Showing papers on "Electrode array published in 2011"

Flexible, foldable, actively multiplexed, high-density electrode array for mapping brain activity in vivo

Abstract: Arrays of electrodes for recording and stimulating the brain are used throughout clinical medicine and basic neuroscience research, yet are unable to sample large areas of the brain while maintaining high spatial resolution because of the need to individually wire each passive sensor at the electrode-tissue interface. To overcome this constraint, we developed new devices that integrate ultrathin and flexible silicon nanomembrane transistors into the electrode array, enabling new dense arrays of thousands of amplified and multiplexed sensors that are connected using fewer wires. We used this system to record spatial properties of cat brain activity in vivo, including sleep spindles, single-trial visual evoked responses and electrographic seizures. We found that seizures may manifest as recurrent spiral waves that propagate in the neocortex. The developments reported here herald a new generation of diagnostic and therapeutic brain-machine interface devices.

Dual-function transducer for a touch panel

Abstract: A dual-function transducer for a force-sensitive multi-touch screen, including a first substrate, at least one electrode array formed on the first substrate, the at least one electrode array including first and second electrodes, and circuitry operatively coupled to the first and second electrodes. The circuitry is configured to measure first and second electrical properties of the electrode array, the first and second electrical properties corresponding to touch and strain, respectively.

A Power-Efficient Neural Tissue Stimulator With Energy Recovery

Abstract: This paper presents a power-efficient neural stimulator integrated circuit, designed to take advantage of our understanding of iridium-oxide electrode impedance. It efficiently creates a programmable set of voltage supplies directly from a secondary power telemetry coil, then switches the target electrode sequentially through the voltage steps. This sequence of voltages mimics the voltage of the electrode under the constant current drive, resulting in approximately constant current without the voltage drop of the more commonly used linear current source. This method sacrifices some precision, but drastically reduces the series losses seen in traditional current sources and attains power savings of 53%-66% compared to these designs. The proof-of-concept circuit consumes 125 μW per electrode and was fabricated in a 1.5-μm CMOS process, in a die area of 4.76 mm2.

First long term in vivo study on subdurally implanted Micro-ECoG electrodes, manufactured with a novel laser technology

Abstract: A novel computer aided manufacturing (CAM) method for electrocorticography (ECoG) microelectrodes was developed to be able to manufacture small, high density microelectrode arrays based on laser-structuring medical grade silicone rubber and high purity platinum. With this manufacturing process, we plan to target clinical applications, such as presurgical epilepsy monitoring, functional imaging during cerebral tumor resections and brain-computer interface control in paralysed patients, in the near future. This paper describes the manufacturing, implantation and long-term behaviour of such an electrode array. In detail, we implanted 8-channel electrode arrays subdurally over rat cerebral cortex over a period of up to 25 weeks. Our primary objective was to ascertain the electrode’s stability over time, and to analyse the host response in vivo. For this purpose, impedance measurements were carried out at regular intervals over the first 18 weeks of the implantation period. The impedances changed between day 4 and day 7 after implantation, and then remained stable until the end of the implantation period, in accordance with typical behaviour of chronically implanted microelectrodes. A post-mortem histological examination was made to assess the tissue reaction due to the implantation. A mild, chronically granulated inflammation was found in the area of the implant, which was essentially restricted to the leptomeninges. Overall, these findings suggest that the concept of the presented ECoG-electrodes is promising for use in long-term implantations.

A Hermetic Wireless Subretinal Neurostimulator for Vision Prostheses

Abstract: A miniaturized, hermetically encased, wirelessly operated retinal prosthesis has been developed for preclinical studies in the Yucatan minipig, and includes several design improvements over our previously reported device. The prosthesis attaches conformally to the outside of the eye and electrically drives a microfabricated thin-film polyimide array of sputtered iridium oxide film electrodes. This array is implanted into the subretinal space using a customized ab externo surgical technique. The implanted device includes a hermetic titanium case containing a 15-channel stimulator chip and discrete circuit components. Feedthroughs in the case connect the stimulator chip to secondary power and data receiving coils on the eye and to the electrode array under the retina. Long-term in vitro pulse testing of the electrodes projected a lifetime consistent with typical devices in industry. The final assembly was tested in vitro to verify wireless operation of the system in physiological saline using a custom RF transmitter and primary coils. Stimulation pulse strength, duration, and frequency were programmed wirelessly from a Peripheral Component Interconnect eXtensions for Instrumentation (PXI) computer. Operation of the retinal implant has been verified in two pigs for up to five and a half months by detecting stimulus artifacts generated by the implanted device.

Resolution of the Epiretinal Prosthesis is not Limited by Electrode Size

Abstract: Epiretinal prostheses for the blind bypass diseased photosensitive cells in the retina, directly stimulating retinal neurons electrically and evoking signals that are relayed to the brain. Current clinical implants have few electrodes and provide limited visual acuity. Acuity may be improved by identifying electrode array design features and operational details that enhance or interfere with visual percept formation. We labeled all retinal ganglion cells in whole mount retina with a calcium reporter and then measured the number and pattern of cells responding, over a range of electrode diameters and stimulus durations. Span of the response scaled with electrode diameter for electrodes 60 μm and larger. Short stimulation pulse widths selectively activated cells nearest the electrode. Our measurements in the salamander retina suggest that the spatial resolution is 150 μm, which on a human retina is equivalent to 0.55° of human visual field and corresponding Snellen acuity of 20/660. Reading large print could be possible with such a prosthesis.

Dual mode capacitive touch panel

Abstract: A dual mode capacitive touch panel includes a sensor substrate, an electrode layer comprising an array of sensor electrodes arranged over the sensor substrate, the array of sensor electrodes including a plurality of drive electrodes and a plurality of sense electrodes, each sensor electrode corresponding to a location on the sensor substrate, and a shield layer arranged over and spaced apart from the electrode layer. The shield layer includes a predetermined resistance that permits transmission of an electric field at a first frequency and prevents transmission of an electric field at a second frequency, wherein a spacing between the shield layer and the electrode layer is deformable as a result of a force applied to the shield layer due to a user touch, wherein the deformation alters a capacitance between the shield layer and a sensor electrode of the array. A controller is operatively coupled to the array of sensor electrodes, the controller configured to drive the shield layer and at least some sensor electrodes of the sensor electrode array at the first frequency in a first mode to measure a location of an object relative to the sensor substrate, and drive the shield and the at least some sensor electrodes at a second frequency in a second mode different from the first mode to measure a force applied toward the sensor substrate.

Carbon nanofiber electrode array for electrochemical detection of dopamine using fast scan cyclic voltammetry

Abstract: A carbon nanofiber (CNF) electrode array was integrated with the Wireless Instantaneous Neurotransmitter Concentration Sensor System (WINCS) for the detection of dopamine using fast scan cyclic voltammetry (FSCV). Dopamine detection performance by CNF arrays was comparable to that of traditional carbon fiber microelectrodes (CFMs), demonstrating that CNF arrays can be utilized as an alternative carbon electrode for neurochemical monitoring.

Electric crosstalk impairs spatial resolution of multi-electrode arrays in retinal implants

Abstract: Active multi-electrode arrays are used in vision prostheses, including optic nerve cuffs and cortical and retinal implants for stimulation of neural tissue. For retinal implants, arrays with up to 1500 electrodes are used in clinical trials. The ability to convey information with high spatial resolution is critical for these applications. To assess the extent to which spatial resolution is impaired by electric crosstalk, finite-element simulation of electric field distribution in a simplified passive tissue model of the retina is performed. The effects of electrode size, electrode spacing, distance to target cells, and electrode return configuration (monopolar, tripolar, hexagonal) on spatial resolution is investigated in the form of a mathematical model of electric field distribution. Results show that spatial resolution is impaired with increased distance from the electrode array to the target cells. This effect can be partly compensated by non-monopolar electrode configurations and larger electrode diameters, albeit at the expense of lower pixel densities due to larger covering areas by each stimulation electrode. In applications where multi-electrode arrays can be brought into close proximity to target cells, as presumably with epiretinal implants, smaller electrodes in monopolar configuration can provide the highest spatial resolution. However, if the implantation site is further from the target cells, as is the case in suprachoroidal approaches, hexagonally guarded electrode return configurations can convey higher spatial resolution.

Combined electric acoustic stimulation with the PULSARCI(100) implant system using the FLEX(EAS) electrode array.

Abstract: Conclusion: This study demonstrates that electric acoustic stimulation (EAS) using the FLEXEAS electrode is a successful treatment method for preservation and stability of low frequency hearing, and results in significant improvements in speech perception. Objectives: Low frequency hearing preservation and stability following EAS surgery with the MED-EL PULSARCI100 implant using the FLEXEAS electrode and a combined processor in a multicentre setting, and the efficacy and benefits of EAS over time were evaluated. Methods: Eighteen subjects with normal to moderate hearing loss in the low frequencies and severe to profound hearing loss in the high frequencies were implanted with the FLEXEAS electrode. Implantation was performed by different surgeons at three participating centres using either the cochleostomy or the round window approach. Preoperatively, hearing was measured; a battery of speech perception tests and the subjective benefit questionnaire were administered. These tests were subsequentl...

Multiple factors may influence the performance of a visual prosthesis based on intracortical microstimulation: Nonhuman primate behavioural experimentation

Abstract: We hypothesize that a visual prosthesis capable of evoking high-resolution visual perceptions can be produced using high-electrode-count arrays of penetrating microelectrodes implanted into the primary visual cortex of a blind human subject. To explore this hypothesis, and as a prelude to human psychophysical experiments, we have conducted a set of experiments in primary visual cortex (V1) of non-human primates using chronically implanted Utah Electrode Arrays (UEAs). The electrical and recording properties of implanted electrodes, the high-resolution visuotopic organization of V1, and the stimulation levels required to evoke behavioural responses were measured. The impedances of stimulated electrodes were found to drop significantly immediately following stimulation sessions, but these post-stimulation impedances returned to pre-stimulation values by the next experimental session. Two months of periodic microstimulation at currents of up to 96 µA did not impair the mapping of receptive fields from local field potentials or multi-unit activity, or impact behavioural visual thresholds of light stimuli that excited regions of V1 that were implanted with UEAs. These results demonstrate that microstimulation at the levels used did not cause functional impairment of the electrode array or the neural tissue. However, microstimulation with current levels ranging from 18 to 76 µA (46 ± 19 µA, mean ± std) was able to elicit behavioural responses on eight out of 82 systematically stimulated electrodes. We suggest that the ability of microstimulation to evoke phosphenes and elicit a subsequent behavioural response may depend on several factors: the location of the electrode tips within the cortical layers of V1, distance of the electrode tips to neuronal somata, and the inability of nonhuman primates to recognize and respond to a generalized set of evoked percepts.

Detection of word-line leakage in memory arrays: current based approach

Abstract: Techniques and corresponding circuitry are presented for the detection of wordline leakage in a memory array. In an exemplary embodiment, a capacitive voltage divider is used to translate the high voltage drop to low voltage drop that can be compared with a reference voltage to determine the voltage drop due to leakage. An on-chip self calibration method can help assure the accuracy of this technique for detecting leakage limit. In other embodiments, the current drawn by a reference array, where a high voltage is applied to the array with all wordlines non-selected, is compared to the current drawn by an array where the high voltage is applied and one or more selected wordlines. In these current based embodiments, the reference array can be a different array, or the same array as that one selected for testing.

A transparent μECoG array for simultaneous recording and optogenetic stimulation

Abstract: In this paper we report for the first time the design, fabrication and characterization of an optically transparent electrode array for micro-electrocorticography. We present a 49-channel μECoG array with an electrode pitch of 800 μm and a 16-channel linear μECoG array with an electrode pitch of 200 μm. The backing material was Parylene C. Transparent, sputtered indium tin oxide (ITO) was used in conjunction with e-beam evaporated gold to fabricate the electrodes. We provide electrochemical impedance characterization and light transmission data for the fabricated devices.

Addressable electrode array device with IDA electrodes for high-throughput detection

Abstract: An electrochemical device is proposed for high-throughput electrochemical detection that consists of 32 row and 32 column electrodes on a single glass substrate. The row and column electrodes are connected to interdigitated array (IDA) electrodes to form 1024 (32 × 32) addressable sensor points in the device. Electrochemical responses from each of the 1024 sensors were successfully acquired on the device within 1 min using redox cycling at individual IDA electrodes, which ensures application of the device to comprehensive, high-throughput electrochemical detection for enzyme-linked immunosorbent assay (ELISA), reporter gene assay for monitoring gene expressions, and DNA analysis.

Antenna and wireless IC device

Abstract: An antenna and a wireless IC device that includes the antenna are provided for which the manufacturing process is simple and for which there is a low probability of a poor connection occurring between a feeder portion and a radiation electrode. An antenna includes a radiation electrode that is provided on a main surface of an insulator board, a ground electrode and/or a counter electrode that is arranged so as to oppose the radiation electrode, and a magnetic field electrode that is connected to the radiation electrode through a connection portion. The magnetic field electrode is defined by line-shaped electrodes and feeds a signal to the radiation electrode from a feeder portion defined by ends of the line-shaped electrodes through the magnetic field electrode.

Development and evaluation of the modiolar research array – multi-centre collaborative study in human temporal bones

Abstract: ObjectiveMulti-centre collaborative study to develop and refine the design of a prototype thin perimodiolar cochlear implant electrode array and to assess feasibility for use in human subjects. Study DesignMulti-centre temporal bone insertion studies. Materials and MethodsThe modiolar research array (MRA) is a thin pre-curved electrode that is held straight for initial insertion with an external sheath rather than an internal stylet. Between November 2006 and February 2009, six iterations of electrode design were studied in 21 separate insertion studies in which 140 electrode insertions were performed in 85 human temporal bones by 12 surgeons. These studies aimed at addressing four fundamental questions related to the electrode concept, being: (1) Could a sheath result in additional intra-cochlear trauma? (2) Could a sheath accommodate variations in cochlea size and anatomies? (3) Could a sheath be inserted via the round window? and (4) Could a sheath be safely removed once the electrode had been inserted...

Two-phase AC electrothermal fluidic pumping in a coplanar asymmetric electrode array

Abstract: The ability to achieve fast fluid flow yet maintain a relatively low temperature rise is important for AC electrothermal (ACET) micropumping, especially in applications such as bioMEMS and lab-on-a-chip systems. In this paper, we propose a two-phase ACET fluidic micropump using a coplanar asymmetric electrode array. The proposed structure applies a two-phase AC voltage, i.e., voltage of phase 0°/180°, to the narrow electrodes while the wide electrodes are at ground potential. Numerical simulation demonstrates that this simple coplanar electrode configuration can achieve at least 25% faster fluid flow rates than using a single AC signal. By selecting certain design parameters, a two-phase ACET structure can achieve up to 50% faster fluid flow rates than a corresponding single-phase structure. The simple two-phase AC signal sources are easily produced by using inverter buffers, which is a considerable improvement compared to the multi-phase AC signals required by other electrokinetic micropumping methods, such as traveling wave structures.

Three-dimensional electrode arrays for retinal prostheses: modeling, geometry optimization and experimental validation.

Abstract: Three-dimensional electrode geometries were proposed to increase the spatial resolution in retinal prostheses aiming at restoring vision in blind patients. We report here the results from a study in which finite-element modeling was used to design and optimize three-dimensional electrode geometries. Proposed implants exhibit an array of well-like shapes containing stimulating electrodes at their bottom, while the common return grid electrode surrounds each well on the implant top surface. Extending stimulating electrodes and/or the grid return electrode on the walls of the cavities was also considered. The goal of the optimization was to find model parameters that maximize the focalization of electrical stimulation, and therefore the spatial resolution of the electrode array. The results showed that electrode geometries with a well depth of 30 µm yield a tenfold increase in selectivity compared to the planar structures of similar electrode dimensions. Electrode array prototypes were microfabricated and implanted in dystrophic rats to determine if the tissue would behave as hypothesized in the model. Histological examination showed that retinal bipolar cells integrate the electrode well, creating isolated cell clusters. The modeling analysis showed that the stimulation current is confounded within the electrode well, leading to selective electrical stimulation of the individual bipolar cell clusters and thereby to electrode arrays with higher spatial resolution.

Input device with force sensing and haptic response

Abstract: Devices and methods are provided that facilitate improved input device performance. The devices and methods utilize a first electrode disposed on a first substrate, a second electrode coupled to a first side of a piezoelectric material and a third electrode coupled to a second side of the piezoelectric material. The second electrode and the third electrode are configured to facilitate actuation of the piezoelectric material, while the first electrode and the second electrode define at least part of a variable capacitance that facilitates force determination. A spacing element is coupled to the first substrate and defines a spacing between the first electrode and the second electrode. A transmission element is coupled to the third electrode and configured such that a force biasing the transmission element causes the second electrode to deflect relative to the first electrode, thus changing the variable capacitance.

Potential benefits from deeply inserted cochlear implant electrodes.

Abstract: Objectives This review examines evidence for potential benefits of using cochlear implant electrodes that extend into the apical regions of the cochlea. Most cochlear implant systems use electrode arrays that extend 1 to 1.5 turns from the basal cochleostomy, but one manufacturer (MED-EL GmbH) uses an electrode array that is considerably longer. The fundamental rationale for using electrodes extending toward the apex of the cochlea is to provide additional low-pitched auditory percepts and thereby increase the spectral information available to the user. Several experimental long arrays have also been produced by other manufacturers to assess potential benefits of this approach. Design In addition to assessing the effects of deeply inserted electrodes on performance, this review examines several underlying and associated issues, including cochlear anatomy, electrode design, surgical considerations (including insertion trauma), and pitch scaling trials. Where possible, the aim is to draw conclusions regarding the potential from apical electrodes in general, rather than relating to the performance of specific and current devices. Results Imaging studies indicate that currently available electrode arrays rarely extend more than two turns into the cochlea, the mean insertion angle for full insertions of the MED-EL electrodes being about 630°. This is considerably shorter than the total length of the cochlea and more closely approximates the length of the spiral ganglion. Anatomical considerations, and some modelling studies, suggest that fabrication of even longer electrodes is unlikely to provide additional spectral information. The issue of potential benefit from the most apical electrodes, therefore, is whether they are able to selectively stimulate discrete and tonotopically ordered neural populations near the apex of the spiral ganglion, where the ganglion cells are closely grouped. Pitch scaling studies, using the MED-EL and experimental long arrays, suggest that this is achieved in many cases, but that a significant number of individuals show evidence of pitch confusions or reversals among the most apical electrodes, presumably reducing potential performance benefit and presenting challenges for processor programming. Conclusions Benefits in terms of speech recognition and other performance measures are less clear. Several studies have indicated that deactivation of apical electrodes results in poorer speech recognition performance, but these have been mostly acute studies where the subjects have been accustomed to the full complement of electrodes, thus making interpretation difficult. Some chronic studies have suggested that apical electrodes do provide additional performance benefit, but others have shown performance improvement after deactivating some of the apical electrodes. Whether or not deeply inserted electrodes can offer performance benefits, there is evidence that currently available designs tend to produce more intracochlear trauma than shorter arrays, in terms of loss of residual acoustic hearing and reduction of the neural substrate. This may have important long-term consequences for the user. Furthermore, as it is possible that subjects with better low-frequency residual hearing are more likely to benefit from the inclusion of apical electrodes, there may be a potential clinical dilemma as the same subjects are those most likely to benefit from bimodal electroacoustic stimulation, requiring a relatively shallow insertion.

Nanocavity electrode array for recording from electrogenic cells

Abstract: We present a new nanocavity device for highly localized on-chip recordings of action potentials from individual cells in a network. Microelectrode recordings have become the method of choice for recording extracellular action potentials from high density cultures or slices. Nevertheless, interfacing individual cells of a network with high resolution still remains challenging due to an insufficient coupling of the signal to small electrodes, exhibiting diameters below 10 µm. We show that this problem can be overcome by a new type of sensor that features an electrode, which is accessed via a small aperture and a nanosized cavity. Thus, the properties of large electrodes are combined with a high local resolution and a good seal resistance at the interface. Fabrication of the device can be performed with state-of-the-art clean room technology and sacrificial layer etching allowing integration of the devices into sensor arrays. We demonstrate the capability of such an array by recording the propagation of action potentials in a network of cardiomyocyte-like cells.

Device and method for positioning an electrode in tissue

Abstract: A device for positioning an electrode in tissue includes: a lead body having a distal portion; an electrode array coupled to the lead distal portion; an anchoring element having an anchor tip and being operable in a first configuration in which the anchor tip is retracted within the lead and in a second configuration in which the anchor tip is extended outside the lead and configured to fixate within the tissue; and a displacement mechanism that is actuated to bias the electrode array or the anchoring element toward the tissue. A method for positioning an electrode in tissue includes: navigating, to the tissue, a lead with an electrode array, an anchoring element with a distal anchor tip, and a displacement mechanism; biasing the electrode array and anchoring element towards the tissue with the displacement mechanism; and deploying the anchoring element, and verifying fixation of the anchor tip within the tissue.

Solid-state imaging device, method of fabricating solid-state imaging device, method of driving solid-state imaging device, and electronic apparatus

Abstract: A solid-state imaging device includes a substrate, a photoelectric conversion element provided on the light incidence side of the substrate and including a photoelectric conversion film sandwiched between a first electrode provided separately for each of pixels, and a second electrode provided opposite the first electrode, the photoelectric conversion film being made of an organic material or an inorganic material and generating a signal charge according to the quantity of incident light, an amplifier transistor having an amplifier gate electrode connected to the first electrode, and a voltage control circuit that is connected to the second electrode, and supplies a desired voltage to the second electrode.

Nanoscale metal oxide resistive switching element

Abstract: A non-volatile memory device structure. The non-volatile memory device structure comprises a first electrode formed from a first metal material, a resistive switching element overlying the first electrode. The resistive switching element comprises a metal oxide material characterized by one or more oxygen deficient sites. The device includes a second electrode overlying the resistive switching layer, the second electrode being formed from a second metal material. The second electrode is made from a noble metal. The one or more oxygen deficient sites are caused to migrate from one of the first electrode or the second electrode towards the other electrode upon a voltage applied to the first electrode or the second electrode. The device can have a continuous change in resistance upon applying a continuous voltage ramp, suitable for an analog device. Alternatively, the device can have a sharp change in resistance upon applying the continuous voltage ramp, suitable for a digital device.

Electric field driving display device

Abstract: It is an object to provide an electric field driving display device capable of displaying a high quality image and to provide an electric field driving display device in which residual images in an outline of a pixel is prevented from occurring. An insulating film is formed over a second electrode and a plurality of first electrodes are provided over the insulating film. Each of the first electrodes is electrically connected to the second electrode. The second electrode is provided to partly overlap a region between the adjacent two first electrodes. In other words, viewing the top and the bottom of the display device, the adjacent first electrodes are provided apart from each other and the second electrode is provided to embed a space between the adjacent first electrodes.

System and method for performing electrical impedance tomography

Abstract: An electrode array, a system and a method for reconstructing the distribution of electrical properties within a multi-material object. One embodiment includes electrodes arranged along a three-dimensional helical path to provide one or more helical arrays and circuitry to measure signals for calculating a conductivity or admittivity distribution representative of the interior of the structure. Image data may be obtained which is representative of the multi-material region.

Active matrix device and method of driving the same

Abstract: An active matrix electrowetting on dielectric (AM-EWOD) device includes a plurality of array elements configured to manipulate one or more droplets of fluid on an array, each of the array elements including a corresponding array element circuit. Each array element circuit includes a top substrate electrode and a drive electrode between which the one or more droplets may be positioned, with an insulator layer being interposed between the one or more droplets and the drive electrode; and write circuitry configured to write data to the array element. At least some of the array element circuits include measure circuitry configured to detect a pinhole defect in the insulator layer.

Electrode for medical applications, system having an electrode, and method for producing an electrode

Abstract: The invention relates to an electrode for medical applications for neuromodulation and/or nerve stimulation and/or neurological signal detection, which electrode can be compressed and expanded in order to insert same into a hollow organ of a body and is or can be coupled to a current supply. The electrode has a compressible and expandable lattice structure (10) comprising lattice webs (11), which form cells (22), wherein the lattice structure (10) is or can be coupled to the current supply and forms at least one electrically conductive region (12) and at least one electrically insulated region (13).

Electroporation electrode configuration and methods

Abstract: Provided herein are the concept that "singularity-based configuration" electrodes design and method can produce in an ionic substance local high electric fields with low potential differences between electrodes. The singularity-based configuration described here includes: an anode electrode; a cathode electrode; and an insulator disposed between the anode electrode and the cathode electrode. The singularity-based electrode design concept refers to electrodes in which the anode and cathode are adjacent to each other, placed essentially co-planar and are separated by an insulator. The essentially co-planar anode/insulator/cathode configuration bound one surface of the volume of interest and produce desired electric fields locally, i.e., in the vicinity of the interface between the anode and cathode. In an ideal configuration, the interface dimension between the anode and the cathode tends to zero and becomes a point of singularity.