A phase change no pipeline diagram mentions
In tissue, charge is carried by ions; in the wire, by electrons. The electrode is where that conversion happens, and it is not free. The instant metal touches electrolyte, ions rearrange and a structured region forms: the electrical double layer. Everything the electrode does — its impedance, its DC offset, its noise, its ability to inject charge — flows from this interface.
The electrical double layer
Ions of opposite sign accumulate at the metal surface. The compact layer of specifically-adsorbed ions is the Helmholtz layer, backed by a diffuse Gouy–Chapman cloud (together, the Stern model). Electrically this behaves like a capacitor of remarkably high specific capacitance — on the order of 10–20 µF/cm² for a smooth metal — because the charge-separation distance is molecular (~nm).
At equilibrium the interface also sits at a material-specific half-cell potential — a DC offset (from tens of mV up to about a volt versus a reference) that has nothing to do with the brain but which your front-end must reject.
Faradaic vs capacitive charge transfer
Charge crosses the interface two ways. Capacitive (non-faradaic): charging and discharging the double layer, with no chemical reaction — fully reversible, ideal for recording and for safe stimulation. Faradaic: electrons actually cross via redox reactions; these can be reversible (for example the valence changes of iridium oxide) or irreversible (electrode corrosion, water electrolysis, gas and pH shifts that damage tissue).
The Randles equivalent circuit
The standard lumped model is the Randles circuit: a solution / spreading resistance R_s in series with a parallel pair — the double-layer capacitance C_{dl} and the charge-transfer resistance R_{ct} — plus a Warburg element for diffusion when it matters.
Z(\omega) = R_s + \dfrac{R_{ct}}{1 + j\omega R_{ct} C_{dl}}Randles impedance (Warburg neglected). As ω→0, Z→R_s+R_ct; as ω→∞, Z→R_s.
A simple circuit model of the electrode–tissue boundary. At low frequencies the double-layer capacitor blocks current, so you feel the full resistance; at high frequencies the capacitor 'shorts out', leaving only the spreading resistance. It captures how the electrode's opposition to current changes with frequency.
- Z(\omega)
- Total impedance — the frequency-dependent opposition to current.
- R_s
- The spreading (solution) resistance of the surrounding tissue.
- R_{ct}
- Charge-transfer resistance — how hard it is to push current chemically across the interface.
- C_{dl}
- The double-layer capacitance at the interface.
R_s is the tissue spreading resistance (set by geometry and σ); C_{dl} dominates across the frequencies we record; R_{ct} governs the DC and faradaic leak.
Why a real electrode is not an ideal capacitor: the CPE
Measured interfaces almost never behave as a pure capacitor. Surface roughness, porosity and a distribution of time constants make the phase 'stick' somewhere between 0° and −90°. We capture this with a constant phase element (CPE).
Z_{CPE}(\omega) = \dfrac{1}{Q\,(j\omega)^{n}},\qquad 0 < n \le 1CPE impedance; n = 1 is an ideal capacitor, n < 1 the depressed-semicircle behavior real electrodes show.
Real electrodes aren't perfect capacitors — their impedance falls with frequency at an odd, in-between rate. The constant-phase element captures this with an exponent n: at n = 1 it's an ideal capacitor, and below 1 it matches the 'squashed' behavior that rough, porous surfaces actually show.
- Q
- A capacitance-like magnitude for the CPE.
- n
- The exponent between 0 and 1 setting how 'ideal' the capacitor is.
- \omega
- The angular frequency of the signal.
- n=1
- Recovers a perfect capacitor; n < 1 gives real-electrode behavior.
Fitting R_s, R_{ct}, Q, n to a measured spectrum is exactly what electrode impedance spectroscopy does — the subject of the next guide.