# A new memristor for implementation of reliable spiking neuron models

# A trivial implementation of a memristive analog spiking neuron, for less than $1

**Jiaming WU and Marcelo Rozenberg***

Paris-Saclay University, CNRS Laboratory of Solid-State Physics, Orsay, France
marcelo.rozenberg@universite-paris-saclay.fr

## Abstract

Memristors are finding widespread applications in neuromorphic circuits due to their unique memory effect in resistance. Remarkably, non-volatile memristors can be used to mimic artificial synapses, while volatile memristors can be used to implement spiking neurons. An important example of the latter are the memristive neurons (aka neuristors) based on the unique properties of the quantum materials known as Mott insulators. However, fabricating memristors is an ongoing challenge in material science, especially for transition metal oxides and chalcogenides. Despite intense work, obtaining reliable devices with consistent properties is an open and difficult problem that still hinders their adoption. In recent years, we have been developing an alternative approach introducing a new volatile memristive device (M device) trivially made with a thyristor and a resistor, two inexpensive out-of-the-shelf electronic components. The key non-trivial insight was to recognize that our device exhibits a memristive current-voltage characteristics that is qualitatively the same as that of the challenging Mott insulators. Here we describe why our device is a memristor and how it can be used to trivially implement a spiking neuron. We describe in detail two simple circuits, which we argue are minimal, that implement a tonic spiking and a bursting neuron model. Remarkably, we demonstrate that the behavior of our memristive neuron has a fascinating dynamical behavior, which is quite reminiscent of biological neurons. Our neuron also offers a new exciting technological platform. Its simplicity and low cost is ideal for the implementation of electroceutical medical devices for the treatment of neuropathologies, such as epilepsy and Parkinson's disease. In addition, our neuron can also be implemented in VLSI CMOS technology, for the fabrication of large Artificial Spiking Neural Networks for efficient low-power applications in Artificial Intelligence.

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**Keywords:** memristor, neuromorphic, spiking neuron, bursting

# 1. Introduction. The Mott connection.

Since the theory of memristors was proposed in the 1970s[1][2], these devices have been regarded as the fourth fundamental circuit element and have attracted much attention. However, that theory has also provoked some debate [DiVentra]. Besides that debate, the physical phenomenon of a resistance whose value depends on the past history of the current that has circulated thought it, is well established. The potential technological interest has been the initial driver for the massively increasing number of publications that are produced [4]. Memristors have been already applied in various areas, such as resistive random access memories (ReRAM), in-memory computing (IMC), and neuromorphic hardware [5][6][7][8]. The application in neuromorphic technology has gained particularly widespread attention in the AI. This is motivated by the possibility of using memristive arrays to process massive data at extremely low power consumption [9][10][11].

Neuromorphic computing is a computational approach inspired by the structure and function of the neural network of human brain [12][13]. The basic components of neural networks are neurons that are interconnected by synapses [14][17]. The former can be viewed as computational units that use the information that is stored in the latter. Hence, there is intimate physical proximity between CPU and memory, which is in drastic contrast with the traditional von Neumann computer architecture. In neuromorphic hardware, non-volatile memristors are usually used to implement the functions of synapses [18][19][20][21], while volatile memristors are used to implement the artificial neurons [22][23][24][25]. One of the most prominent type of materials that are adopted for fabricating memristors are transition metal oxides [27]. Some of the most popular non-volatile resistive switching compounds are TiO2, HfO2 and Ta2O5, which are finding their way into large crossbar arrays [strukov]. On the other hand, volatile compounds for neuristors are NbO2 [williams] and specially vanadates [27], such as VO2 [del Valle nature], V2O3 [del valle science] and V3O5 [ada PRX]. However, a major challenge that the implementation of this type of neuromorphic devices still face is their lack of reliability and systematic reproducibility of their characteristics [22][26][27]. This issue is significantly more severe for the neuristors, i.e the electronic spiking neurons, which are based on Mott quantum materials. The key property of the so called Mott insulators that allows them to implement spiking neurons is that they exhibit a spectacular insulator-to-metal transition as function of temperature. In essence, upon application of short and intense voltage pulses, the Mott insulator accumulates the effect of the electric stress until, suddenly, its resistance collapses. As it does, a surge of current flows through the device. That surge is assimilated to the emission of an action potential [stoliar AFM 2017]. We should also mention that another aspect that complicates the situation further is that the theoretical description of Mott materials is also challenging [RMP DMFT]. This is even more so when the materials are pushed away from equilibrium, under the action of strong electric pulsing. Therefore, to provide qualitative understanding of the physics, phenomenological descriptions are a useful [Rocco PRApp]. Thus, despite much current effort, the implementation of reliable Mott insulator devices to implement artificial spiking neurons remains a significant problem of material science.
In this context, in this book chapter we describe our recent efforts to propose a new volatile memristive device that permits a practical, simple, affordable and reliable implementation of an electronic spiking neuron [28][29][30]. Our device is composed of inexpensive out-of-the-shelf components, namely a thyristor and a resistor, which will be referred to as the "M device" in the following (M for Marcelo). The key insight that enable this breakthrough was to realize that the thyristor has a I-V characteristics [wikipedia] that resembles closely that of Mott insulator materials, such as VO2, NbO2, and GaTa4Se8 [23][31][32][33]. However, the thyristor is a three terminal device, while the Mott devices only one. Hence, a second key insight was to realize that analogy with the Mott I-V characteristics can be made complete by simply connect the gate and anode terminals of the thyristor with a resistance. In the Fig.XX below we show the qualitative agreement between the I-V characteristics of NbO2 and our M device.

<figure><img src="image_2.png" /><figcaption>Fig.XX caption. Notice the hysteresis effect.</figcaption></figure>

To fully demonstrate this full qualitative agreement between the Mott and the M device behaviors, we show in the Fig.YY below that both systems when connected in parallel with a capacitor produce the same type of periodic oscillations which resemble the periodic neuronal tonic spiking.

<figure><img src="image_3.png" /><figcaption>Fig.YY caption. Spiking in V305 (from Ada PRX) and spiking in the M device.</figcaption></figure>

In the present work we shall describe in detail how our M device volatile memristor is a simple solution that allows us to build reliable artificial neuron circuits in a very cost-effective and straightforward manner. We shall first describe a basic memristive spiking neuron (MSN), and then extend it to implement a memristive spiking bursting neuron (MSBN) with a more complex dynamical behaviors. We shall explore the full parameter space on the MSBN, obtaining a phase diagram of neuronal behaviors. Surprisingly, the phase diagram exploration allowed us to discover various

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qualitatively different spiking modes that bear striking similarities with those observed in real bursting neurons, rendering unexpected biological relevance to our neuron circuit model. This suggests exciting direction for future developments, such as the implementation of electroceuticals for the medical treatment of neurodegenerative diseases, such as Parkinson's [35] and epilepsy.

Since the thyristor is a conventional electronic device, one may also extend the memristive circuit to the realm of very-large-scale-integrated CMOS circuits (VLSI) [34]. This enables the fabrication of massive numbers of spiking neurons and adopt it as a platform of large artificial neural networks for Artificial Intelligence applications.

## 2. The memristive device based on the Thyristor

The thyristor, also known as the Silicon-Controlled Rectifier (SCR), is a conventional semiconductor current control device invented in the 1950s [36]. It is composed of p-type and n-type materials arranged in a p-n-p-n four-layer structure. Figure 1 (a) and (b) show the schematic and circuit symbol of the thyristor, which has three terminals: Anode, Cathode, and Gate (A, K, G).

The working principle of a thyristor can be most simply described as a diode with a threshold. A diode is a p-n junction that exhibits low resistance in the forward direction (say, p to n for positive bias) but experiences a significantly high resistance (low conductance) when subjected to a reverse voltage bias. Therefore, a positively biased thyristor can be regarded as two p-n junctions in direct where two p-n junctions are in direct (J1 and J3 in Figure 1a), and one in inverse (J2). Therefore, when a forward voltage is applied between Anode and Cathode, and no external voltage applied to the Gate, J1 and J3 would be able to conduct. However, J2 being reverse-biased, creates a large depletion region that is difficult for carriers to overcome (Figure 1 (a)). Thus the thyristor is at a very low conductance state. By applying bias to the Gate, the electrons/holes forming the depletion region can be removed, so that the depletion layer becomes thinner (Figure 1a). Thus, there are in principle two ways to switch on the device, one by applying sufficient positive bias so that a current is forced through the depletion region (as in an inverse bias breakdown of a diode) or by injecting current into the gate. Indeed, when the Gate current (or voltage) reaches a sufficiently high level, the depletion layer becomes "flooded", and the device switches to the high conductive state. These two phenomena are intertwined, so we can think of a threshold AK voltage that is controlled by the GK current (see Figure 1d).
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Figure 1. (a) Schematic of the thyristor with p-n-p-n four-layer structure, and the changes of the width of the depletion layer at the J2 junction under different external conditions. (b) Circuit symbol of the thyristor. (c) Schematic of the M-device, consist of one thyristor and one resistor connecting the Anode and Gate of the thyristor. (d) characteristic I-V of a thyristor in forward bias. Initially the thyristor is in the off-state as there is a depletion layer in the inverse junction J2. The gate current (or applied voltage) controls the threshold switch-on value. Note that the I-V displays a negative differential resistance, which is key for the excitability of the memristive neuron device. In addition, there is also hysteresis, which is crucial to produce the spiking behavior in contrast to simple periodic oscillations (see Section 3).

Based on this property of the thyristor, a key insight is that one may connect the Gate directly to the Anode through a resistor *R*<sub>gate</sub>, so to form a two-terminal device that we shall call "M-device" (Figure 1 (c)). In this way, without the need for an additional control of the Gate bias, when the (external) voltage is increased at the Anode, the voltage (and current) at the Gate will also increase. Thus, by controlling the value of the *R*<sub>gate</sub> we may tune the resistive switching I-V characteristics of the two-terminal device (Figure 2b,c). When applied voltage V<sub>AK</sub> is small, the Gate voltage

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$V_{GK}$ (and $I_{GK}$) is also small, then the M-device is at high resistance as the depletion layer of the thyristor is in place. As the voltage $V_{AK}$ increases, the current injected into the J2 through the Gate also increases so that the depletion layer starts to thin out. At a certain applied voltage the M device reaches a threshold where the depletion layer suddenly collapses and the AK resistance of the M-device commutes, switching from the high resistance (low conductance) to the low resistance (high conductance) (Figure 2 (a) blue arrow).

Another key property worth noting of the M-device characteristics is the hysteresis, which originates in the “holding” current of the thyristor. Once the thyristor is at a high conductance state, a large current density is flowing through the formerly depleted region. Hence, even the $V_{AK}$ falls below the threshold, the high conductive state remains self-sustained as the current density prevents the depletion layer to re-form. As the $V_{AK}$ is further decreased, the current density through the M-device also decreases. Eventually that current density may decrease below small value called the holding current ($I_{hold}$) where the depletion layer forms again. This produces the switching back of the M-device to the low conductance state (Figure 2 (a) red arrow).

These mechanisms enable the memristive characteristic of the M-device, which is shown in the measured I-V curve in Figure 2 (a) [NCE]. It performs the behavior of a “pinched” hysteresis loop [chua], with the blue arrow indicating the transition from the low-conductance state to the high-conductance state at a threshold voltage, and the red arrow indicating the reverse transition at $I_{hold}$. This I-V curve carries qualitative similarities to the I-V loops of volatile Mott insulators [32].

<figure><img src="image_10.png" /><figcaption>Figure 2. (a) The measured I-V characteristics of the M-device with $R_{gate}$ = 100K Ω. (b) The I-V curves of M-devices with different $R_{gate}$ values. [[There are two panels with the same $R_{gate}$ value 150K]] (c) Linear relation between the $R_{gate}$ and the threshold voltage of the M-device. (The I-V curves are measured with a 1KΩ resistor in series with the M-device, and driven by voltage wave generator. The thyristors used in the experiments are the SCR from STMicroelectronics with reference number P0118MA 2AL3.)</figcaption></figure>

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In addition, from the engineering point of view, the threshold voltage ($V_{th}$) for the M-device to switch from high resistance to low resistance, a key parameter of the memristive spiking neuron model described in the next section, is easily adjustable. Figure 2 (b) shows that by increasing the value of the $R_{gate}$, the $V_{th}$ also increases correspondingly. Moreover, there exists a linear relationship between the $R_{gate}$ and the $V_{th}$, granting it with excellent engineering controllability.

As the thyristor and resistor are both stable, inexpensive, and out-of-the-shelf electronic components, they make the M-device a reliable and cost-effective volatile memristive device.

### 3. The Memristive Spiking Neuron Model

In this section, we describe the implementation of an analogue spiking neuron model exploiting the volatile memristive properties of the M-device. We shall first introduce a basic circuit configuration that implements a leaky-integrate-and-fire model neuron. We shall then extend the basic model to implement a bursting neuron with rich dynamic characteristics. An important point to make is that while a mathematical model is defined by a set of equations, the present neuron models are defined by their hardware implementation. In other words, *the circuit is the model*.

A typical biological neuron is schematically represented in Figure 3 (a). The neuron receives input current signals through the dendrites, and generates electric action potential pulses (spikes) that travel down the axon and generate the output to other (downstream) neurons. A molecular-level mechanism of the spiking neuron can be roughly described by looking at the membrane of the neuron cell depicted in Figure 3 (b). The cell body of the neuron is surrounded by a membrane, which is a lipid bilayer containing various types of embedded protein structures [37]. The lipid bilayer membrane is an excellent insulator, preventing the free flow of ions (such as sodium and potassium ions) between the two sides of the cell membrane. However, there are also proteins that form many ion channels across the cell membrane to actively transport or passively allow the passage of ions under certain conditions. Therefore, from a functional perspective, the lipid bilayer can be represented as a capacitor $C_m$, while the ion channels can be represented by several non-linearly conductances whose values depend on the membrane voltage $g(V)$. The latter are also called voltage-gated conductance channels and in our neuron model they are implemented by the M-device, as we shall discuss next.

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<figure><img src="image_13.png" /><figcaption>Figure 3. Schematic of (a) a biological neuron, (b) its membrane structure with lipid bilayer and ion channels, and (c) an equivalent circuit of the integrate-and-fire model, where we only show one conductance channel for simplicity.</figcaption></figure>

The very simplified working principle of such a system can be described as an Integrate-and-Fire (IF) neuron model, as shown in the Figure 3 (c). The input currents to the neuron, the charges (ions in the biological case) are first "integrated" in the membrane capacitor $C_m$. The resulting voltage of the $C_m$ (membrane potential $V_m$) increases until it reaches some certain threshold value, where the conductance channel $g(V)$ becomes conductive. Then the $C_m$ rapidly discharges through the conductance channel, i.e. "fires" a strong current or voltage spike, which is called the action potential. This basic mechanism is also present in some more biologically realistic models such as in the Hodgkin-Huxley model or, in its simplified version, the Morris-Lecar model [15][16][17][38], where multiple ion channels may be considered, with different voltage-gated channels represented by respective non-linear conductances. Regardless of their simplicity or complexity, all those models share a similar mathematical structure, which is defined by a set of equations of the form:

$$C_m \frac{dV_m}{dt} = I_{ion}(t) + I_{in}(t) = \sum_k I_k(t) + I_{in}(t) \quad (1.)$$

$$I_k(t) = g_k(V_m, S_k)V_m \quad (2.)$$

where $I_k$s are the channel currents and $g_k(V_m, S_k)$ is the respective voltage-gated conductances. The variable $S_k$ indicates the "state" of the conductance.

In an innovative hardware implementation that is receiving significant attention, a type of qunatum materials called Mott insulators are used to implement the non-

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linearity required in a voltage-gated channels [22][23]. The key insight in the present approach is, as demonstrated in Section 2, that the M-devices exhibit I-V characteristics that are analogue to those of Mott insulators. Specifically, the volatile history-dependent conductivity or memristance. Therefore, M-devices exhibit the key feature required to implement the conductance channels of Eq. (1) and (2) above. Moreover, in contrast to the Mott materials which remain a challenge to reliably fabricate and to theoretically understand, our M-device is solely implemented in inexpensive conventional off-the-shelf electronic components.

Figure 4 (a) shows the circuit implementation (i.e. definition) of a Memristive Spiking Neuron (MSN). As can be seen in Figure 4 (c), the basic MSN circuit realizes a (Leaky) Integrate-and-Fire neuron model. In this model, $C_m$ plays the role of the lipid bilayer while the M-device act as a (leaky) ion channel. For convenience, in order to transform the current spikes produced by the sudden discharge of the $C_m$ into a voltage spike signal $V_{out}$, we adopt a small $R_{load}$ (47Ω) in series with the M-device.

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Figure 4. (a) Schematic of the Memristive Spiking Neuron (MSN) circuit based on the M-device. (b) A typical spike generated by the MSN, and the corresponding non-linear conductance changes of the M-device during the spike event (adapted from [28]) (c) Demonstration of the MSN integrate-and-fire function under a train of input pulses. The top panel shows the leaky integration of the current in the $C_m$. The dotted line indicate the threshold potential. As $V_m$ reaches the threshold a spike is emitted, as shown in the mid panel. The bottom panel shows the train of input synaptic current pulses. The data are obtained with $C_m = 10\mu F$, $R_{gate} = 100K\Omega$, $R_{load} = 47\Omega$, and $I_{in} = 92.4\mu A$.

Under the excitation of a constant current $I_{in}$, the MSN circuit emits a regular succession the electric spikes, which is called tonic spiking. One, such a spike is shown in the top right panel of Figure 4 (b). During the spike generation, a conductance switch of the M-device is clearly observed, as shown in the main panel of Figure 4 (b). The colored dots with numbers indicate the conductance state of the M-device which are correlated to the respective colored dots along the spike trace in the top-right panel. When the M-device switches from low conductance to high conductance (dot 1 to dot 2), a significant increase in $V_{out}$ is observed, ie the spike is emitted. Then the $C_m$ rapidly discharges and the current going through the M-device decrease to $I_{hold}$ (dot 2 to dot 3, and also see Figure 2). Finally, the M-device switches back to low conductance (dot 3 to dot 4). Note that the difference between slow charge time (integrate) and fast discharge (fire), corresponds to the dramatic change in the RC constant of the circuit due to the conductance change of the M-device.

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The Integrate-and-Fire behavior is also very intuitive observed when the input is a train of synaptic current pulses, as seen in Figure 4 (c). The pulse inputs are (leaky) integrated in the $C_m$ one by one, until the membrane potential $V_m$ reaches certain threshold, then a spike is generated and $V_m$ is reset. One important thing to note here is that during the time between two input pulses there is no input current and the membrane potential shows a small decrease. This is the leaky integration, which is due to the fact that the M-device is not an ideal switch (i.e. the low conductance is not open circuit). The small charge leakage through the M-device is controlled by the value of R that is connected between the anode and the gate of the thyristor (see Section 2). Hence, the MSN circuit may be better characterized as a realization of the Leaky-Integrate-and-Fire (LIF) model, which will be further elaborated in the following discussion.

From this description, one can readily understand that the spike frequency (firing rate) should depend on the input current. For instance, upon the excitation with a constant current $I_{in}$, the membrane potential $V_m$ of an ideal IF neuron linearly increases with time, as $V_m(t) = \frac{l_{in}t}{c_m}$, and suddenly discharges with a spike when $V_m(t) = V_{th}$. Therefore, an IF neuron model emits spikes at a frequency $f(I_{in}) = \frac{l_{in}}{c_m v_{th}}$, which is linear in the applied $I_{in}$. This behavior is qualitatively well reproduced by our MSN model, as shown in Figure 5.

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Figure 5. Spiking frequency's dependence on input current $I_{in}$. In the middle region, the curve follows an approximate linear behavior (blue dashed line) similar as the ideal IF mode. Deviations from linearity occur near the excitation thresholds $I_{in}^{min}$ and $I_{in}^{max}$. The red curve indicates the fitted frequency versus input relationship of a LIF model [ermentrout book]. Right panel shows experimental data of the current dependent spike rate of a biological neuron (exp) that shows the same qualitative behavior of our MSN model [reproduced from [Bianchi et al J Comp Neurosc 2012]. The sudden decrease of spike rate with increasing current is called the "depolarization block".

However, it is worth noting that such a linear relationship between spiking frequency and $I_{in}$ is not applicable across the entire $I_{in}$ range. As shown in Figure 5, deviations from this simple linear relationship occur when $I_{in}$ is too small or too large. In Figure 5, we denote by $I_{in}^{min}$ and $I_{in}^{max}$ the lower and upper thresholds of excitability, namely they define the current interval where spike emission is triggered in the MSN model. Biological neurons also have qualitatively similar thresholds. The region beneath the lower one may be associated to the so called "hyperpolarized" state, while the region above the upper threshold may be associated to the

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"depolarized" one. Another consequence of these two boundaries is that there exists a maximal spike frequency rate that can be sustained by the neuron, also qualitatively similar to that present in biological neurons, as shown in Figure 5 (b). At the maximal frequency we obtain a minimal inter-spike time interval which may be considered as a refractory period of the neuron.

The reason for the departure from the linear behavior at $I_{in}^{min}$ is due to the leakage of the M-device. As explained before, the conductance of the M-device is not zero when it is at the low conductance state, which results in a leak current. Thus, the $I_{in}^{min}$ is due to the minimal current $I_{in}$ that is required to overcome the leakage, so as to enable the charge integration in $C_m$ and eventually emit a spike. In the LIF model, $f(I_{in})$ and $I_{in}$ are no longer linearly related, but they follow a logarithmic relationship: [17][38]

$$f(I_{in}) \propto -\frac{1}{\log\left(1 - \frac{I_{in}^{min}}{I_{in}}\right)} \quad (3.)$$

We observe in Figure 5, the good quality of the fit using the expression of the LIF model (3). Those data also demonstrates the type 1 excitability of the MSN model, which refers to systems that initiate spiking from zero frequency [17][38].

As for the deviation from linear behavior at $I_{max}$, it is mainly related to the holding current. More specifically, it happens when the $I_{in}$ is greater than $I_{hold}$. In this case, once the M-device switch to the high conductance state (dot 1 to dot 2 in Figure 4 (b)), the $C_m$ is continuously discharging but the charge is replaced by the relatively large $I_{in}$. Thus, for $I_{in} > I_{hold}$, the M-device never gets the chance to switch back to its low conductance state (dot 3 to dot 4 in Figure 4 (b)). The single spike is a phasic firing, which is followed by the neuron accommodating to a continuous flow of a large current through the M-device. This high-current state is reminiscent of the depolarization block observed in biological neurons (see Figure 5(b)) [39].

## 4. The Memristive Bursting Neuron Model

To search for more complex neuronal dynamics, we extended the MSN into a Memristive Spiking Bursting Neuron (MSBN) model by simply introducing a second time constant in the circuit model. This is inspired by the two-compartment models (Figure 6 (a)) from theoretical neuroscience such as Pinsky-Rinzel model [40]. We introduced a second capacitor $C_s$ to the MSN in parallel with the load resistor $R_s$ as shown in Figure 6 (b) The resulting MSBN model exhibits, in addition to the simple spiking behavior (referring to Tonic Spiking (TS) shown in Figure 6) three new spiking modes: Fast Spiking (FS) and two types of Intrinsic Bursting (IB Type1 and IB Type2) as displayed in Figure 6.

Thus, the MSBN has, in addition to the first time constant in the circuit defined by the $\tau_m = R_{gate}C_m$, a second time constant, $\tau_s = R_sC_s$. Then, similarly to the Pinsky-

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Rinzel model [40], we associate each part of the circuit to two compartments, namely a dendrite compartment and a soma compartment, respectively. The increase in dimensionality of this model is considered a necessary condition for generating intrinsic bursting [38]. More specifically, $C_s$ and its voltage $V_s$ introduce an additional modulation to the $I_{ion}$ in Equation 1, leading to the following model equations:

$$C_m \frac{dV_m}{dt} = I_{ion}(g, \tau_s) + I_{in} \quad (4.)$$

$$I_{ion} = g(V_m, S)(V_m - V_s) \quad (5.)$$

$$C_s \frac{dV_s}{dt} = I_{ion} - \frac{C_s}{\tau_s} V_s \quad (6.)$$

Note that the last equation in this set shares the same form as the dynamic equation for the [Ca]-current in the theoretical model of Bursting neurons [38].

The MSBN has, nevertheless, some differences from those theoretical mathematical models. In those models, the second time constant $\tau_s$ is usually greater than $\tau_m$, so as to slowly modulate the tonic spike generation by driving the neuron in-and-out of its excitability range. In contrast, in the MSBN we shall see that $\tau_s$ is much smaller than $\tau_m$. Another difference is that in our case only one compartment, the "dendrite", is excitable, while the "soma" is passive as it merely introduces a second timescale [cite Idan Segev, Wilfrid Rall, Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations, Trends in Neurosciences, 21, 453 (1998)]. Finally, in the two-compartment model there is a conductance connecting dendrite and soma compartments $g_c$, while in the MSBN this conductance is taken to be infinite, namely the dendrite and soma are assumed to be directly connected. This known as electrotonic coupling [40].

An important point to make is that, as mentioned in the previous section, the spiking frequency is a function of $C_m$, which allows for a very simple control of the timescales of neuronal behaviors. In Figure 6 (d), we demonstrate how the spiking traces of the MSBN can straightforwardly be tuned to biological realistic timescales by modulating $C_m$. This feature is very attractive for potential applications in brain-machine-interfaces and neuroprosthetics.

The MSBN is a remarkably compact circuit comprising only five conventional electronic components, yet its emergent dynamics is complex and biomimetic. Figure 6 (c) illustrates four distinct neuronal behaviors generated by the MSBN, including Tonic Spiking (TS), Fast Spiking (FS), Intrinsic Bursting type 1 (IB1), and Intrinsic Bursting type 2 (IB2). A qualitative but more concrete discussion of the mechanisms underlying these neuronal behaviors will be explored in the next section after obtaining the neuronal behavior phase diagram for MSBN. Notably, as we shall discuss later on, the neuronal behaviors generated by the MSBN are qualitatively similar to those observed in some biological neurons, such as nigral dopamine neurons under the effect of neurotoxins.

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<figure><img src="image_21.png" /><figcaption>Figure 6. (a) Schematic representation of a two-compartment neuron model. (b) Memristive Spiking Bursting Neuron (MSBN) inspired by the two-compartment model. (c) Four neuronal behaviors generated by the MSBN, including tonic spiking, fast spiking, intrinsic bursting type 1 and type 2. (d) The same spiking and bursting traces generated in a biological realistic timescale by tuning the value of $C_m$ [NCE paper]</figcaption></figure>

## 5. The Phase Diagram And Connections To Neuroscience

To better understand the origin of the various spiking states of the MSBN, we explore the phase diagram of the model. We find convenient to do this in terms of the variables input current $I_{in}$ and the time constant of the soma compartment $\tau_s$ ($C_s$ fixed, and $\tau_s$ varied by changing $R_s$). In the limit of $\tau_s$ small we recover the basic MSN that we discussed before, that shows only tonic spiking (see Figures 4 and 5).

Figure 7 illustrates the full phase diagram. The red, blue, green and yellow regions represent the location of the four different neuronal behaviors shown above. The gray region stands for the quiescent state, which appears when $I_{in}$ is too small or too big with respect to the onsets of excitability ($I_{in}^{min}$ and $I_{in}^{max}$), as discussed in Section 3. Examples of the spiking and bursting traces are shown on the right panels, with their respective locations indicated in the phase diagram. All four colored regions in the phase diagram are relatively large, so each neuronal behavior may be easily realized, without the need of any fine tuning.

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Figure 7. Left: Phase diagram of the Memristive Spiking Bursting Neuron circuit, with the two variables being the input current and the time-constant of the soma-compartment ($τ_s$). The four regions correspond to the 4 spiking types: tonic spiking (TS), fast spiking (FS), intrinsic bursting type 1 (IB1) and intrinsic bursting type 2 (IB2). Right: measured traces in selected locations of the phase diagram. They illustrate the effects produced by changing the *lin* (first column) or *ts* (third column) with respect to the states shown in the second column. The corresponding positions of all traces in the phase diagram are marked with numbered white squares, with (*ts*, *lin*) [mathematic font??] indicated in each numbered panel.

The phase diagram gives us further insights on the mechanism operating in the MSBN neuron model. For example, on the left part of the phase diagram, where $τ_s$ is small, the MSBN only shows Tonic Spiking, as it approaches the MSN model. This tells us that when the soma time-constant is relatively small, its impact on the dendrite time constant ($τ_m ≈ R_{gate}C_m$) is minimal. *keeping the entire circuit essentially in the original state of MSN*. In other words, the effect of the "small" soma can be considered as a small perturbation to the large (excitable) dendrite body, without affecting the generation of tonic spiking. [cite Idan Segev, Wilfrid Rall, Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations, Trends in Neurosciences, 21, 453 (1998)]

As $τ_s$ gradually increases, complex neuronal behaviors emerge. Interestingly, during the phase transition from TS to other spiking states, a distinct vertical boundary appears at a certain value of $τ_s$. This implies that the transition from TS to other behaviors is primarily controlled by $τ_s$ and rather independent of $I_{in}$. In other words, the emergence of other states occur when $τ_s$ can no longer be considered a negligible perturbation.

We consider the transition from TS to FS as an example to describe the mechanism by which other spiking states emerge. Let's for the moment disregard the influence of $C_s$ and let's assume that $R_s$ is relatively large with $R_M^{low} << R_s << R_M^{high}$. Initially, we start injecting current into the MSBN, with the M-device in a high-resistance (low conductance) state. Then, the membrane voltage $V_m$ mostly drops across the M-device as $R_M^{high}$ and $R_s$ form a voltage divider. As $C_m$ becomes gradually charged, the membrane voltage $V_m$ reaches the threshold $V_{th}$ and provokes the M-device

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switching to the low-resistance state $R_M^{low}$. This would produce a large current from the discharge of $C_m$, which is the spike generation mechanism. However, and in contrast to the normal spike production mechanism in the basic MSN, now $R_s$ is taken relatively large, $R_s \gg R_M^{low}$, we have a different situation. In fact, as soon as the M-device switches to low resistance the membrane voltage on the M-device gets dramatically reduced to close to zero. This is because the voltage divisor has drastically changed and all the $V_m$ now drops $R_s$. This has severe consequence on the stability of the spike emission, because the negligible voltage on the M-device would immediately switch it back to the high resistance state, aborting the spike emission. As soon as the M-device switches back, it finds itself again above $V_{th}$ so it should switch back to low resistance. Hence, the emission mechanism became now a unstable state due to the increase of $R_s$. It is not hard to see that this instability may be cured by introducing a finite $C_s$. The (short) time-constant $\tau_s = RsCs$, $\tau_s \ll \tau_m$ allows a "portion" of the spike to be emitted during the short interval $\tau_s$, during which the $R_s$ is effectively shorted by $C_s$. Hence we shall see the fast spiking behavior emerge, with the FS frequency $\sim 1/\tau_s$.

The understanding of the mechanism of IB1 and IB2 is not so straightforward. We have provided a mathematical description of its dynamics elsewhere [L. Fernandez et al, submitted]. Nevertheless, we can qualitatively see that these behaviors are associated to the upper and lower current excitability thresholds, that correspond to $I_{in}^{min}$ and $I_{in}^{max}$. Hence we can qualitatively understand that these complex intermittent spiking modes are emerging from the interaction of two instabilities that are in competition. Namely, (i) large-$R_s$/low-$I_{in}$ (IB1) and (ii) large-$R_s$/high-$I_{in}$ (IB2), which also explain the "triangular" shape of those two regions of the phase diagram (green and yellow).

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Figure 8. [use the old figure that we had with the actual experimental data, let the editors worry about the copyright permits :) and adjust the caption to the old caption that we had in that version of the manuscript] The neuronal behavior evolutional paths explored in the phase diagram. (a) Two evolution paths achieved by changing τs and Iin, which qualitatively reproduced the neuronal behavior changing in Nigral Dopamine neuron under the application of neurotoxins (Apamin+TTX+TEA) [41]. (b) The evolution path achieved by solely increasing Iin, where the same neuronal behavior changing is observed in pre-Bötzinger respiratory neurons with increasing electric stimulus [42].

While our MSBN is not a bio-realistic model, it is a striking observation that it is bio-mimetic. Indeed, we discovered surprising similarities between the spiking modes generated by our circuit and those measured in traces of biological bursting neurons under the action of different neurotoxins that affect their normal spiking behavior. This is shown in Figure 8, where we directly compare MSBN traces to those from biological bursting neurons. In panel (a) we compare the different traces obtained in nigral dopamine neurons, which start from the normal (control) state and then become severely affected by exposure to neurotoxins: first apamin and then apamin+TTX+TEA [41]. These neurotoxins are found to provoke two different set of pathological spiking behaviors, as shown in the Figure 8a. Interestingly all the different spiking behaviors seen in the nigral dopamine neurons can be qualitatively found in our phase diagram, as we show in the figure. It is a very interesting observation that we may then associate a "path" in the phase diagram to the action of neurotoxins. This may have a potentially useful application as a guide for neurologist to understand the origin of pathological spiking behaviors.

Perhaps even more striking is our second example shown in panel (b). In this case the biological neurons are another type of bursters: pre-Bötzinger respiratory neurons [42]. In this case the experiment consisted in rendering the neuron quiescent by application of a neurotoxin. Then by sole application of input current the neuron can be reactivated. Upon increasing input current in small steps, it is observed that the neuron emits an increasing number of short spike trains. Eventually, at high enough input current the trains become a continuous fast spiking. Quite strikingly we find that this same behavior can be captured by our MSBN. In fact, as shown in the Figure 8b, we observe how by moving vertically in the phase diagram, ie, solely increasing the input current, the MSBN evolves from the quiescent state through a series of bursting states with increasing number of spike trains, to a final fast spiking, in excellent qualitative agreement with the behavior of the biological pre-Bötzinger neuron [42].

Conclusions

In this work, we described in detail a novel volatile memristive electronic component, the M-device, which we synthetize out of two conventional out-of-the-shelf electronic components (a thyristor and a resistor).

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The M-device exhibits pinched hysteresis and bears a striking qualitative similarity to the I-V behavior of many Mott insulators, which are quantum materials emerging as candidates for future neurocomputing hardware [Janod review, del Valle review, Hoffmann review, wei Yi Nat com VO2, william's nat mat neuristor]. Like Mott neurons, our device has extreme simplicity enabling straightforward biomimetic spiking neuron behavior. Unlike Mott neurons, which are still hard to fabricate and function in a consistent and reliable manner, our M-device is extremely reliable and affordable allowing for very cost-effective implementations. While neurons made out of discrete components present an evident limit to the practical number of neurons that one may envision in a neural network (up to hundreds), they provide a flexibility in circuit design that is unachievable in CMOS VLSI systems, which require long design times and fabrication. In any case, the M-device can be straightforwardly be ported to CMOS VLSI technology [P Stoliar et al 2022 *J. Phys. Commun.* 6 021001], while carrying along the conceptual simplicity provided by the memristive device.

In recent years, Neuromorphic technology is becoming a significant application arena for volatile memristors [Janod review, del Valle review, Hoffmann review, wei Yi Nat com VO2, william's nat mat neuristor]. Based on the M-device, we demonstrated here a simple Memristive Spiking Neuron (MSN) circuit, which provides a physical hardware implementation of the functionality of the popular Leaky Integrate-and-Fire (LIF) model, in fully analogic electronic. This is a concrete realization of the concept "the circuit is the model". Furthermore, we also demonstrated how this concept can be exploited following the insights from theoretical neuroscience models that indicate that a simple spiking neuron model requires an additional time-scale to produce bursting behavior. That was directly implemented in the second circuit model, the Memristive Spiking Bursting Neuron (MSBN), that we introduced and analyzed here. Our simple 4-component MSBN was found to exhibit spiking and bursting behaviors that bear striking similarities to biological bursting neurons.

This simplicity enabled us the full exploration of the phase diagram of neuronal behavior of the MSBN. We made explicit contact with two notable examples of biological bursting neurons: Nigral Dopamine and pre-Bötzinger respiratory. Specifically we were able to "follow the path" in the phase diagram of their pathological spiking evolution under the action of neurotoxins. These amazing connection may open up the way for potential unprecedented novel approaches to understanding pathological states such as those in Parkinson's and epilepsy, or even new ways for treating neurodegenerative diseases, though implementations in the field of neuroprosthetics and neuro-implants.

## Acknowledgments

We acknowledge support from the French ANR "MoMA" project ANR-19-CE30-0020.

## Conflict of Interest

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The authors declare no conflict of interest.

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