I realized that I had never developed an intuitive understanding of what electricity actually is.
The first variable that really bothered me in computational neuroscience was V, the membrane voltage.
Mathematically, it was clear. Physically, it was still blurry: I was still interpreting voltage according to the conventional understanding I had been taught.
In the Hodgkin-Huxley equations, V evolves in time, drives currents, interacts with conductances, and produces spikes. I could follow all the lines. But at some point I realized I was manipulating voltage without having a real picture of what voltage is. Not “potential difference.” That is the kind of definition that explains nothing, it just moves the question one step further.
So I decided to follow V downward, as far as I could.
At the standard level, electricity is described with a familiar vocabulary: charge, field, voltage, current. A charge placed in a field feels a force, F = qE. Voltage is energy per unit charge. Current is the movement of charge. This is already sufficient to model a membrane, and the majority of neuroscience does not need more than this. But it rests on one word I could not actually define: charge.
The textbook answer felt like entering a story at the middle: some particles carry positive charge, others negative, like charges repel.
What changed things for me was understanding that modern physics does not begin with small labeled balls. It begins with fields. An electron is a ripple in the electron field; a photon is a ripple in the electromagnetic field. They are distinct fields that interact with each other. And charge, seen from this angle, is not a substance contained inside the electron. It is the way one field couples to the other.
That single sentence reorganized the picture for me.
Going one level deeper: each field carries something like an internal phase, an invisible dial. The absolute position of this dial is not observable. If every dial were rotated by the same amount, nothing physical would change. This invariance is a symmetry. For electromagnetism, the relevant one is called U(1), which is essentially the symmetry of rotating an internal phase.
Making this symmetry local is where things get strange. Instead of rotating every dial everywhere by the same amount, imagine each point in space is free to choose its own setting. Here the dial points one way, a little further it points differently. But then a problem appears: how do you compare a phase at one point with a phase at another?
Think of every city maintaining its own clock, each free to decide where noon is placed on the face. To travel between two cities, you need a rule to translate one clock into the other. Otherwise you cannot know if something truly changed, or if you simply crossed into a different local convention. The electromagnetic potential plays exactly this role. The electric and magnetic fields are what you observe when this translation rule varies from place to place.
Charge is not a substance. It is a way of coupling to an electromagnetic connection. Positive and negative are not two mysterious fluids. They are opposite ways of responding to the same underlying field.
From there, the familiar layer becomes readable again. F = qE: a positive charge moves in the direction of the field, a negative charge in the opposite direction, and attraction and repulsion follow naturally. Electric potential becomes an energy landscape, and voltage is simply a difference of height in that landscape. A battery does not create electrons. It maintains a difference of potential. A wire contains no electric fluid, only charges that are willing to move when a field acts on them.
In a neuron, the charges in movement are not electrons in a metal. They are ions: sodium, potassium, calcium, chloride, moving through water on both sides of a thin membrane. The cell maintains different concentrations on each side. A resting potential close to -70 mV means the interior of the neuron sits about 70 millivolts below the exterior. When a channel opens, two things act on the ion simultaneously: the electric field, and the concentration gradient that pushes it to spread out. The point where these two forces exactly compensate each other, for a given ion, is its Nernst potential. Voltage in a neuron never exists alone. It is always entangled with concentration, permeability, and which channels happen to be open at a given moment.
When voltage and channel states evolve together in the right conditions, the membrane enters the regenerative process we call an action potential.
This is what I try to remember when I read Hodgkin-Huxley now. V is not an abstract state variable. It is a physical quantity: the membrane voltage. The current terms are not only mathematical flows; they represent ions crossing a biological structure that regulates them with a precision that is, frankly, difficult to believe.
I do not need quantum field theory to simulate a neuron. Conductance-based models and cable theory operate far above it, and that is fine. But I wanted an orientation. I wanted to understand what the variables I was working with are actually pointing toward.
The answer turned out to be a single line across scales:
charge → fields and forces → voltage → ionic currents → spikes → computation
This is not a claim that quantum fields explain cognition. They do not. It is only a way of remembering that V is not a symbol that fell from nowhere. It compresses a long chain of physical reality, and now I can see most of that chain.
That is where I want this to begin: not with the brain as an abstract computational system, but with the physical language in which its computation is written.