**Tags**

brain, combinatorics, graph theory, math, neurons, neuroscience, patterns

T.N

tnaithani96@gmail.com

August 1, 2020**Abstract**

While reading about brain,this problem struck me. How many mem-

ories are possible in Brain given N number of neurons?. A memory in

brain is a result of neurons fired in a particular sequence.If every pattern

of Neuron firing corresponded to a new memory, which can have any func-

tions from playing music to building sentences. How many memories can

the brain can have, is it infinite or a finite number.**Introduction**

The crux of this article is to find a solution to the problem of how many memories

can the brain have or how many unique Patterns are possible for a given number

of Neurons or Nodes. I would move forward with taking calling Neurons as

Nodes. With a given number of Nodes how many connections are possible?

Since for any given Number of Nodes we can know total number of edges will

be E. Where E is,

E = N (N − 1)/2

Where N is number of Nodes

The above examples are connected graphs of 5 and 4 nodes with number of

edges: n(n − 1)/2

1Theorem

Now considering there are N nodes in brain how many unique patterns will be

position,Under the condition that we need atleast one active connection between

2 neurons to trigger a specific memory.

Example of patterns in 3 Node system

The above are the patterns formed by a 3 Node system. The approach we can take while making patterns like these is to consider every Edge as an unique space.

For patterns in row one we are selecting one of the 3 spaces so the patterns

that could be formed are 31

Similarly,Patterns in row 2 are selecting two of 3 unique places so they are

3

2 .

So, the total number of Patterns can are

In the above case we have had 3 unique spaces, What if there are N nodes.

The total number of unique spaces for a N node system is N (N − 1)/2 The new

number of patterns are

The important detail we missed till now is, since these are uniques spaces(Space

between 2 nodes/neurons) the 2 differnt directions are possible. Since, The sequence of firing can trigger different patterns.

Even though the patterns have same connections but the direction of each

edge makes these 3 different configurations.

Each active edge can have one of the two direction. When taking directions

into account the total number of patterns form changes to the following:

Where m denotes minimum number of active connections threshold for any memory**Conclusion**

The answer to how many memories or patterns can be formed in brain with N

neurons are:

Where m is the minimum connection restraint

This is the total number of patterns, I read that there are 90 Billion neurons

in brain so you can calculate the Patterns by putting N = 90 ∗ 10 9 . Which

would given a finite answer though a big one.

Reference

Linear algebra and it’s application by Gilbert Strang