<p>You might want to take a look at the work about conceptual spaces by Peter Gärdenfors, e.g. <a href="https://rads.stackoverflow.com/amzn/click/0262572192" rel="nofollow noreferrer">https://www.amazon.com/Conceptual-Spaces-Geometry-Thought-Press/dp/0262572192</a></p>

<p>Also, the work about semantic pointers might be interesting here: <a href="https://www.ncbi.nlm.nih.gov/pubmed/26235459" rel="nofollow noreferrer">https://www.ncbi.nlm.nih.gov/pubmed/26235459</a></p>

<p>My opinion with regard to this is, that using set theory to represent concepts is a fruitful approach, but there needs to be a precise definition of the "space" we are talking about as well as if and how it is possible to define some metric on this space or a higher structure, e.g. if it is a vector space. This metric must not only take into account set overlaps but also principal phenomena corresponding to what we know about neurobiology, e.g. classical/operant conditioning etc. </p>

<p>Let's try to build this up systematically (Note, that I do not use a fuzzy set definition here but it might provide some more flexibility and thanks for putting me on that track ;)): We could define the set of all neurons </p>

<pre><code>N = {n_1,n_2,...,n_m}
</code></pre>

<p>and the set of all possible states of a neuron (avoiding, for now, a precise definition of what a state is. For example, a state could be a frequency or some pattern of action potentials, e.g. rate/temporal/population code or just a discrete value of the membrane potential)</p>

<pre><code>S={s_1,s_2,...,s_l}
</code></pre>

<p>The Cartesian product set <code>N x S</code> will give us the set of all neurons in all possible states <code>C_0</code>: </p>

<pre><code>C_0={n_1(s_1),n_1(s_2),n_2(s_1),n_2(s_2),...,n_m(s_l)}
</code></pre>

<p>Then let's define a <em>concept space</em> as a mapping <code>f1</code> from <code>C_0</code> to a new set <code>C_1</code></p>

<pre><code>C_1={{n_1(s_1),n_3(s_3),n_4(s_1),n_5(s_8)},{n_1(s_1),n_2(s_3),n_5(s_9)},...}={c1_1,c1_2,...,c1_x}
</code></pre>

<p>For example, an element (<code>c1_x</code>) of <code>C_1</code> could be a visual representation of a house or some sound or smell or just a point or a line. This is what Thagard &amp; Steward (2014) defined as a semantic pointer and it shares some similarity with Tononi's (e.g. 2016) approach of a qualia space (although approached from a different (set theoretic) angle since Tononi uses information theory). This also is similar to Gardenförs' concept of a conceptual space, although he did not make his definitions mathematically precise. Because strictly, <code>C_1</code> still has no structure, i.e. is not a vector space or more generally a metric space. There still has to be defined some distance function, which is not easy to be found. </p>

<p>Now, it is possible to generate more and more sets representing more and more complex concepts by consecutive mappings <code>f1 to fn</code>. That is we can generate a new set <code>C_2</code> consisting of elements of <code>C_1</code>:</p>

<pre><code>C_2={{c1_1,c1_3,c1_5,c1_9},{c1_7,c1_1,c1_8},...}={c2_1,c2_2,...,c2_x}
</code></pre>

<p>Imagine the element <code>c2_1</code> consisting of four lines with coordinates of length <code>l1 to l4</code> each of them in a specific angle <code>a1 to a4</code> such that they create a square. Or take <code>c2_2</code> as three lines with length <code>l1 to l3</code> and angles <code>a1 to a3</code> such that they create a triangle.</p>

<p>This process can be repeated to generate more and more abstract concepts. I think it is easy to understand it when considering how letters are combined to words and words form sentences and sentences form even more abstract meanings.</p>

<p>There are now many questions that arise, e.g. are the mappings injective, surjective or bijective? Do inverse mappings exist? What is a suitable metric that is in accord with neurobiological and experimental data? Can a higher structure e.g. vector space be defined?</p>

<p>Finally, I can try to answer your questions: </p>

<ol>
<li><p>Based on the above definition one can assume that more general concept e.g. a cube is formed by a set of lines, i.e. a set of semantic pointers forms another set of semantic pointers. For example, a complex movement with the hand was initially learned by combining different simple movements. Or take learning to speak a word for example.</p></li>
<li><p>Also based on the definition above, there must not be an overlap between sets at all! Take two very different semantic pointers for example, like barking and the visual representation of a dog. Whereas one set of semantic pointers might represent the sound, the other may represent the visual aspect, but they might not share the same neurons. But all of them together create a concept of a dog. So here again, the definition of similarity seems to be not just a simple set overlap metric (e.g. Szymkiewicz-Simpson coefficient or Jaccard index). </p></li>
<li><p>This depends on the definition of important. If you define important as activated more often together, then one could just assume a simple (e.g. Hebbian or STDP) learning rule which strengthens the connection between neurons. But I doubt that more neurons will be included in the concept except when the concept is updated/broadened by new experiences. </p></li>
</ol>

<p>Hope this helps a bit! I'm trying to build a more general understanding of conceptual spaced based on set theory for a while and I always welcome discussions, critique or collaborations :).</p>

<p>Sources: </p>

<p>-Thagard, Paul, and Terrence C. Stewart. "Two theories of consciousness: Semantic pointer competition vs. information integration." Consciousness and Cognition 30 (2014): 73-90.</p>

<p>-Tononi, Giulio. "Consciousness as integrated information: a provisional manifesto." The Biological Bulletin (2016).</p>

CC BY-SA 3.0
2017-09-04T11:29:28.327Z
Thanks for the elaborate answer that gives me something to read and think about. Maybe you want to have a look at another question of mine: https://cogsci.stackexchange.com/questions/18048/hypothetical-working-and-function-of-grandmother-cells
Hans-Peter Stricker
CC BY-SA 3.0
2017-09-04T11:41:41.473Z
@HansStricker: You are welcome, let me know if something did not become clear. Gonna take a look at your other question and see if I can contribute something. If you are interested in occasional exchange or discussions about this topic, let me know and we can exchange e-mails (also from germany btw ;)).
holistic
CC BY-SA 3.0
2017-09-04T11:08:28.827Z
2017-09-04T13:26:28.550Z
2017-09-04T13:26:28.550Z
holistic
<p>It seems generally agreed upon that semantic <a href="https://www.kfki.hu/~csdori/Concepts.pdf" rel="noreferrer">concepts</a> are distributedly represented by groups and groups of groups of neurons spread over the cortex.</p>

<p>Disregarding the localizations, shapes and connectivity of these groups of neurons, I tried to consider them as sets (in the mathematical sense). Let $S(A)$ be the (probably fuzzy) set corresponding to concept $A$.</p>

<p>The only thing that I would take for granted is that for very similar semantic concepts $A$, $A'$ their corresponding sets of neurons $S(A), S(A')$ do overlap strongly. Otherwise the model of distributed representation would not make much sense to me.</p>

<p>Other answers aren't so obvious:</p>

<ol>
<li><p>Does a more <em>general</em> concept $A$ use more neurons than a more specific concept $B \subset A$ - or less? Or does it depend (on what)?</p></li>
<li><p>How do the sets of neurons of two concepts $A, B$ with $B \subset A$ overlap? Is maybe one a (fuzzy) subset of the other? More generally: How are semantical or extensional relations between two concepts reflected by set theoretic relations between their corresponding sets of neurons? (My intuitive guess is: $S(A)$ is larger than $S(B)$, and $S(B)$ is contained in $S(A)$, i.e. $S(B)\subset S(A)$. But this may be simply wrong.)</p></li>
<li><p>Does a more <em>important</em> concept $A$ (being used more often) use more neurons than a more rarely used.</p></li>
</ol>

<p>I wonder if there is any evidence (empircal or theoretical) that sheds some light on these questions.</p>

<p><em>[If you think, this question is devoid of scientific meaning or otherwise ill-posed, please let me know why.]</em></p>

CC BY-SA 3.0
2017-08-31T13:40:36.087Z
Your question may be related to the following, https://cogsci.stackexchange.com/questions/17451/how-does-the-brain-process-generalizations/17503#17503, although it has yet to receive a fully satisfying answer.
Robin Kramer-ten Have
CC BY-SA 3.0
2017-08-31T13:55:23.093Z
Your question might also be related to this question about biologically plausible knowledge representation which does have some satisfying answers https://cogsci.stackexchange.com/q/494/4397
Seanny123
<neuroscience><cognitive-neuroscience><theoretical-neuroscience>
Conceptual representations in brain by distributed groups of neurons
Hans-Peter Stricker
# Conceptual representations in brain by distributed groups of neurons

(Asked by: username_0 on 2017-08-31)

It seems generally agreed upon that semantic [concepts](https://www.kfki.hu/~csdori/Concepts.pdf) are distributedly represented by groups and groups of groups of neurons spread over the cortex.

Disregarding the localizations, shapes and connectivity of these groups of neurons, I tried to consider them as sets (in the mathematical sense). Let $S(A)$ be the (probably fuzzy) set corresponding to concept $A$.

The only thing that I would take for granted is that for very similar semantic concepts $A$, $A'$ their corresponding sets of neurons $S(A), S(A')$ do overlap strongly. Otherwise the model of distributed representation would not make much sense to me.

Other answers aren't so obvious:

1. Does a more *general* concept $A$ use more neurons than a more specific concept $B \subset A$ - or less? Or does it depend (on what)?
2. How do the sets of neurons of two concepts $A, B$ with $B \subset A$ overlap? Is maybe one a (fuzzy) subset of the other? More generally: How are semantical or extensional relations between two concepts reflected by set theoretic relations between their corresponding sets of neurons? (My intuitive guess is: $S(A)$ is larger than $S(B)$, and $S(B)$ is contained in $S(A)$, i.e. $S(B)\subset S(A)$. But this may be simply wrong.)
3. Does a more *important* concept $A$ (being used more often) use more neurons than a more rarely used.

I wonder if there is any evidence (empircal or theoretical) that sheds some light on these questions.

*[If you think, this question is devoid of scientific meaning or otherwise ill-posed, please let me know why.]*

**Tags:** neuroscience, cognitive-neuroscience, theoretical-neuroscience

> **Comments:**
> **username_1**: Your question may be related to the following, https://cogsci.stackexchange.com/questions/17451/how-does-the-brain-process-generalizations/17503#17503, although it has yet to receive a fully satisfying answer.  
> **username_2**: Your question might also be related to this question about biologically plausible knowledge representation which does have some satisfying answers https://cogsci.stackexchange.com/q/494/4397  

---

## 1 answers

### Answer by username_3 on 2017-09-04. Score: 3 | Accepted answer

You might want to take a look at the work about conceptual spaces by Peter Gärdenfors, e.g. https://www.amazon.com/Conceptual-Spaces-Geometry-Thought-Press/dp/0262572192
Also, the work about semantic pointers might be interesting here: https://www.ncbi.nlm.nih.gov/pubmed/26235459
My opinion with regard to this is, that using set theory to represent concepts is a fruitful approach, but there needs to be a precise definition of the "space" we are talking about as well as if and how it is possible to define some metric on this space or a higher structure, e.g. if it is a vector space. This metric must not only take into account set overlaps but also principal phenomena corresponding to what we know about neurobiology, e.g. classical/operant conditioning etc. 
Let's try to build this up systematically (Note, that I do not use a fuzzy set definition here but it might provide some more flexibility and thanks for putting me on that track ;)): We could define the set of all neurons 
N = {n_1,n_2,...,n_m}

and the set of all possible states of a neuron (avoiding, for now, a precise definition of what a state is. For example, a state could be a frequency or some pattern of action potentials, e.g. rate/temporal/population code or just a discrete value of the membrane potential)
S={s_1,s_2,...,s_l}

The Cartesian product set N x S will give us the set of all neurons in all possible states C_0: 
C_0={n_1(s_1),n_1(s_2),n_2(s_1),n_2(s_2),...,n_m(s_l)}

Then let's define a concept space as a mapping f1 from C_0 to a new set C_1
C_1={{n_1(s_1),n_3(s_3),n_4(s_1),n_5(s_8)},{n_1(s_1),n_2(s_3),n_5(s_9)},...}={c1_1,c1_2,...,c1_x}

For example, an element (c1_x) of C_1 could be a visual representation of a house or some sound or smell or just a point or a line. This is what Thagard & Steward (2014) defined as a semantic pointer and it shares some similarity with Tononi's (e.g. 2016) approach of a qualia space (although approached from a different (set theoretic) angle since Tononi uses information theory). This also is similar to Gardenförs' concept of a conceptual space, although he did not make his definitions mathematically precise. Because strictly, C_1 still has no structure, i.e. is not a vector space or more generally a metric space. There still has to be defined some distance function, which is not easy to be found. 
Now, it is possible to generate more and more sets representing more and more complex concepts by consecutive mappings f1 to fn. That is we can generate a new set C_2 consisting of elements of C_1:
C_2={{c1_1,c1_3,c1_5,c1_9},{c1_7,c1_1,c1_8},...}={c2_1,c2_2,...,c2_x}

Imagine the element c2_1 consisting of four lines with coordinates of length l1 to l4 each of them in a specific angle a1 to a4 such that they create a square. Or take c2_2 as three lines with length l1 to l3 and angles a1 to a3 such that they create a triangle.
This process can be repeated to generate more and more abstract concepts. I think it is easy to understand it when considering how letters are combined to words and words form sentences and sentences form even more abstract meanings.
There are now many questions that arise, e.g. are the mappings injective, surjective or bijective? Do inverse mappings exist? What is a suitable metric that is in accord with neurobiological and experimental data? Can a higher structure e.g. vector space be defined?
Finally, I can try to answer your questions: 

Based on the above definition one can assume that more general concept e.g. a cube is formed by a set of lines, i.e. a set of semantic pointers forms another set of semantic pointers. For example, a complex movement with the hand was initially learned by combining different simple movements. Or take learning to speak a word for example.
Also based on the definition above, there must not be an overlap between sets at all! Take two very different semantic pointers for example, like barking and the visual representation of a dog. Whereas one set of semantic pointers might represent the sound, the other may represent the visual aspect, but they might not share the same neurons. But all of them together create a concept of a dog. So here again, the definition of similarity seems to be not just a simple set overlap metric (e.g. Szymkiewicz-Simpson coefficient or Jaccard index). 
This depends on the definition of important. If you define important as activated more often together, then one could just assume a simple (e.g. Hebbian or STDP) learning rule which strengthens the connection between neurons. But I doubt that more neurons will be included in the concept except when the concept is updated/broadened by new experiences. 

Hope this helps a bit! I'm trying to build a more general understanding of conceptual spaced based on set theory for a while and I always welcome discussions, critique or collaborations :).
Sources: 
-Thagard, Paul, and Terrence C. Stewart. "Two theories of consciousness: Semantic pointer competition vs. information integration." Consciousness and Cognition 30 (2014): 73-90.
-Tononi, Giulio. "Consciousness as integrated information: a provisional manifesto." The Biological Bulletin (2016).

> **Comments:**
> **username_0**: Thanks for the elaborate answer that gives me something to read and think about. Maybe you want to have a look at another question of mine: https://cogsci.stackexchange.com/questions/18048/hypothetical-working-and-function-of-grandmother-cells  
> **username_3**: @HansStricker: You are welcome, let me know if something did not become clear. Gonna take a look at your other question and see if I can contribute something. If you are interested in occasional exchange or discussions about this topic, let me know and we can exchange e-mails (also from germany btw ;)).  

---

