Distortion-Free Sensing of Neural Activity Using Graphene Transistors.

Graphene solution-gated field-effect transistors (g-SGFETs) are promising sensing devices to transduce electrochemical potential signals in an electrolyte bath. However, distortion mechanisms in g-SGFET, which can affect signals of large amplitude or high frequency, have not been evaluated. Here, a detailed characterization and modeling of the harmonic distortion and non-ideal frequency response in g-SGFETs is presented. This accurate description of the input-output relation of the g-SGFETs allows to define the voltage- and frequency-dependent transfer functions, which can be used to correct distortions in the transduced signals. The effect of signal distortion and its subsequent calibration are shown for different types of electrophysiological signals, spanning from large amplitude and low frequency cortical spreading depression events to low amplitude and high frequency action potentials. The thorough description of the distortion mechanisms presented in this article demonstrates that g-SGFETs can be used as distortion-free signal transducers not only for neural sensing, but also for a broader range of applications in which g-SGFET sensors are used.


Introduction
Graphene has attracted much attention for its use in active sensors due to its high carrier mobility [1] and chemical stability. [2] Many applications have been explored, and more are under study, in which graphene is used to transduce the physical property of interest into an electrical signal. Prominent examples include biochemical sensors, [3][4][5] gas sensors, [6,7] pH sensors, [8,9] ion sensors [10] or transducers of electrical potential for neural interfaces. [11][12][13] The latter, has recently attracted increasing attention due to the potential of graphene solution-gated fieldeffect transistors to record infra-slow [14] brain activity with a high spatial resolution. [15] This capability opens the door to full-band neural activity sensing; from large amplitude infra-slow oscillations to small amplitude but faster signals such as local field potentials or action potentials.
In g-SGFETs, the graphene channel is in contact with an electrolyte, which comprises the gate of the g-SGFET (see Figure 1a). The electrochemical potential in the electrolyte can therefore be regarded as the gate-to-source voltage ( ), which couples with the channel through the graphene-electrolyte interface capacitance ( ). [16,17] The electric field at the interface produces a change in the number of charge carriers in the graphene channel and therefore a variation in the drain-to-source current ( − ) for a constant drain-to-source bias ( ). These current changes are proportional to the signal at the gate ( − ) and to the transconductance ( ) (see Figure 1b), which represents the input-output relation of the g-SGFET, also referred to as its transfer function. The in g-SGFETs is, in a first approximation, proportional to the electrical mobility of graphene and per unit area; [18] therefore presenting no dependence on the area of the active channel. This is the main cause for the improved sensitivity of g-SGFETs in the infra-slow frequency band in comparison with conventional microelectrodes. [19] Despite the great potential of g-SGFETs for full-band neural activity monitoring and other applications in sensing, the signal distortion introduced by these transducers remains a key aspect, which has been so far largely underestimated. Signal distortion can appear in any amplifying system due several reasons; including voltage dependent gain, which leads to harmonic distortion, frequency dependent gain that leads to different amplification of various frequency components and phase distortion due to a non-null imaginary part of the gain. In the specific case of the g-SGFET, a voltage or frequency dependence of will lead to signal distortion. In the first place, non-linearities in the − curves, lead to a dependence with and therefore to harmonic distortion of the signal (see Figure 1b and 1c). The amplitude of the harmonics introduced can ideally be derived from the Taylor expansion of the -curves.
This description has been previously presented and evaluated for a GFET analytical model. [20] However, theoretically predicted parameters have never been compared to experimentally measured distortion levels in GFETs. Validating the relation between non-linearities in thecurves and harmonic distortion is critical to understand and mitigate its effect on the sensing capabilities of g-SGFETs. On the other hand, the transfer function or also shows a dependence with the signal frequency ( ), as illustrated in Figure 1c, and a non-null imaginary part, introducing frequency response distortion. A detailed characterization of the frequency response is the first step towards the derivation of a complete analytical transfer function to describe the distortions introduced by a non-ideal frequency response in g-SGFETs.
While harmonic distortion is expected to affect more strongly signals of large amplitude, the frequency response is expected to affect more prominently signals with higher frequency components. Due to the wide range of amplitudes and frequencies found in sensing of full-band neural activity, this application is expected to be particularly sensitive to signal distortion. To show the impact of non-ideal response on the sensing capabilities of g-SGFETs, we have measured full-band electrophysiological activity on the rat cortex and in a cell culture. Figure   1d illustrates the effect of distortion on the different signals measured, from large amplitude and low frequency cortical spreading depression events (Figure 1d-top) to high frequency and low amplitude action potentials (Figure 1d-bottom). In order to recover the equivalent signal at the gate ( − ), the measured − has to be calibrated by using the inverse of the transfer function. If voltage and frequency dependence of the transfer function, as well as its imaginary part are taken into account, the inverse transfer function will correct the distortions in the output signal, leading to the distortion-free input signal. In order to anticipate changes in the transfer function of the g-SGFET under different conditions and prevent defective calibration, the phenomena involved in signal distortion must be properly understood.
The characterization and modelling of the g-SGFET response reported in this work comprises an important step towards the mitigation of signal distortion in g-SGFETs. On the one hand, the theoretical understanding provided, represents a solid basis for the improvement of the linearity and frequency dependence of the g-SGFET response. On the other hand, the calibration procedure presented represents a method for the acquisition of quasi-distortion-free signals using g-SGFET transducers, with especial potential for the sensing of full-band neural activity.

Results and discussion
In this section, the characterization of harmonic distortion and frequency response of g-SGFETs is presented. The measurements have been performed using g-SGFETs on a flexible polyimide substrate (see Experimental Section for details on the fabrication procedure). The devices have been immersed in a phosphate buffered solution (PBS) bath and the gate bias has been applied using an Ag/AgCl reference electrode. The stationary response of the g-SGFETs ( − ) and their dynamic response ( − ) have been characterized under different bias conditions (see Figure 1e). The quasi-DC current and the alternating current signals have been decoupled in order to adapt their dynamic range to fill the full scale of the analog-to-digital converters (see Experimental Section for more details).

Characterization of harmonic distortion
Harmonic distortion, caused by the voltage dependence of the transfer function in g-SGFET is the result of non-linearities in the -curves. Ideally, GFETs present a particularly linear -relation around their conduction minimum, [21] also referred to as the charge neutrality point (CNP). However, there are various factors that can introduce non-linearities in the response of GFETs. [18] For instance, the resistance between the metal contacts and the graphene channel limits the conductance of the transistor, which can eventually result in a curvedrelation away from the CNP. On the other hand, the conductance minimum at the CNP should ideally be perfectly sharp, only limited by thermal fluctuations of charge carriers. However, any inhomogeneous doping of the graphene channel, for instance due to substrate inhomogeneities, can lead to a broadening of the CNP. [22] Harmonic distortion is commonly characterized by applying test signal and measuring the higher order harmonics generated by signal distortion. In Figure 1c, harmonic distortion is represented in the frequency domain where harmonics at frequencies multiple of can be distinguished. In this section, the harmonic distortion introduced by the g-SGFETs is experimentally determined following this methodology, i.e. applying a pure tone signal at the gate ( − ) and measuring the amplitude of higher order harmonics. Besides, harmonic distortion is also calculated from the Taylor expansion of the stationary − curves. The 0 th order term of the Taylor expansion corresponds to the DC current − and the 1 st order term represents the linear conversion of the signal at the gate. The 2 nd and 3 rd order terms of the Taylor expansion represent the 2 nd and 3 rd order harmonics produced by the curvature and changes in the curvature of the vs curves respectively. Comparing the harmonic distortion levels obtained following these two approaches can be used to validate the stationary description of harmonic distortion, which is critical to correct harmonic distortion in a calibration process.
In Figure 2a and 2b this comparison is shown for signals of different amplitudes ( = 2, 4 and 8 ) in a wide range for a frequency of 10Hz. An excellent agreement between the measured and calculated values is shown for the 1 st and 2 nd order harmonics in the whole and range. For the 3 rd order harmonic this validation is only shown for the = 8 due to the contribution from the 1/ noise of the g-SGFET, which masks the low amplitude 3 rd harmonic for small values (see Figure 2b and Supporting Information S1).
Once we have validated the calculation of harmonic distortion by the Taylor expansion of the − curves, it is possible to calculate the signal-to-distortion ratio (SDR) (i.e. the 1 st order harmonic amplitude over the sum of 2 nd and 3 rd order harmonics) for any combination of and bias conditions(see Figure 2c). The SDR is a figure of merit that can be used to evaluate the impact of distortion on the signal quality. The SDR presents a maximum at − ≈ ±100 , where the linearity of the -curve is maximal. The SDR exhibits a minimum close to the CNP due to the abrupt drop of transconductance and the increase of the 2 nd order harmonic amplitude. Besides, Figure 2c also reveals that the SDR increases with decreasing values. This dependence is explained by the linear, quadratic and cubic relationship of the 1 st , 2 nd and 3 rd order harmonics, respectively with (see Supporting Information S1).
Another useful way to represent the impact of the harmonic distortion on the signal quality is to compare it to the intrinsic 1/ noise of the g-SGFET, which is usually considered to limit the signal quality in g-SGFETs. [13,23] Figure 2d shows the distortion-to-noise ratio (DNR) for the noise integrated in a one-decade frequency band (i.e. 4-40Hz). The DNR presents its minimum at − ≈ ±100 , its maximum at the CNP and it decreases, as expected, with . The 0dB contour line indicates the parameters for which distortion and noise will contribute with the same power to the degradation of signal quality (see Supporting Information S1 for more details). These results highlight the importance of choosing a right gate bias to minimize the impact of harmonic distortion on the signal quality, especially for large amplitude signals.

Characterization and modelling of the frequency response
In addition to harmonic distortion, non-idealities in the frequency response can also limit the recording capabilities of g-SGFETs. Deviations from a frequency independent transconductance might appear due to for instance a slow charging of the graphene-electrolyte interface capacitance or a slow filling and depletion of trap states in the graphene environment. [24,25] In addition, displacement currents through parasitic capacitances at the graphene-electrolyte interface can also result in the attenuation of the signal. [26] Here, we have characterized the frequency dependence of the magnitude and phase of the g-SGFETs transfer function. The aim of this section is to propose and validate an equivalent circuit and its analytic transfer function, which fully describes the frequency response of the g-SGFETs in a wide range of signal frequencies and under different experimental conditions. Having such a complete model, a calibration method could be proposed to correct signal distortions introduced by the frequency response of g-SGFETs.
For this purpose, the response of g-SGFETs to input signals ranging from 0.4 to 200 has been measured (see Experimental Section for more details). This characterization has been performed for different electrolyte concentrations in order to show the effect that changing the electrolyte (e.g. to physiological medium) could cause on the frequency response. Figure 3a shows the experimentally obtained module and phase of the transfer function (Bode diagram) compared with the extracted from the stationary -curves. The Bode diagrams in Having identified these three regimes of frequency response, it is now possible to propose a small signal model of the g-SGFET (see Figure 3b and 3c). In the proposed equivalent circuit, the graphene-electrolyte interface is not modelled as an ideal capacitor, as done in standard models, [18] but instead as a constant phase element ( − ) which is commonly required to properly describe the response of a double-layer capacitance. [27,28] Although CPEs are often used in electrochemical impedance modelling, its physical origin is still a topic under discussion. [29,30] In the particular case of the graphene-electrolyte interface it has been previously reported in impedance measurements [28] . The equivalent circuit in Figure The first factor on the right hand side of Equation (1) corresponds to the intrinsic transconductance, and it presents a frequency dependence that results from the non-ideal capacitive behavior of the interface, [31] i.e. / | = − ( ) (see Supporting Information S2 for more details). The second factor in Equation (1) results from the contribution of the resistance in series ( ) with the g-SGFET (i.e. plus ) and in parallel with the transistor, which are responsible for the attenuation of the signal at the interface with frequency [18] (see Supporting Information S2 and S3). The last term in Equation (1) has to be included to account for the measured increase of the transconductance at high frequencies.
When a signal is applied at the gate, the − and elements represent a direct current path between gate and the g-SGFET contacts. The impedance of these CPE elements characterizes this capacitive current, which adds-up to the drain-to-source current. This contribution is not result of the field-effect coupling, and can be therefore regarded as an effective transconductance ( ) term. Above a certain frequency, this capacitive contribution to the current measured at the drain is expected to dominate the frequency response, as shown in Figure 3a. The frequency dependence of the terms in Equation (1)   the derived transfer function can be used to set a calibration procedure to cancel signal distortions introduced by the non-ideal frequency response of the g-SGFETs.

Calibration of harmonic distortion using the non-linear stationarycurve
We have previously discussed that harmonic distortion can degrade the signal quality significantly, especially for signals with large amplitudes and for not optimum gate bias conditions. Under these conditions, correction of the harmonic distortion by a proper calibration methodology can be critical. The conversion of measured signals in current ( − ) back to an equivalent voltage at the gate ( − ) is commonly done using a constant factor, i.e. the voltage-independent of the GFETs, which is estimated at the bias point of operation.
However, following this approach harmonic distortion is rescaled together with the original signal, but not corrected. In order to correct harmonic distortion we propose to interpolate the measured − into the − curves (see Supporting Information S5).
In order to validate this calibration procedure, which takes into account the voltage dependence of , we have quantified the attenuation of the 2 nd harmonic after calibration. Figure 4a shows the 2 nd harmonic of a pure tone signal applied at the gate after calibration using a constant or by interpolating into the − curves. Using the interpolation method, we can obtain a very effective attenuation of the distortion; confirmed by the observation that the In order to show the impact of harmonic distortion on relevant electrophysiologic data, CSD events have been recorded in a headfixed rat model (see Experimental Section and reference [15]). These signals are specially prone to harmonic distortion because of their large amplitude.

Calibration of LFP activity and APs using the inverse transfer function
The dependence of on the non-ideal double layer capacitance of the graphene-electrolyte interface makes the response of the g-SGFETs to change with frequency even at low frequencies. Small deviations from ideality can cause relatively large changes in the amplitude and phase. Signals showing a wide band activity such as local field potentials, with frequency components ranging from ~1 to ~200 , [32,33] will present a more pronounced distortion of the signal shape produced by different amplification of various frequency components. In addition, phase distortion might cause defective phase determination in LFPs. [19] To show the impact of non-ideal frequency response on actual physiologic data, LFPs have been recorded in-vivo on the rat cortex using epicortical g-SGFET flexible probes (see Experimental Section) and action potentials have been recorded from a HL-1 cell culture on a g-SGFET array fabricated on a SiO2 rigid substrate (see Experimental Section).
In order to calibrate the electrophysiological data, the frequency response of the g-SGFETs under in-vivo conditions and in the culture medium must be known. Figure 4c shows The impact of such distortions can be shown by calibrating the recorded signals using the inverse of the fitted ( ). Figure 4d shows a LFP signal in the (1-10Hz frequency band) calibrated using a frequency independent and using the frequency dependent transfer function of the g-SGFETs in the in-vivo conditions. The non-ideal frequency response causes changes in the signal amplitude as well as phase shifts of ~10 degrees. Similarly, Figure 4e shows an average of 15 action potentials recorded in an in-vitro cardiomyocyte culture. In the case of the APs, neglecting the effect of frequency response leads to a significant error in the amplitude determination (an understimation of ~30% of the actual amplitude) due to the high frequency of these signals.
Finally, loss at high frequencies will cause loss of signal-to-noise ratio that cannot be corrected by calibration. Figure 4f shows the effect of loss on the power spectral density (PSD) of the calibrated noise ( ). The understimation in the noise power made by using a constant for calibration can be quantified by integrating the PSD in different frequency bandwidths (see Figure 4f), leading to different values of equivalent noise at the gate ( − ).
These results emphasize the importance of understanding and improving the frequency response of g-SGFETs, especially for sensing high frequency and low amplitude signals. It is also shown the importance of characterizing the frequency response of the g-SGFETs in the environment used for the final application, in our case on the brain cortex, for a proper calibration of the signal.

Conclusion
In summary, this work provides a detailed characterization and modelling of the signal distortion mechanisms in g-SGFETs, describing their impact on the recording capabilities of these devices. The effect of distortion on electrophysiological signals has been evaluated in a wide frequency and amplitude range. It has been shown that non-linearities in the − curves can have a significant impact on the amplitude and shape determination of large amplitude signals, such as spreading depression events. This distortion can be corrected by a calibration procedure which takes into account the voltage dependence of . On the other hand, the non-ideal frequency response of g-SGFETs results in a significant effect on the amplitude determination of LFPs and APs as well as on the phase extraction for LFPs. These distortions can be calibrated using the frequency dependent transfer function presented in this work, following a similar approach as previously proposed for microelectrodes when a frequency independent gain cannot be obtained. [19] In spite of performing a proper calibration of the signal, conventional microelectrodes introduce dramatic SNR loss at infra-slow frequencies. [19] Alternatively, g-SGFETs present a high at low frequencies, allowing to record infra-slow neural activity with a high sensitivity. This high slightly drops as the frequency is increased, therefore producing a progressive loss of SNR.
Eliminating the frequency dependence of is thus a clear strategy towards improving the already outstanding sensitivity of g-SGFETs at high frequencies. The understanding of the g-SGFET response provided in this work sets the ground to their frequency response as well as to mitigate harmonic distortion by operating the devices at optimum bias conditions. Finally, and more importantly, the calibration procedures presented in this work have been shown effective to correct the impact of signal distortion on the recording capabilities of g-SGFETs.
These results demonstrate the potential of g-SGFETs as a highly sensitive, distortion-free platform, not only for full-band neural sensing, but also for a broader range of applications in which g-SGFET sensors are used.

Experimental Section
Fabrication of g-SGFETs: Arrays of g-SGFETs were fabricated on a 10 μm thick polyimide (PI-2611, HD MicroSystems) film spin coated on a Si/SiO2 4" wafer and baked at 350°. A first metal layer (10nm Ti/100nm Au) was deposited by electron-beam vapour and then structured by a lift-off process. Afterwards, the graphene grown by chemical vapour deposition on Cu was transferred (process done by Graphenea s.a.). Graphene was then patterned by oxygen plasma (50 sccm, 300W for 1min) in a reactive ion etching (RIE) after protecting the graphene in the channel region with HIPR 6512 (FujiFilm) positive photoresist.
After the graphene etching, a second metal layer was patterned on the contacts following the same procedure as for the first layer. The lift-off was followed by an annealing in ultra-high vacuum consisting on a temperature ramp from room temperature to 300°C. Subsequently, the transistors were insulated with a 3-µm-thick photodefinable SU-8 epoxy photoresist (SU-8 2005 Microchem), keeping uncovered the active area of the transistors channel and the contacting pads. The polyimide substrate was structured in a reactive ion etching process using a thick AZ9260 positive photoresist (Clariant) layer as an etching mask. The neural probes were then peeled off from the wafer and placed in a zero insertion force connector to be interfaced with our custom electronic instrumentation. Finally, the devices were rinsed for 2 minutes in ethanol to eliminate remaining resist residues on the graphene channel.

Electrical characterization in PBS:
To measure the DC and AC accurately, the electrical current from drain to source is pre-amplified in a first amplification stage with a "low" 10 4 gain. The pre-amplified signal is then low-pass filtered and subtracted from the pre-amplified signal, thus canceling its low frequency (i.e. DC level) components. The resulting signal can therefore be further amplified and low-pass (anti-aliasing) filtered in a second stage with an additional 10 2 gain factor. A custom built circuit was used for the amplification of the current signals, which were digitalized using a NI DAQCard in all characterization procedures in-vitro.

Recording of CSD events in a rat model and calibration of harmonic distortion:
The procedure followed has been thoroughly described previously [15] . The data from one of the rats used in this previous study has been used for evaluating the effect of harmonic distortion on infra-slow CSD signals. The recordings were performed using a custom built data acquisition system with capabilities to switch between the output of the 1 st and the 2 nd amplification stages. The former was used for the acquisition of LFP activity while the latter was used for the determination of the curves needed for calibration.
Characterization of frequency response in-vivo for calibration of LFP: The frequency response of the g-SGFETs has been observed to depend on the ion concentration in the electrolyte. For this reason, the frequency response has to be properly determined in-vivo for subsequent calibration of detected neural activity.
To obtain the frequency response in-vivo, two Pt-Ir wires were implanted at either side of the g-SGFET array. The two Pt-Ir wires were then connected to a current signal generator to perform a bipolar current stimulation at different frequencies. The current signal generator was decoupled from the recording system in order to avoid any current to leak through the g- and after a few minutes' incubation at 37ºC the enzymatic activity was neutralized using twice the volume of culture media. The solution containing the cells was centrifuged at 120 g for 2 minutes and the supernatant was aspired before re-suspending the cells on fresh culture media.
The cells were seeded then on the desired substrate. To prepare them for culture, the g-SGFET were sterilized in 70% ethanol for 15 minutes, rinsed several times with PBS and coated with fibronectin/gelatine for at least 2 hours. Then the fibronectin/gelatine was aspired and the device was rinsed again with PBS. Experiments were performed 48 to 72 hours after subculture.
All chemicals are bought from Sigma Aldrich unless otherwise noted. The action potentials recording was performed using the home-built system schematized in Figure 1e and described before. The signals were digitalized from the second amplification (band-pass filtered) stage.
The transistors were operated at a constant source-drain and gate voltage. Signals were recorded at a sampling frequency of Fs = 20 kHz and digitally band-pass filtered from 55 Hz to 4 kHz after digitalization. The same setup used for the characterisation of the g-SGFETs in a PBS bath was used for the recording of electrophysiological signals from the cell culture. The output from the 1 st amplification stage was used for the determination of the transfer characteristics while the output from the 2 nd stage was used for the recording of recording of action potentials. The frequency response was determined following the same approach as for its characterization in a PBS bath.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.

S1. Comparison of calculated harmonic distortion with experimental data
As mentioned in the introduction of the main text, harmonic distortion introduced by the GFETs can, in principle, be calculated from the Taylor expansion of the − curves (Equation (S1)). The expansion to the 3 rd order can be expressed as follows: where represents the amplitude of the small signal applied at the gate − (see Figure   1b) in the main text. Here the agreement for the other measured signal amplitudes ( = 4 and 2 ) is also shown for a wide range. The noise level (in this case at 2 or frequency of the second harmonic) is displayed in Figure S1a and S1b. Figure S1c shows, the rms noise integrated in the 4-40Hz bandwidth, which was used to calculate the DNR displayed in Figure 2d. Figure   S1d shows the S/(D+N) ratio, a figure of merit that can be defined to show the impact of distortion on the signal quality for a wide − range. The rms noise shown in Figure S1c was also used to calculate the S/(D+N) ratio. c) The integrated noise in the 4-40 Hz is shown. These values were used to calculate the DNR in Figure 2d of the main text and Figure S1d. d) Signal to distortion plus noise in a wide − range.

S2. Derivation of analytic transfer function
The derivation of the analytic transfer function describing the frequency response of the g-SGFETs is summarized in the main text. Here, the frequency dependence of the terms in Equation (1) of the main text are expressed explicitly.
The first factor in Equation (1) ( / | ), referred to as the intrinsic transconductance of the g-SGFET expresses how an electrical potential at the graphene-electrolyte interface couples with the channel conductivity, producing changes in the drain-to-source current. This term is therefore proportional to the capacitance of the interface, which is usually considered independent of the frequency, as expressed in Equation (S2).
where and stand for the width and length of the channel respectively, and µ for the charge carriers electrical mobility. If the interface capacitance does not present an ideal capacitive behavior, but a constant phase element (CPE)-like response, a frequency dependent equivalent capacitance can be derived. Equating the imaginary part of the CPE impedance with the impedance of a pure capacitance the following equivalent capacitance ( ) has been presented previously: [1]  impedance could be dominated by either leakage through the metal contacts accidentally exposed to the electrolyte or through the capacitive coupling of the metal tracks with the electrolyte through the 2 thick passivation layer. A similar explanation to this cut-off frequency has been reported before. [2] This attenuation factor or voltage divider can be therefore expressed as: where the factor of 2 multiplying comes from the definition of − and in the equivalent circuit shown in Figure 3 Table S1: Parameters extracted from the Bode plot fitting in Figure 3a of the main text.

S3. Effect of leakage through metal contacts exposed to the electrolyte
The signal attenuation at high frequencies due to the voltage divider between − and might be affected by the parasitic capacitance from a poor insulation of the metal contacts of the g-SGFETs. In order to determine whether this is the case or the impedance of the grapheneelectrolyte interface dominates the attenuation in the transconductance, two sets of devices, with and without a part of the metal contacts exposed to the electrolyte have been characterized.
The access length ( ), which is defined as the distance between the end of the metalgraphene contacts and the uninsulated graphene channel can be defined as negative (i.e. -2 ) in the first case and positive (1.5 ) in the second. In Figure S2, the frequency response for these two types of device is shown. First, it can be observed that at low frequencies (below 10 ) the CPE behavior dominates the frequency response with a = 0.89 (PBS concentration was equal to 0.5mM). In a second place, it can be observed that in the 3 − 40 band, the attenuation due to the previously described voltage divider dominates the frequency response. In this frequency range, the attenuation is more pronounced for the = −2 due to increased parasitic currents.
It is interesting to note that despite the voluntary exposure of the metal contacts to the electrolyte, the frequency response is not dominated by the drop term in the frequency bandwidth of interest for neural sensing (i.e. < 10 ). The action potentials shown in Figure   4e in the main text were recorded with the devices corresponding to the Bode plot in Figure   S2b

S4. Effect of impurities on the frequency response
It has been previously speculated that impurities on the graphene channel might play a crucial role in the CPE behavior of the graphene-electrolyte interface. [3] This dependence on the surface charge could also explain the dependence of the CPE behavior with the ionic concentration in the environment. Figure S3a shows the frequency response of a macroscopic g-SGFETs where the graphene was not processed after the transfer (i.e. the channel of the g-SGFET consists of the complete graphene layer transferred on the substrate and the passivation was done manually using epoxy). This device shows a flat frequency response, with a phase of −180° (corresponding to the holes conduction regime). The increase of the magnitude and phase for frequencies above 10 can be attributed to the capacitive currents described in the main text, which present a much larger contribution in this case due to the large area of the device (i.e. 5 5 ). The effect of contamination can also be observed from the effect of cleaning with ethanol the finalized devices fabricated using photolithographic means (see Figure S3b). The pronounced loss of magnitude due to contamination is followed by an increased hysteresis in the − curves and a doping of graphene which results in an increased minimum number of charge carriers and displacement of the CNP. The sign of the hysteresis (i.e. increase/decrease of the current for an increasing/degreasing voltage) confirms that it is caused by a slow charging of the interface capacitance. [4,5] a b c Figure S3: a) Bode representation of the macroscopic (unprocessesed) g-SGFET frequency response. b) Magnitude of the over frequency for the processed g-SGFETs before and after cleaning with ethanol (n=4). c) Effect of cleaning with ethanol on the − curves of the g-SGFETs. The filled area in all graphs indicates the standard deviation (n=4).

S5. Correction of harmonic distortion by interpolation into the − curve
The schematic shown in Figure S4a illustrates the interpolation approach used to calibrate the signals. The distorted signal (dashed-red line) when interpolated into the − curve is rescaled and the harmonic distortion simultaneously corrected. The resulting signal is shown by the solid-red line. The effect of calibration on the amplitude of the second harmonic is shown in Figure S4b for two different values. The three vertical lines in Figure S4b indicate the bias conditions for the spectra shown in Figure S4c. In Figure S4c it is possible to see that the increase of amplitude at 2 around the CNP is due to calibrated noise level with small transconductance. The effectivity of the calibration method is also confirmed by showing the SDR (after calibration) on top of the -parameter map from Figure 2c in the main text.
It is shown that the SDR is approximately flat over a wide range and its dependence with is effectively canceled.
The measured CSD events have been calibrated following the interpolation method.
Afterwards, the calibrated signal corresponding to a quasi-distortion free signal has been artificially distorted for different values to show the effect of bias on the signal distortion (see Figure S4e). It is possible to observe a transition from a signal amplitude underestimation (for far away from the CNP) to a signal amplitude overestimation (close to the CNP). Figure   S4f shows The high SDR ratio close to the CNP appears due to an increase in the equivalent noise at the gate rather than an increase of higher orders. d) The SDR shown in Figure 2d is now plotted with the SDR after calibration with the non-linear − curves. e) Effect of harmonic distortion on the recording of CSDs for different bias conditions.
f) The error in the amplitude of the CSD due to harmonic distortion is shown for different values. The colored dots indicate the bias conditions corresponding to the CSD curves in part e).

S6. Limits in the static model of harmonic distortion
The slow charging of the graphene-electrolyte interface capacitance leads to a hysteretic response of the − curve as shown in Figure S3c. The impact of the hysteresis on the harmonic signal distortion, which is related to a non-ideal frequency response, can be better evaluated by measuring harmonic distortion at different frequencies.
To evaluate the impact of the non-ideal frequency response on the harmonic distortion, the comparison between measured harmonic distortion and calculated values from the Taylor expansion of the − curve has been performed for a 1 pure tone. Figure S5a shows the result of this comparison; it is possible to observe a discrepancy between the DC and AC-derived values. This deviation for the 1 st harmonic (corresponding to the applied signal) is proportional to the drop observed in the magnitude of the frequency dependent transfer function (∆ shown in Figure S5a and S5b). Interestingly, the 2 nd harmonic and the 3 rd harmonic drop by the same factor (∆ ) squared and cubed respectively. This attenuation corresponds to divide the signal at the interface ( ) by a factor ∆ . The close match between the actual distortion and the values extracted from the DC characteristics after correcting the signal amplitude indicates that the hysteretic response does not change significantly the harmonic distortion ( Figure S5c), but only the effective signal at the gate .
This deviation from the stationary response, however, has a small impact for the calibration of

S7. Numerical implementation of the analytic transfer function
To implement the transfer function of fractional order it has to be approximated as a higher (and integer) order system which mantains a constant phase over a certain bandwidth. [6] Figure S6 shows the approximation of the transfer function using Oustaloup's method which synthetizes the TF by the recursive distribution of (2 + 1) zeros and poles. [7] This method gives an accurate description of the non-integer order TF within a defined bandwidth [ , ℎ ] (see Figure S6). In addition, it has the advantage over other modelling tools that its digital implementation for real-time applications is given in a straight forward manner from its finite number of zeros and poles.