A Toy Model of Human Intelligence: Neurons, Connections, Learning and Speed

No throat-clearing. I want to borrow the language of neural networks and use it to sketch a model of biological intelligence.

The word sketch matters. This is a thought experiment, not a validated model of the brain. The variables below are not things a neuroscientist can simply read from a scanner, and the equations have not been fitted to human data. I still find the exercise useful because it forces a vague word, intelligence, into parts that can be argued with.

Four variables

Let the model begin with four quantities:

• N(x,t): the neurons available for use across position and time.

• C(x,t): the connections already established among them.

• S(x,t): the rate at which useful connections can be formed, a rough stand-in for learning.

• V(x,t): the speed at which signals travel through the active pathways.

The neural-network analogy is tempting.

N resembles parameter capacity. More available units might enlarge the range of patterns a system can represent.

C resembles the learned weights and active routes in a trained network. It stands for the working structure built from knowledge and experience.

S resembles training speed. It tries to capture plasticity, adaptation and the ability to build a new internal model.

V resembles inference speed. In a biological system it would depend on far more than one number, including axon properties, myelination, neurotransmission, network topology and the route a task actually uses.

Already the analogy is under strain. That is fine. An analogy becomes interesting at the point where it starts to fail.

A dynamic system

The original model described the four variables with a set of coupled differential equations:

∂N/∂t = α∇²N − βN + f(x,t)
∂C/∂t = γS − δC
∂S/∂t = η∇²S + κC − μS
∂V/∂t = λ∇²V + ρN + σC − ξV

The coefficients play intuitive roles.

α and η describe spatial spread. β, δ, μ and ξ represent different forms of loss or decay. γ links learning speed to new connections. κ represents feedback from established structure into further learning. ρ and σ connect available neurons and existing connections to transmission speed. f(x,t) is everything arriving from outside the system: stimulation, environment, injury, nutrition, education and whatever else the toy model has failed to name.

I would not claim that neurons literally diffuse through an adult brain according to the first equation, or that intelligence can be measured by solving this system. The equations are scaffolding. Their purpose is to say that capacity, structure, learning and speed change together, and that gain always comes with decay.

Turning the variables into one score

The bluntest possible definition is a weighted functional:

I(t) = ∫[w₁N(x,t) + w₂C(x,t) + w₃S(x,t) + w₄V(x,t)]dx

This says that intelligence is some combination of capacity, established structure, learning rate and processing speed.

It is also obviously incomplete.

The weights would vary by task and person. A chess problem, a poem, a negotiation and a motor response do not use the same brain systems in the same way. Connection count says nothing about which connections exist, whether they are excitatory or inhibitory, or whether the network has a useful architecture. More neurons do not mechanically produce more intelligence. Glial cells, metabolism, development, embodiment, memory, attention and emotion are missing too.

So why write the equation at all?

Because it separates four things that everyday language keeps collapsing into one.

A person can know a great deal and learn slowly. Another can know little and update quickly. Someone may reason deeply but respond slowly. Someone else may react with startling speed inside a narrow domain. Calling all of them simply smart hides the mechanism.

What the model is good for

The model is useful as a set of questions:

• What is the person's current representational capacity for this task?

• What useful structure has already been built?

• How quickly can that structure change when the world changes?

• How quickly can information move through the relevant pathways?

It also suggests that intelligence should be treated dynamically. A snapshot misses learning. A test score misses the cost of producing the score. A fast answer misses whether the internal model is shallow or deep.

The honest conclusion is smaller than the original ambition. I have not produced a mathematical definition of human intelligence. I have produced a four-variable toy model that makes one point clearly: intelligence is not stored knowledge alone. It is capacity, structure, change and speed, interacting over time.

That is enough for a beginning.