The brain is not a finished circuit diagram. It is a pot of soup that never stops simmering. This paper uses four partial differential equations to express 'smartness' as something that flows through time.
We usually measure intelligence with a score: an IQ of 120 or an IQ of 90. But a score is a snapshot, while the brain is plainly not static. You at three and you at thirty use the same brain, yet it is not the same machine.
The paper proposes a completely different approach:do not assign intelligence a number; give it a system of equations. It borrows the reaction-diffusion equations that Turing used in 1952 to study zebra patterns. Those equations describe chemical substances reacting and spreading at the same time until patterns emerge spontaneously. The author argues that intelligence grows in much the same way.
From this perspective, intelligence is not a value but a process. It is not what you possess, but how your nervous system is evolving at this moment.
The paper's first claim is that intelligence is not a single whole, but four quantities that can be measured separately and can also change separately. Each evolves through time and space, and none is dispensable.
The scale of your hardware. It determines the theoretical ceiling: how much information can be held at once and how complex a problem can be processed. In AI, its counterpart is the model'stotal parameter count。
What you can actually use at this moment. Experience, knowledge and skills all reside in connections. In AI, its counterpart after training isweights and effective activation paths。
How quickly you learn something new. This is synaptic plasticity, and it is also the source of creativity: connecting two places that have never been connected before. In AI, its counterpart istraining speed。
How quickly you respond. It depends on the degree of myelination and the efficiency of neurotransmission. It is pure hardware performance. In AI, its counterpart isinference latency。
| Biological brain | Artificial neural network | |
|---|---|---|
| Total available neurons | → | Total parameter count: determines the network's expressive capacity and the upper limit of problem complexity |
| Total established connections | → | Weights and effective activation paths: the network's current working model, in which knowledge and experience are sedimented |
| Rate of connection formation | → | Training speed: corresponds to the efficiency of synaptogenesis and determines adaptability |
| Signal conduction velocity | → | Inference latency: determined by underlying hardware performance |
The table also explains why the author believes that present-day AI has taken a wrong turn:the parameter count and structure of a neural network are frozen once training ends, while none of these four quantities in a biological brain ever stops changing, even for a day.
They look impenetrable, but every term can be stated in plain language.Select any term, and its meaning will appear below.
Each term corresponds to a concrete event in the brain. Together, they form the model's complete account of how intelligence changes over time.
With the four variables in place, the paper defines four forms of intelligence. Each is an integral, meaning that performance across the whole brain and the whole period is accumulated.
Capacity multiplied by structure. How many neurons are available, and how many pathways have they built? This represents the complexity of what you can think about now.
Plasticity accumulated over time. It measures not what you know now, but what you can still become.
Run more connections, faster, with fewer neurons. N in the denominator is crucial: an efficient brain does not rely on brute force.
The product of all four quantities. Multiplication means thatif any one component falls to zero, the whole index falls to zero. This is the sharpest statement in the model.
Below is a simplified numerical solution to the equations.All three scenarios begin with the same twenty-year-old brain, and are plotted against the same vertical scale, so their curve heights can be compared directly. When scenarios change, a grey dashed line retains the lifelong learner as the reference.
All four indicators use the lifelong learner as a baseline of 100. This is a simplified simulation intended to aid understanding, not a representation of real biological data.
Switch to 'Stop learning' and the most striking feature is this:the violet N and amber V barely fall, while the magenta C and cyan P collapse. The hardware remains, the neurons are still there and the response is not especially slow. What has stopped is the formation of new connections, and learning speed reaches zero first. P is sustained by positive feedback from C, while C grows through P, so the two descend together. The composite intelligence index at the end is the product of all four quantities. If any one approaches zero, so does the whole.
The author identifies its weaknesses explicitly in the conclusion. This may be the most valuable part of the entire paper.
Every equation currently uses linear coupling, while real neurons are all-or-nothing. Below a threshold they do not respond at all; above it they fire completely. Synaptic growth also has a ceiling and cannot increase without limit. A closer biological model needs threshold functions and saturation terms, similar to Sigmoid or ReLU in artificial networks.
The brain accounts for only 2 per cent of body weight but consumes about 20 per cent of the body's energy. High connection density and fast conduction both consume ATP, while the body's energy budget is finite. Omitting this term amounts to assuming that intelligence is free. Only with a metabolic cost can the model simulate the real trade-off between cognitive performance and survival.
The author's proposed route forward is to add these two components, study the equations' stability under different boundary conditions and validate the model through numerical simulation. How far this route can go is, as the author writes at the beginning, for time to decide.