A hierarchy of time-scales and the brain

Abstract

In this paper, we suggest that cortical anatomy recapitulates the temporal hierarchy that is inherent in the dynamics of environmental states. Many aspects of brain function can be understood in terms of a hierarchy of temporal scales at which representations of the environment evolve. The lowest level of this hierarchy corresponds to fast fluctuations associated with sensory processing, whereas the highest levels encode slow contextual changes in the environment, under which faster representations unfold. First, we describe a mathematical model that exploits the temporal structure of fast sensory input to track the slower trajectories of their underlying causes. This model of sensory encoding or perceptual inference establishes a proof of concept that slowly changing neuronal states can encode the paths or trajectories of faster sensory states. We then review empirical evidence that suggests that a temporal hierarchy is recapitulated in the macroscopic organization of the cortex. This anatomic-temporal hierarchy provides a comprehensive framework for understanding cortical function: the specific time-scale that engages a cortical area can be inferred by its location along a rostro-caudal gradient, which reflects the anatomical distance from primary sensory areas. This is most evident in the prefrontal cortex, where complex functions can be explained as operations on representations of the environment that change slowly. The framework provides predictions about, and principled constraints on, cortical structure-function relationships, which can be tested by manipulating the time-scales of sensory input.

Figure 1. Data and states, over two seconds, generated by a two-level birdsong model.

(A) At the first level, there are two outputs (i.e., data) (left: blue and green solid line) and three hidden states of a Lorenz attractor (right: blue, green, and red solid line). The second level is also a Lorenz attractor that evolves at a time-scale that is one magnitude slower than the first. At the second level, the causal state (left: blue solid line) serves as control parameter (Rayleigh number) of the first-level attractor, and is governed by the hidden states at the second level (right: blue, green, and red solid line). The red dotted lines (top left) indicate the observation error on the output. (B) Sonogram (time-frequency representation) constructed from model output. High intensities represent time-frequency locations with greater power.

Figure 2. Dynamic online inversion of the data presented in Figure 1.

Observed data (see Figure 1) are now shown as black, dotted lines, and the model predictions as solid, coloured lines. (A) The 90% confidence interval around the conditional means is shown in grey. The prediction error (i.e., difference between observation and model prediction) is indicated by red dotted lines. (B) Simulated local field potentials (LFPs) caused by the prediction error time series of both levels. See text for their simulation. Red: LFPs at first level, dark red: LFP at second level.

Figure 3. Dynamic online inversion of surprising input.

The sensory data presented in Figure 1 were set to zero at 1.4 seconds, see also Figure 2. (A) The first-level dynamics return to zero after a transition period of ca. 100 ms. We plotted the hidden states and the causal state as dotted lines, for the uninterrupted song. The second-level increases its conditional uncertainty and no longer constrains the first-level dynamics. (B) Sonogram constructed from output. (C) Simulated LFPs of both levels. The red arrow indicates time point of largest prediction error due to interruption.

Figure 4. Single-level model dynamic online inversion of the data presented inFigures 1 and 3.

(A) The single-level model can explain the data (no song interruption) well. (B) The single-level model quickly approaches the zero line after an interruption at 1.4 seconds. (C) Simulated LFPs for model inversion in (A). (D) Simulated LFPs for model inversion in (B).

Figure 5. Comparison of single- and two-level model inversion of high-noise birdsong data.

We show only the output of each model and the causal state of the two-level model. (A) The two-level model can explain the data relatively well, although it misses the third syllable. (B) The single-level model is unable to predict the data at all.