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Edited by: Themis Prodromakis, University of Southampton, United Kingdom

Reviewed by: Adnan Mehonic, University College London, United Kingdom; Erika Covi, Politecnico di Milano, Italy

This article was submitted to Neuromorphic Engineering, a section of the journal Frontiers in Neuroscience

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Information in the central nervous system (CNS) is conducted via electrical signals known as action potentials and is encoded in time. Several neurological disorders including depression, Attention Deficit Hyperactivity Disorder (ADHD), originate in faulty brain signaling frequencies. Here, we present a Hodgkin-Huxley model analog for a strongly correlated VO_{2} artificial neuron system that undergoes an electrically-driven insulator-metal transition. We demonstrate that tuning of the insulating phase resistance in VO_{2} threshold switch circuits can enable direct mimicry of neuronal origins of disorders in the CNS. The results introduce use of circuits based on quantum materials as complementary to model animal studies for neuroscience, especially when precise measurements of local electrical properties or competing parallel paths for conduction in complex neural circuits can be a challenge to identify onset of breakdown or diagnose early symptoms of disease.

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Action potentials (AP) are generated in neurons and propagated to other neurons via synapses (Hodgkin and Huxley,

Wu. et al. reported that the dysfunctional calcium channel in mutant mouse model is associated with the hypokalemic periodic paralysis which is a form of paroxysmal weakness that occurs in motor neuron disease (Wu,

Understanding their origins and the mechanisms to minimize damage to neural pathways is a principal area of study in neuroscience. However, diagnosis of neurological disorder at the molecular level is challenging (Brown et al., _{2} that undergoes an electrically-driven

Oxides have been studied for electronic devices such as resonant tunneling diodes, single-electron transistors, and steep slope switches (Mannhart and Schlom, _{2}. The strongly correlated VO_{2} artificial neuron system can undergo an electrically driven IMT akin to the excitable membrane in the biological neuron. Changes in composition of the material synergistically modifies the ground state resistivity, IMT strength defined as resistance ratio in the two phases as well as the threshold voltage required for initiating a phase change. Such material property is designed to capture the Intrinsic Membrane Excitability (IME) in biological neurons, which refers to a neuron's propensity for generating action potential at a given input. Building on this fundamental concept, we demonstrate neuronal function mimicking a vast range of neuron types found in animal brains and simulate an archetypal monosynaptic circuit (e.g., the

VO_{2} thin films of 200 nm thickness were deposited on SiO_{2}/Si by reactive sputtering at 775 K. The stoichiometry and IMT transition strength in VO_{2} is controlled by the oxygen partial pressure in the sputtering chamber. The IMT occurs at a critical temperature T_{c}. IMT transition strength (R_{ins}/R_{met}) is defined by the ratio of high resistance state (R_{ins}, measured at room temperature) and low resistance state (R_{met}, measured at above critical transition temperature). In VO_{2}, T_{c} is 67°C. The low resistance state is taken at 120°C that is significantly higher than T_{c}. Our film growth experiments have shown controllable thermal IMT strength variation from R_{ins}/R_{met} >10^{5} to R_{ins}/R_{met} = 1 (complete loss of IMT characteristic) (Ha et al., _{ins} and R_{met} are respectively, the resistivity for the insulating state and metallic state, and is characterized by temperature-dependent Hall measurement. The IMT strength can be controlled by substrate temperature during film deposition, oxygen partial pressure during growth, and the choice of substrates (Savo et al., _{2} thin film with IMT strength of 2 × 10^{5} (Lin et al.,

After VO_{2} thin film growth, we fabricated lateral device for artificial neuron circuit testing. Electron-beam lithography (EBL) was used to define the length of the VO_{2} device, L. As shown in Figure _{2}. A “neck-down” design for the contact were used as illustrated in Figure _{2} into transition and reduces the voltage required to trigger the transition (Lin et al., _{2} under the experimental condition is largely un-controllable. Therefore, we use a model that has been calibrated with experiment with a circuit simulation approach to derive a systematic understanding for the impact of VO_{2} resistive state on neuron behavior. The model is discussed in the following sessions.

Experimental details for the VO_{2} device. _{2} devices under DC probing. _{2} device with a “neck-down” layout. The “neck-down” layout leads to lower forward critical transition voltage (V_{c}) and lower power in the switching operation.

All experiments were carried out at room temperature. The DC sweeping and current-clamp are both performed using Keysight B1500A. Waveforms for the current-clamp experiment were acquired by Keysight Digital Oscilloscope DSO9104A. For DC sweeping, a current compliance is set to 5 mA. For current-clamp response, the IMT device is protected by a series resistance through a circuit board so that the measurement can be repeated reliably by avoiding excess heating and burnout.

Figure _{2} devices under DC probing and Figure _{2} device with a “neck-down” layout. The “neck-down” layout is used to minimize the volume of VO_{2} that undergoes transition (Lin et al., _{c}) and lower power in the switching operation. The spacing between two contacts is L = 200 nm for the device being studied in this work. Figure

Excessive bias stress to the VO_{2} device can result in non-reversible damage to the material which is manifested in a permeant change in critical transition voltage under DC measurement. This can happen when the device is subjected to a bias outside the safe operating criteria. Two forms of non-reversible damages are shown in Figure _{c} caused by an increase of the HRS resistance (+ΔR). The current drops over the whole range of applied voltage in the second sweep. Figure _{c} which is the indication of a drop in the HRS resistance (–ΔR). The current is higher over the span of applied voltage.

VO_{2} device under excessive DC stress experiencing non-reversible change in critical transition voltage. Two consecutive sweeps are applied to the VO_{2} device with the first sweep stresses the device. _{c} is caused by an increase of HRS resistance, +ΔR. _{c} is the indication of a drop in HRS resistance, –ΔR.

Figure _{2} neuron. The membrane of the biological neuron (Figure _{m}, and transmembrane conductance, G_{m}. G_{m} is the sum of various ion channels conductance and it can go through a reversible _{in}. The output AP waveforms depend on the input current, the electrical and geometric parameters of the cell, and the environment such as temperature. The HH model and its parameters are described later in Sec. II.A.

The biological neuron and analogous VO_{2} neuron. _{M}) and a membrane conductance (G_{M}) that can go through _{M}) and the membrane conductance (G_{M}) oscillate. The AP can propagate along the axon and transmit signal to the other connected neurons. The myelin sheath surrounding the axon of some neuron cells can enhance the speed at which impulses propagate. _{2} device and a capacitor are used to construct the VO_{2} artificial neuron circuit. The VO_{2} material exhibits a reversible electrothermal _{2} neuron output node and VO_{2} conductance oscillate, similar to that of the biological neuron. The insulating-state resistance can be changed when VO_{2} degrades, and this feature is utilized to model spike-timing related neural disorders. Here +ΔR represents an increase of resistance and –ΔR represents a drop in resistance.

The VO_{2} device and analogous VO_{2} neuron circuit are shown in Figure _{2} is well-known for its reversible _{2} artificial neuron is a circuit that comprises, a minimum of, only two components, the capacitor C_{o} and the conductor (i.e. resistor) G_{VO} as shown in Figure _{in} starts, the VO_{2} neuron exhibits an oscillatory behavior similar to that in the biological neuron. The model and experiment for standalone VO_{2} devices are discussed, respectively in Sec. II.B and Sec. II.C. The VO_{2} neuron circuit is described in Sec. II.D.

The complete circuit schematic for a patch of the neuron membrane with the HH model is illustrated in Figure ^{+} channel, the K^{+} channel and the leakage channel.

Full schematic of biological neuron that contains two ion channels, leaky capacitive membranes. The equations the form the Hodgkin-Huxley model are shown in the Equations 1–3.

The basic mechanisms in the HH model contains ion transport and transmission lines for Action Potential (AP) propagation. The key equations are shown in Equations 1–3. Equation 1 relates the membrane current density I_{IN} to membrane potential V_{m}. The area-normalized membrane capacitance is C_{m}. Two ion channels with the leaky conductance are included in the model. Their conductance is denoted as G_{Na}, G_{K}, and G_{L}. The Nernst equilibrium potential in Equation 2 relates extracellular and intracellular ion concentrations, respectively denoted as C^{i} and C°. Through Equation 2, the Nernst equilibrium potentials V_{Na} and V_{K} can be obtained for the given Na^{+} and K^{+} concentrations. The molar gas constant R and Faraday's constant F are physical constants. Finally, the propagation of AP along z direction is described by the core conductor equation in Equation 3. It couples the voltage and current along a cylindrical cell where the resistances per unit length inside and outside the cell are, respectively, r_{i} and r_{o}. The cylindrical cell has diameter a. The baseline values of the parameters in the HH model are listed in Table

Baseline input parameters for the Hodgkin-Huxley model.

Conductance of Sodium channel | G_{Na}(mS/cm2) |
120 |

Conductance of Potassium channel | G_{K}(mS/cm^{2}) |
36 |

Leakage conductance | G_{L}(mS/cm^{2}) |
0.3 |

Extracellular Sodium concentration | C |
500 |

Intracellular Sodium concentration | C |
50 |

Extracellular Potassium concentration | C |
20 |

Intracellular Potassium concentration | C |
400 |

Membrane capacitance | I |

Figure

Simulated results from the Hodgkin-Huxley model.

The model for electrothermal IMT is first introduced in Lin et al. (_{th}, density ρ_{o}, and thermal conductivity

There are two origins of heat flux. Firstly, Joule heating results in incoming heat flux to the medium. The power generated by Joule heat follows Ohm's law, and for a unit volume it is:

where _{r} is the temperature-dependent resistivity of the IMT. Secondly, the outgoing heat flux is generated by convective heat loss, modeled by the effective convective heat transfer coefficient _{a}, and IMT's surface to volume ratio _{p}/A where _{p} is the cross sectional perimeter. The power dissipated through side wall heat convection is:

Taking into account the heat fluxes, the differential equation for heat transfer is shown in Equation 8.

As shown in Figure _{c}. As the boundary condition, _{c} is assumed to be long enough so that the value of _{c} has negligible impact – _{c} should be significantly longer than the heat diffusion length. Equation 8 is solved using a numerical method: forward difference for the time domain and central difference for spatial domain. In the spatial domain, the IMT bar is discretized into segment of length _{r}(_{IMT} is obtained by integrating the resistance of all segments:

The current through the IMT can then be obtained by

Where _{S} is the series resistance. The IMT resistivity as a function of temperature follows a look-up table of resistivity vs. temperature as measured in the experimental VO_{2} device. A typical example of the resistivity vs. temperature is shown in Figure _{H}, the metal-state resistivity ρ_{L}, and the critical transition temperature (T_{c}). The baseline values of other physical parameters are listed in Table

Full schematic of an IMT device with length L between two metal contacts. The device is discretized into segment of dx, etch with its own temperature and resistivity. Heat conduction is along x direction while convective heat loss through the side wall.

Input parameters for the coupled electrical-thermal model IMT model.

Thermal conductivity | K (W/K-m) | 6 |

Specific heat | C_{th} (J/K-kg) |
690 |

Effective convective heat transfer coefficient | h (W/K-m^{2}) |
10 |

High resistivity state | ρ_{rH} (Ω-m) |
10–^{3} |

Low resistivity state | ρ_{rL} (Ω-m) |
10–^{5} |

Density | ρ_{d} (kg/m^{3}) |
4 × l0^{3} |

Cross sec′ area | A (m2) | 1 × 10^{−12} |

Length | L (m) | 5 × l0^{−7} |

Ambient temperature | T_{a}(K) |
300 |

Series resistance | R_{s} (Ω) |
70 |

The complete VO_{2} neuron circuit is shown in Figure

The complete VO_{2} neuron circuit. The whole circuit contains three elements: a capacitor as well as the VO_{2} device with a sensing resistor in series. The output current is sensed by the sensing resistor. The voltage at the capacitor node is denoted as V_{o}. It is also the input node for the injected current.

The model is focuses on the material properties of VO_{2} that emulate biological neuron functions. A series resistance R_{s} is added in series with the VO_{2} for two reasons. First, it limits the current when the VO_{2} device transitions to the metallic state, and ensures reliable switching. The safe operating design follows the theoretical guideline derived in Lin et al. (

The four stages in one spike cycle in the VO_{2} neuron and the corresponding experimental output waveforms. _{2} is at HRS. Voltage at across the capacitor V_{o} is increasing. This is stage 1. When V_{o} reaches V_{c}, VO_{2} become metallic and it discharges the capacitor. An instantaneous large current spike appears at the output. V_{o} drops sharply. It is stage 2, fire, which is followed by stage 3, refractoriness (refractory period). In stage 3 the VO_{2} remains in its LRS for some time. Any input current will be drained to ground without integrating to the capacitor. After the refractory period, the neuron resets and is ready for another spike cycle (stage 4). _{o} for the 4 stages.

One example of the simulation result is shown in Figure _{2} stays in low resistance state for a finite period (Figure _{2}. The charge is not integrated. As a result, inputs with two pulse durations generates the same firing patterns (Figure

Simulated waveform for current-clamp VO_{2} neuron. _{2} resistance and

Post-firing refractoriness is another important feature in both biological and VO_{2} neurons. A second AP is difficult to be produced immediately following the occurrence of an AP when the cell is regarded to be refractory (Weiss, _{2} element remains at a temperature above the critical temperature for a time, ~τ_{th}+τ_{el}, where τ_{th} and τ_{el} are the thermal and electrical time constants, respectively. τ_{th} is related to the thermal mass and heat dissipation. For the electrical time constant τ_{el}, it is given by _{met}_{o}+_{s}_{o}. R_{met} is the metallic-state resistance and R_{s} is the series resistance. Usually, R_{met} is much >R_{s} in normal operation (Lin et al., _{s}C_{o.} In addition, the continuous high input current can keep the VO_{2} in LRS for longer time. During this period, the VO_{2} element remains in metallic state and new input charge is continuously discharged without being integrated in the capacitor C_{o}. Our coupled electrothermal model captures this process and can be used to design the “refractory period” in the VO_{2} neuron circuit. Subsequently, the VO_{2} element resets and starts another integrate-and-fire cycle. The steps mimic the electrically excitable membrane in neuron cells.

In biological neurons, the inter-spiking interval (ISI) is defined as the time interval between two adjacent spikes (Fadool et al.,

Healthy vs. degenerative fast-firing and slow-firing neurons, and VO_{2} analogy. _{2} neuron model, the intact VO_{2} neuron exhibits oscillatory behavior at a constant input current. _{2} device is degraded and its insulating state resistance increases (+ΔR). Such degenerative VO_{2} neuron results in longer ISI. _{2} device is degraded by decreasing its insulating state resistance increases (–ΔR). The leakier VO_{2} neuron results in shorter ISI. _{2} neuron model.

Similar characteristics can be observed in the VO_{2} neurons. The simulation results for three VO_{2} neurons are shown in Figure _{2} neuron for baseline reference, and Figures _{2} neuron is defined in the same way as for the case of a biological neuron. The AP frequency reduces if the VO_{2} undergoes a +ΔR degradation, and vice versa. Analytically, the value for ISI can be derived from the VO_{2} neuron parameters as t_{ISI} = C_{o}V_{c}/I_{in} where V_{c} is the critical voltage to trigger an insulator-to-metal transition in the VO_{2} device under DC I-V measurement and I_{in} is the input current. V_{c} is related to the HRS resistance of the VO_{2} device. Definition of V_{c} is illustrated in Figure _{c} (ΔV_{c}) due to electrical-stress-induced resistance degradation in the VO_{2} (Figure _{c} be positive or negative depending on the degradation mechanism. Positive ΔV_{c} indicates an increase in the VO_{2} HRS resistance, and vice versa.

The pathologically-altered spike timing is linked to other serious degenerative diseases. For example, certain neuromuscular disorders and motor neuron disease (MND) are resulted from the ionic leakage of degenerating membrane and increase of rest conductance (Younger,

Healthy vs. degenerative leaky neurons, and VO_{2} artificial neurons _{Na}, G_{K}, and G_{Leak} are respectively the Na^{+} conductance, K^{+} conductance and leakage conductance through the membrane, while G_{M} is the the sum of the conductance _{Leak}, while G_{K} and G_{Na} remain unchanged. No AP spike is observed _{2} neuron, the VO_{2} device goes through an insulator-to-metal transition _{2} neuron with excess leakage (–ΔR) _{ins}/R_{met}) illustrates the reduction of resistance of the insulating state by ~100 _{ins}/R_{met}) and input stimulus (I_{in}). Reduction of insulating-state resistance narrows the neuron operating region for a given input stimulus _{in} = 200, 300 and 400 μA in the contour plot _{2} neuron model.

The increase of conductance in a degenerative, leaky neuron can be modeled in a straightforward manner in the VO_{2} circuit. The resistance vs. temperature of the VO_{2} device is normalized to the low resistance state, R_{met}. Figures _{in}: (a), VO_{2} neuron with R_{ins}/R_{met} = 10^{5} fires regularly and demonstrates an insulator-to-metal transition (b) while the neuron with R_{ins}/R_{met} = 10^{3} fails to fire. The change of VO_{2} properties is shown in Figure

To provide a systematical perspective on the design of VO_{2} neurons to mimic the corresponding neural disorder, Figure _{ins} and I_{in} and is shown as a contour plot. The dark red color is the case where the combination of low input current and small R_{ins} results in failure in spike generation (t_{ISI} → ∞). At a given I_{in}, t_{ISI} decreases as R_{ins} drops. When R_{ins} drops to a critical value, the VO_{2} neuron fails to fire (Figure

The AP pulse width is another distinctive characteristic related to timing in different kinds of mammalian central neurons (Bean, _{w} change by 10X. The pulse width can be simulated in the VO_{2} neuron by altering the LRS resistance, R_{met}, in the VO_{2} devices according to t_{w} = C_{o}R_{met}. The VO_{2} neuron with short pulses of 0.2 μs and 2 μs are shown in Figures _{2} neurons (both simulations and experiments). The AP pulse width can range from a few 100 μs−10 ms which can be matched by the VO_{2} neuron with appropriate capacitance.

Diversity in AP pulse width across biological neurons, and VO_{2} artificial neuron analogy _{w} is taken at full width half maximum _{2} neuron with short pulse of 0.2 μs. Experiment and simulation show good agreement. The pulse width control is achieved by changing resistance in the circuit _{2} neuron with long pulse of 2 μs _{2} neuron and the range spans biological neuron studies reported in the neuroscience literature.

The HH model has been proven to be useful as a fundamental description of neuron behavior (Hodgkin and Huxley, _{2} neuron study to emulate such spike rate histogram. Figure _{2} neuron for six input currents. The variation of ISI at low input current (I_{IN} = 110 μA) is significant. Such variation reduces as input current increases. The statistical behaviors are related to the cycle-to-cycle operation in VO_{2} device. These characteristics mimic the behavior of biological neurons under various current stimulus.

Spike rate (ISI) histogram of VO_{2} neuron for six input currents. In each case, about 500–1000 spike events are measured. The ISI follows the Gaussian distribution. At increased input current, the variance and the mean value of ISI both decreased. These characteristics can mimic the behavior of biological neurons under various current stimulus.

We further extend this concept to two-stage cascading neuron circuits in Figure _{x}.

VO_{2} monosynaptic neuron circuit.

We emulate a monosynaptic circuit that corresponds to the well-known _{2} in Neuron 1 are different. Both neurons are initially at rest. Their temperatures are at equilibrium with the environment and is below the critical transition temperature T_{c}. At t = 0, a current is injected to Neuron 1 (see Figure _{ins}/R_{met}, The output current as a function of time in Figure _{diff} = 0.6 μs) for the signal to propagate between the two VO_{2} neurons. The neuron 1 in case B has a lower HRS resistance. Premature spike in Neuron 1 results in a weak spike and it cannot trigger a spike in Neuron 2 (Figure

Demonstration of degenerative Neuron 1 leading to the failed signal reception for Neuron 2 in a monosynaptic circuit

VO_{2} based circuits can emulate neuronal function and disorders. By carefully varying the electrical properties of the ground state resistance of the artificial neuron, we can precisely identify thresholds for firing and signal propagation that present an analogy to neuronal activity in the brain. While the present study has focused on VO_{2} as a model system, a vast range of threshold switching Mott semiconductors can further be explored in the future.

JL and SR developed the method, carried out the data analysis, wrote the manuscript. JL and SG fabricated and characterized the VO_{2} devices. JL developed the VO_{2} model and HH neuron model.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This work was performed, in part, at the Center for Nanoscale Materials, a U.S. Department of Energy Office of Science User Facility. Use of the Center for Nanoscale Materials, an Office of Science user facility, was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. Aspects of the device work was supported by the National Science Foundation under grant 1640081, and the Nanoelectronics Research Corporation (NERC), a wholly owned subsidiary of the Semiconductor Research Corporation (SRC), through Extremely Energy Efficient Collective Electronics (EXCEL), an SRC-NRI Nanoelectronics Research Initiative under Research Task ID 2698.001. S.R. acknowledges the support by ARO W911NF-16-1-0289 and ONR N00014-16-1-2398. The authors acknowledge K. V. L. V. Achari for providing vanadium dioxide film samples.

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