It takes two neurons to ride a bicycle (2004)(fermatslibrary.com)
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It takes two neurons to ride a bicycle (2004)
https://fermatslibrary.com/s/it-takes-two-neurons-to-ride-a-bicycle#email-newsletter
53 comments
> Riding a real bicycle isn't just about steering and leaning a bit. You need to propel the bicycle a certain amount.
My impression from my vague recollection of learning to ride a bike (and the scattered times over the years I've seen others try) is that a pretty huge part of the battle is just learning to trust that going faster is literally how to avoid falling over, when the natural inclination of someone worried about that is to slow down.
My impression from my vague recollection of learning to ride a bike (and the scattered times over the years I've seen others try) is that a pretty huge part of the battle is just learning to trust that going faster is literally how to avoid falling over, when the natural inclination of someone worried about that is to slow down.
>None of them made significant use of the speed—they all managed to control the bicycleusing just the handlebars.
I think is where it refers to it.
I think is where it refers to it.
Two to steer is still impressive. If we added in balance and pedaling/braking I wonder what the count would raise to then.
I'm don't think it's possible to start a bike by pedalling with zero forward momentum. You will fall over. You need to kick off - start pedalling with the bike already moving forward. So you're right, and a third neuron is certainly not sufficient. You need legs, too, and arms, and a torso, and motor neurons, and respiration/metabolism. Clearly, the paper has no practical application; if you need to ride a bike, it's far cheaper to hire a human to do it.
https://www.youtube.com/shorts/zwBW2Akw1P0
Possible, but more complex than most appreciate. Pushing a pedal down shifts the rider's center of gravity over that pedal, requiring the bike to lean in the opposite direction to maintain a straight line. This done by the rider counter-steering to command that angle. Watch in the video how the front wheel and lean angle alternates left-right in time with the pedal pushes. Once at speed, the gyroscopic forces of the front wheel mean the rider doesn't need as much lean angle, so the wobbles get less and less. Compare an accelerating motorcycle where the rider doesn't shift weight and therefore doesn't need to wobble the front tire.
Possible, but more complex than most appreciate. Pushing a pedal down shifts the rider's center of gravity over that pedal, requiring the bike to lean in the opposite direction to maintain a straight line. This done by the rider counter-steering to command that angle. Watch in the video how the front wheel and lean angle alternates left-right in time with the pedal pushes. Once at speed, the gyroscopic forces of the front wheel mean the rider doesn't need as much lean angle, so the wobbles get less and less. Compare an accelerating motorcycle where the rider doesn't shift weight and therefore doesn't need to wobble the front tire.
If you orient the pedal so it's at about 9 o'clock, and hop on, the downforce will produce sufficient forward motion for steerage. Of course you can also push off with your other foot, but it's not essential. But that's another neuron. ;-)
This looks like they simply reinvented PID control. The inputs to the beyond are desired states minus actual states, which is basically how PID works.
I assume you mean proportional–integral–derivative control. https://en.wikipedia.org/wiki/PID_controller
The useful insight is not "compare desired state to actual state"; it's deciding which state to control
No, the bicycle is unstable. PID doesn't work well there.
In addition, it is controlling a coupled 3D system (which is unstable). This is much more than 3 PID controllers.
In addition, it is controlling a coupled 3D system (which is unstable). This is much more than 3 PID controllers.
PID works fine if you parametrise it right, which is what this paper does. Consider the variety of inverted pendulums etc. that are used as as examples to teach PID control.
Bicycles are stable. In fact you need zero neurons to run a bicycle, all it needs is "power" and a large enough circle to run in.
If you can't use a PID for unstable things, why does cruise control exist for motorcycles?
> No, the bicycle is unstable. PID doesn't work well there.
This is just totally wrong. Stabilizing an unstable system is usually the first goal of controller design. Different systems do require different controllers, but pid is _very often_ perfectly adequate. It’s probably the most used controller structure in general.
This is just totally wrong. Stabilizing an unstable system is usually the first goal of controller design. Different systems do require different controllers, but pid is _very often_ perfectly adequate. It’s probably the most used controller structure in general.
Nice article, but the methods they used seem more like they just hand wrote a function for the task and called the function neurons based on how it was implemented. It is encouraging though that a simple network can be found for a complicated task like this, kind of like the Tiny Recursive Model that came out last year.
I think that's basically right. The "neurons" framing feels a bit loose here; it's more like a very compact hand-designed controller expressed in neural-network-ish terms
That was my take, too. You could control a bike with just one linear neuron with one input, if you pass in the correct steering torque. Still a fun paper.
Footnote 1 mentions:
Actually, the title of this paper is unproven. We have not ruled out the possibility that a single neuron could ride a bicycle.
Actually, the title of this paper is unproven. We have not ruled out the possibility that a single neuron could ride a bicycle.
(2004)
Previously:
- https://news.ycombinator.com/item?id=19196664 (25 comments)
- https://news.ycombinator.com/item?id=16215130 (88 comments)
Previously:
- https://news.ycombinator.com/item?id=19196664 (25 comments)
- https://news.ycombinator.com/item?id=16215130 (88 comments)
I had fun reading this. Thanks for sharing.
With dendritic compartments, this seems like a waste of a perfectly good neuron that we could productively use elsewhere. ;)
Note that a SINGLE neuron can compute nonlinear functions like XOR.
Shameless plug: If anyone is interested, I did a post a while back on how neurons can act as logic gates:
https://blog.typeobject.com/posts/2025-neural-logic-gates/
This article builds on the first and creates a half adder out of neurons:
https://blog.typeobject.com/posts/2026-timing-is-the-bit/
With dendritic compartments, this seems like a waste of a perfectly good neuron that we could productively use elsewhere. ;)
Note that a SINGLE neuron can compute nonlinear functions like XOR.
Shameless plug: If anyone is interested, I did a post a while back on how neurons can act as logic gates:
https://blog.typeobject.com/posts/2025-neural-logic-gates/
This article builds on the first and creates a half adder out of neurons:
https://blog.typeobject.com/posts/2026-timing-is-the-bit/
Research question: does it make sense to make a new family of logic gates using neurons? My intuition says there is a rich texture/fabric to uncover here. The best analogy on hand right now is legos: rather than 2-knotch legos [standard gates like NAND, XOR] what about some sort of new, irreducible gates that are bigger "legos"? Been a while since I played with logic gates but my intuition says there is something lurking below the surface. A new class of irreducible gates, maybe cross-connections? Like compacted multilayer gates? Think SHA-512, how certain bits feed into different layers of the "puzzle". Optimistic this thought-amalgam serves you in your continued research :)
Yes!
I started going down the path of building a ripple carry adder already (which seems to work fine). Then I was going to try for a full on ALU, then some sort of ISA that sits on top of it all.
I have no idea what the end result will look like if it all comes together. Hopefully I'll find some weird primitives along the way. :D
It's very hand-wavy, but I'm kinda hoping I can somehow have a machine manually constructed out of neurons that can naturally interact with one built with looser hebbian learning rules.
I started going down the path of building a ripple carry adder already (which seems to work fine). Then I was going to try for a full on ALU, then some sort of ISA that sits on top of it all.
I have no idea what the end result will look like if it all comes together. Hopefully I'll find some weird primitives along the way. :D
It's very hand-wavy, but I'm kinda hoping I can somehow have a machine manually constructed out of neurons that can naturally interact with one built with looser hebbian learning rules.
The ISA could be really cool, having lots of "combo" commands that might reduce program length dramatically. Think ADD and MULT and SHIFT all in one command, to give a simple analogy.
On the interaction, one system uses a clock signal / metronome and the other is all cascades. The clock signal is like a metronome calibrated to the duration of the longest cascade = "critical path." It seems clear that these can interact smoothly, as one simply has the training wheels of the clock, while the other is about progression-via-propagation.
On the interaction, one system uses a clock signal / metronome and the other is all cascades. The clock signal is like a metronome calibrated to the duration of the longest cascade = "critical path." It seems clear that these can interact smoothly, as one simply has the training wheels of the clock, while the other is about progression-via-propagation.
The other day I saw a guy descending a downgrade, standing on the top tube of his bike and not holding the bars, with outstretched arms.
I asked how many neurons he was was using: zero, it turns out.
I asked how many neurons he was was using: zero, it turns out.
>> The actions only differ in how the handlebars are pushed at the first instant: pushed left, pushed right, or not touched.
Have the authors ever ridden a bicycle/motorcycle? The handlebars are not the primary controls. As evidence, I say watch this clip. Handlebars are not needed for cornering. Into a 45* lean angle, standing up on the pegs. Hands are optional.
https://www.youtube.com/shorts/Gyt9DLfYOdU
Have the authors ever ridden a bicycle/motorcycle? The handlebars are not the primary controls. As evidence, I say watch this clip. Handlebars are not needed for cornering. Into a 45* lean angle, standing up on the pegs. Hands are optional.
https://www.youtube.com/shorts/Gyt9DLfYOdU
> We do not have great insight as to how we ride a bicycle, and we do not have
much useful advice for someone who is learning.
I indeed balked at this, finding both of those sentences wildly incorrect, as someone both having been taught as well as having taught multiple people myself.
Also: https://ciechanow.ski/bicycle/
It seems that it is something that is forever doomed to be forgotten and then rediscovered over and over.
I indeed balked at this, finding both of those sentences wildly incorrect, as someone both having been taught as well as having taught multiple people myself.
Also: https://ciechanow.ski/bicycle/
It seems that it is something that is forever doomed to be forgotten and then rediscovered over and over.
What we forget is that two-wheeled vehicles drive themselves. There is a stable feedback loop built into the geometry, akin to an aircraft with a good dihedral wing. You have to force the machine to corner, then back off to allow it to do its thing. See the rider in red in the above video. He leans the bike to initiate the turn but then is actually leaning the other way once the bike is cornering/slowing due to the gyroscopic forces now pushing the bike deeper into the corner. Cornering a two-wheeled object is vastly more complex than handlebar pushes.
https://en.wikipedia.org/wiki/Countersteering https://en.wikipedia.org/wiki/Dihedral_(aeronautics)
https://en.wikipedia.org/wiki/Countersteering https://en.wikipedia.org/wiki/Dihedral_(aeronautics)
The author is using an oversimplified model of a bicycle without a rider. This hypothetical bicycle robot has no way to shift the center of gravity and has to rely on handlebars to indirectly control the lean angle.
Of course any realistic bike+rider model would have a lot more degrees of freedom and require a lot more neurons to control.
Of course any realistic bike+rider model would have a lot more degrees of freedom and require a lot more neurons to control.
I would guess this is also applicable to a segway or any balancing vehicle? Balancing is a matter of steering/driving in the opposite direction of a lean, while steering or moving is adding a bias to where 'upright' is to move in a certain direction without falling over. It's just a segway goes back/forward where a bike goes left/right.
So can we have self-driving bicycles?
Self-balancing bicycles, sure. Self-driving bicycles that navigate city streets safely are a much larger problem.
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Recumbent bike with lidar and maps? Sign me up.
Yes and they'll have one of those wetware computers on board
> U-2200, a non-corporeal entity claiming to be the prehistoric Johorean god of forgetting how to ride a bicycle, engages Quinn in a conversation, suggesting she take a month off in Barbados, drink alcohol, or resign from the Organization.
- There Is No Antimemetics Division
- There Is No Antimemetics Division
Video from the paper: https://www.youtube.com/watch?v=zvi4rZQDyJE
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What about drawing a pelican riding a bicycle?
My neurons still don't get themselves: What kind of processing happens INSIDE neurons?
> The output of the first neuron is fed into the second neuron, whose outputis connected to an actuator which applies the specified amount of torque to the handlebars. As inputs to the network, we provide the desired heading θ_d, as well as the current heading θ and the degree to which the bicycle is currently leaning γ, along with their derivatives
˙θ and ˙γ.
It's somewhat important to consider the inputs, because if you want to make a classifier that can classify "inside circle vs outside circle" but the network needs to derive the nonlinearity itself, then you end up needing a more complex network
Eg on the playground^, see how many neurons you need to train a circle without using more than x1 and x2?
And yet, if you give the network x1^2 and x2^2, it can solve it with minimal additional neurons.
^ https://playground.tensorflow.org/#activation=tanh&batchSize...
It's somewhat important to consider the inputs, because if you want to make a classifier that can classify "inside circle vs outside circle" but the network needs to derive the nonlinearity itself, then you end up needing a more complex network
Eg on the playground^, see how many neurons you need to train a circle without using more than x1 and x2?
And yet, if you give the network x1^2 and x2^2, it can solve it with minimal additional neurons.
^ https://playground.tensorflow.org/#activation=tanh&batchSize...
The instability ink-lines look like a flower blooming.
Observation: 2 neurons, 2 wheels. One for each?
Observation: 2 neurons, 2 wheels. One for each?
Sadly not quite one per wheel, though that would make the title even better
Figure 2 is beautiful!
This is an interesting way to think about how to get to a minimal form of a complex system.
A friend in college told me of a research project that had managed to balance a simulated inverted pendulum in 2D using 25 neurons and back propagation. But I had done this exact problem with conventional state space controls using only 5 summations (the equivalent of 5 neurons).
After I finish patting myself on the back, you then wonder what it would take for that 25 neuron solution to keep optimizing down the theoretical 5 neuron solution? The article is an interesting approach to that problem.
The paper they reference used 3456 input neurons and 9 output neurons, with no hidden nodes. They designed their input and output differently, so it's not a direct comparison. The optimized solution has 17 inputs, 2 outputs, and 2 hidden nodes. That's a massive level of optimization.
A friend in college told me of a research project that had managed to balance a simulated inverted pendulum in 2D using 25 neurons and back propagation. But I had done this exact problem with conventional state space controls using only 5 summations (the equivalent of 5 neurons).
After I finish patting myself on the back, you then wonder what it would take for that 25 neuron solution to keep optimizing down the theoretical 5 neuron solution? The article is an interesting approach to that problem.
The paper they reference used 3456 input neurons and 9 output neurons, with no hidden nodes. They designed their input and output differently, so it's not a direct comparison. The optimized solution has 17 inputs, 2 outputs, and 2 hidden nodes. That's a massive level of optimization.
it takes around 7
This reminds me of a sexist joke. But seriously... This must be much much more... Even I noticed that for example you can lean forward to have better turning curve
now make this one-bit quantized
The simulation is so simplified that I see no terms for the control of pedaling. Riding a real bicycle isn't just about steering and leaning a bit. You need to propel the bicycle a certain amount.
The paper buries this in the following:
They say 'damping instabilities' but it is way more than that, because as anyone who has learned to ride a bike knows, the hard part is getting started at that zero point of forward velocity - how to apply torque to the crank at the same time as compensating with the steering to balance at such low momentum. It's not a trivial solution to 'damping instabilities' when getting going in the first place is the most difficult part (as any 5 year old child will demonstrate).