4 ms·

It takes two neurons to ride a bicycle (2004)

127 points by malshe 5mo ago

fintler 5mo ago
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/ 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/ https://blog.typeobject.com/posts/2026-timing-is-the-bit/
- shomp 5mo ago
  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 :)
  - fintler 5mo ago
    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.
    - shomp 5mo ago
      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.
ebhn 5mo ago
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.
- KolibriFly 5mo ago
  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
- taneq 5mo ago
  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.
  - malshe 5mo ago
    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.
hyperhello 5mo ago
So can we have self-driving bicycles?
- deleted 5mo ago
  [deleted]
- soupspaces 5mo ago
  Recumbent bike with lidar and maps? Sign me up.
- onesingleblast 5mo ago
  Yes and they'll have one of those wetware computers on board
- KolibriFly 5mo ago
  Self-balancing bicycles, sure. Self-driving bicycles that navigate city streets safely are a much larger problem.
shomp 5mo ago
The instability ink-lines look like a flower blooming. Observation: 2 neurons, 2 wheels. One for each?
- KolibriFly 5mo ago
  Sadly not quite one per wheel, though that would make the title even better
wrsh07 5mo ago
> 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=10&dataset=circle&regDataset=reg-plane&learningRate=0.03&regularizationRate=0&noise=0&networkShape=1,1&seed=0.07205&showTestData=false&discretize=false&percTrainData=50&x=false&y=false&xTimesY=false&xSquared=true&ySquared=true&cosX=false&sinX=false&cosY=false&sinY=false&collectStats=false&problem=classification&initZero=false&hideText=false https://playground.tensorflow.org/#activation=tanh&batchSize...
klas_segeljakt 5mo ago
What about drawing a pelican riding a bicycle?
Razengan 5mo ago
My neurons still don't get themselves: What kind of processing happens INSIDE neurons?
overfits-ai 5mo ago
[flagged]
actinium226 5mo ago
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.
- KolibriFly 5mo ago
  The useful insight is not "compare desired state to actual state"; it's deciding which state to control
- gpvos 5mo ago
  I assume you mean proportional–integral–derivative control. https://en.wikipedia.org/wiki/PID_controller https://en.wikipedia.org/wiki/PID_controller
- dchristian 5mo ago
  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.
  - taneq 5mo ago
    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.
  - literalAardvark 5mo ago
    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.
  - malfist 5mo ago
    If you can't use a PID for unstable things, why does cruise control exist for motorcycles?
  - pinkmuffinere 5mo ago
    > 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.
gpvos 5mo ago
(2004) Previously: - https://news.ycombinator.com/item?id=19196664 https://news.ycombinator.com/item?id=19196664 (25 comments) - https://news.ycombinator.com/item?id=16215130 https://news.ycombinator.com/item?id=16215130 (88 comments)
mk_stjames 5mo ago
It seems like it should say "It takes Two Neurons to Steer an already moving Bicycle". 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: >Although the two-neuron network controller works well for a range of speeds, one thing the controller does not do is to try to dampen the instabilities that can arise when riding too slowly or in too sharp of a turn. (This would probably require a third neuron that isdedicated to this task.) 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).
- fwipsy 5mo ago
  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.
  - CDRdude 5mo ago
    You absolutely can start a bike by pedaling with no forward momentum. You can see it when someone starts pedaling again after a track stand.
    - fwipsy 5mo ago
      So now we need enough neurons to do a track stand too?? I don't have the neurons for that...
  - sandworm101 5mo ago
thatxliner 5mo ago
now make this one-bit quantized
KolibriFly 5mo ago
[flagged]
BiraIgnacio 5mo ago
> 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
cnees 5mo ago
Figure 2 is beautiful!
sandworm101 5mo ago
>> 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 https://www.youtube.com/shorts/Gyt9DLfYOdU
- lloeki 5mo ago
  > 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/ https://ciechanow.ski/bicycle/ It seems that it is something that is forever doomed to be forgotten and then rediscovered over and over.
  - sandworm101 5mo ago
    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/Countersteering https://en.wikipedia.org/wiki/Dihedral_(aeronautics) https://en.wikipedia.org/wiki/Dihedral_(aeronautics)
- orbital-decay 5mo ago
  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.
deleted 5mo ago
[deleted]
p0w3n3d 5mo ago
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
dchristian 5mo ago
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.
mangomanai 5mo ago
it takes around 7
ryankrage77 5mo ago
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.
kazinator 5mo ago
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.
kushagra1211 5mo ago
[flagged]
vismit2000 5mo ago
Video from the paper: https://www.youtube.com/watch?v=zvi4rZQDyJE https://www.youtube.com/watch?v=zvi4rZQDyJE

127 points by malshe 5mo ago

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/ 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/ https://blog.typeobject.com/posts/2026-timing-is-the-bit/

shomp 5mo ago
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 :)
- fintler 5mo ago
  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.
  - shomp 5mo ago
    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.

ebhn 5mo ago

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.

KolibriFly 5mo ago
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
taneq 5mo ago
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.
- malshe 5mo ago
  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.

hyperhello 5mo ago

So can we have self-driving bicycles?

deleted 5mo ago
[deleted]
soupspaces 5mo ago
Recumbent bike with lidar and maps? Sign me up.
onesingleblast 5mo ago
Yes and they'll have one of those wetware computers on board
KolibriFly 5mo ago
Self-balancing bicycles, sure. Self-driving bicycles that navigate city streets safely are a much larger problem.

shomp 5mo ago

The instability ink-lines look like a flower blooming. Observation: 2 neurons, 2 wheels. One for each?

KolibriFly 5mo ago
Sadly not quite one per wheel, though that would make the title even better

wrsh07 5mo ago

> 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=10&dataset=circle&regDataset=reg-plane&learningRate=0.03&regularizationRate=0&noise=0&networkShape=1,1&seed=0.07205&showTestData=false&discretize=false&percTrainData=50&x=false&y=false&xTimesY=false&xSquared=true&ySquared=true&cosX=false&sinX=false&cosY=false&sinY=false&collectStats=false&problem=classification&initZero=false&hideText=false https://playground.tensorflow.org/#activation=tanh&batchSize...

klas_segeljakt 5mo ago

What about drawing a pelican riding a bicycle?

Razengan 5mo ago

My neurons still don't get themselves: What kind of processing happens INSIDE neurons?

overfits-ai 5mo ago

[flagged]

actinium226 5mo ago

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.

KolibriFly 5mo ago
The useful insight is not "compare desired state to actual state"; it's deciding which state to control
gpvos 5mo ago
I assume you mean proportional–integral–derivative control. https://en.wikipedia.org/wiki/PID_controller https://en.wikipedia.org/wiki/PID_controller
dchristian 5mo ago
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.
- taneq 5mo ago
  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.
- literalAardvark 5mo ago
  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.
- malfist 5mo ago
  If you can't use a PID for unstable things, why does cruise control exist for motorcycles?
- pinkmuffinere 5mo ago
  > 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.

gpvos 5mo ago

(2004) Previously: - https://news.ycombinator.com/item?id=19196664 https://news.ycombinator.com/item?id=19196664 (25 comments) - https://news.ycombinator.com/item?id=16215130 https://news.ycombinator.com/item?id=16215130 (88 comments)

mk_stjames 5mo ago

It seems like it should say "It takes Two Neurons to Steer an already moving Bicycle". 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: >Although the two-neuron network controller works well for a range of speeds, one thing the controller does not do is to try to dampen the instabilities that can arise when riding too slowly or in too sharp of a turn. (This would probably require a third neuron that isdedicated to this task.) 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).

fwipsy 5mo ago
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.
- CDRdude 5mo ago
  You absolutely can start a bike by pedaling with no forward momentum. You can see it when someone starts pedaling again after a track stand.
  - fwipsy 5mo ago
    So now we need enough neurons to do a track stand too?? I don't have the neurons for that...
- sandworm101 5mo ago

thatxliner 5mo ago

now make this one-bit quantized

KolibriFly 5mo ago

[flagged]

BiraIgnacio 5mo ago

> 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

cnees 5mo ago

Figure 2 is beautiful!

sandworm101 5mo ago

>> 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 https://www.youtube.com/shorts/Gyt9DLfYOdU

lloeki 5mo ago
> 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/ https://ciechanow.ski/bicycle/ It seems that it is something that is forever doomed to be forgotten and then rediscovered over and over.
- sandworm101 5mo ago
  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/Countersteering https://en.wikipedia.org/wiki/Dihedral_(aeronautics) https://en.wikipedia.org/wiki/Dihedral_(aeronautics)
orbital-decay 5mo ago
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.

deleted 5mo ago

[deleted]

p0w3n3d 5mo ago

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

dchristian 5mo ago

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.

mangomanai 5mo ago

it takes around 7

ryankrage77 5mo ago

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.

kazinator 5mo ago

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.

kushagra1211 5mo ago

[flagged]

vismit2000 5mo ago

Video from the paper: https://www.youtube.com/watch?v=zvi4rZQDyJE https://www.youtube.com/watch?v=zvi4rZQDyJE