Discrete Sequential Information Coding: Heteroclinic Cognitive Dynamics

Abstract

Discrete sequential information coding is a key mechanism that transforms complex cognitive brain activity into a low-dimensional dynamical process based on the sequential switching among finite numbers of patterns. The storage size of the corresponding process is large because of the permutation capacity as a function of control signals in ensembles of these patterns. Extracting low-dimensional functional dynamics from multiple large-scale neural populations is a central problem both in neuro- and cognitive- sciences. Experimental results in the last decade represent a solid base for the creation of low-dimensional models of different cognitive functions and allow moving toward a dynamical theory of consciousness. We discuss here a methodology to build simple kinetic equations that can be the mathematical skeleton of this theory. Models of the corresponding discrete information processing can be designed using the following dynamical principles: (i) clusterization of the neural activity in space and time and formation of information patterns; (ii) robustness of the sequential dynamics based on heteroclinic chains of metastable clusters; and (iii) sensitivity of such sequential dynamics to intrinsic and external informational signals. We analyze sequential discrete coding based on winnerless competition low-frequency dynamics. Under such dynamics, entrainment, and heteroclinic coordination leads to a large variety of coding regimes that are invariant in time.

Keywords: control of episodic memory retrieval; heteroclinic binding; hierarchical cognitive networks; information patterns; metastable state brain dynamics.

Figure 1

(A) Stable heteroclinic channel (SHC), an invariant topological construction. An SHC is a set of metastable states sequentially connected by unstable seperatrices. The robustness of such channel means that trajectories in the neighborhood of the sequence of separatrices do not leave it until the end of the channel is reached (Rabinovich et al., 2012a, 2015). (B) This panel shows heteroclinic channels representing a recurrent cognitive-emotion interaction—the dotted trajectories illustrate that the interruption of cognitive performance by emotion, which can happen at any cognitive stage. Adapted from Rabinovich and Varona (2017). (C) Time series of sequential switching of emotional and cognitive modalities from model (1)–(2), (Rabinovich et al., 2010b).

Figure 2

Transition from multistability to WLC dynamics in models with different time scales with connection asymmetry as the control parameter. The figure illustrates the bifurcation toward the birth of a heteroclinic cycle in a Lotka–Volterra model (A–C) and in a H–H model (D–H). (A,D) represent multistable dynamics (stable fixed points indicated in red correspond to the attractors). (B) and (E) represent an intermediate case before the annihilation of the stable fixed points (saddles are indicated in blue). (C,F) represent the heteroclinic cycle that emerges after the saddle node bifurcation. (G,H) represent the time series corresponding to transient heteroclinic dynamics and a robust heteroclinic cycle in the H-H model. Adapted from Nowotny and Rabinovich (2007), Rabinovich and Varona (2011).

Figure 3

Sequential memory—Binding dynamics of 3-modality events. (A) Illustration of an ensemble of 18 competitors fluctuating on three functional communities: each of them is responsible for the processing of different informational modalities. In this example, all connections are inhibitory as characterized by the WLC matrices in the model (1)–(2) with W = const., i.e., focused attention. (B) Illustration of the heteroclinic network in the model phase space, where $Q_i^l$ is the saddle i in the modality l. (C) Mutual modulations of different modalitity sequences, as shown by the projection of a trajectory on a 3-dimensional space. Colored regions point out the vicinity of the metastable states. One can see that this complex trajectory spends some time in a neighborhood of one modality and goes to the next modality afterwards (Afraimovich et al., 2015).

Figure 4

Model of chunking in winnerless competitive networks. (A) Illustration of a 3-layer network for hierarchical chunking. (B) Time series of the sequences of the three-level hierarchy—items are grouped in chunks; these chunks form 3 superchunks of 6 elements each displaying reproducible dynamics according to the model (1)–(2). Different colors correspond to different items inside each group (switching the color means moving from the previous item to the next one). Adapted from Rabinovich et al. (2014).

Figure 5

Modulation of retrieval dynamics of episodic memory in heteroclinic networks. (A) Networks inside the circles represent the intrinsic architecture of interacting envelope patterns. Ω characterizes the frequency of the external forcing. (B) Complex structure of the heteroclinic synchronization: Period T corresponds to the cyclic winnerless competition switching between episodes X, Y, and Z and is depicted as a function of the forcing frequency Ω under a small Gaussian noise with zero correlation in Equation (1). The region in between synchronized regimes displays chaos (Rabinovich et al., 2006a).

Figure 6

(A) An illustration of a 3-modality heteroclinic network and one of the associated trajectories in its vicinity corresponding to the binding process. (B) Joint dynamics of two temporally coordinated binded modalities (Rabinovich et al., 2010a). The time series is plotted with a color-code representing the evolution of time to favor the visualization.

Figure 7

Representation of the visual and auditory sequential information exchange in musical improvisation. Adapted from Walton et al. (2015), (see also Kugler and Turvey, 1987).