am@i3s.unice.fr
Résumé
This paper aims to build an image coding system based on the model of the mammalian retina. The retina is the light-sensitive layer of tissue located on the inner coat of the eye **and** it is responsible for vision. Inspired by the way the retina handles **and** compresses the visual information **and** based on previous studies we aim to build **and** analy- tically study a retinal-inspired image quantizer, based on the **Leaky** **Integrate**-**and**-**Fire** (LIF) model, a neural mo- del according to which function the ganglion cells of the Ganglionic retinal layer that is responsible for the visual data compression. In order to have a more concrete view of the encoder’s behavior, in our experiments, we make use of the spatiotemporal decomposition layers provided by ex- tensive previous studies on a previous retinal layer, the Ou- ter Plexiform Layer (OPL). The decomposition layers pro- duced by the OPL, are being encoded using our LIF image encoder **and** then, they are reconstructed to observe the en- coder’s efficiency.

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am@i3s.unice.fr
R´esum´e – Dans cet article, nous pr´esentons un quantificateur bas´e sur le mod`ele neuronal **Leaky**-**Integrate** **and** **Fire** (LIF). Le LIF est un mod`ele simplifi´e du fonctionnement des cellules ganglionnaires. Les cellules ganglionnaires sont plac´ees dans la couche de la r´etine responsable du codage de l’information visuelle, avant qu’elle ne soit transmise au cerveau `a travers le nerf optique. En g´en´eral, le LIF quantifie les valeurs d’intensit´e selon une valeur seuil, un temps d’observation donn´e, la pr´esence ou non d’une p´eriode r´efractaire dans le neurone et les param`etres R et C caract´erisant la r´esistance et la capacit´e du mod`ele neuronal. En variant la valeur du seuil et du temps d’observation, nous avons test´e exp´erimentalement le comportement du quantificateur LIF `a un signal d’entr´ee donn´e et nous pr´esentons les r´esultats et la comparaison avec le quantificateur scalaire uniforme et le quantificateur Lloyd.

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Figure 1: Simulation of the LIF model. A) Time evolution of the membrane potential. B) The panel illus- trates the arrival times of impulses, so-called Poisson spike train. The red dots correspond to discontinuities induced by the jump process. The parameters are: h = 0.2, v r = 0.1 **and** Poisson rate 100.
neuroscience. We focus our investigation on the existence **and** properties of the steady state measure of a PDE that arises for the description of an excitatory network of **leaky** **integrate**-**and**-**fire** (LIF) neurons. The LIF model is a well-established neuron model within the neuroscience community [ 43 ]. It consists of an ordinary differential equation that describes the subthreshold dynamics of a neuron membrane’s potential. The equation is endowed with a discontinuous reset mechanism to account for the onset of an action potential. Whenever the membrane potential reaches the firing threshold, the neuron initiates an action potential **and** the membrane potential is reset, see [ 10 ] for a review **and** [ 1 , 9 ] for historical consideration. In its normalized form, the LIF model reads

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Mathematical studies of synchronization have been carried out first in deterministic neural models. For instance, in Mirollo **and** Strogatz ( 1990 ), the authors prove that in fully-connected **and** totally excitatory **leaky** **integrate** **and** **fire** neural network models the synchronization occurs for almost all initial state. When weak interactions are as- sumed, a large class of models can be reduced to canonical systems of phase coupled oscillators ( Hoppensteadt **and** Izhikevich , 1997 ; Izhikevich , 1999a ). This formalism allows to study the existence **and** the stability of synchronized solutions in weakly cou- pled general networks ( Izhikevich , 1999b ), including networks containing inhibitory synapses **and** considering synaptic delays ( Van Vreeswijk , 1996 ; Van Vreeswijk et al. ,

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the encoding of the visual data. The Ganglionic layer consists of the ganglion cells, a type of neuron that compresses visual information according to the **Leaky** **Integrate**-**and**-**Fire** (LIF) neural model which encodes intensity values into spikes. Un- der the main belief that nature performs in an optimal way, **and** based on previous works on the OPL filtering in [1], we built a quantization system making use of the LIF properties to compress images already filtered by the OPL. Unlike the already existing static encoding algorithms, this quantization scheme encodes images in a dynamic way **and** then using an inverse function the encoded information provides an estima- tion of the initial image.

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Index Terms—Neuro-inspired quantization, uniform scalar quantizer, **Leaky** **Integrate**-**and**-**Fire** (LIF), spikes, entropy.
I. I NTRODUCTION
Over the last few years, a lot of efforts have been devoted to the visual perception. While these efforts were initially focused on the understanding of the visual system as a quality assessment metric that perceives the visual stimulus, their scope has been widened today trying to mimic the processing mechanisms of the visual system in order to build bio- inspired coding/decoding algorithms. These new algorithms are expected to provide qualitative results which are pleasant to the human eyes. In addition, the characteristic properties of the visual system such as the dynamic processing of the visual stimulus, its plasticity **and** its ability to generate a very sparse but informative code of spikes [1] seems to be beneficial to the progress of the state-of-the-art compression algorithms which are currently based on computationally greedy mechanisms like motion estimation [2].

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1 Introduction
The use of simple models proves often very helpful to identify **and** characterize the mechanisms underlying general dynamical phenomena. Computational neuroscience is a field where this approach is potentially very powerful, given the myriad of inter- actions involved in the functioning of the mammalian brain [ 1 , 2 ]. However, setting the appropriate level of simplicity is not a priori obvious. A particularly enlightening example is the reproduction of the background neural activity. Most of the numerical **and** theoretical studies are based on the so-called rate models, where each neuron is characterized by a single coarse-grained variable representing the strength of the ongoing activity [ 3 , 4 ]. However, it is well known that neurons work by emitting single spikes, so that it is more natural to represent them as (nonlinear) oscillators. This is, indeed the philosophy adopted by many studies based on pulse coupled units, such as **leaky** **integrate**-**and**-**fire** (LIF) neurons [ 5 ]. Accordingly, a general question arises as to whether the two approaches are consistent with one another **and**, in particular, to what extent spiking neurons reproduce the scenario observed in rate models [ 6 , 7 ]. In this paper we look at the evolution of the so-called balanced networks, where excitatory **and** inhibitory interactions compensate each other [ 8 ], since the single- neuron dynamics is rather irregular **and** reminiscent of the background neural activity.

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Abstract—This paper introduces a novel coding/decoding mechanism that mimics one of the most important properties of the human visual system: its ability to enhance the visual perception quality in time. In other words, the brain takes advantage of time to process **and** clarify the details of the visual scene. This characteristic is yet to be considered by the state-of-the-art quantization mechanisms that process the visual information regardless the duration of time it appears in the visual scene. We propose a compression architecture built of neuroscience models; it first uses the **leaky** **integrate**- **and**-**fire** (LIF) model to transform the visual stimulus into a spike train **and** then it combines two different kinds of spike interpretation mechanisms (SIM), the time-SIM **and** the rate- SIM for the encoding of the spike train. The time-SIM allows a high quality interpretation of the neural code **and** the rate-SIM allows a simple decoding mechanism by counting the spikes. For that reason, the proposed mechanisms is called Dual-SIM quantizer (Dual-SIMQ). We show that (i) the time-dependency of Dual-SIMQ automatically controls the reconstruction accuracy of the visual stimulus, (ii) the numerical comparison of Dual- SIMQ to the state-of-the-art shows that the performance of the proposed algorithm is similar to the uniform quantization schema while it approximates the optimal behavior of the non-uniform quantization schema **and** (iii) from the perceptual point of view the reconstruction quality using the Dual-SIMQ is higher than the state-of-the-art.

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quences of rigorous **and** exact results obtained in [ 1 ], char- acterizing the statistics of spike trains in a network of **leaky** **Integrate**-**and**-**Fire** neurons, where time is discrete **and** where neurons are subject to noise, without restriction on the synaptic weights connectivity. The main result is that spike trains statistics are characterized by a Gibbs dis- tribution, whose potential is explicitly computable. This es- tablishes, on one hand, a rigorous ground for the current in- vestigations attempting to characterize real spike trains data with Gibbs distributions, such as the Ising-like distribution [ 2 ], using the maximal entropy principle. However, it tran- spires from the present analysis that the Ising model might be a rather weak approximation. Indeed, the Gibbs poten- tial (the formal “Hamiltonian”) is the log of the so-called “conditional intensity” (the probability that a neuron fires given the past of the whole network [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ]). But, in the present example, this probability has an infinite memory, **and** the corresponding process is non-Markovian (resp. the Gibbs potential has infinite range). Moreover, causality implies that the conditional intensity does not de- pend on the state of the neurons at the same time, ruling out the Ising model as a candidate for an exact characterization of spike trains statistics. However, Markovian approxima- tions can be proposed whose degree of approximation can be rigorously controlled. In this setting, Ising model ap- pears as the “next step” after the Bernoulli model (inde- pendent neurons) since it introduces spatial pairwise cor- relations, but not time correlations. The range of validity of this approximation is discussed together with possible approaches allowing to introduce time correlations, with algorithmic extensions.

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Key words: **Integrate** **and** **fire** models; noise; neural networks; blow-up; spontaneous activity; Mathematics Subject Classification: 35K60, 82C31, 92B20
1 Introduction
The Network Noisy **Leaky** **Integrate** **and** **Fire** (NNLIF in short) model is certainly one of the simplest self-contained mean field equation for neural networks. It describes, at time t, the probability p(v, t) to find a neuron at a voltage v, assuming each individual neuron follows a simple **integrate** **and** **fire** dynamics **and** the coupling changes the current. **Integrate** **and** **Fire** models,for a single neuron or a population of neurons, with or without noise, have been used very widely [1, 4, 20, 23, 14, 15], compared to experimental data [2, 21] **and** qualitative properties have been studied [3, 4, 22, 10]. Many references can be found in surveys **and** books, see [11, 24, 12] among others. However its mathematical structure is still poorly understood **and** very few results are available concerning its solutions. For instance, very recent results are large time existence for the inhibitory case [7], short time existence of smooth solutions for the excitatory case [8] **and** global existence for the model when firing neurons induce finite jumps [9]. Another striking mathematical property is that for fully excitatory networks, the system blows-up in finite time; this holds for any initial data for a large enough network connectivity **and** for any connectivity

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′′
2 h tends towards infinity
when h → +∞ then relation (4) can be verified only if r 23 has a pole (r 2 1 cannot have
one since the field is always propagative in the upper medium). This means that the structure may support a **leaky** mode only if the interface between medium 2 **and** 3 can support a guided mode. It is now well-known that such an interface actually supports a surface mode [17,18] which can, depending on media 2 **and** 3, be backward (resp. forward) corresponding to a pole under the real axis (resp. above the real axis but on the other Riemann sheet). The **leaky** wave always has the same propagation direction as the surface mode, whatever the thickness of the slab, as shown figure 4. In the case of a forward **leaky** wave, only the zero belongs to the first Riemann sheet, just under the real axis. The pole shown figure 4 belongs to the other Riemann sheet.

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Abstract—We present a Bessel-beam launcher based on a **leaky** radial waveguide consisting of a capacitive sheet over a ground plane that supports higher-order **leaky** modes. A propagating Bessel beam is generated above the radiating waveguide. The Bessel beam is Transverse Magnetic (TM) polarized with a vertical component of electric field that is a zeroth-order Bessel function of the first kind. A higher-order **leaky**-wave mode is used to reduce losses at millimeter waves **and**, at the same time, avoid the thin dielectric layers used in previously proposed lower-order **leaky**-wave Bessel launchers. Closed-form design equations are provided for the proposed structure. In addition, the operating bandwidth of the launcher is defined using dispersion analysis. Near-field measurements of a prototype operating in the fre- quency range 38-39.5 GHz validate the concept. The measured launcher generates a Bessel beam with a stable spot size of about 4.3 mm (0.57 λ) over a non-diffractive range of about 16.4 mm (2.2 λ), within about a 4% fractional bandwidth.

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7.1.2 Open Source Packages **and** Data
All packages, frameworks, **and** data in this prototype are open source. While this decision makes sense so that the prototype can be used by anyone **and** be easily extended, this also caused several problems. Primarily, finding the correct packages that actually work proved to be very difficult. The Leaflet packages **and** extensions were particularly troublesome. Leaflet itself is very well maintained **and** has very readable documentation that is kept up to date. However, there are a surplus of third party Leaflet extensions, which are all separately maintained to various degrees. For instance, I tested four different marker customizing packages before finding one that functions correctly. I had this problem multiple times when developing this prototype, but was able to either find or create a functioning extension in each case, so future developers will not have to go through the same process.

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The Reactive Model-based Programming Language (RMPL) (Ingham, Ragno, **and** Williams 2001) is an object- oriented language that allows a domain to be structured through an object hierarchy with subclasses **and** multiple in- heritance. It combines a system model with a control model, using state-based, procedural control **and** temporal represen- tations. The system model specifies nominal as well as fail- ure state transitions with hierarchical constraints. The con- trol model uses standard reactive programming constructs. RMPL programs are transformed into Temporal Plan Net- works (TPN)(Williams **and** Abramson 2001), an extension of Simple Temporal Networks with symbolic constraints **and** decision nodes. Temporal reasoning consists in finding a path, i.e., a plan, in the TPN that meets the constraints. The execution of generated plans allows for online choices (Conrad, Shah, **and** Williams 2009). TPNs are extended with error recovery, temporal flexibility, **and** conditional execu- tion based on the state of the world (Effinger, Williams, **and** Hofmann 2010). Primitive tasks are specified with distri- butions of their likely durations. A probabilistic sampling algorithm finds an execution guaranteed to succeed with a given probability. Probabilistic TPN are introduced in (San- tana **and** Williams 2014) with the notions of weak **and** strong consistency. (Levine **and** Williams 2014) add the notion of uncertainty to TPNs for contingent decisions taken by the environment or another agent. The acting system adapts the execution to observations **and** predictions based on the plan. RMPL **and** subsequent developments have been illustrated

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For the publisher’s version, please access the DOI link below./ Pour consulter la version de l’éditeur, utilisez le lien DOI ci-dessous.
https://doi.org/10.4224/40001681
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mi sono oggi così chiari grazie all’accuratezza delle tue spiegazioni. Grazie per avermi fatto fatto acquisire padronanza di certe tematiche.
Thank you David R. Jackson. After spending four months in Houston, I have understood the reason why Guido defined it human-unfriendly! It is even more true that I do not think it does exist in the world any other place where one can hope to learn something about **leaky** waves, because of you. Since the very first day I was really impressed by your intuition about physical phenomena. However, what I will really miss is our Running Wednesdays/Saturdays. Thank you for always treating me as a friend rather than as a student.

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with the large scale development of CCS have so far been researched in a few studies (ibidem, p. 306). To our knowledge theoretical studies are even fewer.
The objective of this paper is to try to elucidate some theoretical features of optimal CCS policies with **leaky** reservoirs **and** specically the dynamics of the shadow cost of both carbon stocks **and** their relation with the mining rent of the nonrenewable resource, determining the long run relative competitiveness of coal **and** solar energies. The paper has to be seen as mainly exploratory. To conduct the inquiry we adopt the most simple model permitting to isolate the dynamics of captured CO 2 , leakage **and** atmospheric pollution.

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2.3. Experiment 1
Each block started with a written instruction displayed on the screen. In Experiment 1, participants were instructed about the volatility of the environment ( Fig. 1 ). Participants received the instruction: “The rules in this block will probably change” (volati- lity instruction) in half of the blocks, **and** the instruction “The rules in this block will probably remain stable ” (stability instruction) in the other half. Rule reversals occurred in 2/3 of the volatility-in- struction blocks **and** 1/3 of the stability-instruction blocks, with these probabilities made explicit to the subjects. The use of probabilistic instructions ensured that participants had to pay at- tention to the feedback **and** be engaged with the task regardless which instruction they had received. It also allowed us to measure the behavioural effects of instructions on adaptation. Because there was at most one rule reversal per block, we were able to measure the effects of instructions over a large number of trials, i.e., all trials that preceded the rule reversal. For all blocks in the experiment, pre-rule reversal trials differ in no parameter other than instruction. In each trial, participants had to press one of two keys ( ‘f’ **and** ‘h’ on a standard keyboard) with their left or right index ﬁnger in response to the image of a familiar object on the screen ( Fig. 1 , for a detailed description). The images were scaled so that they did not exceed 150 pixels in either width or height. There were two objects in each block, **and** new objects appeared in each block. A left-hand keypress was the initially correct response for one of the objects, **and** a right-hand keypress was the correct response for the other. Participants could only determine this in- itial mapping using feedback in a trial-**and**-error approach. Feed- back contingencies were probabilistic, speci ﬁcally being con- tingent on the correctness of the response in 75% of all trials: If participants implemented the correct mapping, they received positive feedback (a green smiley) in 75% of the trials **and** negative feedback (a red sad face) in 25% of the trials. For incorrect re- sponses, participants received negative feedback in 75% of the trials **and** positive feedback in 25% of the trials. Failures to respond within a time limit of 2000 ms from stimulus onset were followed by a white, crossed-out face. Participants were told about the probabilistic feedback **and** knew that they had to **integrate** feed- back over a number of trials to learn the correct mapping **and** to detect rule reversals.

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respectively. In the present work, we carry out the analysis of the MLC fibre using the finite element method (FEM), which is considered as one of the most powerful **and** versatile methods in many branches of engineering to solve complicated problems. General equations can be easily solved in an approximate manner by using FEM as numerical tool. In the finite element approach, the problem domain can be suitably divided into several triangles of different shapes **and** sizes. This makes the FEM preferable when compared with the finite difference method (FDM) which not only uses inefficient regular spaced meshing but also cannot represent curved dielectric interfaces adequately. Introduction of perfectly matched layer (PML) to the FEM can also enable the calculation of leakage loss of the fibre. In this paper we have carried out FEM analysis of an MLC fibre such as that proposed in [6] **and** have compared our results with those obtained by the TMM. An excellent agreement on effective indices **and** the leakage losses has been obtained between both methods. We have also studied the bending loss of the fibre by employing PML **and** have found that the fibre suffers from small bending loss (0.006 dB/m) at the operating wavelength (1.55 µ m) for the bending radius 10 cm. The analysis has also been extended to a small-elliptical-core MLC fibre **and** the influence of the MLC on the birefringence **and** bending loss of the fibre has been investigated. We have shown that the introduction of the MLC to an otherwise standard elliptical-core fibre can reduce the birefringence. The bending loss of such a fibre is sensitive to polarization **and** the sensitivity can be controlled by the cladding profile. The study should be useful in the designs of MLC fibres for high power optical amplifiers, high power lasers, gain-equalization of optical amplifiers **and** dispersion compensation.

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Dr. M.A . S ulta n is a sen ior research officer in th e **Fire** Risk Man agem en t Program of IRC.
1. Th e p roject w as su p p orted by a con sortiu m th at in clu d ed Can ad a Mortgage an d Hou sin g Corp oration (CMHC); Forin tek Can ad a Corp oration ; Gyp su m Man u factu rers of Can ad a (GMC); In stitu te for Research in Con stru ction , Nation al Research Cou n cil Can ad a; New Hom e Warran ty Program s of On tario, Alberta, British Colu m bia an d Yu kon ; On tario Min istry of Hou sin g; Ow en s Corn in g Fiberglas Can ad a In c.; an d Roxu l In c. In d ivid u als from th ese organ i- zation s form ed a steerin g com m ittee.

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