A Predictive Coding Account for Chaotic Itinerancy

Abstract

As a phenomenon in dynamical systems allowing autonomous switching between stable behaviors, chaotic itinerancy has gained interest in neurorobotics research. In this study, we draw a connection between this phenomenon and the predictive coding theory by showing how a recurrent neural network implementing predictive coding can generate neural trajectories similar to chaotic itinerancy in the presence of input noise. We propose two scenarios generating random and past-independent attractor switching trajectories using our model.

This work was funded by the CY Cergy-Paris University Foundation (Facebook grant) and partially by Labex MME-DII, France (ANR11-LBX-0023-01).

Appendices

A Free-Energy Derivations

In this section, we provide the derivations for Eq. 5. We start from the following probabilistic graphical model:

$$\begin{aligned} p(\mathbf {c}) = \sum _{k=1}^p \pi _k \mathcal {N}(\mathbf {c} ; \mathbf {\mu }_k, \sigma _c^2 \mathbb {I}_p) \end{aligned}$$

(11)

$$\begin{aligned} p(\mathbf {h}|\mathbf {c}) = \mathcal {N}(\mathbf {h} ; f(\mathbf {c}, \mathbf {h}_{t-1}); \sigma _h^2 \mathbb {I}_n) \end{aligned}$$

(12)

$$\begin{aligned} p(\mathbf {x}|\mathbf {h}) = \mathcal {N}(\mathbf {x} ; g(\mathbf {h}); \sigma _x^2 \mathbb {I}_2) \end{aligned}$$

(13)

Where f and g correspond to the top-down predictions described respectively in Eq. 2 and 4. Note that here, $\mathbf {c}$, $\mathbf {h}$ and $\mathbf {x}$ denote random variables, and should not be confused with the variables of the computation model presented in the main text. Since free-energy will be used to perform inference on the hidden variables, and that it’s not possible to update the past hidden variable $\mathbf {h}_{t-1}$, we consider it as a parameter and not a random variable of the probabilistic model. We only perform inference on $\mathbf {c}$ and $\mathbf {h} = \mathbf {h_t}$.

We introduce approximate posterior density functions $q(\mathbf {h})$ and $q(\mathbf {c})$ that are assumed to be Gaussian distributions of means $\mathbf {m}_h$ and $\mathbf {m}_c$. Given a target for $\mathbf {x}$, denoted $\mathbf {x^*}$, the variational free energy is defined as:

$$\begin{aligned} \mathcal {E}(\mathbf {x*}, \mathbf {m}_h,\mathbf {m}_c)&= \text {KL}(q(\mathbf {c}, \mathbf {h}) || p(\mathbf {c}, \mathbf {h}, \mathbf {x^*})) \end{aligned}$$

(14)

$$\begin{aligned}&= - \mathbb {E}_q[\log p(\mathbf {c}, \mathbf {h}, \mathbf {x^*})] + \mathbb {E}_q[\log q(\mathbf {c}, \mathbf {h})] \end{aligned}$$

(15)

The second term of Eq. 15 is the entropy of the approximate posterior distribution, and using the Gaussian assumption, does not depend on $\mathbf {m}_h$ and $\mathbf {m}_c$. As such, this term is of no interest for the derivation of the update rule of $\mathbf {m}_h$ and $\mathbf {m}_c$, and is replaced by the constant $C_1$ in the remaining of the derivations. Using the Gaussian assumption, we can also find simplified derivations for the first term of Eq. 15, and grouping the terms not depending on $\mathbf {m}_h$ and $\mathbf {m}_c$ under the constant $C_2$, we have the following result:

$$\begin{aligned} \mathcal {E}(\mathbf {x*}, \mathbf {m}_h,\mathbf {m}_c)&= -\log p(\mathbf {x^*}|\mathbf {h}) - \log p(\mathbf {m}_h|\mathbf {c}) - \log p(\mathbf {m}_c) + C_1 + C_2 \end{aligned}$$

(16)

$$\begin{aligned}&= \frac{(\mathbf {x^*} - g(\mathbf {h}))^2}{2\sigma _x^2} + \frac{(\mathbf {m}_h - f(\mathbf {c}, \mathbf {h}_{t-1}))^2}{2\sigma _h^2} - \log p(\mathbf {m}_c) + C \end{aligned}$$

(17)

where $C = C_1 + C_2 + C_3$ and $C_3$ corresponds to the additional terms obtained when developing $\log p(\mathbf {x^*}|\mathbf {h})$ and $\log p(\mathbf {m}_h|\mathbf {c})$.

[1] provides more detailed derivations and a deeper hindsight on the subject.

B Linked Videos

Here is the link to a video showing animated example trajectories in modes A and B (https://youtu.be/LRJQr8RmeCY).