Multiple Trace Theory - Mathematical Formulation - Matrix Definition of Traces

Matrix Definition of Traces

By assigning numerical values to all possible attributes, it is convenient to construct a column vector representation of each encoded item. This vector representation can also be fed into computational models of the brain like neural networks, which take as inputs vectorial "memories" and simulate their biological encoding through neurons.

Formally, one can denote an encoded memory by numerical assignments to all of its possible attributes. If two items are perceived to have the same color or experienced in the same context, the numbers denoting their color and contextual attributes, respectively, will be relatively close. Suppose we encode a total of L attributes anytime we see an object. Then, when a memory is encoded, it can be written as m₁ with L total numerical entries in a column vector:

A subset of the L attributes will be devoted to contextual attributes, a subset to physical attributes, and so on. One underlying assumption of multiple trace theory is that, when we construct multiple memories, we organize the attributes in the same order. Thus, we can similarly define vectors m₂, m₃, ..., m_n to account for n total encoded memories. Multiple trace theory states that these memories come together in our brain to form a memory matrix from the simple concatenation of the individual memories:

$\mathbf{M} = \begin{bmatrix} \mathbf{m_1} & \mathbf{m_2} & \mathbf{m_3} & \cdots & \mathbf{m_n} \end{bmatrix} = \begin{bmatrix} m_{1}(1) & m_{2}(1) & m_{3}(1) & \cdots & m_{n}(1) \\ m_{1}(2) & m_{2}(2) & m_{3}(2) & \cdots & m_{n}(2) \\ \vdots & \vdots & \vdots & \vdots & \vdots \\m_{1}(L) & m_{2}(L) & m_{3}(L) & \cdots & m_{n}(L)\end{bmatrix}$ .

For L total attributes and n total memories, M will have L rows and n columns. Note that, although the n traces are combined into a large memory matrix, each trace is individually accessible as a column in this matrix.

In this formulation, the n different memories are made to be more or less independent of each other. However, items presented in some setting together will become tangentially associated by the similarity of their context vectors. If multiple items are made associated with each other and intentionally encoded in that manner, say an item a and an item b, then the memory for these two can be constructed, with each having k attributes as follows:

Read more about this topic: Multiple Trace Theory, Mathematical Formulation

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