Excerpt from the Proceedings of the COMSOL Conference 2010 Paris

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1 Excerpt fro the Proceedings of the COMSOL Conference 21 Paris Modeling of Retinal Electrical Stiulation Using a Micro Electrode Array Coupled with the Gouy-Chapan Electrical Double Layer Model to Investigate Stiulation Efficiency F.Dupont,1, R.Scapolan 1, J-F.Bêche 1, C.Condeine 1, M.Belleville 1, P.Pha 1 1 CEA, LETI, Minatec, Grenoble, France Corresponding author: 17 rue des artyrs 3854 Grenoble Cedex, florent.dupont@cea.fr Abstract: Electrical stiulation for retinal iplant has known significant iproveents in the last decades with any iplantations and experientations. The ability to create better controlled and better adapted stiulation signals to increase efficiency is a ajor objective. In this paper, we present a odel describing cell excitability to electrical stiuli and its nuerical fraework, based on COMSOL Multiphysics 3.5a. The atheatical odel is based on the AC Electrokinetic Coplex Equation coupled to the Gouy-Chapan odel of the Electrical Double Layer (EDL). The EDL is fored at the interface between the electrode surface and the extracellular ediu. In order to validate those atheatical odels, we copared nuerical siulations of retinal nine-eleent Platinu Micro Electrode Array (MEA) with ipedance spectroscopy easureents for sinusoidal low aplitude stiuli. Thanks to the positive outcoe of the coparison, we were able to start the evaluation of different and ore coplex stiuli wavefors using in particular the Fast Fourier Transfor algorith (FFT). The retinal MEA used during the easureent capaign was anufactured at CEA. Keywords: Electrical Siulation, Retinal Ganglion Cell, Electrical Double Layer, Finite Eleent Method, COMSOL 1 Introduction Blindness issues due to Retinis Pigentosa (RP) and Age Related Macular Degeneration (AMD) are diseases affecting the retina and leading to loss of vision. Cures of different types (drugs, surgery, etc.) are being investigated and several attepts at using electrical stiulation have already been ade [1]. Most of the studied cases have seen about 3% of the Retinal Ganglion Cells (RGC) reain functional even years after RP or AMD screening. Retinal prostheses are generally ade of an extraocular part recording visual iages and of an intraocular part providing electrical stiuli to excitable RGC. The inner device generally consists in an integrated circuit delivering electrical signals to an MEA. Besides daage associated to echanical shocks, any studies have shown that electrical stiulation patterns can lead to peranent daage caused by irreversible electrocheical reactions. Thus, biphasic charge-balanced wavefors are preferred to onophasic wavefors when stiulation is applied to single electrodes. Daage caused by electrical stiulation is related to the stiulation pulse s aplitude and frequency, but ore iportant, to the charge density and the charge per phase [2]. Those findings have a direct ipact on the electrode design and on the electrical wavefor used for the stiulation. Likewise, fro an electrical point of view, efficiency of a stiulation syste is also related to the design of the electrodes (pattern and individual diensions) and the shape of the applied wavefors. Optiization of any electrodewavefor syste can be conducted through odeling incorporating eleents describing the biological response. Soe geoetry aspects have already been investigated with nuerical siulation of different brain electrode types using FEM odeling [3]. This study deals with the design diensions for 2D/3D electrodes to ake stiulations ore efficient and safer. Cantrell & al [4] defined different stiulation wavefors based on well known shapes such as Gaussian and sinus. They copared their efficiencies with the epirical square wavefor ostly used by clinicians. Theses coparisons are ade in order to show an iproveent on the aount of charges needed to elicit tissue responses without daaging tissue. Other approaches consisting in an iproveent of efficiency by taking into account the spatio-teporal aspect of stiuli are also being investigated: Valente & al. [5] explained that soe phased arrays stiuli could

2 odify the tissue response behavior. This odeling study is only applied to Deep Brain Stiulation (DBS). Those iproveents are leading to new ways in designing stiulators and prosthetic devices. The previously entioned studies [3-5] are exaples showing any ways of studying the electrical stiulation in living tissues and yet presenting odeling as the starting point. Our objective is to propose a nuerical fraework which will allow us to optiize our electrodes geoetries and the electrical wavefor in order to design ore efficient and safe retinal prosthesis. In this paper, we present the atheatical odels chosen for representing the living atter and the ipedance easureents done to assert the validity of those choices. 2 Ruling equations: Hodgkin-Huxley and AC Coplex Electrokinetic The odel has to represent four contributions: (i) the applied stiulus, (ii) the EDL fored at the electrode surface, (iii) the bulk part of the extracellular ediu and the RGC seen as real dielectrics, (iv) the RGC specific electrical activity. Applied stiuli are non sinusoidal. We use the FFT approach and decopose this teporal signal into a su of sinusoids (see section 3). 2.1 Modeling of cells and extracellular ediu bulk RGC and the bulk part of the extracellular ediu are real dielectrics of conductivity and perittivity. For sinusoidal stiuli and under the linear assuption, the electrical potential distribution is described by the electrokinetic coplex equation [6]:.( V ) = σ (1) Whereσ is the coplex conductivity and V the agnitude of the coplex electrical potential. Associated boundaries conditions (BCs) are of euann hoogeneous type to represent insulated boundaries: σ V. n = (2) n r is the outer noral vector at insulated boundary. MEA contacts polarization induces a capacitive behavior of the electrode-electrolyte interface. Counter-ions of the electrolyte are attracted to the charged electrode and create a capacitive Electrical Double Layer (EDL). Considered as a thin layer approxiation due to its very sall wih (Debye length λ is about d 8n for the extracellular ediu), the EDL is odeled with the following Robin boundary condition [6]: σ V. n = iω C GC V (3) Where V is the potential drop across the EDL and C GC the Gouy-Chapan EDL capacitance: ε eau C GC = (4) λd For the particular case of a saline solution without the presence of retinal tissues (in vitro condition), equation (1) coupled to BC (3) should behave like an electrical RC series circuit. In this study, we first use the GC EDL linear odel to represent the EDL capacitance. Many ore realistic non linear odels for the EDL are available in literature: the replaceent of C GC in BC (3) is possible if needed [7]. 2.2 Modeling of RGC electrical activity eural inforation transfer in the inner retina is perfored by Action Potentials (AP), also referred to as spikes. An AP is a shortlasting event in which the electrical transebrane potential rapidly rises and falls, following a stereotyped wavefor. APs occur in several types of anial excitable cells including RGCs (neuron type). Electrical stiulation of excitable cells either increases the AP firing rate (increase of activity) or inhibits it. Hodgkin Huxley (HH) equations describe how APs in neurons are initiated and propagated. This set of nonlinear Ordinary Differential Equations (ODE) approxiates the electrical characteristics of neurons [8]. A diffusion ter was added to take account for the AP propagation along the axon: 2 a V 2 2ρ x i = C dv + g a 3 h( V E 4 + g n ( V E ) + g ( V E ) (5) K K L L a )

3 d dh dn = α ( 1 ) (6) β = α ( 1 h) h (7) h β h = α ( 1 n) n (8) n β n inforation of the sinusoid of frequency. The odule versus frequency F sapling plot is displayed in figure 1 for various signal wavefors. Those selected wavefors are typical stiulation eleentary patterns. 166 haronics Where I, V and C are respectively transebrane current, voltage and capacitance. α and k β are opening and closing tie k constants of ebrane ionic channels and g are k conductivities associated to those channels. As a first step, RGC activity can be odeled by those HH equations even if soe authors proposed ore refined odels [8]. 3 The Fast Fourier Transfor for non sinusoidal stiuli To be able to use the coplex foralis of the electrokinetic equation (1), only sinusoidal signals can be used as stiuli. However, ost of the epirical stiuli used are periodic but non sinusoidal. A possible way to accoodate this liitation is to decopose a discrete teporal signal into a finite su of sinusoids with discrete frequencies. The coputationally efficient FFT is a widely used algorith to calculate Discrete Fourier Transfor (DFT). In order to benefit fro the FFT efficiency, the nuber of saples () ust be a power of two. This transfor is the link between the discrete teporal and the discrete frequency signals representation. The whole FFT algorith is largely and exhaustively explained in the original text [9]. Any T periodic signal x(t) with a copletely arbitrary shape can be sapled respecting the Shannon theore T sapling T 2 to give: x[ n] = x( ntsapling ), n [, 1] (9) If (2), it is possible to add zeros at the end of the sapled signal x[n] to coplete the sequence without altering the results [9]. X[ ] = FFT[ x] = 1 n= x[ n] e 2πjn (1) The Fourier coefficients X[] are coplex values that contain both phase and aplitude Figure 1. Different types of FFT input wavefors with a display of the spectral inforation obtained after algorith processing (One-sided spectru) This figure ephasizes the fact that a different wavefor eans a different spectral content. Thus the nuber of haronics is varying fro 379 (biphasic square signal) to (asyetric biphasic square signal). The FFT is a reversible transfor allowing recreation of x[n] fro X[] through the FFT -1 transfor: 1 2πjn 1 1 x[ n] = FFT [ X ] = X[ ] e (11) = 379 haronics haronics

4 Finally, there are two essential points of interest in the use of the FFT algorith for us: (i) The decoposition in sinusoidal eleents is notably atching with the underlying hypothesis when using the coplex electrokinetic equation. In fact, those sinusoidal eleents are eigen functions of the linear filter represented by GC Capacitance and the bulk resistance which eans that the output will preserve inputs sinusoidal characteristics. (ii) The enabled reverse transfor leads to the possibility to decopose and recopose every arbitrary teporal wavefor. 4 Use of COMSOL Multiphysics 4.1 COMSOL Multiphysics interface The AC Electrokinetic Coplex Equation (1) coupled to the GC capacitance EDL odel was ipleented in the Partial Differencial Equations (PDE) odule. The equation is solved using a stationary direct linear solver and the solution is extracted as an M-file. The language description of the FEM solution allows the paraeter sweep such as the signal frequency and aplitude. The insertion of those varying paraeters in a COMSOL script brings flexibility to siulations and post-processing. 4.2 MATLAB It is therefore possible to ipleent the following ethod starting with the M-file extracted COMSOL script in order to provide its inputs and to process its outputs. The entire script is written and ran as a MATLAB file: Creation of the signal -points FFT of the signal (equation 1) Loop on positive frequencies (/2+1) Solve in COMSOL Saving of strategic values Switch one-sided to two-sided spectru _points ifft of the signal (equation 11) Post-processing Multiple fields can be asserted in the script such as geoetry diensions, boundaries esh or post processing values. The recording of several points of the current and voltage wavefor can be perfored using the postint, posteval and 'postcrossplot functions. Finally, the switch to two-sided spectru and the FFT -1 function use ( ifft ) lead to teporal signals reconstruction and display. 5 Validation of nuerical siulations with ipedance spectroscopy The nuerical odel was solved for a Micro Electrode Array (MEA) ade of nine Platinu contacts (Ø=4µ), nine counter electrodes (CE, wih=1µ) and a general counter electrode (GCE, wih=2µ). This MEA was developed and anufactured at CEA. MEA Contacts MEA General CE MEA CE Figure 2. Picture of the MEA-RETIE showing the nine platinu circular contacts and the counterelectrodes. Ipedance easureents were first done using the GCE as the working electrode. A illietric Platinu circular electrode, iersed in the electrolyte, was used as the CE. The test setup used for the ipedance easureents was coposed of a VMP-2 potentiostat (Bio-logic copany [1]), electronic switch array to select both the MEA contacts and the CEs. The MEA bathes into a Phosphate Buffered Solution (PBS) solution. The PBS is a water-based solution containing several types of ions such as sodiu, potassiu and chloride. The osolarity and the ion concentrations of PBS usually atch those of the huan body. The conductivity of PBS was easured at 1.6 S/ at roo teperature (25 C). The potentiostat was used in Potentio Electrocheical Ipedance Spectroscopy (PEIS) ode. This ode executes a frequency sweep at the desired threshold and displays the ipedance response. The working electrode (WE) can be either a contact, or a CE, or the GCE or any electrode present in the electrolyte.

5 A a coplex forulation showed that it is just a first-order approxiation [6]. Finally, it ust be said that the potensiostat VMP-2 upper frequency liit is 1 MHz so we assue that the phase drop is ostly due to the setup liits (unreliable above.5mhz). 6 Liitations of the fraework 6.1 Spectru Magnitude Threshold B Figure 3. Ipedance Bode plots for 2V peak to peak (red), coparison to nuerical results (blue). The GCE is used as the working electrode and the CE is a circular illietric electrode iersed in PBS. A/ ipedance odule. B/ ipedance phase. As already said, the electrode-electrolyte syste is assiilated to a RC series circuit. The R resistance corresponds to the high-frequency plateau. The odule plot (A) shows that easured bulk resistance is underestiated by nuerical siulations. At low frequency, the ain effect is due to the capacitance though there is still a isatch between easureents ad calculations. This ight coe fro the WE geoetry which is not identical between siulations and the actual one. Indeed a post study shows that the etching of all contacts and CE was non coplete. As a result, the resistance seen at the interface was underestiated and the capacitance surface was diinished. Additionally, the GC odel assued the EDL capacitance to be constant and its use inside When the one-sided spectru is selected in the script, it is possible to deterine a threshold which will reject the low aplitude peaks fro the solving loop. As noted on figure 1 the wavefor shape is directly influencing the nuber of spectru haronics and in turn the loop iterations nuber. The objective is to optiize the coputation tie without degrading the signal quality. In figure 4, plots A and B show how the signal response quality decreases with the threshold increase. Accuracy validation is the key to properly interpret the future results of our experiental patterns. With the A plot, we estiate that a threshold over 1-5 AU (Arbitrary Units) will contain too any oscillations to give proper results. With the B plot, for a threshold over 1-6 AU, the curve becoes too polluted by unbalanced sinusoids. The haronics aount given on figure 1 was calculated for a null threshold. With threshold of 1-5 AU for both case A and B, the spectral content is reduced fro 379 to 36 haronics for biphasic square signal, fro 166 to 96 for onophasic square signal and fro to 793 haronics for non syetric square signal. The nuber of coputations is respectively reduced of 91%, 94% and 95%.

6 A A B B C Figure 4. Current wavefor spectra according to threshold values. A/ Observation of the wavefor shape for ono and biphasic square signal B/ Observation of the wavefor shape for asyetric biphasic square signal. 6.2 Working and Counter Electrode Mesh In the M-File, iterations can be conducted on the boundary axial eleent size to odify the esh resolution. Total current conservation provides inforation about the esh quality. The objective is to find an acceptable esh threshold to reduce the coputation tie. The central contact and the GCE are respectively used as WE and CE for this calculation siulation. Figure 5. Current spectra according to esh size. A/ Observation of the crossing current at the WE (negative current) B/ Observation at the CE (positive current) C/ Difference to appreciate the right esh. We can verify the high pass filter behavior of the electrode-electrolyte syste. In both A and B, the current begins to significantly cross the electrode-electrolyte interface around 1Hz. A sall unexpected drop is observed above 2kHz, which can be attributed to the nueric odel. As displayed in figure 5, the finer the esh is, the higher the current is (A and B) and

7 the saller the error is (C). The 1-µ esh value (axiu eleent size) sees to be the best coproise providing both high accuracy and reasonable coputation tie. 7 Retinal Ganglion Cell Model Data was collected at key geoetry points to be evaluated during the post-processing phase. Thus the teporal voltage signal was used as a real stiulus and its capability to initiate an AP will be evaluated. A classic Heaviside step function was chosen to evaluate the ipleentation of this teporal odel. easureents. Coputation tie optiization can be achieved by setting thresholds on the signal spectral density (nuber of haronics to be considered) and on the esh resolution. After verifying the accuracy of our procedure with siple rectangular wavefors, the fraework is now ready to accept arbitrary signal wavefors in order to investigate stiulation safety and efficiency. Acknowledgent The authors would like to thank Mr Sadok Gharbi, Mr Stéphane Bonnet and Mr Laurent Gerfault fro the CEA-LETI for helpful discussions and suggestions and Mr Fabien Sauter for the lending of the MEA-RETIE. References Axon s axis Figure 6. Trans-ebrane AP response to a 1-V Heaviside step function stiulus in the iddle of the axon, 1s after stiulation. Figure 6 shows that the 1-µ-radius axon s response to a Heaviside step function stiulus applied in the iddle of the axon. As expected, this artificial stiulation leads to the creation of two identical APs propagating in opposite directions at 1.1.s Conclusion In order to obtain a nuerical fraework to investigate the response of the electrodeelectrolyte-cell syste to arbitrary electrical stiulation wavefors, two odels were ipleented (electrokinetic coplex and HH equations). The FFT algorith was used to decopose the coplex-variable stiulation patterns in frequency-agnitude vectors. The odel s siulation results were copared with ipedance easureents. The anufacturing process of the RETIE-MEA is still aturing and soe probles were pointed out by the discrepancies between our odel and the 1. W. Mokwa, Artificial retinas,coprehensive Microsystes, Oxford: ElsevierScience Eds. Gianchandani, Tabata, Vol.3, D. B. McCreery, W. F. Agnew, T. G. H. Yuen, et al. Charge density and charge per phase as cofactors in neural injury induced by electrical stiulation, IEEE Trans Bioed Eng, (37), M. O. Heuschkel, M. Fejtl, M. Raggenbass, D. Bertrand, P. Renaud, A three diensional ultielectrode array for ulti-site stiulation and recording in acute brain slices, Journal of euroscience Methods, , D. R. Cantrell, J. D. Troy, Extracellular Stiulation of Mouse Retinal Ganglion Cells with on-rectangular Voltage-Controlled Wavefors, Annual International IEEE EMBS Conference V. Valente, A. Deosthenous, R. Bayford, Application of Phased Array Systes to Deep Brain Stiulation, IEEE International Conference Electronics Circuits and Systes P. Pha, R. Scapolan, C. Rubeck and F. Dupont, Search for a Suitable uerical Model for Electrical Stiulation: Fro the Electric Double Layer to Electrokinetics, Confrontation with Ipedance Measureents, subitted proceedings of the European Cosol Conference, Paris France, M. S. Kilic & al, Steric effects in the dynaics of electrolytes at large applied voltages. I. Double-layer charging, Physical Review E, 75, J. Malivuo, R. Plonsey, Bioelectroagnetis, Ed. Oxford University Press, J. W. Cooley, J. W. Tukey, An Algorith for The MachineCoputation of Fourier Series, Matheatics of Coputation, p , Bio-logic copany,