Impedance

Every interface between metal and tissue is an impedance, and almost every constraint on recording or stimulating something in the body turns out to be a statement about it. Impedance, $Z(\omega)$, is voltage over current at one frequency: a magnitude and a phase. Both disciplines draw it with the same symbols; they differ over what carries the current.

What an electrode or a coil presents

An electrode hands current from electrons to ions by charging the double layer at its surface, a capacitance, or by reduction and oxidation, a charge-transfer resistance. In parallel, behind the electrolyte’s resistance, they form the simplest version of the circuit traced to Dolin and Ershler (1940) and Randles (1947).

Its real part makes thermal noise as a resistor would, and the whole forms a divider with a voltage amplifier’s input, stray capacitance included. Shrinking a microelectrode-array site worsens both; Neuropixels sites, 12 µm square, are porous titanium nitride, whose pores add surface for the double layer, holding them to about 150 kΩ at 1 kHz in saline. Measuring current reverses the rule: Neher reckoned that resolving a picoamp to 10% over a 1 kHz bandwidth needs a source of about 2 GΩ or more. The seal sits in parallel with the patch, so it too had to reach gigaohms: the patch clamp’s gigaseal, found by chance around 1980.

A TMS coil makes no electrical contact and is nearly pure inductance, typically 16 µH against 25 mΩ. The induced field is proportional to the voltage across it; in a conventional stimulator, current rises for a quarter period of the coil ringing with its capacitor, $\tfrac{\pi}{2}\sqrt{LC}$ for inductance $L$ and capacitance $C$. To excite a neuron, briefer pulses need more voltage, longer ones more current.

What a membrane presents

Measured with alternating current by Curtis and Cole, and under the voltage clamp, squid membrane is about 1 µF/cm² with the Hodgkin–Huxley conductances leaking across it: a parallel RC whose time constant $\tau = R_m C_m$, a unit area’s resistance times its capacitance, is independent of area. The capacitor is an idealisation here, as it is at an electrode: Curtis and Cole measured an average phase angle of 76°, not a pure capacitor’s 90°.

In cable theory a branch’s input conductance goes as the 3/2 power of its diameter, so a thin dendrite gives a synapse a high input resistance: a large depolarisation where it lands, a smaller one at the soma (Rall and Rinzel, 1973). Measuring in vivo and in vitro, Oostendorp and colleagues put the skull’s conductivity at about a fifteenth of brain’s in 2000, against the eightieth then common in source-localisation models; either way, the skull blurs EEG.

Where the leak becomes chemistry

They meet in a circuit. In 1907 Lapicque modelled the membrane, like a polarised electrode, as a capacitor — then added a leak across it and a resistance in series: the simplified Randles topology.

They part at the leak. Ions cross a membrane through channels unchanged; across an electrode the leak is chemistry, and pushed past water electrolysis it makes gas and swings pH. Stimulators therefore balance each pulse’s charge; cochlear implants put a capacitor in series with every contact. Smooth platinum takes about 35–100 µC/cm² in saline before electrolysis. A coil is exempt: its current starts and ends at zero, so by Faraday’s law the current it induces is balanced.

Yet with platinum or iridium oxide the first damage appears to come from overdriven neurons, not electrochemistry: tissue is damaged at charge densities below the electrolysis limits measured in saline. Those limits may be up to ten times lower in tissue, so chemistry is not cleared. Either way the working limit is empirical. In 1992 Robert Shannon drew a line across McCreery and colleagues’ 1990 cat-cortex data, between the conditions that had damaged tissue and those that had not: $\log D = k - \log Q$, in base-10 logs, for charge density $D$ in µC/cm², charge per phase $Q$ in µC and an adjustable constant $k$. For large electrodes the line is usually drawn at $k \approx 1.85$, damage lying above it. The first deep brain stimulator approved in the US, Medtronic’s Activa Tremor Control System in 1997, carried a 30 µC/cm² ceiling, which on its 0.06 cm² contacts sits near $k = 1.75$.