Je Remercie

Pr Nathalie Rolland

Dr Cécile Delacour

François Danneville

Kévin Carpentier

Ilias Sourikopoulos

Laurent Clavier

Dimitri

Vanessa Avramovic

Sylvie Lepilliet

Sophie Eliet

Etienne Okada

Bernard Verbeke

Ezgi Guillaume

Astrid Sara

Riad

Kevin

Fuanki Laura Louis

Anne-Sophie Aymeric

j'ai eu le plaisir de travailler et profiter de leurs expériences dans une ambiance chaleureuse

Ce travail a été effectué au sein des deux laboratoires CNRS, l'Institut d'Électronique de Microélectronique et de Nanotechnologie (IEMN) et l'Institut de de Recherche sur les Composants logiciels et matériels pour l'Information et la Communication Avancée (IRCICA).

Je souhaite tout d'abord remercier et exprimer ma profonde reconnaissance à mon directeur de thèse. Pr. Alain Cappy, Professeur émérite à l'Université de Lille, pour son expertise reconnue mondialement, sa disponibilité sans faille, sa rigueur scientifique, sa passion communicative, sa thématique de recherche et son excellente pédagogie qui m'ont permis d'évoluer scientifiquement et humainement. Je tiens à le remercier de nouveau pour ces quatre années passées à ses côtés.

Je souhaite remercier Pr. Virginie Hoel, Professeur à l'Université de Lille, ma codirectrice de thèse qui m'a permis de rejoindre en 2014 la thématique dirigée par Pr.Cappy, intitulée bio-inspirée et qui en 2018 est devenue le groupe « Spine ». Ayant confiance en mes capacités scientifiques lors de mon stage de Master 1, elle m'a demandé de suivre plusieurs projets qui ont abouti à cette magnifique thèse.

Chapter 1.

Introduction

he context and challenges of this thesis are described in the chapter 1. A brief introduction of the information processing and its origin are presented. After highlighting the challenges to overcome and the various ways to perform the data processing, we present our investigated solution, which relies on hardware brain inspired approach. An overview of the neuromorphic computing and the major projects in this thematic are also presented in the introduction. Finally, the objective and challenges of this thesis are highlighted.

Information processing 1.Introduction to information processing

There are various ways to process the information. One is realized using manmade devices such as computers and the other one is accomplished by living systems for instance the brain. The neuronal architecture and the coding used in the brain to process the information differ from the conventional computers. The brain is fascinating and a powerful living system showing remarkable performance in areas as varied as the complexity of the processed information, the fault tolerance, learning, energy consumption and regeneration of the signal. Representing 2% of the weight of a human body and consuming 20% of the overall average power, the brain is revealed the most complex and efficient information processor known today. There is a major interest to study the difference between brain and conventional computing in order to bridge neuroscience and electronic engineering to develop artificial neural networks. This is the purpose of the following sections.

History of information processing in hardware

Alan Mathison Turing contributed to several areas such as cognitive science, artificial intelligence and mainly to computer science [START_REF] Beavers | Alan Turing : Mathematical Mechanist[END_REF]. His theoretical work in 1936, on the digital computing machine, is now referred as to universal Turing machine [START_REF] Turing | On computable numbers, with an application to the Entscheidungsprob-lem[END_REF], on which the conventional computer is based. Fundamental idea of a universal stored-program computing machine came from Turing and was promoted in the United States by John von Neumann and in England by Max Newman. By the end of 1945, several groups, including Turing, were devising plans for an electronic stored program universal digital computer (a Turing machine in hardware). His talent in the information processing and computers was demonstrated when he broke Naval Enigma and was the principal designer of the "bombe" : a high-speed code-breaking machine. In 1945, he designed the Automatic Computing Engine (ACE). Turing and his group pioneered the science of computer programming, writing a library of sophisticated mathematical programs for ACE. Finally, the world's first working electronic storedprogram digital computer was developed by Newman in 1948, nevertheless the concept of Turing machine was a fundamental influence on this achievement (Copeland, 2004). T Since, the progress in electronic and nanotechnology (e.g., transistors, integrated circuits, memories, microprocessors) leads to the development of many versions of computers. These developments provided the programmable electronic machines (computers) that are ubiquitous today.

The information processing is combined of computation and communication. Communication moves operands and results around while computation combines operands to produce results (Boahen, 2017). In general, information processing can be described by two categories of analog and digital. Analog computer (see the upper left of (Boahen, 2017).

Challenges to overcome

In the previous paragraph, we have presented briefly the conventional computer and the basis of information processing. In this section, some of the most important challenges of information processing and computers are presented. The quantity of data used, transported and created by sensors and end-user devices (from PCs to tablets and smartphones to digital TV), constitutes the digital universe. This one is growing exponentially and needs new procedures and systems to deal with data.

Exponential growth of Big Data

In our modern era, information processing and the Internet are omnipresent. Social media, sensors, enterprise data are all participating to the generation of enormous amount of data. According to the International Data Corporation (IDC), this amount of data created each year has grown exponentially. It reached 2.8 zettabytes in 2012 and will be around 40 zettabytes in 2020 (see Figure 1.2). (Noor, 2014).

Figure 1.2 Exponential evolution of Big Data

The Big data are stored in vast data centers (see Figure 1.3) with enormous power consumption. Data centers consume about 3% of the global electricity supply and are accounting for about 2 % of total greenhouse gas emissions. In 2015, the world's data centers used 416.2 Terawatt-hours of electricity, which was higher than the UK's total consumption of about 300 Terawatt-hours. USA data centers consumed about 70 billion Kilowatt-hours of electricity in 2014 representing 2% of the country's total energy consumption. That is equivalent to the amount consumed energy by about 6.4 million average American homes this year. Finally, researchers reported that in 2025, all the world data centers would consume 1/5 of earth's power. The increasing gap between the quantity of generated information and the computing possibilities are one of the challenges of the 21st century. Moreover, this implies an increase of the power consumption to transfer and process the data, which represents another major challenge of the 21st century.

Limitation of current information processing devices

The current computing is based on Von Neumann architecture, binary logic and the CMOS technology. The Von Neumann (V.N.) architecture drawback known as the "Von Neumann bottleneck" is due to the separation between processor and memory. These different parts of the microprocessor are connected by communication pathways called buses. Therefore, there are multiple paths and constant data traveling across these buses resulting in the bottleneck. The memory units also benefit of the technological improvement like microprocessors but the address decoding/reading/writing data steps are difficult to accelerate. The memory is not able to deliver information as fast as the processor is able of handling this information. This phenomenon is strongly accentuated when the architecture contains parallel multi-cores. Furthermore, in the paper entitled "The chips are down for Moore's law" published in February 2016, Mitchell Waldrop declares, "the semiconductor industry will soon abandon its pursuit of Moore's law. Now things could get a lot more interesting" (Waldrop, 2016). This sentence shows clearly that the current microprocessor computing technology has now reached some limits. Another point is the power dissipation of microprocessors, even if the integration density continues its exponential growth (which seems difficult due to the end of the Moore law), the circuit speed has been kept constant in order to limit the power dissipation at 100 W/cm 2 (see Figure 1.4). Figure 1. 4 (a) The parallel, distributed architecture of the brain is different from the sequential, centralized Von Neumann architecture of today's computers. The trend of increasing power density and clock frequencies of processors is headed away from the brain's operating point [START_REF] Merolla | A million spiking-neuron integrated circuit with a scalable communication network and interface[END_REF].

Hence, both architecture (V.N.) and devices used in the current information processors are reaching physical limits. Carver Mead declared about the binary coding that: "We lose a factor of about 100 because, the way we build digital hardware, the capacitance of the gate is only a very small fraction of the capacitance of the node. The node is mostly wire, so we spend most of our energy charging up the wires and not the gate. We use far more than one transistor to do an operation; in a typical implementation, we switch about 10 000 transistors to do one operation." (Mead, 1990). In addition, computers, champion for calculation of complex equation, have some difficulties to classify/organize/recognize data/patterns in the gray scale. Hence, it is time to propose new paradigms of information processing able to reduce the energy consumption in a drastic way while improving the performances. Artificial neural networks (ANN), computing in the same way as the human brain, could be the answer. The need of intelligent computers has pushed researchers to develop the neuromorphic computing domain. This part will be presented in following paragraphs with a brief introduction of the need of artificial intelligence.

Need of artificial intelligence

Artificial Intelligence (AI) could be defined as the desire to emulate the human cognitive behavior in order to develop intelligent machines. Many topics are concerned by AI such as self-driving cars, drones and robots. Associated with the appropriate learning rules, artificial neuron networks could be used in the frame of these applications to realize associative memory, video classifiers or cognitive autonomous robots. In this context, the technology used to build brain like processors most be energy efficient, real time computing and able to analyze, communicate and learn at the human brain speed and level. Current AI technologies offer real time computing chips, but with high-energy consumption and without human intelligence, which cannot be used for intelligent embedded systems.

Conclusion on possible solutions

The end of Moore Law for CMOS technology, one of the three pillars of current processors impact all the semiconductor industry of 400 billion dollars (WSTS, 2017) sales and had changed the direction of research. The 'International Technology Roadmap for Semiconductors' (ITRS) is becoming 'International Roadmap for Devices and Systems' (IRDC) implying investigation in both new devices and new systems. In fact, to overcome the challenges mentioned before, three approaches can be investigated, first is to keep current information processing architecture and to replace the CMOS by another revolutionary device. This way has not yet produced any satisfying result. The second approach consists to change both the architecture and the device of the current information processing. Research on quantum computers is an example of this second option where quantum cubes play the role of switches and quantum computers replace the conventional computer. This approach can take a long time, as the researcher must overcome many challenges since the quantum cube can operate at low temperature, which is not yet adapted to our environment. The third option is to keep the CMOS technology and change the computation architecture.

Investigated solution

In conclusion, based on the needs and the challenges to overcome, the alternative solution is to propose new paradigms for information processing. In this task, we have chosen to take as a model, the most efficient and powerful information processor, the brain.

Human Brain

The human brain has a massively parallel and reconfigurable architecture, composed of, 10 11 neurons and 10 15 synapses [START_REF] Vuillaume | Synaptic electronics[END_REF]. The brain has 300 million synaptic connections in each cm 3 , which represents the number of stars in the Milky Way. The brain, with its total 300 billion cells, is able to perform a cognitive task by consuming only around 20W that is 10 4 times less than the power consumption of multicore based supercomputers. The brain is a fascinating information processor. As you are reading these words, your brain is performing multiple complex cognitive tasks consuming only 20W during a millisecond. Moreover, it is capable of imagination, innovation that is beyond any supercomputer capabilities. This enigmatic organ has been studied over hundreds of years and still need to be investigated.

Brain information processing and coding

Based on neuroscience research until now, it is known that the brain uses electrical impulses called spikes to transfer information. A parallel is made between the 0 and 1 in computer science and the spikes in the brain. In fact, the absence of spike is considered like 0 and the generation of a spike in the brain as 1. It seems that the brain uses both the timing of a spike (time coding) and the frequency of a spike train (rate coding) to analyze information. The brain remains a mysterious living system, which has proved his efficiency in information processing. It is the key element of the neuromorphic computing and artificial intelligence.

Neuromorphic computing 1.4.1 Neuromorphic computing origins

In 1990, Carver Mead invented the "neuromorphic computing" term (Mead, 1990). He referred to very large-scale integration with analog components that mimicked biological neuronal systems as "neuromorphic" systems. The use of neuromorphic systems for information processing is the goal of neuromorphic computing. More recently, the term of neuromorphic computing includes implementations that are based on brain-inspired architecture [START_REF] Schuman | A Survey of Neuromorphic Computing and Neural Networks in Hardware[END_REF]. The history of brain-inspired computing starts in 1943 with the Warren McCulloch and Walter Pitts neuron. Their work was the first step toward the artificial neural networks (ANNs) since a neural network could be simply defined as connected neurons. The McCulloch-Pitts neuron model is a simplified mathematical model used to represent neural properties [START_REF] Mcculloch | A logical calculus of ideas immanent in nervous activity[END_REF]. It is based on the existence of a threshold voltage for the activation of an action potential. Using their model, it is possible to figure out most Boolean logic function. Their neurons operated under the following assumptions. They are binary devices as the input and output is either zero or one. They have fixed threshold. They receive inputs from excitatory synapses with identical weight. Inhibitory inputs have an absolute veto on excitatory input, if there is even one inhibitory input, the neuron will not fire (absolute inhibition). At each time step, the neurons are simultaneously updated by their inputs (inhibitory and excitatory synapses). In 1949, Donald Hebb, considered as the father of neural networks, introduced the Hebbian learning rule, which provides the foundation of modern neural network (Wang & Raj, 2017). In 1958, Frank Rosenblatt introduced the first perceptron (Wang & Raj, 2017). The difference between the perceptron of Rosenblatt and the neuron of McCulloch-Pitts was: the synaptic weights (can take positive or negative value) and the neurons thresholds were variable. There is no absolute inhibitory synapse. The most important difference is that the perceptron had a learning rule. In 1970, Marvin L. Minsky and Seymour A. Papert published a book entitled "Perceptrons: an introduction to computational geometry" which acknowledge the perceptron's strengths while also showing major limitations [START_REF] Olazaran | Deep Brain Stimulation of Frontal Lobe Networks to Treat Alzheimer's Disease[END_REF]. Critics of this book state that since a single artificial neuron (one perceptron) is not able of implementing a XOR function, hence larger networks have the same issue and this technic must be dropped. It is known that this book was the source of many controversies in the field of neural networks and artificial intelligence. Since many researchers have continued the research on the neural network, a summary of this evolution is presented in Table 1.1. (Wang & Raj 2017).

Table 1.1 Major contribution to ANNs development

Researchers from a variety of fields, such as materials science, neuroscience, electrical engineering, computer engineering, and computer science are considered as the neuromorphic computing community. In our domain, the neuromorphic systems can be considered in two categories: software and hardware systems. Artificial neural networks (ANNs) used by Google to classify images or playing games with AlphaGo is some of the applications of the software field of neuromorphic computing. To perform this task, the ANNs use new algorithms allowing these networks to be trained and/or learn on their own. The hardware neuromorphic systems are made of neuroinspired devices and architecture. As an example, Truenorth chip is a neuromorphic CMOS integrated circuit produced by IBM in 2014 [START_REF] Merolla | A million spiking-neuron integrated circuit with a scalable communication network and interface[END_REF]. With over a million of programmable neurons and over 268 million programmable synapses, it overcomes the Von Neumann bottleneck, by handling the memory, computation and communication in each of the 4096 cores of this Truenorth chip. However, the energy consumption remains still high.

It is worth mentioning that several international research programs such as SyNAPSE, the Brain initiative, the Human Brain Project, etc. are investing billions of dollars in neuromorphic computing and systems. In the following paragraphs, we will present three of these projects.

Neuromorphic computing major projects

A. The SyNAPSE project SyNAPSE (Systems of Neuromorphic Adaptive Plastic Scalable Electronics) is a 21 million dollars project funded by DARPA (DARPA, 2013). The goal of this project launched in 2008, was to design a new computer architecture that mimics the brain. It had plans to combine the fields of nanoscience, neuroscience and supercomputers with the goal of developing devices with cognitive aspects. IBM's Truenorth chip has been developed in the frame of this project.

B. The Brain initiative project

The brain initiative project was launched in 2013 by the president Obama who declared, "It will spark innovation, create jobs, stimulate the economy, treat autism, Alzheimer's, epilepsy and schizophrenia, not to mention stroke, traumatic brain injury and posttraumatic stress disorder" (Obama, 2013). The total investment was about 3.8 billion dollars with the contribution of three federal agencies, the National Institutes of Health NIH ($40 million), the National Science Foundation NSF ($20 million) and the Defense Advanced Research Projects Agency DARPA ($50 million). Many research institutes have also contributed to this project with 60 million dollars from the Allen Institute, 30 million dollars for Brain Science, 30 million dollars from the Howard Hughes Medical Institute, 28 million dollars from the Salk Institute for Biological Studies and 4 million dollars from the Kavli Foundation. The timeline of the project is ten years.

C. The Human Brain Project

On the other side of the Atlantic, on 28 January 2013, the European Commission announced its decision to finance the Human Brain Project, a 10-year project that is estimated to cost 1.19 billion euros (HBP, 2013). This project aims to bring together all the human brain current knowledge in order to rebuild it, piece by piece, on the strength of computer models and simulations. These two European and American projects are not competitors, but rather complementary. As Richard Walker, a spokesperson for the Human Brain Project, points out, "all the data produced by the American project will be in the public domain, and that is a lot of results that can be incorporated into the Human Brain Project models" (Figaro, 2013).

Understanding the human brain is one of the greatest scientific challenges facing the 21st century. If we can rise to this challenge, we can gain profound insights into what makes us human, develop new treatments for brain diseases and build revolutionary computing technologies. In the frame of our work, two main applications linked to neuromorphic systems are developed. This will be discussed in the next section.

Neuromorphic computing and systems, interests and applications

Neuromorphic computers will gather several advantages such as perform complex calculations faster, more power efficiently, and on a smaller footprint than traditional Von Neumann architectures. These characteristics provide compelling reasons for developing hardware which employs neuromorphic architectures [START_REF] Schuman | A Survey of Neuromorphic Computing and Neural Networks in Hardware[END_REF].

This neuro-inspired field will impact two domains, information processors or computers and the human health care. Neuromorphic systems will be used in both cases to overcome their current limitation and increase their performances.

In the case of computers, you may imagine a smart machine helping you during the day-to-day activities. This intelligent computer communicates, learns, make decisions, and even advice you based on its resources. Watson is an example of this kind of artificial intelligence [START_REF] Ferrucci | Building Watson: An Overview of the DeepQA Project[END_REF].

In the case of human health care, millions of people die due to neurological diseases all over the world [START_REF] Feigin | Global, regional, and national burden of neurological disorders during 1990-2015: a systematic analysis for the Global Burden of Disease Study 2015[END_REF]. The idea is to fabricate a neuromorphic system to stimulate or even replace the defective brain area. For example, DARPA has launched the project RAM (Restore Active Memory-(DARPA, 2014)) to restore the lost memory of the injured soldiers. The first results obtained in the frame of this project are reported in [START_REF] Hampson | Developing a hippocampal neural prosthetic to facilitate human memory encoding and recall[END_REF]. They have demonstrated the first successful implementation in humans of a proof-of-concept system for restoring memory function by facilitating memory encoding using the patient's own neural codes. This idea can also be extended to an improvement of a healthy brain in terms of memory and learning, improving our intelligence.

For neurological diseases such as the epilepsy disease, we can imagine an intelligent neuromorphic system, playing the role of a "brain pacemaker" which detects the brain crises and stimulates in vivo the specific brain area to neutralize the epileptic crises. BBC has reported in January 2018 a brain pacemaker used for Alzheimer affected patients (Scharre et al., 2018).

In 2015, a primate with spinal cord injury regained control of its paralyzed leg with the help of a neuro-prosthetic system called the "brain-spine interface" that bypassed the lesion, restoring communication between the brain and the region of the spinal cord. They interface legs motor cortex activity with epidural electrical stimulation protocols to establish a brain-spine interface that alleviated gait deficits after a spinal cord injury in non-human primates [START_REF] Capogrosso | A brain-spine interface alleviating gait deficits after spinal cord injury in primates[END_REF].

As described in this section, neuromorphic systems address two major applications one in information processing and second in health care. These applications demonstrate the interest of the research community and validate the investment in this field.

Objective and challenges of thesis

In order to achieve an efficient neuromorphic system, we have chosen the bottom-up approach. This approach corresponds to fabricate an artificial neuron and synapse before fabricating an artificial brain inspired system. In this context, the main goal of this thesis was to design and fabricate an ultra-low power artificial neuron. This is a multidisciplinary work requiring both knowledge in neuroscience and electronic field. The challenges to overcome are:

1) Investigate and understand the complex functionality of a living neuron, enabling an efficient artificial neuron's design.

2) Design a simple novel architecture with optimized area, enabling the fabrication of neuromorphic VLSI systems.

3) Design and fabrication of analog spiking artificial neuron, using conventional CMOS technology, enabling a fast dissemination to the industry to fulfill current applications.

4) Obtaining ultra-low power consumption for the artificial neuron, responding to the problematic concerning the energy consumption of current information processing systems.

5) Investigate the noise within the artificial neuron, allowing a comparison to the biological sources of noise in a living neuron.

Chapter 2.

Neurons modeling

n this chapter, the different part of a neuron cell will be identified and the functionality of each of these specialized regions will be briefly explained. Next, an overview of mathematical models, which attempt to describe the behavior of the neuron cell, will be presented. Finally, three biophysical models and their simulation results will be described. These models will be used for the design of an ultra-low power artificial neuron, which is the basic element of future artificial neural networks.

Neuron and neuronal membrane

Glial and neuron cell

All organs of the human body are composed of cells. The specific functions of cells and their interactions determine the functions of organs. This is also true for the most sophisticated organ, the human brain. We can define two main types of cells in the human brain, glial cells and neurons. This chapter will emphasize on the structure of different nervous types of cells, mostly neurons in our case. Glial cells fill the space between the neurons maintaining the homeostasis process, make myelin and play a role in supporting and protecting nervous tissue providing nutrients and oxygen. They eliminate dead cells and fight pathogens. The exact role of glial cells remain still obscure, and it is generally admitted that neurons play a more important role, in particular regarding their contribution to produce and transmit the electrical impulses. This is the 'neuron doctrine' a discovery due to decisive neuro-anatomical work of Cajal (Andres- Barquin, 2001). There is an impressive number of neurons in the human brain, about 100 billion. Neurons are excitable cells and specialized in the reception, integration and transmission of information. Information flows in the form of action potential also called spike. All the activity of our nervous system is based on the transmission of these nerve impulses between neurons. There are various types of neurons, which differ in morphology and functionality (Figure 2.1) [START_REF] Lodish | Overview of Neuron Structure and Function[END_REF]. (c) Sensory neuron [START_REF] Lodish | Overview of Neuron Structure and Function[END_REF].

I

A typical neuron is composed of a cell body and different extensions differing in functionality and size. Indeed, four main parts can be identified for one neuron: cell body also called soma, axon, dendrites and axonal terminals known as synapses (Figure 2.2). All these specialized regions constitute a very dense network that allows permanent and coordinated exchanges between neurons.

Neuron cell body and axon

As we look into the cell body, we can find the cell nucleus (Figure 2.2), which contains the DNA. The cell body is the place of synthesis of almost all neuronal proteins and membranes molecules [START_REF] Lodish | Overview of Neuron Structure and Function[END_REF].

Almost every neuron has a single axon (Figure 2.2). The diameters of the axon can vary from micrometers in certain nerves of the human brain to millimeters in the nerves of squid [START_REF] Lodish | Overview of Neuron Structure and Function[END_REF]. The main function of axon can be assimilated to a oneway road, which allows the conduction of spikes from the cell body to the axon terminals.

Neuron dendrites and synapses

The cell body is surrounded by multiple branches, which play the role of a messenger receiver. Generally known as dendrites (Figure 2.2), they receive chemical signals from other neurons, convert it to electrical impulses, and finally transmit them into the cell body.

The information transfer between two neurons takes place in a specialized region called synapse (Figure 2.2). Synapses are located between an axon terminal of a neuron and a dendrite of an adjacent neuron. Generally, there are two types of synapses: chemical and electrical. The chemical synapses send impulses via neurotransmitters. The electrical synapses are like tunnels; ions are transferred via hydrophilic channels also called gap junction channels (Figure 2.3).

Neuronal membrane and ion channels

The intercellular connections, synapses, consist of a very thin cleft located between two fragile membranes. In fact, each membrane surrounds the neuron and forms the outer boundary of the neuron cell. Thanks to the fluidity of the membrane, the neurotransmitters diffuse across the synaptic cleft allowing the transmission of nerve impulses. When the neurotransmitter binds to the receptor, changes in the ion permeability of the postsynaptic plasma membrane is involved, leading to the membrane electric potential changes at this location. Depending on the type of postsynaptic cell, this electric fluctuation can induce an action potential for a neuron, a contraction in a muscle or even hormone secretion in case of a gland cell.

The creation and propagation of action potential is based on the ions flow across their plasma membranes highlighting that the neuronal membrane plays a fundamental role in the nervous system (Figure 2.2). As mentioned before by delimiting the neuron cell, it contains the cytoplasm inside the neuron and maintains certain substances outside the neuron. This neuronal membrane is about 5 nm thick. It is composed of two layers of lipid molecules with many protein structures embedded in the membrane. Some of these proteins called ion transporters or pump channels maintain a gradient [START_REF] Bear | Neurosciences, A la découverte du cerveau[END_REF], i.e. a difference in ionic concentrations between the intracellular space and the extracellular space of the neuron cell. Other specific proteins form ion channels, which select certain substances that can penetrate through the neuronal membrane (Figure 2.4). Both ion transporters and ion channels are necessary to describe the generation and propagation of action potential. Right: simplified schematic of the neuronal membrane with principal channels and principal ionic concentrations (Na + and K + ) [START_REF] Bear | Neurosciences, A la découverte du cerveau[END_REF].

Neuronal membrane and the action potential

In this part, we will describe the variation of the action potential according to the exchange of ionic charges through the neuronal membrane. It is worth mentioning that the difference in electric potential between the interior and exterior of a neuron cell is called the membrane potential. This term will be used in the next paragraphs. At rest, there is a negative potential difference around -65mV between the intracellular surface of the neuron membrane and its extracellular surface called the resting potential [START_REF] Bear | Neurosciences, A la découverte du cerveau[END_REF]. This resting potential results from a concentration gradient of sodium ions Na + and potassium ions K + between the inside and the outside of the neuron. In fact, in resting state Na+ ions have a higher concentration in the extracellular space than the intracellular space. On the contrary, K + ions have a higher concentration in the intracellular space than the extracellular space. For each ion, the reversal potential (also known as Nernst potential) is the membrane potential for which there is no net flow of that particular ion from one side to another side of the neuronal membrane. At equilibrium the Nernst potential for potassium and sodium ions are respectively EK=-77mVand ENa=+50mV.

The action potential is characterized by a fast and localized modification of the permeability of the neuron's membrane: Na + enters into the cell using ion channels selectively permeable to sodium. The membrane potential takes then a positive value, about +40 mV, close to the Nernst potential of sodium (ENa). It is illustrated as the depolarization phenomena in Then, potassium ions (K + ) leave very quickly the cell, passing through dedicated ion channels. The membrane potential decreases to a lower value than the resting potential, which defined the repolarization and then hyperpolarization sequences. Finally, the membrane potential reaches its initial resting state. All these variations of the action potential last only a few milliseconds in living cells.

This localized and transient nerve impulse, which is mostly generated in a region of the cell body called axon hillock, propagates along the neuron axon. In fact, action potentials can originate not only at the axon hillock, but also in the axon initial segment, 30 to 40μm from the soma and close to the first myelinated segment. In some neurons, the action potential even originates at the first node of Ranvier (Figure 2.6), where sodium channels are highly concentrated. For both myelinated and unmyelinated axons, once the action potential begins in the axon, it is not only propagating toward the nerve terminals but also back into the soma and dendrites (Caldwell, 2010). The myelination of the axon impacts the propagation speed of an action potential. In fact the unmyelinated axon conduction velocities range from about 0.5 to 10m/s, while the myelinated axons can conduct at velocities of up to 150m/s (Purves et al, 2004). Previously, the different regions of a neuron cell were presented. The generation and propagation of spike due to the mechanism of ionic channels in the neuronal membrane have been defined. In the next section, mathematical models of the neuronal membrane will be described.

State of the art: spiking neuron models

In many fields, science tries to explain the existing phenomena using physical equations. This is the case for the neuronal membrane where many scientists try to explain the generation of spike using more or less complex mathematical models. Several neuronal membrane models have been proposed over the time. These models, also called spiking neuron models, are a mathematical description of the generation of spikes in cells as a function of the properties of their membrane. They are designed to accurately describe biological processes. Investigated in 1907 by Lapicque, the Integrate-and-fire (I&F) is one of the most widely used models in computational neuron science. Also known as the simplest neuron model, it describes relation between the current and the membrane potential. When an excitatory current is applied, the membrane voltage increases until it reaches a threshold. Since I&F, more complex models had been developed: leaky integrate and fire, I&F with adaptation, I&F or burst, quadratic I&F. However, they are not complete enough to describe with a good accuracy the complexity of spiking behavior of individual neurons in response to excitatory current pulses. Models such as FitzHugh Nagumo in 1961, Morris-Lecar in 1981, Hindmarsh Rose in 1984, Wilson and Izhikevich in 2003, are more efficient to reproduce the neurocomputational properties of biological spiking neurons. In 2004, a comparison of these neuronal models has been investigated by Izikevich (Izhikevich, 2004) (Figure 2.7). Huxley model (HH) proposed in 1952 [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF]. It is the first model to describe the generation of one spike based on the experimental studies of the neuronal membrane. HH model will be presented in the next section. Several mathematical models have been proposed after the Hodgkin-Huxley model in order to describe the spiking activity in living membranes. These refined models are also based on channel-based equations. Most of these models were designed by introducing extra-ionic channels and by adding corresponding terms to the original HH equation.

In this work, three neuronal membrane models were investigated. First, the widely known and pioneer Hodgkin and Huxley model, which studies the properties of the giant squid axon. Second, the Wei model which proposes a precise model for the human neuron cell to describe the unification of neuronal spikes. Finally, the Morris-Lecar model attracted our attention as it gathers both criteria of biophysically meaningful and simplicity. A description of these models and the simulation results are presented in the next section. As shown in Figure 2.9, they were also the first ones who observed one potential action by introducing an electrode into the giant squid axon. The 500μm diameter of this axon allows the rapid conduction (the speed increases as the square root of the diameter) of action potentials driving a fast response. [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF].

Investigation of biophysical neuron models

The non-linear differential equations of Hodgkin and Huxley constitute the reference model for any other scientist seeking to understand the operation of a neuron. This model is important not only because the parameters have a biophysical meaning and are measurable, but also because they allow us to study issues related to synaptic integration, dendritic cable filtering, the effects of dendritic morphology, the interaction between ionic currents, and other issues related to the dynamics of a neuron and its neuronal membrane.

The Hodgkin and Huxley model is defined by four differential equations coupled to each other. The target of this paragraph is to explain the link between HH equations and their biological meaning.

Based on the HH electrical circuit shown in Figure 2.10, the total current (𝐼) is the result of two main contributions that are the charge of the membrane capacitance (𝐶 𝑀 ) and the ion currents (𝐼 𝑖𝑜𝑛 ) crossing the membrane through the ion channels (Eq. Based on HH equations previously described the numeric resolution of this model with Matlab software, was investigated. The Matlab code of HH model was found in the literature (Siciliano, 2012). The constant parameters as well as the equations are extracted from [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF] and the results are presented in the next paragraph.

Results of HH model: threshold excitatory current

The excitability of the neuron is defined as its ability to produce a spike in response to a stimulus. The generation of spike depends on the excitatory current and its duration, which follows the principle of excitability of the neuron. we define the threshold excitatory current as the required current to generate only one spike for a given time. From this basic behavior, the excitability of the squid's neuronal membrane was investigated by increasing the threshold current amplitude and its time duration (Figure 2.12). In fact, this study was driven to monitor the number of spikes generated in the time domain versus the injected excitatory current.

A B C Figure 2.12 Excitability property of the neuronal membrane for HH model.

The response of the HH model, determined at the threshold excitatory current Iex = 7μA/cm 2 during Ts = 5ms (Figure 2.12.A), was kept as a reference. As we increase Iex to 50μA/cm 2 and Ts to 14ms, a second spike is generated (Figure 2.12.B). A third partial spike is observed for Iex = 100μA/cm 2 and Ts = 14ms (Figure 2.12.C). This study demonstrates the influence of the amplitude and the duration of the pulsed excitatory current in the generation of spikes. To complete this work, we have investigated different pulsed currents, in terms of amplitude and pulsed duration, and conclude that the number of charges induced into the neuronal membrane is the main criteria to obtain a spike train. A detailed result will be presented in a next section (2.3.2.3 Injected charge).

Monte Carlo comparison

In the frame of a collaboration with a team of Salamanca University, a stochastic model based on the Monte Carlo (MC) technique was also used to solve HH equations [START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF]. MC model is based on the probabilities for the different ions to cross the cell membrane. Sodium, potassium and leakage channels are introduced in the model according to the Hodgkin and Huxley equations. As it can be seen in As Vm reaches its peak value, around +40mV, sodium channels start to close (m 3 h decreases, Figure 2.13.c), the slow potassium channels are completely open and potassium ions rush out from the intracellular space to the extracellular space and the voltage quickly returns to its resting value. As the potassium channels are closing slowly, potassium ions continue to leave the cell, resulting in a negative overshoot called hyperpolarization. The resting membrane potential is slowly restored thanks to the diffusion and leakage channel (ICl). During the action potential generation and a short time after the recovery of the steady state, it is impossible to stimulate the neuronal membrane in order to fire again. In fact, the period from the initiation of the spike to immediately after the peak is referred as the absolute refractory period. During this time, the cell cannot produce any new action potential. The generation of one spike obtained by the Matlab and MC resolution of HH model seems to be exactly the same, despite the stochasticity in the MC model. However, a closer look at the MC results corresponding to a zoom of INa and IK (inset in Figure 2.13.b) reveals the presence of fluctuations in the ion currents that is the signature of a channel noise [START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF]. It is interesting to note that the same results have been obtained with Matlab and MC for the resolution of HH model. . In all cases, the excitatory current starts at 10ms [START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF].

As described in this section, Hodgkin and Huxley succeeded in describing the generation of one spike based on the mechanism of ionic channels present in the neuronal membrane. In their model, the neuronal membrane and the neuron cell is a simplified version of the reality. In the next section, we will investigate the Wei model also based on HH model but with a more detailed description of the neuronal cell.

Limitation of Hodgkin Huxley model and introduction to Wei model

Proposed in 2014, Wei model based on HH model describes several types of generation of spikes, such as epileptic state named "seizure", stable state called "steady stream" and cerebral attack labeled "stroke" [START_REF] Wei | Oxygen and seizure dynamics: II. Computational modeling[END_REF]. This model demonstrates that these pathological dynamics, as well as normal spiking behavior of neurons, share a unified dynamic underpinning. They extend the HH formalism to uncover a unification of neuronal membrane dynamics. The main difference between Wei model and HH model lies in their different hypothesis, which is reflected in their respective equations. Table 2.1 compares the main equations of Wei model (Eq.2.5) with HH model equations (Eq.2.4).

Main equations of Wei model Eq.2.5

Main equations of HH model Eq.2.4 As shown in Table 2.1, the Wei model is essentially based on HH model. The variation of the membrane voltage (dVm/dt) is due to similar ionic currents in both models. The gating variable in both models have also the same equations. However, and contrary to HH model, the Wei model uses differential equations to describe the variation of the ionic concentrations. For simplicity, we present Eq.2.6, which describes only the potassium concentration outside the cell (𝑁𝐾 0 + ) and inside the cell (𝑁𝐾 𝑖 + ). Similar equations describe the variation of sodium (𝑁𝑁𝑎 0 + , 𝑁𝑁𝑎 𝑖 + ) and leakage concentration (𝑁𝐶𝑙 0 + , 𝑁𝐶𝑙 𝑖 + ) (see annex).

𝐶 𝑑𝑉 𝑚 𝑑𝑡 = -𝐼 𝑁𝑎 -𝐼 𝐾 -𝐼 𝐶𝑙 - 𝐼 𝑝𝑢𝑚𝑝 𝛾 𝐼 𝐾 = 𝐺 𝐾 𝑛 4 (𝑉 𝑚 -𝐸 𝐾 ) + 𝐺 𝐾𝐿 (𝑉 𝑚 -𝐸 𝐾 ) 𝐼 𝑁𝑎 = 𝐺 𝑁𝑎 𝑚 3 ℎ(𝑉 𝑚 -𝐸 𝑁𝑎 ) + 𝐺 𝑁𝑎𝐿 (𝑉 𝑚 -𝐸 𝑁𝑎 ) 𝐼 𝐶𝑙 = 𝐺 𝐶𝑙𝐿 (𝑉 𝑚 -𝐸 𝐶𝑙 ) 𝑑𝑞 𝑑𝑡 = 𝛼 𝑞 (1 -𝑞) -𝛽 𝑞 𝑞 , 𝑞 = 𝑚, ℎ, 𝑛 𝐶 𝑀 𝑑𝑉 𝑚 𝑑𝑡 = -𝐼 𝑁𝑎 -𝐼 𝐾 -𝐼 𝑙 𝐼 𝐾 = 𝑔̅ 𝐾 𝑛 4 (𝑉 𝑚 -𝐸 𝐾 ) 𝐼 𝑁𝑎 = 𝑔̅ 𝑁𝑎 𝑚 3 ℎ(𝑉 𝑚 -𝐸 𝑁𝑎 ) 𝐼 𝑙 = 𝑔̅ 𝑙 (𝑉 𝑚 -𝐸 𝑙 ) 𝑑𝑞 𝑑𝑡 = 𝛼 𝑞 (1 -𝑞) -𝛽 𝑞 𝑞 , 𝑞 = 𝑚, ℎ, 𝑛
𝑑𝑁𝐾 0 + 𝑑𝑡 = 1 𝜏 (𝛾𝛽𝐼 𝐾 -2𝛽𝐼 𝑝𝑢𝑚𝑝 -𝐼 𝑑𝑖𝑓𝑓 -𝐼 𝑔𝑙𝑖𝑎 -2𝐼 𝑔𝑙𝑖𝑎𝑝𝑢𝑚𝑝 + 𝛽𝐼 𝑘𝑐𝑐2 + 𝛽𝐼 𝑛𝑘𝑐𝑐1 )𝜈 0 𝑑𝑁𝐾 𝑖 + 𝑑𝑡 = 1 𝜏 (-𝛾𝐼 𝐾 + 2𝐼 𝑝𝑢𝑚𝑝 -𝐼 𝑘𝑐𝑐2 -𝐼 𝑛𝑘𝑐𝑐1 )𝜈 𝑖 Eq.2.6
These novel equations describing the variation of the ion concentrations proposed by Wei model allows us to study the individual charge conservation, which was not the case for the HH model (global charge conservation). A look into the Investigation of individual charge conservation with Wei model and global charge conservation with HH model enables us to obtain a profound knowledge of the neuronal membrane which is essential for the design of an artificial neuron used in neuroinspired circuits. In fact, these models, which increase in complexity, give us an overview of what is the reality of a biological neuron and set us the important criteria to take in account to design a neuro-inspired system.

To facilitate the dissemination of their results, the MATLAB code required to reproduce the full model was available and provided by Wei and co-authors. Several modification of the Matlab code has been performed in order to obtain a simple neuroinspired model. We removed the effects of osmotic pressure, which is not essential for our target. In fact, it is not necessary to reproduce all the neuronal phenomena and pathological dynamics to design a neuro-inspired circuit. This work targets the main and normal biological mechanisms of a neuron cell and the efficient way to reproduce it with nano-electronic devices. As mentioned before, Wei model can reproduce different spiking states, seizure and spreading depression and normal spike train.

According to this, we redefined the variation of intracellular and the extracellular volume to reproduce the normal spiking state.

Results of Wei model: threshold excitatory current

Our first study showed that Wei model can generate one spike with Iex = 7μA/cm 2 during 7ms (Figure 2.16).

Figure 2.16 Generation of one spike and gating variables with Wei model. Up:

One spike is generated for Iex = 7μA/cm 2 and Ts = 7ms. Down: Gating variable m 3 h (red) and n 4 (green) for Iex = 7μA/cm 2 and Ts = 7ms.

However, due to the nonlinear differential equations used in the HH and Wei models, it is difficult to determine precisely the threshold current. As it can be seen in (Figure 2.17), the Wei model generates one spike for different threshold excitatory currents.

(Figure 2.17.up) shows the generation of one spike with Iex = 5μA/cm 2 and Ts = 7ms while (Figure 2.17.down) illustrates one spike with Iex = 7μA/cm 2 and Ts = 5ms. Based on these results, it can be concluded that the threshold edge is not well defined. This point needs to be more deeply investigated and another way is needed, to establish the boundary. In order to precisely define this value, the effect of injected charge was investigated for the Wei model. This choice is based on how the neuronal membrane of the soma changes from a resting state to a spiking state, brought by a slight depolarization of the cell body and the charges collected from dendrites.

Injected charge

As demonstrated before, since it is difficult to determine the threshold excitation of the neuron as a function of Iex or Ts, another approach is to determine the threshold excitation by the number of injected charges through the membrane (Q), which is defined as follows:

Q = Iex. Ts Eq.2.7
In this study, the excitatory current Iex and its duration Ts will vary while the injected charges through the membrane, is kept constant. A constant value Q of 49nC/cm 2 obtained with Iex = 7μA/cm 2 and Ts = 7ms was chosen as a reference. As Iex varies from 3.5μA/cm 2 , to 7μA/cm 2 and 14μA/cm 2 and Ts varies from 14ms, 7ms and 3.5ms respectively, the neuron generates one spike for these three cases. It can be concluded that the variation of Iex and Ts does not influence the generation of spike, as the number of charges remains constant (49nC/cm 2 ). Thus it seems that the excitability of the neuron is related to the quantity of injected charges Q and therefore depends on the product of the excitation current (Iex) by the excitation time (Ts). This new conclusion leads us to investigate the excitability of the neuron when Q is variable. Hence, we define the on-state (Q1S) corresponding to the minimal quantity of charges for which the neuron spikes only once. The work was carried out for different excitatory current values. As shown in Figure 2.18, Iex varies from 2μA/cm 2 to 8μA/cm 2 by steps of 1μA/cm 2 . The number of charges decreases as the excitatory current increases. Furthermore, as the excitatory current increases the pulsed duration needed to generate one spike decreases. This study highlights that the minimum number of charges to obtain one spike is around 21nC/cm 2 . As a major conclusion, based on this study, we can determine various excitatory currents to generate only one spike. In fact, a wide range of values can be used for Iex and Ts parameters of the pulsed excitatory current.

Power and energy consumption

One of the main objectives of this thesis is to propose an ultra-low power artificial neuron. Therefore, two different methods to determine the power and energy consumption of a neuron cell will be presented. The first idea consists in estimating the power and energy consumption of one neuron based on the total brain consumption, while the second consists in calculating the power and energy consumption based on ionic currents obtained by HH model and Wei model.

Power consumption per spike based on brain consumption

The former method is presented in this paragraph. It is known that the brain represents less than 2% of a person's weight and consumes 20% of the body energy [START_REF] Drubach | The Brain Explained[END_REF]. In addition, the average power consumption of an adult is about 100W (Rigden, 1996). Hence, the total consumption of the human brain is around 20W. The brain has 10 11 neurons and the overall brain consumption for one second is 20J, hence each neuron will consume 200pJ. In case of fast brain activities, each neuron spikes at an average frequency of 20Hz, corresponding to the average beta wave frequency (Priyanka, 2016). Based on the 20Hz frequency value, we can easily deduce that the energy per spike is around 10pJ/spike. This simple calculation corresponds to the estimation of energy per spike for a biological neuron from human cortex reported from (Poon and Zhou, 2011) and (Lennie, 2003).

The second method is based on our previous results concerning the excitability of the neuronal cell. We have investigated in detail the generation of one spike and its corresponding ionic currents (𝐼 𝑁𝑎 and 𝐼 𝐾 ) in HH model and Wei model. Based on these ionic currents, we can estimate the power and the energy consumption per spike, which is presented in the following section.

Power consumption per spike based on ionic currents

According to the equations presented in the Table 2.2, we define sodium power (𝑃 𝑁𝑎 ) and potassium power (𝑃 𝐾 ) for HH model and Wei model. The power consumption for each models are based on the ionic currents (𝐼 𝑁𝑎 and 𝐼 𝐾 ), the membrane potential (𝑉 𝑚 ) and Nernst potentials (𝐸 𝑁𝑎 and 𝐸 𝐾 ). Obviously, the overall dissipated power (𝑃 𝑑 ), is obtained by the addition of 𝑃 𝑁𝑎 and 𝑃 𝐾 .

𝑃 𝑊𝑒𝑖 , 𝐻𝐻 (𝑛𝑊. 𝑐𝑚 -2 ) The Nernst potentials are constant in the HH model, while the Nernst potentials in the Wei model vary as function of the dynamic of the ionic concentrations (annex). The resulting power of HH and Wei models are illustrated in Figure 2.19. We observe similar shapes for 𝑃 𝐾 . For both models, the 𝑃 𝑑 occurs during 2ms. However the shape of 𝑃 𝑁𝑎 differs between the two models having an impact on 𝑃 𝑑 . In addition, the powers amplitudes in HH model are much higher than in Wei model. We performed a trapeze type integration of 𝑃 𝑁𝑎 and 𝑃 𝐾 and multiplied these results to the surface of the neuronal membrane to determine the energy consumption per spike.

𝑃 𝑁𝑎 = 𝐼 𝑁𝑎 (𝑉 𝑚 -𝐸 𝑁𝑎 ) 𝑃 𝐾 = 𝐼 𝐾 (𝑉 𝑚 -𝐸 𝐾 ) 𝑃 𝑑 = 𝐼 𝑁𝑎 (𝑉 -𝐸 𝑁𝑎 ) + 𝐼 𝐾 (𝑉 -𝐸 𝐾 )
To define the surface of the neuronal membrane, we consider the shape of the cell body to be spherical with a 20μm diameter [START_REF] Bear | Neurosciences, A la découverte du cerveau[END_REF]. We obtain a neuronal surface of 10 -6 cm 2 . Hence, the energy consumption per spike for HH model is 20 fJ/spike and 3 fJ/spike for Wei model. The energy per spike obtained with the Wei model is less than the value obtained with HH model due to the lower amplitude of each ionic power.

Conclusion on HH and Wei models

HH model and Wei model reproduce the excitability of the neuronal membrane. For both models, we have determined a specific excitatory current for the generation of only one spike. Additionally, we have investigated the influence of the number of injected charges on the generation of one spike in the Wei model. We have performed the analyses of power and energy consumption of both models, when the neuron spikes only once. As described previously both HH and Wei model are based on four nonlinear differential equations. These complex models are very useful to calibrate the electronic artificial neuron design but due to the high number of parameters, they lead to complex electrical circuits with considerable silicon surface [START_REF] Yu | Analog VLSI biophysical neurons and synapses with programmable membrane channel kinetics[END_REF]. Thus, a simpler and also biophysically meaningful model, the Morris-Lecar (ML) model was investigated. Next section will introduce the Morris Lecar model and some results from the simulation. It can be noticed that the ML model spike frequency is around 8Hz for Iex = 90pA/cm 2 . As we increase the excitatory current from 120pA/cm 2 to 200pA/cm 2 , the spike frequency increases from 14Hz to 16Hz. It saturates around 16Hz for Iex between 150pA/cm 2 and 200pA/cm 2 . The ML model does not spike for 250pA/cm 2 < Iex < 350pA/cm 2 . This type of "Frequency versus current" curve correspond to class-1 neural excitability, in which action potentials can be generated with arbitrarily low frequency that increases in accordance with the excitatory current (Izhikevich, 2007). We can conclude that ML model validates the excitability behavior of the neuronal membrane and also reproduces the spiking activity of a neuronal cell. Hence, this model can be used as a guideline since it is simple enough to be reproduced by a conventional electrical circuit. ML model will be more investigated in chapter 3. The analogy between the ML equations and the circuit proposed for the artificial neuron will be explained.

Morris-Lecar model

Conclusion

In this part, the various specialized regions of a neuron cell were presented. The main region responsible in the generation of action potential, the neuronal membrane was described. The dynamics of ionic channels present in the neuronal membrane can indeed explain the generation of spikes. Ionic channel mechanisms can be described with mathematical equations known as neuronal model. An overview of various neuronal models was presented in this chapter. In order to understand the complex behavior of the neuronal membrane, three of these biophysical models HH, Wei and Morris-Lecar were investigated. Each of these neuronal models was studied through simulation by Matlab software (and Monte Carlo simulation for HH model). Based on results presented in this chapter, the excitability mechanisms of the neuron were specified. A precise modulation of the generation of spikes as a function of the charge density was performed. This allows us to estimate the power and energy efficiency per spike for HH and Wei model.

Annex Hodgkin Huxley model

Ionic current:

The ionic current is composed of sodium current (𝐼 𝑁𝑎 ), potassium current (𝐼 𝐾 ) and leakage current (𝐼 𝑙 ). Their expression are presented as follow

𝐼 𝑁𝑎 = 𝑔 𝑁𝑎 (𝑉 𝑚 -𝐸 𝑁𝑎 ) 𝐼 𝐾 = 𝑔 𝐾 (𝑉 𝑚 -𝐸 𝐾 ) 𝐼 𝑙 = 𝑔 𝑙 (𝑉 𝑚 -𝐸 𝑙 )

Constant parameters:

The following values are taken from the publication of

Sodium conductance:

The sodium conductance depends on gating variables m and h.

𝑔 𝑁𝑎 = 𝑔̅ 𝑁𝑎 𝑚 3 ℎ 𝑑𝑚 𝑑𝑡 = 𝛼 𝑚 (1 -𝑚) -𝛽 𝑚 𝑚 𝑑ℎ 𝑑𝑡 = 𝛼 ℎ (1 -ℎ) -𝛽 ℎ ℎ

Gating variables and transition rate constants:

Gating variable n is a dimensionless quantity, comprise between 0 and 1, which corresponds to potassium channel activation. Gating variable m is a dimensionless quantity between 0 and 1, which correspond to sodium channel activation. 𝛼 𝑚 and 𝛽 𝑚 represent the transition rate constants.

Gating variable h is a dimensionless quantity between 0 and 1, which correspond to sodium channel inactivation. 𝛼 ℎ and 𝛽 ℎ represent the transition rate constants.

Wei model

Potassium, sodium and chloride concentrations:

The concentration of each ion type is continuously updated by integrating the relevant ion currents and fluxes. As an example, the rate of change of the number of intracellular

Sodium concentrations in Wei model

The dynamics of the number of intracellular Cl -,

𝑑𝑁𝐶𝑙 𝑖 - 𝑑𝑡
, is a function of ICl, Inkcc1, and Ikcc2.

𝑑𝑁𝐶𝑙 𝑖 - 𝑑𝑡 = 1 𝜏 (𝛾𝐼 𝐶𝑙𝐿 -𝐼 𝑘𝑐𝑐2 -2𝐼 𝑛𝑘𝑐𝑐1 )𝜈 𝑖 𝑑𝑁𝐶𝑙 0 - 𝑑𝑡 = 1 𝜏 (-𝛾𝛽𝐼 𝐶𝑙𝐿 + 𝛽𝐼 𝑘𝑐𝑐2 + 2𝛽𝐼 𝑛𝑘𝑐𝑐1 )𝜈 0

Chloride concentrations in Wei model

In these equations τ = 1000 is used to convert second to millisecond. νi and νo are the intracellular and extracellular volumes, respectively. β = νi/νo is the ratio of intra-/extracellular volume.

Nernst potential:

The reversal potentials of Na + (ENa), K + (EK), and Cl -(ECl) are given by Nernst equations:

𝐸 𝑁𝑎 =

Analog artificial neuron

n order to design an original and innovative analog artificial neuron, it was necessary to verify the existing solutions at the beginning of this work. To fulfill this task, a state-of-the-art of stand-alone artificial neurons in terms of power, energy dissipation, and occupied area was provided and presented in the first part of this chapter.

In the second part, the analogy between the artificial neuron and the Morris-Lecar model will be demonstrated. This artificial neuron can emulate with a good accuracy the behavior of living neuron, which we will be referring as "biomimetic neuron". It can also, instead of mimicking the biology, provide higher frequency spike rates to target large scale bio-inspired networks for the information processing applications. This version of the artificial neuron is designated "fast neuron". The biomimetic neuron and fast neuron as well as their design parameters will be presented in the third and the fourth part of this chapter.

To design an artificial neural network, it is necessary to include synapses between neurons, as in biology. Following this idea, we have connected these different artificial neurons with simple synapses described in the third part of this chapter to investigate phenomena encountered in human brain such as the burst phenomena, which is presented in the fifth part.

All these bio-inspired circuits have been fabricated within two different chips designated as WetWire and GreyMatter in the TSMC 65nm CMOS technology. We performed the characterization of these chips at IEMN laboratory and we will present corresponding results in the fifth and sixth part of this chapter. We will conclude on the performances of these proposed neuro-inspired circuits and give some perspectives for the following chips.

State of the art of artificial neuron

In the scope of neuromorphic, computing, different ways are used to fulfill this task such as hardware neural networks. Hardware devices, designed to realize artificial neural network (ANN) architectures and associated learning algorithms especially taking advantage of the inherent parallelism in the neural processing are referred as hardware neural networks (HNN) [START_REF] Misra | Artificial neural networks in hardware: A survey of two decades of progress[END_REF]. The choice of the technology is driven by the application and the aimed performances. It can range from silicon neuron [START_REF] Indiveri | Neuromorphic silicon neuron circuits[END_REF] to all memristor neuron and synapse [START_REF] Pantazi | All-memristive neuromorphic computing with level-tuned neurons[END_REF]) (La [START_REF] Barbera | Filamentary switching: Synaptic plasticity through device volatility[END_REF] and even spintronic oscillator neuron [START_REF] Torrejon | Neuromorphic computing with nanoscale spintronic oscillators[END_REF]. These components often I 3.1 State of the art of artificial neuron present several strengths as well as several constraints such as complicated circuits, immature technology and reliability issues. These basic blocks contribute to the hardware realization of artificial neural network (HNN). Therefore, artificial neuron represents one of the main building blocks for implementing neuromorphic systems [START_REF] Indiveri | Neuromorphic silicon neuron circuits[END_REF].

In our case, an artificial neuron is defined as a hardware implementation of the biological neuron. We focus on silicon artificial neuron based on complementary metal oxide semiconductor abbreviated as CMOS technology. This technology is predominant in micro-and nanoelectronics circuits as microprocessors, microcontrollers and other digital logic circuits. CMOS technology is also widely used for analog circuits such as image sensors, data convertors and so on.

Beyond these industrialized applications, CMOS technology allows a high-density integration of logic functions on a chip and became the most used technology to be implemented in very large-scale integration (VLSI) chips. The use of standard industrial technology was an important point to reduce the fabrication time of artificial neurons using foundries such as Taiwan Semiconductor Manufacturing Company (TSMC) and to scale up neural networks with the integration of more than 1000 neurons in a hardware neuronal network.

Moreover, it is possible to envisage a hybrid system where a CMOS neuron is connected to memristive synapses as illustrated in Analog spiking hardware approach is used to develop general purpose ultra-low power and large-scale neural systems. Different laboratories have worked on stand-alone analog neuron. The state-of-the-art analog artificial neurons based on their performances in terms of power, energy consumption and area are presented in Table 3 As shown in Table 3.1, the common neuron type is Leaky Integrated Fire (LIFpresented in the chapter 2), which represents one of the simplest and most widely used for such implementations. Based on results reported in Table 3.1, we can conclude that the most energy efficient stand-alone neuron was proposed by Cruz-Albrecht in 2012.

It consumes 40pW and 0.4pJ/spike with a core area of 442 µm 2 .

These artificial neurons have been investigated. Based on their performances an ultra low power artificial neuron has been proposed, with several versions and will be presented in the next section.

Design of artificial neuron

There are specific applications such as streaming video compression, which demand high volume adaptive real-time processing and learning of large datasets in reasonable time and necessitate the use of energy-efficient HNN [START_REF] Misra | Artificial neural networks in hardware: A survey of two decades of progress[END_REF]. To meet different applications, several versions of the artificial neuron have been designed in this work. The biomimetic neuron mimics the behavior of living neuron by reproducing biological action potential and time constants. The fast neuron is designed to generate higher spiking frequencies with minimal core area. These versions of the artificial neuron and their corresponding circuits will be explained in the following sections.

The artificial neuron is based on ML model, which is constituted of two non-linear differential equations associated with exponential functions. These equations can be represented by expressions based on current node summation, as per Kirchhoff's Current Law (KCL). These equations, Eq.3.1 to Eq.3.5 have been investigated in chapter 2.

The link between the HH model and electronic circuits is not a trivial matter due to the use of four non-linear differential equations. On the contrary, the ML model is much more attractive for this purpose containing only two state variables: the membrane voltage Vm and the potassium gating variable n (Eq.3.1 to Eq.3.5).

Based on the analyzes presented in chapter 2, we have concluded that the main channels responsible for generating the spike are the sodium and potassium channels. The dynamics of each ionic channel (Na + and K + ), can be reproduced by a single transistor (see Figure 3.2) biased in sub-threshold regime [START_REF] Farquhar | A bio-physically inspired silicon neuron[END_REF] ))] Eq.3.

3

𝑛 𝑠𝑠 (𝑉 𝑚 ) = 1 2 [1 + 𝑇𝑎𝑛ℎ ( 𝑉 𝑚 -𝑉 3 𝑉 4
))] Eq.3.4

𝜆(𝑉 𝑚 ) = 𝜆 0 𝐶𝑜𝑠ℎ ( 𝑉 𝑚 -𝑉 3 2𝑉 4
)) Eq.3.5 In this approach, we replace the calcium channel defined in the original Morris-Lecar model, by the sodium channel in our artificial neuron represented in Figure 3.3. The membrane capacitance (Cm) is charged through a PMOS transistor modeling the sodium channel (MPNa) and discharged through an NMOS transistor modeling the potassium channel (MNK) and the leakage conductance (GL). The leak conductance is in fact an NMOS transistor, which enables us to adjust the artificial neuron stability. We will discuss this point in the next section. A positive feedback loop (through MPNa and MP1/MN1) implements a pull-up network for the creation of Vm. The negative feedback loop, which implements a pull-down network, is defined through MNK and the two cascaded inverters MP 2 /MN 2 and MP 3 /MN 3 . The time constant of the negative-feedback loop is set by capacitance CK.

Excitatory and inhibitory synapses

The excitatory current Iex is assumed to be provided by a synaptic circuit which in the most rudimentary form can be implemented by a single transistor current source.

Therefor for the artificial neuron the synaptic circuit is a PMOS transistor. In fact, a PMOS transistor can model excitatory synapses through additional sodium channels, while conversely, inhibitory synapses would be represented by additional potassium channels and NMOS transistors (see Figure 3.4). As widely supported by many works (Arthur and Boahen, 2006) in order to introduce plasticity, the excitatory and inhibitory post synaptic currents can be controlled by interposing "weight" transistors (VWexc is the weight for excitatory synapse and VWinh is the weight for inhibitory synapse). In conclusion, the excitation of the artificial neuron is produced by the PMOS transistor, which is considered as the input of the artificial neuron. The membrane voltage of the artificial neuron (Vm) is connected to a buffer which is considered as the output of the artificial neuron. This part will be more developed in the "fabrication" section.

In the following paragraphs the artificial neuron circuit will be analyzed. This circuit analysis is divided in three parts: the drain current model, the static properties of the inverters in subthreshold operation and finally the circuit response equation.

Drain current model in the sub-threshold regime

All the transistors used in the artificial neuron are assumed to operate in the deep subthreshold regime, with a supply voltage lower than (or equal to) 200mV. Therefore, the drain current will have an exponential relation with gate-to-source voltage Vgs. Hence, the following expression is suited to model the drain current:

𝐼 𝑑𝑠 = 𝐺 0 exp ( 𝑉 𝑔𝑠 𝜂𝑉 𝑡 ) 𝑉 𝑑𝑠 Eq.3.6
where G0 represents the device conductance with Gn0 for NMOS and Gp0 for PMOS;

Vds is the drain-to-source voltages, 𝜂 is the subthreshold slope factor and Vt is the thermal voltage. The NMOS and PMOS ideality factors are assumed to have the same value.

As it will be proved, the use of such a simple model provides a comfortable medium for demonstrating the correlation between the circuit response and the ML model equations sufficiently. In addition to that, it allows a straightforward circuit analysis that is essential for the implementation effort in the CMOS technology.

Static properties of inverters in sub-threshold

The inverters play a central role in the artificial neuron performances. In fact, they are the elementary circuits used to reproduce the hyperbolic tangent function (Tanh) present in the Morris-Lecar model (see Eq.3.3 and Eq.3.4).

𝑚 𝑠𝑠 (𝑉 𝑚 ) = 1 2 [1 + 𝑇𝑎𝑛ℎ ( 𝑉 𝑚 -𝑉 1 𝑉 2
))] Eq.3.

3

𝑛 𝑠𝑠 (𝑉 𝑚 ) = 1 2 [1 + 𝑇𝑎𝑛ℎ ( 𝑉 𝑚 -𝑉 3 𝑉 4
))] Eq.3.4

A CMOS inverter operating in the sub-threshold regime is illustrated in Eq.3.7

The Tanh in Eq.3.7 can be simplified by using the following expressions where the ratio of conductances (Gn0/Gp0) appears:

𝑉 𝑖𝑠𝑣 ≅ 𝑉 𝑖𝑛 | (𝑉 𝑜𝑢𝑡 =0) = - 𝜂𝑉 𝑡 2 𝑙𝑛 ( 𝐺 𝑛0 𝐺 𝑝0 ) Eq.3.8
As shown in Eq.3.8, Visv represents the input voltage Vin when Vout = 0. It can be either positive or negative depending on the conductance ratio.

In order to simplify Eq.3.7, we also use the parameter Vq defined as follows:

𝑉 𝑞 = 𝑉 𝑖𝑛 -𝑉 𝑖𝑠𝑣 Eq.3.9

The output voltage described in Eq.3.7, shows two important properties of the proposed artificial neuron. Primarily, the switching voltages of inverters in this circuit are controlled by the conductance ratio. For example, with a ratio of 10 between the NMOS and PMOS conductance, the switching threshold can be shifted around 50mV. Hence, we can tune the required voltage shifts V1 and V3 in Eq.3.3 and Eq.3.4. Secondarily, in order to match the behavior of artificial neuron, the presence of voltage gain is necessary. As it has been shown by (Mead, 1989), the high value of the sub-threshold slope of MOS transistors (larger than 60 mV/dec) yields a much lower current-voltage slope than the one in biology. Hence, to mimic the neuron dynamic and reach the correct circuit operation, it is necessary to benefit of the gain brings by the presence of inverters.

In Eq.3.7, the maximum voltage gain Av will be obtained for Vout = 0. This can be expressed as:

𝐴 𝑣 = 𝑑𝑉 𝑜𝑢𝑡 𝑑𝑉 𝑖𝑛 | 𝑉 𝑖𝑛 =𝑉 𝑖𝑠𝑣 = - 𝑉 𝑑 𝜂𝑉 𝑡 Eq.3.10
According to Eq.3.10, the magnitude of Av can be larger than one for Vd > 𝜂Vt . For an 𝜂Vt about 35mV to 40mV, a significant voltage gain can be provided when Vd > 70-80mV. It is worth mentioning that this supply voltage value is close to the Nernst potentials encountered in biological cells. This last observation reveals that the artificial neuron can be implemented to operate under extremely low supply bias.

The circuit response analysis

In this part, we will demonstrate the analogy between the Morris-Lecar equations and the artificial neuron response. The artificial neuron and the two summing nodes VGK and Vm are presented in For the sake of simplicity, we consider a symmetrical supply voltage and neglect the transistor capacitances. The application of KCL on Vm and VGK nodes allows us to obtain the following equations:

𝐼 𝐶𝑚 = 𝐼 𝑁𝑎 -𝐼 𝐾 + 𝐼 𝑒𝑥 -𝐼 𝐿 Eq.3.11 𝐼 𝐶𝑘 = 𝐼 𝑝2 -𝐼 𝑛2 Eq.3.12
Where ICm and ICk are respectively the current through the membrane and the potassium capacitances. ICm is defined by the drain current of MPNa (INa), the drain current of MNK (IK), the excitatory current (Iex) and the leakage current (IL). In the same way, ICk results from the drain current of MP2 (Ip2) and the drain current of MN2 (In2). In the above equations, we replace all the drain currents by the Eq.3.6 and we obtain the following equations: The static power and the energy dissipation of the artificial neuron are described in the following paragraph and results will be presented in the next section. The static power is the product of the supply voltage (+/-Vd) by the leakage currents flowing through the inverters, the leak conductance (if present), the MPNa and MPK. The dynamic power is defined as the additional energy dissipation during spiking activity. It is related to the charge and discharge of the capacitances Cm and Ck.

In conclusion, we have demonstrated the analogy between the artificial neuron response and the Morris-Lecar model. Our approach is based on the use of a simple sub-threshold drain current model (Eq.3.6), the assumption of equal slope factors for PMOS and NMOS and a linearization step. This analytical model was used as a guideline for circuit design and was complemented by more rigorous circuit simulations presented in the next section.

Design of biomimetic neuron

In order to design the biomimetic neuron and to define its design parameters, we had to choose the appropriate bioinspired neuronal model, which enables us to obtain a simple circuit with state-of-the-art performance (energy, power consumption and area). For example, Rasche and Douglas (Rasche & Douglas, 2000) describe an analog implementation of Hodgkin-Huxley model with 30 adjustable parameters, which required 4mm 2 area for a single neuron. This does not correspond to our objectives.

In chapter 2, we have investigated neuronal models with four state variables (Hodgkin Huxley model and Wei model) and two state variables (Morris-Lecar). In order to design a simple circuit, we have chosen the Morris-Lecar model to be the principal neuronal model and we developed the artificial neuron based on this neuronal model. Nevertheless, as we wanted to reproduce also the biological aspect of a living neuron, we choose the Wei model as my reference model. The Wei model enables us to extract the main parameters involved in the human neuron spiking behavior. In the next sections, we will describe my methodology to use both models leading to the design of the biomimetic artificial neuron.

Design of biomimetic neuron: comparison with the Wei model

For some applications, the neuron dynamics should be compatible with the biological neuron one. The Wei model allows describing a wide range of neuronal activities. It unifies neuronal dynamics from usual spikes to seizures, enabling our understanding of the brain and defining what is essential in the control of pathological states [START_REF] Wei | Oxygen and seizure dynamics: II. Computational modeling[END_REF]. Two main criteria have been investigated, the shape of the spike and the excitatory response of the biological neuron. First, parameters like spike width (SW), peak width (PW), threshold voltage (Vth), resting potential (Vrest), absolute refractory period (ARp), spiking frequency (F) and peak-to-peak amplitude (Vpp) were extracted from the reference model proposed by Wei et al [START_REF] Wei | Oxygen and seizure dynamics: II. Computational modeling[END_REF]. The entire design of the biomimetic neuron relies on these parameters. Clearly, the biomimetic neuron must represent the same characteristic.

The second criterion is to understand the excitatory behavior seen in the biological neuron. This point was investigated in chapter 2 by varying the amplitude (Iex) and the width (Ts) of the excitatory current. It was verified that the combination of these parameters resulted in a similar spiking frequency and enabled a precise modulation of the number of spikes.

Thus, we concluded that the excitability of the neuron depends on the amount of injected charge (Q) and this parameter was chosen to compare the Wei model with the artificial neuron. Since the Wei model parameters are defined by unit area, while those of the circuit are defined for a whole neuron, a normalization coefficient is needed to compare the different models. This coefficient is defined as follows:

𝜉 = 𝐼 𝑒𝑥 𝐶 𝑚 ⁄ Eq.3.23
where Iex and Cm are respectively the excitatory current and the membrane capacitance. Using these two different criteria, we adjust the design of the artificial neuron, in order to obtain the same behavior of a biological neuron and designed the biomimetic neuron.

Two circuits have been designed for the biomimetic neuron, one with 6 transistors and another one with 8 transistors. The shape of sodium current (INa) and potassium current (IK) as well as their spike response are illustrated in In the previous section, we have presented the analogy between the artificial neuron and ML neuronal model. We simulated the biomimetic circuit with LTspice and Cadence software. The results of these simulations have been compared with the Wei model, which enabled us to choose the set of design parameters for the biomimetic neuron. These parameters are presented in the Table 3.2 andTable 3 As mentioned before in these circuits the excitatory current (Iex) is produced by a PMOS transistor with a gate length of Lg = 65nm and a gate width of w = 120nm. The main difference between these two biomimetic circuit beside the number of used transistors remains in the flexibility to modulate the threshold voltage. The threshold voltage is modulated with the first inverter (MP1/MN1). In the circuit with 6 transistors the first inverter is also connected to the potassium transistor (MNK). The circuit with 8 transistors enables an independent modulation of the threshold voltage due to the presence of a third inverter (MP3/MN3) connected to the potassium transistor. Hence it offers the possibility to adjust with accuracy the threshold voltage of the spike around -50mV, without impacting the potassium transistor. Hence, we have chosen the design of the biomimetic neuron with 8 transistors to be fabricated and from here now we will analyze the results from this biomimetic neuron.

Comparison between biological neuron and biomimetic circuit

In this part, we will compare the response of the biomimetic neuron (8 transistors) to the Matlab resolution of the Wei model for the same normalization coefficient ξ = 20A/F. The Wei model simulation will be referred as the biological neuron. As shown in Figure 3.9, one spike is obtained for both neurons. In this case, the amount of injected charge for the biological neuron and the biomimetic neuron are respectively Q = 35nC/cm 2 and Q' = 2.5fC. This number of charges is obtained for two different excitatory currents: Iex = 20μA/cm 2 during Ts = 1.75ms in the case of the biological neuron and Iex' = 1pA during Ts' = 2.5ms for the biomimetic neuron. This first result indicates that similar behaviors are observed and the study was continued with higher injected charges. As illustrated in Figure 3.10, we observe the generation of two spikes. In this case the amount of injected charge for the biological neuron and the biomimetic neuron are respectively Q = 175nC/cm 2 and Q' = 5fC. The corresponding excitatory current for the biological neuron is Iex = 20μA/cm 2 during Ts = 8.75ms and Iex' = 1pA during Ts' = 5ms for the biomimetic neuron. A good agreement is also noticed in this case. As illustrated in Figure 3.9 to Figure 3.11 similar responses are obtained for both biomimetic neuron and biological neuron. For both neurons, the ratio of excitatory current and membrane capacitance, i.e. the normalizing coefficient, is 𝜉 =20A/F. Therefore, we demonstrate the precise modulation of the number of spikes for both biomimetic artificial and biological neuron with the same normalizing coefficient. As shown in the above figures, as we increase the amount of injected charge (Q and Q') the number of spikes increases. This validates the second criteria mentioned before: the amount of the injected charge plays a central role in the excitatory behavior of the neuron.

In the next paragraph, we will compare the extracted parameters from the spike obtained with the biological neuron to the spike obtained with the biomimetic neuron. As reported in Table 3.4, the resting potential (Vrest) and the threshold voltage (Vth) for both neurons are respectively -70mV and -50 mV. We obtain the same values with the cadence simulation of the biomimetic neuron. The design of the biomimetic neuron is flexible as we can adjust the parameters presented in the Table 3.4. As mentioned before, Vrest is modulated with the leak conductance, the Vth is adjust with the first inverter, Vpp varies with the drain bias and ARp depends on the excitatory current.

Parameters

The peak width (PW) is defined at the threshold voltage (Vth) of the neuron while the spike width (SW) is defined at the minimum point (Minpoint) of the spike (see Figure 3.12).

Figure 3.12 Spike, peak width (PW) and spike width (SW).

A comparison of PW and SW of the biological neuron and the biomimetic neuron is reported in Table 3.5. As shown in Table 3.5, the peak width and the spike width of both neurons are similar. These close comparisons between these parameters validate the similarity between the biomimetic neuron and the biological neuron in terms of spike shape and excitatory response to an external stimulation. Therefore, we validate the first criteria mentioned previously; reproduce the exact shape of the spike based on defined parameters. These results validate the design of the biomimetic neuron patterned from Wei model.

Design of fast neuron

As shown in the previous section for biomimetic neuron, we choose to take in account two different criteria in order to generate action potentials similar to those in living neuron. For the design of the fast neuron, our purpose was to achieve a higher spiking frequency rate and a better energy efficiency with a compact design in order to build networks with several thousands of artificial neurons. These performances are obtained by trading off the biological accuracy for higher frequency and reduced silicon area. These targets were reached by reducing the value of the capacitances Cm and CK of one order of magnitude. Reducing the capacitances values affect the time constant of the spike and the frequency of the circuit. A satisfying spike shape was obtained even though we removed the leak and the third inverter of the artificial neuron. The reduction of the number of inverters will reduce the design flexibility of the artificial neuron concerning the threshold voltage as described before in the biomimetic section. These modifications are the main characteristics of the fast neuron. The fast neuron is composed of few elementary devices: two inverters (MP1/MN1 and MP2/MN2), sodium and potassium transistors (MPNa and MNK), membrane and potassium capacitances (Cm and CK). The schematic of the fast neuron as well as the design parameters are illustrated in Table 3 The fast neuron with design parameters presented in Table 3.6 was simulated and reproduces the excitability of the neuron with a reduced silicon footprint and a higher frequency performance. The shape of the generated spikes and the excitability of the neuron are still satisfying as illustrated in The spike width for the fast neuron is SW = 16μs. The peak width is defined at the threshold voltage (in this case Vth = -20mV), which corresponds to PW = 5μs. We observe a three orders of magnitude difference between SW and PW of the fast neuron versus the biomimetic ones (In the biomimetic neuron: PW = 1.25ms and SW = 3.44ms for three spikes). The resting potential Vrest is -90 mV. We observe a 20mV difference as we compare the Vrest of the fast neuron (-90mV) to the Vrest of biomimetic neuron (-70mV). There is 30mV difference between the Vth of the fast neuron (-20mV) and the Vth of the biomimetic neuron (-50mV). Therefore, the shape of the spike is slightly different from the biological spike. Nevertheless, the excitatory response remains the same.

The fabricated biomimetic neuron with the design parameters presented in Table 3.2 and the fast neuron with the design parameters presented in Table 3.6 as well as their corresponding performances will be presented in the following sections.

Fabrication of artificial neurons

The neuro-inspired circuits described in the previous sections have been designed and fabricated using TSMC 65nm CMOS process in the LP option. The test chip was called WetWire and its dimensions were 1.2mm x 2.1mm. The WetWire chip contains a variety of neuron circuits including different designs of the biomimetic (with and without leak conductance) and fast neuron, different versions of connected biomimetic to biomimetic neuron and finally connected biomimetic neuron to fast neuron called also tonic-Burst circuit. The connection of these neurons is made through excitatory and inhibitory synapses. Without accounting for the output driving buffer or protection circuitry, the area occupied by the biomimetic neuron is 200μm 2 . This value decreases radically for the fast neuron with 35μm 2 due to its low Cm and CK values. In fact, the capacitors dominate the area utilization, 65% for the fast and 70% in the biomimetic neuron. The fabricated chip is shown in VDDANA and ISSANA are pads to bias the artificial neuron's buffer. The output signal is observed from the VOUT pad, which corresponds to the membrane voltage (Vmem). These eight pads (except the VOUT) are biased in order to characterize the neuron circuit. The detailed set up of the test bench will be presented in the next section.

Characterization of Wet Wire chip

Test bench of WetWire chip

The experimental test bench as shown in Figure 3.17 was constituted of several instruments. For the neuron excitation, we used a Keysight 33500B series waveform generator, which biases the VIN pad. The Keithley 2636A system source meter has been used to bias VSS. Two Agilents SMU (E5273A and E6263A) containing two channels, are used to bias VDD, VDD_T, VDDANA and ISSANA pads. The Agilent SMU supplies used for biasing these pads also enabled us to perform the measurement of the average current with nominal 500fA accuracy. The values used for biasing the biomimetic and fast neuron are reported in the The VIN varies between 0 to 200mV. For VIN = 0, the PMOS transistor is open and thus the neuron spikes with a maximal frequency. In contrary, for VIN = 200mV, the PMOS transistor is completely closed and the neuron does not spike. The output signal issued from the buffer was monitored by ROHDE & SCHWARZ oscilloscope 500MHz 5GSa/s, in order to perform frequency, amplitude and spike width measurements.

The WetWire test bench.

Under probe measurements. Biomimetic neuron under probes.

Two Agilents SMU (E5273A and E6263A) connected to VDD, VDD_T, VDDANA and ISSANA pads.

One Keithley 2636A connected to VSS.

Pulsed characterization of biomimetic and fast neuron

In this part, we will demonstrate experimentally the generation of spikes with the biomimetic and the fast neuron and compare the biomimetic spike to the spike obtained by Wei model. The aim of this part is to verify that the injection of different amount of charges leads to the generation of different number of spikes from the artificial neuron. The input excitation, VIN is defined with two levels as follows: VIN_High corresponds to the maximal value of the pulse and VIN_low is the minimal value of the pulse. The Keysight generator enables us to define the duration of this pulse (Tp).

VIN_High_200mV Vin_Low_36mV Hence, we can conclude that the overall injected charge increases, which modulates precisely the generation of spikes as we expected. In fact, as described in chapter 2, we have demonstrated the precise modulation of the number of spikes based on the injected charge, which is also demonstrated experimentally in this part. In this section, we have demonstrated the ability of these artificial neurons to be externally excited using an amount of charge Q, as in biology.

Vin_High_200mV Vin_Low 25mV

Comparison of biomimetic pulsed characterization with Wei model

To validate the design of a biomimetic neuron, which provides physiological spikes, we will compare the spikes of the biomimetic neuron to the ones issued from Wei model. In Figure 3.22, the spike of the Wei model (Matlab simulation) is compared to the spike obtained with the fabricated biomimetic neuron (Fab_circuit). In both cases, the excitation pulse is applied during the same period T'' = 10ms. The excitatory currents applied in to the simulation and in to the Fab_circuit are respectively Iex = 20μA/cm 2 and Iex' = 1pA. Based on these values, we defined a normalization coefficient, as the ratio of the excitatory current and the membrane capacitance. We obtain 20A/F for both cases with Cm = 1μF/cm 2 for the simulation and Cm' = 50fF for the Fab_circuit. The peak-to-peak amplitude of the spike from the Wei model varies and have a maximal value of Vppmax= 120mV and minimal value of Vppmin= 97mV. For the biomimetic neuron (Fab_circuit), the peak-to-peak amplitude remains constant at Vpp = 122mV. The peak-to-peak values are thus similar. The spike width in both cases are SW = 4ms. Before presenting in detail the performance of the biomimetic and the fast neuron, we will present the results issued from the oscillatory neuron and the tonic burst circuit of the WetWire chip.

Characterization of oscillatory neuron

The oscillatory neuron (or unstable neuron) is designed to spike without any excitation, as it does not have a resting state due to the absence of an inhibitory synapse.

Characterization of tonic-burst circuit

The so-called tonic-burst circuit will generate a particular spiking mode of neurons called bursting (Fox, Rotstein, & Nadim, 2016). It leads to a particular generated pattern consisting of fast spiking separated by intervals of quiescence. In order to emulate the bursting mode, we consider the schematic shown in Figure 3.24.

Figure 3.24 Schematic for the tonic-burst circuit.

This tonic burst schematic is based on the association of a biomimetic neuron (referred as "low frequency" -LF-Neuron in Figure 3.24) with a Fast neuron (referred to high frequency -HF-Neuron in Figure 3.

24).

A key point stands in the different dynamics obtained by the HF and LF neurons. The HF neuron is connected to the LF neuron through an excitatory synapse. However, the LF neuron is connected to the HF neuron through an inhibitory synapse. This tonic-burst circuit behaves following this principle. First, the HF neuron is firing when an excitatory current is applied. Then, due to the connection between HF and LF neuron through an excitatory synapse, the membrane capacitance of the LF neuron integrates the synaptic current induced by every HF neuron spike. When the LF neuron membrane potential reaches its threshold, it starts to fire with spike duration much higher than the HF neuron ones. Because of the inhibitory synapse between these two neurons, the HF neuron is quickly inhibited. Finally, when the LF neuron repolarizes, the HF neuron depolarizes and fires again. This cycle holds as long as HF neuron is excited (see Figure 3.24). This circuit was implemented on WetWire chip under the name of " tonic_burst_h_top_DUT". The biases for this circuit are reported in the Table 3.7.

ISSANA_BF ISSANA_HF VDD VDD_T VDDANA VSS VIN_HF

-42μA -42μA 200mV 200mV 300mV 0V <200mV
Table 3.7 Bias values for the tonic-burst circuit also identified as tonic_burst_h_top_DUT on the Wetwire chip.

The spike generation of HF and LF neurons as a function of the input excitation VIN_HF is shown in

Biomimetic and fast neuron performance

This part is dedicated to the study of the biomimetic and fast neurons performance. In order to perform this analysis, both the spike frequency and the corresponding average power consumption (including the static and dynamic parts) have been measured as a function of the excitation current. The static power refers to a zero-excitation condition, while dynamic power consumption is deduced from the consumption under constant excitation, which induces a spiking mode. The excitability of the biomimetic neuron is shown in Figure 3.27. The minimal observed frequency is around 20Hz. As expected, the firing rate and power consumption increase with the excitatory current. For Iex = 120pA, a maximal spike frequency of 1.2kHz was obtained with a total dissipated power of 90pW. According to (Izhikevich, 2007), this biomimetic neuron is categorized as a Type I neuron. The energy efficiency of the circuit is deduced from the power dissipation and frequency measurements. In Figure 3.28, it is plotted as a function of the excitation current for two cases: (i) when the whole average power (static and dynamic -blue line) is taken into account and (ii) when the standby power (i.e., power consumption at zero excitation current) is subtracted (red line). For excitation current higher than 30pA, the dissipated energy per spike is roughly constant and therefore independent of the spike frequency. An energy efficiency value of 40fJ/spike is obtained when considering the dynamic power consumption. It is worth mentioning that this value is several orders of magnitude lower than the energy efficiency of actual biological neuron, as it is estimated around the 10pJ range, from the ATP consumption in (Attwell & Laughlin, 2001) (Lennie, 2003) (Poon & Zhou, 2011) and our estimation in chapter 2. This low energy dissipation obtained for the proposed artificial neuron can be interesting for spiking neuron networks applications needing the integration of a large number of neurons.

Concerning the fast neuron, a spike frequency as high as 26kHz was obtained with a total power consumption of 105pW for excitation current Iex = 150pA. The power consumption and spike frequency are illustrated in Figure 3.29 as a function of the excitation current. As already discussed, the main objective of this work was to minimize the energy dissipation: the circuit simulation predicted a value in the fJ/spike range. The energy efficiency was determined from measurements of the total dissipated power along with the spike frequency. The energy efficiency of the fast neuron is illustrated as a function of the excitation current in The curves demonstrate that the energy efficiency does not significantly depend on the output spiking frequency and the experimental dissipation reaches values as low as 3fJ/spike when only the dynamic power is considered.

Introduction to GreyMatter chip

The 'GreyMatter' chip has been fabricated in TSMC 65nm technology in order to enable a precise characterization of the elementary circuits of the artificial neurons. As shown in Figure 3.31, it contains various elementary circuits: inverters with different widths, buffer of biomimetic neuron, buffer of fast neuron, a standalone biomimetic neuron and fast neuron without ESD protection. The artificial neuron without ESD protection allows the accurate estimation of the current induced into the circuit. In this case, unlike the WetWire chip, we have a direct access to the exact value of the excitation current. For the WetWire chip, we had to de-embed the value of the induced current due to the diodes protections in order to estimate the exact value of the excitatory current. We have used the GreyMatter's biomimetic and fast neuron to perform a noise analysis, which will be presented in chapter 4. In this section, we will present the buffer characterization, which will allow us to characterize the artificial neurons in chapter 4. investigation of this point was out of the scope of this thesis and we will not detail this point.

Conclusion

In this chapter, we have presented the state-of-the-art standalone artificial neurons in order to evaluate the performance of the existing versions. We analyzed the neuronal membrane, which enables us to design and fabricate two standalone artificial neurons with state of the art performances in terms of power consumption, energy efficiency and area as reported in Table 3.9.

An energy efficiency (excluding the DC power) of 3fJ/spike for the fast neuron and 40fJ/spike for the biomimetic neuron has been extracted. An improvement of two orders of magnitude below the state of the art energy efficiency in (Cruz-Albrecht, Yung, & Srinivasa, 2012) has been achieved, associated to a silicon area decreasing by more than one order of magnitude. This is a very encouraging result in the road to diminish the power consumption of spiking neural networks.

Table 3.9 State-of-the-art performances of reported stand-alone artificial neurons. The energy efficiency is presented including the DC power.

We have connected these ultra-low power artificial neurons with specific synapses and investigated the burst-spiking mode. We have characterized the WetWire and GreyMatter chips. The output spike of the biomimetic neuron was similar to the one in the biological neuron as we expected. In contrary, the frequency obtained for both neurons circuits was less than expected from cadence simulations. In fact, we expected to obtain 1MHz for the fast neuron, which wasn't achieved. The fast neuron was retro simulated in cadence and the excitatory current obtained was higher than the current obtained experimentally. This point will be investigated in future work by studying the sub-threshold NMOS and PMOS transistors of the GreyMatter chip. We have also observed a variability of 70% for the frequency in the spiking train delivered by the fast neuron. This point will also be investigated in future work. Nevertheless, this variability can be interesting for specific spiking neural network, where a sparse system needs to be investigated. Obviously, this variability must be controlled for applications, which require a precise frequency and a uniform spiking neural network. We have also observed an increase of the experimental excitatory current due to the presence of the room light, therefore all the measurements have been performed in the dark environment to avoid this fluctuation.

Chapter 4.

Noise in artificial neurons

Noise in brain

npredictable fluctuations and disturbances that are not part of a signal, called noise, are present in the brain. It is known that the brain is a noisy environment with different sources of noise. In fact, stochastic behavior is observed at different stages in the nervous systems (see Figure 4.1) [START_REF] Faisal | Noise in the nervous system[END_REF]. Bottom stages such as voltage-gated ion channels and neurotransmitter release mechanisms (see In the behavioral task shown (catching a ball), the nervous system has to act in the presence of noise in sensing, information processing and movement [START_REF] Faisal | Noise in the nervous system[END_REF].

U

For instance, in 2015, Stanford neuroscientists revealed that noisy neurons are critical for learning [START_REF] Engel | Choice-correlated activity fluctuations underlie learning of neuronal category representation[END_REF]. Based on these findings, it is obvious that noise plays an important role in the brain function. Hence, in this chapter, the impact of the noise on the artificial neuron response will be investigated. First, the membrane voltage fluctuations of the biomimetic and fast neuron will be studied. Second, the response of the biomimetic neuron to an excitatory synaptic noise will be analyzed. Finally, the stochastic resonance phenomena within the biomimetic neuron will be investigated.

Membrane voltage fluctuations of the artificial neurons 4.2.1 Noise and membrane voltage fluctuations

Noise in neurons causes membrane voltage fluctuations (MVF) even in the absence of synaptic inputs. The most dominant source of such noise is channel noise [START_REF] Faisal | Noise in the nervous system[END_REF]. Channel noise is defined as the thermodynamic fluctuations in the gating channel ions which produce probabilistic gating behavior (Faisal, White, & Laughlin, 2005). In this section, the membrane voltage fluctuations of the artificial neuron at resting state (when the neuron does not spike) will be presented. This phenomenon has been widely investigated in biological neurons [START_REF] Destexhe | Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons[END_REF] [START_REF] Richardson | Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons[END_REF]. A standard deviation (SD) of 4mV has been observed for the MVF of the biological neuron [START_REF] Destexhe | Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons[END_REF].

The aim of this study is to compare the artificial neuron and the biological neuron in terms of membrane voltage fluctuations. Another objective is to show that the artificial neuron source of noise in the resting state is mainly thermal. Hence, assuming an equivalent circuit of the neuron composed by a conductance in parallel with a capacitance, the standard deviation of the membrane voltage fluctuations, SD, will be simply given by: 𝑆𝐷 = √ 𝐾𝑇 𝐶 𝑚 Eq.4.1

In this expression, SD is the RMS noise voltage while K, T and Cm are the Boltzmann constant (K = 1.38 10 -23 J.K -1 ), the room temperature (T = 300K) and the membrane capacitance respectively. Based on Eq.4.1, we calculated the thermal noise standard deviation for the biological neuron, artificial biomimetic and artificial fast neuron (see Table4.1).

Neuron Cm SD

Biological neuron 100pF 6.43μV

Artificial biomimetic neuron 50fF 0.27mV

Artificial fast neuron 4fF 1.01mV The thermal noise in the biological neuron with a 100pF membrane capacitance is less than 10μV, which is much lower than the voltage fluctuations measured on biological neurons. Studies have shown that the membrane voltage noise in the case of cultured neuronal cells is about 0.1mV (Diba, 2004). In this case, only few synapses are connected to the neuron, and this noise could be referred to as the intrinsic noise of the cell. If a higher number of synapses are connected to the cell, the total neuron noise is estimated to be around few millivolts (4mV) as reported by [START_REF] Destexhe | Fluctuating synaptic conductances recreate in vivo-like activity in neocortical neurons[END_REF] where the number of synaptic connections of one neuron is around 10 4 .

These experimental data show that (i) the membrane voltage noise in biological neurons is very high as compared with fluctuations due to thermal noise, and (ii) a significant part of the membrane voltage noise in living neurons is due to the synaptic noise, that is to say the noise generated by the synaptic activity.

Artificial neuron membrane voltage fluctuations test bench

The artificial neuron of the WetWire chip and the artificial neuron of GreyMatter chip contains 8 pads (see The NI through the LabVIEW software enables us to acquire data in a variable period of time. This point will be detailed in the next section concerning the synaptic noise analysis where the acquisition time becomes a critical issue. Different Matlab codes were developed to analyze the data of the membrane voltage fluctuations.

Artificial neuron membrane voltage fluctuations at resting state

In this section, we will present the mean and standard deviation of the membrane voltage for the biomimetic and fast neuron at different resting states. Four cases are defined to characterize and determine specifically the membrane voltage fluctuations at resting state: disconnected neuron (case 1), disconnected trans-conductance (case 2), connected neuron (case 3) and neuron off (case 4). These different cases are also defined to experimentally investigate the influence of the buffer. In the following paragraphs, we will explain the biasing conditions used for each of these cases:

Case 1: Disconnected neuron

To obtain the disconnected neuron, the artificial neuron (biomimetic or fast) is off, by disconnecting VIN, VDD_T and VDD (VIN= VDD_T= VDD= NC). The VSS remains at 0V. The buffer is on, with VDDANA = 300mV and ISSANA = -400μA for the fast neuron and VDDANA = 300mV and ISSANA = -43μA for the biomimetic neuron. The artificial neuron does not generate any spikes, as it is not biased. For simplification, we will refer to this biasing as case 1.

Case 2: Disconnected trans-conductance

As described in chapter 3, the trans-conductance is a PMOS transistor that injects an excitatory current to the artificial neuron when VIN and VDD_T are correctly biased. In this case, VIN and VDD_T are open (VIN = VDD_T = NC). The buffer is on. The remaining pads of the artificial neuron are VDD = VSS = 0V. In this case, the neuron does not generate any spikes. For simplification, we will refer to this biasing as case 2.

Case 3: Connected neuron

To obtain the connected neuron VIN and VDD_T are open (VIN = VDD_T = NC). The buffer is on. The artificial neuron is biased with VDD = 200mV and VSS = 0V. In this case, the neuron does not generate any spikes. For simplification, we will refer to this biasing as case 3. The corresponding mean and standard deviation for each case of the biomimetic neuron are extracted from the PDF and are presented in Table 4.2.

The membrane voltage mean value varies from -183mV in the case 1, when only the buffer is on, to -180mV in the case 4, where the neuron is biased without any excitation current. The standard deviation varies from 0.36mV in the case 1 to 0.23mV in the case 4. As we compare these experimental values to the theoretical standard deviation value (0.27mV for the biomimetic neuron presented in Table 4.1), we can conclude that as expected (see Table 4.1) the biomimetic membrane voltage fluctuations is mainly thermal.

Additionally, the standard deviation observed in biological neuron (4mV) is higher than the one in the biomimetic neuron. Based on this experiment, we can also conclude that in the resting state the observed noise in the artificial neuron is due to the thermal noise.

The impact of the buffer noise has been also investigated using the noise matrix technique. We demonstrated that the noise of the buffer is negligible compared to the noise of the artificial neuron. 4.3, the membrane voltage mean value of the fast neuron varies from -114mV in the case 1 to -105mV in the case 4. The standard deviation is around 0.4mV. This experimental standard deviation value is slightly less than the theoretical standard deviation calculated from the Eq.4.1 of 1mV. corresponds to disconnected biases, in all these cases the buffer is on with VDDANA = 300mV and ISSANA = -400μA. In conclusion, for the fast neuron, the standard deviation mostly does not change for these different cases. However, the membrane voltage mean value is more impacted than the standard deviation, with 10mV variation between case 1 and case 4.

Case

The PDF of case 1, for both fast neuron and biomimetic neurons, corresponds to the minimum mean voltage value compared to other cases. The mean value increases for both neurons respectively from case 2, case 3 to case 4 where the maximum mean value of the membrane voltage is obtained.

Artificial neuron near threshold membrane voltage fluctuations

It is interesting to investigate the membrane voltage fluctuations of the artificial neuron near its threshold. The near threshold state is obtained by biasing the trans-conductance with VINmin, which is defined as the minimum value for VIN when the neuron does not spike. This value for the biomimetic neuron is around VINmin = 69.8mV. We have compared the probability density function for VIN = 180mV to the VINmin as illustrated in For VIN = 180mV, we observe a mean and standard deviation of respectively -180.5mV and 0.23mV. This standard deviation corresponds to 192K in terms of temperature of noise (Eq.2.1). For VINmin = 69.8mV, both mean and standard deviation, increase to respectively -174.2mV and 0.40mV. This standard deviation corresponds to 580K in terms of temperature of noise.

To investigate this effect on our artificial neuron, this synaptic noise was reproduced by adding a Gaussian noise on the excitatory synapse. As explain in Chapter 3, the excitatory synapse of the artificial neuron is the PMOS trans-conductance. The Gaussian noise is applied on the gate of PMOS. Due to the non-linearity of the transconductance (sub-threshold mode), the drain-source noise current (I(t)) is composed of a DC and variable part as follows:

𝐼(𝑡) = 𝐼 0 + 𝜎 𝐼 𝐵(𝑡) Eq.4.2

where σ I is the standard deviation and B(t) is a Gaussian process defined as follows:

𝐸[𝐵(𝑡)] = 0 Eq.4.

3

𝐸[𝐵(𝑡)𝐵(𝑡 -τ)] = exp (- τ τ c ) Eq.4.4 𝐹𝑐 = 1 / (2π. τ c )
Eq.4.5

where Fc is the cutoff frequency of the Gaussian noise source. This cutoff frequency was chosen (Fc = 100kHz) to be larger than the neuron cutoff frequency (Fn = 2kHz), hence the Gaussian noise is approximately a white noise. This noise current (I(t)) is then integrated by the membrane capacitance of the artificial neuron as follows:

𝑉(𝑡) = 1 𝐶 ∫ 𝐼(𝑡)𝑑𝑡 = 1 𝐶 (𝐼 0 𝑡 + σ I 𝑊(𝑡)) 𝑡 0
Eq.4.6

where W(t) is the standard Wiener process that is defined as follows:

𝑊(𝑡) = ∫ 𝐵(𝑡)𝑑𝑡 𝑡 0
Eq.4.7

In fact, V(t) can be considered as a Brownian motion with drift and therefore:

𝜇 𝑣 = 𝐸[𝑉(𝑡)] = 𝐼 0 𝑡 Eq.4.8 σ 𝑣 = 𝑣𝑎𝑟(𝑉(𝑡)) = 2σ 𝐼 2 τ c 𝑡 𝐶 2
Eq.4.9

Let T th be the first time V (t) reaches the spike threshold value Vth, then:

It can be shown (Capasso, 2011) that the hitting time Tth follows the inverse Gaussian distribution hence: 4.11 In this chapter, the membrane voltage and the inter spike interval (ISI) distributions of the artificial neuron under a Gaussian noise excitation will be investigated.

𝑇 𝑡ℎ ~𝐼𝐺( 𝐶 𝑉 𝑡ℎ 𝐼 0 , 𝐶 2 𝑉 𝑡ℎ 2 2σ I 2 𝜏 𝑐 )} Eq.

Input noise characterization

We have characterized the generated noise from the Keysight 33500B to validate its Gaussian nature. In fact, the Keysight will be used as the source of the noise and it is necessary to verify the nature of its generated noise signal.

The input noise has a bandwidth of 100kHz with an offset value of 20mV. This noise was analyzed for two acquisition times window: first, short acquisition time of 13s and second, long acquisition time of 30min. For each acquisition time two level of noise was investigated: Vin Noise = 28.40mVrms which is considered as a low level of noise for our artificial neuron and Vin Noise = 45.45mVrms considered as high level of noise. We used a sampling frequency of 200kHz in the LabVIEW program to extract the noise.

For the short acquisition, the noise was extracted during 13s over 11 trials. To investigate the effect of time on the noise, we have extracted the 11th trials one hour after the acquisition of the 10th trials. We calculated the half width at half maximum of each of these probability density functions. We found that it corresponds to the injected RMS value, as expected. For example, for Vin Noise = 45.45mVrms, the half maximum of the PDF (max value of PDF multiplied by e -1/2 ) is about 36. Based on this value, the half width at the half maximum is about 45mVrms, which corresponds to the input noise.

We have extracted several parameters: the mean value (M), the standard deviation (SD), the variance (VAR) and the median (MED) for each trial. These parameters remain constant with same value for each trial. The mean and median are equal to 0.02V, which corresponds to the 20mV offset value, of the injected noise. The standard deviation is around 4.2mV and the variance varies around 17μV. As we increase the noise signal from 28.40mVrms to 45.45mVrms, the standard deviation increases from 4.2 mV to 6.7mV and the variance changes from 17μV to 45μV. The mean and median remain around 0.02V.

To perform the long acquisition, we have extracted 180 consecutive trials (the acquisition time of each trial is 10s) of the noise and rebuild the noise signal which has an acquisition time of 30min. Same values are obtained for the maximum value of the PDF, during 30min and 10s. The extracted noise parameters (mean, standard deviation, variance and median) of these two levels of noise also remain constant during the 30min acquisition. These parameters have the same values as the parameters obtained for the short acquisition.

Based on these results, we conclude that the generated noise is a Gaussian noise.

As we compare the parameters obtained from the short acquisition to the long acquisition, we conclude that the acquisition time does not impact the input noise. This study validates that, the Gaussian noise generated from the Keysight 33500B remains uniform in time, which is an important point, as we will see in next paragraphs. In the next paragraphs, we will study the response of the artificial neuron to a synaptic noise of a Gaussian nature.

Biomimetic neuron DC characterization

In order to analyze experimentally the response of the artificial neuron to a synaptic noise excitation, a DC characterization was first carried out. Due to the variability observed before, we define a set of criteria to choose the best circuit for the noise analysis. The DC characterization enables us to extract parameters, which allow an accurate noise measurement. The DC characterization has been carried out for 30 biomimetic circuits present on the GreyMatter chip. Four parameters of each artificial neuron have been considered in the DC characterization as follows:

1) The maximal frequency (Fmax)

2) The membrane voltage (VOUT) when VIN = 0V

3) The peak-to-peak amplitude of the spike (mVpp) 4) The minimal value for VIN which the neuron does not generate spike (VINmin)

Based on these four parameters, we define the best circuit for a biomimetic neuron if it presents:

1) Fmax < 2kHz 2) VIN = 0V => VOUT = Vrest (the VOUT is at resting state).

3) 100mV ≤ mVpp ≤ 200mV 4) VINmin ≥10mV

Based on the DC characterization, only 10 biomimetic neurons displayed these four satisfying parameters and therefore were chosen to carry out the noise analysis.

Synaptic noise test bench

The For these biasing values, the noise measurement has been carried out for two different categories: short acquisition, long acquisition. The sampling rate of 5kHz that corresponds to a sampling time of 200μs is used for these categories.

The time acquisition of the membrane voltage is 13s for the short acquisition. This time increases to 30min for the long acquisition. In fact, the biomimetic neuron spike train is extremely stochastic as shown in Figure 4.13 for low levels of injected noise. For this specific case, we have observed interspike intervals (ISI) as long as 40s. Hence, we developed a LabVIEW program to enable us to increase the time acquisition according to the observed ISI and the level of injected noise.

Several analyses have been performed for these two categories. First, the spike extraction with different time duration, using the National Instrument (NI PXIe-1073).

Second, the extracted spikes are analyzed with developed Matlab codes to first detect the spike and then extract the interspike intervals known as ISI. Third, different ISI parameters such as the mean, the standard deviation and the variance are extracted. Forth, the histogram and the probability density function of the obtained ISI are plotted for different noise levels. Finally, different distribution fits such as gamma, inverse Gaussian, exponential and Poisson are investigated and compared to obtained results.

Spike detection and ISI

For the noise analysis, we extract the membrane voltage also called spike train of the biomimetic neuron for different levels of excitatory Gaussian noise. The spike train enables us to detect the occurrence time of each spike, also called spike detection, in the spike train. The ISI parameters obtained from the injected noise on the PMOS gate and the PMOS source are similar. Hence, we will focus on the injected noise on the PMOS gate in the next paragraphs. The mean (M), standard deviation (SD), variance (VAR), scale parameter (Θ) and coefficient of variation (CV) decrease as the noise level increases. The shape parameter (K) increase as the noise levels increase, it saturates around 10 in both cases.

In this experiment, the first level of noise for both cases was 25mVrms but as shown in Due to this poor statistic for this low noise level, we have developed another LabVIEW program in order to increase the acquisition time, which will be presented in the next section.

Long acquisition

The long acquisition characterization enables us to obtain an improved statistic for the low level of noise. In order to obtain this long acquisition, we have developed a new LabVIEW program to acquire 180 consecutive spike trains of 10s. We have also developed a Matlab code in order to rebuild the totality of these 180 spike trains. For each level of noise, we have reported in Table 4.4 the number of spike, the number of ISI, the bin width and the number of bins for an acquisition time of 30min. We have extracted the ISI parameters of each of these spike train as shown in (Pachitariu, Brody, Jun, & Holmes, n.d.).

Beyond a certain level of noise considered as high level of noise (30.68mVrms for the PMOS gate and 28.40mVrms for the PMOS source) the artificial neurons response tends to the inverse Gaussian distribution. This point is put in evidence in Figure 4.29 when a maximum noise of Vin Noise = 28.40mVrms is applied to the PMOS source of the biomimetic neuron.

Simulation of biomimetic neuron response to the synaptic noise

We have analyzed the spike train of the biomimetic neuron for different level of Gaussian noise using LTspice simulations. It enables us to extract the injected current to the membrane capacitance of the biomimetic neuron for different level of injected noise on the PMOS gate. The Gaussian noise level varies from 27mVrms to 100mVrms. To obtain a Gaussian noise in LTspice, we use the random function (rand (time*1e6)) and injected to the PMOS gate of the biomimetic neuron. For instance, to simulate a 27mVrms Gaussian noise, we use the following expression (E. The acquisition time, spike frequency, average current and RMS current are reported in Table 4.7 for distinctive level of noise. The inverse Gaussian and Gamma distribution seems to correspond to the PDF of obtained ISI for different level of noise expected the very low level of noise (27mVrms).

Stochastic resonance in the artificial neuron 4.4.1 Stochastic resonance history

After studying the membrane voltage fluctuations, which corresponds to the noise of artificial neuron in resting state and the response of the artificial neuron to an excitatory synaptic noise, we will investigate in this section the phenomenon of the stochastic resonance.

Historically, stochastic resonance has been defined as a paradoxical phenomenon as induced noise enhances the overall performance of the nonlinear system. Since 1980, when stochastic resonance was introduced in statistical physics, the concept has evolved and has been observed throughout diverse scientific fields, ranging from biological systems to nonlinear physical systems [START_REF] Mcdonnell | What Is Stochastic Resonance? Definitions, Misconceptions, Debates, and Its Relevance to Biology[END_REF]. Stochastic Resonance (SR) has been studied in three different fashions, the first focused on physical and mathematical descriptions of SR (Jung, 1993), the second investigated SR in electronic systems and its potential applications (Harmer, Davis, & Abbott, 2002), and the last have dedicated their research to study SR in biology [START_REF] Faisal | Noise in the nervous system[END_REF], [START_REF] Mcdonnell | [END_REF]. It is worth mentioning, combining these different aspects of SR phenomenon opens tantalizing perspectives, from the development of new families of sensors to brain research enabling bioinspired processing and medical applications. From a neuroscience point of view, SR is ubiquitous in neural systems and plays a major role in facilitating the information processing. These stochastic resonance features have been demonstrated both experimentally and theoretically from the complete organism (crayfish, paddlefish and human) to neural networks and the unitary cell (shark multimodal sensory cell, hippocampal model, cat cortex model) [START_REF] Mcdonnell | [END_REF].

Simulation response of biomimetic neuron to stochastic resonance

The SR phenomenon is observed with LTspice simulation with a resolution time of 10μs. In fact to obtain the SR phenomena it is necessary to combine a sub-threshold signal (VS) (here sinusoidal signal) with a broadband noise (VN) (see For this low level of noise, the artificial neuron generates only one spike for each negative period of VS. As we increase the level of noise (VSN), the number of spike will increase. This part will be presented in detail in the next section (experimental investigation of SR). For this experiment, the Keysight is used to generate both a sub-threshold sinusoidal signal (VS) of weak amplitude and weak frequency (10mVpp, 10 Hz) with an offset value of 110mV and a broadband noise (VN). The resulting combined signal is called VSN and is injected to the input of the neuron, which is the gate of the PMOS transconductance (see Figure 4.36). The output of neuron (VOUT), which corresponds to the membrane voltage after the buffer, is connected to R&S oscilloscope. In this condition, and without significant noise amplitude, the biomimetic neuron stayed in its resting state since VSN was below than the firing threshold as shown in These results clearly highlight the stochastic resonance phenomenon as observed in biology (Hänggi, 2002).

Based on the recording of VOUT for 1s which corresponds to ten times the period of VS, the statistical firing probability, as defined in [START_REF] Chen | Real-time simulation of biologically realistic stochastic neurons in VLSI[END_REF] is obtained by dividing the 1s recording in 10 parts. Each part is divided in 10 and the corresponding number of spikes is counted. The number of spikes is summed for each 0.1s and illustrated in In order to define the error probability of the biomimetic neuron, Vm was recorded for various levels of induced noise during 1s. Each period of sub-threshold signal VS is divided in two states: the active state and resting state. The active state corresponds to the biomimetic neuron excitation for the half-negative period of VS. The active state is referred as 1 in Figure 4.38. The resting state is defined when biomimetic neuron is excited during the half-positive period of VS. The resting state is referred as 0 in The initial state is defined when biomimetic neuron generates only one spike in the active state of VS. When increasing VN, the number of generated spikes is rising (13 spikes for VN= 26mVrms). It is obvious that the overall power consumption is mainly due to biomimetic neuron's standby power; such a low value demonstrates the biomimetic neuron's power efficiency for detection applications. The energy efficiency, for initial detection (number one in Figure 4.41), was estimated from the dynamic power (subtracting the standby power); in this case the biomimetic neuron encodes subthreshold sinusoidal frequency of 10Hz; energy efficiency lower than 100fJ/spike is achieved. Such energy efficiency, compared to other stand alone artificial neuron (see chapter 3), is roughly one order magnitude lower than the state of the art.

Conclusion and perspectives on stochastic resonance

This work investigated and demonstrated the classical stochastic resonance phenomenon experimentally within the context of a biologically realistic artificial neuron. Ultra-low power consumption has been demonstrated while performing detection activity (between 54pW and 57pW). Considering more natural signals [START_REF] Faisal | Noise in the nervous system[END_REF], [START_REF] Duan | Stochastic resonance with colored noise for neural signal detection[END_REF] (such as electrical fields generated by predators or prey, lights, sounds, etc.) and biologically relevant sources of noise (synaptic noise, motor noise, colored noise, etc.) will enable to evolve stochastic facilitation and information processing within the biomimetic neuron. Furthermore, producing connected biomimetic neuron networks which demonstrate SR can be very promising and is currently under study. Finally, biomimetic neuron's biological features imply brand-new investigation of SR phenomenon, where a hybrid system of connected artificial neurons to biological neurons is studied, and aiming new biomedical applications.

General conclusion

ature remains the best source of innovation. Many researchers have been inspired by nature and fabricate novel devices and systems to ensure a better life. In the 21st century, a growing need of intelligent systems with highenergy efficiency is emerging as described in chapter 1. The most important challenge is to continue to increase the computing performance of our computers while controlling their energy consumption. The heating of the processors not only jeopardizes their efficiency, but could also, if the current trend continues, create an energy shortage. Information and communication technologies consume nearly 10% of the worldwide energy generated and the cloud represents the fifth most electricityconsuming country after the USA, China, Russia and Japan. This energy consumption will not stop growing in the coming years due to the increase of generated data, which must be processed and stored. Moreover, the problem of the processors power dissipation has induced manufacturers since 2005 to take an important step: stop increasing the clock frequency of processors and turn to multi-core architectures. Finally, the miniaturization of components will inexorably lead to the emergence of defects that cannot be avoided.

Based on these issues and the need of ultra-low power intelligent systems, we have investigated one of the principal unities of the brain, the neuronal cell and build the first block of an intelligent neuromorphic system, the artificial neuron. This work is a step toward very large-scale hardware networks of neuronal circuits known as Hardware Neuronal Networks (HNN). We proposed in this work a high-energy efficient and simple artificial neuron with a conventional CMOS technology to be the foundation of a future high energy efficient HNN. This thesis is a multidisciplinary work; it targets the design and fabrication of an ultralow power artificial neuron with 65nm CMOS technology. In order to achieve this goal, the chapter 2 is dedicated to understand the electrophysiology of a neuron cell. We verified the excitability of the neuron based on different neuronal membrane models and studied the spikes generation as a function of the charge density. This study allowed us to estimate the power and energy efficiency per spike for neuronal models such as Hodgkin Huxley (HH) and Wei model.

Both HH and Wei model are based on four nonlinear differential equations (four state variables). These complex models lead to complex electrical circuits with considerable silicon surface. Thus, we investigated in chapter 2 the biophysically meaningful Morris-Lecar model as it represents a simpler model with only two nonlinear differential equations (two state variables).

N

The chapter 3 presents the analogy between the two states variable Morris-Lecar model and the designed artificial neuron. The four states variables Wei model enabled us to adjust the parameters of this artificial neuron to obtain a bio-inspired neuron referred as the biomimetic neuron. Additionally, in the chapter 3, we present a simple version of the artificial neuron called the fast neuron. The characterization of various manufactured artificial neurons and neuronal circuits such as the oscillatory neuron and the tonic burst neuronal circuit is also presented in chapter 3.

It is worth mentioning that the biomimetic neuron core occupies 200μm 2 and a maximal spike frequency of 1.2kHz was obtained with a total dissipated power of 90pW. An energy efficiency (including the DC power) of 78.3fJ/spike is obtained for the biomimetic neuron, reaching values as low as 40fJ/spike when only the dynamic power is considered.

The fast neuron has a higher frequency and smaller area in comparison to the biomimetic neuron. It reaches a spike frequency as high as 26kHz with a core area of 35μm 2 . The total power consumption of the fast neuron is 105pW. An energy efficiency (including the DC power) of 4fJ/spike is extracted and the power dissipation reaches values as low as 3fJ/spike when only the dynamic power is considered.

Based on these reported values, state-of-the-art performances were achieved in this work, as the artificial neuron is 1000 times more energy efficient than a living neuron. Additionally, it represents the state-of-the-art in terms of power, energy efficiency and area compared to previous works on the stand alone artificial neuron.

In chapter 4, various distinct noise analysis based on the noise phenomena observed in one living neuron were investigated. In fact, the membrane voltage fluctuation, the synaptic noise and the stochastic resonance, which occurs in the brain neuronal system, were studied within our artificial neuron.

Concerning the membrane voltage fluctuation investigation, we conclude that: (i) the membrane voltage noise in biological neurons is very high as compared with fluctuations due to thermal noise, and (ii) that a significant part of the membrane voltage noise in living neurons is due to the synaptic noise. This remains true as we compare the membrane voltage fluctuation of the artificial neuron to the living neuron.

Concerning the synaptic noise analysis, we injected a Gaussian noise to the PMOS excitatory transconductance of the biomimetic neuron to reproduce the synaptic noise both experimentally and in Ltspice simulation. Different spike trains were obtained from the biomimetic neuron as a function of different levels of injected Gaussian noise. We extracted the interspike intervals (ISI) from these spike trains and their probability density function (PDF) for short and long acquisition time windows. Based on the experimental data it seems that the PDF of ISI follows different distributions as the level of noise increase. For low levels of noise, it seems to follow the Exponential and the Gamma distribution while for higher level of noise it seems to follow the Inverse Gaussian distribution. Based on the simulation data and for high level of noise, the PDF of ISI follows Gamma distribution as well as Inverse Gaussian distribution.

Concerning the stochastic resonance analysis, we demonstrated experimentally the classical stochastic resonance phenomenon within the biomimetic neuron. Ultra-low power consumption has been obtained in the frame of detection activity (between 54pW and 57pW). The energy efficiency was estimated from only the dynamic power and it is lower than 100fJ/spike for the initial detection (when the neuron spike only once for the negative period of the sinusoidal signal). The artificial neuron high-energy efficiency is an important point for detection applications.

Perspectives

The perspectives of this work are divided in two categories, the computing and the biomedical aspect.

For the computing aspect, and precisely for the hardware neural networks (HNNs) part, the aim is to develop an energy efficient neuromorphic system, able to communicate, learn and take decision like the brain. This hardware neural networks target the neuroinspired computation of information with low costs and high-energy efficiency compared to artificial neural networks (ANNs). In this context, we have developed an artificial neuron with high-energy efficiency and small footprint enabling a high level of integration in future hardware neural networks. We have observed 70% variability in our artificial neuron due to the use of CMOS transistors in sub-threshold regime. This variability must be taken in account for future designs of the hardware neural networks. Additionally, future work on artificial synapses regarding the learning and the memory must be carried out to obtain intelligent HNNs.

From the biomedical aspect, the aim as mentioned in the chapter 1 is to fabricate a neuromorphic system, which is able to stimulate (preventing epileptic crises and slowing the Alzheimer disease) or even replace the defective brain area in case of injuries and neurodegenerative diseases. In this context, the perspective of this work is to connect the artificial neuron (the biomimetic neuron) to a living one and study the possibility of their communication while fabricating the interface system. Moreover, investigating the noise in this hybrid system is necessary due to the noisy activity of the living neuron cell. The biological features of the biomimetic neuron enable a brandnew investigation of SR phenomenon where a hybrid system of connected artificial neurons to biological neurons is studied, aiming new biomedical applications. Furthermore, design and fabricate HNNs that communicates, stimulate and even inject drugs to the specific neuronal network of a human brain can be imagined as a long-term perspective.

Figure 1 . 1 )

 11 is the first generation of computer where the computation and the communication was carried out with an analog signal. Current computers are digital computers (see the lower right of Figure 1.1), which use digital signals to compute as well as to communicate. The brain (see the lower left of Figure 1.1), known as the most energy-efficient computer, uses analog signals to compute and digital signals to communicate. In fact, the addition of information at synaptic level is considered as analog computing in the brain. The propagation of information in form of electrical impulses also called spikes is considered as digital signals enabling the communication in the brain. The fourth category (see the upper right of Figure 1.1), which consists of using digital signal to compute and using analog signal to communicate, is not yet explored.

Figure 1 . 1

 11 Figure 1.1 Signal choices for communication and computation(Boahen, 2017).

Figure 1 . 3

 13 Figure 1.3 Left: The Facebook server hall in the city of Lulea, Sweden. Right: The facilities of the Google data center in Taiwan.

Figure 2 . 1

 21 Figure 2.1 Different types of neurons. (a) Multipolar interneurons. (b) Motor neuron.(c) Sensory neuron[START_REF] Lodish | Overview of Neuron Structure and Function[END_REF].

Figure 2 . 2

 22 Figure 2.2 Neuron internal and external parts. Neuron, neuronal membrane, nucleus of cell body, axon, synapse and dendrite. (Bear et al, 2007).

Figure 2 . 3

 23 Figure 2.3 Different types of synapses. (A) Electronic synapse. (B) Chemicalsynapse.(Purves et al, 2004)

Figure 2 . 4

 24 Figure 2.4 Neuronal membrane. Left: comparison of ion transporters with ion channels. Right: simplified schematic of the neuronal membrane with principal channels and principal ionic concentrations (Na + and K + )[START_REF] Bear | Neurosciences, A la découverte du cerveau[END_REF].

Figure 2 Figure 2 . 5

 225 Figure 2.5 The evolution of membrane potential. (A) Neuron and electrodes to stimulate and record the neuronal membrane. (B) Injected current and (C) the membrane potential evolution in time (Purves et al, 2004).

Figure 2 . 6

 26 Figure 2.6 Propagation of action potential in the neuronal membrane. Left:(A) Myelinated axon. (B) Action potential propagation. Right: (A) Action potential at t = 1ms. (B) Action potential at t = 1.5ms. (A) Action potential at t = 2ms (Purves et al, 2004).

Figure 2 . 7

 27 Figure 2.7 Comparison of spiking neuronal models (Izhikevich, 2004).

2. 3 . 1

 31 Hodgkin and Huxley model In this section, the Hodgkin Huxley model, called HH model, will be investigated. Hodgkin, Huxley and Eccles received the 1963 Nobel Prize in Physiology or Medicine for their discoveries concerning the ionic mechanisms involved in the initiation and propagation of the spike in the peripheral and central portions of the nerve cell membrane (Figure 2.8) (Schwiening, 2012).

Figure 2 . 8

 28 Figure 2.8 The cover of the 1963 Nobel Prize Program. Huxley, left, looks on as Hodgkin adjusts a brand-new Tektronix 502A oscilloscope (Schwiening, 2012).

Figure 2 . 9

 29 Figure 2.9 Intracellular recording of the squid giant axon action potential. Left: Loligo forbesi, the 60cm long squid. Right: (A) Photomicrograph of an electrode inside a squid giant axon. (B) The first intracellular recording of an action potential. The sine wave under the spike has a frequency of 500 Hz (Schwiening, 2012).

Figure 2 .

 2 Figure 2.10 The electrical circuit proposed by Hodgkin and Huxley representing the neuronal membrane. Currents, potassium conductance (gK) and sodium conductance (gNa) varies with the membrane potential. The other components have constant values[START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF].

Figure 2 .

 2 11.Left shows a threshold excitatory current with 7μA/cm 2 amplitude (Iex) and 5ms pulsed duration (Ts). For this excitation, the variation of ionic currents are presented versus time in Figure2.11.Right. The maximum sodium current (INaMax) is around -782μA/cm 2 and the maximum potassium current (IKMax) is around 822μA/cm 2 . The leakage current (IL) is negligible compared to both INa and IK.

Figure 2 .

 2 Figure 2.11 Generation of one spike with HH model. Left: Membrane voltage for a pulsed excitatory current with Iex = 7μA/cm 2 and Ts = 5ms. Right: Ionics currents, INa IK and IL.

Figure 2 . 13 ,

 213 Matlab and Monte Carlo simulations of HH equations are in good agreement.

Figure 2 .

 2 Figure 2.13 Comparison between MC and Matlab solver of HH model for. (a) Membrane voltage Vm. (b) Ionics currents INa, IK and ICl. (c) Sodium (m 3 h) and potassium (n 4 ) gating variables. For all cases an excitation of Iapp = 7μA/cm 2 and Ts = 5ms starting at 10ms is used[START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF].

Figure 2 .

 2 Figure2.13 represents the time variation of membrane voltage, ionic currents, sodium (m 3 h) and potassium (n 4 ) gating variables in response to a pulsed excitatory current (Iapp) applied at 10ms with 7μA/cm2 amplitude and 5ms of duration. During the first 10ms, where Iapp is equal to zero, the neuronal membrane is in a steady state. At the onset of Iapp, sodium channels open more rapidly than potassium channels (m3h increases, Figure2.13.c). It induces the flow of sodium ions into the cell, resulting in a negative INa (Figure2.13.b), which leads to the increase of Vm (Figure2.13.a). The rise of Vm will result to the slow opening of potassium channels (shifted n 4 increases, Figure2.13.c) leading to the exit of potassium ions from the intracellular space to extracellular space and a positive IK arises (Figure2.13.b). As Vm reaches its peak value, around +40mV, sodium channels start to close (m 3 h decreases, Figure2.13.c), the slow potassium channels are completely open and potassium ions rush out from the intracellular space to the extracellular space and the voltage quickly returns to its resting value. As the potassium channels are closing slowly, potassium ions continue to leave the cell, resulting in a negative overshoot called hyperpolarization. The resting membrane potential is slowly restored thanks to the diffusion and leakage channel (ICl). During the action potential generation and a short time after the recovery of the steady state, it is impossible to stimulate the neuronal membrane in order to fire again. In fact, the period from the initiation of the spike to immediately after the peak is referred as the absolute refractory period. During this time, the cell cannot produce any new action potential. The generation of one spike obtained by the Matlab and MC resolution of HH model seems to be exactly the same, despite the stochasticity in the MC model. However, a closer look at the MC results corresponding to a zoom of INa and IK (inset in Figure2.13.b) reveals the presence of fluctuations in the ion currents that is the signature of a channel noise[START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF]. It is interesting to note that the same results have been obtained with Matlab and MC for the resolution of HH model.

Figure 2 .

 2 Figure 2.14 MC time evolution of various neuronal membrane parameters. (a) Membrane voltage Vm for Iapp= 7μA/cm 2 and Ts = 14ms. (b) Ionic currents for Iapp = 7μA/cm 2 and Ts = 14ms. (c) Membrane voltage Vm for Iapp = 50μA/cm 2 and Ts = 14ms. (d) Ionic currents for Iapp = 50μA/cm 2 and Ts = 14ms. In all cases, the excitatory current starts at 10ms[START_REF] Vasallo | Monte Carlo model for action potentials in bioinspired devices[END_REF].

Figure 2 .

 2 Figure 2.15 Comparison of HH model and Wei model schematic for the neuron cell (Wei et al, 2014).

Figure 2 .

 2 Figure 2.17 Generation of one spike with Wei model. Figure.2.17.up: Vm is generated for Iex = 5μA/cm 2 and Ts = 7ms. Figure.2.17.down: Vm is generated for Iex = 7μA/cm 2 and Ts = 5ms.

Figure 2 .

 2 Figure 2.18 Variation of injected charges as a function of different excitatory currents.

Figure 2 .

 2 Figure 2.19 Comparison of power consumption in HH model and Wei model. In both models 𝑷 𝑵𝒂 (red), 𝑷 𝑲 (green) and 𝑷 𝒅 (blue) are in nW/cm 2 . Left: Power consumption in HH model for Iex = 7μA/cm 2 during Ts = 14ms (same as Ts = 7ms). Right: Power consumption in Wei model for Iex = 7μA/cm 2 during Ts = 7ms.

Figure 2 .

 2 Figure 2.20 Response of ML model to different excitatory current. ML response to a continuous excitatory current: 0, 100 and 200pA/cm 2 .

Figure 2 .

 2 Figure 2.17 The F-I curve of ML model.

Figure 3 . 1 (

 31 [START_REF]Memristive devices for computing[END_REF].

Figure 3 . 1

 31 Figure 3.1 Hybrid CMOS memristor circuit. Memristive devices correspond to artificial synapses and CMOS circuits correspond to artificial neurons (Yang et al, 2013).

Figure 3 . 2

 32 Figure 3.2 General schematic of artificial membrane circuit.

Figure 3 . 3

 33 Figure 3.3 Artificial neuron circuit based on the ML model.

Figure 3 . 4

 34 Figure 3.4 Indicative synaptic circuits.

Figure 3 . 5 .Figure 3 . 5

 3535 Figure 3.5 Inverter biasing conditions used in the analysis.

Figure 3

 3

Figure 3 . 6

 36 Figure 3.6 Artificial neuron circuit analyses.

Figure 3 .

 3 7 (biomimetic with 6 transistors) and Figure 3.8 (biomimetic with 8 transistor). Both reproduce the spike shape observed in the biological neuron. However, a difference is observed in the output response while the same excitatory current (Iex = 1pA, Ts = 10ms, Q = 10fC and 𝜉 = 80A/F) is used. The biomimetic neuron with 8 transistors (see Figure 3.8) produces three spikes while the biomimetic neuron with 6 transistors produces one spike (see Figure 3.7).

Figure 3 . 7

 37 Figure 3.7 Response of the biomimetic neuron with 6 transistors to a pulsed current. With Iex = 1pA and Ts = 10ms. I(VddK1) corresponds to the potassium current (IK); I(Vddna1) corresponds to the sodium current (INa). The action potential is in blue V(vmem6Tr) and the excitatory current is I(Iex1) in pink.

Figure 3 . 8

 38 Figure 3.8 Response of the biomimetic neuron with 8 transistors to a pulsed current. With Iex = 1pA and Ts = 10ms. I(Ik) corresponds to the potassium current (IK) and I(Ina) corresponds to the sodium current (INa). The action potential is in blue V(vmem) and the excitatory current I(Iex) is in pink.

Figure 3 . 9

 39 Figure 3.9 One spike in response to an excitatory pulse. Biological neuron (up) response to injected charge of Q = 35nC/cm 2 and biomimetic neuron response (down) to injected charge of Q' = 2.5fC pulse of excitatory current with ξ = 20A/F. One spikes is generated for both neurons.

Figure 3 .

 3 Figure 3.10 Two spikes in response to an excitatory pulse. Biological neuron response (up) to injected charge Q = 175nC/cm 2 and biomimetic neuron response (down) to injected charge of Q' = 5fC a pulse of excitatory current with ξ = 20A/F. Two spikes are generated for both neurons.

Figure 3 .

 3 Figure 3.11 Three spikes in response to an excitatory pulse. Biological neuron response (up) to injected charge Q = 200nC/cm 2 and biomimetic neuron response (down) to injected charge of Q' = 10fC to a pulse of excitatory current with ξ = 20A/F. Three spikes are generated for both neurons.

Figure 3 .

 3 13. The spike amplitude is between -100mV and 60mV with following biasing for the fast neuron circuit: VDD = 100mV, VSS = -100mV and the excitatory current of Iex = 40pA during Ts = 200μs.

Figure 3 .

 3 Figure 3.13 The Fast neuron circuit response to a pulsed excitation. Spike train for an excitatory current of Iex=40pA during Ts=200μs.

Figure 3 .Figure 3

 33 14. The biomimetic neuron and fast neuron (referred as simplified neuron) are illustrated on the photographed of WetWire chip (Die photographs of the fabricated WetWire chip. The highlighted DUTs, each one sized 450 x 250μm, refer to the two neuron circuit variants: biomimetic (core size: 200 μm 2 ) and fast or simplified (core size: 35 μm 2 ).

Figure 3 .

 3 14.b: all 18 circuits of the WetWire chip with their respective input and output pads.

Figure 3 .

 3 14.c: fabricated biomimetic neuron and its 8 pads. All these circuits were designed with external biasing pads as shown in Figure 3.14.b. The excitation was implemented with on-chip trans-conductance, a PMOS transistor (w = 120nm Lg = 65nm). This trans-conductance is externally biased through VIN and VDD_T pads (see Figure 3.14.b and Figure 3.16). The VIN pad is connected to the gate of the PMOS as the VDD_T pad is connected to the source of the PMOS transistor (see Figure 3.16). The output signal was monitored through an on-chip unity gain output buffer designed to ensure that the frequency response of the neuron circuit would not be affected (see Figure 3.15).

Figure 3 .

 3 Figure 3.15 Left: Buffer for the biomimetic neuron Right: Buffer for the fast neuron.The output buffer featured independent DC supply through the VDDANA and ISSANA pads (see Figure3.15 and Figure3.16). These independent DC supplies enable accurate power consumption measurements while the neuron's bias pins were not ESD protected. We will describe the result of these characterizations in the following sections. In summary, as shown in the Figure3.16, the VIN pad corresponds to the excitatory input, connected to the gate of the PMOS trans-conductance. VDD and VSS are respectively the drain and the source bias of the artificial neuron. The VDD_T pad corresponds to the source of the PMOS trans-conductance.

Figure 3 .

 3 Figure 3.16 The fabricated biomimetic neuron and fast neuron (left). The schematic of biomimetic neuron and fast neuron (right) connected to the buffer. As shown, 7 pads must be biased in order to characterize these artificial neurons.

Figure 3 .

 3 Figure 3.17 Photographs of the test bench.

Figure 3 .Figure 3 .

 33 Figure 3.18 Biomimetic neuron circuit generates one spike Tp = 4ms.

Figure 3 .Figure 3 .

 33 Figure 3.20 Biomimetic neuron circuit generates three spikes Tp = 10ms.

Figure 3 .

 3 Figure 3.22 Comparison between Wei model and fabricated biomimetic circuit. Left: Wei model. Right: biomimetic circuit (Fab_circuit).

  The bias of the oscillatory neuron pads are VDD = 200mV, VSS = 0mV, VDDANA = 300mV, ISSANA = -400μA, VDD_T = 200mV, VIN = 0V. The resulting spikes shown in Figure 3.21 have a frequency of 3.42kHz and peak-to-peak amplitude of 182mV.

Figure 3 .

 3 Figure 3.23 Spike train of oscillatory neuron.

Figure 3 .

 3 25. The output spikes of HF and LF neuron are represented by Vout_HF and Vout_LF, respectively.

Figure 3 . 6

 36 Figure 3.25 Experimental spikes for the HF and LF neurons used in neuronal networks to emulate bursting mode. The first signal is VIN_HF, the second one is Vout_HF and the third one is Vout_BF. The number of spikes of HF neuron increases as VIN_HF decreases.

Figure 3 .

 3 Figure 3.26 Spikes in tonic burst circuit. Number of spikes as a function of VIN_HF.

Figure 3 .

 3 Figure 3.27 Spike frequency and power consumption over the excitation current for the biomimetic neuron.

Figure 3 .

 3 Figure 3.28 Energy efficiency vs. excitation current for the biomimetic neuron when the DC power is included or not from energy estimation.

Figure 3 .

 3 Figure 3.29 Spike frequency and power consumption vs. excitation current for the fast neuron.

Figure 3

 3

Figure 3 .

 3 Figure 3.30 Energy efficiency vs. excitation current measured for the fast neuron.

Figure 3 .

 3 Figure 3.31 'GreyMatter' chip with 9 circuits.

Figure 4 .

 4 1.b) as well as top stages like motor neurons, where noise impacts the entire neuron response (see Figure 4.1.c), constitute these different stages. This noisy activity of neurons introduces some stochastic properties of the brain information processing. This stochastic behavior impacts many aspects of brain function, including probabilistic decision-making, perception, memory recall, shortterm memory, attention, and even creativity (Rolls & Deco, 2010).

Figure 4 . 1

 41 Figure 4.1 Overview of the behavioral loop and the stages at which noise is present in the nervous system. a. Sources of sensory noise include the transduction of signals. This is exemplified here by a photoreceptor and its signal amplification cascade. b. Sources of cellular noise include the ion channels of excitable membranes, synaptic transmission and network interactions. c. Sources of motor noise include motor neurons and muscle.In the behavioral task shown (catching a ball), the nervous system has to act in the presence of noise in sensing, information processing and movement[START_REF] Faisal | Noise in the nervous system[END_REF].

Figure 4 . 2 .

 42 up). As shown in the schematic (see Figure4.2.down) the artificial neuron is biased via VDD and VSS, the excitation PMOS transistor is biased via VDD_T, VIN and the buffer is biased via VDDANA, ISSANA and VSS. The membrane voltage VOUT is observed at the buffet output.

Figure 4 . 2

 42 Figure 4.2 Artificial neurons circuits and schematic.Figure 4.2.up: The artificial neuron of GreyMatter chip (left) and WetWire chip (right).Figure 4.2.down: The biomimetic neuron pads (left) and its schematic (right).

Figure 4 . 2 .

 42 Figure 4.2 Artificial neurons circuits and schematic.Figure 4.2.up: The artificial neuron of GreyMatter chip (left) and WetWire chip (right).Figure 4.2.down: The biomimetic neuron pads (left) and its schematic (right).

Figure 4 . 2 .

 42 Figure 4.2 Artificial neurons circuits and schematic.Figure 4.2.up: The artificial neuron of GreyMatter chip (left) and WetWire chip (right).Figure 4.2.down: The biomimetic neuron pads (left) and its schematic (right).

Figure 4 . 3

 43 Figure 4.3 Test bench of the membrane voltage fluctuations.

Case 4 :

 4 Neuron offIn this case, the biasing of the artificial neuron (biomimetic or fast) are VIN = VDD_T = VDD = 200mV and VSS = 0V. The buffer is set to be on. Obviously for this set up the trans-conductance is completely closed (due to VIN = VDD_T = 200mV) and the neuron does not generate any spikes. For simplification, we will refer to this biasing as case 4. The test bench and instruments used for these four cases remain the same. The test bench with the pads value in the case 4 for the biomimetic neuron is illustrated in the Figure 4.4.

Figure 4 . 4

 44 Figure 4.4 Membrane voltage fluctuations test bench in case 4 for the biomimetic neuron. We have applied these four cases to the biomimetic neuron and extracted the probability density function (PDF) as a function of the membrane voltage (VOUT) in Figure 4.5.

Figure 4 . 5

 45 Figure 4.5 Probability density function of the biomimetic neuron at different resting states. Bio 1, Bio 2, Bio 3 and Bio 4 correspond to the respective case 1, case 2, case 3 and case 4 biasing.

Figure 4 . 6

 46 Figure 4.6 Probability density function of the fast neuron at different resting states. fast 1, fast 2, fast 3 and fast 4 corresponds to the respective case 1, case 2, case 3 and case 4 biasing.

Figure 4 . 6

 46 Figure 4.6 shows the probability density function of the fast neuron for different cases. As shown in Figure 4.6 and Table4.3, the membrane voltage mean value of the fast neuron varies from -114mV in the case 1 to -105mV in the case 4. The standard deviation is around 0.4mV. This experimental standard deviation value is slightly less than the theoretical standard deviation calculated from the Eq.4.1 of 1mV.

Figure 4

 4

Figure 4 . 7

 47 Figure 4.7 Probability density function at near threshold for biomimetic neuron.

Figure 4 .Figure 4 . 9

 449 Figure 4.9 Histogram and PDF for short acquisition.Figure 4.9.up: 11 trials superimposed for Vin Noise = 28.40mVrms.Figure 4.9.down: 11 trials superimposed for Vin Noise = 45.45mVrms.

Figure 4 .

 4 Figure 4.9 Histogram and PDF for short acquisition.Figure 4.9.up: 11 trials superimposed for Vin Noise = 28.40mVrms.Figure 4.9.down: 11 trials superimposed for Vin Noise = 45.45mVrms.

Figure 4 .

 4 10.up shows the number of occurrences and the PDF of the noise for Vin Noise = 28.40mVrms during 30min.

Figure 4 .

 4 10.down shows the number of occurrences and the PDF of the noise for Vin Noise = 45.45mVrms during 30min.

Figure 4 .

 4 Figure 4.10 Histogram and PDF for long acquisition.Figure 4.10.up: Vin Noise = 28.40mVrms during 30min Figure 4.10.down: Vin Noise = 45.45mVrms during 30min

  synaptic noise test bench is composed of a Keithley 4200-scs, a Keysight 33500B, a ROHDE & SCHWARZ oscilloscope RTM2054 500MHZ-5GSa/s and a National Instrument NI PXIe-1073 (see Figure 4.12). The Keithley 4200-scs is connected to VDD, VDD_T, VDDANA, ISSANA and VSS. The VDD_T and VDD are equal to 200mV. The buffer is in on state with VDDANA = 610mV and ISSANA = -58μA. The VSS is set to zero (see Figure 4.12).The Keysight 33500B is connected to the input of the biomimetic neuron (PMOS gate) to generate the Gaussian noise (Vin Noise). The Gaussian noise has an offset value equal to the VINmin. As explained in the previous section, the VINmin is extracted from the DC characterization.

Figure 4 .

 4 Figure 4.12 Synaptic noise test bench.

Figure 4 .

 4 Figure 4.13 illustrates the response of the biomimetic neuron (VOUT) to the Gaussian noise (Vin Noise) during 200ms. The orange curve is the injected Gaussian noise with Vin Noise = 10mVrms and an offset value of 20mV. For this low level of noise the biomimetic neuron response is extremely variable. The generated stochastic spike shown in Figure 4.13, has an amplitude of VOUT = 124mVpp and a 24Hz frequency, which varies according to the acquisition time.

Figure 4 .

 4 Figure 4.13 Response of biomimetic neuron to the low level of Gaussian noise.

Figure 4 .

 4 Figure 4.14 Response of biomimetic neuron to the high level of Gaussian noise.

Figure 4 .

 4 15.up shows the spike train for an injected noise of 19.31mVrms and Figure 4.15.down shows the spike detection during 6s.

Figure 4 .

 4 Figure 4.15 Spike train and ISI.Figure 4.15.up: Resulting spike train for Vin Noise = 19.31mVrms.Figure 4.15.down: Detected spikes with red lines.

Figure 4 .

 4 Figure 4.15 Spike train and ISI.Figure 4.15.up: Resulting spike train for Vin Noise = 19.31mVrms.Figure 4.15.down: Detected spikes with red lines.

Figure 4 .

 4 Figure 4.17 ISI parameters corresponding to the injected noise to the PMOS source for an acquisition time =10s.

Figure 4 .

 4 18 (noise on PMOS gate) and Figure 4.19 (noise on PMOS source) this level of noise is not enough to generate spikes.

Figure 4 .

 4 Figure 4.18 Spike train for Vin Noise = 25mVrms on PMOS gate.

Figure 4 .

 4 Figure 4.19 Spike train for Vin Noise = 25mVrms on PMOS source.

Figure 4 .

 4 Figure 4.20 Spike train for Vin Noise = 27.27mVrms on PMOS gate.

Figure 4 .

 4 21.a shows the spike train of a biomimetic neuron for high levels of noise during 120s. Each change of color corresponds to one spike train of 10s as illustrated in Figure 4.21.a.

Figure 4 .

 4 Figure 4.21.a Rebuild spike train over 120s.

Figure 4 .

 4 Figure 4.21.b is obtained from another biomimetic neuron and shows the rebuild spike train of 180 consecutive spike trains excited by a low level of noise (Vin Noise = 27.27mVrms) during 30min.

Figure 4 .

 4 Figure 4.21.b Spike train for Vin Noise= 27.27mVrms on PMOS gate during 30min.

Figure 4 .

 4 Figure 4.22 Rebuild spike train for different noise levels. The Vin noise varies from 27.27, 28.40, 29.54 to 30.68, respectively.

Figure 4

 4

Figure 4 .

 4 Figure 4.23 Experimental ISI parameters for different levels of noise injected to the PMOS gate during 30min.

Figure 4 .

 4 Figure 4.24 Experimental ISI parameters for different levels of noise on PMOS gate during 30min.

Figure 4 .

 4 Figure 4.25.a and Figure 4.25.b illustrates the spike train obtained with Vin Noise = 29.54mVrms during 30min and the probability density function (PDF) of ISI corresponding to this spike train.Figure 4.25.c shows the PDF of ISI during 10s.

Figure 4 .

 4 Figure 4.25.a and Figure 4.25.b illustrates the spike train obtained with Vin Noise = 29.54mVrms during 30min and the probability density function (PDF) of ISI corresponding to this spike train.Figure 4.25.c shows the PDF of ISI during 10s.

Figure 4 .

 4 26 compares the probability density function of ISI to these distributions for Vin Noise = 29.54mVrms.

Figure 4 .

 4 Figure 4.26 Comparison of PFD of ISI to several distributions. Vin Noise = 29.54mVrms on PMOS gate during 30min.

Figure 4 .

 4 Figure 4.27 Comparison of PFD of ISI to several distributions. Vin Noise = 30.68mVrms on PMOS gate during 30min.

Figure 4 .

 4 Figure 4.28 Gamma distribution (Pachitariu, Brody, Jun, & Holmes, n.d.).

Figure 4 .

 4 Figure 4.29 Comparison of PFD of ISI to several distributions. Vin Noise = 28.40mVrms is injected on PMOS source during 30min.

  4.1) in LTspice: Vin Noise = Vrms * SQRT(12/6) * ( rand(time*1e6) + rand(A+ time*1e6) + rand(B+ time*1e6) + rand(C+ time*1e6) + rand(D+ time*1e6) + rand(E+ time*1e6) -3) E.4.1Where A, B, C, D and E are respectively 1.1e9, 2.07e9, 3.05e9, 4.06e9 and 5.08e9. The Vrms is the RMS noise value (here Vrms = 27mVrms). This Gaussian noise has a 150mV offset value, which is the minimum value of Vin (injected signal to PMOS gate) when the neuron does not generate spikes. The response of the biomimetic neuron to a 27mVrms noise and the average injected current is illustrated during 100ms in Figure4.30. Only one spike was generated during 100ms.

Figure 4 .

 4 Figure 4.30 Response of the biomimetic neuron (blue curve) under Vin Noise = 27mVrms (green curve) and injected average current of 738.73fA (red curve). This figure is obtained after 100ms.

Figure 4 .

 4 Figure 4.31 shows the extracted ISI parameters of the biomimetic spike train for Vin Noise =27mVrms to Vin Noise=100mVrms.

Figure 4 .

 4 Figure 4.31 ISI parameters obtained with simulation.

Figure 4 .

 4 Figure 4.32 and Figure 4.33 compare the PDF of ISI for Vin Noise = 27mVrms and Vin Noise = 100mVrms with different distributions (same figure for other level of noise are presented in annex).

Figure 4 .

 4 Figure 4.32 Comparison of PDF of ISI obtained with simulation to several distribution functions. Vin Noise = 27mVrms is injected on PMOS gate during 60s.

Figure 4 .

 4 Figure 4.33 Comparison of PDF of ISI obtained with simulation to several distribution functions. Vin Noise = 100mVrms is injected on PMOS gate during 5s.

Figure 4 .

 4 Figure 4.34 μ and λ obtained from the Matlab fit.

Figure 4 .

 4 36).The resulting combined signal (VSN) is injected to the PMOS gate of the transconductance (see Figure4.36). In Figure4.35 the simulated combined signal VSN = 6mVrms with VS = 10mVpp (10Hz frequency and an offset of 150mV) is illustrated.

Figure 4 .

 4 Figure 4.35 LTspice simulation of SR.

Figure 4 .

 4 Figure 4.36 Test bench of stochastic resonance.

Figure 4 .

 4 37.a.A spiking firing pattern arose as VN increased with a timing distribution that varies as a function of the sinusoidal signal rhythm and noise magnitude (Figure4.37.b). Increasing, furthermore, VN increased the sensitivity of the biomimetic neuron to the weak excitatory signal and its response was enhanced as shown in Figure4.37.c.

Figure 4 .

 4 37.d. From the extracted histogram, the signature of the deterministic sinusoidal signal can be recovered. biomimetic neuron response for VS = 10mVpp with 10Hz frequency and different VN and VSN levels. a) VN= 11.36mVrms, VSN= 5.28mVrms. b) VN= 45.45mVrms, VSN= 13.50mVrms. c) VN = 68.18mVrms, VSN = 19.85mVrms. d) Statistical firing probability of biomimetic neuron in response to sinusoidal wave. The noise and signal amplitude are respectively VN = 600mVpp and VS =10mVpp.

4. 4 . 4

 44 Stochastic resonance and biomimetic neuron error probability Stochastic resonance is observed under sinusoidal signal (see Figure 4.38.Left) as well as square signal (Figure 4.38.Right).

Figure 4 Figure 4 .

 44 Figure 4.38 Detecting states: Left: sinusoidal signal detected using stochastic resonance phenomena with error. Right: square signal detected using stochastic resonance phenomena without error.

Figure 4 . 4 . 4 . 5

 4445 Figure 4.39 Error probability of biomimetic neuron as a function sinusoidal signal+noise level.

Figure 4 .

 4 Figure 4.40 SNR as a function of noise levels.

Figure 4 .

 4 Figure 4.41 Power consumption as a function of added noise level for different detection states.

  𝑉 𝑚 ) and the sodium Nernst potential (𝐸 𝑁𝑎 ). This definition is also true for 𝐼 𝑙 (expression of 𝐼 𝑁𝑎 , 𝐼 𝐾 and 𝐼 𝑙 in annex).Hodgkin and Huxley experiment suggests that 𝑔 𝐾 and 𝑔 𝑁𝑎 , varies with the membrane potential and time; but 𝐸 𝑁𝑎 , 𝐸 𝐾 , 𝐸 𝑙 , 𝐶 𝑀 and 𝑔 𝑙 can be considered as constant (values in annex).

	Sodium conductance 𝑔 𝑁𝑎 (expression in annex) is defined in the same way except that
	it depends on other gating variables m and h (Eq.2.4). Where m is associated with
	sodium channel activation and h corresponds to the sodium channel inactivation. The
	expression of their respective 𝛼 and 𝛽 are presented in annex.
	Finally, by replacing the expression of each conductance (𝑔 𝐾 , 𝑔 𝑁𝑎 and 𝑔 𝑙 ) in their
	respective ionic current (𝐼 𝐾 , 𝐼 𝑁𝑎 and 𝐼 𝑙 ) and gathering the three gating variables (n, m
	and h) we can obtain the four main equations of HH model (Eq.2.4) (Hodgkin and
	Huxley, 1952).
			2.1).
	The ion currents result from the addition of sodium and potassium currents 𝐼 = 𝐶 𝑀 𝑑𝑉 𝑚 𝑑𝑡 + 𝑔̅ 𝐾 𝑛 4 (𝑉 𝑚 -𝐸 𝐾 ) + 𝑔̅ 𝑁𝑎 𝑚 3 ℎ(𝑉 𝑚 -𝐸 𝑁𝑎 ) + 𝑔 𝑙 (𝑉 𝑚 -𝐸 𝑙 ) (𝐼 𝑁𝑎 𝑎𝑛𝑑 𝐼 𝐾 ), and a leakage current (𝐼 𝑙 ) (Eq.2.1). 𝐼 = 𝐶 𝑀 𝑑𝑉 𝑚 𝑑𝑡 + 𝐼 𝑖𝑜𝑛 Each of these currents (𝐼 𝑁𝑎 , 𝐼 𝐾 and 𝐼 𝑙 ) is determined by a driving force that can be 𝑑𝑡 = 𝛼 ℎ (1 -ℎ) -𝛽 ℎ ℎ 𝑑ℎ 𝐼 𝑖𝑜𝑛 = 𝐼 𝑁𝑎 + 𝐼 𝐾 + 𝐼 𝑙 Eq.2.1 𝑑𝑛 = 𝛼 𝑛 (1 -𝑛) -𝛽 𝑛 𝑛 𝑑𝑡 𝑑𝑡 = 𝛼 𝑚 (1 -𝑚) -𝛽 𝑚 𝑚 𝑑𝑚 Eq.2.4
	conveniently modeled as an electrical potential difference and a coefficient
	permeability that has the dimension of a conductance. As shown in Eq.2.2, the
	potassium current (𝐼 𝐾 ) is equal to the potassium conductance (𝑔 𝐾 ) multiplied by the
	difference between the membrane potential (Vm) and the equilibrium potential of the
	potassium ion also known as Nernst potential ( 𝐸 𝐾 ) (Hodgkin and Huxley, 1952).
	Sodium current (𝐼 𝑁𝑎 ) also depends on the sodium conductance (𝑔 𝑁𝑎 ), the membrane
	potential (E or 𝐼 𝐾 = 𝑔 𝐾 (𝑉 𝑚 -𝐸 𝐾 )	Eq.2.2
	The potassium conductance 𝑔 𝐾 is defined by the Eq.2.3, where 𝑔̅ 𝐾 is the maximal
	potassium conductance, 𝑛 is a gating variable ranging between 0 and 1. The gating
	variable n is defined by a first order differential equation depending on two rate
	constants 𝛼 𝑛 and 𝛽 𝑛 , which varies only with 𝑉𝑚 (annex).
		𝑔 𝐾 = 𝑔̅ 𝐾 𝑛 4
	𝑑𝑛 𝑑𝑡	= 𝛼 𝑛 (1 -𝑛) -𝛽 𝑛 𝑛	Eq.2.3

Table 2 .1 Comparison of Wei model's equations with HH model's equations.

 2

Table 2

 2

.1, also shows the presence of an additional current 𝐼 𝑝𝑢𝑚𝑝 in Wei model. The Nernst potential defined as constant in HH model (see annex) is more accurate in Wei model since it varies with time. This is due to the Nernst potential definition

Table 2 .2 Power consumption for HH and Wei model.

 2

  Catherine Morris and Harold Lecar have investigated the Barnacle muscle fibers. They proposed in 1981 a model with only two states variables known as the Morris-Lecar (ML) model[START_REF] Morris | Voltage oscillations in the barnacle giant muscle fiber[END_REF]. This model describes the voltage oscillations in the Barnacle giant muscle fiber based on the membrane voltage 𝑉 𝑚 and the potassium gating variable 𝑛 . As shown in Eq.2.8, ML model is a system of two non-linear differential equations associated with exponential functions.In these equations, 𝐶 𝑚 is the membrane capacitance, 𝐼 𝑒𝑥 is the excitatory current, 𝐸 𝐾 , 𝐸 𝐶𝑎 and 𝐸 𝐿 are the ion equilibrium potentials, 𝐺 𝐶𝑎 𝐺 𝐾 and 𝐺 𝐿 are the calcium (same as sodium in HH and Wei model), potassium and leak conductance. The steady-state potassium and sodium gating variables are respectively 𝑛 𝑠𝑠 and 𝑚 𝑠𝑠 while 𝜆 0 is the reference frequency. Finally, 𝑉 1 to 𝑉 4 are fitting parameters that can serve for tuning the dynamic properties in order to represent different systems of interest.

	𝐶 𝑚	𝑑𝑉 𝑚
		𝑑𝑛 𝑑𝑡	= 𝜆(𝑉 𝑚 )(𝑛 𝑠𝑠 (𝑉 𝑚 ) -𝑛)
		𝑚 𝑠𝑠 (𝑉 𝑚 ) =	1 2	[1 + 𝑇𝑎𝑛ℎ (	𝑉 𝑚 -𝑉 1 𝑉 2	))]	Eq.2.8
		𝑛 𝑠𝑠 (𝑉 𝑚 ) =	1 2	[1 + 𝑇𝑎𝑛ℎ (	𝑉 𝑚 -𝑉 3 𝑉 4	))]
		𝜆(𝑉 𝑚 ) = 𝜆 0 𝐶𝑜𝑠ℎ (	𝑉 𝑚 -𝑉 3 2𝑉 4	))

𝑑𝑡

= 𝐼 𝑒𝑥 -𝐺 𝐶𝑎 𝑚 𝑠𝑠 (𝑉 𝑚 )(𝑉 𝑚 -𝐸 𝐶𝑎 ) -𝐺 𝑘 𝑛(𝑉 𝑚 -𝐸 𝐾 ) -𝐺 𝐿 (𝑉 𝑚 -𝐸 𝐿 )

We have simulated the

Morris-Lecar (Waterloo, 2013)

model also called "ML original" with an available Matlab code. The parameters of "ML original" model are extracted from

[START_REF] Morris | Voltage oscillations in the barnacle giant muscle fiber[END_REF]

publication. First, the response of this model in time domain to different constant excitatory current (Iex) was investigated, as shown in Figure

2

.20.

  [START_REF] Hodgkin | A quantitative description of membrane current and its application to conduction and excitation in nerve[END_REF] They are implemented in HH code and simulated with Matlab software.

	𝐶 𝑀 ( µ𝐹. 𝑐𝑚 -2 )	1.0
	𝐸 𝑁𝑎 ( 𝑚𝑉 )	+55
	𝐸 𝐾 ( 𝑚𝑉 )	-72
	𝐸 𝑙 ( 𝑚𝑉 )	-49.4
	𝑔̅ 𝑁𝑎 ( 𝑚. 𝑚ℎ𝑜. 𝑐𝑚 -2 )	120
	𝑔̅ 𝐾 ( 𝑚. 𝑚ℎ𝑜. 𝑐𝑚 -2 )	36
	𝑔 𝑙 ( 𝑚. 𝑚ℎ𝑜. 𝑐𝑚 -2 )	0.3

  The constant 𝛼 𝑛 determines the rate at which a gating channel switch from closed state to open state, while 𝛽

𝑛 is a rate at which it switches from an open state to a closed one. The transition rate constants 𝛼 𝑛 and 𝛽 𝑛 are related to Vm through rather complex exponential functions.

  2𝛽𝐼 𝑝𝑢𝑚𝑝 -𝐼 𝑑𝑖𝑓𝑓 -𝐼 𝑔𝑙𝑖𝑎 -2𝐼 𝑔𝑙𝑖𝑎𝑝𝑢𝑚𝑝 + 𝛽𝐼 𝑘𝑐𝑐2 + 𝛽𝐼 𝑛𝑘𝑐𝑐1 )𝜈 0

	and Ikcc2.
				𝑑𝑁𝐾 𝑖 + 𝑑𝑡	=	1 𝜏	𝑑𝑛 𝑑𝑡 (-𝛾𝐼 𝐾 + 2𝐼 𝑝𝑢𝑚𝑝 -𝐼 𝑘𝑐𝑐2 -𝐼 𝑛𝑘𝑐𝑐1 )𝜈 𝑖 = 𝛼 𝑛 (1 -𝑛) -𝛽 𝑛 𝑛
	𝑑𝑁𝐾 0 + 𝑑𝑡	=	1 𝜏	𝛼 𝑛 (𝑉 𝑚 ) = 𝛽 𝑛 (𝑉 𝑚 ) = 0.125exp (-0.01(𝑉 𝑚 -10) 1 -exp [ -(𝑉 𝑚 -10) 10 𝑉 𝑚 10 ) (𝛾𝛽𝐼 𝐾 -Potassium concentrations in Wei model ]
	Gating variable n and transition rate constants 𝛼 𝑛 and 𝛽 𝑛 The intracellular Na + ion number dynamics, + 𝑑𝑁𝑁𝑎 𝑖 , is modeled based on the 𝑑𝑡 membrane Na + current (INa), Ipump, and Inkcc1.
				𝑑𝑁𝑁𝑎 𝑖 + 𝑑𝑡	=	1 𝜏	(-𝛾𝐼 𝑁𝑎 -3𝐼 𝑝𝑢𝑚𝑝 -𝐼 𝑛𝑘𝑐𝑐1 )𝜈 𝑖
				𝑑𝑁𝑁𝑎 0 + 𝑑𝑡	=	1 𝜏	(-𝛾𝛽𝐼 𝑁𝑎 + 3𝛽𝐼 𝑝𝑢𝑚𝑝 + 𝛽𝐼 𝑛𝑘𝑐𝑐1 )𝜈 0
					𝑑𝑚 𝑑𝑡	= 𝛼 𝑚 (1 -𝑚) -𝛽 𝑚 𝑚
					𝛼 𝑚 (𝑉 𝑚 ) =	0.1(𝑉 𝑚 -25) exp [ -(𝑉 𝑚 -25) 10 ] -1
					𝛽 𝑚 (𝑉 𝑚 ) = 4exp (-	𝑉 𝑚 18	)
				Gating variable m and transition rate constants 𝛼 𝑚 and 𝛽 𝑚
					𝑑ℎ 𝑑𝑡	= 𝛼 ℎ (1 -ℎ) -𝛽 ℎ ℎ
					𝛼 ℎ (𝑉 𝑚 ) = 0.07exp (-	𝑉 𝑚 20	)
					𝛽 ℎ (𝑉 𝑚 ) =	1 + exp [	1 -(𝑉 𝑚 -30) 10	]
				Gating variable h and transition rate constants 𝛼 ℎ and 𝛽 ℎ

K + ions, 𝑑𝑁𝐾 𝑖 + 𝑑𝑡

, is a function of IK and Ipump, as well as cotransport currents Inkcc1

  The units and description of all parameters used in Wei model are summarized in the following table.

	Constant parameters:
	Parameters	Units		Description
	𝐶	1µ𝐹. 𝑐𝑚 -2		Membrane capacitance
	𝐺 𝑁𝑎	30𝑚𝑆. 𝑐𝑚 -2		Maximal conductance of sodium current
	𝐺 𝐾	25𝑚𝑆. 𝑐𝑚 -2		Maximal conductance of potassium current
	𝐺 𝑁𝑎𝐿	0.0247𝑚𝑆. 𝑐𝑚 -2		Conductance of leak sodium current
	𝐺 𝐾𝐿	0.05𝑚𝑆. 𝑐𝑚 -2		Conductance of leak potassium current
	0.1𝑚𝑆. 𝑐𝑚 -2 7 Chapter 3. 𝐺 𝐶𝑙𝐿 𝛽 0		Conductance of leak chloride current Ratio of the initial intra-/extracellular volume
	𝜌 max	0.8 𝑚𝑀/𝑠		Maximal Na/K pump rate
	𝐺 𝑔𝑙𝑖𝑎,𝑚𝑎𝑥	5 𝑚𝑀/𝑠		Maximal glia uptake strength of potassium
	𝜀 𝑘,𝑚𝑎𝑥	0.25𝑠 -1		Maximal potassium diffusion rate
	[𝐾 + ] 𝑏𝑎𝑡ℎ	3.5𝑔. 𝑚𝑜𝑙 -1		Normal bath potassium concentration
	𝜀 0	0.17𝑠 -1		Oxygen diffusion rate
	𝛼	5.3𝑔. 𝑚𝑜𝑙 -1		Conversion factor
	[0 2 ] 𝑏𝑎𝑡ℎ	32𝑚𝑔. 𝐿 -1		Normal bath oxygen concentration
	𝑈 𝑘𝑐𝑐2	0.3𝑚𝑀/𝑠		Maximal KCC2 cotransporter strength
	𝑈 𝑛𝑘𝑐𝑐1	0.1𝑚𝑀/𝑠		Maximal NKCC1 cotransporter strength
		26.64 ln (	[𝑁𝑎 + ] 0 [𝑁𝑎 + ] 𝑖	)
		𝐸 𝐾 = 26.64 ln (	[𝐾 + ] 0 [𝐾 + ] 𝑖	)
		𝐸 𝐶𝑙 = 26.64 ln (	[𝐶𝑙 -] 𝑖 [𝐶𝑙 -] 0	)

where

[.]

i and

[.]

o represent concentrations inside and outside the cell, respectively.

  .1.

	References Neuron	CMOS	Core	Spiking	Power	Energy
		type	node	Area	frequency		(pJ/Spike)
				(µm 2 )	(Hz)
	Indiveri et	LIF	0.35	2573	200	-	900
	al., 2006		µm
	Wijekoon	LIF	0.35	2800	10 6	8-40	8.5-9
	and Dudek,		µm			µW
	2008
	Basu and	Saddle	0.35	2740	100	1.74	17.4
	Hasler,		µm			nW
	2010
	Joubert et	LIF	65 nm	538	1.9 10 6	78 µW	41
	al., 2012
	Cruz-	LIF	90 nm	442	100	40 pW	0.4
	Albrecht et
	al., 2012

Table 3 .1 State-of-the-art performance of reported stand-alone artificial neurons.

 3

  [START_REF] Hynna & Boahen | Thermodynamically equivalent silicon models of voltage-dependent ion channels[END_REF]. 𝐺 𝐶𝑎 𝑚 𝑠𝑠 (𝑉 𝑚 )(𝑉 𝑚 -𝐸 𝐶𝑎 ) -𝐺 𝑘 𝑛(𝑉 𝑚 -𝐸 𝐾 ) -𝐺 𝐿 (𝑉 𝑚 -𝐸 𝐿 )

	𝐶 𝑚	𝑑𝑉 𝑚 𝑑𝑡	= 𝐼 𝑒𝑥 -Eq.3.1
			𝑑𝑛 𝑑𝑡	= 𝜆(𝑉 𝑚 )(𝑛 𝑠𝑠 (𝑉 𝑚 ) -𝑛)	Eq.3.2
			𝑚 𝑠𝑠 (𝑉 𝑚 ) =	1 2	[1 + 𝑇𝑎𝑛ℎ (	𝑉 𝑚 -𝑉 1 𝑉 2

  As shown in Figure3.5 the input and output voltages are Vin and Vout. The supply voltages are presented by Vd and -Vd. By applying the Kirchhoff's Current Law for the output node of this inverter (see Figure3.5) and expressing currents (Ip and In) as in Eq.3.6, the output voltage can be expressed as following:

	𝑉 𝑜𝑢𝑡 = -𝑉 𝑑 . 𝑇𝑎𝑛ℎ [ 𝜂𝑉 𝑡 𝑉 𝑖𝑛	+	1 2	ln (	𝐺 𝑛0 𝐺 𝑝0	)] = -𝑉 𝑑 . 𝑇𝑎𝑛ℎ [	𝑉 𝑖𝑛 -𝑉 𝑖𝑠𝑣 𝜂𝑉 𝑡	]
	= -𝑉 𝑑 . 𝑇𝑎𝑛ℎ [	𝑉 𝑞 𝜂𝑉 𝑡	]

  𝑉 𝑚 + 𝑉 𝑑 ) + 𝐼 𝑒𝑥 -𝐼 𝐿 (𝑉 𝑚 ) VGNa and VGK correspond to the gate voltages of MPNa and MNK. VGNa is the output of the first inverter (MP1/MN1) and Vout3 is the output of the third inverter (MP3/MN3). An input/output relation such as Eq.3.7 relates both VGNa and Vout3 to Vm. The Eq.3.14 could also be rewritten as follows: VGK (Eq.3.15) is similar to that of the Morris-Lecar model. In particular, the time dynamics of VGK are defined by the reference frequency 𝜆(Vm), which is related to the circuit parameters through Eq.3.16. The Eq.3.13 could be rewritten as the first differential equation of the Morris-Lecar model by introducing the parameters mss(Vm) and n :VGNa and VGK are between -Vd and +Vd therefore mss(Vm) and n are between 0 and 1. We observe a good correlation between the mss(Vm) function of the artificial neuron and the Tanh function of the Morris-Lecar model.

	𝐶 𝑚 𝑚 𝑠𝑠 (𝑉 𝑚 ) = exp (-𝑑𝑉 𝑚 𝑑𝑡 = 𝐺 𝑁𝑎 exp ( 𝑉 𝑑 -𝑉 𝐺𝑁𝑎 𝜂𝑉 𝑡 -𝐺 𝐾 exp ( 𝑉 𝐺𝐾 +𝑉 𝑑 𝜂𝑉 𝑡 ) (Eq.3.13 ) (𝑉 𝑑 -𝑉 𝑚 ) 𝑉 𝐺𝑁𝑎 (𝑉 𝑚 ) + 𝑉 𝑑 𝜂𝑉 𝑡 ) Eq.3.20
	𝐶 𝑘	𝑑𝑉 𝐺𝐾 𝑑𝑡	= 𝐺 𝑝2 𝑒𝑥𝑝 ( -𝐺 𝑛2 exp ( 𝑉 𝑑 -𝑉 𝑜𝑢𝑡3 𝜂𝑉 𝑡 𝜂𝑉 𝑡 𝑉 𝑜𝑢𝑡3 + 𝑉 𝑑 ) (𝑉 𝑑 -𝑉 𝐺𝐾 ) ) (V d + V GK )	Eq.3.14
	As shown in Eq.3.19, n is simply related to VGK. Assuming a small VGK, the first order
	𝑑𝑡 approximation of n would be: = 𝜆(𝑉 𝑚 ) [𝑉 𝑑 𝑇𝑎𝑛ℎ ( where 𝑑𝑉 𝐺𝐾 𝑛 = 𝑒𝑥𝑝 ( -𝑉 𝑑 𝜂𝑉 𝑡 ) exp ( 𝑉 𝐺𝐾 𝜂𝑉 𝑡 ) ≈ exp (-𝑉 𝑞2 𝜂𝑉 𝑡 ) -𝑉 𝐺𝐾 ] 𝑉 𝑑 𝜂𝑉 𝑡 ) (1 +	𝑉 𝐺𝐾 𝜂𝑉 𝑡	)	Eq.3.15 Eq.3.21
	with the reference frequency 𝜆(Vm) :
	𝜆(𝑉 𝑚 ) =	2. 𝐶𝑜𝑠ℎ (	𝑉 𝑞2 𝜂𝑉 𝑡	𝐶 𝑘 ) . 𝑒𝑥𝑝 (	𝑉 𝑑 𝜂𝑉 𝑡	) . √ 𝐺 𝑛2 𝐺 𝑝2	Eq.3.16
	and parameter Vq2, based on Eq.3.9:
	𝑉 𝑞2 = 𝑉 𝑖𝑛2 -𝑉 𝑖𝑠𝑣2 = 𝑉 𝑜𝑢𝑡3 +	𝜂𝑉 𝑡 2	𝑙𝑛 (	𝐺 𝑛2 𝐺 𝑝2	)	Eq.3.17
	the differential equation for 𝐶 𝑚 𝑑𝑉 𝑚 𝑑𝑡 = 𝐺 𝑁𝑎 𝑒𝑥𝑝 ( 2𝑉 𝑑 ) 𝑚 𝑠𝑠 (𝑉 𝑚 )(𝑉 𝑑 -𝑉 𝑚 ) 𝜂𝑉 𝑡 -𝐺 𝐾 𝑒𝑥𝑝 ( 2𝑉 𝑑 𝜂𝑉 𝑡 ) 𝑛 (𝑉 𝑚 + 𝑉 𝑑 ) + 𝐼 𝑒𝑥 -𝐼 𝐿 (𝑉 𝑚 )	Eq.3.18
	with the potassium gating variable, n:
				𝑛 = 𝑒𝑥𝑝 (	𝑉 𝐺𝐾 -𝑉 𝑑 𝜂𝑉 𝑡	)	Eq.3.19
	and,

Eq.3.21 demonstrates a linear relation between n and VGK, which allows us to replace VGK by n and rewrite Eq.3.15, as follows:

𝑑𝑛 𝑑𝑡 = 𝜆(𝑉 𝑚 )[𝑛 𝑠𝑠 (𝑉 𝑚 ) -𝑛] Eq.3.22

Hence, the equation of ICK of the artificial neuron (Eq.3.14) matches the second differential equation (Eq.3.2) of the Morris-Lecar model. Based on Eq.3.13 and Eq.3.14 we can conclude that the dynamic behavior of the artificial neuron is described by the currents INa, IK, Ip2, In2 and the two time constants, which are associated with Cm and Ck.

.3.

	MP1	MP2	MP3	MPNa
	400nm 580nm 120nm 600nm
	MN1	MN2	MN3	MNK
	120nm 120nm 650nm 1.83μm
	Cm		CK
	50fF		100fF	Biomimetic neuron circuit with 8
					transistors

Table 3 .2 Biomimetic circuit parameters with 8 transistors.

 3 All Transistor have a gate length of Lg = 65nm.

	MP1	MP2	MPNa
	720nm	720nm	1.44μm
	MN1	MN2	MNK
	120nm	120nm	1.44μm
	Cm		CK
	50fF		100fF	Biomimetic neuron circuit with 6
				transistors

Table 3 .3 Biomimetic circuit parameters with 6 transistors

 3

. All Transistor have a gate length of Lg = 65nm.

Table 3 .4 Spiking characteristics for constant

 3

		Vrest (mV)	Vth (mV)	Vpp_max(mV)	ARp (ms)
	Biological neuron	-70	-50	120	4
	Biomimetic neuron	Adjustable	Adjustable	Adjustable	Adjustable
		-70	-50	120	3.5

ξ. The spike generated from biological neuron (Wei neuron) is compared to the one from biomimetic neuron. Same normalization coefficient ξ = 20A/F is used for both neurons.

Table 3 .6: Fast circuit parameters with 6 transistors

 3

	MP1	MP2	MPNa
	300nm	360nm	400nm
	MN1	MN2	MNK
	600nm	120nm	1.2μm
	Cm		CK
	4fF		8fF	Fast neuron circuit with 6 transistors

.6. . All transistor have a gate length of Lg = 65nm.

Table 3 .6.

 3

	Biasing Pads VIN	VDD	VDD_T VDDANA ISSANA VSS
	Biomimetic	≤200mV 200mV 200mV	300mV	-43μA	0V
	neuron
	Fast neuron ≤200mV 200mV 200mV	300mV	-400μA	0V

Table 3 .6 Bias for the biomimetic and fast neuron of the Wet Wire chip.

 3

Table 4 .1: Membrane capacitance and the theoretical standard deviation of biomimetic and fast neuron.

 4

Table 4 .2 The four resting state cases, mean and standard deviation of membrane voltage fluctuations for the biomimetic neuron. Not

 4 The same analyses have been performed for the fast neuron and results are presented in Figure 4.6 and Table 4.3.

	Case	VIN	VDD	VDD_T	Mean	Standard Deviation
		(mV)	(mV)	(mV)	(mV)	(mV)
	1	NC	NC	NC	-183.2	0.36
	2	NC	0	NC	-181.6	0.35
	3	NC	200	NC	-180.6	0.37
	4	200	200	200	-180.6	0.23
						Connected (NC)
	corresponds to disconnected biases, in all these cases the buffer is on with VDDANA
	= 300mV and ISSANA = -43μA.

Table 4 .3 The four resting state cases, mean and standard deviation of the membrane voltage fluctuations for the fast neuron.

 4 Not Connected (NC)

		VIN	VDD	VDD_T	Mean	Standard Deviation
		(mV)	(mV)	(mV)	(mV)	(mV)
	1	NC	NC	NC	-114,2	_
	2	NC	0	NC	-106,9	0,39
	3	NC	200	NC	-105,6	0,41
	4	200	200	200	-105,6	0,41

Table 4 .4 Spike train data for long acquisition.

 4

	The Vin noise varies

Table 4 .7 Spike train and current parameters obtained with simulation.

 4

	Gaussien Noise
	on PMOS gate	Acquisition	Spike	Average of	RMS of
	(mVrms)	time	frequency	current	current
	27	60s	3Hz to 30Hz	738.73fA	813.65fA
	30	30s	40Hz	783.23fA	881.02fA
	40	25s	176.09Hz	982.12fA	1.21pA
	50	20s	252.3Hz	1.25pA	1.76pA
	60	15s	355Hz	1.63pA	2.67pA
	70	12s	412.32Hz	2.11pA	3.94pA
	80	10s	505.90Hz	2.65pA	5.63pA
	90	8s	758.85Hz	3.20pA	9.15pA
	100	5s	834Hz	4.11pA	17.9pA

2.2 State of the art: spiking neuron models

Remerciements

In order to determine the correct biasing of the biomimetic and fast neuron buffer, we varied the input of the buffer (VIN) from 0 to 200mV with a step of 50mV and extracted the mean value of the output of the buffer (VOUT). This analyze has been performed for various values of VDDANA and ISSANA (see Figure 3.32).

Figure 3.32 Biomimetic buffer output for various biasing.

The black line is obtained with VDDANA = 400mV, ISSANA = -43μA.The blue line is obtained with VDDANA = 300mV, ISSANA = -43μA. The red line is obtained with VDDANA = 200mV, ISSANA = -43μA.

In both cases, we had to adjust the biasing of the buffer in order to achieve the output gain of one. Based on this method we have reported in

ISSANA=-43µA

For the fast neuron the VINmin is 58mV, the probability density function of the fast neuron is illustrated in For VIN = 180mV, we observe a mean and standard deviation of respectively -104.5mV and 0.36mV. This standard deviation corresponds to approximately 38K in terms of temperature of noise. For VINmin = 58mV both mean and standard deviation, increase to respectively -62.7mV and 1mV. This standard deviation corresponds to approximately 252K in terms of temperature of noise. In this case, the standard deviation of the fast neuron corresponds to the value of 1mV obtained by the theory. These experiments demonstrate that as both biomimetic and fast neuron approach their threshold, their mean and standard deviation increases. However, the low noise temperature obtained in the case of the fast neuron remains obscure: the uncertainty of the capacitance value and the validity of a simple C-G circuit for such a complicated circuit could be investigated for a deeper analysis of these results.

Synaptic noise in artificial neuron

In the brain, neurons are subjected to an intense synaptic bombardment [START_REF] Destexhe | The high-conductance state of neocortical neurons in vivo[END_REF]. The synaptic noise is composed of many uncorrelated Poisson pulse trains and some correlated ones. Based on the central limit theorem (CLT), which establishes that in most situations when independent random variables are added their properly normalized sum tends toward a normal distribution, we can assume that the sum of injected noise to a neuron cell is equivalent to a Gaussian voltage noise.

Short acquisition

In this part, the injected Gaussian noise to the PMOS gate varies from 29.55mVrms to 51.1mVrms during 10s. As described before, we have extracted the spike train of each noise level and extracted its corresponding ISI. For each noise level, we define six parameters of the ISI as follows:  M : is the mean value of the ISI.  SD : is the standard deviation of ISI.  VAR : is the variance of ISI.  K: is the ratio of mean to standard deviation of ISI, known as the shape parameter.  Θ : is the ratio of the mean to the shape parameter, known as the scale parameter.  CV: is the ratio of the standard deviation to the mean value of ISI, known as the coefficient of variation.

These parameters can determine the distribution function of the ISI. For instance in a Poisson process, the coefficient of variation is CV = 1 (Pachitariu, Brody, Jun, & Holmes, n.d.).

The shape parameter K and the scale parameter Θ are used to define the gamma distribution.

We will also introduce other parameters to define the inverse Gaussian distribution in the following paragraphs. These parameters of ISI are illustrated for each noise level in In this part, the injected Gaussian noise on the PMOS source varies from 27.27mVrms to 51.1mVrms during 10s. The ISI parameters are illustrated in Figure 4.17.

Annex

Simulation results of biomimetic response to the synaptic noise: Chapter 5.

Conclusion and perspectives