Traditional extensions of the Nyquist-Johnson formula for thermal fluctuations in nonlinear dissipative elements have often led to thermodynamically inconsistent models and sparked long-standing debates about the proper interpretation of stochastic differential equations. In this work, we show that it is possible to derive, for each of the main stochastic interpretations, a Gaussian white-noise model for nonlinear dissipative elements at thermal equilibrium that fully complies with the fundamental principles of thermodynamics. The resulting models reproduce the Gibbs (Maxwell-Boltzmann) distribution and ensure zero-mean voltages and currents, thereby resolving the Brillouin paradox and maintaining consistency with the second law of thermodynamics. Furthermore, we demonstrate that these models satisfy additional thermodynamic requirements, including positive entropy production during transients and zero net heat exchange between dissipative elements at equilibrium.

We investigate multistability, noise-induced transitions, and stochastic resonance in a second-order nonlinear oscillator incorporating a nonvolatile memristive device. The memristor provides a programmable nonlinear conductance, enabling bistable dynamics with two asymptotically stable equilibrium points separated by a saddle. Under periodic excitation, the system exhibits coexisting limit cycles, period-doubling cascades, boundary crises, and transitions to chaos. Lyapunov exponent analysis reveals repeated crossings of the edge-of-chaos regime, where the largest nonzero exponent approaches zero, marking a balance between stability and sensitivity to perturbations. The effects of additive Gaussian white noise are analyzed by reformulating the dynamics in terms of an effective potential landscape, where noise induces random transitions between coexisting attractors. Transition rates are accurately described in the weak-noise regime by the Eyring–Kramers formula. When periodic forcing and noise act jointly, the system exhibits stochastic resonance, with optimal synchronization occurring when the forcing period matches the mean noise-induced transition time. These results demonstrate that memristor-based nonlinear circuits naturally operate near critical dynamical regimes and provide a compact hardware platform for studying noise-assisted computation and edge-of-chaos dynamics in neuromorphic systems.

The electrical engineering community, interested to develop bio-inspired circuits, approaching the efficiency of the neural networks, is searching passionately for accurate yet simple electronic neurons, or neuristors for short. In recent years, the advent of volatile memristor devices, typically referred to as threshold switches, which admit a negative differential resistance under suitable polarization, similarly as the sodium and potassium ion channels across neuronal axon membranes, has opened up new exciting opportunities in neuromorphic circuit design, enabling innovative analogue electronic cells, capable to reproduce closely the intricate dynamical behaviors of biological neurons without requiring a disproportionate use of resources. The study, presented in this manuscript, achieves an important milestone in this area of research, demonstrating, through a circuit design approach based upon concepts and techniques from Dynamical System Theory, how to leverage the rich dynamics of a threshold switch, capable to boost a periodic sine-wave current signal of infinitesimal amplitude, while acting as a source of local energy, when poised on a suitable bias point, lying along the negative differential resistance branch of the respective S-shaped DC current-voltage characteristic, to induce, one after the other, the three fundamental bifurcations, governing the evolution of an electrical voltage spike from birth to extinction via the All-to-None effect across a biological axon membrane under a reverse sweep in the net synaptic current, according to the fourth-order Hodgkin-Huxley neuron model, in a second-order three-element circuit of unprecedented simplicity, as the current, generated by a DC source, appearing in parallel to a linear capacitor as well as to the volatile locally-active memristor, is subject to a monotonic increase.

This manuscript sheds light into the fundamental importance of the Principles of Local Activity and Edge of Chaos for the future design of innovative circuits, which, employing biomimetic memristive devices, are ideally suited for the development of energy-efficient artificially-intelligent technical systems. The focus of the work is the design of a Second-Order Reactance-Less Oscillator, across which oscillations may develop if and only if at least one of its two different volatile thermally-activated memristor physical realizations is biased along a negative differential resistance branch of the respective DC locus, which turns it into a source of local energy. Very importantly, the proposed cell is first found to lock in the oscillatory mode out of a local Hopf Supercritical Bifurcation when its design parameters are chosen from the Edge of Chaos region, providing clear evidence for the high degree of excitability it acquires as a result.

Stability of ultra-low-voltage SRAM bitcells in retention mode is threatened by two types of uncertainty: process variability and intrinsic noise. While variability dominates the failure probability, noise-induced bit flips in weakened bitcells lead to dynamic instability. We study both effects jointly in a unified SPICE simulation framework. Starting from a synthetic representation of process variations introduced in a previous work, we identify the cases of poor noise immunity that require thorough noise analyses. Relying on a rigorous and systematic methodology, we simulate them in the time domain so as to emulate a true data retention operation. Short times to failure, unacceptable for a practical ultra-low-power memory system application, are recorded. The transient bit-flip mechanism is analyzed and a dynamic failure criterion involving the unstable steady state is established. We conclude that, beyond static variability, the dynamic noise inflates defectiveness among SRAM bitcells. Then, a stochastic nonlinear model, fully characterizable from conventional deterministic SPICE simulations, is presented. We then leverage it to efficiently and accurately predict the mean time to failure with an analytical Eyring-Kramers formula, recently extended to account for the varying-noise behavior of nonlinear systems.

Weakly Connected Oscillatory Networks (WCONs) are bio-inspired models which exhibit associative memory properties and can be exploited for information processing. It has been shown that the nonlinear dynamics of WCONs can be reduced to equations for the phase variable if oscillators admit stable limit cycles with nearly identical periods. Moreover, if connections are symmetric, the phase deviation equation admits a gradient formulation establishing a one-to-one correspondence between phase equilibria, limit cycle of the WCON and minima of the system’s potential function. The overall objective of this work is to provide a simulated WCON based on memristive connections and Van der Pol oscillators that exploits the device mem-conductance programmability to implement a novel local supervised learning algorithm for gradient models: Equilibrium Propagation (EP). Simulations of the phase dynamics of the WCON system trained with EP show that the retrieval accuracy of the proposed novel design outperforms the current state-of-the-art performance obtained with the Hebbian learning.

We present a novel phase-amplitude model for noisy oscillators described by Itô stochastic differential equations. The model is completely rigorous and it holds for any value of the noise intensity. The phase and amplitude equations depend on the choice of an appropriate set of basis vectors. We show that using Floquet's basis, a phase-amplitude description is obtained analogous to others, previously proposed. We also show how, using moment closure techniques, information on the expected angular frequency, oscillation amplitude and amplitude variance can be obtained from the phase-amplitude model without solving the equations explicitly.

Locally Connected Oscillatory Networks (LCONs) are a special class of Cellular Neural Networks (CNNs) where each cell (neuron) exhibits time periodic behavior. In this paper we investigate the dynamics of LCONs whose neurons exhibit the coexistence of a stable equilibrium point and a stable limit cycle. We consider a constant external stimulus applied to each neuron, which influences the neuron's own natural frequency. We show that new interesting dynamics, namely synchronous oscillations of various amplitudes, may arise due to the interaction between different kind of attractors. We also show that neurons subjected to different external stimuli are able to synchronize if their local couplings are strong enough.

Networks of coupled nonlinear oscillators are popular mathematical models in many areas of applied sciences. The most successful approach for their analysis is based on phase modeling, founded on the idea to represent each oscillator by a phase variable. Phase models have been analyzed with wealth of details and in a plethora of different variants, but little research has been made in view of the reduction of a physical system to the corresponding phase model. In this paper we propose a technique to obtain the phase model corresponding to a given network. Examples based on Stuart-Landau and van der Pol oscillators are presented.

An equivalent circuit is presented for a quantum system composed of two spin 1/2 particles. Such a circuit shows that the entire dynamics of the system, including single-particle and two-particle annihilation and creation, as well as the single particle transitions between ground and excited states, can be described as the superposition of the variables of two uncoupled resonant circuits.

The existence and the stability of waves and phase locked oscillations in lattices composed by oscillators with delayed interactions is investigated. In the neighborhood of a multiple Hopf bifurcation, the equations governing the dynamics of the whole network reduce to an amplitude-phase model, reducing the research of phase locked oscillations to the prospecting of equilibrium points. The stability of the solutions is determined analytically and the possible coexistence of waves and phase locked oscillations is shown.

The global dynamics of an artificial neural network composed by oscillators with delays is investigated. Using center manifold reduction and normal form theory, the equation governing the whole network dynamics is reduced to an amplitude-phase model (i.e. a set of coupled differential equations describing the evolution of both the amplitudes and the phases of the oscillators). The analysis of a network with a simple architecture reveals that different kind of phase locked oscillations is admissible, and the possible coexistence of in-phase and anti-phase locked solutions.

The steady state performance of a two-state quantum system interacting with a thermal bath at a fixed temperature and with a classical electromagnetic wave is analyzed by the harmonic balance technique. Thus the time-variant equations can be solved to any chosen approximation and the corrections to be brought to the classical RWA approximation to take into account the effect of the counter-rotating field can be calculated.

Many studies in neuroscience have shown that nonlinear dynamic networks represent a bio-inspired model for information and image processing. Recent studies on the thalamo-cortical system have shown that weakly connected oscillatory networks, forced by an external input, have the capability of modelling the architecture of a neurocomputer. In particular they have associative properties and can be exploited for dynamic pattern recognition. In this manuscript the global dynamic behavior of such networks is investigated. In case of weak coupling, their main dynamic features are revealed by the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling). Firstly a very accurate analytic expression of the phase deviation equation is derived, via the joint application of the describing function technique and of Malkin's theorem. Furthermore, a complete analysis of the phase-deviation equation shows that the proposed technique can be effectively exploited for designing dynamic associative memories