This is slightly modified from the section of our book (Neural Engineering, MIT Press) that describes each of the three principles of the NEF using equations.

### Three principles of neural engineering quantified

In order for the principles of neural engineering to underwrite a theory of neurobiology, they must be quantitatively expressed. As we have shown that each of scalars, vectors, and functions can be treated as vector representation (in the book), we only present these principles as they relate to vectors.

#### Principle 1

Neural representations are defined by the combination of nonlinear encoding and weighted linear decoding.

Neural encoding is defined by

$$\sum_n \delta(t - t_{in}) = G_i \left[\alpha_i \langle \tilde{\phi_i} \mathbf{x}(t) \rangle_m + J_i^{bias} \right]$$

Neural decoding is defined by

$$\mathbf{\hat{x}}(t) = \sum_i a_i(\mathbf{x}(t)) \phi_i^{\mathbf{x}},$$

where

\begin{align} a_i(\mathbf{x}(t)) &= \sum_n h_i(t) * \delta(t - t_{in}) \\ &= \sum_n h_i (t - t_{in}). \end{align}

In both cases, $i$ indexes neurons in population and $n$ indexes spikes transmitted from one population to another.

#### Principle 2

Transformations of neural representations are functions of the variable that is represented by the population. Transformations are determined using an alternately weighted linear decoding.

Assuming the encoding in principle 1, we can estimate a function of $\mathbf{x}(t)$ as

$$\hat{f}(\mathbf{x}(t)) = \sum_i a_i (\mathbf{x}(t)) \phi_i^f,$$

where $a_i(\mathbf{x}(t))$ is defined as before. The only difference between this decoding and the representational decoding are the decoders themselves.

#### Principle 3

Neural dynamics are characterized by considering neural representations as control theoretic state variables. Thus, the dynamics of neurobiological systems can be analyzed using control theory.

Allowing $\mathbf{x}(t)$ to be a state variable and $\mathbf{u}(t)$ to be the input, we have the following general expression for the encoding:

$$\sum_n \delta(t - t_{in}) = G_i \left[ \alpha_i \left\langle \tilde{\phi_i} \left( h_i(t) * \left[ \mathbf{A}' \mathbf{x}(t) + \mathbf{B}' \mathbf{u}(t) \right] \right) \right\rangle_m + J_i^{bias} \right]$$

To get a better understanding of how these principles can be used together, please refer to the various examples on the website. For several simple applications, see this paper.