**Topic**: How are concepts represented by the brain?

Computers can represent concepts as programs, in a chosen programming language. But how about brains?

**Questions**:

- How is a concept represented?
- How does a directed, weighted, dynamic graph represent/form a memory/concept?
- What is a memory?
- interrelated
- reproducible (with stimulus or without stimulus)
- distinguishable
- corresponds to a sampleable distribution
- hierarchical

**Concrete requirements for memory**

- Similar response for similar stimuli
- Distinguishable
- Hierarchical (concepts of concepts)

# Simple Connectome Model

Neurons are modeled as a directed, weighted and dynamic graph:

= strength of edge (i, j)

= activation level of a neuron i

Suppose W is fixed and only X changes with time t. Then,

= edge weight

activation level i at time t

If we use a linear threshold function for :

At equilibrium, it converges to (eigenvector of X).

This model (with a linear threshold function) is not satisfactory because no matter which stimulus we give, it converges to the same X!

There are 2 detailed models in the literature:

- Neuroidal Model (Valiant)
- Cell Assemblies (Hebb)

**Neuroidal Model**

Concept is stored as an “item” (a subset of neurons).

Each concept is memorized as a subset of neurons of size r, and if k out of r neurons fire, that concept is recalled.

Using this model, we can memorize up to number of concepts.

Since each concept should be distinguishable, overlaps between subsets should be small!

**Assumption**:

- Base graph is random () and support is fixed (weights will be changed).
- Output is random
*JOIN*&*LINK*operations

How do we represent hierarchical concepts in neuroidal model?

If is a concept that is composed of and , we want to fire when and both fire.

**Use union**:- is simply an union of and
- Problem: concept size doubles for every union (not stable)!

**Create another subset of size r**:- We want to set up so that if k neurons in fire and k neurons in fire, then k neurons in also fire.
- Pick (“recruit”) neurons that is connected to both and
- P(neuron l fires when fires) = P (on r tosses of p biased coin, we get at least k heads) = q(r,p,k). Since it should happen for both and , it should be . We want

**Cell Assemblies**

A concept is stored as an “assembly” of highly interconnected neurons. Because of such high interconnectivity, some assembly member neurons can activate the entire assembly.

**Hypothesis**:

- Reader neurons
- Rules (neural syntax)
- “Synapsembles”: weights are dynamically changing all the time

**Model**:

Suppose an external stimulus X(0) is given. Then,

How should we change weights W? We should strengthen the connection between two neurons if both keep firing:

Also, we normalize the pre-synaptic weights at each neuron by keeping the sum of all incoming weights at 1.

Note that changes depending on both and both fire.