Dynamic Memristor Model

• Dynamic memristor models are state-space descriptions that couple terminal responses with evolving internal variables to encode the history of excitation.
• They incorporate diverse mechanisms such as ionic drift, tunneling-gap dynamics, vacancy transport, and nonlinear window functions to represent device behavior.
• These models underpin applications in analog memory, threshold switching, and neuromorphic computing, offering actionable insights for circuit design.

A dynamic memristor model is a state-space description of a memory resistor in which terminal current and voltage are coupled to one or more internal variables whose evolution encodes the history of excitation. In the general memristive-system form, the dynamics are written as

$\frac{dx}{dt}=f(x,u,t), \qquad y=g(x,u,t)u,$

and, for circuit modeling, often as

$$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$

Across the literature, the internal state may represent a doped-region width, a tunneling gap, a depletion width, floating-gate charge, oxygen-vacancy populations, or a higher-dimensional flux state. That modeling choice determines whether the device exhibits linear drift, thresholded switching, volatility, asymmetry, bistability, or coupled ionic-electronic dynamics (Parajuli et al., 2019, Bou et al., 2024).

1. Formal structure and state variables

The common feature of dynamic memristor models is the separation between a terminal relation and an internal-state equation. In voltage-controlled form, one widely used template is

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$

while a standard Chua-type formulation writes

$$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$

Extended current-controlled formulations also appear, for example

$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$

in memristor circuits used to emulate nonlinear dynamical systems (Bou et al., 2024, Habart et al., 5 Mar 2026, Itoh, 2019).

The state variable is not universal. In different model classes it may be a normalized doped-region width $w/D$, a filament-electrode gap $x_f(t)$, a depletion width $x_d(t)$, a floating-gate charge $Q_{FG}$, oxygen-vacancy densities $N_{V_O^{2+}}, N_{V_O^{2-}}, N_0$, or carrier/vacancy fields $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$0. This diversity is not cosmetic: it determines which physical mechanism is taken as primary, and therefore which dynamic phenomena can be represented.

Model family	State variable(s)	Defining dynamic mechanism
Drift/state-variable ODE models	$$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$1, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$2, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$3	Ionic drift, relaxation, threshold kinetics
Barrier and filamentary models	$$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$4, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$5, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$6, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$7	Tunneling-gap motion, Schottky-barrier evolution, floating-gate charge dynamics
Defect-kinetic and transport models	$$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$8, $$\frac{dw}{dt}=f(w,v_M,t), \qquad i_M=M(w,v_M,t)v_M.$$9	Vacancy chemistry, trap occupancy, drift-diffusion-Poisson coupling
Computing-oriented volatile models	$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$0	Volatility kernels, saturation, eligibility traces, synaptic-like plasticity

A recurring misconception is that a dynamic memristor model must be a one-state first-order ODE. The published literature includes first-order drift laws, second-order oscillator models, coupled rate equations, and full PDE systems. In that sense, “dynamic memristor model” denotes a modeling class rather than a single constitutive law.

2. Drift-based first-order models and analytic classifications

The simplest canonical family is the drift-based state-variable model. In the linear boundary-drift formulation, the memristance is

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$1

with state evolution

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$2

This linearity makes the internal state directly proportional to current, which is why the Strukov model is described as useful for analog applications requiring predictable incremental updates. By contrast, the Yang model uses a voltage-driven law,

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$3

so the exponent $$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$4 controls the degree of nonlinearity: $$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$5 reduces to a Strukov-like linear case, whereas higher odd $$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$6 values preserve switching symmetry while increasing nonlinearity (Parajuli et al., 2019).

Boundary effects motivated nonlinear window functions. In the nonlinear ionic-drift framework for the $$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$7 memristor,

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$8

with Joglekar’s window

$$I_{\text{tot}} = G(u,w)u, \qquad T_{\text{rel}}\frac{dw}{dt}=H(u,w),$$9

This suppresses drift near $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$0 and $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$1. The resulting equation of state can be written implicitly in terms of the input flux $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$2 and solved analytically with hypergeometric functions. That analysis introduces the “Characteristic Curve of State” $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$3, which determines operation point, waveform, and saturation level, and shows that identical inputs can generate different $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$4-$$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$5 orbital dynamics under different initial conditions. The same work argues that nonlinear ionic drift belongs to an Abel differential-equation class rather than a Bernoulli class (Cai et al., 2011).

For the HP memristor under voltage drive, however, the linear-drift dynamics do admit a Bernoulli reduction. The explicit current response can be written as

$$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$6

and the resulting hysteresis is governed by a single dimensionless lumped parameter $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$7, or by $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$8 after waveform-specific normalization. This provides a formal quantitative hysteresis measure rather than a purely graphical one (Georgiou et al., 2010).

The main technical limitation of windowed first-order drift models is boundary handling. The Joglekar window can lock the state at a boundary when current reverses, whereas the Biolek window makes the active boundary current-direction dependent and removes that convergence problem while also supporting asymmetric $$i = G(\mathbf{x})\,v,\qquad \dot{\mathbf{x}} = f(\mathbf{x},v).$$9-$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$0 characteristics (McDonald et al., 2010).

3. Barrier, filamentary, and second-order dynamic models

A different line of work abandons the moving-boundary abstraction in favor of explicit barrier dynamics. In the filamentary mem-resistor model, the system state is defined by two coupled barriers: a dynamic tunneling barrier associated with the filament-electrode gap $$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$1, and a dynamic Schottky barrier associated with the depletion width $$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$2. The filament tip is treated as a moving mass, and the linearized gap dynamics take a damped-driven oscillator form. Total current density combines tunneling, Schottky, and capacitive components,

$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$3

so the model explicitly separates filament-region transport from non-filament-region transport (Mouttet, 2011).

The related harmonic-oscillation model for ionic mem-resistors makes ion inertia, collision damping, and ion–ion repulsion explicit. Starting from Newton’s law, it arrives at

$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$4

which is interpreted as a driven damped harmonic oscillator for the mean ionic position or depletion width. In that formulation, hysteresis is tied to resonance and phase lag, and can vanish far from resonance. This is a major departure from the first-order view in which pinched hysteresis is treated as the defining signature irrespective of inertial or capacitive effects (&&&10&&&).

The Pickett model is the most nonlinear member of the three-model comparison between Strukov, Yang, and Pickett. It represents a Pt/TiO$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$5/Pt device as an ohmic resistor in series with a tunneling barrier whose width changes under bias. Separate ON and OFF state equations, nested exponentials, and hyperbolic-sine terms produce strongly nonlinear and asymmetric switching, including different ON-to-OFF and OFF-to-ON times and polarity-dependent thresholds. The same comparison concludes that this model is less suitable for linear analog storage and more relevant for logic-style or threshold-based switching applications (Parajuli et al., 2019).

Floating-gate memristors instantiate yet another dynamic state. In the Y-Flash model, the memory variable is the floating-gate charge $$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$6, evolving as

$$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$7

Program is modeled by lucky hot-electron injection, erase by band-to-band-tunneling-assisted hole injection, and the read current is formed by a smooth combination of subthreshold and above-threshold transistor currents. Because the state is a stored charge rather than a resistive boundary, the model supports transient program/erase dynamics, intermediate analog states, and multiple operating configurations in a CMOS-compatible device. The same work stresses that readout is non-ohmic, which is a limitation for direct analog vector-matrix multiplication (Wang et al., 2022).

4. Vacancy chemistry, transport PDEs, and rigorous device-level dynamics

In oxide memristors, the dynamic state can be elevated from a single scalar variable to coupled carrier and defect fields. A three-species drift-diffusion model uses electron density $$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$8, hole density $$v_M=\hat{R}(x,i_M)i_M, \qquad \frac{dx}{dt}=\tilde f(x,i_M),$$9, oxygen-vacancy density $w/D$0, and electric potential $w/D$1, with drift-diffusion equations coupled to Poisson’s equation. In the fast-relaxation limit $w/D$2, electrons and holes reach quasi-equilibrium and the system reduces to a vacancy–potential model. The analytical results are unusually strong for memristor modeling: global existence of weak solutions for the full system in any space dimension, uniform-in-time boundedness of the full and reduced systems in two dimensions, and weak-strong uniqueness for the reduced model. One-dimensional finite-volume simulations reproduce the pinched hysteresis loop in current-voltage characteristics (Jourdana et al., 2022).

A more compact but still physics-based alternative takes oxygen-vacancy concentration and occupancy as the state variables. In that model, the accessible site density is conserved through

$w/D$3

and the occupied-vacancy population evolves through coupled capture, emission, generation, and recombination processes. The total current is decomposed as

$w/D$4

combining band-to-band tunneling, Ohmic conduction, and bound-to-band or trap-assisted contributions, while a self-heating equation accounts for Joule heating and conductive loss. A central claim is that electron occupancy of oxygen-vacancy traps changes the switching kinetics and retention, so resistance depends not only on how many vacancies exist but also on whether they are occupied. The model is implemented in Verilog-A and tested in a $w/D$5 1T1R array (Zeumault et al., 2022).

These transport-oriented models are significant because they reject the assumption that memristive dynamics must be represented by a geometric filament-gap variable. The oxygen-vacancy model explicitly states that it does not make strong assumptions regarding filament geometry, while the three-species PDE model shows that vacancy transport and electrostatics alone are sufficient to recover hysteresis and mathematically well-posed long-time dynamics.

5. Volatility, relaxation, and synaptic-like plasticity

Volatility changes the interpretation of memristive dynamics from nonvolatile storage to fading-memory processing. In a volatile extension of the Strukov model, the state equation becomes

$w/D$6

where $w/D$7 is a decay rate. The same work also gives an equivalent Wiener-model approximation with a linear exponentially decaying state and a static sigmoid nonlinearity. Its central claim is application-specific: in reservoir computing, volatility is not a defect but a necessary source of fading memory, and device variability becomes an asset rather than a liability (Carbajal et al., 2014).

A biology-inspired compact model translates this idea into a single continuous relaxation law. The current is written as a conductance interpolating between low- and high-conductance states, and the internal state obeys

$w/D$8

Set and reset are therefore unified by one voltage-activated relaxation time rather than separate sign-dependent equations. By tuning the voltage dependence of $w/D$9 and $x_f(t)$0, the model reproduces volatile threshold switching, gradual nonvolatile switching, abrupt nonvolatile switching, scan-rate dependence, upper-vertex dependence, and pulse-induced potentiation in halide perovskite and related devices (Bou et al., 2024).

At molecular scale, dynamic memristor models explicitly separate fast electronic transport from slow chemical evolution. One such framework uses a two-channel decomposition,

$x_f(t)$1

with first-order kinetics

$x_f(t)$2

Bias-dependent occupancies derived from Landauer- and Marcus-inspired transport connect fast conduction to slow proton/ion migration or conformational switching. The same model reproduces conductance hysteresis, short-term plasticity, spike-timing-dependent plasticity, and reservoir-computing behavior, with the main design rule that performance is optimized when input frequency and bias mapping range match the intrinsic molecular kinetics (Chen et al., 25 Aug 2025).

The most explicitly modular recent framework separates state dynamics, cumulative conductance, volatility, saturation, and optional STDP-like plasticity. Its internal variables are $x_f(t)$3, with a cumulative conductance derivative

$x_f(t)$4

a volatility stage

$x_f(t)$5

and a saturated output

$x_f(t)$6

Eligibility traces obey

$x_f(t)$7

and update an STDP-like weight variable $x_f(t)$8. The volatility kernel is inspired by linear viscoelasticity, so long-tailed conductance decay is represented through hereditary convolution rather than a single exponential (Habart et al., 5 Mar 2026).

6. Nonlinear dynamics, bifurcations, and learning with dynamic weights

Dynamic memristor models are also used as dynamical-systems primitives rather than only as device fits. Memristor circuits can be rewritten to reproduce nonlinear systems from engineering, physical, chemical, biological, and ecological dynamics. Under periodic forcing they can exhibit periodic, quasi-periodic, non-periodic, and chaotic motion; attractors can be reconstructed from terminal voltage and current; and the memristor may switch between passive and active modes depending on terminal voltage. The same work repeatedly emphasizes sensitivity to initial conditions, parameter choices, and numerical step size $x_f(t)$9, with overflow possible in long simulations (Itoh, 2019).

In memristor-based oscillators with a line of equilibria, the state variable can function as an internal bifurcation coordinate. A smooth or piecewise-smooth memductance $x_d(t)$0 yields pitchfork, transcritical, and saddle-node bifurcations “without parameters,” meaning that the classical branching scenarios occur as trajectories move along the equilibrium manifold rather than under external parameter sweeps. Adding a forgetting term,

$x_d(t)$1

collapses the line of equilibria and removes the without-parameter bifurcation geometry (Korneev et al., 2022).

Time-averaged pulse-driven dynamics can likewise generate multistability in a single device. For a TaO memristor driven by alternating narrow pulses, the reduced averaged equation $x_d(t)$2 can exhibit $x_d(t)$3, $x_d(t)$4, or $x_d(t)$5 stable fixed points depending on pulse amplitudes and widths. In the bistable regime, two stable fixed points are separated by an unstable one, so the long-time state depends on the initial condition and on which side of the separatrix the device starts (Pershin et al., 2019).

The same dynamical viewpoint extends to neuromorphic learning. In memristor neural networks with simultaneous weight and activation dynamics, neuron activations are capacitor voltages $x_d(t)$6 and weights are memductances $x_d(t)$7 and $x_d(t)$8. Under strict passivity, irreducibility of the interconnection structure, and a slope condition ensuring boundedness, almost all initial conditions converge to an equilibrium point. The result explicitly includes the multistable regime needed for content-addressable memory (Marco et al., 28 Jul 2025).

Training dynamics are similarly constrained by device physics in memristor-based equilibrium propagation. Six memristor models—linear ion drift, Joglekar, Biolek, VTEAM, Yakopcic, and MMS—were characterized by their voltage–current hysteresis and embedded in the EBANA framework. The central empirical result is that equilibrium propagation remains robust under nonlinear memristor-driven weight updates provided the resistance range is sufficiently wide, specifically when

$x_d(t)$9

The study therefore identifies conductance dynamic range, rather than nonlinearity alone, as the dominant device-level constraint on convergence (Döll et al., 13 Dec 2025).

Taken together, these models show that the decisive question is not whether a memristor model is dynamic, but which internal state is made dynamic, which timescales are retained, and which application regime is targeted. Linear drift models remain attractive for gradual analog updates; tunably nonlinear and thresholded models are better suited to mixed-signal or logic-style switching; transport and defect-kinetic models connect directly to oxide-device physics; and volatile or plasticity-augmented models are designed for fading-memory computation, synaptic emulation, and dynamic learning systems (Parajuli et al., 2019).