]Bernstein Center for Computational Neuroscience, 10115 Berlin,
Germany]Department of Physics, Institute of Physics, Federal University of Rio
Grande do Sul, Porto Alegre, Brazil

# A Mean-Field Method for Generic Conductance-Based Integrate-and-Fire Neurons
with Finite Timescales

Marcelo P. Becker<EMAIL_ADDRESS>[  Marco A. P. Idiart  [

###### Abstract

The construction of transfer functions in theoretical neuroscience plays an
important role in determining the spiking rate behavior of neurons in
networks. These functions can be obtained through various fitting methods, but
the biological relevance of the parameters is not always clear. However, for
stationary inputs, such functions can be obtained without the adjustment of
free parameters by using mean-field methods. In this work, we expand current
Fokker-Planck approaches to account for the concurrent influence of colored
and multiplicative noise terms on generic conductance-based integrate-and-fire
neurons. We reduce the resulting stochastic system from the application of the
diffusion approximation to a one-dimensional Langevin equation. An effective
Fokker-Planck is then constructed using Fox Theory, which is solved
numerically to obtain the transfer function. The solution is capable of
reproducing the transfer function behavior of simulated neurons across a wide
range of parameters. The method can also be easily extended to account for
different sources of noise with various multiplicative terms, and it can be
used in other types of problems in principle.

††preprint: APS/123-QED

##  I Introduction

The brain is a complex system that organizes itself into structures ranging in
size from fine-scale components [1] to large-scale arrangements involving the
whole organ [2]. These structures are composed of neurons and supportive cells
that interact with each other in complex ways [3], presenting a typical
phenomenon of complex systems. As a result, the study of these systems poses
significant challenges, requiring the development of theoretical models and
frameworks to bridge the multiple scales.

Theoretical models and frameworks are essential to address the multi-scale
challenges found in complex systems. Nature has many examples of such systems,
and condensed matter physics has developed a variety of methods to treat these
types of problems [4, 5, 6, 7]. These methods have been co-opted by
theoretical neuroscientists to understand the complex interactions of neurons
and supportive cells in the brain [8].

Mean-field methods are an effective tool for investigating collective
behaviors in neural networks, as neurons are often influenced by numerous
stochastic inputs. By modeling neurons as simple transfer functions, attractor
dynamics, network oscillations, synchronization, pattern formation, and phase
transitions can be better analyzed [9, 10, 11, 12]. Through mean-field
approaches, a connection can be made between the individual spiking neuron at
the microscopic scale and the population’s rate descriptions at the mesoscopic
scale [12, 13]. Amit and Brunel’s seminal work introduced a typical approach
using the Fokker-Planck mean-field formalism to model the simple leaky
integrate-and-fire model [14].

However, the mean-field treatment of conductance-based integrate-and-fire
neurons introduces a new level of complexity due to the conductance variables,
which increase the dimensionality of the stochastic system. These complexities
can generate measurable differences, such as the temporal correlations
introduced by synaptic filtering, which have been shown to alter the spiking
statistics [15] and the scaling properties of neurons [16]. Dealing with these
temporal correlations presents a challenge for mean-field methods.
Perturbative solutions have been found for a single source of additive
exponentially correlated noise [17, 18]. For linear multiplicative noise, it
is possible to use the same perturbative methods by employing the effective
time-constant approximation [19, 20]. By combining these two methods, a large
set of problems with linear multiplicative terms can be treated. But what
happens if we want to expand those methods to include more generic forms of
neurons that include colored noise with nonlinear multiplicative terms?

Looking back at physics, the presence of temporal correlations in colored
noise processes is known to prevent the construction of an exact Fokker-Planck
equation. Some forms of approximation were derived in the 80s. The best
Fokker-Planck approximation, proposed by Sancho and Lindenberg [21, 22],
proposes a differential form for the diffusion term that is exact in the white
noise limit but only solvable in particular cases. Therefore, perturbative
methods are required in most applications. The Fox theory, which uses
functional calculus to derive an approximation for small correlations [23,
24], yields the same result as the best Fokker-Planck approximation at first
order in the time correlation but diverges in higher orders [25]. Other
methods, such as the projection operator method [26, 25], the adiabatic
elimination procedure used by Jung and Hänggi [27], and the renormalized
operator cumulant expansion of Der [28], have also been developed to deal with
similar problems.

In this work, we adapted the Fox Theory to the case of a conductance-based
integrate-and-fire neuron under the influence of stochastic inputs. We
numerically solved the resulting stationary Fokker-Planck equation and
extracted the transfer function with different assumptions on the boundary
conditions, comparing the results with proper simulations.

##  II General model

We consider the behavior of a point-like generic leaky integrate-and-fire
neuron with conductance-based input embedded in a network of similar units.
The neuron is described by the membrane potential $V$ that follows from

$\tau_{L}\frac{dV}{dt}=-(V-E_{L})-\sum_{i}g_{i}(t)s_{i}(V)(V-E_{i})\,,$ (1)

where $\tau_{L}$ is the membrane time constant, $E_{L}$ is the resting
potential, $E_{i}$ is the reversal potential of the corresponding channel $i$,
and $s_{i}(V)$ is a modulating function that can depend on $V$. It is
important to note that the addition of nonlinear functions of $V$ to the
equation (a quadratic or exponential function for example) is possible in
principle, although we will not deal with this case here. The conductances
$g_{i}(t)$ behave as linear filters of the input signal. Specifically, we have

$\tau_{i}\frac{dg_{i}}{dt}=-g_{i}+w_{i}\sum_{j,k}\delta(t-t_{j}^{k})\,.$ (2)

The summation here is performed over all pre-synaptic sites $j$ and all spikes
$k$ emitted in that site. $w_{i}$’s are the synaptic weights, which are kept
the same for all neurons belonging to the same population.

The membrane potential $V$ evolves according to (1) until it reaches the
threshold $\theta$ when a formal spike is emitted. The potential is then reset
to $V_{r}$ and is not updated for the extent of the refractory interval
$\tau_{r}$.

##  III Mean-field analysis

###  III.1 Conductance

As a starting point, we suppose that the neuron receives inputs from separate
populations of neurons corresponding to different channels in the equation,
each of them making $K_{i}$ connections. We assume that the inputs from each
population come from Poisson rate neurons with fixed rate $\nu_{i}$. If the
number of connections is large ($K_{i}\gg 1$) and the connection weights small
($w_{i}\ll 1$) the diffusion approximation can be used [14, 29] and equation
(2) becomes

$\tau_{i}\frac{dg_{i}}{dt}=-g_{i}+\mu_{i}+\sqrt{\tau_{i}}\sigma_{i}\xi_{i}(t)\,,$
(3)

with

$\displaystyle\mu_{i}=w_{i}K_{i}\nu\tau_{i}\,,$ (4)
$\displaystyle\sigma_{i}^{2}=w_{i}^{2}K_{i}\nu\tau_{i}\,.$ (5)

The $\xi_{i}(t)$’s here are uncorrelated Gaussian variables with zero mean and
unit variance.

###  III.2 Membrane Potential

The membrane potential equation (1) can then be written as

$\frac{dV}{dt}=-\frac{(V-\mu)}{\tau}+\sum_{i}h_{i}(V)\eta_{i}(t)\,,$ (6)

where

$\displaystyle\tau=\tau_{L}/(1+\sum_{i}s_{i}(V)\mu_{i})\,,$ (7)
$\displaystyle\mu=\frac{\tau}{\tau_{L}}(E_{L}+\sum_{i}s_{i}(V)\mu_{i}E_{i})\,,$
(8) $\displaystyle
h_{i}(V)=s_{i}(V)\frac{\sqrt{\tau_{i}}}{\tau_{L}}\sigma_{i}(E_{i}-V)\,,$ (9)

and the noise variables are now

$\eta_{i}(t)=\frac{1}{\tau_{i}}\int_{0}^{t}e^{-\frac{T}{\tau_{i}}}\xi_{i}(t-T)dT\,,$
(10)

with the following correlations

$\langle\;\eta_{i}(t)\;\eta_{j}(t^{\prime})\;\rangle=\frac{1}{2\tau_{i}}e^{-\frac{|t-t^{\prime}|}{\tau_{i}}}\delta_{ij}\;\;.$
(11)

What we have now is a Langevin equation with distinct sources of colored
noise. The $n$ dimensional stochastic system is now reduced to a single SDE.
This happens, however, with the cost that the noise is no longer Markovian,
i.e., the fluctuations at time $t$ depend on the fluctuations at previous
times $t^{\prime}<t$. It makes it impossible to obtain an exact Fokker-Planck
equation since it adds an additional temporal integration. Therefore, in order
to be able to use the Fokker-Planck approach [14], it is necessary to build an
approximate Fokker-Planck equation.

###  III.3 Effective Time-Constant Approximation

In some cases, it might also be useful to simplify the problem by finding an
approximation that allows us to eliminate the multiplicative noise. Observe
that in Eq. (6) the stochastic variable $\eta_{i}(t)$ is multiplied by a
function of $V$ implying that the noise level depends on voltage values. This
complexity can be avoided by using the effective time-constant approximation
[19][20] where the membrane potential is replaced by the equilibrium potential
$\mu$, resulting in

$h_{i}(V)\rightarrow h_{i}(\mu)$

This approximation implies that the modulation of noise can be seen as
dependent on the distance of the equilibrium potential from its reversal
potential, at least at first order. In fact, it can be argued that using this
approximation, when the terms $h_{i}(V)$ are linear, leads to a more
consistent treatment of the problem, since the error generated by this
approach is of the same order as the error introduced by the diffusion
approximation [20]. But the most important fact here is that this approach
simplifies considerably the treatment of the resulting Fokker-Planck equation.
A direct consequence of this is that it lends to more easily interpretable
parameters. We will compare the results with and without the use of this
approximation when it is applicable.

###  III.4 Fox Theory

Temporal correlations in the noise of Langevin equations are known to impede
the construction of an exact Fokker-Planck equation. So our task is to find an
appropriate approximation that leads to a differential equation of the
probability distribution of the membrane potential. From the different options
available, we use here the Fox theory, since it is the one with the most
direct application and is the easiest to generalize for multiple noise
sources. It also possesses some relevant properties for this work. First of
all, it is important to note that the approach followed in this method is non-
perturbative. In fact, under a certain condition, the convergence of the
approximation for $\tau_{i}\rightarrow 0$ is uniform, that is, it converges to
the white noise case for all values of $V$ in the domain of interest [24]. The
uniformity condition is

$1-\tau_{i}\left(W^{\prime}(V)-\frac{h_{i}^{\prime}(V)}{h_{i}(V)}W(V)\right)>0\,,$
(12)

where $W(V)$ is the drift term in the Langevin (in our case $(\mu-V)/\tau$)
and primes indicate derivatives with respect to the argument $V$. This
condition sets a scale for $\tau_{i}$ for which the approximation behaves
reasonably well.

Jung and Hänggi adiabatic method [27] is an approximation that is valid for
small and large values $\tau_{i}$, whose stationary solution agrees with the
stationary solution of the Fox theory. Therefore, even though the resulting
effective Fokker-Planck obtained by the Fox theory was derived for small
$\tau_{i}$ values, the stationary solution is valid also for large $\tau_{i}$
(given that condition (12) is obeyed). The validity for both limits suggests
that the Fox theory is a good interpolation between both stationary results. A
small derivation of the validity of the stationary solution of the Fox theory
for $\tau_{i}\rightarrow\infty$ can also be found in [25].

The application of the Fox Theory to equation (6) results in the effective
Fokker-Planck

$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial
V}\left[W(V)P-\sum_{i}h_{i}(V)\frac{\partial}{\partial V}(S_{i}(V)P)\right]$
(13)

with

$S_{i}(V)=\frac{1}{2}\left[\frac{h_{i}(V)}{1-\tau_{i}(W^{\prime}(V)-\frac{h_{i}^{\prime}(V)}{h_{i}(V)}W(V))}\right]\,$
(14)

The expansion for multiple noise sources can be simply done by using the same
assumptions as in the original papers [23, 24].

###  III.5 Transfer Function

First, we will explore the result of the application of the effective time-
constant approximation. The resulting effective Fokker-Planck is then

$\frac{\partial P(V,t)}{\partial t}=\frac{\partial}{\partial
V}\left[\frac{(V-\mu)}{\tau}P(V,t)\right]+\frac{\sigma_{V}^{2}}{2\tau}\frac{\partial^{2}P(V,t)}{\partial
V^{2}}\,,$ (15)

where

$\sigma_{V}^{2}=\sum_{i}\sigma_{V_{i}}^{2}=\sum_{i}\frac{\tau^{2}}{\tau+\tau_{i}}h_{i}^{2}\,.$
(16)

and the notation was simplified by calling $h_{i}(\mu)=h_{i}$.

The resulting effective Fokker-Planck is the same as the one obtained from the
simpler Langevin

$\frac{dV}{dt}=-\frac{(V-\mu)}{\tau}+\sigma_{V}\;\xi(t)$ (17)

with the zero mean unit variance Gaussian white noise $\xi(t)$.

The independence of the terms in the sum suggests the separation of the full
variance into two independent components. There is, then, a simple
interpretation of the parameters of the effective Fokker-Planck equation. The
first-order term (drift) corresponds to the deterministic drive of the
membrane potential. The second-order term (diffusion) can be seen as the sum
of the variance of the noise sources, where the noise sources are treated as
white. The combination of both approximations (effective time-constant and the
Fox theory), therefore, produces the same behavior as a Langevin system with
white noise and summed variances.

The transfer function can be calculated, resulting in

$\frac{1}{\nu}=\tau_{r}+\tau\sqrt{\pi}\int^{\frac{\theta-\mu}{\sigma_{V}}}_{\frac{V_{r}-\mu}{\sigma_{V}}}e^{x^{2}}(1+\operatorname{erf}(x))dx\,,$
(18)

where $\operatorname{erf}(x)$ is the error function. We can also get the
stationary probability distribution

$\displaystyle P_{S}(V)$ $\displaystyle=$
$\displaystyle\frac{2\nu\tau}{\sigma_{V}}\exp\left(-\frac{(V-\mu)^{2}}{\sigma_{V}^{2}}\right)$
(19)
$\displaystyle\times\int^{\frac{\theta-\mu}{\sigma_{V}}}_{\frac{V-\mu}{\sigma_{V}}}\Theta\left(x-\frac{V_{r}-\mu}{\sigma_{V}}\right)e^{x^{2}}dx\,,$

where $\Theta(x)$ is the Heaviside function. In these results, there is an
implicit assumption of the continuity of the distribution at the threshold,
which implies $P_{S}(\theta)=0$. This assumption is not reasonable, since
perturbative results for single channel conductance-based models show a
discontinuity at that point [17]. We can estimate numerically the value of the
distribution in a procedure that will be described shortly. We will provide,
also, comparison between results with and without the assumption.

###  III.6 Multiplicative Noise

The complicated form of (13) allows only a formal stationary solution. Using
again the continuity of the distribution and standard procedures [29], we
arrive in

$\frac{1}{\nu}=\tau_{r}+\int_{V_{r}}^{\theta}\int_{-\infty}^{x}\frac{e^{F(x)-F(V)}}{\chi(x)}\;dVdx\,,$
(20)

where

$\displaystyle
F(V)=\int\frac{\sum_{i}h_{i}(V)\frac{dS_{i}}{dV}-W(V)}{\chi(V)}\;dV$ (21)

and

$\displaystyle\chi(V)=\sum_{i}h_{i}(V)S_{i}(V)\,.$ (22)

The lower limit of the first interval can be limited by the lowest of the
reversal potentials since the dynamics of the membrane potential cannot be
lower than this value. But in a general sense, the infinity can be written in
place as we did.

This formal solution, unfortunately, is not straightforward to use, since a
closed form for the integrating factor is generally not obtainable. Therefore,
we have to rely on numerical methods.

###  III.7 Numerical Methods and Simulations

Efficient results can be obtained by using the numerical approach developed by
Richardson [30]. It takes advantage of the formal solution to get faster
convergence than a typical Euler integration and uses $P_{S}(\theta)=0$ as an
initial condition for the integration. However, since this assumption is not
well founded in our case, we need to develop a different approach. We opted
for using a double integration procedure, starting with the more reasonable
assumption $P(E_{I})=0$ and integrating forward. This results in an estimated
value for $P_{S}(\theta)$, so that we can backward integrate to obtain the
distribution and the firing rate.

The simulation data was generated using Brian2 [31] treating the input layer
as Poisson neurons. Firing rates were taken as the time average of the spikes
for a period of 10s after waiting for 5s to eliminate transients. The same
strategy was used for the generation of the distribution.

##  IV Conductance-Based Integrate-and-Fire Neuron

We will first apply our method to a simple conductance-based integrate-and-
fire neuron with two input channels: one excitatory $g_{E}(t)$, and one
inhibitory $g_{I}(t)$. The equations describing this system are

$\displaystyle\tau_{L}\frac{dV}{dt}=-(V-E_{L})-\sum_{i=E,I}g_{i}(t)(V-E_{i})\,,$
(23)
$\displaystyle\tau_{E}\frac{dg_{E}}{dt}=-g_{E}+\sum_{j,k}w_{E}\delta(t-t_{j}^{k})\,,$
(24)
$\displaystyle\tau_{I}\frac{dg_{I}}{dt}=-g_{I}+\sum_{j,k}w_{I}\delta(t-t_{j}^{k})\,.$
(25)

The neuron receives input from $K_{E}$ excitatory input neurons and $K_{I}$
inhibitory. Both populations fire at a fixed firing rate $\nu_{i}$. The
parameter values are chosen to be physiologically plausible and are in the
range typically used in simulation works (for instance [32]), see table 1.

Parameter | Value
---|---
$E_{L}$ | -60mV
$E_{E}$ | 0mV
$E_{I}$ | -80mV
$w_{E}$ | {0.1, 0.5}
$w_{I}$ | {0.1, 0.4, 1.0, 10.0}
$\tau_{L}$ | 20ms
$\tau_{E}$ | variable
$\tau_{I}$ | 10ms
$\tau_{R}$ | 2ms
$K_{E}$ | 400
$K_{I}$ | 100
$\theta$ | -50mV
$V_{r}$ | -60mV
$\nu_{i}$ | {5, 20, 50}Hz
Table 1: Table containing the parameters used for the simple Conductance-Based
Integrate-and-Fire model.

To illustrate the validity of the diffusion approximation, for the range of
values used here, we plotted the mean, the standard deviation, and the
skewness of the analytical conductance and the simulated one (figure 1). As
expected by the construction, the expressions for the mean and the standard
deviation are a good representation of the values simulated even when the
synaptic weight $w_{E}$ is large. In contrast, we can see deviations in the
skewness for small values of $\tau_{E}$. This is somewhat expected since the
values of $g(t)$ can’t be negative and the diffusion approximation doesn’t
take this into consideration. For small $\tau_{E}$, we have small $\mu_{E}$
and the Gaussian form of the approximation fails to account for the asymmetric
shape of the distribution with a hard boundary at $g=0$. Therefore, as stated
in [20], the diffusion approximation introduces errors at the third-order
moment of the distribution.

Figure 1: Comparison of the analytical expressions obtained for the statistics
of $g_{E}$ using the diffusion approximation (lines) with simulations
(circles). The analytical expressions are good descriptions of the simulations
for all the range of parameters tested for the first and second moments. The
skewness, however, exhibits a deviation from the Gaussian approximation for
small $\tau_{E}$. Parameters for first column, $w_{E}=0.1$, $w_{I}=0.4$;
second column, $w_{I}=0.8$, $\nu_{i}=5Hz$

The resulting form of the Langevin equation for the Conductance-Based
Integrate-and-Fire is

$\frac{dV}{dt}=-\frac{(V-\mu)}{\tau}+h_{E}(V)\;\eta_{E}(t)+h_{I}(V)\;\eta_{I}(t)\,,$
(26)

where

$\displaystyle\tau=\frac{\tau_{L}}{1+\mu_{E}+\mu_{I}}\,,$
$\displaystyle\mu=\frac{\tau}{\tau_{L}}(E_{L}+\mu_{E}E_{E}+\mu_{I}E_{I})\,,$
$\displaystyle
h_{E,I}(V)=\frac{\sqrt{\tau_{E,I}}}{\tau_{L}}\sigma_{E,I}(E_{E,I}-V)\,.$

We can now use the effective time-constant approximation or deal with the full
multiplicative problem.

###  IV.1 Additive Noise

With the effective time-constant approximation, the Langevin simplifies to

$\frac{dV}{dt}=-\frac{(V-\mu)}{\tau}+h_{E}\eta_{E}(t)+h_{I}\eta_{I}(t)\,,$
(27)

where the constant coefficients are $h_{E}=h_{E}(\mu)$ and $h_{I}=h_{I}(\mu)$.
The application of the Fox Theory results in the transfer function (18) and
the probability distribution (19) with the expression for $\sigma_{V}$ given
by

$\sigma_{V}^{2}=\frac{\tau^{2}}{\tau+\tau_{E}}h_{E}^{2}+\frac{\tau^{2}}{\tau+\tau_{I}}h_{I}^{2}\,.$
(28)

A comparison of the analytical results (calculated numerically) with the
simulations can be seen in Fig.2. Fig.2A and Fig.2C assume $P(\theta)=0$,
while Fig.2B and Fig.2D do not. Good agreement is present for most sets of
parameters tested, the exception being the high inhibition regime
($w_{I}=10$). Fig.2A and 2B, show that the input rate $\nu_{i}$ has little
effect on the stationary potential $\mu$ but changes the noise variance
$\sigma_{V}^{2}$ of the neuron without threshold. Therefore, in this case, the
firing rate behavior is mostly given by the changes in the input variance. A
higher noise variance makes the transition from silent to firing smoother, as
can be seen by the slower convergence of the firing rate curves with higher
input variance.

Figure 2: Comparison of the analytical model (lines) with simulations
(circles) for the simple conductance-based integrate-and-fire neuron using the
effective time-constant approximation. In columns (A) and (B), we use three
different values of input firing rate $\nu_{i}$ as a function of the
excitatory time constant and set $w_{E}=0.1$ and $w_{I}=0.4$. In columns (C)
and (D), we compare three values of inhibitory weights $w_{I}$ and set
$w_{E}=0.5$, $\nu_{i}=5Hz$. In the first row and columns (A) and (C) the mean
potential for a thresholdless model is plotted. Columns (B) and (D) show the
standard deviation of the membrane potential for the same thresholdless model.
In the second row, columns (A) and (C) display the firing rates for the model
with the continuous distribution assumption, and columns (B) and (D) display
the firing rates using the double integration procedure. The absolute errors
for the corresponding models are plotted in the third row.

Figs.2C and 2D explore the effect of varying the inhibitory synaptic weight (
$w_{I}=0.1$, $w_{I}=1.0$, and $w_{I}=10.0$) for a constant excitatory weight
$w_{E}=0.5$. For low inhibition, there is a good agreement between theory and
simulation. But for high inhibition ( $w_{I}=10.0$) the theory produces a
sharp transition of firing rate that is not observed in the simulations. The
discrepancy (see the error in Fig.2C, third row) is larger in the region where
$\mu$ is between $E_{L}$ and $\theta$, that is, in the sub-threshold regime.
In this region, spikes are driven by membrane potential fluctuations. The
sharpness of the transition compared to the data suggests that, for those
parameter values, the model underestimates fluctuations. The high inhibition
case also allows us to see that the $P(\theta)=0$ brings the transition to
lower $\tau_{E}$ values. The better result of the double integration procedure
stems from the improvement of the estimation of the transition region since
the shape of the curve is almost the same.

###  IV.2 Multiplicative Noise

The full-multiplicative noise treatment results in a Fokker-Planck equation
with the form

$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial
V}\left[W(V)P\right]+\sum_{i=E,I}\frac{\partial}{\partial
V}h_{i}(V)\frac{\partial}{\partial V}\left(S_{i}(V)P\right),$ (29)

where the functions $S_{E}(V)$ and $S_{I}(V)$ are given by the generic
expression in (14). The stationary differential equation that needs to be
solved numerically is then

$\frac{\partial P_{s}}{\partial V}+B(V)P_{s}=-\nu H(V)\,,$ (30)

with

$\displaystyle
B(V)=\frac{h_{E}(V)S^{\prime}_{E}(V)+h_{I}(V)S^{\prime}_{I}(V)-W(V)}{\chi(V)}\,,$
(31) $\displaystyle H(V)=\frac{\Theta(V-V_{r})}{\chi(V)}\,,$ (32)
$\displaystyle\chi(V)=h_{E}(V)S_{E}(V)+h_{I}(V)S_{I}(V)\,.$ (33)

We used this approach to solve the stationary Fokker-Planck equation
numerically for the same set of parameters as in the last subsection.No
appreciable differences were found between the additive and multiplicative
models for different input firing rates (see Figs. 3A and 3C). However, for
high inhibition ($w_{I}=10.0$), the full multiplicative treatment with the
continuity assumption ($P(\theta)=0$) produced a worse quantitative result but
with a better overall shape of the curve. This error was mostly corrected when
we dropped this assumption and used the double integration procedure (see Fig.
3D), concentrating on the region where $\mu$ is around the threshold value.

Figure 3: We compare the analytical model with simulations for the simple
conductance-based integrate-and-fire neuron, incorporating full multiplicative
noise. In columns (A) and (B), we vary the input firing rate $\nu_{i}$ as a
function of the excitatory time constant and set $w_{E}=0.1$ and $w_{I}=0.4$.
In columns (C) and (D), we compare different values of inhibitory weight
$w_{I}$ and set $w_{E}=0.5$, $\nu=5Hz$. In the first row, columns (A) and (C)
display the firing rate for the model assuming continuous distribution, and
columns (B) and (D) display the firing rates using the double integration
procedure. The absolute errors for the corresponding models are plotted on the
second row.

###  IV.3 Stationary Probability Distribution

To complete the analysis of the conductance-based integrate-and-fire neuron,
we look at the stationary probability distributions with and without the
effective time-constant approximation and the $P_{S}(\theta)=0$ assumption.
Fig.4 top row corresponds to the parameters of the blue curves that appear in
Figs.2 and 3. Fig.4 bottom row corresponds to the yellow curves of the same
figures. Each column corresponds to progressive values of $\tau_{E}$ (1, 5,
10, 20, 70 ms, respectively). As the excitatory time constant increases, the
voltage distribution evolves from a Gaussian shape with a sharp peak far from
the threshold, towards an increasingly more distorted distribution as it
interacts with the threshold. The area under the curve diminishes (since we
are omitting the refractory period in the graph) corresponding to the higher
firing rate of the neuron.

Comparing the mean-field solutions, we can see that the major difference comes
from the continuity assumption for intermediary values of $\tau_{E}$. It is
clear that the simulated distributions are not continuous. In fact, when in
the mean-driven regime ($\mu>\theta$), the distributions concentrate between
the reset and the threshold. The double integration procedure gives good
results for the distribution in most cases, with an artificial low-end tail
for some values. The difference generated by the effective time-constant
approximation is minor, generating almost superimposed curves in some cases.

Figure 4:  The stationary probability distributions for the Voltage of
conductance-based integrate-and-fire neurons. Column (A) $\tau_{E}=1ms$, (B)
$\tau_{E}=5ms$, (C) $\tau_{E}=10ms$, (D) $\tau_{E}=20ms$, (E) $\tau_{E}=70ms$.
The first row corresponds to blue curves in Fig. 2, that is, $\nu_{i}=5Hz$ and
parameters from table 1. The second row corresponds to the yellow curves on
the same figure ($w_{I}=10$, $w_{E}=5$, and $\nu_{i}=5Hz$). The different
lines correspond to different assumptions in the model, as can be seen in the
legend in column (E).

##  V NMDA Integrate-and-Fire

In the standard conductance-based integrate-and-fire neuron model, there is no
$V$ nonlinearity in the resulting Langevin equation. In principle, our mean-
field method should be able to handle nonlinearities in either the drift term,
the diffusion terms, or both. Here we will introduce a nonlinearity by adding
NMDA channels, which are excitatory channels whose activation depends on the
membrane potential. A convenient way to model its behavior is by adding an
appropriately tuned sigmoidal factor to the conductance term [33]. The
complete model can be written as

$\displaystyle\tau_{L}\frac{dV}{dt}=-(V-E_{L})-(1-\alpha)g_{A}(t)(V-E_{E})-$
$\displaystyle-\alpha s(V)g_{N}(t)(V-E_{E})-g_{I}(t)(V-E_{I})\,,$ (34)

where

$\displaystyle\tau_{i}\frac{dg_{i}}{dt}=-g_{i}+\sum_{j,k}w_{i}\delta(t-t_{j}^{k})\,,$
(35) $\displaystyle s(V)=\frac{1}{1+([\text{Mg}^{2+}]/\gamma)\exp{(-\beta
V)}}\,.$ (36)

Where $i$ can be $A$, $N$, or $I$, representing the AMPA, NMDA, and inhibitory
channels, respectively. We also set $w_{A}=w_{N}=w_{E}$. $s(V)$ is the
sigmoidal modulating function, $[\text{Mg}^{2+}]$ is the concentration of
magnesium ions, and $\gamma$ and $\beta$ are fitting parameters. For $\alpha$
to represent the proportion of AMPA and NMDA channels, we kept the number of
their inputs equal, i.e., $K_{A}=K_{N}=K_{E}$. Like before the input rate
$\nu_{i}$ is the same for all the population. Table 2 displays the values of
all the model parameters.

In the diffusion approximation and reduced to a one-dimensional Langevin
equation, this model produces the following set of equations:

$\frac{dV}{dt}=\frac{(V-\mu(V))}{\tau(V)}+\sum_{i=A,N,I}h_{i}(V)\eta_{i}(t)\,,$
(37)

where

$\displaystyle\tau(V)=\frac{\tau_{L}}{1+(1-\alpha)\mu_{A}+\alpha
s(V)\mu_{N}+\mu_{I}}\,,$
$\displaystyle\mu(V)=\frac{\tau}{\tau_{L}}(E_{L}+(1-\alpha)\mu_{A}E_{E}+\alpha
s(V)\mu_{N}E_{E}+\mu_{I}E_{I})\,,$ $\displaystyle
h_{A}(V)=(1-\alpha)\frac{\sqrt{\tau_{A}}}{\tau_{L}}\sigma_{A}(E_{E}-V)\,,$
$\displaystyle h_{N}(V)=\alpha
s(V)\frac{\sqrt{\tau_{N}}}{\tau_{L}}\sigma_{N}(E_{E}-V)\,,$ $\displaystyle
h_{I}(V)=\frac{\sqrt{\tau_{I}}}{\tau_{L}}\sigma_{I}(E_{I}-V)\,.$

Parameter | Value
---|---
Variable | Value
$\alpha$ | variable
$E_{L}$ | -60mV
$E_{E}$ | 0mV
$E_{I}$ | -80mV
$w_{E}$ | {0.1, 0.5}
$w_{I}$ | {0.1, 0.4, 1.0, 10.0}
$\tau_{L}$ | 20ms
$\tau_{A}$ | 1ms
$\tau_{N}$ | 100ms
$\tau_{I}$ | 10ms
$\tau_{R}$ | 2ms
$K_{E}$ | 400
$K_{I}$ | 100
$\theta$ | -50mV
$V_{r}$ | -60mV
$\nu_{i}$ | {5, 20, 50}Hz
$[\text{Mg}^{2+}]$ | 1mM
$\gamma$ | 3.57mM
$\beta$ | 0.062(mV)-1
Table 2: Table containing the set of parameters used for the NMDA model.
Figure 5: Comparison of the NMDA analytical model with simulations for three
different values of input rate $\nu_{I}$ (columns (A) and (B)) and inhibitory
synaptic weight $w_{I}$ (columns (C) and (D)) as a function of the
interpolation parameter $\alpha$. In (A) and (B), we set $w_{E}=0.1$ and
$w_{I}=0.4$, while in (C) and (D), we set $w_{E}=0.5$ and $\nu_{i}=5$Hz.
Columns (A) and (C) assume the continuity of the distribution, while (B) and
(D) use the double integration procedure. We calculate the error as the
absolute distance between the simulation and analytical results.

Direct use of the effective time-constant approximation is no longer possible,
since $h_{N}(V)$ is no longer linear, and the approximation loses its logic
[20]. However, it is worth mentioning that for the NMDA model, it is possible
to linearize the term $s(V)(V-E_{E})$ around the average membrane potential,
as was done by Brunel and Wang [34]. Since we are trying to see how well the
method performs for non-linear multiplicative noise, we will not perform the
linearization and the approximation.

We start with the Fokker-Planck equation,

$\frac{\partial P}{\partial t}=-\frac{\partial}{\partial
V}\left[W(V)P\right]+\sum_{i=A,N,I}\frac{\partial}{\partial
V}h_{i}(V)\frac{\partial}{\partial V}(S_{i}(V)P)\,,$ (38)

where the functions $S_{A}(V)$, $S_{N}(V)$, and $S_{I}(V)$ are given by (14).
$W(V)$ is no longer linear in $V$ and it results in different functional forms
for the $S$ functions. The linear differential equation in $V$ is then

$\frac{\partial P_{s}}{\partial V}+B(V)P_{s}=-\nu H(V)\,,$ (39)

with coefficients

$\displaystyle
B(V)=\frac{\sum_{i=A,N,I}h_{i}(V)S^{\prime}_{i}(V)-W(V)}{\chi(V)}\,,$ (40)
$\displaystyle H(V)=\frac{\Theta(V-V_{r})}{\chi(V)}\,,$ (41)
$\displaystyle\chi(V)=\sum_{i=A,N,I}h_{i}(V)S_{i}(V)\,.$ (42)

Figure 6: We present the stationary probability distribution for different
values of the interpolation parameter $\alpha$: (A) $\alpha=0$, (B)
$\alpha=0.1$, (C) $\alpha=0.2$, (D) $\alpha=0.5$, and (E) $\alpha=0.9$. The
first row corresponds to the blue curves in Figure 5, with $\nu_{i}=5Hz$ and
parameters from Table 2. The second row corresponds to the yellow curve on the
same figure, with $w_{I}=10$, $w_{E}=5$, and $\nu_{i}=5Hz$. The legend
indicates different assumptions in the model for each line.

We proceed to test this model against simulation data for different input
rates (figure 5A and 5B) and for different inhibitory weights (figure 5C and
5D). Given the highly nonlinear model, it is remarkable how good is the
agreement between the mean-field results and the simulations for the majority
of the cases. The transition region is where most of the error is
concentrated. We can also observe an interesting behavior in this model. The
use of the continuity assumption can result in smaller errors in some
situations, especially when $\alpha\approx 1$. However, in the high inhibition
regime, the double integration procedure produces a more accurate result, with
the calculated firing rate following close to the almost quiescent simulated
neuron. Transitions are then slightly better estimated with the double
integration procedure, which can be important for phase transition
calculations.

A crucial point has to be made regarding the condition for uniform convergence
(12) that is usually used as a metric for good behavior in approximation
methods [24]. For the linear conductance-based models, it is obeyed for all
values. However, the introduction of the NMDA nonlinearity makes this
condition break for a large range of parameters as can be seen by numerical
calculations of (12). Notably, the failure of the uniform convergence
condition didn’t affect the ability of the model to describe the system. But,
for the method to work correctly it is necessary to deal with the divergence
point at
$1-\tau_{i}\left(W^{\prime}(y)-\frac{h_{i}^{\prime}(y)}{h_{i}(y)}W(y)\right)=0$.
We were able to remove the divergence by excluding from the numeric
integration an interval of $\pm 0.5$ around the value of $V$ that contains the
divergence. No significant impact can be observed in the results when using
this procedure, which indicates that the divergence cancels out in the
integration.

Fig 6 displays the estimated probability distributions for the NMDA model. The
top row (corresponding to the blue curve in Figs. 5A and 5B), indicates that
the estimation is good when the distribution don’t interact heavily with the
threshold but degrades when the interaction is significant. We see,
nevertheless, that the continuity assumption causes a large error at the end
of the left tale of the distribution, which doesn’t occur as much with the
double integration procedure. It still overestimates the probability of values
slightly smaller than the reset potential. The bottom row (corresponding to
the yellow curve in Figs. 5C and 5D) shows that the distribution remains
mostly below the reset potential and little interaction with the threshold
happens. The distribution spreads with higher $\alpha$ but the peak barely
moves. The area of the red curve remains higher than the yellow, which helps
explain the higher firing rate observed in 5C.

##  VI Conclusion

We developed a new method for constructing a transfer function for
conductance-based integrate-and-fire neurons. This method is based on the
mean-field Fokker-Planck approach and incorporates colored noise through the
Fox Theory. We reduced the N-dimensional system into a single Langevin
equation with colored and multiplicative noise and used the Fox Theory to
construct an effective Fokker-Planck equation, which was solved to obtain
stationary firing rates. We tested the method on two neuron models with
progressive complexity: a standard conductance-based integrate-and-fire and a
conductance-based integrate-and-fire with nonlinear NMDA channels.

For the standard conductance-based integrate-and-fire neuron, we compared
mean-field results with firing rate data resulting from simulations as a
function of the excitatory time constant. We found good agreement between the
data and mean-field results in most scenarios, but the continuity assumption
generated substantial errors in the transition regions. To correct that, we
developed a double integration procedure that consists in estimating the
probability density at the threshold with the first integration and using this
value to start a backward integration to better estimate the distribution and
the firing rate. We also observed that the effective time-constant
approximation produced sharper transitions and the double integration
procedure translated the curve to better match the transition point.

We then added a nonlinear NMDA channel to the model to test the method’s
effectiveness for nonlinear multiplicative terms. The method produced a good
description of the simulated data even when outside the range of validity
given by equation (12), but required extra care for some sets of parameters.
Additionally, we discovered that in this particular scenario, the application
of the double integration procedure does not always yield superior outcomes
compared to utilizing the continuity assumption. While it generates a more
accurate estimation in the transition region, it slightly underperforms for
high $\alpha$ values.

When the membrane potential mean-field probability distributions are compared
with simulation data, the effects of the continuity assumption become very
clear. As expected, the distributions observed in the simulations are
discontinuous. In fact, the discontinuity can occur not only at the threshold
but also at the reset potential. Therefore, the continuous mean-field solution
does not correctly estimate the distribution in most cases. The double
integration procedure, however, produces better results, with $P(\theta)$
approximating the simulated one more accurately. We can also see that the
effective time-constant approximation generates only small deviations from the
full treatment, and is generally a good approximation.

From the analysis of the probability distribution, it is clear that the double
integration procedure produces good estimations of the distribution at the
threshold limit for most cases. However, when the neuron is in the mean-driven
regime, the discontinuity at the reset potential is not taken into account. An
appropriate study of the correct boundary conditions of the 1-D reduced
Fokker-Planck equation can probably generate better results at the
discontinuity points. Since the area of the distribution is related to the
firing rate, this would probably improve the transfer function resulting from
the method.

We have only tested our method on two types of neuron models, but it has the
potential to be applied to a wider range of models. One possible extension,
mentioned in the methods section, is the introduction of a nonlinear term in
the drift expression. This nonlinear term can represent a quadratic [35] or an
exponential integrate-and-fire neuron [36], for example. It is also possible
to introduce adaptation currents that depend on the spiking time of the
modeled neuron. This would introduce the firing rate in the resulting Langevin
equation, making a self-consistent treatment required.

It is also possible to look for non-stationary solutions when the input
changes over time. For example, if we introduce an oscillatory Poissonian
input, it is possible to construct a Fokker-Planck equation with the Fox
theory, as the noise terms still have the same form but are now modulated by
the input firing rate. However, only the stationary solutions of Fox theory
are valid for all the ranges of time constants [27]. The non-stationary
solutions are valid in the small $\tau_{i}$ limit, and careful consideration
of this fact is necessary when putting the method into practice.

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# Scrapping The Web For Early Wildfire Detection

Mateo Lostanlen
PyroNear
Paris, France
<EMAIL_ADDRESS>
Felix Veith
PyroNear
Paris, France
<EMAIL_ADDRESS>
Cristian Buc
Centro Nacional de Inteligencia Artificial
Santiago, Chile
<EMAIL_ADDRESS>
Valentin Barriere∗
DCC – Universidad de Chile
Centro Nacional de Inteligencia Artificial
Santiago, Chile
<EMAIL_ADDRESS>
Core Contributors

###### Abstract

Early wildfire detection is of the utmost importance to enable rapid response
efforts, and thus minimize the negative impacts of wildfire spreads. To this
end, we present PyroNear2023 , a web-scraping-based dataset composed of videos
of wildfires from a network of cameras that were enhanced with manual
bounding-box-level annotations. Our dataset was filtered based on a strategy
to improve the quality and diversity of the data, reducing the final data to a
set of 10,000 images. We ran experiments using a state-of-the-art object
detection model and found out that the proposed dataset is challenging and its
use in concordance with other public dataset helps to reach higher results
overall. We will make our code and data publicly available.

##  1 Introduction and Related Work

Wildfires have become an increasingly prevalent and devastating natural
disaster worldwide, causing loss of life, destruction of property, and
significant environmental damage. While wildfires have always been a part of
nature’s cycle, human activities and climate change have exacerbated their
frequency and intensity. As such, there is an urgent need to develop and
implement advanced technologies to prevent wildfires, as well as predicting
their behavior.

Given the undeniable link between climate change and the increasing frequency
of wildfires, the urgency of addressing this issue is underlined by
projections from the Intergovernmental Panel on Climate Change (IPCC), which
anticipate a continued rise in the occurrence of extreme events like
wildfires.111https://www.ipcc.ch/report/ar6/syr/downloads/report/IPCC_AR6_SYR_LongerReport.pdf
The urgency of responding to this crisis is augmented by its ramifications for
sustainable development, the environment, and social equity.

Early wildfire detection is of paramount importance in mitigating the
catastrophic consequences of these increasingly prevalent natural disasters,
driven by climate change-induced warmer temperatures and drier conditions
Reidmiller et al. (2018). Recent advancements in artificial intelligence (AI)
and deep learning (DL) techniques have spurred innovative methodologies for
addressing this critical issue, yielding a profusion of methods and datasets
tailored to diverse aspects of early wildfire detection.

Fernandes et al. (2023) demonstrated the use of EfficientDet for automatic
early detection of wildfire smoke with visible-light cameras, emphasizing the
significance of large and representative datasets for training. Their dataset
consists of 14,125 smoke and 21,203 non-smoke images. Their results achieved a
true detection rate of 80.4% and a false-positive rate of 1.13%, outperforming
previous studies focusing on smoke plumes. Yazdi et al. (2022) introduced
Nemo, an open-source benchmark for fine-grained wildfire smoke detection,
tailored to the early incipient stage. They adapted Facebook’s DEtection
TRansformer (DETR) Carion et al. (2020) to wildfire detection, achieving
superior performance in detecting smoke across different object sizes. Their
model detected 97.9% of fires in the incipient stage, outperforming baseline
methods. Dewangan et al. (2022) propose a model with an associated dataset:
SmokeyNet and Fire Ignition Library (FIgLib). The dataset is publicly
available and composed of 25,000 labeled wildfire smoke images as seen from
fixed-view cameras deployed in Southern California. The proposed model relies
on a novel deep learning architecture using spatiotemporal information from
camera imagery for real-time wildfire smoke detection, which outperformed
comparable baselines and even rivaled human performance, demonstrating the
potential for real-time wildfire smoke detection.

Resource-efficient solutions have been explored to extend the applicability of
wildfire detection systems. de Venâncio et al. (2022) proposed an automatic
fire detection system based on deep CNNs suitable for low-power, resource-
constrained devices, achieving significant reductions in computational cost
and memory consumption while maintaining performance. In the same vein, Khan
and Khan (2022) presented "FFireNet," a deep learning-based forest fire
classification method, utilising a small neural network, the MobileNetV2 model
for feature extraction and achieving remarkable accuracy in binary
classification of fire images.

Satellite imagery has been a pivotal data source for early wildfire detection.
Barmpoutis et al. (2020) offered an overview of optical remote sensing
technologies used in early fire warning systems. They conducted an extensive
survey on flame and smoke detection algorithms employed by various systems,
including terrestrial, airborne, and spaceborne-based systems. This review
contributes to future research projects for the development of early warning
fire systems. James et al. (2023) developed an efficient wildfire detection
system utilizing satellite imagery and optimized convolutional neural networks
(CNNs) for resource-constrained devices, using a MobileNet on an Arduino Nano
33 BLE.

Video-based fire detection techniques have emerged as a promising avenue for
early wildfire detection. Jin et al. (2023) provided a comprehensive review of
deep learning-based video fire detection methods, summarizing recent advances
in fire recognition, fire object detection, and fire segmentation using deep
learning approaches. Their review provided insights into the development
prospects of video-based wildfire detection.

de Venâncio et al. (2023) proposed a hybrid method for fire detection based on
spatial and temporal patterns, combining CNN-based visual pattern analysis
with temporal dynamics to reduce false positives in fire detection.
Additionally, Marjani and Mesgari (2023) introduced "FirePred," a hybrid
multi-temporal CNN model for wildfire spread prediction, emphasizing the
importance of considering varying temporal resolutions in fire prediction
models.

These systems have been applied to support wildfire management decisions. Bot
and Borges (2022) conducted a systematic review of applications of machine
learning techniques for wildfire management decision support. Their emphasis
was on summarizing applications across different case studies, machine
learning methods, case study locations, and performance metrics, highlighting
the potential of machine learning in enhancing fire management decision
support.

Concerning the datasets, at first view many of them can be found in the
literature with a focus on wildfire detection. Nevertheless works such as
Toulouse et al. (2017); Sharma et al. (2017); Foggia et al. (2015) are
actually fire detection datasets, containing pictures of fires at an already
advance stage. In this work, as we mainly focus on smoke plumes in order to
detect early wildfires from watchtowers, we discard the (easier) task of fire
detection. In this context, it is notable to remark that only a very few of
the datasets containing annotations for the smoke plume detection are publicly
available.

In general, there are two main sources of videos for smoke plumes detection in
the wild that are available online: HPWREN (2023) (High Performance Wireless
Research & Education Network) and ALERTWildfire (2023). These two sources were
used to create several datasets. Leveraging the camera network of the HPWREN,
(Dewangan et al., 2022; Govil et al., 2020; AIforMankind, 2023) propose
annotated datasets for early wildfire detection, while other works (Schaetzen
et al., 2020; Yazdi et al., 2022; AIforMankind, 2023) propose datasets
obtained from the ALERTWildfire network. Finally, from private sources and not
publicly available, Fernandes et al. (2022) constructed a dataset of 35k
images from Portugal that are annotated in smoke plumes. It is composed of
14,125 images that contain smoke plumes and 21,203 that do not.

##  2 Datasets Collection, Fusion And Annotation

###  2.1 Available Datasets

In the development of an early wildfire detection model, the assembly of a
comprehensive and diverse dataset is crucial. Given the limited number of
Pyronear cameras currently deployed in the field, our effort to gather data
extends to additional sources. This subsection outlines the primary sources of
data and the derivative datasets that have been instrumental in our research.

##### Primary Data Sources

Our data acquisition strategy leverages two main sources:

  * •

HPWREN: Funded by the National Science Foundation, HPWREN is a non-commercial,
high-performance, wide-area, wireless network of Pan-Tilt-Zoom (PTZ) cameras
serving Southern California. It focuses on network research, including the
demonstration and evaluation of its capabilities in wildfire detection.

  * •

ALERTWildfire: A consortium of universities in the western United States
provides access to advanced PTZ fire cameras and tools, aiding firefighters
and first responders in wildfire management, covering extensive regions
spanning Washington, Oregon, Idaho, California, and Nevada. The ALERTWildfire
website222https://www.alertwildfire.org/ grants public access to live feeds
from these cameras.

##### Derived Datasets

From these sources, several projects have proposed datasets that are of
interest to our wildfire detection initiative:

  * •

SmokeFrames: Developed by Schaetzen et al. (2020) this dataset comprises
nearly 50k images sourced from ALERTWildfire. To tailor it to our specific
requirements, we created a subset, SmokeFrames-2.4k, consisting of 2410 images
from 677 different sequences, with an average of 3.6 images per sequence. This
subset includes a significant number of false positives, essential for a
comprehensive wildfire detection model.

  * •

Nemo: The dataset of Yazdi et al. (2022) includes frames extracted from raw
videos of fires captured by ALERTWildfire’s PTZ cameras, encompassing various
stages of fire and smoke development.

  * •

Fuego: Initiated by the Fuego project (Govil et al., 2020), this dataset was
created by manually selecting and annotating images from the HPWREN camera
network, based on historical fire records from Cal Fire. The authors are
claiming 8500 annotated images with a focus on the early phases of fires, but
only a subset of 1661 images are publicly available.

  * •

AiForMankind: Two training datasets emerged from hackathons organized by AI
For Mankind (AIforMankind, 2023), a nonprofit focusing on using AI for social
good. These datasets, combined into one, offer a substantial collection of
annotated images for smoke detection and segmentation.

  * •

FIgLib: Dewangan et al. (2022) propose the Fire Ignition image Library
(FIgLib) which was composed of 24,800 images from South California from 315
different fires. It is the official dataset from the HPWREN.

###  2.2 Creation of the PyroNear2023 Dataset

This section presents the collection of the data, its annotation using a
homemade platform and a summary of the final dataset.

####  2.2.1 Data Acquisition Strategy

Our wildfire detection initiative utilizes the AlertWildfire camera network,
which comprises approximately 130 cameras. It’s crucial to note that the
actual number of operational cameras fluctuates due to occasional
unavailability, the reasons for which are often unclear. Despite these
variances, we ensure comprehensive monitoring.

The core of our data collection is an automated scraping script that interacts
with the AlertWildfire API. This script retrieves images from each camera at
the predetermined frequency of one image per minute, set by AlertWildfire.
While we would prefer a higher frequency—since our cameras can capture an
image every 30 seconds for analysis—this limitation necessitates that we work
with around 1,440 images per camera per day, summing up to about 187,200
images daily across the network.

Our initial filtering stage targets the elimination of nighttime images, as
our current model is designed primarily for daylight image analysis. Nighttime
images are in grayscale due to the cameras switching to infrared mode, and
these are automatically excluded from our dataset.

After filtering out the nighttime images, we perform inference on the
remaining daylight images using our model. This model analyze each image, and
any with a wildfire detection score above 0.2 is marked as a potential fire
event. To ensure comprehensive coverage of potential fire events, we also save
images taken 15 minutes before and after each detected event from the same
camera. This approach helps in capturing a broader timeline around each
potential wildfire incident.

All the images flagged during this process, including both potential wildfire
detections and corresponding time-framed images, are stored for later
annotation. This rich collection, encompassing potential early signs of
wildfires as well as false positives, offers a diverse dataset. This dataset
is invaluable for enhancing the performance and accuracy of our wildfire
detection models, particularly in distinguishing true wildfires from non-
threatening natural occurrences.

In this way, because the fires all grow to a point where they are easily
detectable, we capture all of the events. This collection methods, which is
already based on a trained model, helps to gather a diverse range of images,
from clear instances of wildfires to challenging scenarios that have
historically led to a false detection.

####  2.2.2 Collaborative Annotation Platform

In order to annotate the wildfire data scrapped from the web, we developed a
collaborative annotation tool with custom code in order to streamline the
annotation process. We collected a total of 120,000 annotations in a few month
by leveraging the help of the PyroNear community. It had to answer to a few
constraint, especially the one that as the annotators were all volunteers
using their free time to help developing an open-source dataset and model. We
gave to the volunteers a precise quantity of images, which number was selected
as 150 so that the annotation task would take less than 15 minutes so that it
can be done during a train commute or a break between two activities, and it
kept the cognitive load low in order to avoid mistakes and care the
annotators. Finally, the platform have also been designed to ensure a smooth
and coherent workflow. A snapshot of the platform is visible in Figure 1.

Figure 1: Snapshot of the Computer Vision Annotation Platform for Smoke Plume
Detection.

Each of the images has been annotated by five annotators in order to minimize
the label errors. To validate the quality of the annotationo, we calculated
the inter-annotator agreement using Krippendorff’s $\alpha$ Krippendorff
(2013) with the presence or not of fire in each image, and obtained satisfying
value.

####  2.2.3 Final Dataset: PyroNear2023

The creation of the PyroNear2023 dataset is a carefully orchestrated process,
tailored to suit our current focus on one-image object detection. Initially,
we started with an extensive collection of 120,000 labeled images. With a
5-times cross-labeling approach, this pool was refined down to 24,000 unique
images.

Given our emphasis on one-image object detection, it was crucial to streamline
the dataset to reduce redundancy and enhance model performance. Our reduction
process involved two critical steps:

Selective Removal of Background Images: To maintain a balanced dataset, we
strategically reduced the number of background images. This step ensured that
our dataset did not disproportionately favor non-wildfire scenarios. We aimed
to keep around 15% of background images to achieve a balanced representation.
After this filtering, the dataset was condensed to roughly 10,000 images.

Subsampling to Eliminate Redundancy: Acknowledging the potential for
redundancy, especially in images sourced from video sequences, we implemented
a subsampling strategy. By retaining only one image every 10 minutes, we
effectively minimized repetitive or near-identical images. This approach was
crucial in preserving the diversity of the dataset while ensuring its
relevancy to our one-image object detection focus. The dataset was thus
further refined to 1,096 images, including 951 smoke images.

The final composition of the PyroNear2023 dataset is:

Training Set: 987 images, including 836 smoke images. Validation Set: 109
images, with 88 smoke images. While the current version of the PyroNear2023
dataset is optimized for one-image object detection, our future plans include
the utilization of the entire collection of images. We intend to develop a
temporal model that leverages a series of images for prediction, thus
enhancing the accuracy and robustness of wildfire detection.

##  3 Experiments and Results

In this study, our primary objective is to evaluate the quality of various
datasets by conducting a preliminary optimization process.

###  3.1 Methodology

We use the YOLOv8 model (MMYOLO, 2023), renowned for its proficiency in
diverse detection scenarios. For frugality reasons because of the type of our
task, the small version of the model was chosen for its balance between speed,
size and accuracy. The optimal batch size and number of epochs were found
using a grid search in $\\{50,100\\}$ and $\\{2^{k},k=4...6\\}$.

##### Dataset Splitting Strategy

In preparing our datasets for the model training and validation process, we
were guided by the existing split in the Nemo dataset, where approximately
9.3% of the data was allocated for validation. To maintain consistency across
all datasets and ensure a comparable evaluation framework, we adopted a
similar approach for the other datasets, targeting a close approximation of a
10% split for the validation set. This strategy enables a balanced and uniform
methodology for assessing the performance of our models across different
datasets, ensuring that each dataset is represented fairly in both training
and validation phases.

Dataset | Total Images | Wildfire Images | Train∗ | Validation∗
---|---|---|---|---
AiForMankind | 2935 | 2584 | 2642 / 2305 | 293 / 279
Fuego | 1661 | 1572 | 1495 / 1421 | 166 / 151
Nemo | 2691 | 2570 | 2440 / 2333 | 251 / 237
SmokeFrames-2.4k | 2410 | 976 | 2169 / 906 | 241 / 70
Table 1: Summary of Datasets: Total and Wildfire Images in Training and
Validation Sets. In the columns with ∗ are shown Total/Wildfire images.

##### Metrics

Following past works (Schaetzen et al., 2020; Yazdi et al., 2022; Dewangan et
al., 2022) we use precision, recall, and the F1 score as metrics in order to
validate the different models. We chose not to use the usual object detection
metric such as mean average precision (mAP) as The goal is about correctly
classifying areas in an image as indicating a wildfire or not, wihtout being
able to get the countours of the smoke plumes which can be subjective.

##  4 Results

###  4.1 Single Dataset Evaluation

Dataset | Best Model | Highest F1 Score | Confidence | Batch Size | Epochs
---|---|---|---|---|---
SmokeFrames | SmokeFrames3 | 0.920 | 0.15 | 32 | 50
Nemo | Nemo3 | 0.899 | 0.02 | 64 | 100
AiForMankind | Aiformankind2 | 0.883 | 0.03 | 32 | 50
PyroNear2023 | Wildfire3 | 0.793 | 0.04 | 16 | 100
Fuego | Fuego | 0.623 | 0.02 | 32 | 100
Table 2: Best Performing Model for Each Dataset Sorted by Highest F1 Score
with Corresponding Confidence Threshold, Batch Size, and Epochs. Best models
depends on the hyperparameters.

###  4.2 Cross-Dataset Model Evaluation

Having trained several models on each of our diverse datasets, the next
critical phase of our study involves a rigorous cross-dataset evaluation. This
process is pivotal in determining not only the versatility and robustness of
our models but also their applicability in a wide range of real-world
scenarios.

###  4.3 Selection of Optimal Models and Confidence Thresholds

Our approach begins with the careful selection of the best-performing model
from each dataset. The criterion for this selection is based on a F1 score,
ensuring that each chosen model demonstrates the highest level of accuracy and
reliability within its training domain. Alongside this, we also identify the
optimal confidence threshold. The performances of the model trained on a
combined dataset on the different test sets are shown in Table 4.

Train Dataset | Test Dataset | Precision | Recall | F1 Score
---|---|---|---|---
PyroNear2023 | AiForMankind | 0.4493 | 0.2222 | 0.2974
Fuego | 0.3750 | 0.0397 | 0.0719
Nemo | 0.7386 | 0.7236 | 0.7310
SmokeFrames-2.4k | 0.3056 | 0.9167 | 0.4583
PyroNear2023 | 0.7604 | 0.8295 | 0.7935
Nemo | AiForMankind | 0.9402 | 0.6201 | 0.7473
Fuego | 0.3889 | 0.2781 | 0.3243
Nemo | 0.8528 | 0.9496 | 0.8986
SmokeFrames-2.4k | 0.3503 | 0.9718 | 0.5149
PyroNear2023 | 0.6364 | 0.6292 | 0.6328
AiForMankind | AiForMankind | 0.9186 | 0.8495 | 0.8827
Fuego | 0.7647 | 0.7697 | 0.7672
Nemo | 0.7812 | 0.3151 | 0.4491
SmokeFrames-2.4k | 0.4356 | 0.6111 | 0.5087
PyroNear2023 | 0.7937 | 0.5682 | 0.6623
SmokeFrames | AiForMankind | 0.9750 | 0.4194 | 0.5865
Fuego | 0.5000 | 0.0066 | 0.0131
Nemo | 0.8737 | 0.7004 | 0.7775
SmokeFrames-2.4k | 0.9545 | 0.8873 | 0.9197
PyroNear2023 | 0.5217 | 0.1364 | 0.2162
Fuego | AiForMankind | 0.6370 | 0.6307 | 0.6338
Fuego | 0.8352 | 0.4967 | 0.6230
Nemo | 0.4719 | 0.1743 | 0.2545
SmokeFrames-2.4k | 0.2833 | 0.2329 | 0.2556
PyroNear2023 | 0.4762 | 0.1136 | 0.1835
Table 3: Performance of the best models across different datasets Tested Dataset | Precision | Recall | F1 Score
---|---|---|---
AiForMankind | 0.9565 | 0.9462 | 0.9514
Fuego | 0.9083 | 0.7171 | 0.8015
Nemo | 0.8750 | 0.9156 | 0.8948
SmokeFrames-2.4k | 0.6106 | 0.9718 | 0.7500
PyroNear2023 | 0.7660 | 0.8182 | 0.7912
Combine | 0.8590 | 0.8839 | 0.8713
Table 4: Performance of the ’Combine3’ Model Across Different Datasets at
Confidence Threshold 0.05

##  5 Future Enhancements: Towards PyroNear2024

While the cross-labeling approach used for PyroNear2023 has significantly
contributed to the accuracy of our dataset, it has also led to a substantial
reduction in the number of images we could include. Acknowledging this
limitation, we are currently developing a new methodology for the upcoming
PyroNear2024 dataset, which aims to semi-annotation process.

##### Faster Annotation

We are exploring semi-automatic annotation techniques that will accelerate the
labeling process while maintaining high-quality annotations. By integrating
advanced algorithms with manual oversight, we can swiftly annotate large
volumes of images without compromising on accuracy.

##### Normalization of Annotations

The semi-automatic approach also aims to standardize the annotation process
across different users. This consistency is crucial for ensuring that the
dataset reflects a uniform understanding of wildfire and smoke
characteristics.

##### Reduced Cross-Labeling

With the improved efficiency and consistency brought by semi-automatic
annotation, we anticipate the need for cross-labeling to decrease
significantly. This reduction will enable us to retain a larger portion of the
images initially collected, thereby enriching the PyroNear2024 dataset with a
broader range of data.

These advancements are expected to not only enhance the volume of annotated
data but also to improve the overall quality and representativeness of the
PyroNear2024 dataset. This progression illustrates our commitment to
continuously refining our methodologies in response to the evolving challenges
of wildfire detection and monitoring.

##  6 Conclusion

In this paper we presented PyroNear2023 , a new dataset for smoke plume
detection. We collected it by scrapping online data and an already trained
model in order to get the most challengin examples. We kept the images before
and after every fire event in order to make it usable by sequential models
taking into account the temporality. The dataset was then re-annotated by a
pool of volunteers using an online platform designed for the purpose. We
showed that training using our dataset helps to improve smoke plume detection
models in other public datasets. This data collection and annotation effort
will be pursued in order to extend this dataset to other domains, such as new
landscape and meteorological conditions, and it will be online soon for
research and non-profit purposes.

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##  7 Supplementary Material

###  7.1 Results with the different hyperparameters

Following our comprehensive grid search and optimization process, we present
the detailed results for each dataset in Tables 5 to 10. These tables
encapsulate the outcomes of our hyperparameter tuning, showcasing the optimal
settings that yielded the highest F1 scores for each dataset. The results are
indicative of each dataset’s unique characteristics and the efficacy of the
YOLOv8s model under varying training conditions.

In addition to the tabular data, we also provide F1 score curves for each
dataset, corresponding to the tables. These curves, shown below each table,
visually represent the performance of the models across different confidence
thresholds. The inclusion of F1 curves offers an intuitive understanding of
the model’s classification performance, highlighting the trade-offs between
precision and recall at various thresholds. This graphical representation
complements the tabulated results, providing a more holistic view of the
model’s capabilities in detecting potential wildfire indicators within each
dataset.

Model | Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
Wildfire3 | 0.04 | 0.793 | 0.830 | 0.760 | 16 | 100
Wildfire4 | 0.03 | 0.771 | 0.784 | 0.758 | 16 | 50
Wildfire | 0.08 | 0.764 | 0.682 | 0.870 | 64 | 50
Wildfire6 | 0.03 | 0.747 | 0.705 | 0.795 | 64 | 50
Wildfire2 | 0.09 | 0.759 | 0.682 | 0.857 | 32 | 50
Wildfire5 | 0.18 | 0.726 | 0.602 | 0.914 | 64 | 100
Table 5: Optimal Thresholds and Corresponding F1 Scores, Recall, and Precision for Wildfire Models, Sorted by F1 Score Figure 2: F1 Score by Model at Various Thresholds Model | Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
Nemo3 | 0.02 | 0.899 | 0.950 | 0.853 | 64 | 100
Nemo | 0.03 | 0.893 | 0.929 | 0.860 | 32 | 100
Nemo4 | 0.05 | 0.886 | 0.899 | 0.873 | 64 | 50
Nemo6 | 0.06 | 0.880 | 0.882 | 0.878 | 32 | 50
Nemo2 | 0.17 | 0.883 | 0.873 | 0.892 | 16 | 100
Nemo5 | 0.03 | 0.877 | 0.912 | 0.845 | 64 | 50
Table 6: Optimal Thresholds and Corresponding F1 Scores, Recall, and Precision for Nemo Models, Sorted by F1 Score Figure 3: F1 Score by Model at Various Thresholds Model | Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
Fuego | 0.02 | 0.623 | 0.497 | 0.835 | 32 | 100
Fuego4 | 0.01 | 0.502 | 0.856 | 0.355 | 64 | 50
Fuego5 | 0.01 | 0.458 | 0.574 | 0.380 | 16 | 50
Fuego3 | 0.02 | 0.459 | 0.331 | 0.746 | 64 | 100
Fuego2 | 0.01 | 0.476 | 0.743 | 0.350 | 32 | 50
Fuego6 | 0.04 | 0.337 | 0.270 | 0.451 | 16 | 100
Table 7: Optimal Thresholds and Corresponding F1 Scores, Recall, and Precision for Fuego Models, Sorted by F1 Score Figure 4: F1 Score by Model at Various Thresholds Model | Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
Aiformankind2 | 0.03 | 0.883 | 0.849 | 0.919 | 32 | 50
Aiformankind3 | 0.03 | 0.878 | 0.854 | 0.902 | 16 | 50
Aiformankind5 | 0.05 | 0.868 | 0.810 | 0.934 | 32 | 100
Aiformankind | 0.06 | 0.827 | 0.753 | 0.917 | 64 | 50
Aiformankind6 | 0.01 | 0.835 | 0.864 | 0.809 | 64 | 100
Aiformankind4 | 0.03 | 0.843 | 0.875 | 0.813 | 16 | 100
Table 8: Optimal Thresholds and Corresponding F1 Scores, Recall, and Precision for Aiformankind Models, Sorted by F1 Score Figure 5: F1 Score by Model at Various Thresholds Model | Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
SmokeFrames3 | 0.15 | 0.920 | 0.887 | 0.955 | 32 | 50
SmokeFrames | 0.1 | 0.901 | 0.901 | 0.901 | 16 | 50
SmokeFrames6 | 0.15 | 0.892 | 0.873 | 0.912 | 64 | 100
SmokeFrames5 | 0.13 | 0.887 | 0.887 | 0.887 | 32 | 100
SmokeFrames4 | 0.07 | 0.878 | 0.915 | 0.844 | 64 | 50
SmokeFrames2 | 0.19 | 0.870 | 0.845 | 0.896 | 16 | 100
Table 9: Optimal Thresholds and Corresponding F1 Scores, Recall, and Precision for SmokeFrames Models, Sorted by F1 Score Figure 6: F1 Score by Model at Various Thresholds Model | Best Threshold | F1 Score | Recall | Precision | Batch Size | Epochs
---|---|---|---|---|---|---
Combine3 | 0.05 | 0.871 | 0.884 | 0.859 | 32 | 100
Combine | 0.05 | 0.867 | 0.877 | 0.857 | 64 | 100
Combine2 | 0.05 | 0.867 | 0.877 | 0.857 | 16 | 100
Combine4 | 0.04 | 0.861 | 0.865 | 0.856 | 16 | 50
Combine6 | 0.04 | 0.855 | 0.835 | 0.876 | 64 | 50
Combine5 | 0.11 | 0.851 | 0.807 | 0.901 | 32 | 50
Table 10: Performance Metrics of Combine Models Sorted by F1 Score with
Corresponding Batch Size and Epochs