array}[]{ll}u_{j}+W_{i\rightarrow
j},&\mbox{if}\ j\neq i\\\ 0,&\mbox{if}\ j=i\end{array}\right.,$

$\lambda,\alpha\geq 0$ are positive parameters modelling, respectively, the
strength of electrical synapses and the leakage effect,
$\bar{u}=\frac{1}{N}\sum_{i=1}^{N}u_{i}$ and

######  Assumption 1.

$\varphi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a non-decreasing function
such that $\varphi(0)=0$ and there exists a constant $r>0$ satisfying

$\int\limits_{0}^{2r}{\varphi(u)\over u}du<+\infty.$

We are assuming that there is no external stimuli. This assumption appears in
the condition $\varphi(0)=0$. In addition, from the neurobiological point of
view, it is reasonable to assume that $\varphi$ is a non-decreasing function
since an addition in the membrane potential increases the probability of a
spiking occurs.

The first term in (1) depicts how the chemical synapses are incorporated in
our model. Neurons whose potential is $u$ spike at rate $\varphi(u)$.
Intuitively this means that any initial configuration $u\in\mathbb{R}_{+}^{N}$
of the membrane potentials

$\mathbb{P}(\mathrm{U}(t)=\Delta_{i}(u)|\mathrm{U}(0)=u)=\varphi(u_{i})t+o(t),\
\mbox{as}\ t\rightarrow 0$

Thus, the function $\varphi$ is called firing or spiking rate of the system.

The second and third terms in (1) represent the electrical synapses and the
leak current respectively. They describe the deterministic time evolution of
the system between two consecutive spikes. More specifically, in an interval
of time $[a,b]$, without occurrence of spikes in the whole system, the
membrane potential of neuron $i\in\mathcal{N}$ obeys the following ordinary
differential equation

${d\over
dt}\mathrm{U}_{i}(t)=-\alpha\mathrm{U}_{i}(t)-\lambda(\mathrm{U}_{i}(t)-\bar{\mathrm{U}}(t)).$
(2)

Notice that the first term of the right-hand side of (2) pushes simultaneously
all neurons to the resting state, while the second term tends to attract the
neurons to the average potential.

Our first theorem proves the existence of the process.

######  Theorem 2.1.

Let $\varphi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be any function
satisfying the Assumption 1. For any $N\geq 1$ and any
$u\in\mathbb{R}_{+}^{N}$ there exists a unique strong Markov process
$\mathrm{U}(t)$ taking values in $\mathbb{R}_{+}^{N}$ starting from $u$ whose
generator is given by (1).

###### Proof.

Let $N_{i}(t),\ t\geq 0$, be the simple point process on $R_{+}$ which counts
the jump events of neuron $i\in\mathcal{N}$ up to time $t$. Define
$E_{i}=\sum_{j\neq i}W_{i\rightarrow j}$ and $E=\max_{i\in\mathcal{N}}E_{i}$
and, following De Masi et al. (2014), consider the following random variable,
for all $t>0$,

$k(t)=\sum_{i\in\mathcal{N}}\int_{0}^{t}{\bf
1}\hskip-1.42262pt\\{\mathrm{U}_{i}(s^{-})\leq 2E\\}dN_{i}(s).$

The random variable $k(t)$ counts the number of spikes of neurons whose the
potential is at most $2E.$

Suppose $\mathrm{U}_{i}$ fires at time $t$, in this case

$\bar{\mathrm{U}}(t)=\frac{1}{N}\sum_{j\neq
i}\big{(}\mathrm{U}_{j}(t^{-})+W_{i\rightarrow
j}\big{)}=\bar{\mathrm{U}}(t^{-})+\frac{1}{N}\big{(}E_{i}-\mathrm{U}_{i}(t^{-})\big{)}.$

Now, using the expression of $\bar{\mathrm{U}}(t)$ above and adapting the
proof of Theorem 1 of De Masi et al (2014), we have the following inequalities
for all $t>0$,

$\bar{\mathrm{U}}(t)\leq\bar{\mathrm{U}}(0)+\frac{E}{N}k(t),\ \ EN(t)\leq
N\bar{\mathrm{U}}(0)+2Ek(t)\ \ \mbox{and}\ \
\mathrm{U}^{max}(t)\leq(N+1)\mathrm{U}^{max}(0)+2Ek(t),$

where $\mathrm{U}^{max}(t)=\max_{i\in\mathcal{N}}\mathrm{U}_{i}(t).$

Since we can bound $Ek(t)$ by a Poisson process of intensity $N\varphi(2E)$,
the second inequality above shows that number of jumps of the process is
finite almost surely on any finite time interval. To conclude the proof just
note that the construction of the process can be achieved by gluing together
trajectories given by the deterministic flow between successive jump times.
This procedure is feasible since the number of jumps of the process is finite
on any finite interval.

∎

Now, we shall present an elementary argument which shows that for all leakage
rate $\alpha$ large enough and if the firing rate $\varphi$ is globally
Lipschitz with $\varphi(0)=0$, the system goes extinct. This result was the
starting point of this paper. The idea of this proof was taken from
discussions with Galves and Löcherbach. The result is the following.

######  Theorem 2.2.

For any $N\geq 1$, $\alpha\geq 0$, $\lambda\geq 0$ and $c$-Lipschitz function
$\varphi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ such that $c>0$ and
$\varphi(0)=0$, the following inequality holds, for all $t\geq 0$ and
$\mathrm{U}(0)=u\in\mathbb{R}^{N}_{+}$,

$\mathbb{E}[\bar{\mathrm{U}}(t)]\leq\bar{u}e^{t(\alpha^{*}c-\alpha)},$

where
$\alpha^{*}=\max\limits_{k\in\mathcal{N}}\sum_{j\in\mathcal{N}}W_{j\rightarrow
k}$. In particular, if $\alpha>\alpha^{*}c$, then the process goes extinct.

###### Proof.

For each $i\in\mathcal{N}$, plugging $f=\pi_{i}$ in (1), where $\pi_{i}$ is
the projection onto the $i$-th coordinate, we have

$\displaystyle{d\over
dt}\mathbb{E}[\mathrm{U}_{i}(t)]=\sum_{j\in\mathcal{N}}W_{j\rightarrow
i}\mathbb{E}[\varphi(\mathrm{U}_{j}(t))]-\mathbb{E}[\mathrm{U}_{i}(t)\varphi(\mathrm{U}_{i}(t))]-\alpha\mathbb{E}[\mathrm{U}_{i}(t)]+\lambda\mathbb{E}[\mathrm{U}_{i}(t)-\bar{\mathrm{U}}(t)].$

Summing over all $i\in\mathcal{N}$ and then using that $\varphi$ is a non-
negative $c$-Lipschitz function such that $\varphi(0)=0$, it follows that

$\displaystyle{d\over
dt}\mathbb{E}[\bar{\mathrm{U}}(t)]\leq(\alpha^{*}c-\alpha)\mathbb{E}[\varphi(\bar{\mathrm{U}}(t))].$

Therefore applying the Grownwall’s lemma to the inequality above we finish the
proof. ∎

Even assuming that the firing rate $\varphi$ satisfies only the Assumption 1,
we claim that for any fixed numbers of neurons, the presence of the leak
current is a necessary and sufficient condition for the extinction of the
process. In fact, we shall prove a stronger result. It states that, if there
is the leakage, there will be only a finite numbers of spikes eventually
almost surely. On the other hand, it is shown that, excluding the trivial
initial configuration, the system is ergodic when there is no leakage. In
particular, this results generalize the Theorem 2.2 above.

In order to state our main result, we need to introduce the some extra
notation.

For each neuron $i\in\mathcal{N}$, let
$T^{i}_{1}=\inf\\{s>0:\mathrm{U}_{i}(s)=0\\}$ be the first spiking time of
neuron $i$ and for each $k\geq 2$, let
$T^{i}_{k}=\inf\\{s>T^{i}_{k-1}:\mathrm{U}_{i}(s)=0\\}$ be the $k$-th spiking
time of neuron $i$. Then, the first and the $k$-th spiking time of the system
are defined respectively by

$T_{1}=\inf\limits_{i\in\mathcal{N}}T^{i}_{1}\quad\mbox{and}\quad
T_{k}=\inf\limits_{i\in\mathcal{N},m\geq 1}\\{T^{i}_{k}>T_{m-1}\\},\ \ k\geq
2.$ (3)

Our main theorem is given below.

######  Theorem 2.3.

Let $(\mathrm{U}(t))_{t\geq 0}$ be the Markov process whose the infinitesimal
generator is given by (1) and $T_{k}$ be as defined in (3). Assume that
$\varphi$ satisfies the Assumption 1 with
$r>\max\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{N}}W_{j\rightarrow
i}$. Then for any $\alpha>0$ and $\lambda\geq 0,$

$\mathbb{P}\left(\sum_{k\geq 1}{\bf
1}\hskip-1.42262pt{\\{T_{k}<\infty\\}}<\infty\right)=1.$

######  Corollary 1.

Under the same hypothesis of Theorem 2.3, for all $i\in\mathcal{N}$, it holds

$\lim_{t\rightarrow+\infty}\mathrm{U}_{i}(t)=0\quad a.s.$

In particular, the delta of Dirac at point $0^{N}$, $\delta_{0^{N}}$, is the
unique invariant measure for the process in the presence of the leak current.

It remains to analyse what happens in the long-run behavior of the system in
the absence of the leakage. This is the content of the next result.

######  Theorem 2.4.

Let $(\mathrm{U}(t))_{t\geq 0}$ be the Markov process whose the infinitesimal
generator is given by (1) with $\alpha=0.$ Then the process admits only two
different invariant probability measures either $\delta_{0^{N}}$ or
$\mu\neq\delta_{0^{N}}$. If $\mathrm{U}(0)=(0,\cdots,0)$, the invariant
measure is $\delta_{0^{N}}$, while whatever is the initial configuration
$\mathrm{U}(0)=u\in\mathbb{R}^{N}_{+}\setminus\\{0^{N}\\}$, the invariant
measure is $\mu$.

##  3 Proof of Theorem 2.3

First of all, observe that, from the equation (2), for any time
$t\in[T_{n},T_{n+1})$,

$\mathrm{U}_{i}(t)=\big{(}\mathrm{U}_{i}(T_{n})-\bar{\mathrm{U}}(T_{n})\big{)}e^{-(\alpha+\lambda)(t-T_{n})}+\bar{\mathrm{U}}(T_{n})e^{-\alpha(t-T_{n})}.$
(4)

We shall explore this equation many times.

We start giving a lower bound to the probability of there is no spikes when
the system starts with a initial condition small enough.

######  Proposition 3.1.

Let $(\mathrm{U}(t))_{t\in\mathbb{R}}$ be the Markov process whose the
generator is given by (1),
$\mathrm{U}^{max}(t)=\max\\{\mathrm{U}_{i}(t),i\in\mathcal{N}\\}$ be the
maximum potential of the system at time $t$ and $T_{1}$ as defined in (3).
Suppose that $\varphi$ satisfies the Assumption 1. If $\alpha>0$, then

$\mathbb{P}(T_{1}=\infty\mid\mathrm{U}^{max}(0)<r)\geq
e^{-N\alpha\int_{0}^{2r}{\varphi(u)\over u}du}>0.$

###### Proof.

By the equation (4), we have, for all $0\leq t<T_{1}$, the following
inequalities

$\displaystyle\mathrm{U}^{max}(t)\leq\bar{\mathrm{U}}(0)e^{-\alpha
t}+\mathrm{U}^{max}(0)e^{-\alpha t}\leq 2re^{-\alpha t}.$

Using the inequality above and the non-decreasing assumption on $\varphi$, we
have

$\displaystyle\mathbb{P}(T_{1}>t\mid\mathrm{U}^{max}(0)<r)$
$\displaystyle\geq$
$\displaystyle\exp\left\\{-\int_{0}^{t}N\varphi(\mathrm{U}^{max}(s))ds\right\\}$
(5) $\displaystyle\geq$ $\displaystyle\exp\left\\{{-N\alpha\int_{2re^{-\alpha
t}}^{2r}{\varphi(u)\over u}du}\right\\}.$

Therefore, when $t$ goes to infinite, it follows the result. ∎

######  Proposition 3.2.

Under the same hypothesis of Proposition 3.1. For
$\theta<(\alpha+\lambda)^{-1}$, it holds that

$\displaystyle\mathbb{P}(T_{1}>\theta\mid\mathrm{U}^{max}(0)>r)\leq
e^{-\theta\varphi(\beta)}<1,$

where $\beta=\beta(\theta)=r(1-(\alpha+\lambda)\theta)>0$.

###### Proof.

Indeed,

$\displaystyle\mathbb{P}(T_{1}>\theta\mid\mathrm{U}^{max}(0)>r)$
$\displaystyle\leq$
$\displaystyle\exp\left\\{-\int_{0}^{\theta}\varphi(\mathrm{U}^{max}(s))ds\right\\}.$
(6)

Moreover, by (4) for all $0<s\leq\theta$ is true that

$\displaystyle\mathrm{U}^{max}(s)\geq\mathrm{U}^{max}(0)e^{-(\alpha+\lambda)s}\geq
re^{-(\alpha+\lambda)s}\geq r(1-(\alpha+\lambda)\theta).$ (7)

From (6) and (7) we deduce the desired inequality

$\displaystyle\mathbb{P}(T_{1}>\theta\mid\mathrm{U}^{max}(0)\geq r)\leq
e^{-\theta\varphi(\beta)}<1.$

∎

The next result claims that even when all membrane potentials are large, there
is a positive probability of all potentials become small after a fix time
$T>0$.

######  Proposition 3.3.

Consider $(\mathrm{U}(t))_{t\in\mathbb{R}}$ the Markov process whose the
generator is given by (1), $\mathrm{U}^{max}(t)$ and $T_{1}$ as in proposition
3.1. Suppose that $\varphi$ satisfies the Assumption 1 and $\alpha>0$. If
$r>\max\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{N}}W_{j\rightarrow
i}$, then there exists $T>0$ such that

$\mathbb{P}(\mathrm{U}^{max}(T)<r\mid\mathrm{U}^{max}(0)>r)>0.$

###### Proof.

Define $\tau_{1}=\inf\\{t>0:\mathrm{U}^{max}(t)=0\\}$ and take
$\theta<(\alpha+\lambda)^{-1}$ (this is possible since $\alpha>0$). By the
proposition 3.2, it holds that

$\displaystyle\mathbb{P}(T_{1}<\theta,T_{1}=\tau_{1}\mid\mathrm{U}^{max}(0)>r)\geq\left(1-e^{-\theta\varphi(\beta)}\right){1\over
N}.$

Define $T_{0}=0$, $T=N\theta$ and consider the following sequence of events

$A_{1}=\\{T_{1}<\theta,T_{1}=\tau_{1},\mathrm{U}^{max}(T_{1})<r,T_{2}>T\\},$

and for $l=2\ldots,N$,

$A_{l}=\Big{\\{}\cap_{k=1}^{l}\\{T_{k}<T_{k-1}+\delta,T_{k}=\tau_{k},\\},\cap_{k=1}^{l-1}\\{\mathrm{U}^{max}(T_{k})>r\\},\mathrm{U}^{max}(T_{l})<r,T_{l+1}>T\Big{\\}}.$

The event $A_{k}$ corresponds to the following situation. At the firsts $k-1$
consecutive spikes of the maximum, there always exists at least one neuron
whose potential is larger than $r$. But at $k$-th spike of the maximum all
membrane potentials get less than $r$ and no more spikes happen up to time
$T$.

From the definition of the events $A_{k}$ we have

$\displaystyle\big{\\{}\mathrm{U}^{max}(T)<r\big{\\}}\supseteq\bigcup_{k=1}^{N}A_{k}.$

Wherefore, it suffices to compute the probability of each event $A_{k}$ in the
right side of the inclusion, conditioned to the event
$\\{\mathrm{U}^{max}(0)>r\\}$. Now, defining
$\beta_{0}=\left(1-e^{-\delta\varphi(\beta)}\right){1\over N}$, and
$\beta_{1}=\exp\left\\{{-2Ncr\over\alpha}(1-e^{-\alpha N\delta})\right\\}$, by
the proposition 3.2 and the inequality (5), it follows that

$\displaystyle\mathbb{P}(A_{1}\mid\mathrm{U}^{max}(0)>r)\geq\beta_{0}\beta_{1}\mathbb{P}(\mathrm{U}^{max}(T_{1})<r\mid\mathrm{U}^{max}(0)>r,T_{1}<\delta,T_{1}=\tau_{1}).$

Similarly, we have

$\displaystyle\mathbb{P}(A_{2}\mid\mathrm{U}^{max}(0)>r)\geq\beta_{0}^{2}\beta_{1}\mathbb{P}(\mathrm{U}^{max}(T_{1})>r\mid\mathrm{U}^{max}(0)>r,T_{1}<\delta,T_{1}=\tau_{1})$
$\displaystyle\ \
\times\mathbb{P}(\mathrm{U}^{max}(T_{2})<r\mid\mathrm{U}^{max}(T_{1})>r,T_{2}<T_{1}+\delta,T_{2}=\tau_{2}).$

Thus, summing these two inequalities, we get the following lower bound for
$A_{1}\cup A_{2}$,

$\displaystyle\mathbb{P}(A_{1}\cup
A_{2}\mid\mathrm{U}^{max}(0)>r)\geq\beta_{0}^{2}\beta_{1}\mathbb{P}(\mathrm{U}^{max}(T_{2})<r\mid\mathrm{U}^{max}(T_{1})>r,T_{2}<T_{1}+\delta,T_{2}=\tau_{2}).$

Proceeding in this way for the other terms, we obtain that

$\displaystyle\mathbb{P}(\mathrm{U}^{max}(T)<r\mid\mathrm{U}^{max}(0)>r)$
$\displaystyle\geq\beta_{0}^{N}\beta_{1}\mathbb{P}\left(\mathrm{U}^{max}(T_{N})<r\mid\mathrm{U}^{max}(T_{N-1})>r,\cap_{k=1}^{N}\\{T_{k}<T_{k-1}+\delta,T_{k}=\tau_{k}\\}\right).$

Since
$r>\max\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{N}}W_{j\rightarrow
i}$, we easily see that

$\mathbb{P}\left(\mathrm{U}^{max}(T_{N})<r\mid\mathrm{U}^{max}(T_{N-1})>r,\cap_{k=1}^{N}\\{T_{k}<T_{k-1}+\delta,T_{k}=\tau_{k}\\}\right)=1.$

∎

As a consequence of proposition 3.3 we have

######  Corollary 2.

Le $T>0$ be the positve constant given by the proposition 3.3. Define the
stopping times $R_{1}=\inf\\{n\geq 1;\mathrm{U}^{max}(nT)\leq r\\}$ and
$R_{k}=\inf\\{n\geq R_{n-1};\mathrm{U}^{max}(nT)\leq r\\}$ for all $k\geq 2$.
Then, under the same hypothesis of the proposition 3.3,

$\sum_{k=1}^{\infty}{\bf
1}\hskip-1.42262pt\\{R_{k}<\infty\\}=\infty\quad\mathbb{P}-a.s.$

###### Proof of Theorem 2.3.

For this proof we must define the following stopping times:
$K_{1}=\inf\\{n\geq 1:T_{n}>R_{1}T\\},$ $J_{1}=\inf\\{n\geq
1:R_{n}T>T_{S_{1}}\\}$, and for $k\geq 2$,
$K_{k}=\inf\\{n>J_{k-1}:T_{n}>R_{J_{k-1}}T\\}$ and $J_{k}=\inf\\{n\geq
1:R_{n}T>T_{S_{k-1}}\\}.$ From corollary 2 all these stopping times are finite
almost surely and, in particular, well defined. Now, from Theorem 2.1 and the
definition of $K_{k}$

$\displaystyle
1=\mathbb{P}\left(\bigcap_{k=1}^{\infty}\bigcup_{i=1}^{\infty}\\{T_{i}>R_{k}T\\}\
\left|\
\bigcap_{i=1}^{\infty}\\{T_{i}<\infty\\}\right)\right.=\mathbb{P}\left(\bigcap_{k=1}^{\infty}\\{K_{k}<\infty\\}\
\left|\ \bigcap_{i=1}^{\infty}\\{T_{i}<\infty\\}\right)\right..$

Thus, $\cap_{k=1}^{\infty}\\{T_{k}<\infty\\}$ implies
$\cap_{k=1}^{\infty}\\{K_{k}<\infty\\}$.

On the other hand, by proposition 3.1, one knows that

$\displaystyle\mathbb{P}\left(\bigcap_{k=1}^{n}\\{K_{k}<\infty\\}\right)\leq\Big{(}1-e^{-N\alpha\int_{0}^{2r}{\varphi(u)\over
u}du}\Big{)}^{n},$

which converges to $0$ when $n$ diverges. ∎

##  4 Proof of Theorem 2.4

To simplify the proof of the theorem 4 we shall split it into several steps.
The main part of the argument is to find a positive recurrent regeneration set
$B$ in sense that

  1. (i)

$B$ is positive recurrent and,

  2. (ii)

there exist some $t_{*}>0$, $\epsilon>0$ and a probability measure $\nu$ on
$\mathbb{R}^{N}_{+}$ such that

$P_{t_{*}}(u,A)\geq\varepsilon\nu(A),\ u\in B,$

for all measurable set $A\in\mathcal{B}(\mathbb{R}^{N}_{+}).$

Usually, a Markov processes with a regeneration set are called Harris
recurrent Markov processes. For such Processes an invariant measure always
exits.

In what follows, for any positive real number $a>0$ we use the notation
$R_{a}(x)$, meaning that there exists a constant $l>0$ such that
$|R_{a}(x)|\leq la$ for all $x.$ When a function satisfies such condition it
is called a function of order $a$. Note that we are not specifying the domain
in which the function $R$ is defined on.

For each $\varepsilon>0$ define the following event

$A_{\varepsilon}=\\{(i-1)\varepsilon<T_{i}<i\varepsilon,\ i=1,\cdots,N\\},$

and let $(S_{m})_{m\geq 1}$ be the spiking mark sequence of the system. The
event $\\{S_{m}=i\\}$ means that the $\mathrm{U}_{i}(T_{m})=0.$ Finally,
consider the event

$S=\\{S_{i}=i,i=1,\ldots,N\\}.$

The first lemma below says that, conditioning to the event $A_{\varepsilon}$,
when $\varepsilon$ is sufficiently small the process evolves, modulo an error
of small order, as in the case without gap junction ($\lambda=0$). Before
stating this lemma we need to introduce a finite sequence of potential
configurations.

Consider the sequence $(v(k))_{k=0,\ldots,N}$ with $v(k)\in R_{+}^{N}$ given
by

$v(0)=\Big{(}\sum_{j=2}^{N}W_{j\rightarrow 1},\sum_{j=3}^{N}W_{j\rightarrow
1},\ldots,W_{N\rightarrow N-1},0\Big{)},$

and for $1\leq k\leq N,$ $v_{k}(k)=0$ and for $i\neq k$

$v_{i}(k)={\bf
1}\hskip-1.42262pt_{\\{i>k\\}}\Big{(}v_{i}(0)+\sum_{j=1}^{k}W_{j\rightarrow
i}\Big{)}+{\bf
1}\hskip-1.42262pt_{\\{i<k\\}}\Big{(}\sum\limits_{j=i+1}^{k}W_{j\rightarrow
i}\Big{)}.$

######  Lemma 4.1.

Fix $\delta>0.$ If $\mathrm{U}(0)=u\in B(v(0),\delta)$, then conditioning to
the event $A_{\epsilon}\cap S$, for each $k=1,\cdots,N$ the following
equalities hold:

  1. (i)

$\mathrm{U}_{i}(T_{k})=v_{i}(k)+\sum\limits_{r=i+1}^{k}\lambda(T_{r}-T_{r-1})d_{i}(r-1)+R_{\delta\varepsilon}(T^{k}_{1},u)+R_{\varepsilon^{2}}(T_{1}^{k},u),$
if $i<k;$

  2. (ii)

$\mathrm{U}_{i}(T_{k})=v_{i}(k)+\sum\limits_{r=1}^{k}\lambda(T_{r}-T_{r-1})d_{i}(r-1)+R_{\delta}(u)+R_{\delta\varepsilon}(T^{k}_{1},u)+R_{\varepsilon^{2}}(T_{1}^{k},u),$
if $i>k$;

  3. (iii)

$\bar{\mathrm{U}}(T_{k})=\bar{v}(k)+R_{\delta}(u)+R_{\varepsilon}(T_{1}^{k},u),\
\mbox{if}\ k<N\ \mbox{and}\
\bar{\mathrm{U}}(T_{N})=\bar{v}(0)+R_{\varepsilon}(T_{1}^{N},u),$

where $d_{i}(m)=\bar{v}(m)-v_{i}(m)$, $T_{0}=0$ and
$T_{1}^{k}=(T_{1},\ldots,T_{k}).$ Furthermore, all the partial derivatives of
the remainder functions $R_{\delta\epsilon}(T_{1}^{k},u)$,
$R_{\epsilon^{2}}(T_{1}^{k},u)$ above are either of order $\delta$ or
$\varepsilon.$

###### Proof.

The proof is given by induction on $k$. On the event $A_{\epsilon}\cap S$, we
have that $\mathrm{U}_{1}(T_{1})=0$ and for each $i\neq 1$ and
$\mathrm{U}(0)=u\in B(v(0),\delta),$

$\displaystyle\mathrm{U}_{i}(T_{1})$ $\displaystyle=$ $\displaystyle
W_{1\rightarrow i}+\bar{\mathrm{U}}(0)+(1-\lambda
T_{1}+R_{\varepsilon^{2}}(T_{1}))(v_{i}(0)+R_{\delta}(u)-\bar{\mathrm{U}}(0))$
$\displaystyle=$ $\displaystyle v_{i}(1)+\lambda
T_{1}d_{i}(0)+R_{\delta}(u)+R_{\epsilon\delta}(T_{1},u)+R_{\varepsilon^{2}}(T_{1},u),$

where in the first equality we have used the expansion series of the
exponential function.

Thus, from the expression of $\mathrm{U}_{i}(T_{1})$ above, we conclude that

$\displaystyle\bar{\mathrm{U}}(T_{1})$ $\displaystyle=$
$\displaystyle\bar{v}(1)+\lambda
T_{1}\frac{1}{N}\sum_{i=2}^{N}d_{i}(0)+R_{\delta}(u)+R_{\varepsilon\delta}(T_{1},u)+R_{\varepsilon^{2}}(T_{1},u)$
$\displaystyle=$
$\displaystyle\bar{v}(1)+R_{\delta}(u)+R_{\varepsilon}(T_{1},u).$

In addition, it is easy to check that all remainders functions above have
partial derivatives with respect to $T_{1}$ and all of them are either
$R_{\delta}$ or $R_{\epsilon}$ functions. Therefore (i), (ii) and (iii) it is
verified for $k=1.$

Now, suppose that (i), (ii) and (iii) hold for some fixed $1<k<N$. As before,
on the event $A_{\epsilon}\cap S$, we have $\mathrm{U}_{k+1}(T_{k+1})=0$ and,
for $i<k+1$, by the inductive hypothesis,

$\displaystyle\mathrm{U}_{i}(T_{k+1})$ $\displaystyle=$ $\displaystyle
W_{k+1\rightarrow
i}+\bar{\mathrm{U}}(T_{k})+(1-\lambda(T_{k+1}-T_{k})+R_{\varepsilon^{2}}(T_{k-1},T_{k}))(\mathrm{U}_{i}(T_{k})-\bar{\mathrm{U}}(T_{k}))$
$\displaystyle=$ $\displaystyle
v_{i}(k+1)+\sum_{r=i+1}^{k+1}\lambda(T_{r}-T_{r-1})d_{i}(r-1)+R_{\delta\varepsilon}(T_{1}^{k+1},u)+R_{\varepsilon^{2}}(T_{1}^{k+1},u)$

Using the same argument for the case when $i>k+1$, we get the first equation.
Using again the inductive hypothesis and looking at the expression written in
the first equality of the membrane potential, it is readily seen that the
remainder functions possesses partial derivatives with to $T_{l}$, for
$l=1,\ldots,k+1$ and they are either of order $\delta$ or $\epsilon.$

Finally, summing $\mathrm{U}_{i}(T_{k+1})$ over all neurons $i=1,\ldots,N$

$\displaystyle\bar{\mathrm{U}}(T_{k+1})$ $\displaystyle=$
$\displaystyle\bar{v}(k+1)+R_{\varepsilon}(T_{1}^{k+1},u)+R_{\delta}(u).$

∎

Note that $v(T_{N})=v(0)$, thus, from the previous lemma it follows the

######  Corollary 3.

Under the same assumptions of Lemma 4.1, if $T=N\varepsilon,$ then for each
$i=1,\ldots,N$, the following equality is verified

$\mathrm{U}_{i}(T)=v_{i}(0)+\lambda(T-T_{N})d_{i}(0)+\sum\limits_{r=i+1}^{N}\lambda(T_{r}-T_{r-1})d_{i}(r-1)+R_{\delta\epsilon}(T_{1}^{N},u)+R_{\epsilon^{2}}(T_{1}^{N},u).$

######  Remark 4.1.

In order to simplify the notation, we shall denote the map
$\gamma^{0}:A_{\epsilon}\rightarrow\mathbb{R}_{+}^{N}$ by
$\gamma^{0}(t_{1}^{N})=(\gamma_{1}^{0}(t_{1}^{N}),\ldots,\gamma_{N}^{0}(t_{1}^{N}))$
where for each $i=1,\ldots,N$,

$\gamma_{i}^{0}(t_{1}^{N})=v_{i}(0)+\lambda(T-t_{N})d_{i}(0)+\sum_{r=i+1}^{N}\lambda(t_{r}-t_{r-1})d_{i}(r-1).$

By corollary 3, conditioning to the event $A_{\epsilon}\cap S$, we have the
following representation for all $\mathrm{U}(0)=u\in B(v(0),\delta)$

$\mathrm{U}(T)=\gamma^{0}(T_{1}^{N})+R_{\delta\epsilon}(T_{1}^{N},u)+R_{\epsilon^{2}}(T_{1}^{N},u),$

where both $R_{\delta\epsilon}(T_{1}^{N},u)$ and
$R_{\epsilon^{2}}(T_{1}^{N},u)$ are multivalued functions whose the
$L_{1}$-norms are remainders functions of order $\delta\epsilon$ and
$\epsilon^{2}$ respectively. Moreover, we have the following corollary

######  Corollary 4.

For each $u\in B(v(0),\delta)$, the determinant of the Jacobian of map
$A_{\varepsilon}\ni
t_{1}^{N}\mapsto\gamma_{u}(t_{1}^{N})=\gamma^{0}(t_{1}^{N})+R_{\delta\epsilon}(t_{1}^{N},u)+R_{\epsilon^{2}}(t_{1}^{N},u)$,
is given by

$|J\gamma_{u}(t_{1}^{N})|=\lambda^{N}\prod_{i=1}^{N}\bar{v}(i)+R_{\epsilon}(t_{1}^{N},u)+R_{\delta}(t_{1}^{N},u),$

which is different from zero for $\delta$ and $\varepsilon$ small enough for
all $(t_{1}^{N})\in A_{\epsilon}$ and $u\in B(v(0),\delta)$.

We shall use this representation to show that our process, at time $T$,
satisfies a Harris-type condition (see proposition 4.1). Before proving this
proposition, we need an extra lemma. Here again the non-decreasing assumption
on $\varphi$ is important.

######  Lemma 4.2.

  1. (i)

Let
$f_{u}(t_{1},\ldots,t_{N})=f_{u,(S_{1}=1,\ldots,S_{N}=N)}(t_{1},\ldots,t_{N})$
denote the joint density of $(T_{1},\ldots,T_{N})$ conditioning to the event
$S$, when the starting configuration is $u$. Then, for any
$0<\delta<\min_{i=1,\ldots,N}\\{v_{i}(0)\\}$, there exist a constant $\beta>0$
such that for all $u\in B(v(0),\delta),$

$f_{u}(t_{1},\ldots,t_{N})\geq C_{1},\ \mbox{for}\ (t_{1},\ldots,t_{N})\in
A_{\epsilon}.$

  2. (ii)

For any given $\min_{i=1,\ldots,N}\\{v_{i}(0)\\}>\delta>0$ there exist
$\epsilon>0$ and a constant $C_{2}>0$ such that for $u\in B(v(0),\delta)$, we
have

$P_{T}(u,B(v(0),\delta))\geq C_{2}\epsilon^{N}>0.$

In particular, defining $T_{B(v(0),\delta)}=\inf\\{t>0:\mathrm{U}(t)\in
B(v(0),\delta)\\}$, we have

$\sup_{u\in B(v(0),\delta)}\mathbb{E}_{u}[T_{B(u^{*},\delta)}]\leq{T\over
C_{2}\epsilon^{N}}.$

###### Proof.

(i) Since
$P_{u}(T_{1}>t)=\exp\left[{-\int_{0}^{t}}\sum_{j=1}^{N}\varphi(\mathrm{U}_{j}^{u}(s))ds\right]$,
we immediately see that the density function of $T_{1}$ given that
$\mathrm{U}(0)=u$ is

$f_{u,T_{1}}(t_{1})=\sum_{j=1}^{N}\varphi(\mathrm{U}_{j}^{u}(t_{1}))\exp\left[{-\int_{0}^{t_{1}}}\sum_{j=1}^{N}\varphi(\mathrm{U}_{j}^{u}(s))ds\right],\
\mbox{for}\ t_{1}\geq 0.$ (8)

Since $u\in B(v(0),\delta)$ and $\delta<\min\\{M_{i}\\}$, we know that there
exist positive constants $c_{1}$ and $c_{2}$ such that $c_{1}<u_{j}<c_{2}$ for
all $j=1,\ldots,N$. But then for all $u\in B(v(0),\delta)$ and $j$, we have
that $c_{1}<\mathrm{U}_{j}(t_{1})=\bar{u}(1-e^{-\lambda
t_{1}})+u_{j}e^{-\lambda t_{1}}<c_{2}$. Thus from the previous inequality and
the identity (8) that it follows that $f_{u,T_{1}}(t_{1})\geq
N\varphi(c_{1})e^{-t_{1}N\varphi(c_{2})}.$ Now, from the definition of the
process one easily sees that the density function of the increment $T_{2}$
given $T_{1}=t_{1},S_{1}=1$ and $\mathrm{U}(0)=u$, for $t_{2}>t_{1}$, is

$f_{T_{2}|T_{1}=t_{1},S_{1}=1,u}(t_{2})=\sum_{j=1}^{N}\varphi(\mathrm{U}_{j}^{\Delta_{1}(\mathrm{U}^{u}(t_{1}))}(t_{2}))\exp\left[{-\int_{t_{1}}^{t_{2}}}\sum_{j=1}^{N}\varphi(\mathrm{U}_{j}^{\Delta_{1}(\mathrm{U}^{u}(t_{1}))}(s))ds\right].$
(9)

Note that for $j\neq 1$, $W_{1\rightarrow
j}+c_{1}<\Delta_{1}(\mathrm{U}^{u}(t_{1}))_{j}=W_{1\rightarrow
j}+\mathrm{U}^{u}(t^{-}_{1})<W_{1\rightarrow j}+c_{2}$, so that
$\bar{\mathrm{U}}^{u}(t_{1})>c_{1}{N-1\over N}+{1\over
N}\sum_{j=2}^{N}W_{1\rightarrow j}$. From these two inequalities and using
that the average potential is constant between successive jumps it follows
that there exist positive constants $c_{1}^{1}$ and $c_{2}^{1}$ such that
$f_{T_{2}|T_{1}=t_{1},S_{1}=1,u}(t_{2})\geq\varphi(c^{1}_{1})e^{-(t_{2}-t_{1})N(c^{1}_{2})}$,
for $t_{2}>t_{1}.$ Proceeding in this manner we obtain sequences
$(c_{1}^{n})_{n=1,\ldots,N-1}$ and $(c_{2}^{n})_{n=1,\ldots,N-1}$ satisfying
for $k=2,\ldots,N$,

$f_{T_{k}|T_{k-1}=t_{k-1},S_{k-1}=k-1,\ldots,T_{1}=t_{1},S_{1}=1,u}(t_{k})\geq\varphi(c_{1}^{k-1})e^{-(t_{k}-t_{k-1})N(c^{k-1}_{2})},$

where $t_{k}>t_{k-1}>\ldots>t_{1}\geq 0.$ Thus, we have that, over
$A_{\epsilon}$, the product of these conditional densities is positive. This
concludes the prove of item (i).

(ii) The first part follows immediately from the item (i) above and corollary
3. To prove the second part, we write
$\mathbb{E}[T_{B}(v(0),\delta)]=\int_{t=0}^{\infty}\mathbb{P}(T_{B}(v(0),\delta)>t)dt$
and we note that $\\{T_{B}(v(0),\delta)>t\\}\subset\\{\mathrm{U}(lT)\notin
B(v(0),\delta),l=1,\ldots,\big{\lceil}{t\over T}\big{\rceil}\\}$ for $t>T.$ A
simple application of the Markov property together with the lower bound given
by the first part finishes the proof. ∎

######  Proposition 4.1.

(Harris-type Condition) For any $u\in B(v(0),\delta)$, there exists a non
negative function $h_{u}$ such that for all measurable sets
$A\in\beta(\mathbb{R}_{N}^{+}\setminus\\{0^{N}\\})$,

$P_{T}(u,A)\geq\int_{A}h_{u}(v)dv.$

Moreover, there exist a measurable set $I$, with positive Lebesgue measure,
and a constant $C_{3}>0$ such that $h_{u}(v)\geq C_{3}1_{I}(v)$ for all $u\in
B(v(0),\delta).$

###### Proof.

For each $u\in B(v(0),\delta)$, as in Corollary 4 let us call
$\gamma_{u}:A_{\epsilon}\rightarrow I_{u}$ , the map

$\gamma_{u}(t_{1}^{N})=\gamma^{0}(t_{1}^{N})+R_{\delta\epsilon}(t_{1}^{N})+R_{\epsilon^{2}}(t_{1}^{N}),$

where $I_{u}=\gamma_{u}(A_{\epsilon}).$ From the corollaries 3, 4 and the
remark 4.1, it follows that for each $u\in B(v(0),\delta)$, conditioning to
$A_{\epsilon}\cap S$, the random vector $\mathrm{U}^{u}(T)$ has a density
$h_{u}$, where

$h_{u}(v)=\left\\{\begin{array}[]{ll}f_{u}(g_{u}(v))|Jg_{u}(v)|&,\ \mbox{if}\
v\in I_{u}\\\ 0&\mbox{otherwise},\end{array}\right.$

with $g_{u}:I_{u}\rightarrow A_{\epsilon}$ being the inverse of $\gamma_{u}$.
This concludes the first part of the proposition.

The proof of second part is more delicate and requires some work. From
Corollary 4 and Lemma 4.2 part (i) it suffices to prove that there exists a
set $I$ such that $I\subset\cap_{u\in B(v(0),\delta)}I_{u}.$

Now, consider the event $B_{\epsilon}$ defined by

$B_{\epsilon}=\Big{\\{}(i-1)\varepsilon+\frac{\varepsilon}{4}<T_{i}<i\varepsilon-\frac{\epsilon}{4},\
i=1,\cdots,N\Big{\\}}\cap S,$

define $I=\gamma_{0}(B_{\epsilon})$ and fix $v\in I.$ We want to show that for
all $u\in B(v(0),\delta)$ there is an vector
$t_{1}^{N}=t_{1}^{N}(u)=(t_{1}(u),\ldots,t_{N}(u))\in A_{\epsilon}$ such that
$\gamma_{u}(t_{1}^{N})=v.$ To this end, we introduce the function
$F(s,t_{1}^{N})=v-\gamma_{0}(t_{1}^{N})-s[R_{\delta\epsilon}(t_{1}^{N},u)+R_{\epsilon^{2}}(t_{1}^{N},u)],$
for $s\in[0,1]$ and $t_{1}^{N}\in A_{\epsilon}.$ Note that we need to show the
existence of vector $t_{1}^{N}\in A_{\epsilon}$ such that $F(1,t_{1}^{N})=0.$
This means that we need to study the equation $F(s,t_{1}^{N})=0$ with
$t_{1}^{N}$ as function of $s$. To ease the notation, from now on we will
write $t$ instead of $t_{1}^{N}.$

Note that by the definition of $I$, there exists $t_{0}\in B_{\epsilon}$ such
that $F(0,t_{0})=0.$ Besides, $t=t(s)$ is solution of the equation $F(s,t)=0$
if and only if it satisfies

$0=-D\gamma_{0}(t(s))\cdot{dt(s)\over
ds}-[R_{\delta\epsilon}(t(s),u)+R_{\epsilon^{2}}(t(s),u)]-s[DR_{\delta\epsilon}(t(s),u)+DR_{\epsilon^{2}}(t(s),u)]\cdot{dt(s)\over
ds}$

or equivalently,

$\Big{[}-D\gamma_{0}(t(s))-s\big{(}DR_{\delta\epsilon}(t(s),u)+DR_{\epsilon^{2}}(t(s),u)\big{)}\Big{]}\cdot{dt(s)\over
ds}=R_{\delta\epsilon}(t(s),u)+R_{\epsilon^{2}}(t(s),u),$ (10)

where $Df(\cdot)$ stands for the differential operator of $f$. By corollary 4,
the linear operator inside the brackets is invertible and the function on the
right hand side is of order $\delta\epsilon+\epsilon^{2}$ whose the derivative
is of order $\delta+\epsilon.$ Therefore, it follows that $t=t(s)$ is a
solution of (10) if and only if it is the solution of the ODE of the form

${d\over ds}t(s)=H(s,t(s)),\ t(0)=t_{0},$ (11)

where the derivative of $H$ with respect to $t$ exists and it is of order
$\delta+\epsilon$. In particular, it is limited and therefore the ODE has
unique solution $t=t(s)$ for all $s$ such that $t(s)\in A_{\epsilon}.$ Since
$t(s)$ moves as $H$ which is of order $\delta\epsilon+\epsilon^{2}$ and the
initial condition $t_{0}$ is at a distance of order $\epsilon$ of
$A_{\epsilon}$, we have that $t(1)\in A_{\epsilon}$, decreasing both $\delta$
and $\epsilon$ if necessary. Hence, we have that $I\subset\cap_{u\in B}I_{u}.$
∎

###### Proof of Theorem 4.

Item (ii) of lemma 4.2 implies that the time of $B(v(0),\delta)$ is a positive
recurrent set. Now by poposition 4.1 the Harris-type condition holds once the
process reach $B(v(0),\delta)$, so that in fact $B(v(0),\delta)$ is a positive
recurrent regenerative set. ∎

##  5 Final discussion

The present paper presented a new class of stochastic processes and studied
its asymptotic distribution in absence and presence of the leak current,
without external stimuli in either cases. Similar results have been recently
obtained by Robert and Touboul in [7]. Their model considers a constant
leakage rate without the gap junction term ($\alpha=1$ and $\lambda=0$ in our
context) where the synaptic weights are positive i.i.d random variables with
finite mean. They showed that if $\int_{0}^{1}{\varphi(x)\over x}dx<+\infty$,
then no spike occurs after some finite time. This condition imposed on the
spiking rate $\varphi$ is the same type as ours. They also proved that when
$\varphi(0)>0$ and the synaptic weights are bounded almost surely, the process
admits a unique non trivial invariant measure. In other words, when external
stimuli are considered the system remains active almost surely, contrasting
with non external stimuli case. We did not study this case, nonetheless it
could be treated similarly as we did in Theorem 4.

##  6 Acknowledges

We are indebted to R. Cofré for the discussions which originate this work. We
thank E. Presutti for helpful suggestions and stimulating discussions; A.
Galves and E. Löcherbach for the discussions which led us to the proof of
Theorem 2.2. This article was produced as part of the activities of FAPESP
Center for Neuromathematics (FAPESP grant 2013/ 07699-0). A. Duarte is fully
supported by a CNPq fellowship (grant 141270/2013-6) and G. Ost is fully
supported by a CNPq fellowship (grant 141482/2013-3).

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