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.

Quantitatively, we can write the following general expression for the encoding:

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

Here we have introduced the neural dynamics matrix, $\mathbf{A}'$, and input matrix $\mathbf{B}'$. These matrices define the dynamics of the system, and can be related to the standard dynamics and input matrices in linear control theory using:

\begin{align} \mathbf{A}' &= \tau \mathbf{A} + \mathbf{I} \\ \mathbf{B}' &= \tau \mathbf{B}. \end{align}

The signal $\mathbf{u}(t)$ is the input and $\mathbf{x}(t)$ is the neural population's represented state vector. Note: $*$ indicates convolution.

As noted in section 1.3 (this and all subsequent section references are to the book), we can adapt standard control theory to be useful for modeling neurobiological systems by accounting for intrinsic neuron dynamics. There are a number of features of control theory that make it extremely useful for modeling neurobiological systems. First, the general form of control systems, captured by the state-space equations, can be used to relate to the dynamics of non-biological systems (with which engineers may be more familiar) to the dynamics of neurobiological systems. Second, the engineering community is very familiar with the state-space approach for describing the dynamic properties of physical systems, and thus has many related analytical tools for characterizing such systems. Third, modern control theory can be used to relate the dynamics of external variables, like actual joint angles, to internal variables, like desired joint angles. This demonstrates how one formalism can be used to span internal and external descriptions of behavior.

Adopting this perspective on neural dynamics allows us to develop a characterization of what we call a "generic neural subsystem." This multi-level, quantitative characterization of neural systems serves to unify our discussions of neural representation, transformation, and dynamics (section 8.1.3). Given our previous discussion regarding the importance of nonlinear computations, a focus on standard control theory, which deals mainly with linear dynamics, may seem unwarranted. However, contemporary nonlinear control theory, which may prove more valuable in the long run, depends critically on our current understanding of linear systems. Thus, showing how linear control theory relates to neurobiological systems has the effect of showing how nonlinear control theory relates to neurobiological systems as well. In fact, many of the examples we provide are of nonlinear dynamic systems (see sections 6.5 and 8.2).