Foundations — Neuromorphic computing

This page assumes nothing. Every letter, every squiggle, every word in bold on the parent page is unpacked below, in an order where each idea rests only on the ones before it. Read top to bottom once and the parent note will read like plain English.

1. Voltage $V$ — "how hard the charge is pushing"

Picture a water tank. The height of the water is the pressure at the bottom of the tank: fill it higher and water squirts out harder. Voltage is exactly that height, but for electric charge instead of water.

The topic needs $V$ because a neuron's whole job is to let a voltage build up and then react when it gets high enough. In the parent note the neuron's voltage is called the membrane potential $V(t)$ — the little $(t)$ just means "$V$ measured at time $t$", i.e. the water height right now, which changes as time passes.

2. Current $I$ — "how much charge flows per second"

Back to the water tank: current is the flow rate of water through a pipe (litres per second). Voltage is the pressure; current is the actual flow that pressure produces.

The parent note feeds a neuron an input current $I(t)$ — a stream of charge arriving from other neurons. That inflow is what raises the voltage.

3. Charge $Q$ — "the amount of electricity itself"

Charge is the water itself; current is how fast charge moves. Written precisely: $I=\frac{dQ}{dt}$ which reads "current is the rate of change of charge" — exactly parallel to "flow rate is how fast the amount of water changes." We will need this relation the moment we ask how fast a capacitor fills.

4. Resistance $R$ and the leak current $I_{\text{leak}}$

The neuron's membrane is dotted with ion channels — tiny holes that let charge leak back out. More leak = the voltage falls faster. In the RC-circuit picture this leak is drawn as a resistor.

5. Capacitance $C$ and the capacitor current $C\frac{dV}{dt}$

The neuron's membrane is two thin layers with charge sitting on either side: that is physically a capacitor. Pour current in and the "water level" (voltage) rises.

Put the resistor (leak) and capacitor (storage) together and you have an RC circuit. That is the whole electrical skeleton of a neuron, and it is why the parent links RC circuits and Memristors and ReRAM — memristors are the hardware that plays the role of an adjustable resistor/synapse.

6. The derivative $\frac{dV}{dt}$ — "how fast the level is changing"

Here a piece of maths enters, so we earn it before using it.

Why do we need this tool and not just algebra? Because the neuron's voltage is not a fixed number — it is always moving. To describe motion (rising, falling, how fast) you need the language of rates of change, and that language is the derivative. Asking "will $V$ reach the threshold?" is asking "how is $V$ moving right now?", which only $\frac{dV}{dt}$ can answer.

Picture the water level as a curve on a graph, time going right, height going up. At any moment:

• level going up → slope positive → $\frac{dV}{dt}>0$,
• level flat → slope zero → $\frac{dV}{dt}=0$,
• level falling (leaking) → slope negative → $\frac{dV}{dt}<0$.

7. The exponential $e^{-t/\tau }$ — "steady fading", and how the charging curve is born

Another maths tool. We earn it too.

Why this exact shape and not, say, a straight line down? Because a leaking bucket leaks faster when fuller (more pressure) and slower when nearly empty. A quantity whose fall speed is proportional to its own size traces out exactly the $e^{-t/\tau }$ curve — no other function does. That is why the parent's leaky voltage decays as $e^{-t/{\tau }_m}$, never as a straight line.

Two mirror-image uses appear in the parent:

• Charging up toward a target: $V(t)=RI\left(1-e^{-t/{\tau }_m}\right)$ — rises fast, then eases in.
• Fading learning (used later in Section 9): $e^{-\Delta t/{\tau }_{+}}$ — a big change if spikes were close, shrinking as the gap grows.

8. The threshold $V_{th}$, reset, spike, and the refractory period

Picture the water tank with an overflow lip at height $V_{th}$. Fill past the lip and it dumps a splash (the spike) and drops back down. Everything below the lip is smooth analog charging; the lip is the one nonlinear switch that makes the neuron spiking instead of a plain analog circuit. This links Spiking Neural Networks (SNN).

9. Spike-Timing-Dependent Plasticity: the weight $w$, $\Delta$, and STDP

This is the language of learning. The sign of $\Delta t$ decides whether the weight grows ($\Delta w>0$, potentiation) or shrinks ($\Delta w<0$, depression), and the exponential (Section 7) decides by how much.

10. Big-number notation — $10^{11}$, milli, nano, mega

The parent throws around $10^{11}$ neurons, $\text{M}\Omega$, $\text{nF}$, $\text{mV}$. Decode once:

How these foundations feed the topic

Equipment checklist

Cover the right side and answer each before revealing:

Back to Neuromorphic computing · prerequisites: RC circuits, Memristors and ReRAM, Spiking Neural Networks (SNN), Hebbian learning, In-memory computing.