Foundations: matter, mathematics & computation

Begin with a quantity you can measure

A voltage is a difference in electrical potential, not a material flowing through a wire. Current measures charge crossing a boundary per unit time. An ion is an atom or molecule with a net charge; Na+ and K+ carry one positive elementary charge, Ca2+ carries two, and Cl- carries one negative charge. Write the measurement and its unit together. A number such as 20 is not a membrane timescale until you say whether it means seconds or milliseconds.

A useful circuit picture

Imagine a container that can store water while a small opening lets water escape. Storage and leakage compete. A membrane can be approximated as electrical storage plus a leakage pathway, although the water analogy does not explain selective ion channels. Capacitance relates stored charge to voltage; resistance relates voltage difference to current. Their product has units of time. The illustration below separates the circuit quantities so that the algebra has a physical meaning.

From biology to a neural computation

A passive membrane has memory: its voltage depends on recent input rather than only the current sample. A neural network with state can similarly integrate weak evidence across time. Before discussing intelligence, ask whether the model preserves the relevant timescale. A simulator stepping once per second cannot resolve a 20-millisecond process reliably. Numerical convergence is part of the model, not cosmetic cleanup.

Your first derivation

Current conservation gives I = I_leak + I_capacitor. With a leak reference E, I_leak = (V - E)/R and I_capacitor = C dV/dt. Substitute these expressions and multiply by R to obtain the membrane equation used later. This derivation assumes a constant resistance and capacitance; biological conductances can vary with voltage, time and chemical state.

SEE THE RELATIONSHIPS

From structure to behavior.

MAKE THE IDEA PRECISE

The mathematics.

Build the equation, step by step

1. Start with Q = CV: at fixed capacitance, doubling the voltage difference doubles the stored charge. Differentiate with respect to time to obtain the capacitive current C dV/dt.
2. For the parallel leak pathway, the voltage across the resistor is V − E, so its current is (V − E)/R. Add the two currents and equate their sum to the injected current.
3. Multiplying by R gives RC dV/dt = −(V − E) + RI. Define τ = RC. With no input, displacement from E decays exponentially; after one τ, about 36.8% remains.

WORK THROUGH THE NUMBERS

Worked example

Choose an illustrative resistance of 100 megaohms and capacitance of 200 picofarads. Their product is 0.020 seconds = 20 milliseconds. A 0.1-nanoamp current through that resistance gives a 10-millivolt drop. These are chosen example values, not measurements of a particular fly neuron.

A small Python experiment.

Python 3 · Standard library only · Synthetic educational example. Run this in your own notebook or a local file.

R = 100e6       # ohms
C = 200e-12     # farads
I = 0.1e-9      # amperes
print(f"tau = {R*C*1000:.1f} ms")
print(f"I R = {I*R*1000:.1f} mV")

Expected output

tau = 20.0 ms
I R = 10.0 mV

Change one quantity, predict the result, and then run the code again. Explain any difference before adding complexity.

THE NEXT LAYER

Chapter depth map.

This introduction provides a first model. The following chapters are outlines for deeper future editions; their full treatments have not yet been published.

Physical intuition

The mathematics of change

Uncertainty and scientific computation

Read with the original sources.

These readings support the concepts and deeper investigation. The worked numbers and illustrations above are teaching examples.