Neural representations are defined by the combination of nonlinear encoding (exemplified by neuron tuning curves) and weighted linear decoding.

Quantitatively, we can write the encoding:

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

and the decoding:

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


\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 these equations, $a$ is the activity of a neuron, $i$, encoding a vector signal $\mathbf{x}(t)$, in a spike train (represented as a sum of delta functions, with each spike indexed by $n$). The encoding is defined by the nonlinear function, $G$, and parameterized by a gain, $\alpha$, and a bias, $J^{bias}$. The encoding vector $\tilde{\phi}$ determines the neuron's preferred stimulus in the signal (the elements inside the square brackets determine the soma current). An estimate of the signal is recovered through linear population decoding, using the vectors $\phi$ and linear temporal decoding using the filter $h(t)$. These can be combined to give a population-temporal decoder, $\phi_i(t - t_{in})$.

Principle 1 emphasizes the importance of identifying both encoding and decoding when defining neural representation. Moreover, this principle highlights the central assumption that, despite a nonlinear encoding, linear decoding is valid (see Rieke et al. 1997, pp. 76-87). As discussed in detail by Rieke et al., a nonlinear response function like that of typical neurons is, in fact, unrelated to whether or not the resulting signal can be linearly decoded. That is, the nature of the input/output function (i.e., encoding) of a device is independent of the decoder that is needed to estimate its input. This means that a nonlinear encoding could need a linear or nonlinear decoding, and vice versa. This is because the decoding depends on the conditional probability of input given the output and on the statistics of the noise (hence our addendum). Perhaps surprisingly, linear decoding works quite well in many neural systems. Specifically, the additional information gained with nonlinear decoding is generally less than 5%.

Of course, nonlinear decoding is able to do as well or better than linear decoding at extracting information, but the price paid in biological plausibility is generally thought to be quite high (see, e.g., Salinas and Abbott 1994). Furthermore, even if there initially seems to be a case in which nonlinear decoding is employed by a neural system, that decoding may, in the end, be explained by linear decoding. This is because, as we discuss in section 6.3 (this and all subsequent section references are to the book), nonlinear transformations can be performed using linear decoding. Thus, assuming linear decoding at the neuron (or sub-neuron, see section 6.3) level can well be consistent with nonlinear decoding at the network (or neuron) level. So, especially in combination with principle 2, linear decoding is a good candidate for describing neural decoding in general.

It is important to emphasize that analyzing neurons as decoding signals using (optimal) linear or nonlinear filters does not mean that neurons are presumed to explicitly use opti-mal filters. In fact, according to our account, there is no directly observable counterpart to these optimal decoders. Rather, the decoders are embedded in the synaptic weights between neighboring neurons. That is, coupling weights of neighboring neurons indirectly reflect a particular population decoder, but they are not identical to the population decoder, nor can the decoder be unequivocally read-off of the weights. This is because connection weights are determined by both the decoding of incoming signals and the encoding of the outgoing signals (see, e.g., section 6.2). Practically speaking, this means that changing a connection weight both changes the transformation being performed and the tuning curve of the receiving neuron. As is well known from work in artificial neural networks and computational neuroscience, this is exactly what should happen. In essence, the encoding/decoding distinction is not one that neurobiological systems need to respect in order to perform their functions, but it is extremely useful in trying to understand such systems and how they do, in fact, manage to perform those functions.