Front. Phys. Frontiers in Physics Front. Phys. 2296-424X Frontiers Media S.A. 1622487 10.3389/fphy.2025.1622487 Physics Original Research Monophasic and biphasic neurodynamics of bi-S-type locally active memristor Wang et al. 10.3389/fphy.2025.1622487 Wang Xinyi 1 Dong Yujiao 1 * Wang Guangyi 1 2 Zhou Ziyu 1 Mao Yidan 1 Jin Peipei 1 Liang Yan 1 1 Zhejiang Key Laboratory of Intelligent Vehicle Electronics Research, Hangzhou Dianzi University, Hangzhou, China 2 College of Computer and Information Engineering, Qilu Institute of Technology, Jinan, Shandong, China

Edited by: Fei Yu, Changsha University of Science and Technology, China

Reviewed by: Huihai Wang, Central South University, China

Minglin Ma, Xiangtan University, China

Ning Wang, Changzhou University, China

 *Correspondence: Yujiao Dong, yjdong@hdu.edu.cn 30 05 2025 2025 13 1622487 03 05 2025 19 05 2025 Copyright © 2025 Wang, Dong, Wang, Zhou, Mao, Jin and Liang. 2025 Wang, Dong, Wang, Zhou, Mao, Jin and Liang

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Inspired by the energy-efficient information processing of biological neural systems, this paper proposes an artificial memristive neuron to reproduce biological neuronal functions. By leveraging Chua’s unfolding theorem, we establish a bi-S-type locally active memristor mathematical model exhibiting negative differential resistance (NDR), which serve as fingerprints for local activity. A second-order neuronal circuit is constructed to emulate periodic spiking and excitability, while a third-order circuit extends functionality to chaotic oscillations and bursting behaviors. Besides, the constructed neuronal circuit generates biphasic action potential through voltage symmetry modulation, replicating bidirectional signal transmission akin to biological systems. Hardware emulation validates neurodynamics under varying stimuli from theoretical analyses, offering a unit module and theoretical reference for energy-efficient neuromorphic computing network.

 memristor local activity neuron spikes neuromorphic behaviors section-at-acceptance Interdisciplinary Physics 1 Introduction

As information technology rapidly advances, traditional computing architectures face growing limitations in energy efficiency and computational complexity. Against this backdrop, neuromorphic computing has emerged as a novel computing paradigm [1, 2]. Its core concept is to emulate information processing mechanisms of biological systems by constructing brain-like computing structures to achieve energy-efficient computation [3, 4]. This brain-inspired approach demonstrates superior capabilities in adaptive learning, positioning it as a cornerstone for next-generation intelligent systems [5–7]. Central to this technology are neuroelectronic devices that emulate neuronal functions, which are fundamental units in the construction of neuromorphic computing systems [8, 9].

Current neuroelectronic implementations primarily employ Complementary Metal-Oxide-Semiconductor (CMOS) circuits, leveraging mature fabrication techniques to simulate membrane potential dynamics and action potential generation [10, 11]. However, CMOS-based neurons suffer from inherent limitations including complex circuit topologies and elevated power consumption hinder scalability in large neural networks [12]. Memristive devices present an alternative solution through their intrinsic nonlinearity and low-power operation [13], yet conventional passive memristors require auxiliary negative impedance converters to achieve neuronal dynamics, compromising system integration efficiency [14, 15]. These challenges have driven the exploration of locally active memristors (LAMs) [16, 17], whose negative differential resistance (NDR) enables weak signal amplification and action potential generation without external circuitry [18–20].

Recent advancements in LAM-based neuronal modeling demonstrate promising results. The FitzHugh-Nagumo circuit modifications using N-type LAMs successfully replicate biological spiking patterns [21, 22]. Enhanced LAM designs with ultra-robust NDR characteristics further enable hardware implementation of nine distinct neuronal firing modes [4]. However, these advancements remain primarily confined to monophasic action potential emulation. Emerging experimental evidence from sciatic nerve electrophysiology and myocardial fiber studies demonstrates that biphasic potentials constitute fundamental encoding mechanisms in neural systems [23–25], enabling sophisticated information processing [26]. Current neuromorphic platforms predominantly neglect this biphasic paradigm, impeding hardware-level implementation of biologically plausible neural networks.

Memristive neurons exhibit broad application potential in neuromorphic systems, including frequency-based classifiers for animal sound recognition [27], image protection systems [28, 29], and cyclic neural networks with self-adaptive synapses [30]. Notably, their implementation in artificial neural networks has achieved high-precision MNIST digit recognition and effective edge detection in image processing [31].

The structure of this work is as follows: section 2 characterizes the proposed bi-S-type LAM’s nonlinear dynamics; section 3 constructed a second-order neuronal circuit and demonstrates spiking regimes; Section 4 illustrates various monophasic neurodynamics, biphasic spikes, and symmetry behaviors in the third-order memristive neuron. Section 5 gives the circuit simulated validation.

 2 Bi-S-type locally active memristor

Most nanoscale memristors fabricated using various materials exhibit characteristics of generic or extended memristors. Chua’s unfolding theorem provides a systematic method to construct generic memristor models [32]. A generic current-controlled memristor can be defined as v = R m x i = ∑ k = 0 r d k x k i d x d t = f x , i = ∑ k = 0 n α k x k + ∑ k = 0 m β k i k + ∑ k = 0 p ∑ l = 0 q δ k l i k x l where v, i, and x are the voltage, current, and state variable of the memristor, respectively; R m (x) represents memristance; f (x, i) is the state-controlled equation; α k , β k , δ kl , and d k are tunable parameters.

Using Equation 1, we derive a memristor model characterized by v m = R M x i m = d 2 x 2 + d 0 i m = f 1 x , i m d x d t = δ 0 + α 1 x + β 2 i m 2 = f 2 x , i m with parameters: δ 0 = 3 × 104, α 1 = −3 × 103, β 2 = −8 × 107, d 2 = 2, d 0 = 20.

2.1 Fingerprints of locally active memristor

Chua indicates that a pinched hysteresis loop in the voltage-current plane constitutes a definitive memristor signature [33]. The negative differential resistance (NDR) regions on the DC V-I curve serve as critical indicators of local activity in one-port memristors [34]. These are fingerprints of LAMs.

2.1.1 fingerprint 1: pinched hysteresis loop

Let us apply a sinusoidal voltage v = Asin(2πft) with amplitude A = 5 V and frequencies f = 2 kHz, 5 kHz, 200 kHz to the proposed model. The characteristics of input voltage v m and response current i m are depicted in Figure 1A. It shows that the loci plotted on the v m-i m plane is a pinched hysteresis loop at f = 2 kHz (dark red curve). The lobe area decreases progressively with increasing frequency (black curve: f = 5 kHz), collapsing to a linear blue line at f = 200 kHz, confirming memristive behavior.

(A) Pinched hysteresis loops measured from Equation 2 on v m-i m plane for input voltages v m with amplitude A = 5 V and frequencies f = 2 kHz, 5 kHz, 200 kHz; (B) DC V-I curve of the LAM with the shaded NDR regions.

 2.1.2 fingerprint 2: negative differential resistance (NDR) regions

The DC V-I curve (Figure 1B), obtained by sweeping DC currents from −25 mA to 25 mA with the step size of 0.1 mA, reveals two NDR regions (yellow shading) corresponding to local activity. Signal amplification occurs at these operating points where V ∈ [0.385 V, 1.287 V] (I ∈ [9.2 mA, 19.1 mA]) and V ∈ [−1.287 V, −0.385 V] (I ∈ [−19.1 mA, −9.2 mA]).

However, the operating points Q (V, I) of the LAM exhibit instability when biased at V ∈ [–1.287 V, −0.385 V] ∪ [0.385 V, 1.287 V]. As shown in Figure 2 (left inset), three intersections (M0, M1, M2) or (M3, M4, M5) emerge at V = ±1 V: two stable (M1, M2 or M4, M5) and one unstable (M0 or M3), which is verified by the dynamic route with x-dx/dt. Then, stabilization was achieved by adding an appropriate resistor R 0 = 1 kΩ, and the obtained locally active voltages are V ∈ [R 0 I D + V D, R 0 I C + V C] ∪ [R 0 I B + V B, R 0 I A + V A], i.e., V ∈ [−19.485 V, −10.487 V] ∪ [10.487 V, 19.485 V] (Figure 2, right inset). Observe that single stable equilibria emerge under these two operating points.

Stabilization mechanism: (left) unstable equilibria in the memristor without resistors; (right) stabilized operation with series resistor R 0.

 2.2 Small-signal equivalent circuit of LAM

The small-signal equivalent circuit enables nonlinear dynamics prediction at arbitrary operating points. By applying Taylor series expansion to Equation 2 at operating point Q (V Q , I Q ) under sufficiently small signals (ignoring higher-order terms), we obtain △ v = a 11 Q △ x + a 12 Q △ i △ x ˙ = b 11 Q △ x + b 12 Q △ i where Δv, Δi, and Δx denote small perturbations; a 11 = (δf 1/δx)| Q = 2d 2 I Q X Q , a 12 = (δf 1/δi)| Q = d 2 X Q 2 + d 0 = R M(X), b 11 = (δf 2/δx)| Q = α 1, b 12 = (δf 2/δi)| Q = 2β 2 I Q . Here, the differential resistance R D=(Δv/Δi)| Q = a 11 (Δx/Δi) + a 12 = a 12 – (a 11 b 12)/b 11, the equivalent resistance R M = a 12.

Taking the Laplace transform of (Equation 3) yeilds Z s , Q = v ^ s i ^ s = a 11 Q b 12 Q s − b 11 Q + a 12 Q = 1 s b 11 R M − R D + 1 R D − R M + R M = 1 C s s + 1 R s + R M

Figure 3A illustrates the small-signal equivalent circuit of the LAM about an operating point: a parallel R s-C s network in series with R M. Figure 3B shows parameter variations under V ∈ [10 V, 20 V]. Notably, negative capacitances (C s < 0) occur at locally active voltages V ∈ [10.487 V, 19.485 V], which critically determine memristive characteristics for neuronal circuit design. Similar trends hold for V ∈ [–19.485 V, −10.487 V].

Small-signal analysis of the LAM: (A) the small-signal equivalent circuit of the LAM; (B) the values of R M, C s, and R s varying with operating voltages over the range of V ∈ [10 V, 20 V].

 3 LAM-based second-order neuron

To construct a second-order neuronal circuit using the LAM, an external capacitor C 0 is required to compensate for the inductive behavior of the LAM in locally active domains (LADs). The proposed circuit includes excitation and response signals (v in and v out = v C = v m), a biasing resistor R 0, and capacitor C 0, as depicted in Figure 4.

LAM-based second-order neuronal circuit.

Frequency-domain analysis determines C 0. Substituting s = iω into the impedance function Z (s, Q) in Equation 4 yields Z ⁡ i ω , Q = Re Z ⁡ i ω , Q + i Im Z ⁡ i ω , Q = R M + R s 1 + R s 2 C s 2 ω 2 + i − R s 2 C s ω 1 + R s 2 C s 2 ω 2

The resonant frequency ω 0 occurs when Re [Z (iω, Q)] = 0. From Equation 5, the corresponding imaginary part can be calculated. For oscillation initiation, the critical capacitance satisfies: C 0 = 1 ω 0 Im i ω 0 , Q = − R s C s R s + R M

3.1 Composite impedance function

The oscillation condition for the composite neuronal circuit is derived from its impedance function: Z C s , Q = 1 1 Z s , Q + C 0 s = R M s − R D b 11 R M C 0 s 2 + 1 − R D C 0 b 11 s − b 11 with two poles p 1 = R D C 0 b 11 − 1 + 1 − R D C 0 b 11 2 + 4 R M C 0 b 11 2 R M C 0 p 2 = R D C 0 b 11 − 1 − 1 − R D C 0 b 11 2 + 4 R M C 0 b 11 2 R M C 0

Figure 5A maps three operational domains: Locally Passive Domains (LPD, yellow), Unstable Locally Active Domains (RHP, right-half plane, cyan), Stable Locally Active Domains (EOC, edge of chaos, green) based on Equations 6, 7a, 7b. These domains align with the memristor’s LAD and LPD characteristics. RHP requires simultaneous local activity and instability, while EOC demands local activity with asymptotic stability.

(A) The parameter distributions of LAD, LPD, and EOC on V in-C plane; (B) poles trajectories with the change of voltage V in at C = 10 μF.

The pole evolution analysis in Equation 7b under bias voltages v in ∈ [–20.5 V, −9.5 V]∪[9.5 V, 20.5 V] and C = 10 μF reveals dynamic stability transitions, as depicted in Figure 5B. Red and blue curves represent the trajectories of p 1 and p 2, respectively, with oscillation occurring when v in ∈ [–19.03 V, −11.4 V]∪[11.4 V, 19.03 V] (orange region). In this region, at least one pole is in the right-half plane (RHP). However, stability persists when p 1, 2 ∈ LHP (left-half plane). Particularly, Hopf bifurcation emerges at v in = ±11.4 V and ±19.03 V, characterized by conjugate complex pole pairs.

 3.2 Periodic spikes

The state equations of the second-order neuronal circuit in Figure 4 are governed by d x d t = δ 0 + α 1 x + β 2 v m d 2 x 2 + d 0 2 d v m d t = 1 C 0 v in − v m R 0 − v m d 2 x 2 + d 0 where x and v m represent memristor state and membrane potential, respectively.

With C 0 = 10 μF and initial condition [x (0), v m (0)] = (0, 0), distinct neuromorphic behaviors emerge under varying stimuli v in according to Equation 8. For stimuli v in = 9.5 V (LPD) and 10.8 V (EOC), the trajectories converge from the initial point (0, 0) into (v C, x) = (1.27, 8.19) and (1.29, 7.58), respectively, maintaining resting states as shown in Figure 6A. Increasing v in to 18 V (see RHP domain in Figure 5A) triggers sustained periodic spikes with frequency f = 204 Hz, demonstrated by time-domain waveform of v C and limit cycles in the x-v out phase portrait (Figure 6B). We conclude that the neuron maintains quiescence when operating in the LPD or EOC domains, while inducing spikes under locally active operating points. Notably, spiking frequency modulation under varying v in = 12 V, 14.5 V, 16.5 V, and 18.5 V replicates biological neural encoding mechanisms (Figure 6C). Besides, the neuron emulates excitatory and inhibitory response transitions, as illustrated in Figure 6D.

Neuromorphic behaviors under some typical voltages: (A) v in = 9.5 V, 10.8 V, resting states; (B) v in = 18 V, periodic spikes; (C) spiking frequency modulation; (D) excitatory and protective inhibition behaviors.

 4 LAM-based third-order neuron

Second-order neurons cannot simulate complex neurodynamics such as chaos and bursting, then we construct a memristive neuron with third-order complexity, as shown in Figure 7, including an LAM, a capacitor, an inductor, a resistor, and a voltage source.

The circuit schematic of the third-order memristive neuron model.

 4.1 Stability condition

The impedance function Z T (s, Q) of third-order memristive neuron circuit is written as: Z T ⁢ s , Q = 1 1 / Z s , Q + s L + s C = L s 2 + R M − b 11 L s − b 11 R D L C s 3 + R M − L b 11 C s 2 + 1 − C R D b 11 s − b 11 whose three poles are p 1 = − R M − b 11 L 3 L + 3 − q 2 + Δ + 3 − q 2 − Δ p 2 = − R M − b 11 L 3 L + − 1 + 3 i 2 ⁢ 3 − q 2 + Δ + − 1 + 3 i 2 2 ⁢ 3 − q 2 − Δ p 3 = − R M − b 11 L 3 L + − 1 + 3 i 2 2 ⁢ 3 − q 2 + Δ + ω 3 − q 2 − Δ where Δ = R M − b 11 L · 1 − C R D b 11 6 L 2 C − R M − b 11 L 3 27 L 3 + b 11 2 L C 2 + 1 − C R D b 11 3 L C − R M − b 11 L 2 9 L 2 3 , q = − R M − b 11 L 1 − C R D b 11 3 L 2 C + 2 R M − b 11 L 3 27 L 3 − b 11 L C .

Based on Equation 9, the trajectory diagram of poles p 1,2,3 within the range of 8.2 V ≤ V in ≤ 20 V is depicted in Figure 8A, where blue, red, and yellow curves correspond to the trajectories of p 1, p 2, and p 3, respectively, with arrows denoting directionality as v in increases. Oscillatory behavior occurs when Re p > 0, particularly within v in ∈ [11.12 V, 19.38 V] where Hopf bifurcation emerges at v in = 11.12 V and v in = 19.38 V, characterized by conjugate complex poles (Im p = 0). For v in ≤ 11.12 V, all poles reside in the left-half plane (LHP), driving the circuit to stable equilibrium. Conversely, right-half plane (RHP) poles dominate in the oscillatory regime, enabling sustained dynamics. Figures 8B,C confirm this operational range through Lyapunov exponent and bifurcation diagram analysis, demonstrating consistent periodic and chaotic domains in this range under L = 20 mH and C = 10 μF, where the chaotic ranges are v in ∈ [–18.98 V, −18.89 V] ∪[18.89 V, 18.98 V].

(A) Poles diagram with respect to biasing voltages v in when L = 20 mH, C = 10 μF; (B) Lyapunov exponents under L = 20 mH, C = 10 μF; (C) bifurcation diagram with respect to v in under L = 20 mH, C = 10 μF.

 4.2 Monophasic neurodynamics

The third-order LAM-based neuronal circuit in Figure 7 is described by d x d t = δ 0 + α 1 x + β 2 i L 2 d v C d t = 1 C v in − v out R 0 − i L d i L d t = 1 L v out − d 2 x 2 + d 0 i L where x (memristor state), i L (inductor current), and v C (output voltage) define the neuron dynamics.

Under L = 20 mH and C = 10 μF, six monophasic neuromorphic behaviors emerge through parametric control of v in based on Equation 10. At v in = 19.3 V (RHP domain), subthreshold oscillations occur (Figure 9A). Reducing v in to 18.5 V and 18.9 V within the RHP domain induces periodic spikes (Figure 9B) and chaotic dynamics (Figure 9C), respectively. Time-varying stimulation v in = 9.9 t V (t ∈[1.2 s, 2.2 s]) triggers Class II excitability, maintaining constant spiking frequency despite voltage modulation (Figure 9D). For periodic square-wave inputs (T = 0.0625 s, A = 15 V), the neuron exhibits bursting patterns (Figure 9E). Besides, depolarizing after-potentials emerge under the parameter set of C = 0.5 μF, L = 20 mH and v in = 15 V, mimicking post-spike membrane potential modulation (Figure 9F). These results demonstrate voltage-controlled emulation of biological neuronal encoding.

Monophasic neurodynamics under different input voltage v in with C = 10 μF and L = 20 mH: (A) subthreshold oscillation; (B) periodic spiking; (C) chaos; (D) Class II excitability; (E) periodic bursting. (F) depolarizing after-potential with C = 0.5 μF, L = 20 mH and v in = 15 V.

 4.3 Biphasic spikes

The neuronal circuit in Figure 7 generates biphasic action potentials when driven by bipolar square-wave inputs (v in = 16 V, D = 50%). As shown in Figures 10A a T= 10 ms periodic stimulus (blue waveform) induces single-cycle bidirectional spiking, characterized by counterphase positive and negative pulses in the inductor current i L (red waveform). When we increase the period T of the input periodic square wave to 22.22 ms, 33.33 ms, 43.48 ms, 55.56 ms and 66.67 ms, the output waveform changes into two spikes, three spikes, four spikes, five spikes and six spikes in the upward direction and down direction in one period, as shown in Figures 10B–F.

Biphasic action potentials generated by the neuron circuit, when driven by a bipolar periodic square wave with the amplitude v in = 16 V, duty cycles D = 50% and various period T. (A) T = 10.00 ms; (B) T = 22.22 ms; (C) T = 33.33 ms; (D) T = 43.48 ms; (E) T = 55.56 ms; (F) T = 66.67 ms.

 4.4 Symmetric behaviors

The third-order memristive neuron demonstrates voltage-polarity-dependent symmetry in neurodynamic behaviors, originating from the voltage symmetry in Figure 5A. This nonlinear symmetry allows symmetrical action potential generation: positive DC voltages (v in > 0) induce upward-polarized spikes, while negative inputs (v in < 0) produce downward-polarized counterparts, as depicted in Figure 11.

Voltage-polarity-modulated symmetric behaviors: (A) periodic spikes; (B) chaos; (C) resting states; (D) phase portraits of periodic spikes; (E) phase portraits of chaos; (F) phase portraits of resting states.

Under voltage excitation v in = ±18.5 V, the inductor current i L exhibits mirror-symmetric periodic spiking, i.e., upward polarization for positive bias (orange curve) versus downward polarization for negative bias (blue curve) in Figure 11A. Voltage modulation to ±18.9 V induces symmetrical chaotic dynamics with identical Lyapunov exponents but opposing phase-space trajectories, as shown in Figure 11B. Transient behavior analysis reveals bidirectional spike initiation: v in = 19.4 V triggers upward spikes while v in = −19.4 V generates downward equivalents, both returning to symmetrical resting potentials after undergoing 5 m (Figure 11C). The corresponding phase portraits of these three nonlinear behaviors are depicted in Figures 11D–F.

 5 Circuit emulator

The circuit emulator of the memristive neuron with third-order complexity is constructed, as shown in Figure 12, which consists of two operational amplifiers (U1A and U1B), three analog multipliers (U1, U2, and U3), two capacitors (C 0 and C 1), one inductor L, and some resistors.

Circuit emulator schematic of the third-order memristive neuron.

As shown in Figure 12, the equivalent circuit of the locally active memristor comprises four functionally integrated modules: (1) Current sensing module ① monitors emulator input current in real-time, generating proportional output v i ; (2) analog multiplier arrays ② and ④ implement nonlinear term computations; (3) State equation solver ③ converts DC bias V d into memristor state variable x through differential integration, that is, v x = x; (4) Feedback integration completes the loop via R 1. Kirchhoff’s voltage and current laws govern this circuit architecture, yielding three coupled differential equations that mathematically describe electrophysiological dynamics of the memristive neuron, as C 1 d x d t = − V d R 9 + x R f + R w + R z 10 R w R 8 R 7 R 5 R 1 R 4 R 6 + R 7 i L 2 C 0 d v C d t = v in − v C R 0 − i L L d i L d t = v C − R 1 R 5 R 11 10 R 4 R 10 x 2 + R 1 i L where the circuit parameters are R 1 = 20 Ω, R 0 = R 2 = R 4 = R 6 = R 7 = R w = 1 kΩ, R 3 = R 5 = R 8 = R 9 = 10 kΩ, R z = 7 kΩ, R f = 33.3 kΩ, C 1 = 10 nF, C 0 = 9.5 μF, L = 20 mH, and V d = −3 V.

The circuit simulated results calculated via Equation 11 are shown in Figure 13, reproducing key neurodynamics including periodic spikes, class II excitability, self-sustained oscillations, bursting, chaos, and depolarizing after-potential. These results demonstrate quantitative agreement with theoretical predictions.

Circuit simulated neuromorphic dynamics under various biasing voltages.

 6 Conclusion

This work constructs neuronal circuits leveraging a bi-S-type locally active memristor that amplifies weak signals through intrinsic local activity. The designed second-order circuit achieves voltage-modulated periodic spiking and adaptive inhibition, while the third-order extension emulates biological neural dynamics including monophasic and biphasic action potentials, chaos, and bursting, which are driven by memristive symmetry. The study of memristive neurons not only offers essential building blocks for neuromorphic computing architectures but also lays a theoretical reference for the development of more advanced and bio-realistic neural processing systems.

 Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

 Author contributions

XW: Software, Writing – original draft, Formal Analysis, Validation. YD: Methodology, Writing – review and editing, Validation, Software, Funding acquisition. GW: Writing – review and editing, Supervision, Formal Analysis. ZZ: Writing – original draft, Formal Analysis, Software. YM: Writing – review and editing, Visualization. PJ: Investigation, Resources, Writing – review and editing. YL: Formal Analysis, Writing – review and editing, Funding acquisition.

 Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This work was supported in part by the Zhejiang Provincial Natural Science Foundation of China under Grant LQ23F010018 and Y24F010007; in part by the National Natural Science Foundation of China under Grant 62301202 and 62171173.

 Conflict of interest

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.

 Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

 Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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