<p>I would wager <em>yes</em> if you don't constrain path length, but it would depend on how you define a &quot;connection&quot;; these things are not binary in the brain, and you could definitely make graphs for some definitions of &quot;connection&quot; where things are not connected.</p>
<p>For an example of an argument for &quot;no&quot;, you would plausibly have sensory neurons where there is a direct connection x -&gt; y but not in reverse y -&gt; x for some definition of &quot;connection&quot;. Take a photoreceptor cell in the <a href="https://en.wikipedia.org/wiki/Retina" rel="nofollow noreferrer">retina</a> as x, for example. That would leave a one-way node that you can reach the whole brain from but you can't reach from the whole brain. You'll encounter similar one-way &quot;gates&quot; between retinal ganglion cells and the LGN. But, in reality, no connection like this is ever fully one-way and there is always going to be some level of two-way communication across a synapse, even if it's not the primary 'information carrying' path one would ordinarily be interested in. Then the question would be how much you need to limit your scope of what you consider &quot;in the brain&quot; to make the answer &quot;true&quot; again. Given the extent of feedback connectivity in the brain more generally, I think you will find everything reciprocally connected and therefore connected if you're focusing on cortex/thalamus/basal ganglia/cerebellum/etc and not sensory/motor systems specifically.</p>
<p>I don't think proving the &quot;yes&quot; is feasible outside of a simplified model organism except by arguments from probability given the extraordinary number of total connections made, but consider the implication if &quot;yes&quot; were true outside of the one-way gate-like sort of circumstance: you'd have not just some modest disconnection but two or more completely independent functional networks, and that doesn't seem at all possible and has never been described. Even rare conditions like <a href="https://en.wikipedia.org/wiki/Split-brain" rel="nofollow noreferrer">split-brain</a> patients are not <em>completely</em> split. In a split-brain patient the left and right hemispheres are disconnected at the corpus callosum leading to clear psychological separation of information processing, but there are other pathways that would create some connection in the mathematical sense, and once any connection is made across hemispheres the shear number of connections would take over.</p>
<p>Anyways, I think this sort of question is not particularly interesting from a biological perspective because the thresholds (i.e.: what is &quot;connected&quot;?) you choose create trivial solutions in one way or the other. Biologists are going to be more interested with how the answer varies over choice of those thresholds and varies by brain state (e.g., while performing a certain task versus at rest; while conscious versus unconscious) for a given threshold choice/definition of &quot;connected&quot;, and by abstracting the units involved a bit (individual neurons versus assemblies/regions/networks).</p>

CC BY-SA 4.0
2024-07-15T17:27:31.733Z
Thank you for the detailed answer. I was certainly speaking only of connectivity within the brain - i.e. excluding sensory-motor systems. Your insights were illuminating.
lafinur
CC BY-SA 4.0
2024-07-15T17:34:26.903Z
@lafinur It's a bit complicated because some sensory systems like vision and olfactory are really part of the brain itself. So if you want to focus on just parts of the brain it's best to specify what those parts are. I think a lot of non-neuroscientists interested in brain networks might say "brain" when they mean "cortex", for example. Definitely worth spending the 10 minutes to read about brain structures and not be one of those people.
Bryan Krause
CC BY-SA 4.0
2024-07-15T13:50:13.833Z
2024-07-15T15:33:44.917Z
2024-07-15T15:33:44.917Z
Bryan Krause
<p>In mathematics, a graph <span class="math-container">$G = (V, E)$</span> is <em>connected</em> if, given any pair of nodes <span class="math-container">$x, y$</span>, there is a path <span class="math-container">$x\ldots y$</span>.</p>
<p>The brain is of course an intractably large network and can be modeled with a directed graph. However, I don't know if it has been determined, or even if it <em>could</em> be determined, if the brain is connected or not. This is, if given any pair of arbitrary neurons in the brain, there is a path of axons which would lead from the first to the latter element of the pair.</p>
<p>Is this something we know? If we know it, what is the answer? And what does it entail?</p>

<neuroscience><brain><neuroanatomy>
Is the brain's network connected in the graph theoretic sense
lafinur
# Is the brain's network connected in the graph theoretic sense

(Asked by: username_0 on 2024-07-13)

In mathematics, a graph $G = (V, E)$ is *connected* if, given any pair of nodes $x, y$, there is a path $x\ldots y$.

The brain is of course an intractably large network and can be modeled with a directed graph. However, I don't know if it has been determined, or even if it *could* be determined, if the brain is connected or not. This is, if given any pair of arbitrary neurons in the brain, there is a path of axons which would lead from the first to the latter element of the pair.

Is this something we know? If we know it, what is the answer? And what does it entail?

**Tags:** neuroscience, brain, neuroanatomy


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## 1 answers

### Answer by username_1 on 2024-07-15. Score: 2 | Accepted answer

I would wager *yes* if you don't constrain path length, but it would depend on how you define a "connection"; these things are not binary in the brain, and you could definitely make graphs for some definitions of "connection" where things are not connected.

For an example of an argument for "no", you would plausibly have sensory neurons where there is a direct connection x -> y but not in reverse y -> x for some definition of "connection". Take a photoreceptor cell in the [retina](https://en.wikipedia.org/wiki/Retina) as x, for example. That would leave a one-way node that you can reach the whole brain from but you can't reach from the whole brain. You'll encounter similar one-way "gates" between retinal ganglion cells and the LGN. But, in reality, no connection like this is ever fully one-way and there is always going to be some level of two-way communication across a synapse, even if it's not the primary 'information carrying' path one would ordinarily be interested in. Then the question would be how much you need to limit your scope of what you consider "in the brain" to make the answer "true" again. Given the extent of feedback connectivity in the brain more generally, I think you will find everything reciprocally connected and therefore connected if you're focusing on cortex/thalamus/basal ganglia/cerebellum/etc and not sensory/motor systems specifically.

I don't think proving the "yes" is feasible outside of a simplified model organism except by arguments from probability given the extraordinary number of total connections made, but consider the implication if "yes" were true outside of the one-way gate-like sort of circumstance: you'd have not just some modest disconnection but two or more completely independent functional networks, and that doesn't seem at all possible and has never been described. Even rare conditions like [split-brain](https://en.wikipedia.org/wiki/Split-brain) patients are not *completely* split. In a split-brain patient the left and right hemispheres are disconnected at the corpus callosum leading to clear psychological separation of information processing, but there are other pathways that would create some connection in the mathematical sense, and once any connection is made across hemispheres the shear number of connections would take over.

Anyways, I think this sort of question is not particularly interesting from a biological perspective because the thresholds (i.e.: what is "connected"?) you choose create trivial solutions in one way or the other. Biologists are going to be more interested with how the answer varies over choice of those thresholds and varies by brain state (e.g., while performing a certain task versus at rest; while conscious versus unconscious) for a given threshold choice/definition of "connected", and by abstracting the units involved a bit (individual neurons versus assemblies/regions/networks).

> **Comments:**
> **username_0**: Thank you for the detailed answer. I was certainly speaking only of connectivity within the brain - i.e. excluding sensory-motor systems. Your insights were illuminating.  
> **username_1**: @username_0 It's a bit complicated because some sensory systems like vision and olfactory are really part of the brain itself. So if you want to focus on just parts of the brain it's best to specify what those parts are. I think a lot of non-neuroscientists interested in brain networks might say "brain" when they mean "cortex", for example. Definitely worth spending the 10 minutes to read about brain structures and not be one of those people.  

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