Recording and modeling of pinched hysteresis loops, the fingerprint of a memristor, in neurons

Pabst, Oliver; Schülke, Christin; Balasingham, Ilangko; Krauss, Stefan; Glover, Joel Clinton; Martinsen, Ørjan Grøttem

doi:10.1038/s41598-025-21035-0

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Published: 24 October 2025

Recording and modeling of pinched hysteresis loops, the fingerprint of a memristor, in neurons

Oliver Pabst¹^na1,
Christin Schülke^1,2^na1,
Ilangko Balasingham^3,4,
Stefan Krauss^2,5,
Joel Clinton Glover^6,7^na2 &
…
Ørjan Grøttem Martinsen^1,8^na2

Scientific Reports volume 15, Article number: 37212 (2025) Cite this article

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Abstract

A variant of the Hodgkin-Huxley model has been proposed in which time-varying Na⁺ and K⁺ ion conductances are replaced by memristors, based on the proposal that ion channels are memristors. This hypothesis predicts that current-voltage plots of neurons subjected to sinusoidal stimulation should exhibit a pinched hysteresis loop, the fingerprint of a memristor. We tested this using whole-cell patch clamp recordings from human neurons derived from neural progenitor cells and observed pinched hysteresis loops in 16% of all recorded neurons. Recordings with current clamp and voltage clamp with a holding potential of 0 mV yielded a higher success rate than voltage clamp recordings with a holding potential of around − 60 mV. In addition, more neurons exhibited pinched hysteresis loops when the applied voltage or current amplitude was high, and the frequency was low. The recording of pinched hysteresis loops in neurons gives evidence that these contain memristors and this is the first time to be shown experimentally. This is not just an indication that the memristor-based Hodgkin-Huxley model is valid; it also demonstrates that models of neurons based on memristors actually resemble nature. We then subjected simulated neurons expressing Hodgkin-Huxley voltage-gated Na⁺ and K⁺ channels to sinusoidal membrane potential oscillations and found that the presence of pinched hysteresis loops depended strongly on the density of voltage-gated K⁺ channels and on the membrane capacitance but not on the presence of voltage-gated Na⁺ channels. Because Na⁺ channels are expected to be largely inactivated in neurons at a holding potential of 0 mV, these findings together suggest that it is the presence of voltage-gated K⁺ channels that is critical for the observed memristive behavior.

Introduction

A memristor (memory resistor) is a passive electrical circuit element that correlates voltage and current via a state-dependent Ohm’s law. The existence of memristors was predicted by Leon Chua in 1971¹ and confirmed by scientists from Hewlett-Packard Labs in 2008². Chua et al. also predicted that ion channels are memristors^3,4,5. Chua et al. demonstrated that the time-varying sodium (Na⁺) and potassium (K⁺) conductances in the Hodgkin–Huxley model of the action potential⁶ can be replaced by time-invariant memristors⁴. Considering the voltage-gated Na⁺ and K⁺ ion channels not as time-varying conductances but as time-invariant memristors may provide new insights into neuronal function.

The fingerprint of a memristor is a pinched hysteresis loop in the current–voltage (I–V) representation with the pinched point at the origin of the current–voltage plane⁷ (see Fig. 1). This must be consistent for all possible periodic input signals with zero mean⁸. The two branches of the pinched hysteresis loop can cross with different slopes (transversal pinched hysteresis loop) or touch with equal slopes (tangential pinched hysteresis loops). Multiple pinched points are possible as long as one is at the origin of coordinates^7,9.

In biological systems, the movement of ions is likely to be central to the behavior of memristors. The direction of ion movement depends on the direction of the applied electrical field. With a low frequency alternating current (AC) excitation, the ions will move alternatingly in opposite directions. As the frequency increases, the ability of the ions to follow the field reversal will diminish. Thus, memristive properties are more obvious at lower frequencies, which is also reflected by another fingerprint of memristors, namely, that I–V plots tend towards a single-valued function (the measurement becomes linear) as the frequency increases⁹. In addition, the stronger the field strength, the stronger the movement of ions. Increasing the field strength therefore can increase the frequency at which memristive properties can be observed (see example in¹⁰).

The existence of memristors in nature has been demonstrated experimentally. Memristors have been detected in slime molds¹¹, and studies on human skin have identified the sweat ducts¹² and the stratum corneum as memristors¹⁰. It is difficult to perform measurements that selectively target memristive elements in nature. Measurements on human skin, for example, are affected by the capacitive properties of the stratum corneum and a DC offset, both of which alter the shape of the obtained pinched hysteresis loop¹³. The pinched point may thus be shifted away from the origin of the I–V plane^14,15 and the I–V curve can exhibit an elliptical shape due to the capacitive phase shift at higher frequencies (e.g. already evident at 2.5 Hz in human skin¹⁰. The transfer impedance of the electrodes (transition from electron conduction to ion conduction at the electrode-tissue interface) may also affect the measurement.

Here, we used whole-cell patch recording to assess the presence of pinched hysteresis loops in single human neurons differentiated from neural progenitor cells (Fig. 1). These neurons express a number of ion channels inserted into the lipid bilayer of the cell membrane and can thus be modelled as a capacitance (the lipid bilayer) in parallel with conductances (the ion channels, Fig. 1). Some ion channels are voltage-gated and can exhibit complex opening, closing, and inactivation dynamics. Voltages created by ion gradients will add a DC component to the measurements. In whole-cell patch recordings, a glass micropipette electrode has direct electrical contact with the inside of the neuron and can therefore be used to pass current of defined magnitude and temporal dynamics into or out of the neuron. We applied sinusoidal waveforms of different amplitudes and (very) low frequencies (range 0.01–5 Hz), using either current clamp mode (applied current controlled and membrane potential recorded) or voltage clamp mode (applied membrane potential controlled and current recorded). With the low frequencies used, capacitive current is minimized and the current passes preferentially through the ion channels. We have observed pinched hysteresis loops in some neurons, indicating that at least some neurons show this memristive feature. We suggest on this basis that ion channels are likely candidates for the memristive elements⁴.

We further investigated this hypothesis by simulations based on the Hodgkin-Huxley model, in which we varied the densities of voltage-gated Na⁺ and K⁺ channels (to the point of removing them entirely) and the membrane capacitance. We found that simulated neurons also exhibited pinched hysteresis loops, and that this behavior was highly dependent on voltage-gated K⁺ channel density and membrane capacitance but essentially independent of voltage-gated Na⁺ channel density. Pinched hysteresis loops continued to be expressed even in the absence of voltage-gated Na⁺ channels. The location of the pinched point in the I-V curve shifted systematically and reciprocally with variation of voltage-gated K⁺ channel density and membrane capacitance, such that a pinched hysteresis loop was no longer evident at a critically low density of potassium channels or a critically low membrane capacitance. Thus, a pinched hysteresis loop was exhibited within a defined state-space of voltage-gated K⁺ channel density and membrane capacitance, which may explain why only some in vitro neurons showed this property.

The recording of pinched hysteresis loops from neurons gives evidence that these contain memristors and that memristor based neuron models actually resemble nature. By recording and analyzing the pinched hysteresis loops one gains insights about the actual dynamics of the inherent memristors. These findings may be used in neuromorphic computing^17,18, in which examples of memristor-based neurons^19,20,21, memristor-based synapses²² and memristor-based dendrites²³ can be found.

Results

The experiments were performed with neurons differentiated from an immortalized human fetal midbrain neural progenitor cell line. The differentiation success was characterized using marker protein staining. Differentiation led to neuron-like morphology and expression of proteins characteristic of a differentiated neuronal phenotype. We observed a mixture of TH+ (presumed dopaminergic) neurons, other neuron subtypes, undifferentiated progenitors, astrocytes, and potentially other glial cell types (Supplementary Fig. 1). A detailed description of the expressed marker proteins can be found in the Supplementary Information. Bright-field and fluorescence microscopic images of the recorded neurons are presented in Supplementary Fig. 2.

Some neurons exhibit pinched hysteresis loops

We obtained datasets from voltage-clamped membrane potential oscillations centered at −60 mV from 31 neurons, from voltage-clamped membrane potential oscillations centered at 0 mV from 21 neurons, and from current-clamped membrane oscillations from 23 neurons. In total, we obtained 109 data sets from 94 neurons (2 or 3 recording conditions were performed in some neurons). However, the electrode-neuron contact was lost or unstable in 34 data sets, mainly for voltage-clamped oscillations centered at −60 mV. Thus, we ended up with 75 data sets from 63 neurons.

Recordings were typically noisy, especially using voltage clamp, (Fig. 6 and Supplementary Fig. 3), which sometimes made it difficult to identify pinched hysteresis loops. For the data analysis we averaged consecutive samples and developed criteria for deciding whether a neuron exhibited pinched hysteresis loops or not (see the “Sampling frequency and data processing” section in the methods).

Current clamp

Six of 23 neurons recorded with current clamp exhibited clear pinched hysteresis loops (Fig. 2a,b) in multiple measurements. Pinched hysteresis loops were more likely to appear at lower frequencies and higher amplitudes (Table 1). They were markedly asymmetric, with the pinched point shifted to the right side of the plot. For the other 17 neurons (Fig. 2c), we did not obtain pinched hysteresis loops in at least three different measurements (Fig. 2c). As an example of a neuron that did not meet the criteria for exhibiting pitched hysteresis loops, the recording of neuron 3 at 0.01 Hz and amplitude of 8 mV could be misinterpreted as a pinched hysteresis loop. However, closer inspection shows no clear pinched point: the upper branch of the I-V curve (light cyan in the third quadrant and red in the first quadrant) crosses the lower branch (dark cyan in the third quadrant and grey in the first quadrant) twice and is thus disqualified. Although the memristor theory allows for multiple pinched points, in this recording the crossings of the upper and lower branch are merely due to random noise. In all the other recordings of neuron 3, it is obvious that there is no pinched hysteresis loop (see recording at 0.02 Hz in Fig. 2c).

Voltage clamp oscillating around − 60 mV

Under this condition only two neurons out of 31 exhibited pinched hysteresis loops at different frequencies and amplitudes (Fig. 3a,b). The location of the pinched point varied along the y-axis. In the other 29 neurons we did not observe pinched hysteresis loops at different frequencies and amplitudes (Fig. 3c).

Voltage clamp oscillating around 0 mV

Under this condition, four of 21 neurons exhibited pinched hysteresis loops at different frequencies and amplitudes (Fig. 4a,b). The pinched hysteresis loops were quite symmetrical, and the pinched point was close to the origin. Recordings from the remaining 17 neurons were noisy and no pinched hysteresis loops could be discerned (Fig. 4c).

Measurements performed with multiple recording conditions

Two different recording conditions were used on some neurons (n = 10). Two of these 10 neurons exhibited pinched hysteresis loops under both recording conditions. An example is neuron 2, which exhibited pinched hysteresis loops in current clamp (Fig. 2b) and in voltage clamp oscillating around − 60 mV (Fig. 3b). Two of the remaining 8 neurons exhibited pinched hysteresis loops only in current clamp, and the remaining six neurons did not exhibit pinched hysteresis loops under either recording condition. To one neuron, all the three recording conditions were applied but the neuron did not exhibit pinched hysteresis loop under any of these conditions.

In total, pinched hysteresis loops were obtained from 10 out of 63 neurons.

Greater incidence of pinched hysteresis loops at higher amplitudes and lower frequencies

For all three measurement methods, there was a clear trend that the higher the amplitude and the lower the frequency, the more likely it was to obtain pinched hysteresis loops (Table 1). Pinched hysteresis loops were obtained up to 0.2 Hz and even at 0.5 Hz from a single neuron. At higher frequencies, the current-voltage plots had an elliptical shape (Supplementary Fig. 4).

Table 1 Number of neurons in which pinched hysteresis loops were obtained, itemized for each combination of amplitude and frequency used in all three recording methods.

Full size table

In current clamp, pinched hysteresis loops could be obtained at an amplitude as low as 2 pA (one neuron), but were more common at higher amplitudes (6 neurons at 8 pA). In voltage clamp with a holding potential of 0 mV, a pinched hysteresis loop was observed in one neuron already at an amplitude of 4 mV and from more neurons at higher amplitudes (for example, 4 neurons at 16 mV). The success rate of obtaining pinched hysteresis loops in the measurements increased with increasing amplitude and decreasing frequency (see Supplementary Tables 1 and 2). Both parameters have a significant effect on the success rate (see Supplementary Table 3). Depending on the amplitude and frequency, the success rate in current clamp was up to 29% (for amplitudes of 16 pA and 18 pA at 0.1 Hz, see Supplementary Table 1). The maximum success rate for voltage clamp recordings with a holding potential of 0 mV was 20–22% (for amplitudes of 16 mV at 0.01 Hz, 12 and 16 mV at 0.02 Hz, and 16 mV at 0.1 Hz, see Supplementary Table 2). Voltage clamp recordings with a holding potential of around − 60 mV barely revealed any pinched hysteresis loop and the maximum success rate was around 7% (for amplitudes of 8, 12 and 16 mV at 0.01 Hz, and 12 and 16 mV at 0.02 Hz, see Supplementary Table 2).

Pinched hysteresis loops in simulated neurons

To explore the possible dependence of memristive properties on specific ion channels, we assessed simulated neurons created in the widely used Neuron platform (www.neuron.yale.edu/neuron/) that were equipped with Hodgkin-Huxley voltage-gated Na⁺ and K⁺ channels whose density could be varied in over several orders of magnitude. A script was composed (see Supplementary Information) that generated a neuron of size typical for a rodent cortical excitatory neuron with a resting membrane potential of −70 mV, that was stimulated by a sinusoidal current that generated a sinusoidal membrane potential change. We then varied Na⁺ and K⁺ channel density and specific membrane capacitance systematically, and plotted the resultant I–V curves.

We found that neurons with an initial state of physiologically relevant surface conductance densities (0.12 S/cm² for Na⁺ channels, 0.036 S/cm² for K⁺ channels), consistently exhibited pinched hysteresis loops (Fig. 5, Table 2a). Very little change occurred when voltage-gated Na⁺ channel surface conductance density was decreased while voltage-gated K⁺ channel surface conductance density was kept constant. By contrast, when voltage-gated K⁺ channel surface conductance density was decreased while maintaining a constant surface conductance density of voltage-gated Na⁺ channels, the pinch in the hysteresis loop shifted systematically to the right (Supplementary Fig. 5).

We also varied the specific membrane capacitance (C_M = capacitance per membrane area, in units µF/cm²) systematically, while maintaining the initial, physiologically relevant surface conductance densities of voltage-gated Na⁺ and K⁺ channels. This corresponds to a directly proportional change in membrane time constant, tau (= RC), which in physiological terms would imply a greater rate of rise and rate of fall of membrane potential changes as specific membrane capacitance decreases. When specific membrane capacitance was decreased stepwise (Supplementary Fig. 6 and Table 2b), the pinch in the hysteresis loop shifted systematically to the left, such that a pinch was barely evident when the decrease was 10-fold.

Table 2 Pinched hysteresis loop expression of simulated neurons at different densities of voltage-gated Na⁺ and K⁺ channels (a) and membrane capacitance (b), during a 10 Hz imposed oscillating current that generated a voltage Oscillation around a resting membrane potential of −70 mV.

Full size table

Discussion

We observed pinched hysteresis loops in about 16% (10/63) of recorded neurons at one or more of the three measurement conditions (current clamp, voltage clamp at 0 mV or at −60 mV), with some variability both among and within neurons. All observed pinched hysteresis loops had an oblique trajectory with the two branches crossing at different slopes. Pinched hysteresis loops were more prevalent at low frequencies (from 0.01 Hz and up to 0.2 Hz in some cases and up to 0.5 Hz in one case) and at higher amplitudes and were more easily detected in the less noisy current clamp condition. Neurons can be described as a combination of memristors, resistors, capacitors and voltage sources (Fig. 1), all contributing to an electrical measurement. The capacitive properties of the membrane and applied holding potentials can shift the pinched point away from the origin of coordinates. At very low frequencies, the influence of the capacitance is largely eliminated, thus pinched hysteresis loops are more likely to appear. For frequencies of 1 Hz and higher, we observed no pinched hysteresis loops. Although experiments with higher amplitudes could potentially yield pinched hysteresis loops at frequencies over 0.5 Hz, recordings can become unstable, and neurons can even be damaged. Pinched hysteresis loops were obtained at 10 Hz in the simulated neuron, and have been obtained at even higher frequencies in other types of memristors (see for example²⁴).

Memristive properties are related to the movement of ions, with Na⁺, K⁺, Ca²⁺, Mg²⁺, and Cl^- being central in biological systems. Applied voltage or current will force the ions to move. The intracellular space is much more conductive than all ion channels together. In a serial connection, the highest resistance dominates the measurements and thus, ion channels will dominate the measurement and represent the most likely source of the recorded pinched hysteresis loops. This supports the idea that at least some ion channels are memristors as predicted in⁶.

It has been shown that changes in the polarization impedance of electrodes placed in saline solution or apple juice can also result in pinched hysteresis loops^25,26, indicating that the electrodes themselves can become memristive. Recordings with the patch clamp electrodes in medium without neurons (Supplementary Fig. 7) are linear and confirm that the pinched hysteresis loops we observed here derive from the recorded neurons. The access resistance²⁷, which is located just outside the pore opening where ions could conceivably “pile up” as they try to enter the channel, may not be constant. However, this resistance is unlikely to be of the same order of magnitude as the resistance of the channel itself and thus likely does not contribute to the memristive properties exhibited by neurons.

The composition of ion channels in the recorded neurons was not determined and we considered that underlying variability in the expression and activation of ion channels might influence the results, especially since neuronal differentiation was clearly not homogeneous.

To gain more insight into the role that specific ion channels might play in the observed memristive behavior, we therefore turned to a simulation of a neuron with Hodgkin-Huxley voltage-gated Na⁺ and K⁺ channels, which we could vary in density within the membrane. Here, the appearance of a pinched hysteresis loop was very dependent on the density of voltage-gated K⁺ channels but essentially independent of the density of voltage-gated Na⁺ channels. This supports the idea that memristive properties in excitable cells arise through the activity of specific types of ion channels.

Neurons express a variety of non-gated as well as voltage-gated ion channels, which exhibit different kinetics. At the resting potential of −60 mV, non-gated inward rectifying potassium channels are open, and the neuronal cell membrane is predominantly permeable to K⁺ ions. Changes in membrane polarization (depolarization or hyperpolarization) will trigger the opening or closing of voltage-gated ion channels. Rapid changes in transmembrane current can result, which can be net inward or outward depending on the ion selectivity, the concentration gradients of the ions, and the electrical gradient. Upon depolarization of the cell, voltage-gated Na⁺ channels and voltage-gated K⁺ channels open and increase the membrane permeability for both ions, with Na⁺ permeability increasing more rapidly than K⁺ permeability (due to the higher sensitivity of Na⁺ channels to voltage changes). With small depolarizations (of a few mV), only a small fraction of the channels will open.

We can expect that most voltage-gated ion channels will have been opened at a holding potential of 0 mV. Once opened, voltage-gated Na⁺ channels undergo a conformational change that leads to their inactivation (in physical terms, a portion of the channel protein moves to plug the channel pore from the inside). This inactivation can only be removed effectively by repolarizing the membrane to levels near the resting membrane potential. This means that maintaining depolarization at 0 mV will lead to a rapid time-dependent inactivation of the voltage-gated Na⁺ channels, such that they would no longer contribute substantially to the I-V curve.

During oscillations around − 60 mV, many voltage-gated channels would be repeatedly opened and closed. Assuming that the magnitude of the hyperpolarizing phase removed the inactivation of the voltage-gated Na⁺ channels, these would also be repeatedly opened, inactivated, and then revert to a closed state from which they could again be opened. However, voltage-gated Na⁺ channels can also be inactivated before opening in response to a slow depolarization (so called closed-state inactivation)²⁸. Therefore, even oscillating depolarizations of 12 or 16 mV at a holding potential of −60 mV, particularly at the very low frequencies used in this experiment, would likely lead to substantial and sustained Na⁺ channel inactivation. Thus, at both holding potentials, voltage-gated Na⁺ channels will most likely make only a minor contribution due to their inactivation kinetics.

The results of our simulations using the Hodgkin-Huxley model support this. Diminishing density of Na⁺ channels by steps of a factor of two, and even removing them entirely, had no effect on the presence of a pinched hysteresis loop. This does not mean that Na⁺ channels are not memristors. It just means that with the stimulation parameters we have used they do not contribute, likely due to their inactivation kinetics in response to relatively slow or maintained depolarization.

The memristive behavior might, however, be explained by the physiology of rectifying K⁺ channels. Non-gated tandem pore domain K⁺ channels and inward rectifying K⁺ channels are open at the resting membrane potential and are blocked by intracellular Mg²⁺ and polyamine ions upon depolarization. The voltage-gated delayed rectifying K⁺ channels open in response to a depolarization, returning the membrane potential back to the resting potential during an action potential. Voltage-gated delayed rectifying K⁺ channels show little time-dependent inactivation, as opposed to voltage-gated Na⁺ channels and A-type K⁺ channels²⁹. As they do not inactivate in response to a slow polarization change, they are a more likely candidate to explain the appearance of pinched hysteresis loops. The neuron simulations support this, as the pinched hysteresis loop disappeared when the density of voltage-gated K⁺ channels was reduced to sufficiently low levels.

Various types of K⁺ channels are certainly expressed in the human neurons we recorded from and their potential contribution to the memristive behavior could be investigated in future experiments, both in live neurons (isolating specific ion channel types pharmacologically or genetically) or simulated neurons.

The cell line we used expresses channelrhodopsin-2 as an introduced ion channel. Although channelrhodopsin-2 channels are opened efficiently by an excitation spectrum with a peak at 470 nm, some channel openings and resultant leak currents might arise under the broad-spectrum lighting used for viewing the neurons during recording. However, the cell line used was also polyclonal, thus not all cells expressed channelrhodopsin-2. It is therefore unlikely that this introduced ion channel has contributed significantly to the observed memristive behavior.

More targeted studies comparing excitable cell types (such as neurons) to non-excitable cell types would shed light on the issue of whether memristive properties depend on specific ion channel types. Another approach could involve measurements in cells that only express one or a combination of specific ion channel types, such as transgenic Xenopus oocytes.

Summary and outlook

We recorded pinched hysteresis loops, the fingerprint of memristors, from 10 out of 63 neurons, identifying ion channels as the likely membrane elements underlying this behaviour. Based on the simulations using a digital neuron model, we hypothesize that voltage-gated delayed rectifying potassium channels are candidates for endowing actual neurons with memristive properties. The extent to which these and other ion channels contribute could be assessed in the future by selectively blocking specific ion channels pharmacologically, or by studying genetically modified cells that only express a single channel type. In addition, recording single ion channel currents using outside-out or cell-attached patch clamp would provide better resolution of the memristive mechanism.

Our findings provide the first experimental link between ion channels in neurons and memristive behavior. This contributes a new perspective in cellular biophysics including the possibility to apply new analytical and mathematical tools from the memristor theory or to develop new sensor modalities for diagnostic purposes. The memristor itself is an electrical circuit element and memristor-based neuron models like the one shown in Fig. 1 or others could be realized as an electrical circuit, with potential applications in neuromorphic computing and artificial intelligence.

Methods

Cell culture

Experiments were performed on neurons differentiated from the human fetal ventral mesencephalic neural progenitor cell line hVM1-Bcl-XL-GFP-ChR2-mCh. Cell isolation and immortalization of the hVM1 cell line using v-myc have been described by Villa et al.^30,31. Prior research has shown that neuronal differentiation results in a high proportion of tyrosine hydroxylase-expressing, presumed dopaminergic neurons that exhibit electrophysiological activity³². The expression of the anti-apoptotic protein Bcl-XL (basal cell lymphoma – extra-large) by stable transfection with the vector LTR-Bcl-XL-IRES-rhGFP-LTR resulted in increased dopaminergic properties^33,34,35. An optogenetically modified polyclonal cell line expressing Channelrhodopsin-2 under control of the human synapsin 1 gene promoter was created by lentiviral transfection with Syn1-ChR-2(H134R)-mCherry-WPRE³⁶. Cells were cultured in cell culture flasks pre-treated with Geltrex™ (ThermoFisher Scientific, Waltham, MA, USA, A1413301) in growth medium (GM) saturated with 5% CO₂ at 37 °C.

Basic Medium (BM): Dulbecco’s modified Eagle medium/F-12 medium with Glutamax (ThermoFisher Scientific, Waltham, MA, USA, Cat 31331028) supplemented with 0.5% Albumax I (ThermoFisher Scientific, Waltham, MA, USA, Cat 11020039), 5 mM HEPES (ThermoFisher Scientific, Waltham, MA, USA, Cat 15630056), 0.6% glucose (MerckMillipore, Merck, Darmstadt, Germany, BioReagent, Cat G7021), and 1% penicillin/streptomycin (Sigma, Merck, Darmstadt, Germany, Cat P4333). BM was used for cell passaging.

Growth Medium (GM): BM supplemented with N2 supplement (ThermoFisher Scientific, Waltham, MA, USA, Cat 17502048), non-essential amino acids (Ala, Asn, Asp, Glu, Pro; 40 mM each; MerckMillipore, Merck, Darmstadt, Germany, Cat 101007, 101565, 100126, 100291, 107434), epidermal growth factor (EGF) and basic fibroblast growth factor (bFGF/FGF2), 20 ng/ml each (R&D Systems, Minneapolis, MN, USA, Cat 236-EG-200, 233-FB-025).

Differentiation Medium (DM): EGF and bFGF were withdrawn and the medium was supplemented with 2 ng/ml glial-derived neurotrophic factor (GDNF, Peprotech, ThermoFisher Scientific, Waltham, MA, USA, Cat 450-10) and 1 mM db-cAMP (N⁶,2’-O-Dibutyryladenosine 3′,5′-cyclic monophosphate sodium salt, Sigma, Merck, Darmstadt, Germany, Cat D0627-250MG).

Neuronal differentiation

hVM1-Bcl-XL-GFP-ChR2-mCh cells were seeded onto coverslips in GM at a density of 30,000 cells/cm². 24 h after seeding, denoted as differentiation day 0 (DD0, DD = days of differentiation), all GM was replaced by DM. 48 h after seeding (DD1), all DM was replaced with fresh DM. Subsequently, 2/3 of the medium was replaced with fresh DM every second day (DD3, DD5, DD7, etc.).

Immunocytochemistry

Cells seeded on coverslips were fixed in 4% (w/v) paraformaldehyde for 15 min at room temperature and washed with PBS three times. Samples were collected at different time points during differentiation (DD0, DD5, DD15 and DD25). After fixation, coverslips were stored in PBS at 4 °C until staining. Cells were permeabilized and blocked for 1 h with 10% FBS (vol/vol) and 0.1% Triton X-100 (vol/vol) in PBS at room temperature. Samples were washed twice in PBS and incubated in primary antibodies over night at 4 °C. Antibodies were diluted in dilution buffer with 5% FBS (vol/vol) and 0.05% Triton X‐100 (vol/vol) in PBS. Primary antibodies used in this study were NF-H (neurofilament heavy, 1:4000, chicken polyclonal, Biolegend, San Diego, CA, USA, Cat 822601); MAP-2 (microtubule-associated protein 2, 1:1000, rabbit polyclonal, MerckMillipore, Merck, Darmstadt, Germany, Cat AB5622); GFAP (glial fibrillary acidic protein, 1:1000, rabbit polyclonal, Abcam, Cambridge, UK, Cat ab7260); TH (tyrosine hydroxylase, 1:200, mouse monoclonal, Sigma, Merck, Darmstadt, Germany, Cat T1299); NeuN (Neuronal Nuclear Antigen, 1:500, rabbit monoclonal, Abcam, Cambridge, UK, Cat ab177487); Nestin (1:200, mouse monoclonal, MerckMillipore, Merck, Darmstadt, Germany, Cat MAB5326); and synaptophysin I (SYP, 1:500, mouse monoclonal, Synaptic Systems, Göttingen, Germany, Cat 101 011).

After incubation with primary antibodies, cells were washed with PBS three times for 15 min per washing step. The samples were stained with secondary antibodies, diluted in dilution buffer, for 1 h at room temperature. The following secondary antibodies were used: Alexa Fluor 488 goat anti-rabbit (Life Technologies, ThermoFisher Scientific, Waltham, MA, USA, Cat A11008, 1:200); Alexa Fluor 594 goat anti-chicken (Invitrogen, ThermoFisher Scientific, Waltham, MA, USA, Cat A11042, 1:400) and Alexa Fluor 647 donkey anti-mouse (Jackson Immuno Research, West Grove, PA, USA, Cat 715-605-150, 1:400). Nuclei were counterstained using Hoechst 33,258 diluted 1:500 in PBS for 10 min at room temperature. Samples were washed three times in PBS for 3 min each and mounted on glass slides using gelatin-based mounting medium. Images were acquired using a laser scanning confocal microscope Zeiss LSM 700 (Zeiss, Oberkochen, Germany) and analyzed using FIJI/ImageJ (US National Institutes of Health, Bethesda, MD, USA, version 1.54f).

Whole-cell patch clamp recording

Whole-cell patch clamp recordings were performed on several neurons on any given day. All recordings were performed between DD14 and DD47 (DD = days of differentiation). A total of 24 recording sessions on different days over three rounds of culture and differentiation were conducted. During this time period, the NPCs differentiated into neurons that exhibited long, multiply branched neurites, and as a population expressed TH at levels consistent with the dopaminergic phenotype, as expected from prior studies. Recordings were performed at room temperature, which was in the range of 25–28 °C. A standard patch clamp setup was used. Micropipettes were pulled from borosilicate glass capillaries (1.5 mm outer diameter, Sutter Instruments) to a tip diameter of about 1 micron (electrode resistance about 5–7 MΩ) using a Narishige PC-10 (Narishige Group, Tokyo, Japan) electrode puller. The pipette filling solution contained: 140 mM KMeSO₄, 5 mM NaCl, 0.5 mM CaCl₂, 10 mM HEPES, 2 mM EGTA, 2 mM Mg-ATP and was filtered at 0.22 micron prior to use. Alexa Fluor 488 or Alexa Fluor 567 dye (ThermoFisher Scientific, Waltham, MA, USA) was added to the pipette filling solution to enable intracellular labeling of recorded neurons for post-experiment morphological assessment.

For the patch clamp experiment, DM was replaced with HEPES-buffered artificial cerebrospinal fluid (ACSF): 135 mM NaCl, 2 mM KCl, 2 mM CaCl₂, 1 mM MgCl₂, 1 mM MgSO₄∙7H₂O, 10 mM D-glucose, 10 mM HEPES, pH 7.4, osmolarity adjusted to match the osmolarity of DM, approximately 320 mOsm/l.

Recordings were obtained and pre-processed using a MultiClamp 700B amplifier (Molecular Devices, San José, CA, USA) and WinWCP software (University of Strathclyde, Glasgow, UK). Signals were sampled at different frequencies (in the range of 20.1 Hz to 8333 Hz, see below) with a DigiData 1322 A (Molecular Devices, San José, CA, USA). Data analysis was performed with p-CLAMP 10 (Molecular Devices, San José, CA, USA), OriginLab 8 (OriginLab Corp., Northampton, MA, USA) and Matlab 2021 (MathWorks, Natick, MA, USA).

Cells were patched based on neuronal morphology. Upon achieving a good seal of the patch pipette, conventional electrophysiological characterization was performed under current clamp to confirm that cells had a satisfactory resting membrane potential (range: −55 to −62 mV) and the ability to fire action potentials before proceeding to the low-frequency sinusoidal stimulation. At the end of the experimental manipulation, resting membrane potential was re-assessed, and only experiments in which the resting membrane potential had been maintained within the limits indicated above were included in further analysis.

Recording conditions for memristive measurements

Three different recording conditions were used:

1.
Current clamp with sinusoidal oscillations.
2.
Voltage clamp with sinusoidal oscillations centered at −60 mV (roughly emulating the natural resting potential of neurons).
3.
Voltage clamp with sinusoidal oscillations centered at 0 mV (to obtain measurements with zero DC component according to the memristor theory⁸).

The number of measurements that could be performed on any given neuron were limited by the stability of the electrode-neuron contact and how long the neuron survived the recording procedure. Usually only one of the above recording conditions was implemented per neuron. However, we managed to implement two different recording conditions in 10 neurons and all three recording conditions in one neuron. In these cases, the conditions were implemented in the order shown above.

For all measurements, either current or voltage was applied in a sinusoidal waveform at different frequencies (0.01 Hz, 0.02 Hz, 0.05 Hz, 0.1 Hz, 0.2 Hz, 0.5 Hz, 1 Hz, 2 Hz, and 5 Hz) and amplitudes. For voltage clamp, peak amplitudes were 4 mV, 8 mV, and 12 mV for all excitation frequencies and additional measurements with an amplitude of 16 mV were performed for all frequencies except for 0.05 Hz. Additional recordings with amplitudes of 2 mV, 6 mV, 10 mV, 14 mV, and 18 mV were conducted for the frequencies 0.1 Hz, 0.2 Hz, 0.5 Hz, 1 Hz, 2 Hz, and 5 Hz respectively. For current clamp, peak amplitudes of about 2 pA, 4 pA, and 6 pA were used for all frequencies. In addition, recordings with an amplitude of 8 pA were performed for all frequencies except for 0.05 Hz, and additional amplitudes (10 pA, 12 pA, 14 pA, 16 pA, and 18 pA) were used for the frequency of 0.1 Hz only. Measurements were performed in a series, first at the lowest frequency and at successively higher amplitudes, then the next highest frequency and successively higher amplitudes, and so on.

Depending on the imposed frequency, one of two different time windows (10–100 s) was used for the recordings. This means that all measurements at a frequency of 0.01–0.1 Hz were carried out over 1 cycle, those at a frequency of 0.02 Hz and 0.2 Hz were carried out over two cycles, and those at a frequency of 0.05 Hz and 0.5 Hz were carried out over 5 cycles. All measurements at a frequency of 1 Hz were carried out over 5 cycles, those at a frequency of 2 Hz were carried over 10 cycles and those at a frequency of 5 Hz were carried out over 25 cycles.

In the first series of experiments (November, December 2020), only one cell was measured at a time. In a second series of experiments (March 2021), only the recordings with frequencies of 0.01 Hz, 0.02 Hz, and 0.05 Hz were conducted and some of the voltage clamp recordings were performed in parallel on two different neurons using two pipette electrodes (one with holding potential of 0 V and one with holding potential of −60 mV).

Sampling frequency and data processing

Sampling frequency was not standardized throughout the study. Initially (in the first three recording sessions, encompassing 11 neurons), the sampling frequency differed between recordings using 10 s versus 100 s time windows (170.6 Hz and 20.1 Hz, respectively). Thereafter (15 recording sessions encompassing 72 neurons), all voltage clamp recordings were done with a sampling frequency of 8333 Hz and 4902 Hz for the 10 s and 100 s time windows, respectively, and all current clamp recordings were done with a sampling frequency of around 490 Hz independent of the time window.

Recordings, especially using voltage clamp, were typically noisy (see Fig. 6 and Supplementary Fig. 3), which sometimes made it difficult to identify pinched hysteresis loops. To filter noise and for comparison between measurements, all I-V plots were constructed from a restricted number of samples per cycle (around 100). This was achieved by averaging consecutive samples accordingly. Depending on the sampling frequency f_sa and the oscillation frequency f_si, the number of samples per cycle, N, can be achieved by averaging over a number n_a of consecutive samples of each signal by

$$\:{n}_{a}=\frac{{f}_{sa}}{{f}_{si}\cdot\:N\:}.$$

(1)

In other words, each new set of n_a samples of the raw data was averaged to obtain 1 sample in the plotted data. A comparison between raw and averaged data is shown in Figs. 6 and Supplementary Fig. 3. Signal averaging and plotting were performed with Matlab 2021. Another data processing algorithm that we tested for comparison was digital lowpass filtering in Matlab (command ‘lowpass’ with passband frequency, f_cut, of 20 Hz and sampling frequency of 1000 Hz). Both methods led to similar results (Fig. 6), and we decided to use data averaging for further presentation and visual inspection. The plots of all recordings are available (see data availability section) including the original data, the filtered data, and the averaged data (N = 100) plus additional averaged data (N = 39 and N = 500).

All recordings were visually inspected for the presence of pinched hysteresis loops under different recording conditions. Pinched hysteresis loops were considered to be exhibited by the I-V plots if all the following criteria were fulfilled:

There was a clearly visible pinched point. Since other elements like capacitance and holding potential can affect the measurement, the pinched point may be shifted away from the coordinate origin^14,15.
There was a clear separation in the forward and return trajectories of the loop (as seen for Neuron 5 in Fig. 6a). A pinched hysteresis loop can be asymmetric⁷ (recording from Neuron 1 in Fig. 6b represents this situation).
Pinched hysteresis loops were obtained from multiple (at least 3) measurements from each neuron, and there had to be a clear similarity in the orientation (indicated by green arrows in Figs. 2, 3 and 4 of the pinched hysteresis loops obtained from the different measurements.

Noise occasionally caused random crossings between the upper and the lower branch of the I-V plots (Figs. 2c, 3c and 4c, and Supplementary Fig. 3b) and strict application of the above criteria ensured that such recordings were not falsely classified as showing pinched hysteresis loops.

Neuron simulations in the neuron program

To gain more insight, a simulation was set up in the program Neuron 8.2 (Yale University, New Haven, CT, USA), in which a neuronal membrane was provided with Hodgkin-Huxley voltage-gated Na⁺ and K⁺ channels, and with a resting membrane potential that could be defined for each simulation. The specific membrane capacitance and the density of voltage-gated Na⁺ and K⁺ channels could each be varied over several orders of magnitude. The simulation imposed an oscillating current of a preset magnitude, and generated graphs of the resulting voltage response.

Specific membrane capacitance is not highly variable and in mammalian neurons, it is generally considered to lie in the range of 0.5 to 1 µF/cm², with little variation as a function of protein/lipid content³⁷. The presence of myelin or perineuronal nets (PNNs) can however increase the effective thickness of the capacitive element and thereby decrease the overall specific capacitance^38,39. Since myelin is not relevant for neuronal somata, but PNNs are, it is conceivable that some variation in specific membrane capacitance among neurons combined with variable presence of PNNs could affect the appearance of pinched hysteresis loops. We assume that the same effect could be achieved by varying the total capacitance by changing the membrane area, which is equivalent to changing the size of the neuron. However, the model did not allow us to simulate changes in total membrane capacitance.

Data availability

All voltage and current over time plots and corresponding current-voltage plots of all recorded neurons are available with figshare: [https://figshare.com/s/739e8df9cb6baabeef89](https:/figshare.com/s/739e8df9cb6baabeef89) The collection contains 5 different versions of each recording. 1. Original data without any data processing. 2. Averaged plots in which consecutive data points are averaged so that: 2.1. the plots contain N=100 data points per cycle. 2.2. the plots contain N=39 data points per cycle. 2.3. the plots contain N=500 data points per cycle. 3. Digital low pass filtered plots with fcut =20 Hz.

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Acknowledgements

The hVM1 cell line has been kindly provided by the research group of Prof. Alberto Martínez-Serrano (retired). Requests for the use of the cell line for non-commercial purposes can be sent to Prof. Marta Pereira (m.pereira@uam.es) at the Center of Molecular Biology Severo Ochoa (UAM-CSIC), Universidad Autónoma Madrid, Spain. We acknowledge the assistance of Oleksandr Ievglevskyi in performing patch clamp recordings and of Hua Hu for generating code for simulations in Neuron and discussions of simulation results. We also acknowledge use of instruments and facilities at the Norwegian Core Facility for Human Pluripotent Stem Cells of Oslo University Hospital and the University of Oslo. Fig. 1 was created with BioRender.com.

Funding

This project was funded by the European Union Horizon 2020 Programme (H2020-MSCA-ITN-2016) under the Marie Skłodowska-Curie Innovative Training Network, project Training4CRM and Grant Agreement No. 722779. This work was partially supported by the Research Council of Norway through its Centre of Excellence scheme, project number 262613, and by the South-Eastern Norway Regional Health Authority, project number 2016081. This project was partially funded by the Research Council of Norway/FRINATEK project, CIRCLE (project #332422).

Author information

These authors contributed equally: Oliver Pabst and Christin Schülke.
These authors jointly supervised this work: Joel C. Glover and Ørjan Grøttem Martinsen.

Authors and Affiliations

Department of Physics, University of Oslo, Sem Sælands vei 24, Oslo, 0371, Norway
Oliver Pabst, Christin Schülke & Ørjan Grøttem Martinsen
Hybrid Technology Hub – Centre of Excellence, Institute of Basic Medical Sciences, P.O. Box 1110, Blindern, Oslo, 0317, Norway
Christin Schülke & Stefan Krauss
Intervention Center, Oslo University Hospital, Sognsvannsveien 20, Oslo, 0372, Norway
Ilangko Balasingham
Department of Electronic Systems, Norwegian University of Science and Technology, Trondheim, 7491, Norway
Ilangko Balasingham
Department of Immunology and Transfusion Medicine, Oslo University Hospital, P.O. Box 4950, Oslo, 0424, Norway
Stefan Krauss
Laboratory of Neural Development and Optical Recording (NDEVOR), Division of Physiology, Department of Molecular Medicine, University of Oslo, P.O. Box 1103, Blindern, Oslo, 0317, Norway
Joel Clinton Glover
Norwegian Center for Stem Cell Research, Department of Immunology and Transfusion Medicine, Oslo University Hospital, Sognsvannsveien 20, Oslo, 0372, Norway
Joel Clinton Glover
Department of Clinical and Biomedical Engineering, Oslo University Hospital, Sognsvannsveien 20, Oslo, 0372, Norway
Ørjan Grøttem Martinsen

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Oliver Pabst
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Contributions

O.P. analyzed and interpreted the in vitro experiments and wrote the main parts of the paper. C.S. cultured and provided the neurons, performed and evaluated the immunocytochemical staining and fluorescence microscopy, wrote the manuscript sections devoted to these and parts of the discussion, revised the manuscript and created the graphical abstract. J.C.G. performed the neuron simulations, analyzed and interpreted the results of these, wrote the manuscript sections devoted to these and parts of the discussion, contributed to some analysis of the in vitro experiments and revised the manuscript. Ø.G.M. was the initiator of the project. Ø.G.M. and S.K. revised the manuscript. Cell cultivation was done in the laboratory of S.K. and patch clamp experiments were performed in the laboratory of J.C.G. I.B. provided additional resources for the patch clamp experiments and advice on the experimental plan.

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Correspondence to Oliver Pabst.

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Pabst, O., Schülke, C., Balasingham, I. et al. Recording and modeling of pinched hysteresis loops, the fingerprint of a memristor, in neurons. Sci Rep 15, 37212 (2025). https://doi.org/10.1038/s41598-025-21035-0

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Received: 08 July 2025
Accepted: 18 September 2025
Published: 24 October 2025
Version of record: 24 October 2025
DOI: https://doi.org/10.1038/s41598-025-21035-0