^{1}

^{*}

^{2}

^{2}

^{1}

^{*}

^{1}

^{2}

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

Reviewed by: Adnan Mehonic, University College London, United Kingdom; Rune W. Berg, University of Copenhagen, Denmark

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 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.

Artificial neural networks can harness stochasticity in multiple ways to enable a vast class of computationally powerful models. Boltzmann machines and other stochastic neural networks have been shown to outperform their deterministic counterparts by allowing dynamical systems to escape local energy minima. Electronic implementation of such stochastic networks is currently limited to addition of algorithmic noise to digital machines which is inherently inefficient; albeit recent efforts to harness physical noise in devices for stochasticity have shown promise. To succeed in fabricating electronic neuromorphic networks we need experimental evidence of devices with measurable and controllable stochasticity which is complemented with the development of reliable statistical models of such observed stochasticity. Current research literature has sparse evidence of the former and a complete lack of the latter. This motivates the current article where we demonstrate a stochastic neuron using an insulator-metal-transition (IMT) device, based on electrically induced phase-transition, in series with a tunable resistance. We show that an IMT neuron has dynamics similar to a piecewise linear FitzHugh-Nagumo (FHN) neuron and incorporates all characteristics of a spiking neuron in the device phenomena. We experimentally demonstrate spontaneous stochastic spiking along with electrically controllable firing probabilities using Vanadium Dioxide (VO_{2}) based IMT neurons which show a sigmoid-like transfer function. The stochastic spiking is explained by two noise sources - thermal noise and threshold fluctuations, which act as precursors of bifurcation. As such, the IMT neuron is modeled as an Ornstein-Uhlenbeck (OU) process with a fluctuating boundary resulting in transfer curves that closely match experiments. The moments of interspike intervals are calculated analytically by extending the first-passage-time (FPT) models for Ornstein-Uhlenbeck (OU) process to include a fluctuating boundary. We find that the coefficient of variation of interspike intervals depend on the relative proportion of thermal and threshold noise, where threshold noise is the dominant source in the current experimental demonstrations. As one of the first comprehensive studies of a stochastic neuron hardware and its statistical properties, this article would enable efficient implementation of a large class of neuro-mimetic networks and algorithms.

A growing need for efficient machine-learning in autonomous systems coupled with an interest in solving computationally hard optimization problems has led to active research in stochastic models of computing. Optimization techniques (Haykin, _{2}), wherein the inherent physical noise in the dynamics is used to implement stochasticity. The firing probability, and not just the deterministic frequency of oscillations or spikes, is controllable using an electrical signal. We also show that such an IMT neuron has similar dynamics as a piecewise linear FitzHugh-Nagumo (FHN) neuron with thermal noise along with threshold fluctuations as precursors of bifurcation resulting in a sigmoid-like transfer function for the neural firing rates. By analyzing the variance of interspike interval, we determine that for the range of thermal noise present in our experimental demonstrations, threshold fluctuations are responsible for most of the stochasticity compared to thermal noise.

A stochastic IMT neuron is fabricated using relaxation oscillators (Shukla et al., _{2}), in series with a tunable resistance, e.g., transistor (Shukla et al.,

_{2} based IMT spiking neuron circuit consisting of a VO_{2} device in series with a tunable resistance.

The equivalent circuit model for an IMT oscillator is shown in Figure _{v(m/i)} (_{vm} in metallic and _{vi} in insulating state), a series inductance _{h} and _{l}, respectively, with _{h} > _{l}, and the current-voltage relationship of the hysteretic conductance be

where _{i} and

The system dynamics is then given by:

with _{i} and _{o} as shown in Figure

In VO_{2}, IMT, and MIT transitions are orders of magnitude faster than RC time constants for oscillations, as observed in frequency (Kar et al., _{i} × (_{dd} − _{o}). V-I curves for IMT device in the two states metallic (M) and insulating (I) and the load line for series conductance _{o} = _{i}/_{s} for the steady state are shown along with the fixed points of the system _{1} and _{2} in insulating and metallic states respectively. The load line and V-I curves are essentially the nullclines of _{o} and _{i}, respectively. The capacitance- inductance pair delays the transitions and slowly pulls the system toward the fixed points S_{1} and S_{2} even when the IMT device transitions instantaneously. For small _{1} (or _{2}), the IMT device transitions into metallic (or insulating) state changing the fixed point to S_{2} (or _{1}). Two trajectories are shown starting from points A and B each for the system (Equation 1)—one for small

_{i} × (_{dd} − _{o}) for a small _{i}-nullclines of system (1) are shown as solid black lines in the metallic (AB') and insulating (BA') states of the IMT device, and S_{1}S_{2} is the _{o}-nullcline. Depending on the state, the phase space is divided into three vertical regions - I, M and N. In the region N the _{i}-nullclines are dependent on

The model of (Equation 1) is very similar to a piecewise linear caricature of FitzHugh-Nagumo (FHN) neuron model (Gerstner and Kistler,

where ^{3}/3, and _{ext} is the parameter for bifurcation, as opposed to _{s} in Equation (1). In the FHN model, one variable (_{dd} − _{i}, _{i}-nullcline in those regions. In the region N, the difference between _{dd} − _{i}, _{2} neuron is achieved by tuning the load line using a tunable resistance (_{s}), or a series transistor (Figure _{gs}), where one gives rise to spikes while the other results in a resting state.

_{i} × _{dd} − _{o} for two different _{gs} values for spiking and resting states. Bifurcation occurs when a stable points crosses the boundary of region _{dd} − _{o} ∈ [_{l}, _{h}].

Moreover, a single dimensional piecewise approximation of the system can be performed using a dimensionality reduction by replacing the movement along the eigenvector parallel to the x-axis with an instantaneous transition from A to A′, or B to B′. This leaves a 1-dimensional subsystem in M and I each along the V-I curves AB′ and BA′. Experiments using VO_{2} show that the metallic state conductance _{vm} is very high which causes the charging cycle of _{o} to be almost instantaneous (Figure _{o}. The inductance being negligible can be effectively removed and only the capacitance is needed for modeling the 1D subsystem of insulating state (_{i} = _{dd} − _{o}.

Experimental waveforms of VO_{2} based spiking neuron for various _{gs} values (1.78, 1.79, and 1.81 V). A VO_{2} neuron shows almost instantaneous charging (spike) in metallic state.

The two important noise sources which induce stochasticity in an IMT neuron are (a) V_{IMT} (_{h}) fluctuations (Zhang et al., _{t}d_{t} where _{t} is the standard weiner process and _{h} is assumed constant during a spike, but varies from one spike to another. The distribution of _{h} from spike to spike is assumed to be Gaussian or subGaussian whose parameters are estimated from experimental observations of oscillations. If the series transistor always remains in saturation and show linear voltage-current relationship, as is the case in our VO_{2} based experiments, the discharge phase can be described by an Ornstein-Uhlenbeck (OU) process

where μ, θ, and σ are functions of circuit parameters of the series transistor, the IMT device and σ_{t}. The interspike interval is thus the first-passage-time (FPT) of this OU process, but with a fluctuating boundary.

Analytical expressions for the FPT of OU process (with μ = 0) for a constant boundary were derived using the Laplace transform method in Ricciardi and Sato (_{0} and hits a boundary _{f}_{0}), and its _{m}(_{0}). Also, let

where

where ϕ_{k}(

with ρ(

We extend this framework for calculating the FPT statistics with a fluctuating boundary _{h}_{2} based IMT neuron, the 1D subsystem in the insulating phase can be converted in the form of Equation(3) with μ = 0 by translating the origin to the fixed point. If this transformation is _{i} = _{dd} − _{o}), _{h}_{o} = _{l}. The start and end points are B′ and A, respectively in Figure _{h}_{h}_{h}_{imt}_{f}

The moments of _{imt}

where _{h}

IMT devices are fabricated on a 10nm VO_{2} thin film grown by reactive oxide molecular beam epitaxy on (001) TiO_{2} substrate using a Veeco Gen10 system (Tashman et al., _{VO2}). Pd (20 nm)/Au (60 nm) contacts are then deposited by electron beam evaporation and liftoff. The devices are then isolated and the widths (W_{VO2}) are defined using a CF_{4} based dry etch.

The IMT neuron is constructed using an externally connected n-channel MOSFET (ALD110802) and the fabricated VO_{2} device. A prototypical I-V curve is shown in Figure _{o}, limiting the current would not have noticeable effect on spiking statistics of the neuron.

_{2} device exhibits abrupt threshold switching at V_{IMT} and V_{MIT}. The current in the metallic state has been arbitrarily limited to a 200μA compliance current. _{IMT} distribution as a function of the peak current during oscillations (value is set by the MOSFET saturation current). V_{IMT} is extracted from 300+ cycles.

Threshold voltage fluctuations (cycle to cycle) were observed in all devices which were tested (>10). Threshold voltage distribution was estimated using the varying cycle-to-cycle threshold voltages collected from a single device. Thermal noise is not measured directly, but is estimated approximately by matching the simulation waveforms from the circuit model (Figure _{gs} (Figure

_{gs} using the analytical model for different _{h}

First moment of _{imt}

The expansion for ϕ_{k}(_{vh}[ϕ_{k}(α_{h}_{h}

Figure _{gs} for various σ_{t} values and for three distributions of threshold fluctuations. The calculations approximate the experimental observations well for all three _{h} distributions, the closest being EP[3] with σ_{t} = 4.

For higher moments, higher order terms are encountered. For example, in case of the second moment, using Equations(5) and (7), we obtain

with a higher order term _{1}(α_{h}_{2}(α_{h}_{k} term is an infinite sum, we construct a cauchy product expansion for the higher order term using the infinite sum expansions of the constituent ϕ_{k}s and then distribute the expectation over addition. For example, if the ϕ_{k} expansions of ϕ_{1}(_{2}(_{i}) and (∑_{i}), respectively, then the cauchy product expansion of ϕ_{1}(_{2}(_{i}, where _{i} is a function of _{1…i} and _{1…i}, and the expectation 𝔼[ϕ_{1}(_{2}(_{i}]. Since _{i} is a polynomial in _{i}] can be calculated using the moments of

If μ_{imt} and σ_{imt} are the mean and standard deviation of interspike intervals _{imt}_{imt}/μ_{imt}) varies with the relative proportion of the thermal and the threshold induced noise. Figure _{imt}/μ_{imt} (calculated using parameters matched with our VO_{2} experiments) plotted against σ_{t} for various kinds of _{h}_{imt}/μ_{imt} as observed in our VO_{2} experiments is about an order of magnitude more than what would be calculated with only thermal noise using such a neuron, and hence, threshold noise contributes significant stochasticity to the spiking behavior. As the IMT neuron is setup such that the stable point is close to the IMT transition point (Figure _{t} results in high and diverging σ_{imt}/μ_{imt} for any distribution of threshold noise, and σ_{imt}/μ_{imt} reduces with increasing σ_{t} for the range shown. For a Normally distributed _{h} the variance diverges for σ_{t} ≲ 8, but for Exponential Power (EP) distributions with lighter tails, the variance converges for smaller values of σ_{t}. Statistical measurements on experimental data, as indicated in Figure _{imt}/μ_{imt} (dotted line) and σ_{t} (shaded region). We note that EP distributions provide a better approximation of the stochastic nature of experimentally demonstrated VO_{2} neurons as the range of σ_{t} is estimated to be <5.

σ_{imt}/μ_{imt} for the interspike interval plotted against σ_{t} for _{gs} = 1.8V with Constant, Gaussian, and Exponential Power (EP[κ], where κ is the shape factor) distributions of the threshold noise. The experimentally observed σ_{imt}/μ_{imt} for a VO_{2} neuron is shown with a dotted line. The shaded region shows the experimentally estimated range of σ_{t} (σ_{t} < 5).

In this paper, we demonstrate and analyse an IMT based stochastic neuron hardware which relies on both threshold fluctuations and thermal noise as precursors to bifurcation. The IMT neuron emulates the functionality of theoretical neuron models completely by incorporating all neuron characteristics into device phenomena. Unlike other similar efforts, it does not need peripheral circuits alongside the core device circuit (an IMT device and a transistor) to emulate any sub-component of the spiking neuron model like thresholding, reset etc. Moreover, the neuron construction not only utilizes inherent physical noise sources for stochasticity, but also enables control of firing probability using an analog electrical signal—the gate voltage of series transistor. This is different from previous works which control only the deterministic aspect of firing rate like the charging rate. A comparison of spiking neuron hardware characteristics in different works is shown in Table

Comparison of this work (experimental details from Jerry et al.,

_{2}) |
|||||
---|---|---|---|---|---|

Neuron type | Integrate & Fire | Hodgkin Huxley | Integrate & Fire | Integrate & Fire | Piecewise Linear FHN |

Material/Platform | Chalcogenide | Mott insulator NbO_{2} |
MTJ | 0.35 μm CMOS | Vanadium Dioxide (VO_{2}) |

Material phenomenon | Phase Change | IMT | Spin transfer torque (STT) | – | IMT |

Spontaneous spiking using only device | No | Yes | No | – | Yes |

Peripherals needed for spiking | Yes, for spike generation and reset | No | Yes, for spike generation and reset | – | No |

Integration mechanism (I&F) | Heat accumulation | – | Magnetization accumulation | Capacitor charging | Capacitor charging |

Threshold mechanism (I&F) | External reset by measuring conductance | Spontaneous IMT | External reset by detecting magnet flip | Reset using comparator | Spontaneous IMT |

Stochastic | Yes | – | Yes | No | Yes |

Kind of stochasticity (I&F) | Reset potential | – | Differential | – | Threshold and differential |

Source of stochasticity / noise | Melt-quench process | – | Thermal noise | – | IMT threshold fluctuations & Thermal noise |

Control of stochastic firing rate | Only integration rate | – | Only integration rate | Only integration rate | Yes |

Status of experiments | Constant stochasticity, variable integration rate | Deterministic spiking | None | Deterministic spiking | Sigmoidal variation of stochastic firing rates |

Peak current | 750–800 μA | – | 200 μA | ||

Power or Energy/spike | 120 μW | – | 900 pJ / spike | 196 pJ / spike | |

Voltage | 5.5 V | 1.75 V | – | 3.3 V | 0.7 V |

Maximum firing rates | 35–40 KHz | 30 KHz | – | 200 Hz | 30 KHz |

We also show that the neuron dynamics follow a linear “carricature” of the FitzHugh-Nagumo model with intrinsic stochasticity. The analytical models developed in this paper can also faithfully reproduce the experimentally observed transfer curve which is a stochastic property. Such analytical verification of stochastic neuron experiments is one of the first in this work. It is an important result as it indicates reproducibility of stochastic characteristics and helps in creating the pathway toward perfecting these devices. With a growing concensus that stochasticity will play a key role in solving hard computing tasks, we need efficient ways for controlled amplification and conversion of physical noise into a readable and computable form. In this regard, the IMT based neuron represents a promising solution for a stochastic computational element. Such stochastic neurons have the potential to realize bio-mimetic computational kernels that can be employed to solve a large class of optimization and machine-learning problems.

AP worked on the development of theory, simulation frameworks, and mathematical models; MJ worked on the experiments; AR advised AP and participated in the problem formulation; SD advised MJ and also participated in the design of experiments and problem formulations.

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 project was supported by the National Science Foundation under grants 1640081, Expeditions in Computing Award-1317560 and CCF- 1317373, 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 IDs 2698.001 and 2698.002.

_{2}-based selection device for passive resistive random access memory application