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The Mechanism of Extracellular Stimulation of Nerve Cells on an Electrolyte-Oxide-Semiconductor Capacitor

Department of Membrane and Neurophysics, Max Planck Institute for Biochemistry, Martinsried/Munich, Germany

Correspondence: Address reprint requests to Peter Fromherz, Tel.: 49-89-8578-2820; E-mail: fromherz{at}biochem.mpg.de.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS TWO-DOMAIN-STIMULATION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Extracellular excitation of neurons is applied in studies of cultured networks and brain tissue, as well as in neuroprosthetics. We elucidate its mechanism in an electrophysiological approach by comparing voltage-clamp and current-clamp recordings of individual neurons on an insulated planar electrode. Noninvasive stimulation of neurons from pedal ganglia of Lymnaea stagnalis is achieved by defined voltage ramps applied to an electrolyte/HfO₂/silicon capacitor. Effects on the smaller attached cell membrane and the larger free membrane are distinguished in a two-domain-stimulation model. Under current-clamp, we study the polarization that is induced for closed ion channels. Under voltage-clamp, we determine the capacitive gating of ion channels in the attached membrane by falling voltage ramps and for comparison also the gating of all channels by conventional variation of the intracellular voltage. Neuronal excitation is elicited under current-clamp by two mechanisms: Rising voltage ramps depolarize the free membrane such that an action potential is triggered. Falling voltage ramps depolarize the attached membrane such that local ion currents are activated that depolarize the free membrane and trigger an action potential. The electrophysiological analysis of extracellular stimulation in the simple model system is a basis for its systematic optimization in neuronal networks and brain tissue.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS TWO-DOMAIN-STIMULATION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Extracellular stimulation of nerve cells in a tissue is a classical technique in brain research and a fundamental tool in neuroprosthetics (1–3). Electrical current that originates at an electrode creates a gradient of extracellular electrical potential such that an action potential may be elicited. Possible depolarizing and hyperpolarizing effects on different parts of oriented nerve cells with dendrite, soma, and axon were pointed out (4), and were simulated in a theoretical model (5). In recent years, particular attention was given to the extracellular stimulation of cultured neurons on planar metallic electrodes (6–9). Questions considered were: Which part of a neuron is responsible for excitation? What is the role of anodic and cathodic current on attached and nonattached parts of the membrane? What is the contribution of capacitive and of Faradayic current? Are defined currents or defined voltages better to control stimulation?

In a tissue as well as in a dish with cultured neurons, the geometry of numerous cells with a meshwork of dendrites and axons is rather involved such that an electrophysiological characterization of extracellular stimulation is not possible. Furthermore, with common metal electrodes it is difficult to avoid Faradayic currents that give rise to toxic electrochemical reactions. For these reasons, we present here an electrophysiological study of extracellular stimulation in a model system under simple and well-defined conditions:

1. We use an insulated planar electrode without Faradayic current.
   
2. We study neurons that are attached to the electrode without branched dendrites and axons.
   
3. We apply stimulation by defined voltages with rising ramps as well as falling ramps to provide anodic and cathodic stimulation.
   
4. We compare extracellular stimulation in a whole-cell patch-clamp configuration for voltage-clamp and for current-clamp.
   

The cell-electrode geometry is sketched in Fig. 1 A. The attached domain of the plasma membrane is separated from the insulated electrode by a thin layer of electrolyte whereas the free domain is in contact to the bath. Capacitive current across the electrode/electrolyte interface gives rise to a profile of extracellular voltage in the area of adhesion and to a change of the intracellular voltage with respect to the bath on ground potential. A patch-pipette is used to record the response of the neuron—the membrane current under voltage-clamp and the membrane voltage under current-clamp. Under these conditions, we are able to elucidate the effect of extracellular stimulation on the voltage-gated ion channels in the attached as well as in the free domain of the membrane and also the role of gating in either domain for the elicitation of an action potential.

View larger version (17K): [in this window] [in a new window]   FIGURE 1  Nerve cell on a capacitor and two-domain-stimulation (TDS) model. (A) Schematic view, not to scale. The cell diameter is 50 µm, the width of the cleft between cell and chip is 20 nm. Two domains of the plasma membrane are distinguished, an attached domain with area AJ (shaded) in contact to the capacitor and a free domain with area AM–AJ in contact to the bath. A voltage ramp VS(t) applied to the substrate induces an extracellular voltage VJ in the cell-capacitor junction. The resulting change of the intracellular voltage VM and the membrane current IM are measured or controlled with a pipette. (B) Equivalent circuit of TDS model. Substrate, cell-chip contact, membrane in the junction (JM) and free membrane (FM) are characterized by the area-specific parameters of substrate capacitance cS, of membrane capacitance cM, of seal conductance gJ, and of ionic conductances and (reversal voltages ). The parameters of the junction are weighted by the area AJ and the parameters of the free membrane by the area AM–AJ as indicated.

This article is organized as follows. After a section on Materials and Methods, we summarize basic relations of capacitive stimulation under voltage-clamp and current-clamp in terms of a two-domain-stimulation (TDS) model. That chapter serves as a basis for a planning of the experiments and for their interpretation. Then we describe the capacitive polarization of neurons under current-clamp for closed ion channels. Under voltage-clamp, the gating of ion channels in the attached membrane is considered as it is induced by falling voltage ramps. For comparison, the gating of all channels is characterized by variation of the intracellular voltage. Finally, action potentials are elicited under current-clamp by applying falling as well as rising voltage ramps. The mechanism of stimulation is interpreted in terms of the results of capacitive polarization under current-clamp and of the ionic currents under voltage-clamp.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS TWO-DOMAIN-STIMULATION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Chips
We use electrolyte/oxide/silicon capacitors as described in a previous study (10). Quadratic 4 x 4 mm chips were made of p⁺-doped silicon (0.01 cm) with a 1-µm-thick SiO₂ field oxide. Circular capacitors with a diameter of 250 µm are obtained by etching and insulation with 10 nm HfO₂ by atomic layer deposition. The area-specific capacitance in electrolyte is c_S = 1.22 µF/cm² in the accumulation region of silicon beyond a voltage of V_s = 2 V between Si and Ag/AgCl without leakage current up to V_s = 4.5 V. An aluminum layer (200 nm) is evaporated on the back.

We glue the chips on an aperture in the bottom of 35 mm polystyrene culture dishes (Falcon 3001, Becton Dickinson, NJ) and contact them with gilded springs. They are cleaned with hot detergent (5% Tickopur R36, Bandelin, Berlin, Germany), rinsed with Millipore water (Millipore, Billerica, MA), dried in nitrogen and sterilized with UV light for 20 min. They are coated with the laminin fragment YIGSR (16) (No. C-0668, Sigma, Taufkirchen, Germany) by adsorption from a 0.5 mg/ml solution in 10% (v/v) acetic acid for 2 h.

Nerve cells
The isolation of nerve cells from Lymnaea stagnalis follows the procedure described by Syed et al. (15). The central ring of ganglia is isolated from snails with a shell length of 15 mm. After digestion with trypsin (No. T-4665, Sigma; 50 U/3 ml, 20 min at 18°C) and inhibition of the enzyme with soybean inhibitor (No. T-9003, Sigma, 2 mg/ml, 10 min) the tissue around the pedal ganglia is removed. Neuronal somata with parts of the axonal processes are extracted from the A-clusters with a suction pipette (75 µm diameter) attached to a syringe (1750LT, 500 µl, Hamilton, Bonaduz, Switzerland) with a polyethylene tubing. The isolated neurons are kept in defined medium (DM) (18°C, 80% humidity) with plastic dishes coated by adsorption of bovine serum albumin (No. A-9647, Sigma, 3 mg/ml) for 1 h in normal saline (NS). There the axon stump degenerates, leaving an approximate spherical cell body. After one day, the neurons are transferred to a chip with 2 ml DM and placed on the capacitor. DM is made by mixing Leibovitz's L-15 medium with fourfold concentration (without inorganic salts and without L-glutamine, special order, No. 21083027, Invitrogen, Karlsruhe, Germany) and from normal Lymnaea saline (NS) (40 mM NaCl, 1.7 mM KCl, 1.5 mM MgCl₂, 4.1 mM CaCl₂, 10 mM HEPES, pH 7.6 adjusted with NaOH) with fourfold concentration at a ratio 1:1 by dilution with Milli-Q water (Millipore) to final concentrations and supplementing with 25 µg/ml gentamycin sulfate. All reagents were from Sigma unless mentioned otherwise.

Cell-chip adhesion
Chips with neurons are mounted in a microscope (BX50WI, Olympus, Hamburg, Germany) with a differential interference contrast (DIC) inset. The cells are cultured for 1–4 h until the visible adhesion area becomes similar to the diameter of the somata. The distance between chip and attached cell membrane is measured by fluorescence interference contrast (FLIC) microscopy with the hemicyanine dye DiIC₁₈(3) as a fluorescent probe (17). Common silicon chips with SiO₂ terraces are used because FLIC chips with HfO₂ were not available. Cleaning and coating of the FLIC chips follows the protocol for the stimulation chips.

Electrical measurements
We perform experiments at 23 ± 2°C with patch-clamp pipettes in whole-cell configuration to provide for defined extra- and intracellular conditions and also with impaled sharp micropipettes to avoid a washout of cytosolic factors.

The sharp micropipettes are pulled (DMZ Universal puller, Zeitz-Instruments, Martinsried, Germany) from borosilicate glass (TW150F, WPI, Sarasota, FL), filled with saturated K₂SO₄ and contacted with chlorinated silver wires. The experiments are carried out in DM that contains 4.1 mM Ca²⁺. The bath is held at ground potential with a Ag/AgCl electrode (EP05, WPI). The resistances are in the range of 35–60 M. Current-clamp experiments are performed with a bridge amplifier (BA-1S, NPI Electronic, Tamm, Germany).

Micropipettes are pulled from borosilicate glass (GB150T-10, Science Products, Hofheim, Germany) with a three-stage puller (DMZ Universal puller, Zeitz-Instruments) and fire-polished. They are coated with Sylgard (Dow Corning, Midland, MI) and filled with an intracellular solution (ICS) containing 6 mM NaCl, 31 mM KCl, 20 mM KF, 5 mM HEPES, 5 mM EGTA (pH 7.6 adjusted with KOH). DM is replaced by an extracellular recording solution (ECS) without Ca²⁺ containing 40 mM NaCl, 1.7 mM KCl, 5.6 mM MgCl₂, 0.2 mM CdCl₂, 10 mM HEPES (pH 7.6 adjusted with NaOH, specific resistivity _E = 163 cm). An ECS for control experiments without potassium and sodium currents contains 65 mM TEA chloride, 1.5 mM MgCl₂, 5 mM HEPES, 5 mM EGTA (pH 7.6 adjusted with KOH, specific resistivity _E = 165 cm). ECS and ICS are adjusted to an osmolality of 140 mmol/kg with D⁺ glucose. The resistances of the pipettes are in the range of 1.7–2.2 M. Seals are obtained by suction (20 cm water column) for 20 s. After breakthrough by strong suction, the serial resistance is compensated by >70%. Cells are discarded when the seal resistance is <500 M and when the serial resistance is >4 M. Data are recorded with an EPC8 patch-clamp amplifier (HEKA Electronics, Lambrecht, Germany), and filtered at 3 kHz. In some voltage-clamp experiments, a P/4 protocol is used to subtract capacitive and leak currents.

For capacitive stimulation, voltage ramps (waveform generator 33120A, Agilent, Palo Alto, CA) are applied between the chip and the Ag/AgCl electrode in the bath. To determine the capacitive current, the voltage drop at a serial 50 resistor is recorded.

	TWO-DOMAIN-STIMULATION MODEL

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS TWO-DOMAIN-STIMULATION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
Usually the interpretation of electrophysiological experiments relies on three concepts:

1. The electrical potential is constant across the cytoplasm and across the bath.
   
2. The difference of the two electrical potentials—the membrane voltage—is given by the voltage between two Ag/AgCl electrodes in the cytoplasm and in the bath.
   
3. The current through all domains of the membrane is driven by the same membrane voltage.
   

In a situation of extracellular stimulation these assumptions are no longer valid. Current from the stimulation electrode to the bath electrode gives rise to a gradient of the electrical potential in the extracellular electrolyte that depends on the geometry of cell and stimulation electrode as well as on the resistivity of the bath. Even though the cytoplasm may still be isopotential, the membrane voltage depends on the position of the membrane domain in the field of the extracellular potential. Thus the current through different membrane domains is controlled by different voltages.

In the special case of cell adhesion on a planar electrode, we may distinguish

1. The region of adhesion with a large drop of the electrical potential due to the high electrical resistance of the narrow cleft between cell and electrode, and
   
2. The region in the surround of the cell with a minor drop of the electrical potential in the surrounding bath.
   

As a basis for the planning of the experiments and for the discussion of the results, we introduce a two-domain-stimulation (TDS) model that relies on two approximations:

1. The minor drop of the electrical potential in the surround of the cell above the uncovered capacitor is neglected. The extracellular electrical potential near the free membrane is assumed to be probed by the bath electrode.
   
2. The potential profile in the area of adhesion is replaced by a mean extracellular potential. There exists a potential difference—an extracellular voltage—between the attached membrane and the bath. Hence the current through the attached membrane is driven by a different voltage as compared to the current through the free membrane.
   

Deviations between the TDS model and the experimental data may occur if these assumptions are not perfectly valid.

Neuron-capacitor system
We consider an individual nerve cell on oxidized silicon. The attached plasma membrane is separated from the substrate by a thin film of electrolyte as illustrated in 1 1 A. The contact area with the insulating layers of membrane and oxide forms a planar core-coat conductor (14,18). When a changing voltage V_S is applied to the substrate with an area-specific capacitance c_S, current flows along the cell-chip junction with a sheet resistance r_J and across the membrane with an area-specific capacitance c_M. A profile of extracellular voltage V_J arises in the junction as well as a change of the intracellular voltage V_M with respect to the bath at ground potential. A drop of extracellular voltage in the surround of the cell with respect to the bath is neglected.

To account for crucial features of capacitive stimulation, we use a model that describes the core-coat conductor as a single equipotential compartment (12,14,18,19) with a representative extracellular voltage V_J as illustrated in Fig. 1 B. We distinguish two domains of the membrane with a total area A_M: the free membrane with an area A_M–A_J is controlled by the voltage V_M, whereas the attached membrane with an area A_J and a fraction _JM = A_J/A_M is controlled by the voltage V_M–V_J.

The two-compartment stimulation (TDS) model is determined by the capacitance (A_M–A_J)c_M and the ionic conductances of the free membrane and the capacitance A_Jc_M and ionic conductances of the attached membrane (area-specific conductances and ), as well as by the chip capacitance A_Jc_S in the junction and the conductance A_Jg_J from the junction to the bath. The area-specific conductance g_J = _J/r_JA_J is defined in terms of the sheet resistance, the contact area, and a geometry factor that is _J = 8 under stationary conditions and _J = 5.78 for relaxation (see Appendix).

Passive response to voltage ramps
Without ionic membrane conductances (), the balance of electrical current in the cell and in the cell-chip junction is expressed by Eqs. 1 and 2 where I_M is the net membrane current through a pipette:

(1)

(2)

When we apply a voltage ramp with constant slope V_S/t_S to the capacitor under current-clamp with I_M = 0, an extracellular voltage V_J is induced as well as voltage changes V_M and V_JM = (V_M–V_J) across the free and attached membrane. After the onset of a ramp at t = 0, we obtain Eqs. 3 and 4 with a stationary extracellular voltage and a time constant , where is the effective capacitance of the cell per unit area of attachment:

(3)

(4)

Under current-clamp, the voltage change V_M across the free membrane is observed with a micropipette. The polarization change V_JM across the attached membrane follows a similar dynamics, yet with opposite sign and with a larger amplitude. Analogous exponential relations are obtained after the termination of a voltage ramp.

Onset and termination of a voltage ramp are related with capacitive current transients I_JM = A_Ji_JM across the attached membrane where the current density is given by Eq. 5 for t > 0 with a negative sign for the onset and a positive sign for the termination. E.g., the onset of a rising ramp V_S/t_S > 0 implies an inward current i_JM < 0 that leads to a depolarization of the free membrane, whereas its termination is related with an outward current i_JM > 0 that leads to repolarization:

(5)

We can achieve an instantaneous establishment and an instantaneous elimination of a stationary polarization when we superpose a positive or negative voltage step with height to the onset and the termination of a voltage ramp, respectively. The step height must be proportional to the slope of the ramp with (see Appendix).

Under voltage-clamp, there is no voltage change across the upper free membrane with V_M = 0. The capacitive currents through the attached membrane that originate from changes of the cleft voltage V_J are observed as pipette currents. The voltage change V_JM and the capacitive current density i_JM follow Eqs. 4 and 5 when we formally set _JM = 0, also in the expressions for and .

Gating of ion channels under voltage-clamp
A falling voltage ramp V_S/t_S > 0 has a depolarizing effect V_JM > 0 on the attached membrane. At a constant intracellular voltage V_M, it may activate voltage-gated ion channels in the attached membrane with ion channels in the free membrane remaining deactivated. From the current balance in the junction, we obtain Eq. 6 for the transmembrane voltage V_JM where values are the reversal voltages:

(6)

There is a current I_JM = A_Ji_JM through the attached membrane with a capacitive and an ionic component of current density according to Eq. 7:

(7)

If the ion currents through the attached membrane are far smaller than the leak current g_JV_J, they can be neglected in Eq. 6. In that case, the voltage V_JM(t) after the onset of a voltage ramp is approximated by Eq. 4 with a time constant _J: A falling voltage ramp leads to a quasi-stationary depolarization of the attached membrane. That kind of extracellular voltage-clamp is able to activate ion channels in analogy to common intracellular voltage-clamp.

We may compare 1), the ionic current through the attached membrane area A_J that is induced by a falling ramp according to Eq. 8 with the conductances ; and 2), the ionic current through the whole membrane area A_M that is activated by common voltage-clamp according to Eq. 9 with area-specific conductances ):

(8)

(9)

The two currents are proportional to each other, if the depolarization V_JM induced by a falling voltage ramp is equal to the depolarization V_M applied by a pipette. Vice versa, assuming similar voltage-dependent conductances in the attached and total membrane, we can match the ionic membrane currents for extracellular and intracellular voltage-clamp when we fit the depolarization V_JM and the area fraction _JM = A_J/A_M of the attached membrane.

Neuronal excitation under current-clamp
Under current-clamp, rising voltage ramps have a small depolarizing effect on the free membrane whereas falling voltage ramps have a large depolarizing effect on the attached membrane due to the different serial capacitance of the two membrane domains (see Eqs. 3 and 4). Both kinds of stimulation may activate ion channels such that an action potential is elicited.

A rising ramp V_S/t_S > 0 leads to a hyperpolarization of the attached membrane with V_JM < 0 such that neither Na⁺ nor Ca²⁺ channels are activated. In the initial phase of stimulation, the attached membrane plays the role of a passive coupling element between capacitor and free membrane that is depolarized. When we distinguish the fast capacitive polarization and the slow dynamics of ion channels in the free membrane, we obtain Eq. 10 where the capacitive current pulse is given by Eq. 5:

(10)

The situation resembles an intracellular stimulation by a charge pulse that is applied by a pipette. An action potential is elicited if a threshold depolarization V_M > 0 is reached. A relatively large slope of the rising voltage ramp is required, because the stimulating current density is weighted by the small ratio A_J/(A_M–A_J) of attached and free membrane area.

For a falling voltage ramp, the situation is quite different. The attached membrane is depolarized such that Na⁺ or Ca²⁺ channels are activated there, whereas the hyperpolarized free membranes plays the role of a passive load. For the initial phase of stimulation, the dynamics is described by Eq. 11:

(11)

The large capacitance may suppress the dynamics of an action potential in the small activated area of cell adhesion. Thus, we may consider a two-step mechanism of excitation: The primary depolarization of the attached membrane induces an ionic inward current. In a secondary phase, that current depolarizes the whole cell such that the initial hyperpolarization of the free membrane is overcome and an action potential is elicited in the free membrane. We describe the dynamics by Eq. 12 where the primary hyperpolarization of the free membrane is neglected and where the ionic current through the attached membrane is described by Eq. 8:

(12)

The attached membrane plays the role of an active coupling element between capacitor and free membrane that injects ionic current during the applied voltage ramp. An activation of ion channels in the attached membrane is achieved by a relatively small slope of a falling ramp. Nonetheless, it may be difficult to reach the threshold of an action potential because the stimulus current per unit area is again weighted by the small ratio A_J/(A_M–A_J) of attached and free membrane area.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS TWO-DOMAIN-STIMULATION MODEL RESULTS AND DISCUSSION CONCLUSIONS APPENDIX ACKNOWLEDGEMENTS REFERENCES

 
At first, we describe the contact of nerve cells and electrolyte/oxide/semiconductor capacitors, the displacement current through the electrolyte/oxide/semiconductor capacitors, and the passive response of nerve cells with deactivated ion channels. Subsequently, the gating of Na⁺ and K⁺ channels in the attached membrane is studied by capacitive stimulation with falling voltage ramps at constant intracellular voltage and their gating in the whole cell membrane by conventional variation of the intracellular voltage. Finally, action potentials are elicited under current-clamp by rising and falling voltage ramps.

Cell and capacitor
Cell-chip contact
Nerve cells with a diameter of 40–70 µm from the A-clusters of the pedal ganglia are plated in defined medium on an electrolyte/HfO₂/silicon capacitor that is coated with the peptide YIGSR, a fragment from laminin that promotes integrin-specific adhesion. Extended adhesion contacts are formed within a few hours as illustrated by Fig. 2 A. Electrophysiological experiments are performed around the time when the visible area of adhesion has reached the size of the cell body.

View larger version (89K): [in this window] [in a new window]   FIGURE 2  Neuron-chip system. (A) Differential interference contrast (DIC) micrograph of snail neuron on an electrolyte/HfO2/silicon capacitor coated with the peptide YIGSR after 2 h in culture. The neuron forms an adhesion area that extends just beyond the cell body. (B) Fluorescence interference contrast (FLIC) micrograph of snail neuron stained with DiI on a silicon chip with SiO2 terraces of four different heights. From the intensities in the marked areas, a distance of 23.5 ± 0.5 nm is evaluated between lipid membrane and chip.

The measured distance is lower than the value 51 ± 2 nm previously reported for snail neurons on poly-L-lysine (20). The difference may be due to the absence of growth-promoting factors in our system that adsorb to the substrate (21) and enhance the distance, or to the peptide YIGSR that may improve cell adhesion.

With an extracellular resistivity _J, the distance d_J determines the sheet resistance r_J = _J/d_J of the cell-chip junction. For d_J = 22 nm, we obtain r_J = 75 M/, assuming a resistivity _J = 164 M cm of bulk electrolyte (22).

The estimated sheet resistance r_J is rather high due to the small distance of cell and chip. Considering the large size of the cells, the area-specific conductance g_J = _J/r_JA_J with contact area A_J and geometry factor _J becomes very low. As a consequence, we expect an effective capacitive polarization of the cells according to Eqs. 3 and 4 with a stationary extracellular voltage even for a relatively low capacitance c_S of the chip and moderate slopes V_S/t_S of the voltage ramp.

Capacitive chip current
We determine the area-specific current i_S across an electrolyte/HfO₂/silicon capacitor without nerve cells by applying rising and falling voltage ramps above a bias voltage of V_S = 1.5 V with respect to a Ag/AgCl electrode. Fig. 3 shows that there is no significant direct current at the voltages V_S = 1.5–4V before the start of the ramps. Capacitive current appears at the onset of the ramps and disappears at their termination. There is a slow change of current during the ramps. Also, after the ramps there is a slow decay until the resting state without direct current is reached. These slow changes are due to the electronic dynamics of the heterojunction HfO₂/silicon that give rise to a capacitance that depends on voltage and frequency (10).

View larger version (14K): [in this window] [in a new window]   FIGURE 3  Current through HfO2/silicon capacitor in electrolyte without cell. (Top) Applied voltage VS(t) of rising and falling ramps with various slopes and a duration of 4 ms. (Bottom) Area-specific current iS. The current increases with the slope. The average area-specific capacitance is cS = 1.2 µF/cm2 in the voltage range of VS = 1.5–4 V. Deviations from an ideal rectangular response of capacitive current are due to the voltage and frequency dependent capacitance of the HfO2/Si contact.

The capacitors with a diameter of 250 µm are far larger than the diameter of Lymnaea neurons. Thus, there is a capacitive current to the bath around the attached cells. For a current density i_S flowing through a circular electrode of radius a_S to a semi-infinite electrolyte with resistivity _E, the maximum voltage drop is V_E = i_SEa_S as evaluated from the Poisson equation with proper boundary conditions (23). For the highest voltage ramp dV_S/dt = 400 mV/ms used in this study with a_S = 125 µm and _E = 164 cm, the estimated voltage drop around a cell is V_E 1 mV. Direct measurements with a patch-pipette near the capacitor are in good agreement with that prediction (data not shown). That minor voltage drop is neglected in the evaluation of our experiments in accordance with the tenets of the TDS model.

Capacitive polarization of neurons
We test the response of snail neurons to capacitive stimuli under conditions where ion channels are not activated. In the current-clamp mode of whole-cell patch-clamp, we record the change V_M of the intracellular voltage, i.e., the polarization of the upper membrane. In the voltage-clamp mode, we observe the capacitive current I_JM through the attached membrane. The experiments shown in Fig. 4 refer to a neuron with a contact area of A_J = 2500 µm².

View larger version (15K): [in this window] [in a new window]   FIGURE 4  Passive response of snail neuron to capacitive stimulation with rising voltage ramps. (A) Current-clamp experiment starting at an intracellular voltage of –95 mV with a voltage ramp of +100 mV/ms and superposed voltage steps at the start and end of the ramp. The applied voltage (top) is drawn for the experiments without voltage steps and with maximum steps of ±100 mV. The current (bottom) is shown for all voltage steps ±0, 20,...100 mV. (B) Voltage ramps of 50, 100, 150, and 200 mV/ms with matched height of voltage steps (top), voltage changes in current-clamp (center) and currents in voltage-clamp (bottom).

The voltage change across the attached membrane is proportional to the voltage change across the free membrane with according to Eqs. 3 and 4. From the small depolarization and with _JM = 0.15 we evaluate a large hyperpolarizing effect in the attached membrane.

The specific conductance g_J = _J/r_JA_J allows us to determine the sheet resistance r_J of the cell-chip contact. From g_J = 2.6 mS/cm² with a contact area A_J = 2500 µm² estimated from a DIC micrograph, we obtain r_J 385 M/ with _J = 8. That value is five times higher than r_J 75 M/ estimated from the width of the extracellular space with bulk resistivity. We attribute the discrepancy to an enhanced resistivity _J in the junction. A similar effect was observed for erythrocyte ghosts on SiO₂ coated with poly-L-lysine (29,30), whereas bulk resistivity was observed in other systems (22). The high resistivity may be related with the narrow extracellular space in our system (20 nm) and in erythrocyte/polylysine adhesion (10 nm), that is far smaller than for rat neurons on polylysine and HEK293 cells on fibronectin (50–70 nm) (22).

Supercharging
Extracellular and intracellular voltages are settled within the time constant of the cell-chip junction. The slow rise and decay can be accelerated by superposed voltage steps that instantaneously charge and discharge the cell-chip junction. The optimal amplitude of voltage steps is determined by the slope V_S/t_S of the voltage ramps with (see Appendix).

Fig. 4 A shows current-clamp experiments for a slope of V_S/t_S = 100 mV/ms superposed with a set of voltage steps . A change of intracellular voltage that resembles the time course of the capacitive chip current (Fig. 3) is achieved at . That optimum corresponds to a time constant . When we vary the slope of the ramps in a range of 50–200 mV/ms using voltage steps we obtain intracellular depolarizations within fractions of a millisecond as shown in Fig. 4 B. The empirical factor differs from the time constant of relaxation. The discrepancy is due to an intrinsic problem with the one-compartment model of the cell-chip contact to account for stationary and transient responses (see Appendix). As a consequence, the ratio of the two time constants reflects the ratio of the geometrical parameters _J for stationary and transient perturbations with 0.6 ms/0.8 ms 5.78/8.

Passive response under voltage-clamp
Under voltage-clamp at V_M = –95 mV, we measure the capacitive current through the attached membrane for a set of voltage ramps with matched voltage steps. Fig. 4 B shows sharp transients of membrane current at the start and termination of the ramps. They correspond to the current transients expected from Eq. 5 with an acceleration due to the superposed voltage steps. There is no current across the attached membrane during the voltage ramps. In the stationary phases, the capacitive stimulation current flows along the cell-chip junction and keeps a constant extracellular voltage.

Capacitive activation of ion channels
To study ionic currents that are induced by capacitive stimulation, we hold the cell at an intracellular voltage V_M where the channels are deactivated. The pipette current probes the current through ion channels that are opened in the attached membrane. An analogous study of capacitive activation of ion channels in the free membrane is not possible, of course. For comparison, we perform a conventional voltage-clamp experiment by variation of the intracellular voltage V_M to activate all ion channels in the free and attached membrane. In these experiments, we use an extracellular medium with blocked Ca²⁺ currents to simplify the interpretation.

Elimination of capacitive membrane current
The total current I_JM across the attached membrane has a capacitive and an ionic component according to Eq. 7. At first we show how the capacitive component is eliminated. In an example, we apply a falling voltage ramp with a slope of –50 mV/ms and matched supercharging steps at an intracellular voltage of –60 mV (Fig. 5 A). The pipette current (Fig. 5 B) exhibits a fast positive transient at the beginning, a slow response during the ramp and a fast negative transient at the end of the ramp. Subsequently, we hold the intracellular voltage at –120 mV. With the same stimulus, the pipette current shows again fast transients at the start and end of the ramp. The slow signal during the ramp disappears (Fig. 5 B).

View larger version (9K): [in this window] [in a new window]   FIGURE 5  Elimination of the capacitive membrane current for extracellular stimulation under voltage-clamp. (A) Falling voltage ramp VS applied to the capacitor. (B) Pipette current at intracellular voltages VM = –60 mV and VM = –120 mV in normal extracellular medium. (C) Net ion current obtained by subtracting the pipette currents at VM = –60 mV and VM = –120 mV for normal extracellular medium and for a medium with TEA+ replacing Na+.

Ion currents through attached and total membrane
We study the gating of ion channels in the attached membrane at an intracellular voltage of –60 mV by applying falling voltage ramps with slopes of –20...–70 mV/ms and a duration of 20 ms. The net ion currents through the attached membrane are shown in Fig. 6 A. We observe transient inward currents that become faster and shorter when the slope of the ramp is enhanced and delayed outward currents with increasing amplitudes.

View larger version (14K): [in this window] [in a new window]   FIGURE 6  Gating of ion channels by extracellular and intracellular stimulation under voltage-clamp. (A) Capacitive stimulation by falling voltage-ramps with VS/tS = –20, –30,...–70 mV/ms (top) at an intracellular voltage VM = –60 mV and ion currents through the attached membrane (bottom). (B) Depolarizing pulses of intracellular voltage of VM = 25.0, 32.5,...,62.5 mV starting from VM = –60 mV (top) and ion currents through the total cell membrane (bottom). The two sets of currents can be matched with a voltage change across the attached membrane VJM = –1.1 ms x VS/tS for capacitive stimulation and a ratio JM = 0.27 of attached and total membrane area.

Apparently, the waveforms of ion current found with capacitive stimulation by falling voltage ramps closely resemble the ion currents induced by intracellular depolarization. The voltage ramps are able to activate Na⁺ and K⁺ channels in the attached membrane by a depolarization V_JM > 0, in analogy to the effect of depolarizing intracellular voltages V_M > 0 on the total membrane. The experiment implements a kind of extracellular voltage-clamp: the depolarizing voltage, V_JM > 0, is defined by the stimulation ramp according to Eq. 6 when the ionic current through the attached membrane is small compared to the current along the cell-chip junction.

However, we note some systematic differences of the current induced by extracellular as compared to intracellular voltage-clamp in Fig. 6:

1. There is a delay by 1 ms for the initiation of Na⁺ current.
   
2. There is a heterogeneous dynamics for the activation of Na⁺ current.
   
3. There is no slow rise of the delayed K⁺ outward current.
   
4. There are positive tail currents after the stimulus.
   

The first and fourth effect is an artifact of the amplifier: for capacitive stimulation with supercharging, we have to undercompensate the amplifier to avoid oscillations such that a constant intracellular voltage is not instantaneously established. The second effect is due to the nonuniform voltage profile in the extracellular space during a voltage ramp, a feature that is neglected in the TDS model (see Appendix). It corresponds to an imperfect space-clamping for intracellular voltage-clamp. We attribute the third effect to an enhanced K⁺ concentration in the narrow extracellular cleft (28) that lowers the outward current during the ramps.

Scaling of extra and intracellular stimulation
The similar waveforms of ion currents for extracellular and intracellular stimulation indicate that the dynamics of ion conductances in attached and total membrane is similar, as described by Eqs. 8 and 9. The extracellular stimulation affects the attached membrane with ion conductances that are driven by the transmembrane voltage V_JM = V_M–V_J whereas the intracellular stimulation affects the whole membrane with ion conductances driven by the voltage V_M.

Despite the minor difference in the waveforms, we match the two sets of measurements to get a scaling of the falling voltage ramps in terms of the transmembrane voltages with V = –(c_S/g_J)V_S/t_S. From a fit of the data, we obtain c_S/g_J = 1.1 ms and _JM = 0.27. With c_S = 1.2 µF/cm², we evaluate a specific seal conductance g_J = 1.1 mS/cm². These parameters are rather similar to _JM = 0.26 and g_J = 1.8 mS/cm² that are obtained from the passive response of the same cell under current-clamp. The result shows that the concept of an "extracellular voltage-clamp" is adequate, i.e., that capacitive stimulation gives rise to a depolarization of the attached membrane that is determined by the slope of the ramp, the chip capacitance, and the seal conductance, and that it has the same effect on ion channels as intracellular depolarization.

Action potentials
The experiments under voltage-clamp show that voltage-gated ion channels are present in the attached and free domain of the plasma membrane, and they demonstrate an equivalence of falling voltage ramps applied to the capacitor and depolarizing pulses of intracellular voltage. Also under current-clamp, falling voltage ramps have a depolarizing effect on the attached membrane and we expect an activation of ion channels. With rising ramps, we expect that the free membrane is depolarized and that ion channels are activated there. We attempt to elicit action potentials under current-clamp using rising as well as falling voltage ramps in whole-cell patch-clamp configuration. The experiments (Figs. 7–10) are performed under the same conditions as used for voltage-clamp without Ca²⁺ currents. The results are reproducible for different cells (n = 8) with some variability of the threshold for action potentials.

View larger version (9K): [in this window] [in a new window]   FIGURE 7  Capacitive stimulation under current-clamp with rising voltage ramps. (A) Applied voltage with slopes of 140,150,...,190 mV/ms. (B) Response of the intracellular voltage to the six different voltage ramps (solid lines). The shaded lines mark the passive response computed by scaling the response of the first experiment.

View larger version (30K): [in this window] [in a new window]   FIGURE 10  Capacitive stimulation by falling ramps. (A) Current-clamp. Intracellular depolarization at the end of falling voltage ramps as a function of slope and duration of the ramp. (B) Voltage-clamp at VM = –60 mV. Injected ionic charge through attached membrane as a function of slope and stimulation time.

View larger version (9K): [in this window] [in a new window]   FIGURE 8  Threshold for action potentials by capacitive stimulation with rising voltage ramps. The threshold of the slope VS/tS of voltage ramps is plotted versus the duration tS of the ramps. On the right ordinate, the concomitant hyperpolarizing voltage |VJM| across the attached membrane is indicated as estimated from the relation |VJM| = (1–JM)(cS/gJ)VS/tS.

View larger version (8K): [in this window] [in a new window]   FIGURE 9  Capacitive stimulation under current-clamp with falling voltage ramps. (A) Applied voltages with slopes of –25,–35,...,–75 mV/ms. (B) Response of the intracellular voltage to the six different voltage ramps (black lines). The shaded lines mark the passive response computed by scaling the response of the first experiment. In some repetitions of the experiment, the third ramp is sometimes able to elicit an action potential (dashed).

The weakest ramp with a slope of 140 mV/ms causes a positive change of intracellular voltage. The response saturates at V_M = 25 mV and relaxes after the stimulus (Fig. 7 B, first record). It reflects the capacitive depolarization of the free membrane that is mediated by the hyperpolarized attached membrane analogous to Fig. 4. When the slope is enhanced in steps of 10 mV/ms, the passive response becomes larger. For comparison, shaded lines in Fig. 7 B mark the passive signals that are expected from a scaling of the first record. Superposed to the capacitive response, we observe an exponential depolarization during the ramp and a long lasting signal after the stimulus (Fig. 7 B, 150...170 mV/ms). Beyond a threshold of the voltage ramp, an action potential is elicited. For 180 mV/ms, the peak appears after the stimulus during the relaxing capacitive response. For 190 mV/ms, the peak is reached during the stimulus. Its decaying phase is superposed by the relaxing capacitive response.

Action potentials can be elicited by voltage ramps of different duration. The threshold of the slope is plotted in Fig. 8 versus the duration. For long stimuli, the threshold approaches a lower limit of 160 mV/ms. For shorter ramps an enhanced slope must be applied to elicit an action potential. At 2 ms duration, the slope is 400 mV/ms.

Falling voltage ramps
After adjusting the intracellular voltage to –60 ± 1 mV, falling voltage ramps with a duration of 10 ms are applied with slopes of –25...–75 mV/ms, ending at a bias voltage of V_S = 1.5 V (Fig. 9 A). The intracellular voltage V_M(t) is plotted in Fig. 9 B.

For the weakest ramp with a slope of –25 mV/ms, there is a saturating response of V_M –6 mV that relaxes after the stimulus (Fig. 9 B, first record). It is due to a capacitive hyperpolarization of the free membrane. When the slope of the ramp is enhanced to –35 mV/ms, the capacitive hyperpolarization increases. The responses expected from scaling of the passive signal are marked in shading. Superposed, we observe a depolarizing signal during the ramp and a long lasting signal after relaxation of the capacitive response. For larger slopes, the primary capacitive hyperpolarization changes to a depolarization. Sometimes it overcomes the threshold for an action potential (dashed line in Fig. 9 B, –45 mV/ms). When the slope is further enhanced, the depolarization becomes lower again (Fig. 9 B, –55...–75 mV/ms) and an action potential is not elicited.

As a measure for the efficacy of the stimuli, we evaluate the depolarization V_M at the end of falling ramps from a series of experiments where no action potentials are elicited. The result is shown in Fig. 10 A as a contour plot obtained by interpolation. The depolarization increases and decreases with enhanced slope at a certain duration, say 10 ms, and with enhanced duration at a certain slope, say –40 mV/ms. When the ramps become shorter, an enhanced slope must be applied to achieve a certain depolarization. There exists an upper limit of depolarization that is V_M = 20 mV in Fig. 10 A. The neuron is excited only if that limit is above the threshold for an action potential.

Asymmetry
The mechanism of capacitive stimulation with rising and falling ramps seems to be similar, with a mere exchange of depolarization and hyperpolarization of the attached and free membrane. However, we must point out two aspects of asymmetry:

1. There is a structural asymmetry of cell adhesion with a small area of the attached membrane and a large area of the free membrane. As a consequence, the polarizing effects on attached and free membrane have different amplitudes for a given slope of the rising or falling voltage ramp.
   
2. There is an asymmetry of recording. For rising and falling ramps, we probe the polarization of the free membrane.
   

For rising ramps, the observed voltage V_M(t) reflects a primary capacitive depolarization of the free membrane and a secondary depolarization due to activated ion currents in the free membrane. For falling ramps, V_M(t) indicates the primary capacitive hyperpolarization of the free membrane and a secondary depolarization that is caused by ion currents in the attached membrane.

Mechanism with rising voltage ramps
Under voltage-clamp, intracellular depolarization activates ion channels in the free membrane as shown in Fig. 6 B. Under current-clamp, depolarization of the free membrane is induced by a pulse of capacitive inward current at the start of a voltage ramp according to Eq. 5. Sodium channels are activated in the free membrane and the self-enhancing dynamics of an action potential is initiated as described by Eq. 10.

There is a threshold of 160 mV/ms for the slope of rising ramps in the limit of long durations (Fig. 8). With a ratio _JM = 0.22 of attached and total membrane area and a parameter c_S/g_J = 0.6 ms as determined for that particular cell from the passive cell response, we obtain with Eq. 3 a threshold depolarization of V_M = 21 mV in the limit of long durations.

When the depolarizing charge is withdrawn too early by the capacitive current pulse at the termination of the voltage ramp, the formation of the action potential is inhibited. That interference is avoided when the dynamics is accelerated by steeper ramps with enhanced inward current pulses as shown in Fig. 8. When short voltage ramps are used, attention must be paid to the danger of electroporation. Due to the smaller size, the hyperpolarization of the attached membrane is larger than the depolarization of the free membrane by a factor according to Eqs. 3 and 4. With _JM = 0.22, we estimate a voltage change of V_JM = 75 mV in the limit of long ramps. For short pulses, the voltage change approaches 200 mV as indicated in Fig. 8. When long ramps are used to avoid the inhibiting effect of ramp termination, care must be taken that the voltage change across the capacitor V_S = (V_S/t_S)t_S remains below the threshold of Faradayic current.

Mechanism with falling voltage ramps
Under current-clamp, falling voltage ramps depolarize the attached membrane and ion channels are activated in analogy to the voltage-clamp experiments (Fig. 6 A). The small area of the attached membrane has two consequences:

1. Ion channels are activated at lower slopes as compared to the effect of rising ramps on the free membrane, because a larger fraction of the extracellular voltage V_J drops across the small attached membrane (see Eqs. 3 and 4). Fig. 9 B shows that a significant depolarizing effect appears at a slope of –35 mV/ms. Using Eq. 4 with _JM = 0.26 and c_S/g_J = 0.7 ms as obtained from the passive response of that particular cell (same cell as used for the voltage-clamp experiment of Fig. 6), we estimate V_JM = 18 mV.
   
2. In principle, an action potential may be initiated in the attached membrane by a capacitive stimulus in analogy to the effect of a rising ramp on the free membrane. However, such a local excitation is suppressed by the large capacitive load of the hyperpolarized free membrane according to Eq. 11.
   

The dynamics of the intracellular voltage in Fig. 9 B indicates a different mechanism: In a first step, the depolarization of the attached membrane activates a local inward current through activated ion channels. In a second step, that ionic current leads to a depolarization of the whole cell such that the primary capacitive hyperpolarization of the free membrane is overcome. The dynamics is described by Eq. 12. Eventually, the threshold for an action potential is reached. Such a mechanism was postulated also by Buitenweg et al. (8) based on numerical simulations. We confirm the concept of indirect ionic stimulation with falling ramps by comparing the experimental data of current-clamp and voltage-clamp.

In a first step, we evaluate the charge flow through the attached membrane under voltage-clamp (Fig. 6 A). For a certain slope, the injected charge increases with duration due to the Na⁺ current. It decreases when the Na⁺ channels inactivate and the K⁺ outward current dominates. When the slope is enhanced, the charge increases due to faster activation of Na⁺ channels and it decreases when the driving force for Na⁺ is diminished and the K⁺ outward current starts to dominate. The integrated charge injection as a function of slope and duration is shown in Fig. 10 B as a contour plot obtained by interpolation. There is an upper limit of injected charge at a certain duration and slope.

In a second step, we compare the injected charge under voltage-clamp with the depolarization under current-clamp as plotted in Fig. 10 A. The upper limit of 3.5 pAs for charge injection under voltage-clamp corresponds to the upper limit of 20 mV for depolarization. From the charge and the depolarization, we obtain a cell capacitance of 0.175 nF that corresponds to a cell diameter of 75 µm. This evaluation confirms the interpretation that the injection of ionic charge through the attached membrane determines the depolarization of the neuron with falling ramps such that eventually the threshold for an action potential is reached.

Advantages and disadvantages
For capacitive stimulation with rising as well as with falling voltage ramps, the attached membrane is a mediator between the capacitor and the excitable free membrane. With rising ramps, it promotes the injection of depolarizing capacitive current, with falling ramps the injection of depolarizing ionic current. With rising ramps, the stimulation is achieved by the charge pulse at the onset of the ramps. It is hindered for short ramps by the inverted charge pulse at the termination of the ramp. With falling ramps, the stimulation is induced by ionic charge that is injected during the whole duration of the ramp.

Neuronal stimulation by rising voltage ramps can be forced by enhancing the slope. Care must be taken, however, to avoid electroporation of the attached membrane and electrochemical reactions at the electrode when large slopes and large amplitudes are applied for weak coupling of cell and chip. In that respect, neuronal stimulation with falling ramps is more harmless because it is achieved by smaller slopes and amplitudes. However, that mechanism may completely fail to reach the threshold of an action potential if the ionic current through the attached membrane is too small due to a small area or low ion conductances in the attached membrane.

Action potentials for physiological conditions
The nature of capacitive stimulation under physiological conditions is checked for snail neurons in a common culture medium with sharp electrodes to avoid a washout of cytosolic factors that are important for Ca²⁺ currents. For each cell, we apply both rising as well as falling voltage ramps to the capacitor in a range of ±25...95 mV/ms in steps of 10 mV/ms with a duration of 10 ms.

The response of intracellular voltage is shown in Fig. 11. Both, falling and rising voltage ramps are able to trigger action potentials with the typical Ca²⁺ shoulder of A-cluster neurons. With rising ramps (Fig. 11 A), we see a capacitive depolarization of the free membrane for small slopes (traces 1–5 in Fig. 11 B). Excitation is observed beyond a slope of +75 mV/ms. There, the action potential is elicited after relaxation of capacitive polarization. For steeper ramps the peak appears right after the onset of the ramp (last two traces in Fig. 11 B). With falling ramps (Fig. 11 C), we observe a hyperpolarization of the free membrane for the weakest stimulus in Fig. 11 D. An action potential is elicited between a slope of –35 mV/ms and –65 mV/ms. Beyond, the depolarization of the free membrane is too low again.

View larger version (5K): [in this window] [in a new window]   FIGURE 11  Capacitive stimulation for physiological conditions under current-clamp by rising and falling ramps with ±25,35,..., 95 mV/ms. (A) Rising voltage ramps. (B) Response of intracellular voltage. (C) Falling voltage ramps. (D) Response of intracellular voltage.